Brice Saint-Michel

Marc Georgelin

Sylvain Deville

Alain Pocheau

Interaction of multiple particles with a solidification front : from compacted particle layer to particle trapping

The interaction of solidification fronts with objects such as particles, droplets, cells, or bubbles is a phenomenon with many natural and technological occurrences. For an object facing the front, it may yield various fates, from trapping to rejection, with large implications regarding the solidification pattern. However, whereas most situations involve multiple particles interacting with each other and the front, attention has focused almost exclusively on the interaction of a single, isolated object with the front. Here we address experimentally the interaction of multiple particles with a solidification front by performing solidification experiments of a monodisperse particle suspension in a Hele-Shaw cell, with precise control of growth conditions and real-time visualization. We evidence the growth of a particle layer ahead of the front at a close-packing volume fraction and we document its steady state value at various solidification velocities. We then extend single particle models to the situation of multiple particles by taking into account the additional force induced on an entering particle by viscous friction in the compacted particle layer. By a force balance model, this provides an indirect measure of the repelling mean thermomolecular pressure over a particle entering the front. The presence of multiple particles is found to increase it following a reduction of the thickness of the thin liquid film that separates particles and front. We anticipate the findings reported here to provide a relevant basis to understand many complex solidification situations in geophysics, engineering, biology, or food engineering, where multiple objects interact with the front and control the resulting solidification patterns.

I. INTRODUCTION

The interaction of particles with a solidification front is a phenomenon encountered in numerous natural and technological situations, such as the evolution of frozen soils by frost heave or by ice lenses formation [1][2][3][4][5][6][7][8], cryobiology [9,10], food industry [START_REF] Velez-Ruiz | Handbook of food preservation[END_REF] and materials science, from alloy casting in presence of particles [START_REF] Stefanescu | [END_REF][13][14] to the fabrication of bio-inspired composites [15]. When a particle suspension freezes, a solidification front interacts with the dispersed particles by short range thermomolecular forces induced by Van der Waals like interactions [START_REF] Chernov | nikova[END_REF]17]. Depending on the freezing conditions and characteristics of the suspension, this interaction may affect the particle distribution in the frozen phase in a number of ways and thus yield a large variety of segregation patterns, with important implications regarding the effective properties of the resulting solid [18][19][20][21].

In most of the natural and technological occurrences of freezing suspensions, multiple particles interact with each other an with the front, sometimes forming ahead of the front a dense layer of particles referred to as a pile-up [22] or a build-up [START_REF] Stefanescu | [END_REF] of particles. Whereas this layer may directly impact the trapping mechanism, little work has been done so far to understand the interaction of multiple particles with the front due to the inherent complexity of the problem. In particular, the trapping mechanism has long been studied by considering a single, isolated particle interacting with the freeze front, both experimentally [22][23][24][25] and theoretically [13,[START_REF] Chernov | nikova[END_REF]17,[26][27][28][29][30][31][32][33][34][35][36][37].

The objective of this study is to experimentally address the solidification regime of a particle suspension where multiple particles interact with the front, and use its characteristics to improve the modeling of multiple particles/front interaction.

For this, a unidirectional freezing experiment in a Hele-Shaw sample is performed using a monodisperse particle suspension. We focus on the steady state regime where a compacted particle layer of constant width h is pushed at a constant velocity V by the front. As the steady-state width h of this layer is directly linked to the trapping mechanism, its evolution with the velocity V provides a mean to document it and question its modeling in the presence of particles.

We thus build a mechanical model for trapping that takes into account the presence of multiple particles through the additional force exerted on an entering particle by the particle layer, following viscous friction and pressure drop accross it. The particle layer width h then provides an indirect measure of the mean pressure PT exerted by thermomolecular forces on a particle entering the front. The elaboration of this model is facilitated by the absence of frozen fringes and thus by a homogeneous distribution of hydrodynamical resistance throughout the particle layer. Comparison of the resulting mean thermomolecular pressure to that predicted by existing models of single particle/front interaction questions both their relevance and their extension to multiple particles/front interaction.

Section II reports the experimental set-up and the main observations and measurements. The model that links the layer thickness h, the solidification velocity V and the mean thermomolecular pressure on an entering particle PT is exposed in section III. The resulting indirect measure of this pressure is reported in section IV together with a comparison to the value proposed by the single particle models. A discussion about quantitative deviations and the extension of single particle models to particle layers is drawn in Section V.

II. EXPERIMENT A. Experimental set-up

The experimental set-up has been originally designed to perform unidirectional solidification of binary solutions in thin samples [38,39]. In this setup, a sample is pushed at a controlled velocity V in an imposed thermal gradient G, and video microscopy is performed (see Fig. 1) in the vicinity of the solidification interface to follow its evolution.

The thermal field is set by top and bottom heaters and coolers, electronically regulated to better than 0.1 • C by resistive sheets and Peltier devices. Top heater and cooler are designed and fixed to provide altogether a planar surface on which the sample can slide, the thermal expansion of the top heater being anticipated. Bottom heater and cooler are then placed so as to sandwich samples, not by rigid fixation but by using springs, so as to reduce mechanical constraints and allow self-adjustment. An external circulation enables heat to be evacuated from the Peltier devices and also from the set-up sides to minimize thermal boundary perturbations.

The thermal gradient is set by both the gap between heaters and coolers and their temperature difference. Heaters and coolers temperatures are chosen so as to place the melting isotherm in the center of the gap, both to achieve an easier visualization of the solidification interface and to minimize of the thermal gradient dependence on the sample velocity V [38,40]. The gap was 10 mm and the imposed temperatures 20 • C and -20 • C for heaters and coolers and -10 • C for the circulating external flow. A closed dry atmosphere helped avoid condensation and ice formation.

Sample displacement is provided by a pushing device held on a linear track (see Fig. 1). Regular translation of the pusher is produced by a recirculating ball screw mounted on a screw rotated at a controlled rate by a microstepper motor. Vibration sources from the motor rotation are minimized by the large number of steps (200 per turn) and of microsteps (32 per step), together with the generation of Foucault currents to slow rotation at the end of microsteps. This provides 6400 microsteps per turn which, with a 5 mm screw pitch, yields an elementary displacement of 0.8 µm. Modulations of sample velocities on a screw turn reduced to a sinusoidal part with a relative magnitude δV /V less than 3% at most. Regular solidification velocities could thus be applied up to 50 µm•s -1 .

Visualization of a part of the solidification front is achieved by a CCD camera with time-lapse recording. To minimize perturbations in the vicinity of the sample, large frontal distances have been preferred. For this an exploded optical chain with a 50 mm focal lens has been used instead of a microscope. Magnification could allow for the visualization of fields extended from 200 µm to 2 mm with a resolution limited by diffraction to about 0.2 µm. As the camera involved 1024×768 pixels, the numerical resolution was thus in practice above the optical resolution.

Two kinds of observations could be performed: in transmission or in reflection. In transmission, the suspension and the frozen phase appear grey but the particle layer is opaque, which forbids the observation of its constituents. In reflection, the suspension and the frozen phase still appear grey but the particle layer is bright, which allows particles close to the sample surface to be individually followed in real time. As in both cases, the particle layer width was the same, we preferred the observation by reflection to provide access to some particle dynamics. In addition, some post-mortem complementary observations were made by confocal microscopy (Leica SP8, combined to a Leica DM6000 optical microscope) to better visualize particle ordering in the vicinity of a plate.

Samples were made of glass plates sandwiching the suspension. Their dimensions (top glass 100 × 45 × 0.7 mm 3 ; bottom glass 150 × 50 × 0.8 mm 3 ) have been taken long and large enough to allow a large central zone (> 35 mm) free of boundary effects. They were separated by calibrated spacers made of polypropylene sheets. After having filled the samples with the suspension by capillarity, they were sealed with cyanoacrylate and epoxy glue.

For binary solutions, the sample depth was usually chosen close to the natural width of microstructures, about 100µm [38,39,41]. This enabled a single layer of microstructures to settle in the sample, thereby facilitating visualization. Implications of confinement could be studied on dendritic sidebranching by experiment [38] or on cell forms by 3d phase field simulations [42,43]. Both confirmed that microstructure layers grown in a thin, but thick enough, sample were physically equivalent to those growing in an unconfined environment. These findings led us to choose the sample depth large enough compared to both the typical sizes of ice microstructures and the particle diameter. In addition, as multi-layers of microstructures were allowed here, a sample depth of 125 µm has been adopted.

The suspensions, manufactured by Magsphere Inc., contained monodisperse polystyrene spheres of 3 µm diameter and density 1.03, at volume fraction φ 0 = 10% or 20%. Sphericity and monodispersity were confirmed by confocal microscopy. The suspensions were stable over months. Although they might sediment in the samples, the particles stayed disperse enough to be able to redistribute in the sample depth when encountering the solidification front and its adjacent particle layer. Solutal effects were investigated by filtering out the particles using chromatography micro-filters. They led here to a critical velocity of about 10 µm•s -1 for the morphological instability of planar fronts.

The physical parameters of the experiment are given in Tab. I.

FIG. 1. Sketch of the experimental set-up. A thin sample sandwiched between heaters and coolers is pushed at a prescribed velocity by a micro-stepper motor coupled to a linear track. This allows solidification under controlled conditions together with a real-time non-invasive visualization.

B. Observations and measurements

This study focuses on the velocity range 0.5 ≤ V ≤ 20 µm•s -1 with a special attention to velocities below the critical velocity of a planar front (around 10 µm•s -1 ). Therefore, most data will refer to a planar solidification interface. Other velocities and/or particle diameters yield different solidification patterns that will be reported in a forthcoming paper.

A typical steady-state, obtained at V = 3 µm•s -1 , is displayed in Fig. 2. We observe a grey and bright layer surrounded by the frozen phase (bottom) and the liquid phase (top). The layers are separated by sharp boundaries. The frozen phase shows repetitive black striations which refer to a phenomenon that is both time-dependent and uniform along the front. This is reminiscent of the segregation provided by the formation of ice lenses [44], but a more detailed analysis will be required to state this firmly. However, no implication of these striations could be noticed on the rest of the system, especially on the front shape or the width and the organization of the adjacent bright layer. We shall therefore ignore them here.

The layer between the frozen phase and the liquid phase corresponds to the compacted layer of particles studied by Anderson and Worster [7,44]. Its existence indicates that the velocity at which particles enter the frozen phase is smaller than the growth velocity V , thereby requiring an increased particle concentration φ in steady-states to satisfy the particle flux balance in the whole system. This agrees with direct observation from confocal microscopy (Fig. 3) which reveals a closepacking of particles.

In the vicinity of the front, the particle layer displays bright patches with a broad size distribution. As revealed by confocal microscopy, they correspond to ordered clusters of particles near the plates (see Fig. 3) which likely result from the long-range ordering induced by the planar glass surface. The observations reveal that they evolve with time in both shape and size with possible internal reorganization responsible for cluster blinking (see Movie S1, Supporting Information). These clusters, reminiscent of particle crystallization in suspensions [45], are thus likely limited to the vicinity of plates. They therefore correspond to a visual artifact induced by the plate with no prejudice on the remaining particle layer, both in width and depth.

FIG. 2. Steady state at φ0 = 20% and V = 3 µm•s -1 showing a dark frozen phase (bottom) followed by a grey particle layer (middle) and the liquid suspension (top). The frozen phase shows striations reminiscent of ice lenses. The particle layer displays bright patches corresponding to an ordering of the particles in the vicinity of the plates.

The cluster brightness could be related to particles. It thus makes a definite difference with the frozen fringes observed by Anderson and Worster on smaller particles [44]. In particular, their non-uniformity, their timedependence and their disconnection with the front refer to features that do not fit with the current description of the frozen fringes. In addition, no steady bright zones similar to frozen fringes could be observed close to the front. This supports the absence of frozen fringes in this experiment.

Starting from the beginning of solidification, the particle layer grows linearly as displayed in fig. 4 in a series of snapshots and in a spatio-temporal diagram (see Movie S1, Supporting Information). Meanwhile, the frozen phase behind the front appears dark, meaning that no particle enters it, at least close to the sample plates. This is confirmed by transmission observations, where the brightness of the frozen phase proves that it contains no particle in the whole sample depth. As the incoming particle flux in the particle layer is V φ 0 , particle conservation yields V φ 0 = (φ -φ 0 )dh/dt where h denotes the layer thickness, φ 0 the particle concentration in the suspension and φ its mean value in the particle layer. The linear increase of h with t thus enables an indirect measure of the particle concentration φ in the layer. The fit of its growth rate yields φ = 0.634 ± 0.007, close to the random close-packing value φ p = 0.637 and in agreement with observation of clusters of particles in contact with each other (Fig. 3). This determination supports previous assumptions [7,44] and predictions [46] that a layer of particles under such conditions is at the close-packing concentration φ p . It thus confirms that the denomination "compacted layer of particles" is relevant. The growth of the layer ends at a definite layer thickness which is thus representative of the suspension and solidification conditions. As only steady states will be addressed from now on, we shall then still denote their permanent layer thickness h in the remaining. This stop of growth means that the entrance flux of particles in the frozen phase now equals their incoming flux V φ 0 in the particle layer. It thus indicates that the steady state layer width h is intimately linked to the repelling/trapping mechanism of particles by the front. In addition, as it is simple and accurate to measure, it thus stands as an ideal quantity to indirectly probe the particle/front interaction mechanism in this experiment.

Fig. 5 displays steady states obtained at various solidification velocities V . They all show a similar kind of patterns with a particle layer and a frozen phase possibly displaying, at low velocities, bright patches and striations respectively. The layer thickness h shows a continuous decrease with the velocity V . This is documented for two suspension concentrations φ 0 = 0.1 and 0.2 in fig. 6, where no dramatic change in the evolution of h can be noticed for the two data points close or above the critical velocity of destabilization of planar interface, about 10 µm•s -1 here.

The following sections of this article are devoted to understand both this trend and its order of magnitude so as to gain relevant insights on the underlying repelling/trapping mechanism of particles by the solidification front.

III. MECHANICAL MODELING

The particle layer refers to a self-organized steady state in which the repelling/trapping mechanism plays a key role. To access relevant information on it, we model the mechanics of the particle layer, especially viewed from the particles that enter the front. As in steady state the forces applied on the particle equilibrate, the magnitude of the short-range thermomolecular force could then be deduced from the remaining forces.

Three forces apply on a particle close to the solidification front (see Fig. 7) :

i) The thermomolecular force F T exerted by the solidification front on an entering particle. It results from Van der Waals interactions between the particle and the front, through the fluid.

ii) The lubrication force F l on an entering particle. It results from viscous effects in the thin film separating the particle from the front.

iii) The force F µ exerted by the particle layer on an entering particle. It results from pressure drop and viscous stress induced by the fluid flowing across the particle matrix. It is transmitted to an entering particle by contacts along the particle matrix.

Whereas the two former forces push the particle toward the front, the latter repels it. Their resultant will be labeled F. These forces are addressed in more detail below before we can deduce the thermomolecular force from mechanical equilibrium.

A. Mechanical balance

For convenience, we shall consider the average over a particle section πR 2 of the z-components of these three forces and their resultant on the thermal gradient direction e z = G/G. The corresponding pressures will be referred to as Pµ , Pl , PT and P respectively. They thus satisfy the relationship P = Pµ + Pl + PT , where Pµ < 0, Pl < 0 and PT > 0 involve the following expressions :

1. Viscous force F µ , Pµ
In steady states, mass conservation of fluid and of particles yields the following mean velocities for the fluid, v f , and the particles, v p , in the front frame: 8). As they differ, viscous stresses are generated on the particle matrix. Regarding the fluid, they result in a pressure drop ∆p across the particle layer, referred to as cryosuction.

v f = -V (1 -φ 0 )/(1 -φ), v p = -V φ 0 /φ (Fig.
From a mechanical point of view, this pressure drop expresses the rate of momentum flux on the direction e z that is lost by the fluid and gained by the particle matrix, by viscous stresses on the particle surfaces. As the particle layer is close-packed, this momentum is entirely transmitted to the entering particles and results in a mean pressure Pµ on each of them. However, as particles occupy only a part φ of a suspension section, the mean pressure on an entering particle is larger than the pressure drop and reads Pµ = ∆p/φ. In particular, its excess with respect to the pressure drop refers to the viscous stresses that are also undergone by the particle layer, in addition to the viscous drop itself.

Regarding hydrodynamics, the compacted layer of particles behaves like a porous medium. In addition, following the absence of frozen fringes (Sect. II B), it may be considered as homogeneous. Accordingly, its internal viscous dissipation may be modeled by Darcy's equation, µU = -kdP/dz, where µ denotes the liquid viscosity, k the medium permeability and U the flux of liquid volume in the z-direction, viewed in the frame of the particle matrix. Taking into account the mean velocity difference v f -v p between the liquid phase and the particles and the part (1-φ) of a suspension section occupied by the liquid, one gets

U = (1-φ)(v f -v p ) and U = -V (φ -φ 0 )/φ.
Following the indirect measure reported in Sec. II B, the particle concentration φ in the com- pacted layer of particles will be taken as that of random close packing : φ p = 0.64. As the suspension concentration φ 0 is smaller than φ p , one notices that U is negative, which means that liquid flows from the suspension towards the front.

On the other hand, as the suspension is monodisperse and random, its permeability may be modeled by the Kozeny-Carman relation:

k(φ, R) = (1 -φ) 3 /φ 2 R 2 /

Lubrication force F l , Pl

When a particle is displaced with respect to the suspension, some fluid has to fill the freed volume.

For particles adjacent to the front, such a flow occurs in the film that separates them from the front (Fig. 7). Its thinness then induces, by viscous friction, a noticeable pressure drop at the base of particles which results in an attractive force between them and the front. Referred to the cross-section FIG. 8. Sketch of the system that shows the suspension, the compacted layer of particles and the frozen phase. The mean velocities of the fluid and of the particles with respect to the solidification front are denoted v f and vp respectively. In the suspension, they are both equal to the opposite of the solidification velocity. However, in the compacted layer of particles, they differ following the rise of particle volume fraction, thus resulting in viscous friction. As particles are randomly distributed, their section by a plane displays different radii although they actually share the same radius.

of particles, it corresponds to a mean pressure Pl on them.

Computation of Pl may be performed in the framework of Stokes flow and of the lubrication approximation. The former yields this mean pressure to be proportional to the velocity V p of particles with respect to the suspension : Pl = -f V p . The latter approximation yields the prefactor f to be determined with respect to the geometry of the film, i.e. the front shape and the position of the particle with respect to it.

In the following and in most models, attention is focused on steady states of particles adjacent to the front for which the relative velocity V p with respect to the suspension is constant. However, we notice that the value of this velocity differs depending on whether the entering particle is isolated and repelled by the front or immersed in a steady particle layer and thus eventually trapped by the front. The former case is usually considered by models and yields V p = V . The latter case is the one which applies here. It corresponds to

V p = v p + V or, equivalently V p = -U = |U |.

Thermomolecular force F T , PT

The thermomolecular force results from Van der Waals like interactions between the front and a particle. It is normal to the particle surface and inversely proportional to the cube of the distance d between a surface point and the front. It thus gives rise to a thermomolecular pressure P T = A/6π d -3 where the Hamaker constant A depends on materials and solute.

Given the front shape and the position of the particle with respect to the front, the thermomolecular pressure may be integrated along the particle surface to give the mean thermomolecular pressure PT .

Thermodynamical equilibrium

To compute both the lubrication force and the thermomolecular force, the front shape must be determined for a given position of the particle with respect to it. It follows the Gibbs-Thomson relation:

L v t = P T + σκ (1)
where the pressure melting term induced by the difference of density between water and ice has been neglected. Here P T , σ, κ and t = (T M -T I )/T M denote the thermomolecular pressure, the surface tension and curvature of the interface, and its relative undercooling with (T M , P M ) the temperature and pressure of a reference state and T I the solid/liquid interface temperature.

In a thermal gradient where T I ≡ Gz I , this provides a differential equation for the location z I of the interface whose integration yields the front geometry, given the position of the particle with respect to the melting isotherm. Usually, numerical integration is required to determine the front shape and thus the channel geometry, especially when capillary effects are handled [34,37].

B. Mechanical equilibrium

Mechanical equilibrium reads: Pµ + Pl + PT = 0. It applies to particles involving a constant velocity V p , either because they are repelled by the front or because they belong to a steady particle layer displaying continuous trapping. The present experiment refers to the latter case for which V p = |U | and thus Pl = -f |U |. Introducing U c = PT /f , the critical velocity for a single particle (i.e. for h = 0), the mechanical equilibrium of an entering particle then reads:

h µ φk = PT ( 1 |U | - 1 U c ) (2)
where we recall that the permeability k depends on φ and R. This allows the determination of both the mean thermomolecular pressure PT at the trapping transition and the critical velocity U c from the data set of steadystate thicknesses h at various velocities U . In relation (2), we notice that the left hand side depends on the suspension properties φ and R whereas the right hand side seems at first glance to be independent of them. This difference traces back to physics of the repelling/trapping mechanism : part of it (the viscous pressure Pµ ) refers to the suspension whereas the remaining part (lubrication and thermomolecular forces Pl and PT ) refers to the interaction of an entering particle with the front. Thus, whereas the former deals with the whole particle layer, the latter corresponds to short range forces exerted in the very vicinity of an entering particle and formally decoupled from the rest of the suspension. Following this picture, the repelling/trapping mechanism is expected to remain the same, irrespective of the presence of a particle layer, so that PT and U c should be viewed as independent of h. However, we shall show later on, in section V A, that this corresponds to a first order approximation with respect to the particle layer thickness beyond which there exists an indirect, hidden, link between the trapping mechanism and the suspension conditions.

IV. RESULTS AND MEAN THERMOMOLECULAR PRESSURE

We first report the evolution of the width h of the particle layer with respect to the relative velocity U . This will provide an indirect measure of the mean thermomolecular pressure PT . To interpret its magnitude, we then revisit the Rempel-Worster model [35] that refers to the interaction and the capture of a single particle by the front.

A. Particle layer thickness and mean thermomolecular pressure Figure 9 reports the particle layer widths h with respect to the inverse relative velocity 1/|U |. It shows an affine trend beyond U ≈ 1 µm•s -1 followed at lower velocities by a departure towards a saturation. As the latter regime is confined to low velocities, it is much less representative of the suspension solidification than the former regime which englobes all velocities beyond. We shall therefore skip it in the following and postpone its analysis to a forthcoming paper.

Above U ≈ 1 µm•s -1 , the affine evolution (2) followed by data supports the mechanical modeling involving viscous forces. In addition, the use of the relative velocity |U | here instead of V as in figure (6) makes the the two data sets φ 0 = 0.1 and 0.2 collapse on the same master curve. In particular, whereas their fitted slopes agree to within 5% here, they would differ by about 30% if V was used instead. As U corresponds to the relative velocity of the fluid with respect to particles, this supports the hydrodynamic origin of the forces which counterbalance the thermomolecular force.

The slope and the intersection with the x-axis displayed in Figure 9 respectively provide estimates of the mean thermomolecular pressure PT and of the critical ve-locity U c . However whereas the former determination is accurate, the latter is largely sensitive to the distribution and the uncertainty of data so that its determination can only be quite approximate. Data yield PT = 1000±63 Pa and an extremely large and imprecise critical velocity U c which must thus be overlooked.

The value found for PT , a kPa,stands two orders of magnitude below the thermomolecular pressure P T0 at the base of the particle evaluated for a usual expectation of a nanometer thick film: P T0 = A/6πd -3 0 = 690 kPa. To clarify this large difference, we turn attention towards the modeling of the repelling/trapping mechanism proposed by Rempel and Worster [35] and question its prediction regarding the mean thermomolecular pressure PT on an entering particle.

B. Single particle modeling and thermomolecular constraint at critical state

The most recent modelings of the interaction between a single spherical particle and the solidification front [34,35,37] provide the following integral expressions of the z-component of the forces F l and F T on a particle :

F l = 2πR 2 6µR 2 V p π 2 0 sin(θ) cos(θ) θ π 2 sin(φ) d 3 (φ) dφ dθ (3)
F T = 2πR 2 A 6π π 2 0 sin(θ) cos(θ) d 3 (θ) dθ ( 4
)
Here θ denotes the zenith angle with the -e z direction from the particle center and d(θ) the distance of the particle to the front on this direction (see Fig. 10). The determination of F l and F T and thus of the features of the critical state at the trapping transition, requires to derive d(θ) from the Gibbs-Thomson relation (1). When surface tension is neglected, this cannot be handled analytically and requires numerical determinations [35,37].

Unfortunately, these numerical studies report a number of information on the velocity V c p and the minimal distance d c 0 to the front at the critical state, but do not report on the mean thermomolecular force. However, within some assumption regarding the geometry of the channel separating particle and front, we may obtain analytical relationships that provide a link between V c p , d c 0 and F T or PT . We consider below the Rempel and Worster assumption following which the channel between the particle and the front involves a constant width d 0 up to an angle θ 0 beyond which it suddenly dramatically increases. This corresponds to a film climbing along the sphere and relaxing to a plane at a turning point, i.e to a concave-convex front shape around the particle (see Figure 10). This configuration echoes the change of front shape induced by the ratio of thermal conductivities k p /k m between the particle and the medium. In particular, numerical studies by Park, Golovin and Davis [37] have shown that, for our parameters γ = T M σ/L v GR 2 = 2.68 and k p /k m = 0.205, the front shape assumes a concave-convex shape which fits the assumption of the Rempel-Worster model.

Within this constant film thickness assumption, d(θ) = d 0 for θ < θ 0 and d(θ) = ∞ beyond, the integrals (3) and (4) can then be determined. Anticipating a small value of θ 0 , they yield, for θ 0 π/2 :

PT = A 6π θ 2 0 d 3 0 (5) Pl = - 3 2 µR 2 V p θ 4 0 d 3 0 (6)
At mechanical equilibrium, both yield:

θ 2 0 V p = A 6π 2 3µR 2 (7)
On the other hand, the numerical determination of Rempel and Worster [35] yields for the film thickness d c 0 and the velocity V c p = V p (d c 0 ) at the critical state of the trap-ping transition :

V c p = v c p A 6π 2/3 σ 1/3 6µR 4/3 (8)
d c 0 = δ c 0 A 6π R σ 1/3 . (9)
with, for the present parameters of the suspension (γ = 2.68), v c p ≈ 0, 95 and δ c 0 ≈ 0.95. Our goal is now to obtain from this an evaluation of the mean thermomolecular pressure PT at the critical state. For this, we use the relation (7) to determine the critical value θ c 0 of θ 0 which, together with d c 0 , then provides from (5) the value of PT in the critical state. We obtain:

θ c 0 2 = 4 v c p A 6π 1/3 (σR 2 ) -1/3 (10) PT = 4 v c p δ c 0 3 A 6π 1/3 σ 2/3 R -5/3 (11)
Relation ( 11) may be used here to provide an order of magnitude of the mean thermomolecular pressure exerted on a single entering particle in conditions similar to our experiment. It yields PT = 208 Pa with, from relations (8) (9), V c p = 12.29 µm•s -1 and d c 0 = 3.1 nm. The latter value yields a thermomolecular pressure of P T0 = 22.5 kPa at the base of the particle and up to the angular position θ 0 in the present constant thickness model. The large relative difference between PT and P T0 then means that P T0 is indeed applied over a small angle θ 0 . This is confirmed from relation (10): θ c 0 = 0.096 rad in the critical state.

V. DISCUSSION

The experimental results as well as their comparison with the Rempel-Worster model [35], hereafter called RW model, yield both qualitative and quantitative outcomes.

We first notice that the mean pressure Pµ on particles in the compacted layer is lower than the mean thermomolecular pressure PT ≈ 1 kPa. It is thus significantly smaller than the pressure threshold of about 10 kPa [47,48] above which a particle layer would turn aggregated [7,44]. Accordingly, the particle layer, although compact, may be considered as dispersed, i.e. composed of freely moving particles, as confirmed by direct observation (see movie in supplementary material). This supports the modeling of trapping based on individual particles rather than on a cohesive layer.

On a qualitative level, the affine relation of h with 1/|U | supports the relevance, at least at the dominant order, of the mechanical model involving a viscous force proportional to hU , a lubrication force proportional to U and a constant thermomolecular force.

On a quantitative level, the measured mean thermomolecular pressure PT ≈ 1 kPa is found five times greater than in the constant film thickness RW model, PT = 0.2 kPa. However, both appear much smaller than the thermomolecular pressure P T0 = 22.5 kPa exerted in this model at the base of the particle and in the whole constant thickness film. This large difference means that our experiment supports the main feature of the particlefront interaction worked out in the RW model: a very large thermomolecular pressure exerted on a very small part of the particle surface.

However, beyond this agreement, the factor five difference between the magnitudes of the mean thermomolecular pressure deduced from experiment or predicted by the RW model questions the fine modeling of the repelling/trapping mechanism. In particular, an important difference between experiment and model is that our experiment addresses a particle layer whereas the RW model concerns a single isolated particle. This yields in the experiment an additional force on an entering particle that is not involved in the RW model: the viscous force generated on the whole particle layer. We investigate below its implication on critical state features at the trapping transition as well as on the other possible origins of the factor five difference regarding the mean thermomolecular pressure.

We thus address the main differences between the RW model and the experiment so as to interpret their quantitative discrepancies. Some are fundamental: the implication of a particle layer on the film thickness. Some refer to a difference of parameters: the Hamaker constant (or S in [35]) and the difference of thermal conductivity between particles and water. We review them below and determine the sense and the order of magnitude of their implication on PT .

A. Particle layer effect on film thickness

We address here the effect of the particle layer on the characteristics of the critical state of the trapping transition. On a particle adjacent to the front, the mechanical balance of forces yields, from section III A, a resultant mean pressure :

P = - µ φk h|U | -f V p + PT (12)
Following the Gibbs-Thomson relation ( 1), the geometry of the channel separating the entering particle and the front may be parametrized by its thickness d 0 at the base of the particle. Accordingly, the integrals involved in the lubrication and thermomolecular forces (3) (4) appear as functions of d 0 : f ≡ f (d 0 ), PT ≡ PT (d 0 ).

The equilibrium states, P = 0, may refer either to repelling or to trapping. The transition between them corresponds to a critical state. It may be reached on a single particle by increasing the solidification velocity V or, in the present experiment where many particles are involved, by increasing the particle layer h. The criterion for the transition may be obtained by considering the implications of fluctuations of the film thickness parameter d 0 : a reduction of d 0 will enforce trapping if the resultant d P is negative ; it will enforce repelling if d P is positive ; at the transition between these regimes, d P thus vanishes, so that V p = PT /f , where the prime denotes the first derivative with respect to d 0 .

This relation shows that d 0 and V p are linked on the critical states for trapping. It thus applies to the steady states of the particle layer on which the layer thickness h and V p = |U | are related one to the other (see Figure 9). Altogether, this shows h and d 0 are linked together : the additional force exerted on an entering particle by the compacted layer modifies the thickness of the film that separates it from the front. As a consequence, the mean thermomolecular pressure PT on it should raise accordingly.

To evaluate the order of magnitude of this multiple particle effect on d 0 and PT , we may consider the analytical determinations of P T and f derived when surface tension is neglected [34,37]. They correspond to polynomial expressions [START_REF]0 +4/5δ -5 0 +1/3δ -9 0 ), where δ0 stands for the non-dimensional film thickness d0/l with l 4 = A/6π TM /Lv[END_REF] from which V p (d 0 ) = PT /f may be deduced. Denoting, as in section IV B, the critical velocity for single particle trapping by V c p and its critical film thickness by d c 0 , it then appears that, down to V c p /10, the relationship between V p and d 0 may be approached by

V p = V c p [1 -3/2 (1 -d 0 /d c 0 )].
In our experiment, as a vanishing layer thickness was obtained for V ≈ 20 µm•s -1 at φ 0 = 0.2, i.e for U ≈ 14 µm•s -1 , one may estimate V c p at about 15 µm•s -1 . In comparison, the domain on which the determination of PT has been obtained stands at velocities around V c p /5 for which d 0 is decreased by a half, and the mean thermomolecular pressure PT raised by a factor 40.

Although this estimation stands for a vanishing surface tension, it indicates that, in our experiment where a particle layer is involved, the measured mean thermomolecular pressure PT on an entering particle is likely raised by a factor of several tens in comparison to its value on a single isolated particle.

B. Hamaker constant

Models of particle trapping are based on a numerical determination of the front shape from the Gibbs-Thomson relation (1). Three kinds of effects then affect the front position: (i) the effect of the thermal gradient given the front undercooling; (ii) capillary effects on undercooling; and (iii) thermomolecular pressure on undercooling, the solutal effect being neglected. These effects are respectively conveyed by the three variables G, σ and A or equivalently by two nondimensional parameters, γ = σT M /LvGR 2 and either = (AT M /6πL v GR 4 ) 1/4 [34,37] or S = (A/6πσR 2 ) 1/3 [35] depending on studies.

However, the numerical solutions of these models are provided in terms of graphs parametrized by only one of these parameters, γ [34,35,37]. In particular, the RW model [35] on which the analysis of Section IV B relies, provides evolution of non-dimensional velocity and film thickness at S = 10 -3 whereas our experiments are conducted at S = 2.9 × 10 -3 . As S scales as A 1/3 , this may be interpreted as an Hamaker constant 2.9 3 ≈ 25 larger in experiment than in simulations. Equivalently, this means that the net thermomolecular pressure PT that should be expected from a more accurate modeling should be increased accordingly.

In particular, assuming that the non-dimensional variables δ c 0 and v c p are constant under this variation of A, relation [START_REF] Velez-Ruiz | Handbook of food preservation[END_REF] states that PT scales as A 1/3 , i.e. as S , and should therefore be enhanced by a factor 25 1/3 ≈ 2.9.

C. Thermal conductivity differences

Particles are made of polystyrene of a lower thermal conductivity k p = 0.115 than that of water k m = 0.561. The resulting modification of the thermal field in their vicinity tends to reject the solidification front away from the particle and form a bulge below the particle. This would dramatically increase the film thickness d and decrease PT . However, this tendency is partly balanced by capillarity which acts so as to flatten the front. For our experimental parameters (γ = 2.68, k p /k m = 0.205), the outcome appears as a concave-convex interface which follows the particle shape up to a turning point where it relaxes towards the planar front [37]. This provides a film thickness somewhat constant over an angular sector below the particle and beyond which it sharply increases (see Figure 10), in agreement with the geometric features of the modeling considered by Rempel and Worster [35] and used in section IV B.

The implication of a thermal conductivity mismatch has been addressed by Park, Golovin and Davis [37] on both the velocity V c p and the film thickness d c 0 but not the mean thermomolecular force. However, it may be deduced from the former variables within the assumption of a constant film thickness over an angular sector θ 0 , using the analysis provided in section IV B. Following the numerical results of reference [37], the change at γ = 1 of the ratio k p /k m from 1, as implicitly considered up to now, to 0.1 (resp. 10) yields an increase of V c p by a factor 7.8 (resp. 4.3 × 10 -2 ) and of d c 0 by a factor 1.7 (resp. 0.5). Accordingly, for the experimental parameters k p /k m = 0.205, γ ≈ 1, one may estimate, by a power law extrapolation, the corresponding increase factors for V c and d c 0 as α = 3.94 and β = 1.45 respectively. Following relation (7), this implies a reduction of θ 2 0 by the factor α and thus, from (5), a reduction of PT by a factor αβ 3 , i.e. about 12.

D. Synthesis

The value PT = 208 Pa deduced in section IV B from an extrapolation based on the RW model for a single particle needs thus to be corrected by definite factors for several reasons : i) a fundamental reason relative to the particle layer interplay with the critical state features (factor ≈ 40) ; ii) parameter shifts relative to the parameter S (or to the Hamaker constant) in [35] (factor 3) and to thermal conductivity differences between particle and water (factor 12 -1 ). Altogether, this yields a reevaluation of the PT estimate by a factor about 40 × 3/12, i.e. about 10.

This moderately large factor illustrates that several agonist and antagonist phenomena take part in the design of the film which separates the particle and the front and, thus, in the net thermomolecular pressure applied on the particle. The correction factor, of order 10, agrees with the observed difference between the experimental measure of PT (≈ 1.0 kPa) and its primary estimation by modeling (≈ 0.2 kPa). On the other hand, it is weak enough to maintain a large gap between such estimations and that of the thermomolecular pressure at the base of an entering particle (≈ 20 kPa), thereby corroborating that a small part of the particle surface supports the main thermomolecular pressure.

These results support the thickness of the particle layer as a relevant indicator of the physical conditions experienced by the entering particles. Interestingly, we notice that this variable involves a macroscopic magnitude as well as a statistical nature implied by the large number of particles hold in the layer or entering to the front. Altogether, it thus provides a macroscopic probe of the averaged trapping mechanism on a solidification front.

VI. CONCLUSION

This study addressed the solidification of suspensions below the trapping velocity of single isolated particles. Particles were found to accumulate in a particle layer whose thickness growth rate reveals its close-packing. This implies multiple particle interactions both in the particle layer and with the front. At any velocity, particle trapping occurred for a sufficiently large thickness of the layer. It then compensated particle accumulation and made the layer growth stop. We focussed attention on this steady-state layer thickness to provide insights into the multiple particle interactions and the resulting particle trapping mechanism.

Two kinds of multiple particle effects have been identified. In the compacted particle layer, viscous friction on the particle matrix and the associated fluid pressure drop induce an additional force on particles which favors trapping. On the other hand, at the level of a particle entering the front, this additional force leads a reduction of the film that separates particle and front with a corresponding raise of the repelling thermomolecular force exerted by front on particles.

A quantitative study was provided by a mechanical model on particle adjacent to the front which includes the additional force induced by the particle layer. It provided a mean thermomolecular pressure on an entering particle that is low compared to that exerted at its base and large compared to that proposed by models for a single isolated particle. The former feature means that a small part of the particle surface undergoes a large repelling from the front, the film between them thus enlarging quickly beyond. The latter statement was explained by the reduction of film thickness in response to the additional force induced by multiple particle effects.

Altogether, the present study thus evidenced the effects of multiple particle interactions on particle trapping by solidification fronts. In particular, it clarified the formation of a compacted layer of particles ahead of the solidification front and pointed out its role and its effect on the trapping mechanism. For the present particle features and solidification conditions, the solidification front was mainly planar and the particle layer disperse ; for other particle diameters or solidification velocities, some patterning could occur in either the particle layer, the front or the frozen phase and the layer could turn aggregated. As all these events largely result from multiple particles interaction with the front, the findings reported here will thus provide a relevant basis for understanding their origin and their inner mechanisms.

ASSOCIATED CONTENT

Supporting Information

A supporting movie, along with its short description. This material is available free of charge via the Internet at http://pubs.acs.org.

FIG. 3 .

 3 FIG.3. Confocal microscopy image of the compacted layer of particles close to a sample plate. The particle diameter is 1 µm. The particle layer shows close-packed arrangements of particles into both disordered regions or colloidal crystals of relatively large size. This noticeable geometrical order is presumably imposed by the planar glass surface.

FIG. 4 .

 4 FIG. 4. Initial stage of solidification : φ0 = 20%, V = 0.5 µm•s -1 . The corresponding movie is displayed in Supporting Information, Movie S1. (a) -(d) Snapshots taken respectively 220, 3789, 5062 and 13720 s after beginning of solidification.They show, from top to bottom, the suspension (grey zone), the particle layer (bright zone) and the solid phase (dark when no particles are trapped (a to c), grey when some are (d)). (e) Spatio-temporal diagram provided by the juxtaposition at increasing times of a cut of images taken at a fixed position. Space is reported on ordinates over 768 pixels corresponding to 3428 µm. Time is reported on abscissa over 1372 pixels corresponding to 13720 s. The times corresponding to images (a) to (d) are indicated on the top of the diagram. The common slope of the bright lines displayed in the solid phase refers to the advection of embedded particles at sample velocity V . The diagram shows that particles are first repelled, as revealed by the dark zone behind the front. Meanwhile, the particle layer thickness increases until the particle capture by the solidification front sets in (bright lines in the solid phase). The particle layer then reaches a steady state and a constant thickness.

FIG. 5 .FIG. 6 .

 56 FIG. 5. Particle layer (grey zone) in steady states at φ0 = 20% and various front velocities V : 1, 2, 4, 5, 10 and 20 µm•s -1 . The layer thickness decreases with the front velocity. Below V = 10 µm•s -1 , it shows white patches which correspond to clusters of particles on the sample plates (a, b, c, d). As the location of the images is arbitrary, the variation of the front position within the observation frame is meaningless.

45 .FIG. 7 .

 457 FIG.7. Sketch of the forces acting on a particle entering the frozen phase: a thermomolecular repelling force FT exerted by the solidification front (black arrow); a lubrication force F l induced by the liquid flow in the thin film separating the particle and the front (red arrow); a force Fµ exerted by the particle layer on the particle, as a result of the viscous friction induced by a liquid flow through the particle matrix (green dashed arrow). Blue curved arrows symbolize the liquid flow; G recalls the thermal gradient and ez the direction of z-axis. As particles are randomly distributed, their section by a plane displays different radii and few contact points here, although they actually share the same radius and are in contact with each other.

FIG. 9 .

 9 FIG. 9. Evolution of the particle layer thickness h with |U |, the relative velocity of the fluid with respect to particles. The axes refer to the variables µh/φk and 1/|U | to express relation (2) by an affine relation with slope PT.

FIG. 10 .

 10 FIG. 10. Schematic diagram of a particle adjacent to the interface in the constant film thickness assumption where d(θ) ≈ d0 for θ ≤ θ0.

TABLE I .

 I The physical quantities used throughout the manuscript.

ACKNOWLEDGMENTS

The research leading to these results has been supported by the European Research Council under the European Union's Seventh Framework Program (FP7/2007-2013)/ERC grant agreement 278004.


Cecile Noirjean
email: ce.noirjean@gmail.com

Moreno Marcellini

Sylvain Deville
email: sylvain.deville@saint-gobain.com

Thomas E Kodger

Cécile Monteux

Dynamics and ordering of weakly Brownian particles in directional drying

come   L'archive ouverte pluridisciplinaire

I. INTRODUCTION

The drying of colloidal or non-colloidal suspensions is a very common phenomenon with many natural and technological occurrences. Drying and its results are of interest in geophysics (drying of soils), 1 food engineering, 2 paints, 3,4 and ceramic materials science. 5 In the latter, drying is a critical step of many processing and shaping routes for advanced materials. The structural evolution of a typical suspension during drying is the result of the interplay between many parameters, which makes investigations difficult. Of particular interest are the drying patterns and the organisation of particles in the dried deposit, as the development of heterogeneities and cracks in these deposits is critical for many applications. The processing of defect-free colloidal crystals -which could be achieved by drying -is of interest for photonic applications, with a band gap in the visible. 6 Lattice defects, such as planar stacking faults or other heterogeneities, have nevertheless a critical impact on the optical properties of such materials. 7 Understanding the formation mechanism of these structures and their defects during drying is therefore essential.

Various model experiments have thus been developed to understand the fundamentals of drying. In films or sessile droplets, curved fronts are formed and the control of the evaporation conditions is difficult, which result in a variety of drying and crack patterns. 8,9 Loussert et al. 10 developed a method to observe convection and diffusion by using Raman microspectroscopy while drying charged nanoparticles dispersions in thin-film droplet confined between two circular plates. Moreover, in this work the authors provide spatially resolved measurements of the colloid concentration during the drying. Unidirectional drying has been developed as a well-controlled method to investigate the underlying phenomena. In unidirectional drying experiments, the evaporation of the solvent occurs at the open-end of the sample. The solvent flow carries the particles towards the opened solvent/air interface, where they organize into a dense packing [11][12][13][14][15][16] as the progressive accumulation of particles continues.

Two competing phenomena control the drying: the difference of chemical potentials of water at the drying interface and the external humidity, and the removal mechanism of vapour from the drying interface. The latter is almost never precisely controlled. Local fluctuations of humidity in the air at the sample scale -which can hardly be avoided -can have a great impact on the drying behaviour of the system.

The mechanism that builds up the order/disorder deposit at the evaporation interface of drops has been previously identified in the velocity with which the particles impinge on to the pinned interface, 17 and the evaporation rate. 18 The common hypothesis is that particles form polycrystalline (colloidal) crystals for slow evaporation rate, where particles have enough time to reorganize by Brownian motion after hitting the solid region. The same behavior was observed in sessile drops with large (micron-size) particles: disordered structures were obtained for rapid particle velocity. 19 Nevertheless, as it has been shown by Schöpe et al., 20,21 the polydispersity of particles can influence kinetics and structure of the obtained deposits by changing the volume fraction at the order/disorder transition.

The rise of microfluidic chips allows to observe the drying of confined droplets and colloidal dispersions: Ziane et al., in two different works, show that homogeneous drying can be achieved by pervaporation of water through a water-permeable membrane, such that colloidal particles can be concentrated up to the growth of a colloidal crystal in the micro-channels. 14,15 We anticipate that although the size of particles is two orders of magnitude smaller than ours and the geometry completely different, these works show that drying colloidal dispersions of charged hard-spheres in microchannels reveals two different regimes of solidification along the channel. In brief, one can observe a homogeneous concentration dampening crossing the liquid-solid transition toward the closepacking, and the classical growth, observed classically in directional drying, of a water-saturated solid of closepacked particles. 22 In a successive work, Laval et al., 16 by using similar microfluidic device, microfabricated insitu material with predetermined composition. The process strength of the authors' approach relies on the control of the pervaporation process at the microfluidic scale to build materials ranging from colloidal packed beds to polymer composites. In particular Laval et al. present results of how colloidal crystals embedded with gradients and fluorescent barcodes can be fabricated from a dilute dispersion of monodisperse colloids and fluorescent particles. These recent works show that directional drying is a powerful way to build materials with predetermined specifications.

Despite much progress in the understanding of drying, many questions must still be answered. What is the role of the particle size? Does the solid region remains wet during growth? What is the water concentration in the solid region uniform? What are the local drying dynamics? How do particles organize at the interface? And which factors control the dynamics and ordering of particles? Dynamic, particle-scale observations are required to answer some of these questions and in the following we will answer to a few of them.

The organisation of particles is almost always observed by looking at the dried sample, with optical, 23 confocal, 24,25 atomic force, 18 or electron microscopy. 26 The particle velocities were previously measured using markers (large particles), which is not ideal 24 as it only provides an indirect measurement. The dynamics were also assessed indirectly by SANS/SAXS, 27,28 which only provide an average measurement of the particle behaviour.

To progress in the understanding of drying of parti-cle suspensions, we investigate unidirectional drying of weakly Brownian particle suspensions by laser scanning confocal microscopy. 29 Using a high imaging rate (up to 20 Hz) we can follow in situ individual particles dynamics followed by their ordering at the liquid/solid limit. We correlate the particle dynamics to their short/long range ordering organization and find that faster particles velocities yield better ordering and denser packings.

II. MATERIAL AND METHODS

A. Sample preparation

Monodisperse spherical PtBMA/PFEMA particles, 1.1 µm in radius, as measured by microscopy image, were synthesized by dispersion polymerisation with a final density ρ = 1.16 g/cm 3 , and fluorescently tagged with BODIPY 545. 30 The polydispersity index was about 0.1 as measured by Dynamic Light Scattering. They were dispersed in 0.1 mM aqueous solution of Sulforhodamine B (Fluotechnik, France), a fluorescent dye whose absorption/emission peaks are at 566/584 nm in water. The use of this secondary fluorescent dye allows us to follow the receding front of the solvent. The particle concentration of the suspension was Φ=10 vol%. Zeta-potential measurements showed that particles are negatively charged (-30 mV), thus the repulsive force hinders aggregation, and the suspensions are readily and properly dispersed. To ensure an optimal dispersion of particles prior to measurement, the suspensions were sonicated for 1 hour in an ultrasound bath. Because of their density and size, particles settle within a few minutes in the thin sample. The diffusion of particles in the suspension is very limited, thus we define them as weakly-Brownian particles.

To observe the drying process, thin Hele-Shaw cells were prepared as follows: a 8 µL droplet of suspension was placed onto a microscope glass slide (VWR). A square cover slip (2×2 cm 2 and thickness ≈ 170 µm, VWR) was carefully deposited on top of the droplet. The Hele-Shaw cell was sealed on three sides with nail polish. One side of the cell is opened, inducing directional drying (Figure 1). Within the series of measured samples, the apparent as-prepared thicknesses of all the Hele-Shaw cells were in the 10-20 µm range. The thickness within the single Hele-Shaw cell was uniform as we measured the cell thickness in different points of each cell. These values were measured by microscopic images of the vertical cross-section of the samples. Drying occurs at ambient air condition without any control of humidity and temperature. The typical drying time of an entire cell is of several hours, but the formation of the dense zone at the open-end takes about 30 minutes. When air starts to invade the cell and the dense zone stops growing, we observe the formation of labyrinthine structures followed by the appearance of cracks (such events will not be discussed here).

B. Confocal microscopy

To follow in situ and in real-time the dynamic and organization of particles during drying, we used a Leica TCS SP8 (Leica-Microsystems, Mannheim, Germany) confocal laser scanning microscope. We used an external continuous laser source at 488 nm (blue) to scan the sample and a Leica HCX PL APO CS 20× dry objective. The light is collected by a photomultiplier.

The 488 nm laser beam excites the dye (BODIPY) within the particles. The fluorescence is integrated in the 495-520 nm range. A high-speed scanner allows to record up to 20 frames per second at a 512×512 pixels 2 resolution, equivalent to 290×290 µm 2 at the optical magnification 2×. The microscope is controlled in remote by the proprietary Leica LAS-AF software. Due to the time needed to prepare the sample and launch the acquisition, the first images were taken approximately one minute after sealing the sample.

C. Image analysis

The collected series of images were post-processed with Fiji. 31 The only digital manipulation, before any further analysis, was a linear contrast adjustment to distribute the signal in the entire dynamic range. The images were then converted to tiff files for numerical analysis. The velocity of the solid/liquid interface was determined by automated measurement of the interface position, based on the sudden change of fluorescent signal at the interface (Figure 2). The velocities of particles was measured using the TrackPy Python package. 32 We also used a Python code to compute the radial distribution functions, based on the analysis of the particles position obtained with TrackPy.

III. RESULTS AND DISCUSSION

We first describe the sequence of drying events, and how a dense region is formed and grows before air invasion. We then identify the existence of regions with different ordering, and measure the particle dynamics during drying.

A. Drying events

In confined and unidirectional drying, the flow of water towards the open-end of the cell carries particles towards the pinned air/water interface where evaporation occurs. Particles progressively accumulate, building a dense region. We observe a time-delay (on the order of half a hour) between the onset of drying of the dense zone and the dense zone formation, that remains wet as the water flows through it. This phenomenon was already observed by Divry et al. 22 in drying a suspension of latex hard spheres: as these authors, we define the observation of crack as the onset of complete drying.

In particular, while the air/water interface is pinned in constant position, at the open-end of the sample, our observations reveal that the dense zone remains wet as it grows: water flows through it, dragging always more particles. As water flows out of the cell, the same volume should be replaced by, for example, air bubbles or change in thickness: we cannot observe these two phenomena because they would appear far from the point where we observe the flux of particles and the growth of the dense zone.

Particles, driven by the flow, accumulate in closecontact into a dense region (Figure 2(a)). The interface between the dense region and suspension is sharp (Figure 2(b)). The particle concentration goes from the nominal suspension concentration to a nearly dense packing over a distance of only 2-3 particles. The diffusion of particles in the suspension is very limited compared to the induced advection, which is why we defined them weakly-Brownian.

We can already observe in Figure 2(a) that the particle organisation in the dense region is not constant: ordered (hexagonal) and disordered domains can be observed. We outline that the horizontal intensity change is an artifact due to a small tilt of the sample. As the thickness of the optical slice we look at is very thin (≈ 200 nm), a tiny tilt is sufficient to induce a linear gradient of intensity over the entire image. The discontinuity in intensity, assigned to the change in ordering, is instead visible through the vertical lines, which are averaged to yield Figure 2(b). Albeit it is subtle, it is yet visible. Indeed, on Figure 2(b), we observe, at about 100 µm, that the slope changes. We assign this event to the change in ordering. We can rule out that the change in intensity is an artifact of illumination on the sample because we operated the microscope in confocal mode.

The sample is a closed system with limited supply of Although some local reorganization of the particle can occur in the dense zone (described in the text), the ordering of the particles is retained during drying. In this case, the dense and disordered domains observed close to the interface between the dense deposit and the suspension will not turn into a colloidal crystal after further drying. The integrated intensity changes along the horizontal and vertical lines as described in the text. b) The interface between the dense deposit and the suspension is very sharp. We detect this sharp increase to track the interface position and measure its velocity.

suspension. At some point of the drying stage, the pressure drop inside the sample is too high: the air invades the dense deposit (Figure 3). Air invasion occurs too rapidly to be captured in the images. The capillary forces developed during the air invasion pull the particles together in the dense deposit. This locally results in an increase of the crystallinity. In particular, we only observed this behaviour at the liquid/solid interface. In the bulk of the dense deposit, particle ordering is not affected by the air invasion. The structure is probably too compact to further densify, unlike at the edge.

The air invasion also marks the onset of crack growth in the dense deposit (Figure 4). This feature has also been observed during the condensation, by water pervaporation, of charged colloidal particles in microchannels. 14 Cracks propagate well in ordered domains whereas they are deflected and stopped when they reach the disordered ones. Indeed, in the ordered zone, cracks propagate along the grain boundaries that are defects in the crystal. In the disordered zone, we cannot define neither a defect in the crystal structure nor a grain boundary through which the cracks can start and propagate easily. To make easier to understand the picture, we can say that the cracks become indiscernible in the disordered domains.

The air invasion is responsible of another drying regime, resulting in a labyrinthine pattern inside the cell, [33][34][35] which will not be described in the present work.

B. Evidence of alternate order in the dense deposit

At the end of drying and particles flow, the deposit is made of alternating ordered (hexagonal) and disordered (glassy) domains. In the observation plane, the particle density is apparently greater in the ordered domains than in the disordered ones.

Close-up views of the particle organization in the ordered and disordered domains are shown in Figures 5 and6. From these two figures, we estimate the 2D density of particles to be 77% in ordered domains and 62% in disordered ones.

The ordered domain, Figure 5, is made of 2dimensional, long-range ordered, colloidal crystal grains. Defects typical of colloidal crystals can also be observed, such as vacancies, stacking faults, or dislocations. Owing to the well ordered domains, we argue that the ordering should continue in the interior of the domains, in a threedimensional fashion. This fact was already observed in rather similar experiment by post-mortem electron microscope images. 26 We observe that the ordered structure is formed as soon as the particles reach the dense zone. The particles carried by the flow settle in a crystalline, hexagonal organisation at the solid/liquid interface. No further particles reorganization is observable at the current level of magnification (40×) and resolution.

The disordered domains, shown in Figure 6, are free from long-range cracks, as it is expected for a disorder colloidal packing. Both disordered and ordered features were also observe by Ziane et al. 14 in colloidal crystals growth by pervaporations in microfluidic channels. To better quantify the particle order in the dense deposit, we computed the radial distribution function g(d), which represents the probability to find a particle at a distance d from a reference one, at several positions close to the cell edge. Typical radial distribution functions g(d) of both phases are shown in Figure 7.

We find the g(d) confirms, for the ordered case, longrange order extending to several neighbours. The order is limited to the first two neighbours for the disordered case. The g(d) of the ordered domain strengthens the thesis of readily made hexagonal 2-dimensional lattice (made of several grains), whereas the g(d) for the disordered domain is the expected for a randomly packed structure. From these results we cannot infer the actual numerical value of the particle density that is a measure of the three-dimensional packing.

We have shown that directional drying causes the formation of hexagonally ordered domains, and large, dense, and disordered domains. The presence of these two different organizations implies that at least two different mechanisms cause the ordered/disordered arrangements. We hypothesize that the particle collision velocity at the solid/liquid interface is the origin of the mechanism of ordering.

C. Dynamic study of the formation of the dense deposit

The rapid imaging mode of the microscope allows us to individually follow the trajectory of particles during the formation of the dense deposit. From the trajectories we measure individual particles velocities v p so that we may identify the ordering mechanism during the formation of the dense deposit. v p was measured with the particle tracking package TrackPy, 32 which detects the particles, track their position, and compute their velocity. Particles are tracked from about 60 µm from the interface until they reach the interface. As particles are weakly-Brownian, their velocity directly results from the velocity of the water flow towards the interface. There are there-fore little variations in their velocity as the interface is approached.

In the same series of images, we can also determine the interface velocity V of the packed/dilute interface. V is computed by measuring the interface position in each sequence of images. The results of V versus v p are shown in Figure 8 for the ordered domains (colloidal crystal) and in Figure 9 for the disordered domains. Each point drawn in these two figures is represented by at least hundreds of tracked particles: in particular each red point is represented by a population of thousands of particles.

We measured an average vp = 14 ± 2 µm/s for the ordered packing and similarly vp = 9 ± 2 µm/s for the disordered domains. In Figures 8 and9 we represented the distributions of v p that were used to build the V versus v p . We can observe that the distribution are clearly non-gaussian. The interface velocity V is accidentally the same (V ≈ 2 µm/s) for the two types of ordering.

To assess a possible correlation between the V , v p , and order/disorder we propose a simple mass-conservation model, schematically represented in Figure 10.

We suppose that the number dN of particles in the newly formed region of the dense deposit during the time dt is equal to the number of particles arriving from the suspension at the same time, that is:

dN dt = V • Φ d • S = v p • Φ s • S ⇐⇒ V = Φ s Φ d • v p (1)
where Φ d and Φ s are respectively the volume fraction of particles in the dense deposit and in the suspension, and S is the cross-section of the cell, see Figure 10.

Using our model with Φ s = 0.1 in the suspension and Φ d = 0.74 characteristic of a dense crystalline structure, we have drawn the continuous blue lines in Figures 8 and9. Although we cannot really determine the real three-dimensional structure of the dense deposit (unless by breaking the dry sample to image the transversal section), the 2D-image clearly shows the characteristic pattern of a two-dimensional hcp or fcc organization, both of which have the same particle density Φ = 0.74. The prediction of the model (blue line) crosses the largest fraction of data points (red area) of Figure 8. At the spatial length scale of Figure 11 (one hundred of particles per horizontal line) and time-scale, we observe that the particles, once they have reached the densification front, freeze at the contact point and do not locally reorganize or move. This remarkable coincidence might imply that this region forms instantaneously (within a time interval between two frames, which is 0.05 s) as a dense crystalline lattice and it does not become packed afterwards. The precise crystalline nature of the dense packing in 3D is still to identify.

We draw the pink lines in Figure 9 such that it crosses the area with the largest fraction of points (red areas). Within our model, the volume fraction measured by this qualitative approach is Φ g = 0.5. This Φ g value approaches the onset of glassy behaviour (Φ g = 0.58). We note that the phase diagram for hard-sphere particles is quite complex with crystal/liquid coexistance starting at a volume fraction of 0.494 21 continuing to 0.545, and then from 0.545 to 0.74 being crystalline. Additionally, the glass transition occurs at 0.58 and continues to 0.64 which is generally accepted as randomly close packed. To simplify the phase identification in our visual observations, we will consider a phase diagram consisting of either glass or hexagonal lattice. For clarity, the same pink line is replicated in Figure 8. The difference between the measured particle fraction and the onset of glassy behaviour might be due to a different ordering induced by the glass wall. We note in Figure 6 an unfocused bottom layer of particles that allow us to confirm the three-dimensionality of the lattice. However, due to the dense packing, the imaging depth is limited to two layers of particles and we are unable to investigate the particle organization within the bulk volume.

The correlation between particles velocities and order is a counter-intuitive observation after the work of Marín et al. 17 about the drying of water-based suspensions in sessile drops (particle radius in the range of 0.25 -1 µm). The authors elegantly showed that ordering (hcp or bcc/fcc colloidal lattice) is caused by parti-FIG. 10. Mass conservation model used to draw the continuous lines in Figs. 8 and9. S is the cross-section of the cell. Φ d and Φs are respectively the volume fraction of particles in the dense deposit and in the suspension. V is the velocity of the interface (where the deposit grows) and vp is the velocity of particles arriving from the suspension.

cles slowly colliding with the solid/liquid interface. These colloidal particles, due to their Brownian motion, can rearrange into an ordered structures. The colloidal particles that arrive rapidly hamper the rearrangement of the neighbours because they quickly form aggregates.

A phenomenon similar to what we observe has been recently described by Piroird et al. 18 The authors' explanation on how order takes place while drying colloidal silica nanospheres (12 nm radius) in controlled atmosphere, is based on the rate of evaporation of the solvent. For small evaporation rate, particles aggregates form in the suspension that can not order as they reach the interface. During fast drying, the particles have no time to form intermediate aggregates, and impinging singularly onto the solid/liquid interface, they can order in closely packed structures. In our conditions, we never observed any particles aggregation.

We observe that these two reports are apparently in contradiction and predict that particles size is one of the key conditions to observe ordered/disordered particles arrangements in drying experiments. Despite the similarities between the reported works and ours, we reiterate that our samples are unidirectionally dried in a confined Hele-Shaw cell with a single open-end. We believe that the formation of quasi-organized structures is more likely due to the energy necessary to reorganize the particles as they arrive at the interface.

The counter intuitive result that we observe is related to particles that are only weakly Brownian. The Peclet number P e, which is the ratio of diffusion time to hydrodynamic time, P e = 6πη vpR 2 kBT (R = 1.1 µm , radius of the colloidal sphere, η the water viscosity) is 51 and 79 for v p equal to 9 and 14 µm/s respectively. In our system, the particles do not have time to explore configurations by diffusion as they reach the dense deposit. Their motion is controlled by the hydrodynamic flow which drags them to the open end because of the solvent evaporation.

The flow of water through the packed particle deposit generates a pressure gradient that can be estimated using Darcy's law and Kozeny-Carman's equation:

∆P L = kµv p S 2 v φ 2 f (1 -φ f ) 3 ( 2
)
where L is the length of the deposit, φ f the particle volume fraction, µ the viscosity, S v = 3/R the ratio of surface to volume of the particles for ideal spheres, and k a constant whose value is generally 5. 26 This equation shows that a higher velocity leads to a higher pressure gradient and consequently to a higher packing of the particles. It was confirmed experimentally by Inasawa et al. 26 that higher impinging velocities lead to higher volume fraction in the deposit in unidirectional packing. Moreover the pressure acting on the particles may help to overcome the electrostatic repulsions between the particles, leading to short range attraction between the particles. Consequently the authors observe a saturation of the volume fraction in the deposit around φ = 0.5 corresponding to a disordered deposit as attractive particles cannot rearrange as they reach the deposit.

In our case, we find experimentally φ f = 0.74 and v p = 14 µm/s in the ordered domains while for the disordered domains φ f = 0.5 and v p = 9 µm/s . Taking these two sets of values and L = 10 -3 m, we find ∆P = 16 kPa for the ordered domains and ∆P = 0.6 kPa for the disordered domains. The stress pushing on each particle F = 2πr 2 ∆P is of the order of 120 nN for the ordered domains and 5 nN for the disordered domains. We then calculate the DLVO potential taking into account the Van der Waals attraction and electrostatic repulsion. We find that the DLVO barrier is of the order of 12 nN. Therefore the Darcy's stress is sufficient to overcome this barrier in the case of the ordered domains but not for the disordered domains. Consistently, very little particle reorganization occurs in the dense region after its formation (Figure 11) while we show in Figure 12 that once the particles are embedded in the disordered zone they can undergo some Brownian motion but this local motion is insufficient to result in disordered domains rearranging into ordered ones. Water still flows directionally through the dense region, which limits the free movement of the particles and drags them.

The disordered to ordered settlement transition as the particle velocities increase is similar to what has been observed in the formation of crystals with non-colloidal particles. In these cases, an external source of energy, for example ultrasound or mechanical agitation, is necessary to organize grains in a crystal form. 36,37 Likewise, the pressure gradient on the particles due to the solvent flow toward the open side of the cell can then be considered as the source of energy necessary to induce particle organization.

IV. CONCLUSIONS

The drying of particle suspensions is a complex phenomenon where evaporation rate, difference in chemical potential of solvent and environment, and size of colloidal particles play interconnected roles. From our observations we find that confined drying by directional flow behaves differently to drying of colloidal suspension of sessile drop.

By tracking in-situ and in real time the evaporationinduced motion of weakly-Brownian particles close to the open end of a Hele-Shaw cell, we found that the structure of the solid deposit is controlled by the velocity of the particles impinging on the pinned dense zone. In our actual ambient and experimental conditions, we found two particle velocity regimes that allow us to define a particle velocity threshold, only valid for the current experiment. Particles form disordered domains without aggregation on their way to the dense deposit when travelling below this threshold (slow regime). Ordered domains, for which we cannot infer the actual three-dimensional crystalline nature, are formed when the particle velocities are above this threshold (fast velocity branch). Albeit we defined this kinetic threshold about v p ≤ 10 µm/s, we cannot rule out if other factors contribute to the order/disorder organization. It should be stressed that the velocity of the solid/liquid interface does not significantly change for ordered or disordered accretion. A better control of both the evaporation rate in constrained experimental conditions and the Hele-Shaw cell will give a strong impulse to understand the mechanism of densification by drying of colloidal suspensions with strong benefit in material processing routes.

FIG. 1 .

 1 FIG. 1. Sketch of the Hele-Shaw cell with the suspension inside. Three sides are sealed, forcing unidirectional drying on the opened side.

FIG. 2 .

 2 FIG.2. Alternating ordered and disordered domains observed during the formation of the dense zone, close to the opened side of the cell. a) The suspension flows from the top to the bottom toward the air/water interface which is not shown in the picture. Although some local reorganization of the particle can occur in the dense zone (described in the text), the ordering of the particles is retained during drying. In this case, the dense and disordered domains observed close to the interface between the dense deposit and the suspension will not turn into a colloidal crystal after further drying. The integrated intensity changes along the horizontal and vertical lines as described in the text. b) The interface between the dense deposit and the suspension is very sharp. We detect this sharp increase to track the interface position and measure its velocity.

FIG. 3 .

 3 FIG.3. Close-up view of the edge of the dense region during air invasion. In the first frame, water still flows through the dense region (top to bottom), towards the open-end of the sample. The air invasion occurs very rapidly. The capillary forces created during the air invasion compact the structure, whose effect is a higher degree of particle ordering. The dashed line is a guide for the eye to follow the recession of the dense deposit. This behaviour is only observed at the edge of the dense deposit. Far from the edge, the order or disordered configuration of the particles is not affected by the air invasion.

FIG. 5 .

 5 FIG. 5. Close-up view of an ordered domain in a dried sample (no water flows anymore). The dense deposit is composed of close-packed domains with cracks between ordered domains.

FIG. 6 .

 6 FIG.6. Close-up view of of a disordered domain in a dried sample (absence of water flow). The structure is dense and disordered, with an apparent short-range order only.

FIG. 8 .

 8 FIG. 8. Left: Interface velocity V versus particle velocity vp, measured from images series where ordered domains are formed. Colour bar represents the percent fraction of the total data. Continuous lines are drawn from the simple mass conservation model, see Figure 10: blue line for ordered zone, Φ d = 0.74, pink for dense and disordered domains, Φ d = 0.5, see text. Right: histogram of the distribution of vp. The velocity distribution is clearly non-gaussian.

FIG. 11 .

 11 FIG. 11. Close-up view of the dense deposit/suspension interface during the growth of ordered domains. The ordered configuration of particles is formed as the particles arrived at the deposit interface.

ACKNOWLEDGMENTS

The authors acknowledge F. Gobeaux (IRAMIS/NIMBE/LIONS, CEA Saclay) for zeta potential measurements.

FUNDING

The research leading to these results has received funding from the European Research Council under the European Community's Seventh Framework Programme (FP7/2007-2013) Grant Agreement no. 278004 (project FreeCo).