Sylvain Deville

Pierre-Alexis Corinne

Théo

Julie Benjamin

Paolo

Kevin

Quentin

Lucas

Daniel

Alexey Zane Baptiste

Keywords: solidification, solute, inter-particle interactions, heterogeneous systems

This thesis would not have been possible

Introduction

Materials form an integral part of the human society. The importance of materials is such that the human civilization is often divided into ages according to the materials that dominate in the society. The exploration and evolution of materials by mankind started with the Stone Age, and progressed with the Bronze and Iron Age. Today, we have entered into the revolutionary era of engineered materials, where plastics and silicon have prevailed in the past two decades. As the world population continues to rise, the demand for novel and more extravagant materials has become a hallmark of our modern industrialized society. The global expansion, therefore, puts an ever increasing demand on the production of materials with improved physical, chemical, and mechanical properties.

The present approach to material science and engineering is based on the recognition that a material's property (intrinsic or extrinsic) depends on its complex structure. To understand a material, it is therefore necessary to know the material structure at different length scales, such as the atomic-, microscopic-, or macroscopic-scale. A control of the material structure at these diverse scales, where the fabrication route or processing technique plays a major role, aides in the construction of novel hybrid materials. The use of ice-templating or freezecasting for the design of engineered materials has been a topic of interest recently.

The process has seen a surge of interest in the past decades, and represents a unique phenomenon at the intersection of several domains extending beyond the realms of materials science. The basic idea of freeze-casting or ice-templating is the solidification of a suspension to create architectured porosity from the directionally growing solid crystals. The process results in the formation of a green body (before sintering) having a macro-porous structure owing to the sublimation of the solvent crystals after their solidification. Therefore, in the interest of improving properties of freeze-cast materials, an understanding of the suspension solidification phenomenon is very essential. Introduction Solidification, also known as freezing, is the process where a liquid melt undergoes phase transition to form a solid phase. Historically, solidification was used extensively for metal casting and welding processes, but in this modern age it applies to commercially manufactured polymers, ceramics, and even natural biological systems. Great strides have been made in understanding many aspects of solidification. Theoretical models have been developed to simulate actual solid crystal structures, and experiments have been designed and carried out to shed light on the crystal growth. However, a fundamental question still partly unanswered in the solidification of suspensions or alloys is the development of the microstructure in the presence of secondary objects, such as particles or bubbles, in the melt. This phenomenon is also at the core of ice-templating, where ceramic, metallic, or polymer particles dispersed in the solution interact with a solidification front. This is important as the segregation and redistribution of the suspended particles by the advancing front determines the material microstructure. Hence, our present study represents a continuation of the work initiated in LSFC (Saint-Gobain Research Provence), on ice-templating, freezing of colloids, and new architectured materials.

The interaction of foreign objects suspended in a liquid melt with an approaching solid-liquid front is of special interest in nature and in engineering sciences. This phenomenon was initially investigated by geologists and civil engineers studying the interaction of soil particles with an ice-water interface to explain the formation of ice-lenses and frost-heaving. Presently, this subject garners interest in the research of cryopreservation of biological cells, food preservation, ice growth, metallurgy, single-crystal growth, and freeze-casting.

The process is hence encountered in a plethora of situations with significant applications in materials science, physics, or biology. The objects (soft or hard) can have diverse outcomes owing to their confrontation with the solidifying front.

The front can engulf the object, trapping it into the growing crystal, or the front can repel the object, pushing it ahead of itself.

The presence of foreign objects can have a strong influence on the properties of the solidified crystal. In particular, the spatial distribution of the objects, among other parameters, determines the structural and functional properties of the material. It is desired, for example, that the reinforcing second-phase ceramic particles be homogeneously distributed in metal-matrix composites. This can be achieved by an instantaneous engulfment of the objects by the solidliquid interface. In other words, the objects should be oblivious to the presence Introduction of an approaching front. In contrast, the fabrication of porous materials by freeze-casting relies on the complete segregation or clustering of particles by the solidification front. Here, particles must be rejected by the growing crystal and collected between the arms (dendrites) of the solidification front. In single-crystal growth, the rejection of dissolved impurities (gases) and second-phase particles is critical for obtaining pure materials. Therefore, the solidification front should act as a purifying interface (for a partition ratio less than 1), and segregate the impurities at the crystal periphery to eliminate them easily by polishing. Thus, in all of these domains, control of the microstructure resulting from solidification is essential.

The objective of our study is to perform directional solidification experiments to comprehend the mechanisms responsible for the development of a solidification microstructure in the presence of foreign objects. This works focuses on the role of solute on these phenomena, which has been largely underestimated so far. We analyse the interaction of spherical soft objects (droplets and bubbles) and rigid particles (polymers, ceramics, and metals) with an advancing ice-water interface. Our custom cryo-confocal microscopy setup enables a rapid 2D and 3D imaging of the growth of ice crystals, the liquid phase, and the dynamics of objects encountering the solidification front. We characterise and report on the role of solutes while experimentally depicting the evolution of the local solute segregation. In particular, we analyse the impact of solute concentration and their influence on the interface geometry, object dynamics, and the solidified microstructure. We further analyse the fundamental mechanisms of interactions when soft deformable objects are hit by a solidification front. We progress towards more realistic systems by employing a high volume fraction of objects as compared to the isolated systems. Finally, we compare our experimental observations with the past theoretical models to elucidate the different features and propose mechanisms for the contrasting phenomena.

Chapter 1

State of the art 1.1 Introduction

The interaction of objects with a moving solid-liquid interface has been studied both experimentally and theoretically. The first systematic measurements for particle engulfment by an ice-water interface were conducted by Corte [1]. He used particles of different size and shapes to study the influence of an increasing growth rate on the particle capture. The observation of particle pushing by a growing crystal instigated studies to deduce a critical velocity (V c ), the growth rate below which a particle is pushed (V sl < V c ) and above which the particle is engulfed (V sl > V c ) by the growing solid [2]. The theoretical derivation of the critical velocity (V c ) has been treated by numerous studies and the approaches differ mainly in the approximations used. The studies mainly deal with smooth, spherical, and rigid particle of radius R moving ahead of a planar front at a distance of d << R in steady-state directional solidification. A schematic drawing representing an object in the vicinity of a solid-liquid interface with the related parameters is shown in Fig. 1. 1. In this thesis, an emphasis is placed on experimental studies. A detailed background is, however, presented so that the important principles governing the interaction of objects with solidification fronts can be clearly established. I first try to highlight some of the most important theoretical models, which allows us to further understand the isolated object behaviour in the vicinity of a solid-liquid interface. A comprehensive review of the governing equations, the advantages, and the limitations of each predictive model invoked is presented. I discuss the essential mechanisms and parameters influencing the object-front interaction Model sketch, where d is the particle-front gap, G = ∇T is the temperature gradient, κ is the interfacial front curvature (here > 0 for concave geometry), k p , k m are the thermal conductivity of the particle and melt respectively, R is the object radius, T m is the absolute melting temperature, T i is the solidliquid interfacial temperature, and V sl is the solidification velocity. during directional solidification. The impact and role of solutes (or impurities) on modifying the interactions is also explored. I further analyse the experimental techniques employed previously in conducting similar directional solidification studies, and discuss the correlations of the experimentally observed features to the predicted theoretical behaviour. Finally, the complex nature of realistic systems having multiple objects is compared to the ideal isolated object-front interactions.

Theoretical models

Theoretical models

The prediction of a critical velocity in a given system depends on the nature and magnitude of various forces acting on the particle while it is being pushed by the solidification front. At steady-state repulsion, the sum of these forces must be zero, and the critical velocity will be reached when the forces are no longer in equilibrium. Uhlmann et al. [3] were the first to present a derivation of the critical velocity in qualitative agreement with their experimental data. They extended a thermodynamic argument, involving the different surface energies to explain the stability of liquid films, to the present problem of objects interacting with a solidification front. In the case of particle engulfment, the two existing interfaces (particle-liquid and solid-liquid) should be replaced by a single interface (solidparticle). Hence, the thermodynamic criterion for a particle to be engulfed spontaneously by the solidification front is given by

∆σ 0 = σ sp -(σ pl + σ sl ) < 0 (1.1)
where σ sp , σ pl , and σ sl are the interfacial free energies associated with the solid-particle, particle-liquid, and solid-liquid interface respectively. In other words, the particle will be engulfed or pushed depending on the configuration with the lowest energy (see Fig. 1.2). This thermodynamic approach was used extensively by Omenyi and Neumann [5,6]. However, the approach had two significant drawbacks. First, the experimental measurements of the σ sp , solid-particle interfacial energy, were difficult and the theoretical estimations presented various discrepancies. Second, the approach could only be used to make qualitative predictions of engulfment 7 CHAPTER 1. STATE OF THE ART or rejection at infinitely small growth rates (< 1 µm • s -1 ) [7].

Repulsive interfacial force

Uhlmann et al. [3] refined this thermodynamic criterion by assuming that the change in interfacial energy (∆σ) varied with the distance (d ) between the particle and the solidification front as a power law given by

∆σ = ∆σ 0 d 0 d n (1.2)
where ∆σ is the interfacial energy at distance d , ∆σ 0 is as given in Eq.1.1, d 0 is the minimum possible separation distance between the particle and the solid (corresponding to contact) and equal to a few molecular diameters, and n is a positive number chosen equal to 4 or 5 (see Fig. 1.3). This change in interfacial energy was assumed as the driving force for the flow of liquid material (melt) in the particle-front gap. The diffusion of liquid was deemed necessary to maintain a stable liquid film, thereby preventing particlesolid contact and hence, particle engulfment. Hence, the repulsive force preventing the particle engulfment arises from the need to maintain a stable liquid film, which avoids the contact between a particle and an approaching solidification front. An expression for the repulsive force due to interfacial energy can then be 1.2. Theoretical models determined as [3,8]

F σ = 2πR∆σ 0 d 0 d n (1.3)

Attractive viscous force

The attractive force, favoring engulfment of the particle, arises from the viscous drag due to fluid flow around the particle. This force tends to push the object towards the solidification front moving at a velocity V sl . The attractive force was approximated by Uhlmann et al. [3] from the viscous drag on a sphere of radius R and at a distance d (<< R) from the planar interface, using the derivation of Carrier [3];

F η = 6πηV sl R 2 d (1.4)
where η is the viscosity of the melt. To define a critical engulfment velocity (V c ),

Uhlmann et al. [3] used the maximum growth rate at a critical gap thickness (assumed a few molecular diameters) with a steady-state pushing configuration.

Hence, the critical engulfment velocity (V c ) is obtained by balancing both forces, i.e. putting

F σ = F η V c = ∆σ 0 d 3ηR d 0 d n (1.5)
The final expression is thus a function of several unknown interrelated parameters V c = f (d 0 , n) and the origin of repulsive interfacial force remains unclear.

Furthermore, they assumed a negligible temperature gradient (G) and an ideal planar front geometry with no front curvature (κ = 0).

Thermal Conductivity & Interfacial Curvature

The simplified approach of Uhlmann et al. [3] assumed equal thermal conductivity of the melt (k m ), particle (k p ), and solid (k s ), thereby explaining the idealized planar front geometry. However, an object with a thermal conductivity different from the melt (k p = k m ) will modify the local heat flow, thereby modifying the temperature isotherm locally (see Fig. 1.4). As we assume the solid-liquid front to follow the isotherm for the melting point (T m ) of the matrix material, this difference of thermal conductivity can bend the front towards or away from the object, depending on whether the thermal conductivity of the object is higher or CHAPTER 1. STATE OF THE ART lower than the melt. Hence, if a particle has a lower thermal conductivity than the melt (k p /k m < 1), the solidification front bulges towards the object and is convex.

In the opposite case (k p /k m > 1), the solidification front bends away from the particle and is concave [4,9]. The shape and curvature of the solidification front is an important criterion to predict the particle behaviour in the vicinity of a solidifying interface. The curvature determines the magnitude of attractive and repulsive forces acting on an insoluble object [4,10,11,12]. The approximate viscous drag force in the presence of a curvature assumed a priori is given by [4]

F η = 6πηV sl R 2 α 2 d (1.6)
where α = 1 represents a planar interface, α > 1 is a concave interface, and α < 1 is a convex interface. In general, the cusping of the front behind the particle increases the drag force, F η , more than the disjoining force, F σ . Thus, the critical velocity (V c ) may reduce drastically as compared to planar fronts [4,8,11,13].

1.3. Model from Chernov et al.

Model from Chernov et al.

Chernov et al. [10,14] were the first to consider a distortion of the temperature field owing to the differing thermal conductivity of the particle and the melt (k p = k m ). They assumed a paraboloidal front shape a priori to calculate the forces on the particle (see Fig. 1.5). In addition to the rigorous treatment of the temperature field distortion, Chernov et al. [10,14] also tried to explain the origin of the repulsive interfacial forces responsible for the particle pushing. They considered that the particle rejection was due to a disjoining pressure owing to the presence of van der Waals forces. They developed an approach using the non-retarded van der Waals forces varying with the liquid film thickness in the particle-front gap. Their model was relatively successful in predicting the experimental data as compared to the other models [2]. Hence, most of the theoretical studies performed subsequently to model the particle behaviour were based on their approach.

Repulsive disjoining pressure

For pushing to occur, a thin liquid film must always be present between the solidification front and the particle, thereby allowing the crystal to grow. Derjaguin and Kussakov [16] were among the earliest to investigate the stability of thin liquid films. They found that an excess pressure adjacent to the bulk liquid was required to maintain a thin liquid film confined between two parallel plates, which they termed as the disjoining pressure. Therefore, the presence of a stable liquid film implies the existence of a disjoining pressure in the particle-front gap (d ), which is essentially a jump in pressure at the two solid surfaces adjacent to the thin liquid layer [7]. The disjoining pressure depends on the liquid film thickness (d ),

and the type and strength of inter-molecular interactions in the system [17,18].

The disjoining pressure (P T ) owing to dispersion (van der Waals) interactions for a thin film between two unbounded parallel flat surfaces is given as [19] P T = -A sw 6πd 3 (1.7)

where A sw is the Hamaker constant for the system particle-solid separated by a liquid film. For example, A i w p is the Hamaker constant for the ice, water, and particle system, defined as A i w p ≡ A i p + A w w -A i w -A pw . For the ice-waterparticle system, the Hamaker constant ranges from -0.1 to -1 × 10 -20 J [7]. A negative Hamaker constant corresponds to a repulsive force. The total van der Waals force on the particle is calculated by integrating P T over the particle surface.

A detailed description of the methods of computing the disjoining pressure of van der Waals origin has been provided by Patrick Aubourg [7].

Models based on Chernov's approach

The analytical model from Chernov et al. [10,14] was the starting point for the development of more rigorous treatments of the solidification front shape. The progress in computing abilities and reduced cost of numerical simulations facilitated the solution of complete differential equations. Most of the previous models had relied on ad-hoc assumptions for the interface geometry and the thickness of the liquid film separating the particle from the solid. The model from Rempel et al. [20], built upon a similar approach as Chernov et al., made significant improvements to the domain of object-front interactions. They solved explicitly for the entire shape of the front without assuming the geometry a priori (see Fig. 1.6). Further, they employed numerical techniques to predict the particle velocity as a function of its position relative to the spatial temperature field, and approximated the fluid flow in the particle-front gap via lubrication theory.

Figure 1.6: The thick purple line shows the particle behavior that is predicted by Rempel et al. [20], including the complete interfacial curvature, which, for large film thicknesses, is nearly identical to the results obtained when approximating the front as being planar (rightmost magenta curve). The leftmost magenta curve gives the approximate particle velocity when the thermal conductivity of the particle is assumed equal to that of the melt and the solid. The green curve displays the predictions of Chernov et al. [14] for the case where the interface has the form of a paraboloid intersecting with a plane, and the lowest (blue) curve is their prediction for the particle velocity when the front is modeled as a paraboloid, adapted from [21].

CHAPTER 1. STATE OF THE ART

Similar to the approach of Chernov et al. [10], their model accounts for the presence of a liquid film owing to the intermolecular interactions in the gap between the particle and the solid. To solve for the temperature field during solidification in the presence of a small particle (R ≤ 100 µm) they defined an interfacial temperature T i as

T i = T m 1 - λ d ν - σ sl κ L v (1.8)
where T m is the absolute melting temperature, κ is the front curvature, σ sl is the solid-liquid front interfacial energy, L v is the latent heat per unit volume of the solid, d is the film thickness between the particle and the front, and λ is a disjoining pressure length scale characterising the strength of intermolecular interactions that dominate (typically λ is of order 10 -10 m). The exponent ν depends on the type of intermolecular interactions that dominate; ν = 3 represents non-retarded van der Waals interactions, while ν = 2 and ν = 4 is used for retarded van der Waals and long-range electrical interactions [22]. To simplify their approach, Rempel et al. [20] considered the role of repulsive van der Waals interactions (ν = 3) for the derivation of a critical velocity. Hence, they used the disjoining pressure length scale λ which can be defined by

λ = A sw 6πL v 1/3
(1.9)

The effective Hamaker constant A sw can be predicted experimentally or calculated from the dielectric constants of the solid, the liquid, and the particle [20].

Curvature & Interfacial Undercooling

The definition of the interfacial temperature in Eq.1.8 highlights two important phenomena owing to the particle-front interactions. First, the interface curvature impacts the bulk melting temperature of the solid and reduces it by an amount proportional to the κ and the σ sl . This change or lowering of the melting temperature is referred to as the Gibbs-Thomson effect or curvature undercooling.

Second, the disjoining pressure caused by the intermolecular interactions also reduces the bulk melting temperature by an amount proportional to the thickness of the liquid film d . This lowering of the melting point is referred to as the interfacial undercooling. Hence, the equilibrium undercooling T m -T i increases as the liquid film thickness is reduced, instigating the solid-liquid interface to 1.4. Models based on Chernov's approach deviate from the planar T m isotherm, when it approaches an insoluble particle.

In consequence, the interface curvature resulting from this deviation further impacts the equilibrium undercooling. Thus, the interplay between the various interactions determines the shape of the solidification front and hence, the magnitude of forces acting on the particle.

Premelted films

In general, the term premelted films implies the presence of a liquid below the solid bulk melting temperature, T m , in a thin layer adjacent to another bulk phase in thermodynamic equilibrium. The formation of premelted films can be attributed to three phenomena; curvature induced premelting or the Gibbs-Thomson effect, surface or interfacial premelting, and solute or impurity induced premelting.

Curvature induced premelting or the Gibbs-Thomson effect

The curvature induced premelting, as described in section 1.4.1, is owing to the Gibbs-Thomson effect invoked by the solid-liquid interfacial curvature, κ. This phenomenon is commonly encountered in the freezing of porous media (see Fig. 1.7). For example, the water present in the pores of radius R por e in a frozen colloidal suspension can remain liquid and hence, be undercooled (∆T GT ) to a temperature T below its bulk melting temperature T m , given as

∆T GT = T m -T = T m R por e ρ i ∆H f σ i w κ g (1.10)
where σ i w is the ice-water interfacial tension, ρ i is the density of ice, ∆H f is the bulk enthalpy of fusion, and κ g is a geometric prefactor that depends on the porous structure and the wetting properties (κ g = 2 for cylindrical pores and perfect wetting).

Surface or interfacial premelting

The surface or interfacial premelting can occur when the solid is in contact with its vapour phase (called surface premelting), in contact with a similar solid interface with a different geometric orientation (called grain-boundary premelting), and in contact with a foreign substrate such as a particle (called interfacial premelting). The interfacial undercooling ∆T I F for a solid-liquid interface can be given as

∆T I F = T m L v P T (1.11)
where the disjoining pressure P T varies as a function of the premelted film thickness d (see section 1.3.1).

The deformation of the solid-liquid interface and the presence of the liquid film which separates the particle from the solid surface are both manifestations of the phenomenon known as interfacial premelting [22,24]. The liquid present in this film is below the bulk melting temperature T m , and represents a disordered quasi-liquid region formed as a result of the competition between a reduction of interfacial free energy and the expense of maintaining a layer of undercooled liquid [24].

As stated previously, the thickness of the premelted film, d , depends on the equilibrium undercooling, T m -T i (see Eq. interactions [21].

Solute or impurity induced premelting

The addition of a solute or impurity can greatly enhance the phenomenon of premelting through a colligative effect (Raoult's law) lowering the chemical potential of the solvent. As most solids have a lower solubility for solutes as compared to the liquid phase, the solutes rejected during solidification can further enhance the chemical potential difference at the solid-liquid interface [26]. Therefore, a small amount of impurity or solute in the parent liquid can have drastic impacts owing to its segregation during solidification (see section 1.5.2). The thickness of the solute premelted films is highly sensitive to the dopant concentration and is dominated by the colligative effects at sufficiently high solute (or impurity) concentration [27]. The solute (or impurity) induced undercooling (∆T SO ) on a flat substrate (R → ∞) can be given as

∆T SO = R g T 2 m N i L v d (1.12)
where R g is the universal gas constant and N i is the number of moles of solute per unit surface area of the substrate. This equation can be used in ice-waterparticle systems with high solute concentrations, where the particle is larger than a micron and the equilibrium undercooling is greater than 0.05 K . Hence, an increase in the solute concentration increases the premelted film thickness.

The physics of premelting in ice has been investigated in detail owing to motion [26]. Furthermore, the thermal gradient induced fluctuations of the premelted film are known to be at the origin of regelation of particles engulfed in a solid [28]. The thermal or pressure induced regelation can cause migration of small particles (≤ 1 µm) through ice, whereby the particles translate towards the warmer side of the underlying ∇T , owing to a thermomolecular pressure gradient [29] (see Fig. 1.9).

Model from Rempel et al.

Rempel et al. [20] calculated the repulsive interfacial (or thermomolecular) force by integrating the Lifshitz expression for disjoining pressure (see Eq.1.7) over the interfacial surface for a spherical particle. To derive the attractive viscous (or lubrication) force, they first used a lubrication approximation in the particle-front gap to find the volume flux of liquid moving in the premelted film. Subsequently, they calculated a lubrication pressure using conservation of mass since the volume of liquid solidified should be equivalent to the volume of liquid swept in the particle-front gap (premelted film) as the front moves with a velocity V sl . The lubrication (or viscous) force is then calculated by integrating the lubrication pressure over the particle surface. For the critical velocity, they determined that the force is maximum at the base of the particle (θ = 0) where the premelted film is thinnest (see Fig. 1.11).

Hence, they derived a film thickness of the order s R near the base of the particle CHAPTER 1. STATE OF THE ART (θ = 0), diverging towards infinity at θ = π/2, defined as

s = A sw 6πσ sl R 2 1/3 (1.13)
Using typical values for a particle-in-water system, where A sw ≈ 10 -20 J , σ sl ≈ 10 -2 J • m -2 and R ≈ 10 -6 m, we get s ≈ 10 -3 and hence, a minimum film thickness of ≈ 1 nm. The critical velocity, equated by a balance of thermomolecular and lubrication forces at minimum premelted film thickness ( s R), is given as

V c = σ sl A 2 sw 6 5 π 2 η 3 R 4 1/3
(1.14) Hence, a solidification front moving at a velocity V sl will engulf a particle if the characteristic critical velocity is V c < V sl . The critical velocity is a function of the particle radius (V c ∝ R -4/3 ) and interestingly, independent of the temperature gradient (G). Therefore, larger particles tend to be engulfed in the solid at 1.4. Models based on Chernov's approach relatively lower growth rates as compared to smaller particles. The influence of R on V c changes with the type of inter-molecular interactions. V c ∝ R -3/2 for retarded van der Waals interactions, while V c ∝ R -1 for electrical interactions was deduced by their model. The model from Rempel et al. [20] clearly highlighted the steady-state particle velocity as a function of the premelted film thickness. Furthermore, their results demonstrated two steady-state gap thicknesses corresponding to a single particle velocity. The larger of these gap sizes represented a stable pushing of the particle, while the smaller represented an unstable state eventually leading to engulfment. Finally, the model also provided a unique critical velocity, for a given set of solidification parameters, depicted by the maximum of the particle velocity-gap thickness curve [20]. Hence, the incorporation of premelting in the model circumvented the problem of inferring a critical velocity using an arbitrary minimum gap assumption (see Fig. 1.12). However, the main drawback of the model was the assumption of equal thermal conductivity of the particle, the melt, and the crystal. This casts doubt on the predicted shape of the solidification front in the vicinity of an insoluble object (with k p = k m ), as discussed previously in section 1.2.3.

Model from Park et al.

Park et al. [32] developed further on the analytical approach from Chernov et al. [10] by incorporating the role of thermal conductivity mismatch (k p = k m ) in the numerical analysis of the particle-front interactions. Their model was similar to that from Rempel et al. [20] in the use of premelting effects (non-retarded van der Waals interactions) for deducing the repulsive disjoining pressure (see Fig. 1.13). Moreover, they also followed a similar lubrication approximation for estimating the magnitude of attractive viscous drag forces in the particle-front gap. In contrast to Rempel et al. [20], they took into account the influence of bulk flow induced by the density variation upon solidification. This is important as the flow of liquid can be towards (or away) from the solid depending on whether the density of the solidified phase is larger (or smaller) than the melt. The model also included the analysis of a bubble interacting with a solidification front. The major difference between a bubble and an insoluble particle was assumed to be the effect of thermocapillary stress (Marangoni effect) owing to the variation of bubble surface tension with temperature.

The critical velocity (V c ) expression derived for a particle or bubble is given as

V c = 1 6ηR L v A 3 sw G (6π) 3 T m 1/4 u max (γ, k p /k m ) (1.15)
where u max is the maximum of the dimensionless particle velocity and γ is a dimensionless surface energy parameter defined as

γ = T m σ sl L v GR 2 (1.16)
The dimensionless u max varies as a function of the premelted film thickness [32] suggested that a small particleto-melt thermal conductivity ratio promotes repulsion, while particle engulfment is facilitated when the ratio is large, adapted from [32].

thermal conductivity mismatch (k p = k m ) and of solid-liquid interfacial energy (σ sl ) on determining the shape of the solidification front. They concluded that the object behaviour is strongly influenced from the thermal conductivity mismatch between the particle and the melt (k p = k m ). Hence, they suggested that a small particle-to-melt thermal conductivity ratio promotes repulsion, while particle engulfment is facilitated when the ratio is large (see Fig. 1.14). Furthermore, the V c deduced from their model predicts values one order different as compared to that computed from the Rempel et al. [20] model.

Model from Tao et al.

I will now briefly discuss one of the most recent model from Tao et al. [31], employing finite element methods to estimate the particle behaviour. This model uses the numerical solution approach to predict the complete shape of the solidification front in the presence of the premelted liquid layers (see Fig. 1.15). The study was a detailed attempt to accurately resolve the magnitude of forces acting on a particle in close proximity to the solid-liquid interface. This is a complex problem as the solidification front does not remain planar in the vicinity of an insoluble object. The authors further made a comparison of their force magnitude estimations with those obtained from the analytical model of Rempel and Worster [30].

The major differences presented in this study as compared to the previous analytical models were the approximations used to calculate the two forces; the repulsive van der Waals force and the attractive viscous drag force. For estimating the former, the model used the Hamaker's law [33] as it takes into account the geometry of the problem (sphere-plane). The authors argued that the Lifshitz expression (see Eq.1.7) used in previous models [20,30,32,34] had been derived

for parallel plate geometry and hence, overestimates the repulsive forces.

To calculate the drag force, the model did not rely on the lubrication approximation and instead numerically resolved the Navier-Stokes equations for computing the flow throughout the liquid domain. Their study suggested that the lubrication approximation approach is not accurate and underestimates the drag force by 30%. Therefore, the critical velocity predicted by models employing the lubrication approximation (for drag) and/or the Lifshitz expression (for repulsion) have higher values in contrast to their model (see Fig. and -15 °C ) [35]. Polymers, referred to as dispersants, are generally added to the slurry in freeze-casting to prevent flocculation of the suspended ceramic particles and hence, ensure a homogeneous solidified structure [36]. In contrast, the presence of solutes (or rather impurities) in certain situations is deleterious to the properties of the solidifying material. The dissolved gases (especially H 2 and N 2 ) in molten metals can instigate the formation of porosity or blow holes (see Fig. 1.16), thereby serving as stress-concentration sites and induc-CHAPTER 1. STATE OF THE ART ing premature material failure [37]. Inclusions of SiC arising during solidification of multi-crystalline silicon (mc-Si) reduce the solar cell efficiency and can lead to wafer rupture [38].

I will now discuss some of the mechanisms at play owing to the presence of solutes during solidification. This will help us to further understand the impact of solute when objects interact with an advancing solid-liquid interface. The list of solute mechanisms discussed is not exhaustive as the solutes can impact several other parameters including the suspension viscosity, solid-liquid interfacial tension, preferred crystal growth direction, solutal convection, premelted film rheology, however, these phenomena remain highly complex to investigate [23]. A major distinguishing parameter in solidification of pure and impure solutions is the melting temperature. A pure melt solidifies (or freezes) at a constant temperature, while an impure solution solidifies over a range of temperature. The solidification for a solution starts at the liquidus temperature (T l ) and terminates at the solidus temperature (T s ) (see Fig. 1.17). In general, the solubility of a solute is less in the solid phase than in the liquid phase. For example, O 2 gas is more soluble in water as compared to ice [40]. As solidification progresses, the solute is rejected by the growing crystal, thereby corresponding to an increasing con-1.5. Solute effects centration of solute in the remaining liquid [41] (see Fig. 1.18). For steady-state planar growth, where the transport of solute takes place by diffusion only, the concentration field of solute in the liquid is given by

Solute Diffusion

C L = C 0 +C 0 1 -K 0 K 0 exp -V sl D x (1.17)
where C L is the concentration, C 0 is the bulk concentration in the liquid far from the interface, D is the solute diffusion coefficient, x is the distance from the interface, and K 0 is the partition coefficient. This diffusion model from Tiller et al. [41] has simplifying assumptions where the diffusion in solid is negligible, the value of K 0 stays constant during solidification, and the convection in liquid is neglected. However, we suggest that the model can be used to avoid complex calculations in steady-state directional solidification at small growth rates with a constant temperature gradient.

Constitutional Undercooling

During solidification in the presence of objects, the segregation of solutes at the interface is enhanced further by the approaching objects obstructing their diffusion field [12]. Since growth from solutions depends on the concentration gradient of solutes, the liquidus temperature of the melt in the vicinity of the front differs from the liquidus temperature of the bulk liquid far from the front.

As the solute segregates, the liquidus temperature further decreases. This change in liquid composition alters its transformation temperature, referred to as constitutional undercooling [41]. Hence, in other words it is a colligative effect of freezing point depression instigated by the solute segregation and enhanced further between the objects and the solidification front (see Fig. 1.19).

Front morphology

In directional solidification, the advancing solid-liquid front can have predominantly three morphologies; planar, cellular (or columnar), and dendritic (see Fig. 1.20). The constitutional undercooling owing to the solute segregation, in addition to the growth rate and the temperature gradient, determines the front morphology. In general, a high thermal gradient has a stabilising effect, while a solute concentration gradient has a destabilising effect on a steady-state planar front [15]. The planar to columnar front transition is instigated by a growing instability, defined by models such as the Mullins-Sekerka (see Fig. 1.21), where the constitutional undercooling is a necessary criterion [43]. The growth of single crystals requires a planar interface, while casting and welding microstructures are formed under conditions which give rise to cellular and dendritic interfaces [44].

The spatial distribution of the objects in the solidifying material depends on the advancing front morphology [45]. The solute rejection hence plays a major role in determining the final solidification microstructure and the segregation patterns.

Solute segregation

The solute segregation between the object and an approaching solidification front gives rise to solute induced premelted films (see section 1.4.2). This local solute enrichment is of particular importance in understanding several mechanisms that can occur at the solid-liquid interface. The nucleation and growth of bubbles at the front is instigated when the concentration of gases rejected by the growing solid exceeds a certain solubility limit value [40] (see Fig. 1.16). These bubbles can be subsequently engulfed in the crystal leading to macroporosity defects [42]. The rejection of additives during the freezing of biological cells can lead to elevated solute gradients in the remaining liquid. As most biological cells are made of water, they can undergo membrane rupture (or cryoinjury), when subjected to such high osmotic stress owing to the local solute concentration [15,46].

Theoretical models incorporating solute effects

I will now discuss the theoretical models studying the interaction of a foreign object with a solidification front in the presence of solute effects. Most theoretical studies, as shown in section 1.2, only consider solidification of pure materials, whereas many real-life situations of interest involve the solidification of solutions or alloys. Although the solute effects have widespread applicability, yet only a few models have investigated this scenario. In contrast to pure melt models, the solute models need to incorporate the impact of constitutional undercooling owing to the rejected solute segregation. Hence, the solute models essentially need to solve for the thermal and solute concentration field in the particle-front gap to accurately estimate the solidification front shape.

The earliest model solving for a coupled temperature and solute field in the presence of a particle was derived by Pötschke & Rogge [8]. Their approach for estimating the force equilibrium was similar to Chernov et al. [14], using the nonretarded van der Waals interactions for the repulsive force and the viscous drag for the attractive force. They numerically solved for the solute concentration in the particle-front gap and calculated the corresponding liquidus temperature from the equilibrium phase diagram. They further utilised the liquidus temperature as isotherms and predicted the interfacial curvature. To determine a critical velocity, they fitted a function to their numerical solutions using an intermediate range of

V c ≈ 0.053R -1 A sw GDK 0 η|m|C 0 k p k m 1/2 (1.18)
where |m| represents the absolute value of the liquidus slope. The model was one of the first attempts at solving the complex problem of solute field concentration and hence, had several simplifying assumptions. The authors notably ignored the Gibbs-Thomson undercooling induced by the solid-liquid interfacial energy (σ sl ).

Sasikumar & Ramamohan [11] performed a rigorous treatment of the temperature field distortion using numerical methods. The study followed a similar approach of using intermolecular interactions to calculate the force equilibrium.

The model tried to predict the shape of the front using the solute concentration, curvature undercooling, and constitutional undercooling contributions. However, the authors did not provide a critical velocity scaling and only presented qualitative arguments stating that V c ∝ C -1 0 . The authors deduced that the critical velocity for solutions is much lower as compared to pure melts. They further suggested that the particle engulfment in solutions by a planar front may not be likely as the front tends to destabilise before approaching the V c . Thus, the front destabilisation velocity (planar to cellular transition) can be lower than the V c . However, no theoretical proof was provided for these conclusions.

Kim & Rohatgi [12] tried to predict the shape of the front from the ratio of diffusion coefficient of solute to the growth rate (D/V sl ). The authors estimated the force equilibrium using a similar approach as Pötschke & Rogge [8]. However, the model does not provide any scaling for the critical velocity and made qualitative predictions for the shape of the solidification front. The model predicted a concave front curvature at low growth rates (≤ 10 µm • s -1 ) owing to the extensive constitutional undercooling in the particle-front gap. The model also, similar to previous studies, concluded that the critical velocity for alloys or solutions was much smaller than for pure melts. Kao et al. [34] performed a systematic analysis of the behaviour of objects in solidifying binary alloys. They considered a force equilibrium in the presence of ionic solutes in solution, given as

F σ + F η + F g + F DL = 0 (1.19)
where F σ is the repulsive force from van der Waals interactions, F η is the attractive force from the viscous fluid drag, F g is the buoyancy force, and F DL is the doublelayer repulsion force from the ionic solutes. The critical velocity scaling provided by the model in a simplified form can be expressed as

V c ∝ R -1.30 G 0.35 σ 0.30 sl A 0.35 sw η -0.35 D 0.65 (1.20)
Interestingly, the authors deduced that the ionic double layer force was too weak and did not impact the object behaviour. The authors suggested that the double layer had little effect as their length scales were much smaller compared to the van der Waals premelting distances used in their calculations. In agreement with the previous solute models, the V c derived for binary alloys was an order lower than the corresponding pure melts (see Fig. 1.22). The authors suggested that the lower values of V c were due to the increased particle-front gap induced by the constitutional undercooling. Therefore, the repulsive van der Waals was assumed to be considerably reduced owing to the increased particle-front separation.

A different approach towards modelling of solute effects was adapted by 1.5. Solute effects Chang et al. [35]. Here, the authors investigated the interaction of biological cells with an advancing solidification front to emulate cryopreservation. In this level-set model, the numerical simulations calculate the force on a particle without assuming a specific geometry for the problem, thereby improving the scope, range, and accuracy of the predictions. Therefore, the authors estimated the repulsive van der Waals forces between multiple surfaces with arbitrary geometries. Furthermore, the drag forces in the model were computed by a step method to avoid analytical solutions of the lubrication approximation. The model is hence rich in terms of the object behaviour predicted and requires much less approximations in contrast to analytical models. However, the study does not provide a systematic influence of the various solidification parameters on the particle behaviour and remains limited to two-dimensional systems.

Other level-set models, studying the interaction of particles with solidification fronts in presence of impurities, have been developed by Catalina et al. [42] and Yang et al. [47]. These models investigated the interplay between the particlemelt thermal conductivity mismatch (k p /k m = 1) and the role of solute on the shape of the interfacial curvature (concave or convex). The studies deduced that the diffusion of solute plays an important role on the solidification front shape, whereas the thermal conductivity ratio (k p /k m ) does not affect it. Thus, a concave curvature of the front has been predicted theoretically in the presence of solute.

Hence, the major conclusions of the theoretical models incorporating solute effects can be summarized as follows. The solutes are rejected by the growing crystal and segregate in the particle-front gap. This scenario applies to the systems where the solutes are less soluble in the solid phase in contrast to the liquid phase i.e. the partition coefficient K 0 << 1. The segregated solutes induce constitutional undercooling in the solute premelted films owing to their local enrichment, thereby rendering the solidification front concave. The concave geometry of the front facilitates the engulfment of the foreign objects at lower growth rates

(V sol ut i on c < V pur e mel t s c
) owing to the increased drag force (see section 1.2.3) and reduced repulsive van der Waals force. The ionic nature of solute has no impact on the force equilibrium and therefore, the double-layer forces can be neglected.

However, all of these conclusions are theoretical in nature.

CHAPTER 1. STATE OF THE ART

Experimental Results

I have presented till now an overview of the theoretical models developed over the years for pure melts and solutions. A plethora of analytical and numerical approaches predicting the outcome (engulfment or repulsion) of a foreign object interacting with an advancing solidification front were also discussed. I will now try to highlight the experimental studies conducted to understand the object behaviour during solidification and their subsequent correlation(s) to the theoretical models. To clearly distinguish the context and the domain where the mentioned studies were performed, I will discuss the results of hard and soft objects separately. Although a number of theoretical models have been derived, yet the experimental verification of these models remain extremely limited [2,15]. Solidification with objects is a complex phenomenon, where multiple interdependent parameters influence the object-front interaction. An important primary step is to observe, directly or indirectly, these interactions to improve their understanding. The typical features owing to these interactions include the front curvature (concave or convex), object behaviour (engulfment or repulsion), front morphology (planar, cellular, or dendritic), object spatial distribution, object deformation (or not), and premelted films. The essential factors affecting the solidification features are the growth rate V sl , temperature gradient G, particle size R, and the solute concentration C 0 . These factors or parameters need to be regulated efficiently to comprehend their systematic impact on the features obtained (see Fig. 1.23).

Experimental Results

Complex nature of interactions

The solidification features vary in their time and space scales over different orders of magnitude. For example, the liquid premelted film is supposed to be of ≈ 10 nm thickness, while the particle size can typically vary from 10 to 100 µm.

The typical time scale of interest depends on the crystal growth rate (1 ≤ V sl ≤ 40 µm • s -1 ) and the duration over which an object interacts with the interface (0 ≤ t ≤ 100 s). Thus, the major problem associated with studying solidification arises from the need of high space and temporal resolution in conjunction with the corresponding need of attaining the matrix melting temperatures (≈ -10 ≤ T m ≤ 1000 °C ). Therefore, the choice of the experimental observation technique depends largely on the features which needs to be analysed.

Tools used in metallurgy

In metallurgy, a significant number of studies performed in the past have relied on the post-solidification analysis to predict the object behaviour during solidification of particle-reinforced-composites [9,49]. Therefore, the initial observations were mostly qualitative and limited to the specific system utilised for the experiments. Subsequently, the use of low melting point organic analogues (see section 1.6.2) facilitated the in situ observation of object-front interactions with the aid of an optical microscope [7,50,51,52]. More recently, the in situ investigations at high temperatures have been performed for aluminium alloys using an X-ray transmission microscope (XTM) [42,53]. The main limitations include the spatial resolution at the particle level, high energy x-ray induced artefacts [START_REF] Sylvain Deville | Time-lapse, three-dimensional in situ imaging of ice crystal growth in a colloidal silica suspension[END_REF], and the visualization of solute segregation (see Fig. 1.24,1.25).

Tools used in biology

In cryobiology, the interaction of biological cells with an ice-water front have been often investigated using cryo-microscopy [15,[START_REF] Ishiguro | Mechanical interactions between ice crystals and red blood cells during directional solidification[END_REF]. The device is essentially an optical microscope combined with a freezing stage to facilitate imaging of a solidifying aqueous suspension. The advantages include resolution of the extracellular ice growth and the cellular reactions to the surrounding environment.

Körber et al. [START_REF] Ch Körber | Low temperature light microscopy and its application to study freezing in aqueous solutions and biological cell suspensions[END_REF] employed a spectrophotometer in their cryo-microscopy setup to determine the redistribution of solutes (cryo-additives) and the resulting concentration profiles photometrically or densitometrically. A direct visualization of the solute segregation, however, remained evasive. Further progress in this field has been through the design of computer operated slow cooling stages and the use of high optical resolution CCD cameras [START_REF] Bahari | Cryopreservation of adherent mammalian cells on substrate by slow freezing[END_REF]. This facilitates an accurate control over the applied crystal growth rate and the imposed temperature gradient (see Fig. 1.26). However, the imaging is still limited to 2D. demonstrated [START_REF] Dedovets | A temperaturecontrolled stage for laser scanning confocal microscopy and case studies in materials science[END_REF]. The segregation of the rejected fluorophores (from ice) facilitated the resolution of the expected solute concentration profile and helped in elucidating the role of solutes in freezing oil-in-water emulsions [START_REF] Dedovets | Five-dimensional imaging of freezing emulsions with solute effects[END_REF]. Presently, the use of this cryo-confocal microscopy setup, as employed in this thesis, is also helping to establish and correlate the biological cell's physico-chemical environment and the cell-front interactions during their directional freezing [START_REF] Qin | Unveiling cells' local environment during cryopreservation by correlative in situ spatial and thermal analyses[END_REF] (see Fig. 1.30).

Hard Objects (Ceramic, Metal, & Polymer Particles)

A major part of the previous studies investigating particle-front interactions has been to comprehend the development of the material microstructure. Of particular interest, the design and optimization of particle-reinforced metalmatrix-composites and single crystal growth (see Fig. 1.31). The objective of these studies was primarily to deduce the particle capture (or repulsion) to elucidate the spatial distribution of objects in the solidified microstructure [2]. The desired outcome ranges from instantaneous engulfment of particles in metal-matrix- composites to rejection of dissolved impurities in single crystals. Therefore, the studies have essentially focused on the estimation of the critical velocity (V c ), as a function of the various solidification parameters comprising the growth rate (V sl ), particle radius (R), melt viscosity (η), thermal gradient (G), and particle-melt thermal conductivity mismatch (k p /k m = 1).

The experimental studies on particle-front interactions have utilised different systems to study the impact of solidification parameters; particle type (e.g. latex, PMMA, SiC, Al 2 O 3 , Cu, Fe etc. . . ), particle size (3 ≤ R ≤ 1000 µm), growth direction (horizontal or vertical), and matrix type (e.g. Al-Si, camphor, salol, succinonitrile, water, Pb-Sn etc. . . ) [2].

The use of organic matrices instead of metals was mainly for two reasons.

First, most of the organic matrices are transparent in samples of moderate thickness and hence, the particles could be observed while they were being pushed or engulfed by the growing crystal. The other reason for the use of organic matrices is owing to their low melting point (0 ≤ T m ≤ 220 °C ) [7], which facilitates the experiments at laboratory scale. The choice of organic matrices with a low entropy of fusion (∆S f /R g < 2, where R g is the gas constant and ∆S f is the entropy of fusion), such as camphor, succinonitrile-acetone, also presents the benefits of a macroscopically smooth interface (non-faceted growth), similar to most metals. A non-faceted growth implies that the macroscopic interface follows the underlying temperature isotherms. In contrast, the use of high entropy of fusion matrices (∆S f /R g > 2), such as salol, thymol, can result in a macroscopically rough interface (faceted growth, e.g. needle-like). Thus, the prediction of a critical velocity (V c ) can be questionable if the front is faceted as most models are based on a macroscopically planar front [7].

A comprehensive review of the previous experimental work has been performed by Aubourg [7]. Three decades ago, Asthana & Tewari [2] did an extensive examination of the previous experimental results and the reliability of the various theoretical models in predicting the same. Both the studies have concluded that the different experimental data are difficult to compare and hence, the theoretical correlation remains suggestive rather than definitive. This implies that each theoretical model explains only a certain part of the experimental result and hence, cannot be utilised for all systems. The authors have suggested that the complex behaviour is owing to the lack of available experimental details (like growth rate, CHAPTER 1. STATE OF THE ART thermal gradient etc. . . ) and unknown exact particle shape and surface finish.

These parameters are essential for investigation and become critical as the forces are applied on a very limited region near the base of the particle. Furthermore, the experimental studies were unable to rigorously satisfy the assumptions of a particular theoretical model such as steady-state planar front, pure melts, faceted growth, etc. . . [2,7]. Thus, only some qualitative conclusions could be drawn from the previous experimental observations. In general, the critical velocity is a function of the size of the particle and varies approximately as V c ∝ R -α , where α varies between 1 and 3/2 (see Fig. 1.32). The critical velocity is observed to be generally decreasing with an increasing melt viscosity (V c ∝ η -x ). The impact of thermal gradient (G) on the critical velocity could not be estimated accurately. This is mostly owing to a low sensitivity in the regulation of temperature or a lack of suitable setups. Interestingly, none of the previous experimental investigators reported the observation of a premelted 1.6. Experimental Results film, an essential criterion for particle pushing. Finally, it was reported that a sharp transition between engulfment-repulsion at the V c does not exist, rather the particles can be pushed for a certain distance before engulfment (see Fig. 1.33).

This is further referred to as the pushing-engulfment or the repulsion-trapping transition [2]. Thus, the critical velocity appears to be a range of velocities and not a unique value for a given set of solidification parameters. The impact of particle-melt thermal conductivity ratio (k p /k m ) and solutes on the critical velocity presents conflicting results in the existing literature. Zubko et al. [START_REF] Zubko | Reaction of foreign particles with a crystallization front[END_REF] have qualitatively reported the repulsion of thermally insulating particles

(k p /k m < 1), while an engulfment of thermally conducting particles (k p /k m > 1)
in the solidification of metal-matrix system. Although they did not present results on the accurately measured critical velocity for their system. In contrast, Sen et al. [53] have reported the engulfment of thermally insulating particles (k p /k m < 1)

during the solidification of an aluminium-bubble system.

Körber et al. [15] have reported no impact of solute on critical velocity in water-NaMnO 4 solution, in contrast to the theoretically expected decrease in To ascertain the efficiency of a cryopreservation protocol, the biological cell viability is the most important criterion [46]. The cell viability, or population of healthy/living cells when freezing and thawing, is reduced drastically owing to two damage inflicting mechanisms; intracellular ice formation and osmotic injury [START_REF] Mazur | Freezing of living cells: mechanisms and implications[END_REF]. It is generally agreed that the primary cause of damage during freezing is chemical; related to the increased concentration of solutes (rejected during slow freezing) in the unfrozen environment [START_REF] Ishiguro | Mechanical interactions between ice crystals and red blood cells during directional solidification[END_REF]. The cells, mostly made of water, get dehydrated to be in equilibrium with the surrounding osmotic pressure [15,35].

More recently, Mazur et al. [START_REF] Mazur | Freezing of living cells: mechanisms and implications[END_REF] have suggested that the damage to red blood cells could also originate from the mechanical stress induced by the growing ice crystals (see Fig. 1.35).

A comprehensive review of the cryopreservation field has been complied by Körber [15]. Most studies in the domain of cryopreservation have experimentally investigated the freezing rate and the optimum concentration of additives to achieve an improved freeze-thaw cell viability [START_REF] Ishiguro | Mechanical interactions between ice crystals and red blood cells during directional solidification[END_REF][START_REF] Bahari | Cryopreservation of adherent mammalian cells on substrate by slow freezing[END_REF]. However, a systematic study of the cell-front interactions with the impact of corresponding solidification parameters is still lacking. To this effect, a recent study by Qin et al. [START_REF] Qin | Unveiling cells' local environment during cryopreservation by correlative in situ spatial and thermal analyses[END_REF] has demonstrated the importance of quantifying the local solute concentration, based on the approach invoked in this thesis, to help understand the cell viability during directional freezing (see Fig. 1.30).

Bubbles

The study of soft object deformation has been of particular interest in the prediction of pore shape evolution during directional solidification of crystals to avoid or control porosity defects [42]. Moreover, the science of bubble formation and subsequent trapping in ice also has climatic and geophysical applications.

Since gas solubility in solid is usually much less than in the liquid, the rejected gases accumulate ahead of the solidification front. Consequently, nucleation of bubbles at the front can be invoked by the local super-saturation (see Fig. 1.16).

The heterogeneous nucleation of gas bubbles on an advancing solidification front was investigated experimentally by Wei & Lee [START_REF] Wei | Nucleation of bubbles on a solidification front -Experiment and analysis[END_REF]. Their study tried to correlate the concentration of the rejected gases (similar to solutes) to the critical radii of heterogeneously nucleating bubbles. The shape of oxygen and carbon dioxide bubbles engulfed by a growing ice-water interface has been investigated by Wei et al. [START_REF] Wei | Growths of bubble/pore sizes in solid during solidification -An in situ measurement and analysis[END_REF]. Their study reported an initial spherical growth and a solidification-rate controlled elongation. Yoshimura et al. [40] have reported the observation of egg-shaped and cylindrical oxygen bubbles with an increasing growth rate, while using an ice-water system (see Fig. 1.36).

In contrast to the hard objects, the size of bubbles can change owing to the absorption of rejected gases during their interaction with a solidification front. The growth of bubbles at an ice-water interface at a high freezing rate with the presence of a premelted film has been observed by Maeno [START_REF] Norikazu | Air bubble formation in ice crystals[END_REF]. He also reported the formation of a cylindrical gas tube in ice, when the growth rate of the bubble was comparable to the imposed freezing rate. Another distinct behaviour reported in bubbles is their motion relative to the front instigated by a change in the bubble surface tension. This change can be invoked either due to the underlying temperature gradient and/or a solute concentration gradient in the vicinity of the front [32]. This thermo-solutal effect has been investigated experimentally by Wang et al. [START_REF] Wang | Behavior of fine bubbles in front of the solidifying interface[END_REF] who observed the horizontal motion of bubbles towards an ice-water front in the presence of a surfactant. However, a systematic analysis of the interaction of bubbles with an advancing solidification front is yet to be performed.

Droplets

Interaction of a liquid droplet with an advancing solidification front is commonly encountered in the freezing of emulsions. Emulsions are thermodynamically metastable systems containing droplets of one immiscible liquid dispersed in another. Most studies on the freezing of emulsions are related to food engineering and hence, investigate the resistance of food emulsions to freeze-thaw cycling [START_REF] Ghosh | Factors affecting the freeze-thaw stability of emulsions[END_REF]. The major objective in this domain is the preservation of size, shape, and distribution of the liquid droplets to ensure longevity and conserve the original taste as well as texture. Moreover, freezing helps to delay spoilage by microbial growth and enzyme activity [START_REF] Shafiur | Handbook of food preservation[END_REF]. The emulsions being meta-stable tend to demix, a process driven by the interfacial tension γ, i.e. the free energy per unit area of liquid-liquid interface. In addition, they can be destabilised upon thawing, leading to a phase separation that is highly undesirable. Thus, they often require stabilisation by surfactants or solid particles which adsorb onto the liquid-liquid interface and limit phase separation [START_REF] Bernard | Colloidal particles at liquid interfaces[END_REF]. The interaction mechanisms between the droplets and the solidification front could be compared to those encountered in bubbles. The similarities arise from an expected deformation and the presence of a temperature-concentration dependent interfacial tension. However, the realistic food emulsions are quite complex and consist of several components includ-1.6. Experimental Results ing various solutes. Thus, fundamental work on model emulsion systems was certainly required to understand the interactions in these systems.

The directional freezing of monodisperse oil-in-water emulsions to comprehend the droplet-front interactions was performed by Dedovets et al. [START_REF] Dedovets | Five-dimensional imaging of freezing emulsions with solute effects[END_REF] in LSFC. The study observed the impact of solute (surfactant added to stabilise the emulsion) on the behaviour of oil droplets near an ice-water interface. They deduced that the development of the solidification microstructure was controlled by the solute effects; observing droplet motion at typical distances of the order of ≈ 100 µm. Therefore, the authors further suggested the presence of long-range solute forces in contrast to the short-range intermolecular forces discussed previously. Moreover, the presence of solute premelted films between the droplet and the growing ice crystal as well as in the frozen ice phase was also reported (see Fig. 1.37). In conclusion, the study suggested that the freezing behaviour of oil-in-water emulsions could be a analogue of many solidification systems where objects interact with a solidification interface.

The interaction of soft objects is thus relevant to many situations and their behaviour is closely related to the description of hard objects interacting with a solidification front. Although the experimental and theoretical investigation of the soft object systems is much less developed in contrast to the hard particles.

Thus, the lack of attention to this domain and the study from Dedovets et al. [START_REF] Dedovets | Five-dimensional imaging of freezing emulsions with solute effects[END_REF] was the motivation for the work carried out on freezing emulsions during this thesis.

Towards complex. . . more realistic. . . systems

The existing theories and experimental evidences have been primarily derived for isolated, hard, and smooth spherical objects, whereas most real-life applications involve multiple particles with arbitrary shapes and surface roughness.

In realistic systems, the complex nature of the object interactions is increased owing to a higher concentration of the suspended particles. Furthermore, the idealistic assumptions of the theoretical models such as the non-retarded van der Waals forces, absence of solute segregation, and a steady-state planar front interaction are difficult to achieve practically [23]. The transition from a planar to a dendritic front morphology owing to a high particle concentration in composites has been reported theoretically [45,[START_REF] Dutta | Directional dendritic solidification of a composite slurry: Part II. Particle distribution[END_REF]. Progress in studying the spatial distribution, when thousands of particles interact with a solid-liquid front, has been brought about by modern advanced in situ imaging techniques, such as X-ray imaging [START_REF] Spannuth | Particle-scale structure in frozen colloidal suspensions from small-angle x-ray scattering[END_REF], confocal microscopy [START_REF] Dedovets | A temperaturecontrolled stage for laser scanning confocal microscopy and case studies in materials science[END_REF], or the development of numerical modelling techniques, such as molecular dynamics [START_REF] Bouville | Self-assembly of faceted particles triggered by a moving ice front[END_REF][START_REF] Stephen | Structural properties of materials created through freeze casting[END_REF]. However, the underlying questions concerning the evolution of solidified microstructure with the associated complex front morphology remain unresolved. The complex scenario of multiple particles can be partly understood by building on the single-particle models. Isolated particles are rejected at the solid-liquid interface when the growth rate is lower than the critical velocity (V sl < V c ). In 1.7. Towards complex. . . more realistic. . . systems real-life systems, the particles being pushed get accumulated ahead of the front and eventually interact with the other particles, thereby forming a dense particle layer (see Fig. 1.38). The thickness of the particle layer grows until a force equilibrium, similar to isolated systems (F η = F σ ), is reached. Recently, Saint-Michel et al. [START_REF] Saint-Michel | Interaction of Multiple Particles with a Solidification Front: From Compacted Particle Layer to Particle Trapping[END_REF] have proposed a mechanical model to understand these effects of multiple particle interactions on particle trapping by a solidification front. Their study highlights the major differences in the force equilibrium when considering a multi-particle approach. The model suggests that the accumulated particle layer acts as a porous medium and offers resistance to the fluid flow resulting in an additional frictional force (F µ ), which facilitates particle engulfment. Hence, the modified force equilibrium for multiple-particles can be given as

F η +F µ = F σ .
Thus, this reinforces the importance of investigating experimentally the role of particle size distribution, particle-particle interactions and hence, the formation of agglomerates.

Current issues and challenges

In this chapter, we have seen the theoretical background explaining the different isolated object-front interaction models and the definition of the governing equations. I have invoked the various solidification mechanisms, parameters, and features, which influence the behaviour of an individual foreign object in the vicinity of an advancing solidification front. I have also presented a detailed description of the available experimental techniques and their corresponding advantages or limitations in observing the solidification features of interest. Furthermore, the accord as well as the contrasts in the experimental studies and the theoretical predictions were presented. Finally, the complex nature of the realistic systems where multiple objects can additionally interact among themselves was introduced.

The dynamics of interfaces and objects, although an essential aspect of these phenomena, remains extremely difficult to study experimentally. The typical time scale (growth rate of a few µm • s -1 ) and space resolution (particles of a few micrometers) make an in situ characterization almost impossible with the current approaches. To these difficulties can be added the high temperature required for most materials undergoing solidification, especially in the case of metals and single crystals. Furthermore, many of these phenomena occur in the presence of solutes (or impurities), which can segregate at the interface during the growth of a solid. The resulting solute gradients sometimes play an essential role, while their characterization in situ remains problematic. Moreover, the existing theories and experimental evidences have been primarily derived for isolated, hard, and smooth spherical objects, whereas most real-life applications involve multiple particles with arbitrary shapes and surface roughness.

Much of the progress in the domain of object-front interactions has been achieved through numerical simulations and post-solidification analysis [48,[START_REF] Wei | Pore shape development from a bubble captured by a solidification front[END_REF].

In situ experimental evidence of solidification dynamics has been facilitated using transparent analogs with optical microscopes at ambient temperatures [50,52].

However, the use of an optical microscope does not permit three-dimensional imaging, an essential criterion to comprehend these complex phenomena. Moreover, it can be difficult to distinguish the various phases present such as the solid crystals, secondary objects, and solutes (or dopants). Recently, X-ray transmission microscopy has been employed for the in situ evaluation of object-front interactions at elevated temperatures [42,53]. However, the spatial resolution

Scope of the thesis

This doctoral thesis has been written as a compendium of several shorter scientific or academic papers. Therefore, the chapters 2, 3, and 4 are written to respond independently to the different aspects of the existing problems as described over the course of chapter 1.

In chapter 2, we start by studying the simplest system where an isolated object interacts with an approaching ice-water interface, using cryo-confocal microscopy. The objective is to visualize the actual front geometry when an object approaches a solid-liquid interface and correlate the findings to the existing literature. We demonstrate the complete shape of the planar front curvature, for hard particles and soft objects, having a thermal conductivity mismatch with the pure melt. We then compare our experimental data with the theoretical predictions from the Park et al. [32] model, and discuss the distortion of the temperature field in the vicinity of a foreign object. We further explore the role of solute and depict the modifications to the expected front geometry, using the same set of objects, owing to the solute effects. We also report on the impact of object-melt thermal conductivity mismatch for cellular front morphology. We note the dominating effect of solute and discuss its impact on the formation of a thick premelted film. We thus show experimentally the formation of a premelted film and distinguish between the object-front interactions in pure melts and in solute solutions.

In chapter 3, the interaction of spherical oil droplets with a planar ice-water interface is investigated. We demonstrate the systematic influence of three solidification parameters, namely the bulk solute concentration, the growth rate, and the object size, on monodisperse oil-in-water emulsions subjected to directional solidification. The use of soft objects is also to provide insights into the mechanisms at play when a deformable object encounters a solidification front. We first study the in situ solidification dynamics of the droplets in water, far from the solidification front. We measure the typical distance at which the droplets start feeling the presence of an advancing interface and their consequent displacement owing to their dynamic interaction. We compare our results and discuss the observed disparities with the previous theoretical models. We then characterise the deformation of the droplets and depict their shape evolution as they are hit by the approaching solidification front. We attempt to understand the distinct deformation regimes observed and try to relate it to the presence of 1.9. Scope of the thesis a solute-enriched premelted film. As the droplets get engulfed in ice, we report on the dynamic evolution of premelted films and show a droplet deformation unlike the previous observations. The study highlights the dramatic impact of an increasing bulk solute concentration and the absence of theoretical models incorporating such effects.

Finally, in chapter 4, we try to bridge the gap between an ideal isolated object regime and the complex realistic systems, by experimentally studying the interaction of multiple objects with a solid-liquid interface. The goal is to elucidate an in situ development of a material microstructure in the presence of particle-particle interactions and solute effects. We use oil-in-water emulsions as a simplified model analogue system to understand the diverse interactions possible in a complex system. To perform a systematic analysis, we start with a monodisperse oil-in-water emulsion and study the influence of two solidification parameters; droplet size and bulk solute concentration. A modification of the force dynamics following the development of agglomerates, owing to the high particle concentration of the suspension, is depicted experimentally. We then discuss the current theoretical understanding of multiple-particle interactions with regards to engulfment and repulsion, by building on the predictions of the isolated object models. We further demonstrate the solidification of bimodal emulsions, where two colonies of droplets with distinct radii are mixed together.

The influence of an increasing bulk solute concentration is equally investigated with bimodal emulsions. To imitate a realistic situation more closely, we finally investigate the interaction of polydisperse oil droplets and note the formation of droplet clusters. The study helps in visualizing the evolution of complex relevant microstructures, in the presence and absence of solute effects, which are difficult to predict theoretically.

Chapter 2

Objects interacting with solidification fronts: Thermal conductivity effects

Abstract

Objects, such as droplets, bubbles, or particles, interact with a moving solidification front in several material processing routes. The consequences of this confrontation, varying from instantaneous engulfment to complete rejection of suspended objects, regulates the solidified microstructure and thus, the functional properties of the material. In this study, we utilize in situ cryo-confocal microscopy to investigate the interplay between thermal conductivity mismatch (of the object and the melt) and the role of solute, which both impact the geometry of the ice-water solidification front in different manners. First, we experimentally validate the theoretical predictions of physical models on the distortion of thermal fields in the presence of foreign objects. Subsequently, we report on the dominating effect of solute on the front curvature with the dynamic evolution of premelted films. In the case of cellular front morphology, we depict that particles with a lower thermal conductivity than water induce a distortion of the encircling ice crystals, thereby modifying the final microstructure of the solid. Finally, our results demonstrate that to successfully predict and control the solidification microstructure, in the presence of objects, physical models incorporating both thermal and long-range solute effects are required.

Keywords

Introduction

The interaction of solidification fronts with objects (droplets, bubbles, solid particles or biological cells) is a common phenomenon encountered in a plethora of situations, ranging from industrial to natural occurrences, such as the formation of sea ice, growth of single crystals, metallurgy, cryobiology, or food science. The objects (soft or hard) exhibit different types of behavior while interacting with a solidification front, from total rejection to complete or partial engulfment [1].

The dynamics of this interaction influences the solidified microstructure and the mechanical and functional properties of the materials. From homogeneous particle distribution in particle-reinforced metal matrix composites to complete rejection of inclusions or porosity management in castings and growth of single crystals, distinct outcomes may be desired depending on the application. The potential to actively govern the solidification microstructure is thus crucial.

The first experiments on the interaction of particles with a solidifying icewater front were performed by Corte [2] for geological applications. Uhlmann et al. [3] established a criterion of critical velocity (V c ), for a given size of an insoluble object, to predict whether the object will be engulfed (V sl > V c ) or rejected (V sl < V c ), by a freezing front moving at a velocity V sl . Their thin film approach to investigate the engulfment and rejection of particles considered the change in chemical potential of the liquid in the particle-front gap (repulsive) and the viscous drag forces (attractive) to deduce the particle behaviour. The analytical models, utilizing the Uhlmann et al. approach, derive the critical engulfment velocity (V c ) as the maximum particle velocity when the particlefront gap (d ) is minimized and assumed equivalent to a few molecular diameters (d ≤ 10 nm) [4,5,6]. However, these models involve parameters such as change in interfacial energy (solid-object) which are difficult to determine experimentally, and consider idealized planar front geometry in the presence of particles [1].

Chernov et al. [7,8] suggested that the particle behaviour in vicinity of a solidifying front is governed by an interplay between repulsive van der Waals and attractive viscous drag forces. Their study, assuming a paraboloid frontal shape, was the first to consider distortion of the temperature field owing to the differing thermal conductivities of the particle and the melt (k p = k m ). They demonstrated that the shape of the solidification front plays a major role in Rempel et al. [12,13] used the Chernov et al. approach and made significant improvements to the existing analytical models by including the Gibbs-Thomson effect to predict the entire shape of the front. Their model equates the repulsive disjoining forces, originating from inter-molecular interactions, to the attractive viscous forces, emanating from the fluid flow around the particle calculated using a lubrication approximation. The study formulated an analytical relationship for the critical engulfment velocity with the various process parameters such as size of particle (R), temperature gradient (G), and type of inter-molecular interactions.

However, the front shape is predicted by assuming equal thermal conductivity of the particle (k p ) and melt (k m ).

A similar approach of thin-film lubrication approximation was used by Park et al. [14], while incorporating the effects of thermal conductivity difference

(k p /k m = [-1, 1]
). This is important as the difference in thermal conductivity between the particle and the melt should deform the front 10 3 times more than the other parameters [14]. Moreover, the front geometry is the key to determining THERMAL CONDUCTIVITY EFFECTS the forces on the particle and hence predict the particle behaviour [5,6,8,10,15].

An object with a different thermal conductivity from the melt will modify the local heat flow, thereby altering the isotherms locally. As we assume the solid-liquid front to follow the isotherm for the melting point of the matrix material, this difference of thermal conductivity can bend the front towards or away from the object, depending on whether the thermal conductivity of the object is higher or lower than the melt. The previous studies have thus often suggested that if a particle has a lower thermal conductivity than the melt (k p /k m < 1), the solidification front bulges towards the object and may repel it. In the opposite case

(k p /k m > 1)
, the solidification front bends away from the particle and facilitates its engulfment [5,6,16].

The thermal conductivity ratio criterion is not extensive and has been modified in certain experimental predictions of repulsion and engulfment of foreign particles [17]. Few in-situ experimental observations exist to ratify these analytical predictions, owing to the associated small time and space scales, as well as high temperatures of solidification, especially for metals [18,19,20,21,22]. Yemmou et al. [19] presented the first experimental evidence of convex and concave geometries, using an ice-water system, with large (d i amet er > 0.3 cm) Nylon and stainless steel particles. Subsequently, the real-time evolution of solidification front curvature was reported for 500 µm Z r O 2 particles in metal matrix composites by Sen et al. [21], using X-ray transmission microscopy.

The presence of solute (desired or as an impurity) in the melt may also influence the interaction between a particle and a solidification front, as solutes are rejected by the solid (with low partition coefficient) and segregate at the solid-melt interface, leading to a colligative depression of the melting temperature [7,9,10].

The high concentration of solute is further enhanced by the obstruction of solute diffusion away from the front in presence of an insoluble object [15]. The shape of the front conforms to the local solute concentration and hence a concave curvature is predicted by the previous studies [9,10,15,23,24]. Catalina et al. [22] have depicted schematically the change in shape and local solutal segregation in the presence of gas porosity using X-ray transmission microscopy with metallic systems.

Here, we study in a systematic manner the influence of the thermal conductivity of the object on the ice-water solidification front curvature using in situ cryo-confocal microscopy. We report on the front shape evolution using soft objects and rigid particles, representing an extensive range of thermal conductivity

Materials and methods

We perform solidification experiments with dilute aqueous dispersions of droplets, bubbles, and particles, of similar diameter ≈ 50 µm, but varying thermal conductivity (0.04 ≤ k p /k m ≤ 35). We regulate the solidification front velocity (V sl ) by translating a Hele-Shaw cell, under a confocal laser scanning microscope, along a temperature gradient (G) imposed by two Peltier modules, as shown in Fig. 2.2.

The independent selection of the two parameters (V sl & G) in our setup enables a uniform cooling rate and an improved control over the front morphology, compared to the setups where the sample is frozen either at a constant temperature or at a constant cooling rate. In this study, we performed the solidification experiments at a constant front velocity (V sl ) with a stable linear temperature gradient (G).

Materials

Propyl benzoate, TWEEN80, Difluoro2-[1-(3,5-dimethyl-2H-pyrrol-2-ylidene- N)ethyl]-3,5-dimethyl-1H-pyrrolato-Nboron (BODIPY),
= 0.569W • m -1 • K -1 .

Sample Preparation

The oil-in-water emulsions were prepared by using a microfluidic setup (microfluidic starter kit, Micronit Microfluidics, Netherlands) with pressure controlled flow pumps (Fluigent LineUP Flow EZ) and uncoated focused flow droplet generator chips (FF_DROP), with a nozzle diameter of 50 µm. The flow rate of oil and aqueous phases were controlled using Fluigent Flow Unit S (0 -7 µL • mi n -1 ). The oil phase consisted of propyl benzoate with 10 -4 M BODIPY, and the aqueous phase was a 10 -5 M SRhB solution. The air-in-water foams were prepared by using the double syringe technique [25], with a luer adapter as connective, and a liquid fraction of 50 %. For the foams, we bubbled air through liquid perfluorohexane (C 6 F 14 98+%, Alfa Aesar) to obtain a gas composed of air and perfluorohexane vapour. Thus, we could delay coarsening [26] and ensure the bubbles remained stable during the experimental time-scales, typically varying from 4 to 6 hours at V sl = 10 -6 m • s -1 . TWEEN80 was used as a non-ionic surfactant and solute to avoid long-range electrostatic interactions. We used the same surfactant for stabilising both the oil-in-water emulsions and the air-in-water foams. The particle suspensions were thoroughly sonicated (30 -40 mi n) to ensure a homogeneous dispersion. The experiments in the presence of solute were carried out by adding 1w t .% TWEEN80 to the aqueous phase. We used the same solute and concentration for all systems to ensure comparable results. The solutions were frozen in a rectangular Hele-Shaw cell (hei g ht = 100 µm and vol ume = 100 µL), fabricated using two glass slides (Menzel, 24 × 60 mm, thickness 0.13 -0.16 mm), and sealed with nail-polish at one end to avoid evaporation. We do not observe convection effects in our experiments as they are performed in a closed Hele-Shaw cell of small thickness (100 µm) and at low solidification velocity of 1 µm • s -1 with a steady linear temperature gradient of 10 4 K • m -1 .

Freezing Stage

We performed directional freezing experiments, translating the sample along a constant linear temperature gradient of 10 4 K •m -1 , using the cryo-confocal stage described previously [27]. We carried out all of the experiments at a solidification front velocity V sl = 10 -6 m • s -1 , unless stated otherwise. The solidification front tends to appear immobile in the frame of observation, however, in the sample frame, it is the ice solidifying (along x ) at the velocity imposed by the pulling rate of the motor (Micos Pollux Drive VT-80 translation stage PI, USA). We modified the stage by substituting the aluminium plate with a copper serpentine sheet and discarding the silicon carbide honeycomb interface. This led to an improved stability and a more robust cooling setup. The temperature gradient was imposed with two Peltier modules and controlled using TEC-1122 Dual Thermo Electric

Cooling Temperature Controller from Meerstetter Engineering, Switzerland. We can thus decouple and control independently the solidification velocity (V sl ) and the temperature gradient (G).

Imaging & Analysis

The images were acquired through a Leica TCS SP8 confocal laser scanning microscope (Leica Microsystemes SAS, Germany), equipped with 488 nm (blue) and 552 nm (green) lasers. We utilised the microscope at a scanning speed of 600 Her t z, with 1024 × 1024 pi xel s for imaging 775 × 775 µm, resulting in 1.7 s per frame. The high spatial resolution, coupled with rapid screening, enabled us to observe the dynamics of objects interacting with solidification fronts in situ.

We used two photodetectors (PMT) to simultaneously image three phases :

1. BODIPY (λ ex 493 nm ; λ em 504 nm), the dye incorporated into the oil droplets.

The same emission spectrum was utilised to image the fluorescent colloids.

2. SRhB (λ ex 565 nm ; λ em 586 nm), dissolved in water, to image the aqueous phase and the cells boundaries in ice. 3. Ice does not fluoresce as it has very low solubility for solutes [28] and hence, appears black.

The emission spectra of the excited fluorophores was captured using a nonimmersive objective (Leica HCX PL APO CS 20×). The working distance of 590 µm and an insulating foam cover facilitates the minimization of thermal perturbations on the freezing substrate. We used Fiji [29] and Python for image processing and data analysis.

Typical solidification parameters

Parameters Particle in water the melt to the front resulting in a concave depression. Thus, the differing thermal conductivities of the particle and melt result in a distortion of the isotherms away from the horizontal, which is in accordance with the previous theoretical and experimental evidence [6,8,16,21]. In contrast, the presence of a polyethylene (PE) particle does not perturb the shape of the ice-water front owing to its magnitude of thermal conductivity being close to that of water (k p /k m = 0.879, see Fig. 2.3B,C). From Fig. 2.3B, we notice that the shape of the front in vicinity of a spherical object does not change for an air bubble, liquid oil droplet, or a solid particle. This is owing to the fact that the front curvature depends on the ice-water interfacial tension, disjoining pressure, and the particle to melt thermal conductivity ratio [14]. Hence, we do not expect the surface energy of the objects to play a major role.

G (K • m -1 ) 1.0 × 10 4 T m (K ) 273.15 L v (J • m -3 ) 3.03 × 10 8 σ sl (J • m -2 ) 3.0 × 10 -2 V sl (m • s -1 ) 1.0 × 10 -6 λ (m) 2.83 × 10 -10
These observations are in agreement with the numerical model from Park et al. [14]. This model is based on the computation of particle velocity close to the front (d << R, see Fig.

Results and discussion

We can see from Fig. 2.4A that when k p /k m > 1, the front is always concave and has a depression away from the object. While for k p /k m < 1, three curvatures are feasible corresponding to concave, concave-convex, and convex. The domain of concave-convex geometry is limited to higher γ regions while it reduces to a confined zone when k p /k m is close to 1 and γ < 10 -1 . We could not investigate this region as it requires either the use of minute objects (R < 5 µm) or the application of a small temperature gradient (G < 10 2 K • m -1 ), other parameters being constant for the particle-in-water system. While using minute objects is inconceivable as the interfacial curvature would be below the optical spatial resolution, using a smaller temperature gradient would render a planar front thermodynamically unstable owing to the supercooling [30].

The deflection of the horizontal solid-liquid front instigated by the presence of a foreign object is shown in Fig. 2.4B. The values on the plot represent the maximum deflection from the bulk melting-temperature isotherm (T m ), where positive values denote a concave depression and negative values denote a convex bump. Ideally, the absolute value(s) of deflection should increase with a decreasing thermal conductivity ratio for convex curvatures and vice-versa for concave fronts, as predicted previously [6]. However, we report no change in deflection magnitude (for a given front shape) with the thermal conductivity ratio k p /k m , as demonstrated in Fig. 2.4B. This might be explained by the dominant role of solid-liquid interfacial tension (included in γ) near the base of the objects, which tends to flatten the curvature, in contrast to the thermal conductivity effects [14].

Moreover, this contradiction can also originate from the use of theoretical models which ignore the effect of latent heat and assume equal thermal conductivities of the melt and the solid [21].

Interestingly, there seems to be no conspicuous digression in the dynamics of objects with k p /k m < 1 or k p /k m > 1 (Fig. 2.3C), which depict similar behaviour as can be seen in the time-lapse figure (Fig. 2.3A). We do not observe repulsion of neither the soft objects (oil droplet and air bubble) nor the thermally insulating rigid particles in the system investigated, which rather tend to be engulfed instantaneously. This trend is contrary to the previous studies [6,16], where a convex front is assumed to increase the critical engulfment velocity (V c ) and thereby facilitate repulsion of the object through a reduction of lubrication drag forces in the gap between the particle and the front [14]. The engulfment of thermally insulating Z r O 2 particles in Aluminum matrix (k p /k m < 1) has been reported

previously by Sen et al. [21]. However, their study reports the observation of a THERMAL CONDUCTIVITY EFFECTS concave trough as the object approaches the solidification front, irrespective of the particle to melt thermal conductivity ratio. We did not observe the formation of such a concave trough while a thermally insulating particle encountered the ice-water front, as shown in Fig. 2.3A,B. All these predictions and results were obtained so far in pure systems. However, almost all real melt systems consist of solutes in some form or the other, be it desired (for imparting better mechanical and/or physical properties) or undesired (such as impurities, inclusions etc. . . ). It has been shown recently that the dynamics of interaction of an object with a solidification front and the critical role of solutes have been highly underestimated in the studies so far [31]. We cannot use the existing models as they consider a dominant role of thermomolecular forces, which are effective only at very short distances (∼ 10 -10 m) [9,15,24,32],

while solute fields are long range(∼ 10 -4 m) [1,31]. We therefore investigate how solute impacts the interfacial curvature.

Solute effects override thermal conductivity effects

We carried out freezing experiments with the same set of objects (oil droplets, gas bubbles, and particles) in the presence of 1w t .% TWEEN80 (acting as a model solute) added to the aqueous phase. This surfactant, needed to stabilize the bubbles and drops, is used primarily as a model solute as it decreases colligatively the freezing point in all systems. We assume the magnitude of thermal conductivity ratio of the objects to melt (k p /k m ) as shown in Fig. 2.3C remains unmodified.

Thermal conductivity is a material property that depends primarily on temperature, and thus, we expect no significant changes to its value while adding small amounts of solute to water. We observed a convex interfacial curvature in the absence of solute with k p /k m < 1 (see Fig. 2.3B,C), however, in the presence of solute, solely concave curvatures are observed (Fig. 2.5A,B). In the presence of thermally conducting objects (k p /k m > 1, see Fig. 2.5B), the front exhibits a much more pronounced concave depression as compared to the geometry observed in the absence of solutes (Fig. 2.6) [22]. Thus, the solute tends to play a dominant role and renders the solidification front concave regardless of the thermal conductivity ratio, as predicted previously in theory [9,15,22,23,24].

As an insoluble object approaches a solidification front, it obstructs the diffusion field, and leads to a build-up of solutes (rejected by ice) at the front [15]. Since growth from solutions depends on the concentration gradient of solutes at the front, the liquidus temperature of the melt differs from the melting temperature of the bulk composition. The change in liquid composition alters its transformation temperature, referred to as constitutional supercooling [33]. Therefore, the solute-rich region colligatively depresses the freezing point of the aqueous phase. This distortion of the isotherm away from the horizontal manifests itself as the concave depression of the solidification front [9,10,15,23,24]. The presence of solute might lead to a variation of the ice-water interfacial tension (σ sl ), which is complex to evaluate experimentally. However, we believe that the influence of solute, by instigating constitutional supercooling, dominates over the interfacial energy and controls the front curvature, as stated in the past studies [9,15,22,23,24].

The resulting thick film, visible in the high fluorescence crescent (Fig. 2.7)

between the object and the growing ice, is driven by the tendency to reduce the interfacial free energy and known as a premelted film [34,35,36]. The films are thermodynamically stable below their bulk melting temperature and are extremely sensitive to the presence of solutes (known as solute premelting). The solute premelted films are quite evident in these experiments (Fig. 2.7), however, we do not observe any interfacial premelting in the absence or at low concentrations of solute (Fig. 2.3A). The concave curvature of the front facilitates lateral solute diffusion causing the front to accelerate, while engulfing the object with a solute-rich premelted film around it [22,32].

These results depart from the behaviour predicted in the analytical models incorporating solute effects [9,10,15,24]. In these studies, solute fields are expected to affect the curvature of the solidification front and hence the particle behaviour. The models certainly utilise the concave curvature of the front to calculate the modified force equilibrium but still consider a dominant role of inter-molecular forces (d < 10 nm). In our experiments, a thick premelted film (≈ 5 µm) prevents the contact of the object with the growing crystal during engulfment, suggesting an absence of repulsive short-range van der Waals forces. The presence of a thick premelted film is universal and manifests itself in all systems investigated including oil-in-water droplet, air-in-water bubble, and rigid solid THERMAL CONDUCTIVITY EFFECTS particles as shown in Fig. 2.5A.

We believe the long-range solute field interactions (with a length scale of ∼ D/V sl , where D is the diffusion coefficient of the solute and V sl the solidification front velocity) dominate over the short-range thermomolecular forces. This enforces the importance of purely diffusive (solute) effects during solidification as they not only alter the interfacial curvature but also the nature and equilibrium of forces on the particle. Solutes impact the premelted film thickness [35], viscosity, undercooling [13], and can induce phoresis of suspended objects [31,37] during freezing. Therefore, the prediction of the object behaviour when it encounters a solidification front is not trivial. We suggest that the thermal conductivity ratio (k p /k m ) does not affect neither the front morphology nor the particle-front interactions in presence of solute as opposed to pure materials. The diffusion of solute tends to override and dominate the particle-front interaction in impure materials. Hence, the formulation of models, for concluding whether an object will undergo engulfment or rejection, needs to be modified to account for longrange solute field interactions. However, modelling of the solute effects is beyond the scope of this study.

Solute and thermal conductivity effects in cellular growth

We have discussed so far the interfacial geometry for an ideal planar front morphology preceding particle engulfment. We will now focus on the breakdown of the planar front and the subsequent impact of thermal conductivity mismatch (between object, melt and solid), when a particle is engulfed. Most realistic systems present a scenario where a planar shape is thermodynamically not stable [30]. This can be attributed to a high freezing velocity, constitutional supercooling, and the complete rejection of solutes from the solid phase, which produces a severe concentration gradient at the front [33,38]. The resulting perturbations destabilise a steady-state planar front into regularly spaced cells through the propagation of Mullins-Sekerka instability [39]. Although these instabilities have been investigated for a long time, few numerical simulation models [23,40,41,42] and experimental evidence [32] describing the capture of isolated particles by a non-planar (cellular or dendritic) front morphology exist.

Furthermore, these studies discuss the nature and equilibrium of forces pertaining to the induced front curvature before engulfment or entrapment. However, we report on the interplay between thermal conductivity mismatch (of object and matrix) and solute effects before, during, and after engulfment with three At high solidification velocity (V sl = 10 × 10 -6 m • s -1 ) with 1w t .% TWEEN80 in the aqueous phase, we obtain a cellular front morphology with solute-rich cells boundaries (Fig. 2.8). For bubbles (Fig. 2.8A), the front tends to be moderately convex (t ≈ 3 s) with a deflection away from the horizontal, owing to the extremely low thermal conductivity of air, while it transforms into a concave depression in the proximity of the bubble (t > 3 s). For the oil-in-water emulsion (Fig. 2.8B), the solidification front stays at the same ordinate before the droplet impingement THERMAL CONDUCTIVITY EFFECTS (t < 3 s), contrary to the depression in planar morphology (Fig. 2.5A). The stainless steel particle (Fig. 2.8C), having a higher thermal conductivity than water, induces a concave cusping of the front (t ≈ 3 s), which is further enhanced by the constitutional supercooling (t ≈ 6 s).

The solutes are partially segregated into the cell boundaries and hence, the effective concentration gradient at the ice-water front is considerably reduced compared to the previous planar front situations. Cellular fronts redistributes the rejected solutes parallel to the temperature gradient. This diminishes the extent of constitutional supercooling, thereby averting an extensive cusping of the front and preventing a dominant role of solutes before engulfment [22]. Thus, the effective distance between the object and the ice at which solutes modify the interfacial geometry of a cellular front is considerably smaller compared to a planar front morphology [32].

After engulfment, the insulating particles (k p /k m < 1)(Fig. 2.8A,B) deform the cell boundaries in their vicinity (t > 10 s, inset box) and alter the ice tip radius. We believe the thinning of the cells around these particles is essentially related to the local temperature gradients originating from the difference in thermal conductivity between the particle and the encircling crystal. The engulfed objects remain at relatively higher temperatures, in contrast to the surrounding ice crystals, for sufficient amount of time to modify the solidified microstructure. Indeed, the cells boundaries do not undergo geometrical modifications (Fig. 2.8C) with a stainless steel particle, as it is highly thermally conductive (k p /k m > 1), and therefore, homogenises its temperature with the surrounding ice instantaneously [19].

The rapid cryo-confocal microscopy enables us to follow in situ the evolution of the microstructure. The front eventually heals and recovers its original cellular spacing, which is a function of temperature gradient, solidification velocity, and solute concentration gradient [33,39]. The freezing of suspensions with a cellular front is an attractive route for processing porous ceramics, metals, and even composites [43]. An improved understanding of the critical parameters is thus desired to regulate the front morphology and the resulting material properties [44]. The size and spacing of cells is an indispensable attribute controlling the mechanical properties of a polycrystalline material (Hall-Petch relationship) [45,46]. Consequently, the heterogeneous microstructure can be detrimental to an otherwise homogeneous finished product. For example, the object-melt thermal mismatch can lead to a variation in the inter-lamellar spacing around the particle during solidification of particle reinforced alloys [5]. Thus, solutes play a key role in 2.4. Conclusions solidification and an in-depth understanding of their redistribution mechanism is required.

Conclusions

Cryo-confocal microscopy [31] exhibits great promise for the in situ analysis of solidification mechanisms with foreign objects. We have demonstrated the role of a thermal conductivity mismatch of the object and the melt on the distortion of solidification fronts in the absence of solutes. The use of laser fluorescence enabled us to visualize the front geometry and thereby quantify the maximum solid-liquid front deflection induced by the presence of foreign objects. Furthermore, we have discussed the solidification dynamics of an extensive range of insulating and thermally conducting objects, comprising air bubbles, oil droplets, and solid particles.

In the presence of solute, the long-range solute field interactions play a critical role on the front curvature and the evolution of the solidification microstructure.

We have suggested that the solutes impact not only the front geometry but also Chapter 3

Solute effects on dynamics and deformation of freezing droplets Abstract

Soft or rigid objects, suspended in a liquid melt, interact with an advancing solidification front in various industrial and natural processes, such as fabrication of particle-reinforced-composites, growth of crystals, cryopreservation, frost heave, and growth of sea ice. The object-front interaction can have diverse outcomes depending on the solidification parameters and the physical characteristics of the object-melt-solid system. The approaching front can engulf the object immediately, push the object indefinitely in the remaining liquid ahead of itself, or repel the object for some time before capturing it in the growing solid. Hence, the object dynamics relative to the front determine the microstructure as well as the functional properties of the solidified material. The previous studies have extensively investigated the interaction of foreign objects with a moving solidliquid interface in pure melts. However, in most real-life systems, the probability of encountering a pure melt is rare and solutes (or impurities) are almost always present. Here we show experimentally the interaction of spherical oil droplets with a moving planar ice-water interface, while systematically increasing the solute concentration in the bulk liquid, using in situ cryo-confocal microscopy.

We demonstrate that a small amount of solute in the bulk liquid can instigate long-range droplet repulsion, extending over a length scale of 40 to 100 µm, in contrast to the short-range predicted previously (< 1 µm). We report on the droplet deformation, while they are in contact with the ice-water interface, as FREEZING DROPLETS a function of the bulk solute concentration, the droplet size, and the crystal growth rate. Furthermore, we depict the dynamic evolution of solute-enriched premelted films (≈ 5 µm) and the role of solute in instigating the crushing of droplets captured in the growing ice crystal. Our results demonstrate how an increasing concentration of solute in the bulk and its subsequent segregation during solidification can dramatically alter the predicted nature of object-front interactions and the solidification microstructures. We anticipate that our experimental study can serve for the development of theoretical models incorporating solute effects. Finally, we suggest that the freezing of oil-in-water emulsions may serve as an analogue for studying the in situ interaction of foreign objects with an advancing solid-liquid interface in the presence of solute effects.

Keywords: solidification, solute, droplets, deformation, dynamics

Introduction

The interaction of objects with an approaching solid-liquid interface is of special relevance in nature, like frost heave, glacial motion [1], and in engineering sciences, such as food freeze-thaw stability [2], cryopreservation [3,4], metallurgy [5], and crystal growth [6]. This dynamic problem consists of objects, soft or hard, dispersed in a liquid melt interacting with a solid-liquid interface. The objects can be biological cells in cryopreservation [3,4], colloids in freeze-casting [7],

droplets in food preservation [8], gas bubbles in growth of single crystals [6,9] and metallurgy [5], or reinforcing particles in material-matrix-composites [10].

The outcome of the object-interface confrontation determines the solidified microstructure and hence, the functional properties of the solidified material.

Thus, it is essential to understand the underlying mechanisms of solidification (or freezing) to forge the required material microstructure. The object can interact with an advancing solidification interface with diverse outcomes; the object can be engulfed instantaneously upon contact, the object can be pushed ahead in the remaining liquid by the interface indefinitely, or the object may undergo engulfment after being pushed over a certain distance [11].

The dynamics of interaction and spatial distribution of objects are essential features in several material fabrication routes. In particle-reinforced-composites, an instantaneous engulfment in solid phase is desired as it facilitates a homogeneous distribution of the suspended objects [10]. In metallurgy and crystal growth, a continuous repulsion of the impurities by the interface is required to obtain impurity-free-solids [5]. In freeze-casting, a segregation of the colloidal particles by the growing solid enables a porous material structure [7]. The shape of the object (deformed or undeformed) becomes an equally important processing criterion in applications where soft deformable objects (droplets or bubbles) encounter a moving solid-liquid interface. In cryobiology, utmost care should be taken to avoid physical injury inflicted to the cells from the growing ice crystals [12]. In food preservation, freezing is conducted to conserve the size, shape, and distribution of the dispersed droplets, thereby preserving the original taste and appearance [8]. Thus, numerous interaction scenarios, still poorly understood, can exist during solidification.

The role and concentration of solute in the solidifying liquid is often significant and a dominating factor in determining, amongst others, the shape of the solid-liquid interface in the vicinity of the suspended objects [13,14,15]. Solutes, FREEZING DROPLETS render the interfacial curvature concave, thereby promoting engulfment of objects at growth rates lower than those predicted for planar curvatures [16,17].

The solutes can be either desired, like additives (e.g. cryoprotectant used in preservation of biological cells), or be present as an undesired impurity, such as dissolved gases (H 2 ) in liquid metals. The segregation of solutes at the interface is instigated by their relatively low solubility in the solid phase and enhanced further by the approaching objects obstructing their diffusion field [13]. This local solute enrichment is of particular importance in understanding the nucleation and growth of macroporosity in solidifying melts [14], studying the constitutional supercooling with formation of premelted films [18,19], and in determining the osmotic stresses acting on a freezing biological cell [4,12] to give a few examples.

Moreover, the morphology of a solid-liquid interface (planar, columnar, or dendritic), determined by the magnitude of solute concentration gradient build-up ahead of the growing solid, plays a major role on the final microstructure [13,20].

The study of solidification mechanisms such as the object dynamics, object shape evolution, and the local solute concentration are complex and require in situ investigations. The major problem associated with studying solidification in situ arises from the need of high space and temporal resolution in conjunction with elevated temperatures especially for metals. For the object dynamics, past studies have formulated a plethora of analytical and numerical models expressing the outcome (engulfment or rejection) of objects interacting at close distances (< 10 nm) with the solid-liquid interface [16,21,22,23,24]. The models vary in the mathematical formulation of the features taken into account (e.g. the inclusion of object-melt thermal conductivity mismatch, solute effects etc. . . ), while using a similar approach (balance of repulsive and attractive forces between the object and front) to describe the interaction [11]. The study of object deformation has been of particular interest in the prediction of pore shape evolution during directional solidification of crystals to avoid or control porosity defects. Much of the progress in this domain has been achieved through numerical simulations and post-solidification analysis [25,26]. In situ experimental evidence of solidification dynamics (repulsion or engulfment) and shape modification have been facilitated using transparent analogs (e.g. succinonitrile-acetone) with optical microscopes at ambient temperatures [20,27] and X-ray transmission microscope at elevated temperatures [14,28]. However, the volume investigated along with the temporal resolution is limited, while the local solute segregation cannot be visualized. Hence, the tracking of microstructures where objects interact with a 3.2. Methods solid-liquid interface in the presence of solute effects remains challenging.

In light of these uncertainties, we analyse the interaction of spherical oil droplets with an advancing ice-water interface using in situ cryo-confocal microscopy. The objective of our study is to perform directional solidification experiments to comprehend the mechanisms responsible for the development of a solidification microstructure in the presence of foreign objects. Our custom solidification setup enables a small temporal resolution (1.7 s) and high spatial resolution (1024 × 1024 pixels )with laser induced fluorescence to distinguish the rejected solutes and hence, visualize the local solute segregation. To regulate the bulk solute concentration, we prepare oil-in-water emulsions using microfluidics with a varying amount of surfactant Tween 80 (acting as a model solute) in the aqueous phase. We further characterise and report on the in situ droplet dynamics, droplet deformation, and the role of solute in modifying the expected droplet behaviour.

We investigate the impact and consequences of an increasing bulk solute concentration on the mechanisms involved at three different stages: droplets in water (water) far from the solidification front, droplets in contact with an approaching solid-liquid interface, and droplets captured in the growing solid (here ice). For the object-front interaction dynamics, we depict the distinct behaviours of the oil droplets such as instantaneous engulfment and repulsion-engulfment transition. For the object shape evolution while confronting a solid-liquid interface, we correlate the deformation of the oil droplets to the imposed growth rate and the bulk solute concentration. Furthermore, we show the impact of solute on the ice-water meniscus and the evolution of premelted films surrounding the oil droplets in ice. Finally, we depict that the three interaction stages are crucial in determining the object spatial distribution, object shape evolution, and hence, the solidification microstructure. Thus, we suggest that the freezing of oil-in-water emulsions with solute effects can be a versatile approach towards developing an analogue for objects interacting with an advancing solidification front.

Methods

Materials

We purchased the oil (propyl benzoate), surfactant (Tween 80), oil fluorophore We prepared the oil-in-water emulsions using a microfluidic setup, as explained in our previous study [15]. The monodisperse droplets have radii (R 1 , R 2 ) of either 7.2±0.4 µm or 30.9±1.2 µm, as shown in Fig. 3.1. The oil phase consisted of propyl benzoate with 10 -4 M BODIPY to obtain clear imaging of dispersed droplets at 1% laser power. For the aqueous phase, we used 10 -5 M SRhB solution, as self-quenching was reported at concentrations above 2 × 10 -4 M [29]. We added Tween 80 (H LB = 15 [30]), a non-ionic surfactant, to the aqueous phase also acts as a model solute since it colligatively depresses the freezing point of solutions, when its concentration increases locally [15]. We prepared three aqueous solutions with 0.01, 0.1, and 1 w t .% Tween 80 to study the impact of solute on the solidification dynamics and behaviour of oil droplets dispersed in an aqueous phase. The surfactant is added as w t .% of the aqueous solution to have an equal concentration in all the solutions prepared.

Sample Preparation

The concentration of surfactants in the bulk solution at which micelles start forming is known as the cmc. All additional surfactant added above the cmc should form micelles. Individual surfactant molecules that are in the system but are not part of a micelle are called monomers [32]. At 1w t .% of Tween 80 in aqueous solution we are ≈ 600× cmc. Hence, ideally we should have an excess of surfactant at the three concentrations used in the experiments. We assume (as is generally accepted) that the concentration of surfactant added below the cmc remains as dissociated monomers, while the concentration above the cmc results in micelles. The oil-in-water emulsion with the surfactant added in excess to the aqueous phase is represented schematically in Fig. 3.2. The presence of micelles at the given concentration was confirmed by dynamic light scattering (DLS) analysis. A typical micelle size of 9 nm with a corresponding diffusion coefficient of 30 µm 2 • s -1 at 273 K was obtained from the DLS analysis. The prepared emulsions were filled through capillarity and solidified in a rectangular Hele-Shaw cell (height =100 µm and volume =100 µl ). We fabricated the Hele-Shaw cell using two glass slides (Menzel, 24 × 60 mm, thickness 0.13 -0.16 mm), and sealed it with nail-polish at one end to prevent evaporation and leakage.

Imaging & Analysis

We used a Leica TCS SP8 confocal laser scanning microscope (Leica Microsystemes SAS, Germany) equipped with 488 nm (blue), 552 nm (green) lasers and two photodetectors (PMT) for image acquisition. The images were captured for the emission spectra of BODIPY (oil phase) and SRhB (aqueous phase), using a non-immersive objective (Leica HCX PL APO CS 20×) with a working distance of 590 µm. Ice does not fluoresce and hence, we can simultaneously detect three phases (oil droplets, unfrozen aqueous phase, and ice) with two photodetectors. In 2D, we used the microscope at a scanning speed of 600 H z, with 1024 × 1024 pi xel s for imaging 775 × 775 µm, resulting in 1.7 s per frame. We used Fiji [33] for image thresholding in conjunction with Python [34] for image and data analysis.

Freezing Stage

Results & Discussions

We conducted unidirectional solidification experiments, translating the sample cell at a constant velocity (V sl ) along a constant linear temperature gradient (G) of 10 4 K •m -1 , using the cryo-confocal stage described in detail previously [35].

We imposed the temperature with two Peltier modules, and controlled it with high precision (< 0.01 °C ) using TEC-1122 Dual Thermo Electric Cooling Temperature Controller from Meerstetter Engineering, Switzerland. The Peltier elements were separated by a distance of 2 mm to establish a linear temperature gradient along x. The in situ observation of objects interacting with the solid-liquid interface was achieved using a confocal microscope mounted vertically over the gap (2 mm) between the two Peltier modules, as shown in Fig. 3.3. The solidification interface tends to appear immobile in the frame of observation, however, in the sample frame, a solidifying front at the velocity V sl is advancing in the sample (along x). We utilised the VT-80 translation stage (Micos Pollux Drive PI, USA) to impose the rate at the which the sample cell is pulled through the temperature gradient. The rate of translation was verified to be in agreement with the measured solidification velocity (V sl ), using posterior image analysis (er r or < 1 %).

Thus, we can decouple and control independently the solidification velocity (V sl ) and the thermal gradient (G) in our system.

We performed the solidification experiments in the velocity range of 1.0 ≤ V sl ≤ 10.0 µm • s -1 . We wait for 20 -30 mi ns to ensure a steady-state diffusion controlled regime before starting the acquisition at a given solidification velocity. The time needed for a steady-state to establish scales as 2D/V 2 sl ≈ 60 s, where D is the solute diffusion coefficient. We do not expect forced convection in our experiments as they are performed in a closed Hele-Shaw cell of small thickness (100 µm) and at low solidification velocity with a steady linear temperature gradient. The solid-liquid interface is stable over extended time periods (≈ 4 -5 hour s) and the interface does not accelerate or decelerate during the solidification experiments. In addition, we do not observe a transient unsteady regime and the interface morphology is stable for the given experimental parameters.

Results & Discussions

We performed horizontal solidification experiments by displacing a Hele-Shaw cell, containing an oil-in-water emulsion, in a custom cryo-confocal stage, as shown in Fig. 3.3. A typical 2D confocal image of a freezing emulsion with the FREEZING DROPLETS distinct features observed is shown in Fig. 3.4A. The confocal image enables us to distinguish three phases; oil phase in cyan (fluorophore BODIPY), water in colormap viridis (fluorophore SRhB), and a dark ice phase. The solid-liquid phases are delimited by an interface advancing, at a growth rate V sl , under the influence of a linear temperature gradient G = ∇T of 10 4 K •m -1 (see Fig. 3.4A). As solidification progresses, the growing ice phase rejects the dissolved solutes owing to its low solubility (K 0 << 1, where K 0 is the partition coefficient) and hence, appears black. A representation of the rejected solute field in steady-state planar growth is shown in Fig. 3.4A. We also depict the presence of premelted films, which can remain liquid at temperature below the equilibrium melting temperature (T < T m ) and increase in thickness owing to the presence of dissolved solutes [1].

In Fig. 3.4A, we can observe the premelted films between two ice surfaces with different grain orientation as well as around the oil droplets captured in the ice phase.

A typical time-lapse evolution of an isolated oil droplet interacting with the ice-water interface is shown in Fig. 3.4B. The images in Fig. 3.4A and Fig. 3.4B are representative of the freezing of an oil-in-water emulsion with 1w t .% Tween 80 (solute) in the aqueous phase. In this study, we discuss the consequences of the interaction of oil droplets with the solid-liquid interface at three different stages, as described in Fig. 3.4B, while systematically varying the concentration of solute.

First, we investigate the solidification mechanisms at play when the oil droplets are far from the advancing interface. The aim is to study the role of an increasing bulk solute concentration and the different solidification parameters (growth rate, object size) on the droplet dynamics in water (t < 44 s in Fig. 3.4B). Subsequently, we look into the impact of the approaching solid-liquid interface on the droplet shape upon their mutual contact (44 ≤ t ≤ 57 s in Fig. 3.4B). The objective here is to comprehend the object shape evolution in the presence (or not) of solute effects and the corresponding impact of increasing the object size and growth rate. Finally, we analyse the droplets captured in ice and report on the evolution of premelted films with the associated ice-water meniscus (99 ≤ t ≤ 111 s in Fig. 3.4B).

Droplets in water

The ice-water interface is stationary in the frame of observation, while in the sample frame it is in motion at a velocity of V sl along x (see Fig. 3.4A). We calculate the isolated droplet velocity U r , when the ice-water interface is close and investigate how the droplet speed, relative to the front, depends on the solidification parameters. Our Hele-Shaw cell of length 4 cm facilitates an acquisition of 50 --400 isolated droplet velocities in the vicinity of an advancing solidification front and hence, we can deduce the mean droplet velocity as U = 〈U r 〉. A positive magnitude of U implies that the droplet is repelled or pushed by the moving ice-water interface towards the remaining liquid. A detailed description of the analysis performed along with the python script employed can be found in a previous study [19] and in the supplementary data (Fig. 3.14, Fig. 3.15). In Fig. 3.5, we present the mean droplet velocity, U , with the distance to interface for the trajectories investigated at a growth rate V sl of 3 µm • s -1 . We define the distance to interface as 0 µm when the front edge of the droplet comes in contact with the absolute detected position of the ice-water interface.

The droplets in water are unperturbed at distances of ≤ -100 µm and start getting repelled at the approach of the solid-liquid interface. The mean droplet velocity (U ) increases and exhibits a maximum (U max ), when the leading edge of the droplets coincides with the initial position of the growing crystal. As the droplets are captured in ice, their velocity returns to zero. We distinguish the influence of the bulk solute concentration, the droplet size R and the growth rate V sl on the mean droplet velocity U during freezing. U decreases with an increasing growth rate and larger droplet size, while it increases with an increasing amount of bulk solute concentration.

Results & Discussions

We define a characteristic length scale, L v , distance from the interface at which the droplets attain a mean velocity (U ) of 0.1 µm • s -1 , as shown in Fig. 3.6.

We find that the droplets get repelled over distances ranging between 10 and 100 µm, often larger than their diameter, especially for the largest solute concentrations. Smaller droplets with radius R 1 are repelled at greater distances from the interface as compared to the larger R 2 droplets. An increase in the solute concentration leads to higher values of L v , however, increasing the growth rate decreases the characteristic distance L v .

In Fig. 3.7, we report on the evolution of U max with the solute concentration, the droplet size, and the growth rate. Similar to U and L v , we observe an increase of U max owing to the strong influence of an increasing solute concentration. In contrast, an increasing growth rate leads to a decrease of the U max . The smaller R 1 droplets get pushed away from the growing solid at higher velocities as compared to the larger R 2 droplets. To understand the impact of various solidification parameters on the droplet dynamics in water, we depict the interaction time (see Fig. 3.14). This term includes the complete duration where the droplets in water interact with the solidliquid interface, as shown in Fig. 3.8. In general, we notice that the interaction FREEZING DROPLETS time diminishes with an increasing solidification velocity (V sl ) and reduces to zero at higher growth rates (V sl ≥ 5 µm • s -1 ), independent of the solute concentration. An interaction time of 0 s implies that the droplets get engulfed into the ice instantaneously and hence, do not interact with the advancing solid-liquid interface. In contrast, the interaction time increases strongly with the solute concentration at relatively lower growth rates (V sl ≤ 3 µm • s -1 ). Consequently, the droplets undergo considerable horizontal displacement from their expected position in the presence of high solute concentration and at low solidification velocities. We demonstrate the droplet displacement owing to their interaction with an approaching ice-water interface in Fig. 3.9. The droplet displacement, in other words, represents the distance moved by the droplets during the interaction time. We observe that the solute concentration and the imposed growth rate are the two most important criteria for determining the interaction time as well the displacement of a droplet in the presence of an approaching ice-water interface. The results deduced so far emphasize that an increasing amount of solute induces large displacements and repulsion of the droplets over tens of microns away from the interface. Before characterizing the solute concentration profile close to the solidifying front and its influence on the droplet dynamics, let us comment on the previous literature models describing the interaction of a solidification front with particles. Most of these models did not include the influence of solute on the engulfment process. The initial studies performed by Chernov et al. [36,37] laid the foundation for the thermomolecular force equilibrium approach for deriving a critical engulfment velocity (V c ), which is now generally accepted. The particle behaviour in vicinity of a solid-liquid interface is governed by the interplay between repulsive van der Waals and attractive viscous drag forces [36,37]. Rempel et al. [23] developed further on this approach and deduced an analytical model to predict whether an isolated object will be engulfed (V sl < V c ) or rejected (V sl > V c ), by a moving solid-liquid interface. The value of V c is given in table 3.1 for our experimental parameters using the model from Rempel et al.; V c = (σ sl A 2 ow /6 5 π 2 η 3 R 4 ) 1/3 , where

σ sl = 3 × 10 -2 (J • m -2
) is the ice-water interfacial tension, A ow = 1 × 10 -20 (J ) is the oil-water Hamaker constant, and η = 1.8 × 10 -3 (P a • s -1 ) is the dynamical viscosity of water.

In accordance with the model from Rempel et al. [23], isolated R 1 (7.2 ± 0.4) droplets at low solute concentration (0.01 w t .%) exhibit repulsion close to the interface at V sl of 1 µm • s -1 and get engulfed at higher velocities

V sl ≥ 2 µm • s -1 . FREEZING DROPLETS R (µm) V c (µm • s -1 ) 7.2 ± 0.4
1.35 ± 0.11 30.9 ± 1.2 0.19 ± 0.01 Table 3.1: Typical critical velocity for an isolated object encountering an icewater interface, derived from Rempel et al. [23].

However, this theoretical model cannot account for the long-range repulsion between the droplets and the solidification front, observed in our experiments, at high solute concentrations (≥ 0.01 w t .%) for the two droplet sizes investigated.

The results presented so far underline that the droplet dynamics is influenced by the bulk solute concentration, the growth rate and the droplet size. These three parameters control the local gradient of solute concentration close to the solidification front. Indeed solutes present in the melt are rejected by the solid during freezing. The solid phase (here ice) has a low partition coefficient (K 0 ) in contrast to the liquid phase (here water) and hence, the solutes rejected by the solid segregate at the solid-liquid interface, as shown in Fig. 3.10A. In a closed system, the solute concentration close to the solid-liquid interface is therefore higher as compared to the bulk solute concentration far from the interface. We also expect that the local gradient of solute concentration spans over a typical distance L d = D/V sl in the steady-state regime [38].

An increasing growth rate (V sl ) will decrease the distance at which the solute field can be perceived by the droplets and can potentially explain the decreasing values of L v measured (see Fig. 3.6) at high growth rates (V sl ≥ 5 µm •s -1 ). Furthermore, we expect that the size of the droplets, which obstructs the solute diffusion close to the front, will also influence the gradient of solute concentration between the droplets and the front. As already explained in a previous article [19], we suggest that the displacement of the droplets is caused by solute concentration gradients close to the ice-water interface, possibly through a phenomenon called diffusiophoresis [39], according to which solute concentration gradients lead to displacement velocities of the order of microns per seconds, comparable to the droplet velocities that we measure in our study.

To further study the influence of the solute on the droplet dynamics, it would be useful to measure in situ the solute concentration close to the front. However, the solute employed in our experiments, Tween 80, does not fluoresce. Therefore, it is impossible to measure its concentration profile in situ. For steady-state planar growth, the concentration field of rejected solutes in the remaining liquid is given as

C L = C 0 +C 0 1 -K 0 K 0 exp -V sl D x (3.1)
where C L is the concentration, C 0 is the bulk solute concentration in the liquid far from the interface, D is the solute diffusion coefficient, and x is the distance from the interface [38]. To verify whether our experiments are performed in steady-state conditions, we record the fluorescence intensity of the dye, SRhB, which can be obtained easily with the confocal microscope (see Fig. At 3 µm • s -1 , the concentration profile obtained is in agreement with the steady-FREEZING DROPLETS state diffusion profile predicted from Eq.3.1 (see Fig. 3.10B). We note that the steady-state approximation is not valid at higher velocities (not shown) probably because constitutional undercooling modifies the local temperature of the front hence the diffusion coefficient of the dye.

Interestingly, at 3 µm •s -1 , the distance at which the dye concentration profile starts increasing significantly is of the order of 150 µm. This value is much higher than the typical distance L v over which both the large and small droplets get repelled from the interface. In Fig. 3.6, the values of L v range between 40 and 80 µm depending on the solute concentration and the droplet size. Although the solute monomer and the dye have the same diffusion coefficient, it is difficult to know whether the solute monomers behave as the dye, as we are above the cmc of the solutes in all cases and the excess of solute rejected by the ice may incorporate into the micelles rapidly.

An alternative possibility could be that the droplet dynamics is controlled by the concentration gradient of solute micelles instead of the concentration gradient of solute monomers. We measured the diffusion coefficient of the micelles and calculated the corresponding concentration profile using Eq.3.1. As shown in Fig. 3.10B, at a growth rate of 3 µm • s -1 , we obtain a typical distance L d of 30 µm, which is slightly lower than the distance L v measured experimentally at this growth rate.

Droplets at the interface

We now focus on understanding the behaviour of droplets when they come in contact (distance to interface = 0 µm) with the solid-liquid interface. We observe three typical behaviours of oil droplets as they encounter an approaching front, as shown in Fig. 3.11. The droplets can elongate permanently as they get engulfed in the growing ice (Fig. 3.11A), the droplets may deform transiently (t = 13 s) at the ice-water interface and subsequently relax to their original spherical shape as they move further into the ice phase (Fig. 3.11B), or the droplets can remain mostly spherical during their engulfment by the growing crystal (Fig. 3.11C). We notice that the deformation behaviour depends on the droplet size R, the imposed growth rate V sl , and the bulk solute concentration. Therefore, we need to systematically study the effect of these solidification parameters to understand the different types of deformation observed.

In light of the situation, the droplet deformation was estimated from the analysis of 2D shape elongation, as shown in the schematic in Fig. 3.11A, taking the ratio of droplet diameters along x and y. In Fig. 3.12, we depict the mean elongation profiles calculated for 50 to 400 (depending on R and V sl ) droplet interactions in 0.01 w t .% and 1 w t .% solute solution at varying growth rates for two different droplet sizes (R 1 , R 2 ).

We observe from Fig. 3.12A and Fig. 3.12B that the droplets undergo permanent elongation for the two droplet sizes (R 1 , R 2 ), when the bulk solute concentration is 0.01 w t .%. The elongation is ≈ 1.0, representing a circle, when the droplets are in the remaining liquid far from the interface. The droplets start getting elongated as their front edge touches the interface (distance = 0 µm) and their shape transforms into an ellipse (elongation > 1.0). The droplet deformation evolves further and reaches a constant magnitude when the front edge is located at a distance of 2R × El ong at i on. Once the droplets are completely engulfed in the ice, their shape does not evolve any more (Fig. 3.12A,B). Interestingly, we notice that the elongation reduces with an increasing growth rate for both the droplet sizes investigated. However, the maximum elongation for the smaller R 1 droplets is lower as compared to the larger R 2 droplets at the given 0.01 w t .% In Fig. 3.12C, we report the transient deformation of the oil droplets as they confront the ice-water interface with 1 w t .% bulk solute concentration. Here, the oil droplets undergo elongation at the interface (distance =0 µm) but eventually recover their shape as they are completely engulfed in the ice. In contrast, from Fig. 3.12D we notice that the larger R 2 droplets do not undergo any type of deformation at the same solute concentration of 1 w t .%. Hence, the elongation profile of R 2 droplets remains unmodified during the droplet-interface confrontation.

We deduce from these observations that the droplets confronting an approaching interface have distinct behaviours depending on the concentration of solute in the bulk solution. In particular, an increasing solute concentration tends to decrease the droplet elongation significantly. According to the steadystate diffusion model (see Eq.3.1), an increasing bulk solute concentration (C 0 ) implies a higher concentration of the solute segregated (C 0 /K 0 ) at the solid-liquid

Results & Discussions

interface (see Fig. 3.10). The segregation of solute is further enhanced owing to an obstruction of their diffusion field by the foreign objects in the vicinity of the solid-liquid interface [13,40]. The segregated solute, trapped in the layer between the droplet and the interface, induces solute premelting [41,42], as shown in Fig. 3.4B at t = 44 s, in Fig. 3.11B at t = 8 s, and in Fig. 3.11C at t = 3 s. The solute premelting is a colligative effect and occurs as a bulk phenomenon, whereby the equilibrium melting temperature of water is lowered by the presence of another component in the mixture, the solute Tween 80 in our case [38]. The driving force for the premelted films is a lowering of the surface free energy of the system [43].

The premelted films are stable below the solid's bulk melting temperature, T m , and at sufficiently large solute concentrations, the thickness of the premelted films is dominated by the colligative effects [18,41]. Thus, an increase in the thickness of the premelted films should also be instigated by an increasing growth rate owing to the reduced time available for solute diffusion. In our experiments, we observe that the droplet elongation decreases both with an increasing growth rate (see Fig. 3.12A,B) and for higher bulk solute concentrations (see Fig. 3.12C,D).

In consequence, an increase in these two solidification parameters (C 0 and V sl ) also favors the formation of a thick solute segregation enhanced premelted film (see Fig. 3.4B and Fig. 3.11B,C). Hence, we suggest that the deformation of the droplets depends on the thickness of the solute invoked premelted films.

When a liquid droplet is in contact with another fluid, a curvature of the interface generates a Laplace pressure (∆P ), the pressure in the droplet being higher than in the surrounding bulk liquid phase. The lower the radius of curvature the higher the Laplace pressure. In the absence of any external force, the radius of curvature is constant and equal to the radius of the droplets leading to a spherical shape, thereby minimizing the surface energy of the system. In the presence of an external force, such as shear forces, the shape of a droplet is dictated by the balance between Laplace pressure and such external stress. The presence of a liquid-liquid interface between the droplet and the premelted film probably enables the recovery of a spherical shape which is driven by the Laplace pressure of the drop. Indeed the stress generated during the contact of the solid-liquid interface seems to induce the deformation of the premelted film rather than of the droplet. For example, in Fig. 3.11C we observe that the premelted film-ice interface forms a highly curved neck during the droplet engulfment (t = 10 s), while the premelted film-droplet interface depicts a small curvature in comparison.

In the absence of a thick premelted film around the droplet, the stress gener-FREEZING DROPLETS ated during the droplet-interface interactions are balanced by the large Laplace pressure corresponding to a higher elongation and curvature of the droplets (see Fig. 3.11A). Furthermore, we believe the size effect in elongation for the same solidification parameters (C 0 and V sl ) arises from a difference in the associated Laplace pressure (∆P ∝ R -1 ) of the oil droplets. In Fig. 3.12A versus Fig. 3.12B, the smaller R 1 droplets manifest lower elongation owing to their relatively greater Laplace pressure in contrast to the larger R 2 droplets. We have discussed so far the dynamics and consequences of oil droplets interacting with an approaching ice-water interface. In the last section, we investigate the fate of objects and premelted films after their engulfment by the ice front.

Droplets in ice

Previous results [19] have shown how the premelted films at the cells boundaries can destabilize into droplets. Here we focus on two additional behaviours: the crushing of the engulfed oil droplets, and their possible movement by regelation.

We first focus on the shape evolution of the oil droplets after they pass the wa- with the moving ice/water interface.

The mean elongation profiles of the two droplet colonies in Fig. 3.13B emphasizes the monotonic nature of the deformation. This implies that the elongation continues to increase as the droplet moves further into the ice phase. We investigated the nature of the growing ice-water meniscus to understand the origin of the crushing phenomenon. To determine the exact three dimensional shape of ice inside the Hele-Shaw cell, we looked at the orthogonal projections (with y as an invariant) along the sample depth z (see Fig. 3.13C). We can see from the orthogonal view along the vertical cross-section of a 3D image that the ice-water interface is curved.

The crushing phenomenon seems to be instigated by the curved morphology of the advancing ice-water interface. Ideally, one would expect to observe a concave ice-water meniscus owing to the interfacial excess energies between the solid, liquid, and the wall in a glass micro-capillary. The theoretical work of Acker et al. [44] and the experimental study from Liu et al. [45] have highlighted the role of solutes, in addition to the interfacial forces, in modifying the confined freezing behaviour. The sample cell used in our experiments has a depth of 100 µm and hence, we can expect a role of interfacial energies and solutes to be at the origin of the observed ice-water curvature. The solute rejected by the ice gets segregated at the interface and can diffuse over a length of 30 µm at the imposed growth rate.

The increase in fluorescence intensity from the rejected die molecules ahead of the interface helps us to visualize the expected diffusion regime (see 3.13C).

The aqueous solution in the vicinity is thereby enriched with the solutes and exhibits a varying osmotic pressure relative to the pure ice phase (K 0 << 1). Thus, we believe that the osmotic pressure gradient in conjunction with the differing interfacial energies (ice, water, and glass) might induce the observed ice-water FREEZING DROPLETS concave meniscus. The curvature results in an excess of Laplace pressure in the ice phase, which could be at the origin of the observed crushing phenomenon.

Finally, regelation can occur, leading to a displacement of objects engulfed into the ice [46]. The droplets engulfed in ice with 1 w t % solute in solution (see Fig. 3.4B, Fig. 3.13A) are surrounded with a solute-rich premelted film forming a comet-tail segregation [19]. The liquid layer disjoins the droplet from the growing solid and decreases in thickness as the droplets move along the temperature gradient in ice towards an increasing undercooling (∆T = T m -T , where T is the temperature of the substrate) [1]. The thermal fluctuations of the premelted layer are known to be at the origin of regelation of particles engulfed in a solid [47]. The thermal or pressure induced regelation can cause migration of particles through ice, whereby the particles translate towards the warmer side of the underlying ∇T , owing to a thermomolecular pressure gradient [42]. However, limited experimental data are available on regelation. Here, we do not observe regelation of the oil droplets at our experimental time-scales, probably owing to their large diameters. The objects known to exhibit significant regelation are micron-size or smaller. In our scenario, the droplets trapped in ice do not manifest any motion relative to the ice, as depicted in Fig. 3.5. Additional work will be required to investigate in depth the possible regelation of oil droplets. More specifically, the use of smaller droplets (difficult to stabilize) will be required, as well as longer experiments with an immobile ice/water interface, more suitable to detect slow migration velocities.

Conclusions

In the first section, we have investigated the role of solutes on the dynamics of freezing oil droplets using in situ cryo-confocal microscopy. We have reported on the significant magnitude of droplet-front interaction time and the resulting droplet displacement owing to the solute effects. We believe this can be an important criterion for controlling the spatial distribution of objects, especially in multiple object scenarios. The control of the material microstructure has been a topic of interest in alloy solidification (especially in particle-reinforcedcomposites) and for impurity control in castings and single crystal growth. The object dynamics and impact of solute (or impurity) are still complex to quantify as the observation of solidification in situ remains challenging. We have tried to advance towards an in situ quantification of the solute mechanisms at play and further work is required to correlate the dye fluorescence intensity to the absolute solute concentration gradient. The latter can be useful in predicting thermal convection and non steady-state solidification regimes.

In the second section, we have shown that the deformation of droplets at the ice-water interface depends strongly on the growth rate (V sl ) and the corresponding bulk solute concentration (C 0 ). Furthermore, the addition of solute increases the thickness of the observable premelted films, which appears to act as a protection mechanism against the interface initiated droplet deformation.

The local solute environment and deformation are two important criterion for cryopreservation in particular. In cryobiology, the excess of solute causes severe osmotic stresses that can instigate cell membrane rupture and hence, cryoinjury to cells and tissues [4,12]. In food engineering, alterations to the continuous phase concentration or to the shape and size of dispersed droplets is detrimental to the freeze-thaw stability of consumable emulsions [8]. Hence, a complete understanding of the solute redistribution mechanisms along with the associated object deformation at the corresponding freezing conditions is desired. Our multi-dimensional approach highlights the importance of different solidification parameters and the ubiquitous role of solute in dominating the various aspects of object deformation behaviour. Further work is required to accurately estimate the direction and magnitude of forces at the origin of the observed deformation.

In the last section, we have demonstrated the crushing of droplets in the ice phase, when the ice-water meniscus is curved, owing to a low growth rate and high solute concentration. We have also reported an absence of thermal regelation for the oil droplets captured in the ice phase. The study of particle migration in ice is an important topic of research to understand frost heave, glacier motion, and ice-core dating among other technological applications [1]. Recent studies suggest that the presence of impurities or solutes tend to accelerate the regelation of trapped particles in ice. Typical migration velocities of 0.1 µm • s -1 at ∆T of 1 K have been reported for micron sized particles in the presence of impurities [48]. Another recent study highlights the major impact of impurities on the rapid displacement (0.5 cm) over small time scales (120 s) of 1 µm silicon particles trapped in ice [49]. The use of cryo-confocal microscopy with the ability to image the solute segregation has a promising prospect for investigating such mechanisms. We believe the high space and temporal resolution can be used effectively to resolve the dynamics of individual colloidal particles to gain further insight into regelation. FREEZING DROPLETS

In conclusion, we report that the oil droplets undergoing directional solidification feel the impact of solute at every stage, from being in the liquid phase to getting captured by the growing ice-water interface. The solute plays an important role in determining the droplet shape (deformed or not), droplet behaviour (engulfment or rejection), and eventually the droplet spatial distribution. To predict the solidified microstructure, an understanding of the several in-situ mechanisms at play is therefore indispensable. The use of rapid cryo-confocal microscopy facilitates an in situ investigation and quantification of solidification mechanisms with visualization of the local solute segregation. To represent the observations in real-life systems, we need to explore models incorporating interaction dynamics and object behaviour with solute effects. Current theories do not encompass all the factors required for explaining the long-range solute effects on the objects during solidification. We hope our experimental data can serve to improve the existing theoretical models. Finally, we suggest that the freezing of oil-in-water emulsions may serve as an analogue for studying the in situ interaction of foreign objects with an advancing solid-liquid interface in the presence of solute effects. FREEZING DROPLETS

The trends in our data for interaction time seem to be more statistically significant for the solute effects in contrast to the values of U max . This is depicted also in Fig. 3.15, where we distinctly underestimate the value of U max . Chapter 4

Multiple objects interacting with a solidification front Abstract

The interaction of foreign objects suspended in a liquid melt with an advancing solidification front is of special interest in nature (e.g. frost heave) and engineering sciences (e.g. crystal growth). The front can engulf the object, trapping it into the growing crystal, or the front can repel the object, pushing it ahead of itself. Therefore, the object-front confrontation can have a strong influence on the properties of the solidified material. In particular, the spatial distribution of the objects in the resulting microstructure determines the material's structural and functional properties. The past theoretical models and experimental studies have mostly investigated the interaction of isolated, spherical, and hard objects in pure melts. However, the outcome of object-front interactions in complex (more realistic) systems, where multiple objects and solutes are present, is still poorly understood. Here we show the interaction of multiple oil droplets with an ice-water front in the absence and presence of solute effects using in situ cryo-confocal microscopy. We observe the formation of a compact agglomerated layer resulting in a force equilibrium different from the isolated object approach.

We elucidate the role of solute during the evolution of a material microstructure in the presence of foreign objects. We report on how the object size, number of objects, and bulk solute concentration influence the front morphology and the subsequent object spatial distribution. Our results depict how the presence of multiple objects with varying solute concentration can modify the object-front

Introduction

The interaction of objects (soft or hard) with a moving solidification front is a ubiquitous phenomenon with diverse natural and technical occurrences. In nature, the formation and growth of sea ice, frost heave in cold regions [1,2], while the technological incidences include cryobiology [3,4], food engineering [5],

metallurgy [6], growth of single crystals [7,8], and freeze-casting [9]. At the core of these physical processes are objects (particle, droplet, or bubble) encountering an approaching solid-liquid front. A major goal in understanding these systems is to elucidate the outcome of this interaction and predict not only the object behaviour (engulfment or rejection) but also the spatial distribution of objects after solidification. The solidification microstructures obtained owing to this object-front confrontation are vital as they govern the mechanical and functional properties of the solidified material. In metal-matrix-composites, a homogeneous distribution of reinforcing particles improves the structural properties, while a particle segregation along the grain-boundaries is detrimental to their performance [10]. In contrast, an in situ evolution and trapping of gas bubbles in metals and single crystals can lead to their catastrophic failure and hence, a total rejection of bubbles is desired [11].

The first systematic measurements of particle interaction with an advancing ice-water front were conducted by Corte [12]. The objective of his experiments was to study the influence of an increasing growth rate (V sl ) on particle capture by the growing solid. Studies performed subsequently on the interaction of a single isolated object with a solidification front, deduced a critical velocity (V c ), the growth rate below which an object is pushed (V sl < V c ), while the object is engulfed by the growing solid above it (V sl > V c ) [13,14]. The previous studies report that the precise value of V c depends on several factors, comprising of the particle radius R, the temperature gradient G, and the physical characteristics of the particle-melt-crystal system [15,16,17]. In general, V c ∝ R -α where α varies

(1 ≤ α ≤ 3
2 ) owing to an interplay between the viscous drag forces (F η ), which promote capture, and the thermomolecular forces (F σ ), which favor repulsion of the object through a premelted film [16,17,18,19].

A plethora of work has been performed to determine theoretically the magnitude of these opposing forces, which have a major dependence on the shape of the solidification front [10,20,21]. The studies only differ in the theoretical approximations used (e.g. boundary conditions) and mainly deal with an isolated, FRONT smooth, spherical, and rigid particle of radius R moving ahead of a planar front in steady-state (i.e. F η = F σ ) directional solidification [15]. The in situ experimental correlations of these analytical and numerical models are, however, still scarce owing to the numerous complexities associated with solidification, such as multiplicity of space-and timescales [4,11,15,22,23,24,25]. Furthermore, the idealistic assumptions of the theoretical models such as the non-retarded van der Waals forces, absence of solute segregation, and a steady-state planar front interaction are difficult to achieve practically [26]. The advent of accurate finite element models also clearly highlights the disparities in approximations to solidification front shapes and the accuracy of the simulated particle behaviours [27]. The existing theories and experimental evidences have been primarily derived for isolated, hard, and smooth spherical objects, whereas most real-life applications involve multiple particles with arbitrary shapes and surface rough-4.1. Introduction ness. This reinforces the importance of investigating the role of particle size distribution, particle-particle interactions and hence, the formation of agglomerates. Progress in studying the spatial distribution, when thousands of particles interact with a solid-liquid front, has been brought about by modern advanced in situ imaging techniques, such as X-ray imaging [28], confocal microscopy [29], or the development of numerical modelling techniques, such as molecular dynamics [30,31]. Recently, Saint-Michel et al. [32] proposed a mechanical model on the effects of multiple particle interactions on particle trapping by solidification fronts as depicted in Fig. 4.1. Their study highlighted the major differences in force equilibrium when considering a multi-particle approach. The model suggested that the presence of multiple particles can lead to an accumulated particle layer ahead of the solidification front at small growth rates (V sl < V c ).

This particle layer further acts as a porous medium and offers resistance to the fluid flow resulting in an additional frictional force (F µ ), which facilitates particle engulfment. Hence, the modified force equilibrium for multiple-particles can be given as F η + F µ = F σ . Although, these studies help us to bridge the gap between the single particle models and real-life multi-particle systems, yet the underlying questions concerning the evolution of solidified microstructure with the associated complex front morphology remain unresolved.

The presence of solutes, desired or as impurities, impacts the solidification front shape and induces long-range effects (≈ D/V sl , where D is the solute diffusion coefficient), which override the short-range thermomolecular forces (≤ 10 nm). The existing models, incorporating the role of solute, study their influence on the solid-liquid front shape deformation but not on the nature and equilibrium of the repulsive forces [19,21,33,34,35]. In general, the solubility of a solute is less in the solid phase than in the liquid phase. As solidification progresses, the solute is rejected by the growing crystal, thereby corresponding to an increasing concentration of solute in the remaining liquid [36]. In the presence of foreign objects, the segregation of solutes at the front is enhanced further by the approaching objects obstructing their diffusion field [21]. This local solute enrichment influences the object environment (viscosity, undercooling, and phoresis) and is of particular importance in understanding several mechanisms that can occur at the solid-liquid front. The rejection of additives during the freezing of biological cells can inflict membrane rupture (or cryoinjury) owing to the high osmotic stress gradient from the local solute concentration [3,4]. The nucleation and growth of bubbles leading to macroporosity defects is instigated when FRONT the concentration of gases rejected (like solutes) by the growing solid exceeds a certain value [11,37].

The objective of our study is to investigate experimentally the solidification dynamics and spatial distribution of objects in multiple particle regimes. We compare our experimental observations with previous theoretical models to comprehend the critical correlations and disparities in the presence of multiple particles. We perform directional solidification experiments with independently regulated solidification velocity (V sl ) and a linear temperature gradient (G). The use of freezing emulsions as model system for solidification has been promising [38] and we develop further on this approach using in situ cryo-confocal microscopy. We utilise mono-disperse, bimodal emulsions, and subsequently progress towards realistic poly-disperse systems. We demonstrate several important aspects of multiple particle interactions and its consequences on particle redistribution in the solidified microstructure. We increase the complexity by adding solute to the liquid phase, and examine the effects of increasing polydispersity in conjunction with the overriding solute effects. Finally, we illustrate how the systematic approach facilitates to decouple the process parameters impacting the behaviour of an object, while it interacts with a solid-liquid front.

Experimental Methods

Materials

We purchased propyl benzoate, TWEEN 80, Difluoro2-[1-(3,5-dimethyl-2H-pyrrol-2-ylidene-N)ethyl]-3,5-dimethyl-1H-pyrrolato-Nboron (BODIPY), and Sulforhodamine B (SRhB) from Sigma-Aldrich. We utilised 0.45 µm Nylon membrane filters (VWR International) for filtering traces of impurities in the deionized water used for the aqueous phase. We chose propyl benzoate for the oil phase as it has a low melting temperature (T m = -51.6°C ), low solubility in water (0.035 g /100 g ), and similar density to water (ρ oi l = 1.023 g • cm -3 ).

Sample Preparation

We prepared oil-in-water emulsions using a microfluidic setup (microfluidic starter kit, Micronit Microfluidics, Netherlands) with pressure controlled flow pumps (Fluigent LineUP Flow EZ). We used uncoated focused flow droplet generator chips (FF_DROP, Micronit) with nozzle diameter of 10 µm and 50 µm. These 4.2. Experimental Methods microfluidic chips facilitated the preparation of highly monodisperse droplets with radii of 8 µm (R 1 ) and 28 µm (R 2 ), respectively. We ensured a uniform flow rate of oil and aqueous phases using Fluigent Flow Unit S (0 -7 µL/mi n) combined with in-line impurity filters (PEEK 2 µm,VWR International). The oil phase consisted of propyl benzoate with 10 -4 M BODIPY to obtain clear imaging of dispersed droplets at 1% laser power. For the aqueous phase, we used 10 -5 M SRhB solution, as self-quenching was reported at concentrations above 2 × 10 -4 M [39].

We added TWEEN 80 (H LB = 15 [40]), a non-ionic surfactant, to the aqueous phase for stabilising the oil-in-water emulsions. TWEEN 80 (Critical Micellar Concentration = 13 -15 mg • l -1 [41]), acted as a model solute as it depresses the freezing point of solutions alike (colligative property). The bimodal emulsions were synthesized by independently mixing the monodisperse emulsions (R 1 + R 2 ). The polydisperse emulsion was obtained through hand-shaking the aqueous suspension in a 1.5 ml eppendorf vial with 2 vol .% oil phase. The prepared emulsions were filled through capillarity and solidified in a rectangular Hele-Shaw cell (h=100 µm and V=100 µl ). The cell was fabricated using two glass slides (Menzel, 24×60 mm, thickness 0.13-0.16 mm), and sealed with nail-polish at one end to prevent evaporation.

Imaging & Analysis

We used Leica TCS SP8 confocal laser scanning microscope (Leica Microsystemes SAS, Germany), equipped with 488 nm (blue) and 552 nm (green) lasers, for image acquisition. In 2D, we utilised the microscope at a scanning speed of 600 H z, with 1024 × 1024 pi xel s for imaging 775 × 775 µm, resulting in 1.7 s per frame. In 3D, we used a fast resonant mode with 512 × 512 pi xel s for a scanning rate of 0.047 s per frame. We used two photodetectors (PMT) to simultaneously image three phases :

1. BODIPY (λ ex 493 nm ; λ em 504 nm), fluorophore incorporated into the oil droplets.

2. SRhB (λ ex 565 nm ; λ em 586 nm), fluorophore dissolved in water, to image the aqueous phase and the grain boundaries in ice.

3. Ice, does not fluoresce, as it has very low solubility for solutes [42] and hence, appears black. FRONT The emission spectra of the excited fluorophores was captured using a nonimmersive objective (Leica HCX PL APO CS 20×). The objective working distance of 590 µm along with an insulating foam cladding facilitates the minimization of thermal perturbations on the freezing substrate. We used Fiji [43] for image thresholding in conjunction with Python [44] for image and data analysis.

Freezing Stage

We conducted unidirectional solidification experiments, translating the sample cell along a constant linear temperature gradient (G), using the cryo-confocal stage described in detail previously [29]. We imposed the temperature with two Peltier modules, and controlled it with high precision (< 0.01 °C ) using TEC-1122 Dual Thermo Electric Cooling Temperature Controller from Meerstetter Engineering, Switzerland. The Peltier elements were separated by a distance (2 mm) to establish a linear temperature gradient along x. We utilised the VT-80 translation stage (Micos Pollux Drive PI, USA) to impose the rate at the which the sample cell is pulled (V sl ). The rate of translation was verified to be in agreement with the measured solidification velocity using posterior image analysis (er r or < 1 %). Thus, we can control independently the solidification velocity (V sl ) and the thermal gradient (G) in our system. We carried out the experiments in the velocity range of 1 ≤ V sl ≤ 10 µm • s -1 , with sufficient waiting time to establish thermal equilibrium along the sample depth ( z). The in situ observation of object interaction with the solid-liquid front was captured using a confocal microscope placed vertically over the gap (2 mm) between the two Peltier modules. Hence, the solidification front tends to appear immobile in the frame of observation, however, in the sample frame, it is the ice solidifying (along x) at the velocity imposed by the pulling rate of the motor.

Results

Front Morphology

We study the directional solidification of oil-in-water emulsions, translated at a velocity V sl along a linear temperature gradient G, using a confocal microscope, as shown in Fig. 4 In directional solidification, the advancing solid-liquid front can have predominantly three morphologies; planar, cellular (or columnar), and dendritic (Fig. 4.3). The stability and morphology of a front are essentially controlled by the growth rate, the underlying temperature gradient, and the solute concentration FRONT in the melt. The major advantage of our system is that we can independently regulate and accurately maintain a constant solidification velocity (V sl ) as well as temperature gradient (G) over the experimental time-scales. This ensures a steady-state solidification front under a constant cooling rate with the absence of thermal destabilisation.

To establish the domains of different front morphology, we performed experiments using a constant thermal gradient G of 10 4 K • m -1 , as shown in Fig. 4.4. We observe three distinct front morphology (Fig. 4.3,C), in the absence of suspended objects, while solidifying the aqueous phase in a velocity range of 1 ≤ V sl ≤ 10 µm • s -1 , with varying concentrations of solute. In Fig. 4.4, the steady-state planar front is thermodynamically favorable at low growth velocities (V sl ≤ 5 µm•s -1 ) for small solute concentrations (≤ 10 -1 w t .%). As we increase the front velocity or the solute concentration, the steady state planar front develops local growth instabilities with a few grain boundaries, which remain stable during experimental time scales ≈ 40 -60 mi n. At the higher velocities and solute concentrations, the front eventually destabilises into a columnar front morphology. The uniformly spaced lamellae are generated owing to the propagation of a Mullins-Sekerka instability during the solid-liquid phase transformation [45].

The solute rejected by the growing ice front builds up a steep concentration gradient with the bulk solution, over a length scale (≈ D/V sl ) [21,23] . Since growth from solutions depends on the concentration gradient of solutes, the liquidus temperature of the melt in the vicinity of the front differs from the liquidus temperature of the bulk liquid far from the front. As the solute segregates, the liquidus temperature further decreases. This change in liquid composition alters its transformation temperature, referred to as constitutional undercooling [36]. This constitutional undercooling is a necessary criterion for the destabilization of a planar front into a metastable or columnar front [45]. Theoretically, for steady-state planar front growth the solute concentration gradient in the melt scales as ∇C ∝ e -V sl x/D , where x is the distance from the solidification front [36].

Therefore, an increasing growth rate leads to a strong increase of the solute concentration gradient. This explains why the critical bulk solute concentration for the destabilization of a planar front decreases with an increasing growth rate, as reported in Fig. 4.4.

The spatial distribution of the objects in the solidified material is determined by the advancing front morphology obtained at the given experimental conditions [23,46]. In ice-templating, a columnar front is used to segregate the dispersed colloids between the arms of the growing ice crystals [9]. In contrast, the growth of single crystals demands a planar front to ensure a homogeneous structure [7]. We thus feel it is important to investigate the systematic impact of object size, solute concentration, and inter-object interactions on the solidified microstructure with planar and meta-stable fronts. We do not investigate the specific case of columnar and dendritic fronts to avoid comparison, while modifying multiple solidification parameters simultaneously. This implies that we modify only the bulk solute concentration, while keeping a constant growth rate of 1 µm • s -1 and a constant temperature gradient of 10 4 K • m -1 during our FRONT experiments.

Interaction of monodisperse objects

Results

We first examine the interaction of monodisperse oil-in-water emulsions with the solid-liquid front at a growth rate of V sl = 1 µm.s -1 and a temperature gradient of G = 10 4 K • m -1 , as depicted in Fig. 4 In Fig. 4.5A, we observe that the 8 µm (R 1 ) oil droplets with 0.01 w t .% solute concentration are continuously repelled by the ice growing from the bulk suspen- The presence of a similar particle agglomerated layer has been reported

previously [32,47,48], where its steady-state thickness (h) depends primarily on the solidification velocity (h ∝ V -1 sl ). Therefore, the complex scenario of FRONT multiple particles can be partly understood by building on the single-particle models. At 0.01 w t .% concentration, the solute does not play a dominating role and hence, we shall compare this system to the previous theoretical models derived for single isolated objects. In general, when an isolated object is relatively close to the solidification front (< 100 nm), non-retarded van der Waals forces are effective and tend to push the object away from the front. This inter-molecular interaction generates the need to maintain a stable liquid film (to prevent contact) between the object and the growing crystal and results in a repulsive disjoining pressure. The viscous drag force (F η ), emanating from the fluid flow in the particle-front gap to satisfy the mass balance, counteracts the inter-molecular force (F σ ) and favours engulfment of the object. Therefore, the object behaviour (engulfment or rejection) in the vicinity of a solidification front, depends on the equilibrium of the attractive viscous and repulsive van der Waals forces (F η = F σ ) [13,15,16,17,18,19].

We consider the Rempel-Worster (RW) model [16] to predict the droplet behaviour as the model has been deduced for objects of radius R < 10 -4 m. Using this model, we can estimate the characteristic scale of particle size (R c ), at a given solidification velocity V sl , where particles smaller than R c are rejected and particles larger than R c are captured by an advancing front. The characteristic particle size R c , equated by a balance of non-retarded van der Waals and viscous forces (F η = F σ ), can be given as

R c = σ sl A 2 ow 6 5 π 2 η 3 V 3 sl 1/4 (4.1)
where σ sl is the ice-water interfacial tension, A ow is the oil-water Hamaker constant, and η is the dynamical viscosity. We obtain a R c of 9 µm with the typical parameter values (see table 4.2) for an object in the ice-water system. Hence, using the RW model we can account for the initial repulsion of the 8 µm droplets (R 1 < R c ) and an instantaneous engulfment of the 28 µm (R 2 > R c ) droplets.

Isolated droplets are rejected at the ice-water front when their size is smaller than the critical size (R 1 < R c ). In concentrated systems, the droplets being pushed get accumulated ahead of the front and eventually interact with the other droplets, thereby forming a dense object layer (see Fig. 4.1). The thickness of the droplet layer grows until a force equilibrium, similar to isolated systems (F η = F σ ), is reached. Subsequently, we observe a periodic engulfment of droplet mono-layers, when the layer attains a critical thickness of ≈ 80 µm. A similar 4.3. Results observation of particle repulsion-engulfment transition has been reported by the previous studies [32,47,48]. More recently, Saint-Michel et al. [32] have deduced a modified force equilibrium for multi-particle systems, given as F η + F µ = F σ . They have suggested that an additional frictional force F µ is generated by the flow of fluid, through the compact particle layer, toward the ice-water front. In agreement with this model, we believe that the repulsion-trapping of droplets is owing to the fluid flow generated pressure gradient, which pushes the droplets towards the front and favors their engulfment. Thus, we suggest that this additional frictional force F µ , which grows with the height of the compact layer, results in the periodic engulfment of droplets for a critical layer thickness of ≈ 80 µm. We will now discuss the droplet-front interactions with 1 w t .% solute in solution. The 8 µm (R 1 ) droplets in the presence of 1 w t .% solute are pushed by the growing ice and subsequently organise into close-packed clusters (Fig. 4.5B).

Parameters Nominal value G (K • m -1 ) 1.0 × 10 4 T m (K ) 273.15 η (P a • s -1 ) 1.8 × 10 -3 σ sl (J • m -2 ) 3.0 × 10 -2 V sl (m • s -1 ) 1.0 × 10 -6 A ow (J ) 1.0 × 10 -20
The droplet clusters formed are rather non-uniform in their distribution and are heterogeneously scattered over the front in the lateral direction ( y). The droplet clusters are eventually engulfed and incorporate in the growing crystal, while getting textured perpendicular ( x) to the solid-liquid front. This spatial distribution is hence opposite to the parallel mono-layers (along y) obtained in the presence of 0.01 w t .% solute concentration (Fig. 4.5A). In contrast, the 28 µm (R 2 ) droplets get engulfed without repulsion (Fig. 4.5D), and represent similar behaviour as the corresponding large droplets at 0.01 w t .% solute concentration (Fig. 4.5C). We thus observe a distinct variation in the droplet spatial distribution only in the presence of small R 1 droplets.

We believe the formation of these heterogeneous clusters is owing to the local destabilization of the ice front and therefore, instigated by the rejected solute FRONT segregation. The presence of multiple droplets further enhances the local solute concentration by obstructing the solute diffusion field in the bulk liquid [13,21,33,34]. Highly fluorescent zones in the vicinity of the droplets can be clearly seen at the front (Fig. 4.5B,D). These zones are formed owing to the segregation of the fluorescent dye, which is rejected by the growing ice. We expect that the solute, although not fluorescent yet having a similar diffusion coefficient as the dye, segregates in an identical manner at the front. The solute invokes severe constitutional undercooling in the premelted films behind the droplets, thereby inducing a concave curvature of the front [49].

The extensive concave cusping eventually transforms into a pocket with the droplet build-up and results in subsequent entrapment of droplets along the crystal growth direction ( x). Such solute enriched pockets have also been reported by Chang et al. [50] during the directional freezing of biological cells. The premelted films are highly sensitive to the concentration of solute and surround the droplets engulfed by the ice, while the interface heals and rebounds to the horizontal isotherm (T m ). The presence of solute premelted films with thickness of the order ≈ 5 µm leads us to believe that the solute plays a dominating role at such high concentration [51]. In the absence of solute effects, premelted films of thickness ≈ 10 nm have been predicted in the particle-front gap by the past studies [15,16,17].

Our results depict that an augmentation of solute does not seem to influence the previously predicted value (RW model) of critical radius R c for droplet engulfment. Similar to the R 1 droplets in the presence of 0.01 w t .% solute concentration, we report an initial repulsion of the R 1 droplets with 1 w t .% solute in solution.

A few analytical models [20,21,33,34] and numerical simulations [19,35,50] studying the object-front interactions in the presence of solute effects have been derived previously. In theory, the magnitude of van der Waals force, responsible for pushing the particle, is regulated by the particle-front gap. The high concentration of segregated solute increases the particle-front gap by constitutional undercooling and hence, significantly reduces the effect of repulsive van der Waals force. This implies that for a given growth rate, smaller objects can be engulfed much more easily in solutions, in contrast to pure melts (R sol ut i on c < R pur e mel t c

).

Thus, the theoretical models predict a critical radius (R c ) of magnitude much smaller for binary solutions as compared to the pure melts, which is not what we observe experimentally (i.e. R sol ut i on c ≈ R pur e mel t c

). Therefore, we deduce from our experimental observations that R 1 < R c < R 2 is still a valid criterion with We have discussed so far the interaction of monodisperse objects, exhibiting distinct radii (R 1 , R 2 ), with an approaching solid-liquid front. Yet, in many realistic applications, such as fabrication of metal/ceramic matrix composites, we can have a distribution of object sizes. Thus, it is essential in such scenarios to comprehend and predict, the segregation or sorting of particles, during solidification.

In light of these uncertainties, we conducted horizontal unidirectional solidification with the same set of droplets (radii R 1 = 8 µm and R 2 = 28 µm) composed together, at a solidification velocity V sl of 1 µm • s -1 and a constant linear temper-FRONT ature gradient G of 10 4 K • m -1 . To understand the novel scenarios observed, we compare the modes of interactions in bimodal emulsions to the monodisperse regimes at their corresponding solute concentrations (0.01, 1 w t .%). We also depict the associated descriptive schemes in absence (Fig. 4.6A,B,C,D) and presence (Fig. 4.7A,B) of dominating solute effects.

In the presence of 0.01 w t .% solute (Fig. 4.6), we notice two peculiar behaviours of the inter-droplet interaction with the solid-liquid front as follows:

First, as shown in Fig. 4.6C,E the small droplets (R 1 < R c ) form a compacted layer at the planar front as seen in monodisperse emulsions (Fig. 4.6A), which further acts as a barrier layer and provokes the repulsion of a larger droplet (R 2 ).

The latter is pushed ahead for t ≈ 200 s before getting eventually trapped by the growing ice. This suggests that a droplet with a radius greater than the critical value (R 2 > R c ) can still be repelled if it does not interact with the solidification front immediately. In contrast, isolated monodisperse large droplets get engulfed instantaneously without undergoing repulsion-trapping, as shown schematically in Fig. 4.6B and experimentally in Fig. 4.5C.

The second observation (see Fig. temporary opening, at the solid-liquid front, entices a cluster of smaller droplets (R 1 = 8 µm) behind the large droplet (t = 135 s). The droplet cluster, initially isolated, is trapped at the fermeture of the pocket and gets textured perpendicular to the front ( x). This orientation is reminiscent of a solidifying monodisperse emulsion depicted schematically in Fig. 4.7A. The lateral growth of the ice crystals, transverse to the temperature gradient, pushes the droplets together and results in a close-packed microstructure. Thus, a small increase in the amount of solute added has a drastic impact on the solidification microstructure. In the last section, we look at the behaviour of polydisperse oil-in-water emulsions. The low interfacial tension of the oil (propyl benzoate) with water enables us to obtain relatively small polydisperse droplets, varying in size from 5 < R < 30 µm. We utilize in situ cryo-confocal microscopy to acquire 2D and 3D evolution of the microstructure at V sl = 1 µm • s -1 and G = 10 4 K • m -1 . Previously, we have reported on the monodisperse and bimodal size particle interactions between the growing solid and the droplets, which is considerably different from the nature of interactions between the front and an isolated object. To attain a better understanding of complex (more realistic) systems, we will now investigate the confrontation of polydisperse droplets with an advancing solidification front.

The emulsions are prepared with 0.01 w t .% and 1 w t .% solute in solution.

When an object interacts with a solid-liquid front it can be engulfed immediately, pushed irreversibly, or repelled for a certain distance and then engulfed [15].

These behaviours are encountered owing to a critical velocity (V c ), inversely proportional to the particle radius (V c ∝ R -α ), above which an object is engulfed

(V sl > V c
), while below it the object is pushed by the front (V sl < V c ). Conversely, for a given growth rate V sl , there should exist a critical radius R c , where the objects smaller than the critical size should be pushed and vice-versa. Thus, in our experiments with different object sizes one would expect a range of critical velocities or a range of critical radii at the imposed growth rate. Ideally, we expect the engulfment of objects with R y > R c and rejection of objects with R x < R c , where

Conclusions

The size, presence of multiple objects, complex front morphology, and high concentration of solute significantly alter the microstructure developed during directional solidification. We performed solidification experiments to study the droplet rearrangement of monodisperse, bimodal, and polydisperse distributions in the solidified matrix. We have tried to highlight the correlations and disparities of multiple-particle regime with previous isolated single object models [16,19] as well as recent multi-particle models [32], computed both in the absence and presence of solute effects. We have successfully demonstrated in our experiments the distinct behaviour when the size of droplets adheres to the criterion of R 1 < R c < R 2 . Furthermore, we have observed no change in critical radius (R c ) in the presence of solute effects contrary to the predictions of the theoretical models.

The critical radius (R c ) or critical velocity (V c ) is modified by the presence of interdroplet interactions. We depict that the presence of multiple objects can lead to the formation of a segregated microstructure, while the initial suspension is homogeneous. We illustrate for the first time a pushing-engulfment or repulsiontrapping transition in the presence of overriding solute effects. We report no change in the morphology of the solidification front owing to the presence of objects in the melt. We suggest that the volume fraction of objects in the melt is an important criterion to be considered for predicting the object distribution in the solidified microstructure. The solidification of controlled oil-in-water emulsions can help us visualize and model a variety of microstructures by utilizing different colonies of droplets in the presence or absence of solute effects.

Conclusions & Perspectives

We performed this study over a period of three years with the aim of improving our current understanding of the interaction of foreign objects with a moving solidification front. The principal objective was to investigate the mechanisms responsible for the development of a directionally solidified microstructure in the presence of secondary objects with the aide of in situ cryo-confocal microscopy.

We studied the interaction of isolated hard particles (polymers, ceramics, and metals) and soft objects (droplets and bubbles) with an advancing ice-water interface. To explore the role of solutes, we systematically analysed the interfacial curvature, object dynamics, object shape evolution, and the object spatial distribution, in the presence and absence of solute effects. We then progressed towards more realistic systems by employing a high volume fraction of objects in the liquid melt as compared to the isolated systems. We further compared our experimental observations with the past theoretical models to elucidate the different features and proposed mechanisms for the contrasting phenomena.

We now highlight the main conclusions which can be drawn from each individual study and discuss the opportunities for (hopefully) future work.

Objects interacting with solidification fronts: Thermal conductivity effects

In this study, the primary objective was to analyse the shape of the solidification front experimentally, using objects (soft or hard), having a thermal conductivity mismatch with the liquid melt (k p /k m = 1). The shape of the solidification front is an important parameter defining the magnitude of forces acting on an object and hence, plays a major role on the object-front interactions. The macroscopic geometry of the front is essentially a manifestation of the local isotherms and thus, depends on the uniformity of the temperature field at the solid-liquid

Conclusions & Perspectives interface. The previous studies have investigated the role of object-melt thermal conductivity mismatch but nothing systematic with a complete control over the solidification parameters was performed experimentally. A major part of the previous literature is, therefore, composed of theoretical predictions.

The focus of our study was to explore the solidification front shape over a wide range of thermal conductivity ratio (0.04 ≤ k p /k m ≤ 35) by employing polymer, ceramic, metallic rigid particles as well as oil droplets and air bubbles. We first observed and measured the distortion of the ice-water front in the vicinity of the objects with a high spatial resolution owing to our custom cryo-confocal microscopy setup. We noted that for a given curvature (convex or concave), the magnitude of deflection was constant and independent of the nature of the objects. Our observations were consistent with the front curvature predictions of the theoretical model from Park et al., however, the engulfment or repulsion predictions were not in accordance as we noted an engulfment of the objects irrespective of the front curvature.

The next step of our study was to explore the influence of solutes on the front geometry. This was conducted as most real systems consist of solutes (or impurities) and the observation of solute effects in situ remains complex. Interestingly, we deduced that the presence of solute segregation between the object and the ice-water interface dominates over the effects of object-melt thermal conductivity mismatch. The solute instigated the formation of a concave front shape (bending away from the objects) with a curvature magnitude considerably higher than that measured in the absence of solute. This has important implications as the nature and magnitude of forces can get modified with the presence of such dominating solute effects.

In the last section, we depicted the ice-water front shape modification for cellular front morphology. We discussed the modification of the distance at which the solute segregation impacts the front geometry in cellular front morphology as compared to the previous planar front investigations. Furthermore, we demonstrated for the first time that objects, with a thermal conductivity less than water, can distort the neighbouring ice crystals post engulfment during directional solidification. Thus, the solidification microstructure, where objects interact with an advancing front, depends on an interplay between the thermal and solute effects.

Moving forward, an important parameter that can be further investigated is the dimensionless surface energy (γ). The role of this parameter, on modifying In the last section, we identified the deformation of droplets invoked after their engulfment in the solid ice phase. The use of our custom setup provides a high 2D and 3D spatial resolution, which permitted us to analyse the droplet behaviour and the surrounding thickness of premelted films at long distances away from the solidification front. The crushing phenomenon was suggested to be originating from the meniscus of the ice-water interface. We further looked at the motion of droplets in the solid to investigate the possibility of thermal regelation. However, the droplets captured in ice manifested no relative velocity and hence, did not migrate towards the warm side of the underlying temperature gradient. The object migration, premelted films, and particle interaction with the solid phase at low temperatures is investigated especially by geophysicists to understand frost heave, glacier motion, and ice core climate proxies. Thus, we suggest our experimental methodology of using oil-in-water emulsions in the cryo-confocal setup as an analogue for studying such complex phenomena.

In

Multiple Objects Interacting with a Solidification Front

We performed this study to try and bridge the gap between isolated object-front interactions and realistic systems where the complexity increases owing to a high concentration of objects. The objective was to elucidate experimentally the development of a real-life microstructure with multiple objects in the presence or absence of solute effects. Most of the previous studies have focused on predicting the behaviour of isolated, hard, spherical, and rigid particles. Much less attention has therefore been paid to particle-particle interactions and the volume fraction of objects dispersed in the liquid. Moreover, the presence of a solute adds to the complexity as it plays a major role on the front morphology. Therefore, we analysed the interaction of multiple oil droplets, acting as a model analogue, with an advancing ice-water interface.

We first demonstrated the interaction of monodisperse oil droplets, and studied the influence of the droplet size and the bulk solute concentration. We depicted the formation of a compact droplet layer and discussed the presence of an additional frictional force similar to that observed in porous media. The

Conclusions & Perspectives

force equilibrium modification owing to the particle-particle interactions was invoked and compared to that predicted from the isolated object models. We tried to build upon the previous theoretical studies to explain partly the interaction mechanisms observed in our experiments. We noted a significant impact on the texture of the engulfed droplets with a changing solute concentration and object size.

In the second section, we analysed the interaction of bimodal emulsions, where the droplet radii used in the monodisperse regime were mixed together.

We discussed the formation of droplet agglomerates and depicted the repulsion of large droplets owing to the compact layer of smaller droplets being pushed by the solidification front. The addition of a small amount of solute to the bulk liquid resulted in the formation of solute-rich pockets. The presence of pockets instigated droplet clustering, whereby the oil droplets were forced to close pack into agglomerates.

We finally studied the interaction of polydisperse droplets to imitate the reallife systems with a higher accuracy. We reported the formation of agglomerates at the solidification front in the absence of solute effects, while the clusters were observed well ahead of the front in the presence of solute effects. A pushingengulfment or repulsion-trapping transition was deduced in contrast to the previous theoretical models, which suggest an instantaneous engulfment of objects in the presence of solute in solution.

Thus, we demonstrated that a small amount of solute in the parent liquid can have a drastic impact on the evolution of the solidification microstructure.

Furthermore, we suggested that the evolution of such microstructures, where multiple objects in the presence of solute (or impurity) interact with a front having a complex morphology, remain difficult to predict theoretically.

Further progress in this domain will probably be achieved with the development of robust models that can take into account the solidification behaviour of fronts with complex morphologies while encountering thousands of particles of different sizes. This can be brought forth by numerical modelling techniques such as discrete or finite element modelling. The orientation of particles with arbitrary shapes such as rods or discs could be interesting to visualize when they come in contact with a solidification front. The in situ observation of different types of particles (hard or soft) can provide direct observation which can be correlated to improve such predictive models. Ainsi, nous avons démontré qu'une petite quantité de soluté dans le liquide de base peut avoir un impact drastique sur l'évolution de la microstructure de solidification. En outre, nous avons suggéré que l'évolution de telles microstructures, où de multiples objets en présence de soluté (ou d'impureté) interagissent avec un front ayant une morphologie complexe, reste difficile à prévoir théoriquement.

Résumé

De nouveaux progrès dans ce domaine seront probablement réalisés avec le développement de modèles robustes qui peuvent prendre en compte le comportement de solidification des fronts ayant des morphologies complexes lors de la rencontre de milliers de particules de tailles différentes. Cela peut être réalisé grâce à des techniques de modélisation numérique telles que la modélisation par éléments discrets ou finis. L'orientation des particules de formes arbitraires telles que des tiges ou des disques pourrait être intéressante à visualiser lorsqu'elles entrent en contact avec un front de solidification. L'observation in situ de différents types de particules (rigide ou déformable) peut fournir une observation directe qui peut être corrélée pour améliorer ces modèles prédictifs.

List of Figures

LIST OF FIGURES

1.6 The thick purple line shows the particle behavior that is predicted by Rempel et al. [20], including the complete interfacial curvature, which, for large film thicknesses, is nearly identical to the results obtained when approximating the front as being planar (rightmost magenta curve). The leftmost magenta curve gives the approximate particle velocity when the thermal conductivity of the particle is assumed equal to that of the melt and the solid. The green curve displays the predictions of Chernov et al. [14] for the case where the interface has the form of a paraboloid intersecting with a plane, and the lowest (blue) curve is their prediction for the particle velocity when the front is modeled as a paraboloid, adapted from [21]. . . . 1.14 The model from Park et al. [32] suggested that a small particle-tomelt thermal conductivity ratio promotes repulsion, while particle engulfment is facilitated when the ratio is large, adapted from [32]. 23

Figure 1 . 1 :

 11 Figure 1.1: Object interacting with a moving solidification front: principles. Model sketch, where d is the particle-front gap, G = ∇T is the temperature gradient, κ is the interfacial front curvature (here > 0 for concave geometry), k p , k m are the thermal conductivity of the particle and melt respectively, R is the object radius, T m is the absolute melting temperature, T i is the solidliquid interfacial temperature, and V sl is the solidification velocity.

Figure 1 . 2 :

 12 Figure 1.2: Change in interfacial energy when a particle interacts with a solidification front, adapted from [4].

Figure 1 . 3 :

 13 Figure 1.3: Interfacial energy varying with the distance (d ) between the particle and the solidification front, adapted from [3].

Figure 1 . 4 :

 14 Figure 1.4: Solidification front curvature in the vicinity of a particle with a thermal conductivity mismatch with the melt (a) Convex for k p /k m < 1, (b) Horizontal for k p /k m = 1, and (c) Concave for k p /k m > 1, adapted from [4].

Figure 1 . 5 :

 15 Figure 1.5: Paraboloidal front shape (left) considered a priori with distortion (right) of the temperature field (k p = k m ) in the model from Chernov et al [10, 14], adapted from [4, 15].

Figure 1 . 7 :

 17 Figure 1.7: Curvature-induced and interfacial premelted films at an iceparticle-water interface, adapted from [23].

1 . 8 )Figure 1 . 8 :

 1818 Figure 1.8: Thickness of the premelted film, d , depends on the temperature, and determines the magnitude of the repulsive disjoining pressure P T , adapted from [25].

Figure 1 . 9 :

 19 Figure 1.9: Regelation or migration of small particles in a solid towards the warmer side of the temperature gradient ∇T owing to a thermomolecular pressure gradient from the varying thickness of the premelted films, adapted from [27].

Figure 1 . 10 :

 110 Figure 1.10: The physics of premelted films is investigated in the growth of ice lenses with various natural manifestations such as frost heave in partially frozen soils, adapted from [23, 26].

Figure 1 . 11 :

 111 Figure 1.11: Model from Rempel et al. [20] accounts for the presence of a liquid premelted film owing to the inter-molecular interactions in the gap between the particle and the solid, adapted from [30].

Figure 1 . 12 :

 112 Figure 1.12: Under steady-state conditions, the particle velocity is equal to the solidification velocity. (a) When the premelting effect is excluded from the model, the gap thickness decreases monotonically with increasing solidification velocity. A critical velocity for engulfment must be defined via an arbitrary minimum gap thickness. (b) When premelting is included, the Rempel et al. [20] model predicts two solution branches, with two values of gap thickness for a given solidification velocity. A critical engulfment velocity is naturally defined as the maximum solidification velocity, or limit point, of the steady-state solution branches, adapted from [31].

Figure 1 . 13 :

 113 Figure 1.13: The dimensionless particle velocity (u) versus dimensionless particle height (b) varies with the type of inter-molecular interactions ν taken into account. When b is large and negative the edge of the particle is below the plane of the T m isotherm. The exponent ν depends on the type of intermolecular interactions that dominate; ν = 3 represents non-retarded van der Waals interactions, while ν = 2 and ν = 4 is used for retarded van der Waals and long-range electrical interactions [22], adapted from [20].

Figure 1 . 14 :

 114 Figure 1.14: The model from Park et al.[32] suggested that a small particleto-melt thermal conductivity ratio promotes repulsion, while particle engulfment is facilitated when the ratio is large, adapted from[32].

Figure 1 . 15 :

 115 Figure 1.15: Left: The model from Tao et al. [31], employing finite element methods to estimate the particle behaviour. Right: A comparison of their estimations with those obtained from the analytical model of Rempel and Worster [30], adapted from [31].

  1.15). However, the study from Tao et al.[31] used simplifying assumptions such as equal thermal conductivity of the particle, the melt, and the crystal. The model also neglected the Gibbs-Thomson effect and the bulk flow produced by density change so far the interaction of objects with a solidification front in a pure solution. In real-life systems, the probability of encountering a pure melt is rare and solutes (or impurities) are often present. The addition of solute to a solvent may be desired and hence, their presence enhances the mechanical and/or physical properties of the solidified material. Additives, referred to as cryoprotectants, are frequently used to reduce damage inflicted to biological cells during their freezing at extremely low temperatures (ranging between -5

Figure 1 . 16 :

 116 Figure 1.16: Rejection of impurities during solidification (such as dissolved gases) can lead to the undesired nucleation and growth of bubbles at the solidliquid interface, adapted from [15].

Figure 1 . 17 :

 117 Figure 1.17: Binary phase diagram representing solidification for a solution starting at the liquidus temperature (T l ) and terminating at the solidus temperature (T s ). The solute rejected by the growing solid phase creates a diffusion field in the remaining liquid phase. Reproduced with permission from [39].

Figure 1 . 18 :

 118 Figure 1.18: Schematic representation of the comet tail-shaped segregation region in front of the hydrogen pore: (a) Solute diffusion towards the sample centre-line before complete engulfment, and (b) at the time of complete engulfment, a liquid pool of high solute concentration is engulfed, adjacent to the hydrogen bubble, adapted from [42].

Figure 1 . 19 :

 119 Figure 1.19: The segregation of solutes at the interface is enhanced further by the approaching objects obstructing their diffusion field, thereby invoking higher solute concentration behind the foreign objects. Adapted from [12].

1. 5 .Figure 1 . 20 :

 5120 Figure 1.20: Progressive destabilisation of a planar front morphology (a) leading to a dendritic microstructure (f ) during directional solidification, adapted from [39].

Figure 1 . 21 :

 121 Figure 1.21: Schematic depicting the formation of Mullins-Sekerka type instability promoted by constitutional undercooling in the liquid ahead of the solid-liquid interface, adapted from [23].

1. 5 .

 5 Solute effects parameters for R, η, D et c . . . and estimated it as

Figure 1 . 22 :

 122 Figure 1.22: The capture or rejection of particles as a function of particle radius and solidification front speed. Systems above and to the right of the lines (mentioning slope) display particle capturing behaviour, while systems below and to the left display particle-pushing behaviour. There is a significantly sized region where particles may be pushed in a pure system but captured in a binary system. Reproduced from the model of Kao et al. [34].

1. 6 . 1 Figure 1 . 23 :

 61123 Figure 1.23: Bubbles' distribution in the cross section of the sapphire rod crystal as a function of the growth rate (V sl ) (a) V sl = 0.25 mm • mi n -1 , (b) V sl = 1 mm • mi n -1 , (c) V sl = 1.5 mm • mi n -1 , and (d) V sl = 2.5 mm • mi n -1 , reproduced from [48].

Figure 1 .

 1 Figure 1.24: X-ray transmission microscopy in a liquid aluminium matrix to study the interaction and development of gas porosity with a moving solidification interface. The arrow points to the solidification front, reproduced from [42].

Figure 1 . 25 :

 125 Figure 1.25: Use of optical microscopy with low melting point organic analogues to understand object-front interactions. A SiC particle interacting with the solid-liquid interface in a succinonitrile matrix, adapted with permission from [39].

Figure 1 . 26 :Fig. 1 .

 1261 Figure 1.26: Directional solidification of an aqueous (10% DMSO) suspension to evaluate the freeze-thaw viability of biological cells. Ice crystal morphology as a function of the ice front propagation. Left: during the process of freezing. Right: Frozen sample at -20 °C . Scale bar indicates 400 µm. Adapted with permission from [56].

Figure 1 . 27 :

 127 Figure 1.27: Principles of confocal microscopy. Reproduced from [23].

Figure 1 . 28 :

 128 Figure 1.28: Typical confocal microscopy image of ice crystals growing in an aqueous solution that contains a fluorophore (Sulforhodamine B, here). The red corresponds to the liquid water, the black regions in the centres are the crystals. The three-dimensional reconstruction of the crystals surface is shown in the lower picture (front view). Procedure developed by Marcellini et al. [58].

Figure 1 . 29 :

 129 Figure 1.29: Overview of the temperature gradient stage developed by Dedovets et al. [59] for in situ operations under the confocal microscope: A constant temperature gradient is achieved in the gap between the Peltier elements. If the sample is pushed at a velocity V , a solidification front moves at a velocity U f r ont = V . The solidification front is thus always in the centre of the observation windows, which is useful for solidification studies. (A) 3D representation, (B and C) Photographs of front and side views. Reproduced from [59].

Figure 1 . 30 :

 130 Figure 1.30: Cryo-confocal microscopy of yeast cells in alginate solution to correlate their physico-chemical environment to the cell-front interactions. Work performed by Qin et al. [62] using the in situ solidification setup developed in the lab.

Figure 1 . 31 :

 131 Figure 1.31: Schematic representation of the possible outcomes for the microstructure in multi-directional solidification of metal-matrix-composites, as a function of solidification rate, convection level, and particle size. A variety of solidification patterns and microstructures can be obtained. Adapted from [9].

Figure 1 . 32 :

 132 Figure 1.32: The dependence of the critical velocity on the particle radius from a number of experimental studies. A best-fit line is shown running through each of the data sets, where the slope varies from -0.8 to -1.3. The different symbols from (a) to (f ) correspond to the particles in water (copper, tungsten, nylon, and latex), (g) and (h) correspond to polystyrene and silicon carbide particles in succinonitrile melt respectively. The temperature gradient varies from 200 K • m -1 to 10 4 K • m -1 . Reproduced from [30].

Figure 1 . 33 :

 133 Figure 1.33: Sequence of micrographs showing a section across the observation slit as the planar ice-liquid interface propagates through a suspension of latex spheres (diameter 5.7 µm) in water. A pushing-engulfment phenomenon can be envisaged, however, the front does not seem to have a constant growth rate. Time interval between each frame is reported as 30 s. Reproduced from [57].

CHAPTER 1 . 1 . 6 . 3 Figure 1 . 34 :Figure 1 . 35 :

 1163134135 Figure 1.34: Effect of growth rate on the interaction between a bubble and an advancing solid-liquid interface. Reproduced from [64].

Figure 1 . 36 :

 136 Figure 1.36: Typical shape and patterns of oxygen bubbles incorporated in ice. (a) Egg-shaped bubbles. (b) Egg-shaped bubbles and cylindrical bubbles. The bubble shape depends on the freeze front velocity and the concentration of oxygen dissolved in water. Reprinted with permission from [40].

Figure 1 . 37 :

 137 Figure 1.37: Typical interaction between an oil droplet and an ice-water solidification front. Left: Depicting the accumulation of solute and its dynamic redistribution around the droplet. Right: Impact of growth rate on the spatial distribution of droplets in the solidification microstructure. Reprinted with permission from [61].

Figure 1 . 38 :

 138 Figure 1.38: Interaction of multiple particles with a solidification front. A compacted particle layer is built-up ahead of the front at a growth rate of 3 µm • s -1 owing to the particle rejection. Reproduced from [77].

2. 1 .

 1 Fig.2.1. Theoretically, the critical velocity reduces drastically for concave shaped fronts as opposed to ideal planar fronts[6, 9, 10, 11]. Hence, the approach from Chernov et al. is generally accepted and used in most models as the presence of a particle distorts the shape of an approaching solid-liquid front.

Figure 2 . 1 :

 21 Figure 2.1: Objects interacting with a moving solidification front: principles. Model sketch, where d is the particle-front gap, G is the temperature gradient, K is the interfacial front curvature (here > 0 for concave geometry), k p , k m are the thermal conductivity of the particle and melt respectively, R is the object radius, T m is the absolute melting temperature, T i is the solid-liquid front temperature, and V sl is the solidification velocity. © (2020) S. Tyagi et al. (10.6084/m9.figshare.12046560) CC BY 4.0 license https://creativecommons.org/licenses/by/4.0/.

Figure 2 . 2 :

 22 Figure 2.2: Experimental setup for in situ solidification experiments. A thin Hele-Shaw cell containing the particles, bubbles, or droplets in suspension is pulled at a constant velocity (V sl ) through a constant temperature gradient (G) established by Peltier elements. In steady state, the solidification front is thus at a constant position under the microscope objective. © (2020) S. Tyagi et al. (10.6084/m9.figshare.12046560) CC BY 4.0 license https://creativecommons.org/licenses/by/4.0/.

  2.1) and takes into account the balance of hydrodynamic and intermolecular forces, solidification front surface energy, contrast of the particle and melt thermal conductivities, and the flow caused by the density change upon solidification. In particular, the model encompasses an asymptotic analysis in the lubrication approximation and theoretically predicts the curvature of the front as a function of γ and (k p /k m ), shown in Fig.2.4A, where γ is a dimensionless surface energy parameter, γ = (T m σ sl )/(L v GR 2 ), where T m is the bulk melting temperature at constant pressure, L v is the latent heat per unit volume of solid, G is the applied temperature gradient, σ sl is the interfacial tension between the solid front and the melt, of the order 3 × 10 -2 N • m -1 , and R is the radius of the spherical object. Our solidification experiments correspond to γ ≈ 3.5 × 10 -3 .

Figure 2 . 3 :

 23 Figure 2.3: Thermal conductivity effects on interfacial curvature in absence of solute. (A) Typical time lapse for a freezing particle-in-water suspension with a polystyrene (PS) particle. The solidification front develops a bump (t ≈ 17s) in the vicinity of the solid particle with a lower thermal conductivity than water (k p /k m < 1). PS is in cyan, water in colormap viridis (fluorescence bar) while ice is in black. (B) Air-in-water bubble, polystyrene (PS) particle, propyl benzoate (PB) droplet, and poly(methyl methacrylate) (PMMA) particle have a lower thermal conductivity than water and hence, promote a convex curvature of the solidification front. Polyethylene (PE) particle does not affect the curvature. Zirconia (Z r O 2 ) and stainless steel (SS) particles have a higher thermal conductivity than water and hence, result in a concave depression. We added a very small amount (0.01w t .%) of TWEEN80 to stabilise the foam and the emulsion, while no solute was present for the particle suspensions. PS, PB, and PE are in cyan, water in colormap viridis (fluorescence bar) while ice, air, PMMA, zirconia, and stainless steel are in black. Scale bar = 50 µm. (C) Range (log scale) representing thermal conductivity ratio of particle to melt (k p /k m ) used in the experiments. © (2020) S. Tyagi et al. (10.6084/m9.figshare.12046560) CC BY 4.0 license https://creativecommons.org/licenses/by/4.0/.

Figure 2 . 4 :

 24 Figure 2.4: Deflection of interfacial curvature with increasing thermal conductivity ratio (k p /k m ). (A) Three distinct curvatures of the solidification front in the thermal conductivity ratio (k p /k m ) versus γ plane for a spherical particle, adapted from Park et al. [14]. (B) Plot of front deflection versus (k p /k m ) for solidification of spherical objects in absence or at low concentration of solute. © (2020) S. Tyagi et al. (10.6084/m9.figshare.12046560) CC BY 4.0 license https://creativecommons.org/licenses/by/4.0/.

Figure 2 . 5 :

 25 Figure 2.5: Impact of solute on the front curvature. (A) Concave depression of the solidification front with air bubble, polystyrene (PS), propyl benzoate (PB), poly(methyl methacrylate) (PMMA), polyethylene (PE), zirconia (Z r O 2 ), and stainless steel (SS) particles. PS, PB, and PE are in cyan, water in colormap viridis (fluorescence bar) while ice, air, PMMA, zirconia, and stainless steel are in black. Scale bar = 50 µm. (B) Range (log scale) representing thermal conductivity ratio of particle to melt (k p /k m ) used in the experiments remains unmodified on addition of 1w t .% solute to the aqueous phase. © (2020) S. Tyagi et al. (10.6084/m9.figshare.12046560) CC BY 4.0 license https://creativecommons.org/licenses/by/4.0/.

Figure 2 . 6 :

 26 Figure 2.6: Maximum interfacial curvature depression with time observed before engulfment. In absence (A) and presence (B) of solute; 1w t .% TWEEN80, for a zirconia particle with a higher thermal conductivity than water (k p /k m > 1). Scale bar = 20 µm. © (2020) S. Tyagi et al. (10.6084/m9.figshare.12046560) CC BY 4.0 license https://creativecommons.org/licenses/by/4.0/.

Figure 2 . 7 :

 27 Figure 2.7: Thick premelted film at the solid-liquid front in presence of solute. Typical time-lapse for a freezing particle-in-water suspension, with a PS particle, in the presence of solute: 1w t .% TWEEN 80. The solidification front bends away from the solid (t ≈ 41 s) and eventually heals leaving a premelted film around the engulfed particle. PS is in cyan, water in colormap viridis (fluorescence bar) while ice is in black. Scale bar = 50 µm. © (2020) S. Tyagi et al. (10.6084/m9.figshare.12046560) CC BY 4.0 license https://creativecommons.org/licenses/by/4.0/.

2. 3 .

 3 Results and discussion different spherical objects.

Figure 2 . 8 :

 28 Figure 2.8: Deformation of the cell boundaries and blunting of the ice tip radius at the solidification front during object engulfment. (A, B, and C) Time-lapse of oil-in-water emulsion (A) with G = 10 K • mm -1 , air-in-water foam (B) with G = 15 K • mm -1 , and stainless steel particle (SS) (C) with G = 10 K • mm -1 , at V sl = 10 × 10 -6 m • s -1 . Oil is in cyan, water in colormap viridis (fluorescence bar) and ice, air, and SS are in black. Scale bar = 50 µm. © (2020) S. Tyagi et al. (10.6084/m9.figshare.12046560) CC BY 4.0 license https://creativecommons.org/licenses/by/4.0/.

  the nature and magnitude of forces acting on the objects. With cellular front morphology, we have depicted that the persistent local temperature gradients around the engulfed particles (with k p /k m < 1) distort the cell boundaries and can induce a variation of the cell spacing. Our study opens new pathways for the quantification of theoretical parameters which were difficult to measure previously. Finally, we hope our extensive research with ceramic, metal, and polymer objects serves for the formulation of a robust physical model with the incorporation of long-range solute effects.

(

  Difluoro2-[1-(3,5-dimethyl-2H-pyrrol-2-ylidene-N)ethyl]-3,5-dimethyl-1H-pyrro FREEZING DROPLETS lato-Nboron), and aqueous fluorophore (Sulforhodamine B) from Sigma-Aldrich.The fluorophores are referred to as BODIPY (incorporated in oil) and SRhB (incorporated in water) in the study. We cycled the deionized water through 0.45 µm Nylon membrane filters (VWR International) to remove traces of impurities and ensure purity of the emulsions prepared. The solidification experiments were conducted to study the interaction of dispersed phase with the ice-water interface. Hence, we prepared oil-in-water emulsions to ensure the dispersed phase remained liquid, while the continuous phase underwent solidification. Thus, we chose propyl benzoate owing to its low melting temperature (T m = -51.6°C ), low solubility in water (0.035 g /100 g ), and similar density to water (ρ oi l = 1.023 g • cm -3 ).

Figure 3 . 1 :

 31 Figure 3.1: Diameter of monodisperse droplets with radii (R 1 , R 2 ) of 7.2 ± 0.4 µm and 30.9 ± 1.2 µm respectively, generated using microfluidics. The size of droplets remained stable during the experimental time-scale, which varied from 15 mi n to 4 h.

3. 2 .

 2 Methodsto stabilise the oil droplets. The surfactant Tween 80 (cmc = 13 -15 mg • l -1[31])

Figure 3 . 2 :

 32 Figure 3.2: Schematic depicting a monodisperse oil-in-water emulsion with the surfactant Tween 80 and fluorophore SRhB added to the aqueous phase (not to scale). Oil phase is incorporated with the fluorophore BODIPY. The concentration of surfactant below cmc remains as monomers, while the concentration above the cmc forms micelles.

Figure 3 . 3 :

 33 Figure 3.3: Cryo-confocal microscope setup to perform in situ solidification experiments. A Hele-Shaw cell containing an oil-in-water emulsion is pulled at a constant velocity (V sl ) through a constant linear temperature gradient (G), established by two Peltier elements. In steady-state, the solidification interface is at a constant position under the microscope objective. In the sample frame, the interface is moving at a solidification velocity of V sl , imposed by the motor. © (2020) S. Tyagi et al. (10.6084/m9.figshare.12046560) CC BY 4.0 license https://creativecommons.org/licenses/by/4.0/.

Figure 3 . 4 :

 34 Figure 3.4: 2D cryo-confocal image of freezing an oil-in-water emulsion in the presence of 1w t .% Tween 80 in the aqueous phase. (A) Typical features observed for a planar growth at V sl = 2 µm • s -1 (B) Time-lapse evolution of an oil droplet encountering an approaching solid-liquid interface with three distinct regimes of interaction at V sl = 2 µm • s -1 . Ice is in black, oil droplets in cyan, and the aqueous phase is in colormap viridis (fluorescence bar). Scale bar = 100 µm.

Figure 3 . 5 :

 35 Figure 3.5: Droplet dynamics in presence of 0.01, 0.1, and 1w t .% solute in solution, deduced from the droplet trajectories at V sl = 3 µm • s -1 . Mean droplet velocity versus distance to interface for (A) R 1 = 7.2 ± 0.4 µm and (B) R 2 = 30.9 ± 1.2 µm. The droplets accelerate as the solidification front approaches, and decelerate as they are engulfed into the ice.

Figure 3 . 6 :

 36 Figure 3.6: Characteristic length scale L v in the plane of Solute Concentration (w t .%) versus Solidification Velocity (V sl ). Contour plots denoting the magnitude of L v for (A) R 1 = 7.2 ± 0.4 µm and (B) R 2 = 30.9 ± 1.2 µm. Each contour line denotes a constant magnitude of L v equivalent to the value shown on the calibration bar. The distance L v , where droplets in water start getting repelled by the interface, increases significantly with the solute concentration, while it decreases with an increasing growth rate.

Figure 3 . 7 :

 37 Figure 3.7: Maximum mean droplet velocity (U max ) when the droplet front edge coincides with the ice-water interface (distance to interface = 0 µm) for (A) R 1 = 7.2 ± 0.4 µm and (B) R 2 = 30.9 ± 1.2 µm. In general, the magnitude of U max is greater for higher solute concentrations and smaller droplet sizes. An increasing growth rate as well as droplet radius promote a smaller magnitude of U max . The dotted lines are for visualization and do no represent an extrapolation of the results denoted by the circular markers.

Figure 3 . 8 :

 38 Figure 3.8: Interaction time for oil droplets dispersed in water at three distinct solute concentrations. Plot of characteristic interaction time for (A) R 1 = 7.2±0.4 µm and (B) R 2 = 30.9±1.2 µm. Droplets tend to interact or feel the presence of an approaching ice-water interface during 10 to 100 s at high solute concentrations (≥ 0.1 w t .%) and low growth rates (≤ 5 µm • s -1 ). The dotted lines are for visualization and do no represent an extrapolation of the results denoted by the circular markers.

Figure 3 . 9 :

 39 Figure 3.9: Displacement of oil droplets dispersed in water at three distinct solute concentrations, while interacting with the ice-water interface. Plot of droplet displacement for (A) R 1 = 7.2 ± 0.4 µm and (B) R 2 = 30.9 ± 1.2 µm. The interaction of droplets with an advancing ice-water interface can displace them by increasing distances (≥ 10 µm) owing to a high amount of solute (≥ 0.1 w t .%) in the bulk solution. The dotted lines are for visualization and do no represent an extrapolation of the results denoted by the circular markers.

Figure 3 . 10 :

 310 Figure 3.10: Solute rejected by the growing solid phase and segregated at the solid-liquid interface during steady-state directional solidification. (A) Model sketch depicting the concentration profile evolution of a solute (in magenta) in the remaining liquid, rejected by a solidification front advancing at V sl , during steady-state planar growth. (B) Mean fluorescence intensity (magenta) of SRhB, acquired using a confocal microscope, is fit with a theoretical diffusiontype exponential (black line). The predicted steady-state diffusion of SRhB (green line), using the model from Tiller et al. [38], corresponds closely to the experimental data fit. A steady-state planar growth is thus verified. The corresponding steady-state diffusion profile of Tween 80 micelles (blue) shows a significant difference in length scales over which the two molecules (SRhB and Tween 80) diffuse. All data presented was recorded for a growth rate of 3 µm • s -1 .

  3.10A). The concentration profile of SRhB rejected by the ice-water interface, deduced from the fluorescence intensity profile for a growth rate of 3 µm•s -1 is given in Fig.3.10B.

Figure 3 . 11 :

 311 Figure 3.11: Typical time-lapse montage depicting the three types of deformation behaviour of oil droplets while undergoing directional planar front solidification. (A) The droplets elongate at the ice-water interface and remain deformed in the ice phase, scale bar = 50 µm (B) The droplets deform in a transient manner (t = 13 s) at the ice-water interface and recover their shape as they are captured by the growing ice phase, scale bar = 10 µm (C) The droplets do not undergo deformation and preserve their shape during their interaction and further engulfment in the growing crystal, scale bar = 50 µm. Oil is in cyan, water is in colormap viridis (fluorescence bar), and ice is in black.

Figure 3 . 12 :

 312 Figure 3.12: Mean elongation profiles for oil droplets encountering an approaching ice-water interface. Planar solidification front induced permanent deformation of oil droplets dispersed in a solution with (A,B) 0.01 w t .% solute concentration having a size of (A) R 1 = 7.2 ± 0.4 µm (B) R 2 = 30.9 ± 1.2 µm. The transient and no deformation regimes for droplets in (C,D) 1 w t .% solute concentration having a size of (C) R 1 = 7.2 ± 0.4 µm (D) R 2 = 30.9 ± 1.2 µm.

Figure 3 . 13 :

 313 Figure 3.13: Crushing of oil droplets during their engulfment in ice at a growth rate of 1 µm•s -1 with 1 w t % solute in solution. (A) Typical time-lapse montage depicting the crushing of an oil droplet, scale bar = 50 µm (B) Mean elongation profiles of R 1 and R 2 droplets increases monotonically with distance to interface proceeding their capture in ice (C) Orthogonal view (invariant along y) of the ice-water meniscus at the given solidification parameters, scale bar = 20 µm.

112 3 . 3 .

 33 Results & Discussions ter/ice interface. At 1 w t .% solute in the aqueous solution, we observe a distinct deformation behaviour of the droplets, which is different from the elongation reported previously. Here, the droplets undergo a monotonically increasing elongation, whereby the corresponding radii (along x and y) are stretched in magnitude as the droplets progress further in ice. We depict this phenomenon of crushing with a time-lapse montage, highlighting the deformation debut at t > 120 s, in Fig.3.13A. Interestingly, we observe this phenomenon with both R 1 = 7.2 ± 0.4 µm and R 2 = 30.9 ± 1.2 µm droplets, only at a growth rate of 1 µm • s -1 in the presence of 1 w t .% bulk solute concentration. Crushing can thus occur independently of how the droplets interact (no deformation or transient deformation)

Figure 3 . 15 :

 315 Figure 3.15: Deducing the isolated droplet velocity U r and the mean droplet velocity U in the sample frame. In the sample frame, the isolated droplet velocity U r is zero far from the interface, it increases and reaches a maximum when the droplet gets repelled by the interface and subsequently, reduces to zero as the droplet is engulfed in the ice. Experimental conditions for which the curve was recorded: V sl = 3 µm • s -1 , G = ∇T = 10 4 K • m -1 , Droplet size R 1 = 7.2 ± 0.4 µm.

Figure 4 . 1 :

 41 Figure 4.1: Force equilibrium to predict engulfment or rejection of objects in the presence of multiple particles. The particle-particle interactions during solidification can result in the formation of a compact layer, which acts as a porous medium. The subsequent resistance offered to the flow of fluid (blue arrows) towards the growing solid results in an additional frictional force F µ , which favors object engulfment. The modified force equilibrium is hence given as F η + F µ = F σ , where F η is an attractive viscous drag force and F σ is a repulsive interfacial force. ∇T = G is the applied temperature gradient and V sl is the imposed growth rate.

Figure 4 . 2 :

 42 Figure 4.2: Experimental setup for in situ solidification experiments. A thin Hele-Shaw cell containing the droplets in suspension is pulled at a constant velocity (V sl ) through a constant temperature gradient (G) established by Peltier elements. In steady state, the solidification front is thus at a constant position under the microscope objective. © (2020) S. Tyagi et al. (10.6084/m9.figshare.12046560) CC BY 4.0 license https://creativecommons.org/licenses/by/4.0/.

Figure 4 . 3 :

 43 Figure 4.3: Typical confocal images of directional solidification depicting different front morphologies; planar, metastable, and columnar. Confocal image enables us to distinguish the dark solid phase (ice) from the bright aqueous phase (H 2 O). Water is in colormap viridis (fluorescence bar) while ice is in black. Scale bar = 100 µm.

Figure 4 . 4 :

 44 Figure 4.4: Front morphologies obtained during the directional solidification of an ice-water system in the solute concentration versus growth rate (V sl ) plane, at a constant thermal gradient of 10 4 K m -1 . A steady-state planar front (A) gets destabilised partially, with an increasing growth rate and solute concentration, to a metastable front (B) exhibiting a few grain boundaries, and eventually forms a cellular front (C) for the higher solute concentrations (≥ 10 -1 wt.%). Broken lines are drawn to differentiate the three zones distinctly and do not imply an extrapolation of the experimental data.

Figure 4 . 5 :

 45 Figure 4.5: Interaction of monodisperse oil-in-water emulsion with an advancing ice-water front. Solidification induced features and spatial distribution of oil droplets depicting (A) Horizontal mono-layers and stable planar front with 8 µm (R 1 ) at 0.01 w t .% solute, (B) Vertically aligned clusters and local front destabilisation with 8 µm (R 1 ) at 1 w t .% solute, (C) No clusters and stable planar front with 28 µm (R 2 ) at 0.01 w t .% solute, and (D) Rejected solute segregation and local front destabilisation from solute premelted films with 28 µm (R 2 ) at 1 w t .% solute. Oil is in cyan, water in colormap viridis (fluorescence bar) while ice is in black. Scale bar = 100 µm.

  sion, forcing them to build-up as a compacted layer. In our experiment, the height of the close-packed layer increases until ≈ 80 µm and subsequently, the droplets get engulfed by the growing solid. Interestingly, the droplets are engulfed as single layers and fabricate a texture parallel to the solid-liquid front. The droplets though closely packed in the liquid ahead of the front, get separated as they are captured by the growing ice crystal. We believe the lateral gap ( y) between the engulfed droplets in ice is created owing to the elongation induced during their capture. The increase in diameter parallel to the temperature gradient ( x) is compensated by the decreasing width in the lateral direction ( y) and hence, the close-packed droplets get separated in the frozen solid. In contrast to the smaller R 1 droplets (Fig.4.5A), the engulfment of the larger R 2 (28 µm) droplets is instantaneous and we observe no subsequent compact layer formation (Fig.4.5C). The elongation induced owing to the droplet-front interaction is visually prominent during the solidification of the R 2 droplets. The deformation of droplets during solidification will be addressed in a separate communication and is not in the scope of the present study. The steady-state planar front remains stable and does not destabilise neither from the formation of the close packed layer nor during the capture of the different size objects (Fig.4.5A,C).

Figure 4 . 6 :

 46 Figure 4.6: Interaction of bimodal oil-in-water emulsion with an advancing ice-water front with 0.01 w t .% bulk solute concentration. (A,B) Descriptive sketch representing monodisperse emulsion observation as seen previously with (A) 8 µm (R 1 ) and (B) 28 µm (R 2 ) droplets (C,D) Sketch representing the experimental observations with bimodal emulsion, prepared by mixing the 8 µm and 28 µm droplets (E,F) Experimental observation of bimodal emulsions depicting (E) Repulsion instigated by droplets R 1 < R c forming a barrier layer and pushing droplets with R 2 > R c (F) Engulfment instigated of droplets R 1 < R c by droplets with R 2 > R c , thereby breaking the compact layer. The droplet 1 is engulfed instantaneously while droplet 2 hits the compact layer and gets repelled before engulfment. Oil is in cyan, water in colormap viridis (fluorescence bar) while ice is in black. Scale bar =100 µm.

Figure 4 . 7 :

 47 Figure 4.7: Interaction of bimodal oil-in-water emulsion with an advancing ice-water front with 1 w t .% bulk solute concentration. (A) Model sketch elucidating the previous experimental observations with 8 µm monodisperse droplets (R 1 ) with 1 w t .% solute in solution (B,C) Concave cusping of the solidification front is induced by the rejected solute segregation in the object-front gap and results in the formation of a solute-rich pocket. (B) Schematic depicting the formation of a pocket and subsequently induced object clustering with bimodal objects containing R 1 +R 2 objects, where R 1 < R c < R 2 (C) Experimental observation showing engulfment of 8 µm oil droplets in pockets initially induced by 28 µm droplets. The droplets are far apart in water at t = 0 s, get clustered at the fermeture of the solute-rich pocket (t = 135 s), and at t = 172 s get further close-packed owing to the lateral growth (along y) of the ice crystals. Oil is in cyan, water in colormap viridis (fluorescence bar) while ice is in black. Scale bar = 100 µm.

4. 3 . 4 Figure 4 . 8 :

 3448 Figure 4.8: Interaction of polydisperse oil-in-water emulsion with an advancing ice-water front at 0.01 w t .% bulk solute concentration. The droplets with radii smaller than the critical radius (R x < R c ) get pushed by the steady-state planar front and lead to the formation of clusters by blocking the arriving droplets (R x < R c < R y ). Oil is in cyan, water in colormap viridis (fluorescence bar) while ice is in black. Scale bar = 100 µm.

Figure 4 . 9 :

 49 Figure 4.9: Interaction of polydisperse oil-in-water emulsion with an advancing ice-water front at 1 w t .% bulk solute concentration. (A) Droplets 1,2,3,4 form a cluster at the ice-water front, while the droplets in the insets form a cluster ahead of the front. The agglomerates depict a pushing-engulfment transition with local destabilisation of the ice-water front (B) Schematic depicting the experimental observations in (A), where small droplets (R x < R c ) form a barrier and subsequently form close-packed agglomerates with a combination of droplet sizes (R x < R c < R y ) before getting captured in the solid. Oil is in cyan, water in colormap viridis (fluorescence bar) while ice is in black. Scale bar = 100 µm.

  Conclusions & Perspectives at play ahead of the solidification front with the quantification of long-range solute (or impurity) effects.As the solidification front hit the droplets, we further characterised the corresponding 2D shape deformation observed at the different experimental parameters. We depicted three distinct types of deformation; no elongation, transient elongation at the interface, and permanent elongation in ice. The droplet deformation was found to be strongly dependant on the solute concentration and the droplet size. A high amount of solute instigated solute-enriched premelted films were shown experimentally and believed to be at the origin of the shape recovery mechanism. The evaluation of the object deformation and the surrounding solute environment are topics of research in porosity or void formation during crystal growth and for cryopreservation of food emulsions or biological cells. Our study demonstrated the dynamic nature of object interactions and the evolution of premelted films at the solid-liquid interface. These solidification features remain difficult to visualize experimentally while having drastic impacts on the evolution of the solidification microstructure.

  the future, use of a solute exhibiting laser-induced fluorescence can provide further insights into the in situ diffusion field. The fluorescence intensity can be acquired and correlated to the local temperature in the vicinity of the solidification front. The diffusion length of such a solute can be compared with the distance at which the objects feel the repulsion to deduce accurately the presence of a thermo-solutal flow (or not). The use of thermo-responsive polymers for Conclusions & Perspectives which the fluorescence intensity varies with temperature can be envisaged to act as a local temperature probe. The impact of ionic solutes could be investigated to observe additional effects such as the presence of an electrophoresis of colloidal particles. This is interesting as the rejected ions could create in situ gradients owing to their differing solubility and migration velocities in the solid-liquid phase. The direction and magnitude of force acting on soft objects instigating their subsequent deformation or those of premelted films can be investigated by the use of stress-responsive polymer mechanophores. These chemical compounds fluoresce under the solicitation of stress and hence, could be helpful in visualizing the crystal growth pressure and direction.The experimental setup is adapted to study the formation of premelted films at varying solute concentration. Systematic analysis of the premelted film thickness can help in deducing the dominating intermolecular interactions (dispersion or electrostatic) at their origin. The rheology of these premelted films is still largely unknown and we do not know if they behave as Newtonian or non-Newtonian fluids. Measurement of their rheology can be a great stride in the study of objectfront solidification systems as well as contribute to the understanding of several geophysical phenomena.

  Dans la dernière section, nous avons identifié la déformation des gouttelettes après leur engloutissement dans le solide glace. L'utilisation de notre configuration offre une résolution spatiale élevée en 2D et 3D, ce qui nous a permis d'analyser le comportement des gouttelettes et l'épaisseur environnante des films pré-fondus à de grandes distances du front de solidification. Il a été suggéré que le phénomène d'écrasement provenait du ménisque de l'interface glace-eau. Nous Résumé avons ensuite examiné le mouvement des gouttelettes dans le solide pour étudier la possibilité d'un effet thermique. Cependant, les gouttelettes capturées dans la glace ne présentaient aucune vitesse relative et, par conséquent, ne migraient pas vers le côté chaud du gradient de température sous-jacent. La migration des objets, les films pré-fondus et l'interaction des particules avec la phase solide à basse température sont étudiés en particulier par les géophysiciens pour comprendre le soulèvement du gel, le mouvement des glaciers et les indicateurs climatiques des noyaux de glace. Ainsi, nous suggérons notre méthodologie expérimentale d'utilisation d'émulsions huile-dans-eau dans le dispositif cryoconfocal comme analogue pour l'étude de phénomènes complexes. À l'avenir, l'utilisation d'un soluté présentant une fluorescence induite par laser peut fournir des informations supplémentaires sur le champ de diffusion in situ. L'intensité de la fluorescence peut être acquise et corrélée à la température locale au voisinage du front de solidification. La longueur de diffusion d'un tel soluté peut être comparée à la distance à laquelle les objets ressentent la répulsion pour déduire avec précision la présence (ou non) d'un flux thermo-solutal. L'utilisation de polymères thermo-sensibles dont l'intensité de la fluorescence varie avec la température peut être envisagée pour servir de sonde de température locale. L'impact des solutés ioniques pourrait être étudié pour observer des effets supplémentaires tels que la présence d'une électrophorèse de particules colloïdales. Ceci est intéressant car les ions rejetés pourraient créer des gradients in situ en raison de leur solubilité et de leurs vitesses de migration différentes dans la phase solide-liquide. La direction et l'ampleur de la force agissant sur les objets mous provoquant leur déformation ultérieure ou sur ceux des films pré-fondus peuvent être étudiées par l'utilisation de mécanophores polymères sensibles au stress. Ces composés chimiques sont fluorescents lorsqu'ils sont soumis à un stress et peuvent donc être utiles pour visualiser la pression et la direction de la croissance cristalline. Le dispositif expérimental est adapté pour étudier la formation de films pré-fondus à différentes concentrations de solutés. L'analyse systématique de l'épaisseur du film pré-fondu peut aider à déduire les interactions intermoléculaires dominantes (dispersion ou électrostatique) à leur origine. La rhéologie de ces films pré-fondus est encore largement inconnue et nous ne savons pas s'ils se comportent comme des fluides newtoniens ou non newtoniens. La mesure de leur rhéologie peut constituer un grand pas en avant dans l'étude des systèmes de solidification du front d'objet et contribuer à la compréhension de plusieurs 165 Résumé phénomènes géophysiques. Des objets multiples qui interagissent avec un front de solidification Nous avons réalisé cette étude pour essayer de combler le fossé entre les interactions objet isolé et le front et les systèmes réalistes où la complexité augmente en raison d'une forte concentration d'objets. L'objectif était d'élucider expérimentalement le développement d'une microstructure réelle avec de multiples objets en présence ou en l'absence de soluté. La plupart des études précédentes ont porté sur la prédiction du comportement de particules isolées, dures, sphériques et rigides. On s'est donc beaucoup moins intéressé aux interactions entre particules et à la fraction volumique des objets dispersés dans le liquide. De plus, la présence d'un soluté ajoute à la complexité car il joue un rôle majeur sur la morphologie du front. C'est pourquoi nous avons analysé l'interaction de multiples gouttelettes d'huile, agissant comme un modèle analogue, avec une interface glace-eau qui progresse. Nous avons d'abord démontré l'interaction des gouttelettes d'huile monodispersées, et étudié l'influence de la taille des gouttelettes et de la concentration de soluté. Nous avons décrit la formation d'une couche compacte de gouttelettes et discuté de la présence d'une force de frottement supplémentaire similaire à celle observée dans les milieux poreux. La modification de l'équilibre des forces due aux interactions particules-particules a été invoquée et comparée à celle prédite par les modèles d'objets isolés. Nous avons essayé de nous appuyer sur les études théoriques précédentes pour expliquer en partie les mécanismes d'interaction observés dans nos expériences. Nous avons noté un impact significatif sur la texture des gouttelettes englouties avec une modification de la concentration de soluté et de la taille de l'objet.Dans la deuxième section, nous avons analysé l'interaction des émulsions bimodales, où les rayons des gouttelettes utilisées dans le régime monodispersé étaient mélangés. Nous avons discuté de la formation d'agglomérats de gouttelettes et avons décrit la répulsion des grosses gouttelettes en raison de la couche compacte de gouttelettes plus petites qui sont poussées par le front de solidification. L'ajout d'une petite quantité de soluté au liquide a entraîné la formation de poches riches en soluté. La présence de poches a provoqué l'agglomération des gouttelettes, ce qui a forcé les gouttelettes d'huile à se refermer en agglomérats.RésuméNous avons finalement étudié l'interaction des gouttelettes polydispersées pour imiter les systèmes réels avec une plus grande précision. Nous avons signalé la formation d'agglomérats au front de solidification en l'absence de soluté, tandis que les amas ont été observés bien avant le front en présence d'effets de soluté. Une transition répulsion-intégration ou répulsion-piégeage a été déduite, contrairement aux modèles théoriques précédents, qui suggèrent un engouffrement instantané des objets en présence de soluté en solution.

1. 1

 1 Object interacting with a moving solidification front: principles. Model sketch, where d is the particle-front gap, G = ∇T is the temperature gradient, κ is the interfacial front curvature (here > 0 for concave geometry), k p , k m are the thermal conductivity of the particle and melt respectively, R is the object radius, T m is the absolute melting temperature, T i is the solid-liquid interfacial temperature, and V sl is the solidification velocity. . . . . . . . . . . . . . . . . . . . 1.2 Change in interfacial energy when a particle interacts with a solidification front, adapted from [4]. . . . . . . . . . . . . . . . . . . . . . 1.3 Interfacial energy varying with the distance (d ) between the particle and the solidification front, adapted from [3]. . . . . . . . . . . . . . 1.4 Solidification front curvature in the vicinity of a particle with a thermal conductivity mismatch with the melt (a) Convex for k p /k m < 1, (b) Horizontal for k p /k m = 1, and (c) Concave for k p /k m > 1, adapted from [4]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Paraboloidal front shape (left) considered a priori with distortion (right) of the temperature field (k p = k m ) in the model from Chernov et al [10, 14], adapted from [4, 15]. . . . . . . . . . . . . . . . . . . . .

1. 7 1 . 9

 719 Curvature-induced and interfacial premelted films at an ice-particlewater interface, adapted from[23]. . . . . . . . . . . . . . . . . . . . . 1.8 Thickness of the premelted film, d , depends on the temperature, and determines the magnitude of the repulsive disjoining pressure P T , adapted from [25]. . . . . . . . . . . . . . . . . . . . . . . . . . . . Regelation or migration of small particles in a solid towards the warmer side of the temperature gradient ∇T owing to a thermomolecular pressure gradient from the varying thickness of the premelted films, adapted from [27]. . . . . . . . . . . . . . . . . . . . . . 1.10 The physics of premelted films is investigated in the growth of ice lenses with various natural manifestations such as frost heave in partially frozen soils, adapted from [23, 26]. . . . . . . . . . . . . . . 1.11 Model from Rempel et al. [20] accounts for the presence of a liquid premelted film owing to the inter-molecular interactions in the gap between the particle and the solid, adapted from [30]. . . . . . . . . 1.12 Under steady-state conditions, the particle velocity is equal to the solidification velocity. (a) When the premelting effect is excluded from the model, the gap thickness decreases monotonically with increasing solidification velocity. A critical velocity for engulfment must be defined via an arbitrary minimum gap thickness. (b) When premelting is included, the Rempel et al. [20] model predicts two solution branches, with two values of gap thickness for a given solidification velocity. A critical engulfment velocity is naturally defined as the maximum solidification velocity, or limit point, of the steady-state solution branches, adapted from [31]. . . . . . . . . 1.13 The dimensionless particle velocity (u) versus dimensionless particle height (b) varies with the type of inter-molecular interactions ν taken into account. When b is large and negative the edge of the particle is below the plane of the T m isotherm. The exponent ν depends on the type of intermolecular interactions that dominate; ν = 3 represents non-retarded van der Waals interactions, while ν = 2 and ν = 4 is used for retarded van der Waals and long-range electrical interactions [22], adapted from [20]. . . . . . . . . . . . . . 21

1. 15 8 to - 1 . 3 .Figures 2 . 3 74 2. 4 76 3. 6 3 . 7 3 . 8 3 . 9

 1581323744766373839 Photographs of front and side views. Reproduced from[START_REF] Dedovets | A temperaturecontrolled stage for laser scanning confocal microscopy and case studies in materials science[END_REF]. . . . . . 39

Table 2 .

 2

	and Sulforhodamine B

Provence. We purchased fluorescent green polyethylene (1 g • cm -3 , 45 -53 µm) and stainless steel (7 g • cm -3 , 48 -57 µm) microspheres from Cospheric LLC. We ordered monodisperse green-fluorescent polystyrene (48.1 µm) and poly(methyl methacrylate) (PMMA) (50.3 µm) from Micro Particles GmbH. Deionized water, THERMAL CONDUCTIVITY EFFECTS 1: Radius (R), thermal conductivity of the materials (k p ) and their ratio with the aqueous phase (k p /k m ), where k m

Table 2 .2: Typical solidification parameters for an object in front of an ice- water

 2

solidification front. L v , σ sl , and λ adapted from Park et al.[14] 2.3. Results and discussion 2.

3 Results and discussion 2.3.1 Thermal conductivity controls the interfacial curvature

	We observe a pronounced convex curvature (Fig. 2.3A,B) when the objects have
	a lower thermal conductivity than water (k p /k m < 1, see Fig.2.3C) and thus, the
	front tends to diverge towards them as they act as thermal shields causing the
	temperature in the gap behind them to be successively lowered. With zirconia
	(Z r O 2 ) and stainless steel (SS) particles, we observe a concave cusping of the
	front (Fig. 2.3B). The relatively higher thermal conductivity of these particles
	compared to water (k

p /k m > 1, see Fig.

2

.3C) enables a preferential heat flux from

Table 4 .1: Droplet radius (R) and bulk solute concentration

 4 .5. The corresponding droplet size and bulk solute concentration used for the experiments are given in table 4.1.

	Figure 4.5	R	Sol ut e conc.
		(µm)	(w t .%)
	A	8	0.01
	B	8	1.00
	C	28	0.01
	D	28	1.00

(w t .%) in

the aque- ous phase for the monodisperse oil-in-water emulsions.

Table 4 .2: Typical solidification parameters for an object in front of an ice- water solidification front. σ sl adapted from Rempel et al. [16].

 4

Objets en interaction avec le front de solidification : Effets de la conductivité thermique

	Résumé	Résumé
	forces agissant sur un objet et joue donc un rôle majeur dans les interactions en-les objets, dont la conductivité thermique est inférieure à celle de l'eau, peuvent
	tre l'objet et le front. La géométrie du front est essentiellement une manifestation déformer les cristaux de glace voisins après l'engouffrement lors de la solidi-
	des isothermes locales et dépend donc de l'uniformité du champ de température fication directionnelle. Ainsi, la microstructure de solidification, où les objets
	à l'interface solide-liquide. Les études précédentes ont examiné le rôle de la interagissent avec un front en progression, dépend d'une interaction entre les
	non-concordance de la conductivité thermique entre l'objet et la liquide, mais effets thermiques et les effets de soluté.
	rien de systématique avec un contrôle complet des paramètres de solidification À l'avenir, un paramètre important qui pourrait être étudié plus en profondeur
	n'a été réalisé expérimentalement. Une grande partie de la littérature précédente est l'énergie de surface sans dimension (γ). Le rôle de ce paramètre, sur la mod-
	est donc composée de prédictions théoriques. ification de la courbure de l'interface au voisinage des objets, peut être étudié
	L'objectif de notre étude était d'explorer la forme du front de solidification en utilisant des objets ayant un rapport de conductivité thermique identique
	observations expérimentales avec les modèles théoriques passés pour élucider les différentes caractéristiques et proposé des mécanismes pour les phénomènes contrastés. Nous soulignons maintenant les principales conclusions qui peuvent être tirées de chaque étude individuelle et discutons des possibilités de travaux futurs (qui s'éloigne des objets) avec une amplitude de courbure considérablement (nous l'espérons). sur une large gamme de rapports de conductivité thermique (0, 04 ≤ k liquid. Le soluté est à l'origine de la formation d'une forme de l'interface concave par rapport à l'eau (k
	plus élevée que celle mesurée en l'absence de soluté. Ceci a des implications
	importantes car la nature et l'ampleur des forces peuvent être modifiées par la
	présence d'effets dominants du soluté.
	Dans la dernière section, nous avons décrit la modification de la forme du
	Dans cette étude, l'objectif principal était d'analyser expérimentalement la forme front solid (glace) pour la morphologie du front cellulaire. Nous avons discuté
	du front de solidification, en utilisant des objets (mous ou durs), ayant un dé-de la modification de la distance à laquelle la ségrégation du soluté affecte la
	géométrie du front en presence de cellules par rapport aux précédentes études
	sur le front planaire. De plus, nous avons démontré pour la première fois que

Nous avons réalisé cette étude sur une période de trois ans dans le but d'améliorer notre compréhension actuelle de l'interaction des objets étrangers avec un front de solidification en mouvement. L'objectif principal était d'étudier les mécanismes responsables du développement d'une microstructure solidifiée directionnellement en présence d'objets secondaires à l'aide de la microscopie cryoconfocale in situ. Nous avons étudié l'interaction de particules dures isolées (polymères, céramiques et métaux) et d'objets mous (gouttelettes et bulles) avec une interface glace-eau en progression. Pour explorer le rôle des solutés, nous avons systématiquement analysé la courbure interfaciale, la dynamique des objets, l'évolution de la forme des objets et leur distribution spatiale, en présence et en l'absence d'effets de solutés. Nous avons ensuite progressé vers des systèmes plus réalistes en employant une fraction volumique élevée d'objets dans la masse fondue liquide par rapport aux systèmes isolés. Nous avons ensuite comparé nos calage de conductivité thermique avec le liquide en fusion (k p /k m = 1). La forme du front de solidification est un paramètre important qui définit l'ampleur des p /k m ≤ 35) en utilisant des particules polymères, en céramiques, ou métalliques rigides ainsi que des gouttelettes d'huile et des bulles d'air. Nous avons d'abord observé et mesuré la distorsion du front à proximité des objets avec une haute résolution spatiale grâce à notre configuration specifique de microscopie cryo-confocale. Nous avons constaté que pour une courbure donnée (convexe ou concave), l'ampleur de la déflexion était constante et indépendante de la nature de l'objet. Nos observations sont conformes aux prédictions de courbure de l'interface du modèle théorique de Park et al., cependant, les prédictions d'engouffrement ou de répulsion n'étaient pas conformes car nous avons constaté un engouffrement des objets indépendamment de la courbure de l'interface. L'étape suivante de notre étude a consisté à explorer l'influence des solutés sur la géométrie du front. Cette étude a été menée car la plupart des systèmes réels sont constitués de solutés (ou d'impuretés) et l'observation des effets des solutés in situ reste complexe. Il est intéressant de noter que nous avons déduit que la présence d'une ségrégation des solutés entre l'objet et l'interface glace-eau domine sur les effets de la différence de conductivité thermique entre l'objet et la p /k m ) mais avec des rayons différents (R). La nature et l'ampleur des forces peuvent être étudiées pour des objets qui ont une conductivité thermique égale à celle de l'eau (k p /k m = 1). Ici, la forme du front ne joue aucun rôle et donc, l'équilibre des forces peut être estimé analytiquement en l'absence d'effets solutés. Les propriétés de mouillage d'un système liquideparticule-solide peuvent être analysées plus en détail car elles ne sont pas prises en compte dans les modèles théoriques. Cela peut être réalisé en utilisant des particules hydrophobes et hydrophiles, tout en estimant leur vitesse critique dans des conditions expérimentales similaires. Les caractéristiques (épaisseur, rhéologie) des films pré-fondus en raison des différentes propriétés de mouillage des objets peuvent être intéressantes à explorer.

Effets des solutés sur la dynamique et la déformation lors de la congélation des gouttelettes

  Notre étude a démontré la nature dynamique des interactions entre les objets et l'évolution des films préfondus à l'interface solide-liquide. Ces caractéristiques de solidification restent difficiles à visualiser expérimentalement tout en ayant des impacts drastiques sur l'évolution de la microstructure de solidification.

	Résumé
	soluté lorsque le front s'approche des gouttelettes. Nous avons noté qu'une con-
	centration croissante de soluté a un impact drastiques sur l'échelle de longueur
	à laquelle les gouttelettes sont repoussées loin de l'interface qui s'approche. Le
	mouvement des gouttelettes s'est manifesté par une vitesse de répulsion de
	1 µm • s -1 et a été initié à une distance approchant 100 µm pour une concen-
	tration élevée de soluté. Les conséquences de ces effets à longue distance se
	traduisaient par un temps d'interaction élevé et provoquaient un déplacement
	des gouttelettes d'une ampleur bien supérieure à la taille de l'objet. Ces résul-
	tats sont intéressants pour étudier le développement d'une microstructure de
	matériau et pour prédire la distribution spatiale des objets lors de la solidification
	de l'alliage. De plus, les études précédentes ont été principalement dérivées pour
	la prédiction de l'engouffrement ou du rejet d'objets avec des forces à courte
	Cette étude a été réalisée pour comprendre l'impact du soluté (ou de l'impureté)
	sur l'interaction des gouttelettes d'huile avec une interface glace-eau qui pro-
	gresse. Nous avons systématiquement étudié les effets du soluté à trois étapes
	distinctes de la solidification du front planaire : les gouttelettes dans l'eau, les
	gouttelettes à l'interface et les gouttelettes capturées dans la glace. L'objectif
	était d'analyser les modifications provoquées par le soluté ajouté au liquide de
	base, qui est ensuite rejeté et à l'interface, au fur et à mesure de la croissance
	des cristaux de glace. Pour effectuer une analyse approfondie, nous avons utilisé
	deux tailles de gouttelettes différentes et avons progressivement capturé les effets
	d'une concentration croissante de soluté à différents taux de croissance.

Dans un premier temps, nous avons démontré la dynamique des gouttelettes loin de l'interface glace-eau, c'est-à-dire les conséquences de la diffusion du portée (< 1 µm), tandis que notre étude fournit une image complète des mécanismes en jeu avant du front de solidification avec la quantification des effets à longue portée des solutés (ou des impuretés).

Lorsque le front de solidification rencontre les gouttelettes, nous avons caractérisé la déformation de forme 2D correspondante observée aux différents paramètres expérimentaux. Nous avons représenté trois types distincts de déformation : aucun allongement, allongement transitoire à l'interface et allongement permanent dans la glace. La déformation des gouttelettes s'est avérée fortement dépendante de la concentration en soluté et de la taille des gouttelettes. Un film pré-fondu enrichi en soluté a été observé expérimentalement et on pense qu'il est à l'origine du mécanisme de retour à la forme initiale. L'évaluation de la déformation de l'objet et du milieu soluté environnant sont des sujets de recherche sur la porosité pendant la croissance des cristaux et pour la cryopréservation des émulsions alimentaires ou des cellules biologiques.

1.8. Current issues and challenges remains limited and barely sufficient to image the object interactions owing to a low acquisition frequency. In addition, a visualization of the local solute segregation is still lacking. Hence, the tracking of microstructures where objects interact with an advancing solid-liquid interface in the presence of solute effects remains challenging.

Acknowledgements Acknowledgements

The research leading to these results has received funding from the ANRT and Saint-Gobain through a CIFRE fellowship (N • 2017/0774). H.H.'s internship was funded by Saint-Gobain.

Acknowledgements

The research leading to these results has received funding from the ANRT and Saint-Gobain through a CIFRE fellowship (N • 2017/0774).

acknowledge Saint-Gobain for funding this wonderful collaboration. I would particularly like to single out my manager at Saint-Gobain Research Provence, Caroline Tardivat.

Citation

This chapter can be cited as Tyagi, S., Huynh, H., Monteux, C. & Deville, S. Objects interacting with solidification fronts: Thermal and solute effects. Materialia 12, 2020 https://doi.org/10.1016/j.mtla.2020.100802.

Citation

This chapter can be cited as Tyagi, S., Monteux, C. & Deville, S. Multiple objects interacting with a solidification front. Sci Rep 11, 3513 (2021). https://doi.org/10.1038/s41598-021-82713-3

Author contributions

Conflict of interest

The authors declare no conflict of interest.

Supplementary Information

Supplementary Information

We analysed the trajectories of all the oil droplets to obtain their apparent velocity (U ) with varying distance from the interface, using the observation frame of reference. Time and distance 0 correspond to the moment and position where the front edge of each droplet hits the solidification front. In the observation frame, the solidification front is at a constant position, while the droplets move along the x-axis. The sample length of 4 cm corresponds to between 50 to 400 complete droplet trajectories depending on R and V sl as shown in Fig. 3.14. To transpose the data in the sample frame, we averaged the acquired droplet trajectories and performed the analysis as follows; U r = V sl -U as shown in Fig. 3.15.

Figure 3.14: Measurement of interaction time and apparent droplet velocity

U in the observation frame. In the observation frame, far from the solidliquid interface in water the droplet moves at the imposed growth rate U = V sl , as the droplet approaches the interface it gets repelled U = V sl , and as the droplet is engulfed in the ice it doesn't get repelled anymore, thereby recovering U = V sl . The interaction time is the total duration over which a droplet gets repelled by the solidification interface. Relative time is zero when the front edge of the droplet hits the solidification front. Experimental conditions for which the curve was recorded: V sl = 3 µm•s -1 , G = ∇T = 10 4 K •m -1 , Droplet size R 1 = 7.2 ± 0.4 µm.

Results

the critical radius is 9 µm for our experimental parameters. However, we observe the repulsion of miscellaneous droplets irrespective of their radii (R x < R c < R y ), as seen in Fig. 4.8, in the absence of long-range solute effects. The droplet clusters form at the front with the primary layer occupied by the small droplets (R x < R c ), which further creates a barrier and facilitate the repulsion of relatively larger droplets (R y > R c ).

For planar front solidification in the presence of solute, the previous models [19,21,33,34] deduce a critical engulfment velocity which in most cases is an order below the critical velocity for isolated objects in pure materials. The models suggest either an absence of pushing-engulfment transition, or a destabilisation of the steady-state planar front prior to engulfment. In both scenarios, rapid engulfment of the object is suggested as the most favorable outcome, facilitated by an increase in the viscous drag forces. However, the presence of multiple-particle interactions is not incorporated in these models. The experimental evidence are difficult to correlate with owing to varying process parameters and often suggest contradictory results [4,23,24,25]. Körber et al. [4] have reported no impact of solute on critical velocity in water-NaMnO 4 solution, in contrast to the theoretically expected decrease in V c (from pure melts) owing to the impurity effects. On the contrary, Sekhar & Trivedi [23] have demonstrated strong impacts of impurity leading to particle trapping in the directional solidification of succinonitrile-acetone system.

In our experiments, we report a pushing-engulfment or repulsion-trapping transition in the presence of dominating solute effects. From Fig. 4.9A, we observe the presence of droplet clusters generated both at (inset 1,2,3,4 in Fig. 4.9A) and ahead of the front (inset box in Fig. 4.9A). The 3D time-lapse evolution depicts the small droplet (inset arrow Fig. 4.9A) being pushed by the front for ≈ 400 s.

Consequently, the droplets (inset 1,2,3,4 in Fig. 4.9A) are blocked and form a closepacked cluster, eventually engulfed at t = 808 s. We illustrate this scenario in Fig. 4.9B with a model schematic. The morphology of the front does not undergo a transformation in the presence of droplets, as suggested previously [23,52,53], and remains as depicted in the absence of objects (Fig. 4.4). The formation of the cluster ahead of the front is still not clear but we believe the solute phoretic phenomenon such as diffusiophoresis can be at the genesis of such long-range displacement [38]. Although, we require more experiments to conclude effectively the origin of such close-packed clusters.

Author contributions

Conflict of interest

The authors declare no conflict of interest.

Conclusions & Perspectives

the front curvature in the vicinity of the objects, can be studied using objects with equal thermal conductivity mismatch (k p /k m ) with the melt but different radii (R). The nature and magnitude of forces can be explored for objects which have an equal thermal conductivity as the melt (k p /k m = 1). Here, the shape of the front does not play a role and hence, the force equilibrium can be estimated analytically in the absence of solute effects. The wetting properties of a liquid-particle-solid system can be analysed further as they have not been invoked in the theoretical models. This can be achieved by using hydrophobic and hydrophilic particles, while estimating their critical velocity under similar experimental conditions. The characteristics (thickness, rheology) of premelted films owing to the varying object wetting properties can be interesting to explore.

Solute effects on dynamics and deformation of freezing droplets

This study was performed to comprehend the impact of solute (or impurity) on the interaction of oil droplets with an advancing ice-water interface. We systematically investigated the solute effects at three distinct stages of planar front solidification; droplets in water, droplets at the interface, and droplets captured in ice. The objective was to analyse the modifications instigated by the solute added to the parent liquid, which is subsequently rejected and segregated at the interface, as the ice crystal growth occurs. To perform an in-depth analysis, we used two different droplet sizes and progressively captured the effects of an increasing bulk solute concentration at various growth rates.

Initially, we demonstrated the droplet dynamics far from the ice-water interface i.e. the consequences of the diffusing solute as the front approaches the droplets. We noted that an increasing bulk solute concentration has drastic impacts on the length scale at which it can repel or push the droplets away from the approaching interface. The droplet motion was manifested by a repulsive velocity of 1 µm • s -1 and initiated at a distance approaching 100 µm for a high solute concentration. The consequences of such long-range effects were reflected by the high interaction-time and instigated droplet displacement of magnitude much greater than the object size. These findings are of interest for studying the development of a material microstructure and predicting the object spatial distribution in alloy solidification. Furthermore, the previous studies have been mostly derived for predicting engulfment or rejection of objects with short-range forces (< 1 µm), while our study provides a complete picture of the mechanisms

Measurement of interaction time and apparent droplet velocity

U in the observation frame. In the observation frame, far from the solid-liquid interface in water the droplet moves at the imposed growth rate U = V sl , as the droplet approaches the interface it gets repelled U = V sl , and as the droplet is engulfed in the ice it doesn't get repelled anymore, thereby recovering U = V sl . The interaction time is the total duration over which a droplet gets repelled by the solidification interface. Relative time is zero when the front edge of the droplet hits the solidification front. Experimental conditions for which the curve was recorded: In the sample frame, the isolated droplet velocity U r is zero far from the interface, it increases and reaches a maximum when the droplet gets repelled by the interface and subsequently, reduces to zero as the droplet is engulfed in the ice. Experimental conditions for which the curve was recorded: A steady-state planar front (A) gets destabilised partially, with an increasing growth rate and solute concentration, to a metastable front (B) exhibiting a few grain boundaries, and eventually forms a cellular front (C) for the higher solute concentrations (≥ 10 -1 wt.%).

Broken lines are drawn to differentiate the three zones distinctly and do not imply an extrapolation of the experimental data. . . . .

List of Tables

MOTS CLÉS

solidification, interface, soluté, conductivité thermique, microscopie confocale RÉSUMÉ L'interaction du front de solidification avec des objets (gouttelettes, bulles, particules ou cellules biologiques) est un phénomène courant rencontré dans une multitude de situations, allant des procédés industriels aux événements naturels, tels que la formation de glace de mer, la croissance de monocristaux, la métallurgie, la cryobiologie, l'ingéniérie alimentaire ou la purification de l'eau. Ces processus, pourtant bien identifiés, sont complexes et une compréhension approfondie reste à développer. Les objets (rigide ou déformable) peuvent avoir différents comportements lors de leur rencontre avec un front de solidification en mouvement. Le front peut integrer l'objet, le piégeant dans le cristal en croissance, ou le repousser. Les conséquences de cette interaction ont un impact sur la microstructure finale et déterminent notamment les propriétés structurelles et fonctionnelles du matériau solidifié. En outre, la plupart de ces phénomènes se produisent en présence de solutés (ou d'impuretés), qui peuvent être ségrégés à l'interface lors de la solidification.

Dans cette étude, nous analysons l'interaction d'objets sphériques déformable (gouttelettes et bulles) ou de particules rigides avec une front de solidification eau-glace en mouvement. La microscopie cryo-confocale permet une imagerie rapide in situ de la croissance des cristaux de glace, de la phase liquide et de la dynamique des objets qui rencontrent le front de solidification. Nous décrivons l'impact de la concentration des solutés (ou des impuretés) et leur influence sur la géométrie de l'interface, la dynamique des objets et la microstructure de solidification. Nous explorons plusieurs aspects importants des interactions de particules multiples dans des systèmes réalistes et discutons des conséquences de la redistribution des particules par rapport aux systèmes idéaux décrits dans les modèles physiques précédemment développés. Nos résultats montrent comment une concentration croissante de soluté dans le liquide et sa ségrégation lors de la solidification peuvent modifier de façon spectaculaire la nature attendue des interactions entre l'objet et le front et les microstructures de solidification. La congélation des suspensions par microscopie cryo-confocale peut servir d'analogue pour étudier l'interaction in situ d'objets avec une front de solidification en mouvement.

ABSTRACT

The interaction of solidification fronts with objects (droplets, bubbles, particles, or biological cells) is a common phenomenon encountered in a plethora of situations, ranging from industrial to natural occurrences, such as the formation of sea ice, growth of single crystals, metallurgy, cryobiology, food science, and filtration or water purification. The process, though well-known, is complex and an unambiguous quantification of the mechanisms in situ is yet to be developed. The objects (soft or hard) can have diverse outcomes owing to their confrontation with an advancing solid-liquid interface. The front can engulf the object, trapping it into the growing crystal, or the front can repel the object, pushing it ahead of itself.

The consequences of this interaction impacts the final microstructure and hence, determines the structural as well as functional properties of the solidified material. Furthermore, many of these phenomena occur in the presence of solutes (or impurities), which can segregate at the interface during the growth of a solid. In this study, we analyse the interaction of spherical soft objects (droplets and bubbles) and rigid particles with an approaching ice-water interface. Our custom cryo-confocal microscopy setup enables a rapid imaging of the growth of ice crystals, the liquid phase, and the dynamics of objects encountering the solidification front. We depict the impact of solute (or impurity) concentration and their influence on the interface geometry, object dynamics, and the solidified microstructure. We explore several important aspects of multiple particle interactions in realistic systems and discuss the consequences of particle redistribution in comparison to the ideal systems considered by the past physical models. Our results demonstrate how an increasing concentration of solute in the bulk liquid and their subsequent segregation can dramatically alter the predicted nature of object-front interactions and the solidification microstructures. We suggest that the freezing of suspensions using cryo-confocal microscopy may serve as an analogue for studying the in situ interaction of foreign objects with a moving solid-liquid interface. KEYWORDS solidification, interfaces, solute, thermal conductivity, confocal microscopy