Christian Kharif
email: kharif@irphe.univ-mrs.fr

Malek Abid

Julien Touboul

Rogue waves in shallow water in the presence of a vertically sheared current

Two-dimensional rogue wave occurrence in shallow water on a vertically sheared current of constant vorticity is considered. Using Euler equations and Riemann invariants in the shallow water approximation, hyperbolic equations for the surface elevation and the horizontal velocity are derived and closed-form nonlinear evolution equation for the surface elevation is obtained. Following Whitham (1974), a dispersive term is added to this equation using the fully linear dispersion relation. With this new single first-order partial

Introduction

Generally, in coastal and ocean waters, current velocity profiles are established by bottom friction and wind stress at the sea surface, and consequently are vertically varying. Ebb and flood currents due to the tide may have an important effect on water wave properties. In any region where the wind blows, the generated current affects the behavior of the waves. The present work focuses on the nonlinear evolution of two-dimensional gravity waves propagating in shallow water on a shear current which varies linearly with depth. We assume that the directional spread of the wave field is sufficiently narrow to consider unidirectional propagation of the waves.

There are a number of physical mechanisms that focus the wave energy into a small area and produce the occurrence of extreme waves called freak or rogue waves. These events may be due to refraction (presence of variable currents or bottom topography), dispersion (frequency modulation), wave instability (the modulational instability), soliton interactions, crossing seas, etc. For more details on these different mechanisms see the reviews on freak waves by [START_REF] Kharif | Physical mechanisms of the rogue wave phenomenon[END_REF], [START_REF] Kl | Oceanic rogue waves[END_REF], [START_REF] Kharif | Focusing wave group on a current of finite depth[END_REF] [START_REF] Onorato | Rogue waves and their generating mechanisms in different physical contexts[END_REF]. Few studies have been devoted to the occurrence of extreme wave events in shallow water. Among the authors who have investigated rogue wave properties in shallow water, one can cite [START_REF] Pelinovsky | Nonlinear dispersive mechanism of the freak wave formation in shallow water[END_REF] , [START_REF] Kharif | Formation de vagues géantes en eau peu profonde[END_REF], [START_REF] Peterson | Interaction solitons as a possible model for extreme waves in shallow water[END_REF], [START_REF] Kharif | Extreme evaluation and slopes of interacting solitons in shallow water[END_REF], [START_REF] Talipova | Non-gaussian properties of surface elevation in crossing sea states in shallow water[END_REF] and [START_REF] Chambarel | Generation of two-dimensional steep water waves on finite depth with and without wind[END_REF]. [START_REF] Pelinovsky | Numerical modeling of kdv random wave field[END_REF] and [START_REF] Talipova | Non-gaussian properties of surface elevation in crossing sea states in shallow water[END_REF] investigated the statistical properties of rogue waves in shallow water.

To the best of our knowledge, there is no paper on the effect of a vertically sheared current on rogue wave properties apart from that of [START_REF] Touboul | Effect of vorticity on the generation of rogue waves due to dispersive focusing[END_REF] in deep water. We propose to extend this work to the case of shallow water.

Within the framework of the shallow water wave theory [START_REF] Whitham | Linear and nonlinear waves[END_REF] [START_REF] Touboul | Effect of vorticity on the generation of rogue waves due to dispersive focusing[END_REF] ).

Besides, a numerical investigation is carried out by using the KdV equation with constant vorticity whose derivation can be found in the papers by Free-man and Johnson (1970) and [START_REF] Choi | Strongly nonlinear long gravity waves in uniform shear flows[END_REF]. Note that the latter equation can be derived from the generalised Whitham equation with vorticity assuming that the waves are weakly nonlinear and weakly dispersive.

2 Two mathematical formulations

The generalised Whitham equation with vorticity

We consider two-dimensional gravity water waves propagating at the free surface of a vertically sheared current of uniform intensity Ω which is the opposite of the vorticity. The wave train moves along the xaxis and the zaxis is oriented upward opposite to the gravity. The origin z = 0 is the undisturbed free surface and z = -h is the rigid horizontal bottom.

The continuity equation is

u x + w z = 0 (1)
where u and w are the longitudinal and vertical components of the wave induced velocity, respectively. The underlying current is U = U 0 + Ωz where U 0 is the constant surface velocity.

Integrating equation (1) and using the boundary conditions at the free surface and at the bottom we obtain the following equation

η t + ∂ ∂x [u(η + h) + Ω 2 η 2 + U 0 η] = 0 (2)
where u is assumed to be independent of z.

Equation (2) corresponds to mass conservation in shallow water in the presence of constant vorticity.

Under the assumption of hydrostatic pressure, the Euler equation in x-direction is

u t + (u + U 0 + Ωz)u x + Ωw + gη x = 0 (3)
where g is the gravity.

Using the continuity equation and boundary conditions that w satisfies on the bottom and at the free surface, we obtain

w = -(z + h)u x (4)
It follows that the Euler equation becomes

u t + (u + U 0 -Ωh)u x + gη x = 0 (5)
The dynamics of non dispersive shallow water waves on a vertically sheared current of constant vorticity is governed by equations ( 2) and ( 5) that admit a pair of Riemann invariants. These Riemann invariants which are derived analytically allows us to express the longitudinal component of the wave induced velocity u(x, t) as a function of the elevation η. Finally, equations ( 2) and ( 5) can be reduced to the following single nonlinear partial differential equation for η

η t + U 0 - Ωh 2 + 2 g(η + h) + Ω 2 (η + h) 2 /4 -gh + Ω 2 h 2 /4 + g Ω ln 1 + Ω 2g Ωη + 2( g(η + h) + Ω 2 (η + h) 2 /4 -gh + Ω 2 h 2 /4) 1 + Ω g ( Ωh 2 + gh + Ω 2 h 2 /4) η x = 0 (6)
This equation is fully nonlinear and describes the spatio-temporal evolution of hyperbolic water waves propagating rightwards in shallow water in the presence of constant vorticity.

Following [START_REF] Whitham | Linear and nonlinear waves[END_REF], full linear dispersion is introduced heuristically

η t + U 0 - Ωh 2 + 2 g(η + h) + Ω 2 (η + h) 2 /4 -gh + Ω 2 h 2 /4 + g Ω ln 1 + Ω 2g Ωη + 2( g(η + h) + Ω 2 (η + h) 2 /4 -gh + Ω 2 h 2 /4) 1 + Ω g ( Ωh 2 + gh + Ω 2 h 2 /4) η x + K * η x = 0 (7)
where K * η x is a convolution product. The kernel K is given as the inverse Fourier transform of the fully linear dispersion relation of gravity waves in finite depth in the presence of constant vorticity

Ω: K = F -1 (c) with c = gh tanh(kh) kh Ω 2 tanh(kh) 4gk + 1 - Ω tanh(kh) 2k √ gh
Equation ( 7) governs the propagation of nonlinear long gravity waves in a fully linear dispersive medium. For Ω = 0 and U 0 = 0, (6) reduces to generalised equation (13.97) of [START_REF] Whitham | Linear and nonlinear waves[END_REF].

For weakly nonlinear (η/h ≪ 1) and weakly dispersive (kh ≪ 1) waves, equation ( 7) reduces to the KdV equation with vorticity derived by [START_REF] Freeman | Shallow water waves on shear flows[END_REF] and [START_REF] Choi | Strongly nonlinear long gravity waves in uniform shear flows[END_REF] who used multiple scale methods, different to the approach used herein. To set the KdV equation in dimensionless form, h and h/g are chosen as reference length and reference time which corresponds to h = 1 and g = 1. The equation reads

η t + c 0 (Ω)η x + c 1 (Ω)ηη x + c 2 (Ω)η xxx = 0 (8) with c 0 = U 0 - Ω 2 + 1 + Ω 2 /4 , c 1 = 3 + Ω 2 √ 4 + Ω 2 , c 2 = 2 + Ω 2 -Ω √ 4 + Ω 2 6 √ 4 + Ω 2
The equations ( 6), ( 7) and ( 8) are solved numerically in a periodic domain of length 2L. The length L is chosen O(400δ) where δ is a characteristic length scale of the initial condition. The number of grid points is N x = 2 12 . Spatial derivatives are computed in the Fourier space and nonlinear terms in the physical space. The link between the two spaces is made by the Fast Fourier

Transform. For the time integration, a splitting technique is used. The equations ( 6), ( 7) and ( 8) could be written as

η t + L + N = 0, (9)
where L and N are linear and nonlinear differential operators in η, respectively.

Note that in general the operators L and N do not commute. If the initial condition is η 0 , the exact solution of the previous equation is

η(t) = e -(L+N )t η 0 . ( 10
)
This equation is discretized as follows. Let t n = n∆t. We have

η(t n ) = e -(L+N )n∆t η 0 = (e -L∆t/2 e -N ∆t e -L∆t/2 ) n η 0 + O(∆t 2 ), (11)
and the scheme is globally second order in time. The operator e -L∆t/2 is computed exactly in the Fourier space. However, the operator e -N ∆t is approximated using a Runge-Kutta scheme of order 4. The time step is chosen as ∆t = 0.005. Furthermore, the efficiency and accuracy of the numerical method has been checked against the nonlinear analytical solution of the St-Venant equations for the dam-break problem in the absence of current and vorticity (Ω = 0 and U 0 = 0). For U 0 = 0 and Ω = 0 equation ( 6) reduces to

H t + (3 gH -2 gh)H x = 0, with H = η + h. ( 12
)
For t > 0, the nonlinear analytical solution of equation ( 12) is

H(x, t) = h, u(x, t) = 0; x t ≥ gh H(x, t) = h 9 2 + x √ gh t 2 , u(x, t) = -2 3 √ gh -x t ; -2 gh ≤ x t ≤ gh H(x, t) = 0, u(x, t) = 0; x t ≤ -2 gh (13)
At time t = 0 the initial condition is H(x, 0) = h(1+tanh(2x))/2 and u(x, 0) = 0 everywhere. A numerical simulation of equation ( 12) has been carried out with g = 1 and h = 1. The numerical and analytical surface profiles at t = 0 and after the dam has broken are plotted in figure 1.

Within the framework of the KdV equation in the presence of vorticity, we have also checked that solitary waves are propagated with the right velocity that depends on Ω.

The boundary Integral Element Method

The problem considered here is identical to the one described in the previous section. It is two dimensional, and the current field is assumed to be steady, constant in the horizontal direction, and to vary linearly with depth, The three dimensional interaction of water waves propagating obliquely in the assumed current are not considered here. The vorticity within the flow is thus constant, as previously mentioned. It is straightforward that such current, associated with hydrostatic pressure P (z) = p 0 -gz is solution of the Euler equations when considering a problem of constant depth. This will allow to seek for wavy perturbations (u(x, z, t), v(x, z, t)) associated with the pressure field p(x, z, t). The total flow fields are then given by ũ(x, z, t) = u(x, z, t) + U (z), ṽ(x, z, t) = v(x, z, t) and ( 15)

U (z) = U 0 + Ωz. ( 14
p(x, z, t) = p(x, z, t) + P (z).

Using this decomposition, the Euler equations might reduce to

u t + (U + u) u x + vU z + vu z = - p x ρ and (16)
v t + (U + u) v x + vv z + g = - p z ρ , (17)
which has to be fulfilled together with the continuity equation

u x + v z = 0. ( 18
)
As it is demonstrated in [START_REF] Simmen | Steady deep-water waves on a linear shear current[END_REF], and more recently in [START_REF] Nwogu | Interaction of finite-amplitude waves with vertically sheared current fields[END_REF], the wavy perturbations propagating in such current conditions are irrotational.

Indeed, since the second derivative of the background current U zz is nil, the vorticity conservation equation involves no source term, and the vorticity field does not exchange any vorticity with the wavy perturbations. Thus, we might introduce a velocity potential φ(x, z, t) from which derive the perturbation induced velocities (∇φ = (u, v)). It has to be emphasized that the continuity equation ( 18) is automatically satisfied if the velocity potential is solution of Laplace's equation

∆φ = 0. ( 19
)
The kinematic free surface condition might also be expressed, and if (X, Z)

denotes the location of a particle at the free surface, this condition might be expressed

dX dt = u and dZ dt = v -U (η) ∂η ∂x , (20)
where d/dt refers to the material derivative d/dt = ∂/∂t + u∂/∂x + v∂/∂z, and Z = η(x, t). Now, a stream function ψ can also be introduced, so that (∂ψ/∂z, -∂ψ/∂x) = (u, v). The Euler equations ( 16) and ( 17) can now be integrated in space, and it comes

∂φ ∂t + U (z) ∂φ ∂x + ∇φ 2 2 -Ωψ + gz = - p ρ (21)
When applied to the free surface, where the pressure is constant, this equation provides the classical dynamic boundary condition. Introducing the material derivative used in the kinematic condition , this condition reduces to

dφ dt + U (η) ∂φ ∂x - ∇φ 2 2 -Ωψ + gη = 0, (22)
At this point, the knowledge of the stream function ψ at the free surface is still needed. Hopefully, one can notice the relationship

∂ψ ∂τ = - ∂φ ∂n , (23)
where (τ , n) refer respectively to the tangential and normal vectors at the free surface. Thus, the stream function ψ can be evaluated at the free surface as soon as the normal derivative of the velocity potential is known.

Furthermore, if equations ( 20), ( 22) and ( 23) refer to the boundary condition at the free surface, the fluid domain still has to be closed. This is done by using impermeability conditions on the bottom boundary condition, located at z = -h, h being used as the reference length (i.e. h = 1) and on the vertical boundary conditions, located respectively at x = 0 and x = 200.

The numerical approach used here has already been implemented and used successfully in the framework of focusing wave groups in the presence of uniform current [START_REF] Touboul | Nonlinear focusing wave groups on current[END_REF]; Merkoune et al ( 2013)). The extension allowing to take constant vorticity into account was presented in Touboul and

Kharif ( 2016) together with a validation of the approach. It is based on a Boundary Integral Element Method (BIEM) coupled with a Mixed Euler Lagrange (MEL) procedure. At each time step, the Green's second identity is discretized to solve numerically the Laplace equation ( 19). Thus, the potential and its normal derivative are known numerically, and the stream function ψ can be deduced by integration of equation ( 23) along the free surface. This numerical integration is performed in the up-wave direction, starting from the down-wave end of the basin, and using zero as initial value. Then, the time stepping is performed by numerical integration of equations ( 20) and ( 22) using a fourth order Runge & Kutta scheme. Full details of the implementation can be found in [START_REF] Touboul | Two-dimensional direct numerical simulations of the dynamics of rogue waves under wind action[END_REF]. In every simulations, the total number of points considered at the free surface was N f s = 1000, while the total number of points used on the solid boundaries was N bo = 600. The time step used for the simulations was dt = 0.01.

Initial condition

Both numerical approaches described in previous subsections were initialised with the same intial condition. Following the approach described in [START_REF] Kharif | Formation de vagues géantes en eau peu profonde[END_REF]; [START_REF] Pelinovsky | Nonlinear dispersive mechanism of the freak wave formation in shallow water[END_REF], the initial condition is obtained numerically.

A Gaussian initial wave, with no initial velocity, is allowed to collapse under gravity. This simulation is run in the absence of current and vorticity, using the BIEM. Two radiated wave trains, propagating in opposite directions, are generated. The wave group propagating in the (-x) direction is isolated, and space-time coordinates are reverted. This allows the generation of a focusing wave group in shallow water conditions. For the numerical simulations considered here, the initial gaussian elevation has a maximum amplitude a = h where h still being the reference length, and a width σ = 2h.

The wave train considered is used as initial condition for both numerical approaches. The surface elevation of this focusing wave group is used as initial condition for the generalised-Whitham equation with vorticity, and for the KdV equation with vorticity as well. Both elevations and velocity potential are required to initialise the BIEM.

The dimensionless value of the maximum surface elevation of the wave group obtained, η max (t = 0), is 0.0715. The dynamics of this wave packet is illustrated in figure 2, in the framework of BIEM simulations. The initial wave packet is propagated, and the effects of both nonlinearity and dispersion lead to the formation of a high wave.

Results and discussion

Among the rogue wave properties, a particular attention is paid to the amplification factor of the maximum surface elevation, defined as η max (t)/η max (t = tendency that is an increase of the maximum of amplification with Ω. The focusing time t f obtained with both models are very close. On the opposite, the KdV equation underestimates the maximum value of the amplification factor and the focusing time t f as well. In figures 6 and 7, the BIEM shows for negative values of the shear Ω first a reduction of the maximum surface elevation and then an amplification. This attenuation of the maximum of the surface elevation does not occur for the generalised Whitham and KdV equations. We define as extreme wave events or rogue waves those in the group whose surface elevation satisfies η max (t = 0)/η max (t) ≥ 2. In that way, we can introduce the rogue wave lifetime which is the duration of the extreme wave event. In figure 9 is shown this duration as a function of Ω. For positive values of the shear Ω the rogue wave duration is increased whereas it is the opposite for negative values. The numerical simulations carried out with all the approaches have shown that the presence of vorticity modifies the rogue wave properties significantly.

The maximum of amplification factor of the surface elevation increases as the shear intensity of the current increases. The lifetime of extreme wave event follows the same tendency.

  proposed a generalised equation governing the evolution of fully nonlinear waves satisfying the full linear dispersion. The Whitham equation may be derived from the previous generalised Whitham equation assuming that the waves are weakly nonlinear. The Whitham equation and the KdV equation which have the same nonlinear term differ from each other by the dispersive term. Very recently, Hur and Johnson (2015) have considered a modified Whitham equation taking account of constant vorticity. Very recently, Kharif and Abid (2017) have proposed a new model derived from the Euler equations for water waves propagating on a vertically sheared current of constant vorticity in shallow water. The heuristic introduction of dispersion allows the study of strongly nonlinear two-dimensional long gravity waves in the presence of vorticity. Consequently, this new equation extends to waves propagating in the presence of vorticity the generalised Whitham equation. Two different approaches are used to investigate rogue waves propagating in shallow water on a shear current of constant vorticity: the generalised Whitham equation with vorticity and the Boundary Integral Element Method (BIEM) which allows the study of fully nonlinear dispersive water waves on arbitrary depth in the presence of vorticity (see

Fig. 1

 1 Fig. 1 Dam-break: comparison between analytical (solid line) and numerical solutions (•) after the dam has broken. The dashed line represents the initial condition at t = 0

0).Fig. 2

 2 Fig.2Surface elevation of the focusing waves group evolving from initial condition (t/T = 0) to rogue wave occurence (t/T = 75), before defocusing (t/T = 150).

Fig. 3 (Fig. 4 (Fig. 5 (Fig. 6 (Fig. 7 (Fig. 8 (Fig. 9 (

 3456789 Fig. 3 (color online) Time evolution of the amplification factor without vorticity effect (Ω = 0). Generalised Whitam equation (blue solid line), BIEM (red solid line) and KdV equation (black solid line)

Acknowledgements The authors would like to thank the French DGA, who supported this work through the ANR grant ANR-13-ASTR-0007.


Audrey Lasalle
email: lasalle.audrey@gmail.com

Christian Guizard

Jérôme Leloup

Sylvain Deville

Eric Maire

Agnès Bogner

Catherine Gauthier

Jérôme Adrien

Loïc Courtois

Ice templating of porous alumina: supercooling and crystal growth during the initial freezing regime

We investigate the ice-templating behaviour of alumina suspensions by in-situ X-rays radiography and tomography. We focus here on the formation and structure of the transitional zone, which takes place during the initials instants of freezing. For many applications, this part is undesirable since the resulting porosity is heterogeneous, in size, morphology and orientation. We investigate the influence of the composition of alumina suspensions on the formation of the transitional zone. Alumina particles are dispersed by three different dispersants, in various quantities, or by chlorhydric acid. We show that the height and the morphology of the transitional zone are determined by the growth of large dendritic ice-crystals growing in a supercooled state, and growing much faster than the cellular freezing front. When the freezing temperature decreases, the degree of supercooling increases. This results in a faster freezing front velocity and increases the dimensions of the transitional zone. It is therefore possible to adjust the dimensions of the transitional zone by changing the composition of alumina suspensions. The counter-ion Na + has the most dramatic influence on the freezing temperature of suspensions, yielding a predominance of cellular ice crystals instead of the usual lamellar crystals.

Introduction

Ice-templating is a shaping process used to obtain porous materials by freezing a suspension of particles and extracting the ice-crystals by freeze-drying. The potential interest for ceramic material was discovered by Fukasawa in 2001 [START_REF] Fukasawa | Synthesis of porous ceramics with complex pore structure by freeze-dry processing[END_REF] . This process consists in pouring a ceramic suspension into a mold usually frozen by the bottom. Ice crystals grow unidirectionally according to the thermal gradient, through the suspension, rejecting and packing particles between them. Ice crystals are removed by sublimation to preserve the scaffolds of concentrated particles. A porous green body is thus obtained where the porosity corresponds to the previous ice-crystals. The green body can then be consolidated by sintering [START_REF] Deville | Freeze-Casting of Porous Ceramics: A Review of Current Achievements and Issues[END_REF] . Different pore sizes (from 1 to 100 µm), porosity (from 10 to 90%) and pore morphologies can be obtained. Such materials are currently considered for biomedical application [START_REF] Yoon | Aligned porous alumina ceramics with high compressive strengths for bone tissue engineering[END_REF] , active substance delivery [START_REF] Szepes | Freeze-casting technique in the development of solid drug delivery systems[END_REF] , acoustic insulation 5 SOFC [START_REF] Cable | Regenerative Performance of the NASA Symmetrical Solid Oxide Fuel Cell Design[END_REF][START_REF] Cable | A symmetrical, planar SOFC design for NASA's high specific power density requirements[END_REF] or piezoelectric applications [START_REF] Lee | Fabrication of Porous PZT-PZN Piezoelectric Ceramics With High Hydrostatic Figure of Merits Using Camphene-Based Freeze Casting[END_REF] .

Freezing occurs in two stages: an initial transitory regime [START_REF] Deville | In Situ X-Ray Radiography and Tomography Observations of the Solidification of Aqueous Alumina Particle Suspensions. Part I: Initial Instants[END_REF] followed by a steady state [START_REF] Deville | In situ X-Ray Radiography and Tomography Observations of the Solidification of Aqueous Alumina Particles Suspensions. Part II: Steady State[END_REF] . During the initial instants of freezing, the freezing front advances quickly. The microstructure obtained is characterized by two populations of ice crystals: R-crystals (R for randomly oriented) and Z-crystals (for lamellar and oriented ice -crystals). Because of their different orientation, these two populations of crystals do not concentrate ceramic particles with the same efficiency [START_REF] Deville | In Situ X-Ray Radiography and Tomography Observations of the Solidification of Aqueous Alumina Particle Suspensions. Part I: Initial Instants[END_REF] . We have previously shown that the sudden extinction of the R-crystals marks the end of the transitional zone, leaving only lamellar crystals. This corresponds to the beginning of the steady state of the freezing.

For many applications, a continuous porosity is sought. The presence of a graded structure, corresponding to this transitional zone, can be problematic. For instance, in the SOFC design currently investigated by NASA [START_REF] Cable | A symmetrical, planar SOFC design for NASA's high specific power density requirements[END_REF] , the transitional zone limits the efficiency of the infiltration of the porous electrodes obtained by ice-templating.

Several parameters were identified to have an influence over this transitory regime, such as the particle size 11 , the shape of cooling ramp (linear, constant, parabolic) [START_REF] Bareggi | Dynamics of the freezing front during the solidification of a colloidal alumina aqueous suspensions: in situ Xray radiography, tomography and modeling[END_REF] or the application of an electric field (ref: Z. Yumin, H. Luyang, and H. Jiecai, "Preparation of a Dense/Porous BiLayered Ceramic by Applying an Electric Field During Freeze Casting," J. Am. Ceram. Soc., 92 [8] 1874-1876 (2009)). In the present paper, we show that the characteristics of this transitory regime are controlled by the initial supercooling conditions. We assess the influence of the composition of alumina suspensions, stabilized with various dispersants. We use X-rays radiography and tomography to respectively observe the freezing front behaviour and characterize the resulting fro-zen microstructure. Although the 1.4 µm resolution does not permit to resolve individual particles, it is high enough to provide an accurate observation of ice-crystals and concentrated particles regions.

Experimental

Suspensions preparation

Alumina powder (Ceralox SPA 0.5, Sasol, Tucson, AZ, USA), D50 = 0.3 µm, with a specific surface area of 8m².g -1 , is dispersed in distilled water with an organic dispersant or with chlorhydric acid. Alumina content is always 32vol%, unless specified. Three sorts of suspensions are prepared. Some contains an ammonium polymethacrylate as dispersant (Darvan CN, Vanderbilt, Norwalk, CT, USA), others contains a sodium polymethacrylate (Darvan 7 Ns, Vanderbilt, Norwalk, CT, USA), and the third type of suspension is dispersed by an ammonium polyacrylate (Darvan 821A, Vanderbilt, Norwalk, CT, USA). These organic dispersants are respectively referred in text as D[NH4 + ], D[Na + ] and d[NH4 + ]. They are introduced with a quantity varying from 0.2 to 2wt%. Suspensions prepared with chlorhydric acid (HCl) contain 3.1 10 -2 or 6 10 -2 mol.L -1 of HCl. Dispersant (or acid) is stirred with distilled water during 30 min and then the alumina powder is added. Alumina suspensions are ball-milled during 40 h, de-aired before being ice-templated. The dispersant quantity is expressed in weight percent related to the mass of dried powder. The ice-templated alumina suspensions are summarized in the table 1.

Freezing set-up

Suspensions are introduced into a polypropylene mold of 3 mm of diameter, with a syringe. A particular attention is paid to not introduce air-bubbles in suspension. The mold is on a cooper finger frozen by the bottom by a nitrogen liquid flux pumped into a Dewar. The cooling rate is imposed by the liquid nitrogen flow and the temperature profile is registered during experiment, thanks to a thermocouple located near the copper finger surface. The set-up of freezing was previously described [START_REF] Deville | In Situ X-Ray Radiography and Tomography Observations of the Solidification of Aqueous Alumina Particle Suspensions. Part I: Initial Instants[END_REF] . Our measurements have shown that alumina suspensions are frozen at a cooling rate of 3 to 5.3 °C.min -1 .

Freezing temperature measurement

Two techniques provide the freezing temperature. Cold-Differential Scanning Calorimetry (DSC Q100, TA Instruments, USA) was used on a constant 15 µL volume, introduced into a capsule. The capsule is cooled at 2 or 5°C/min. We consider that the freezing temperature corresponds to the beginning of the endothermic peak and not to the temperature corresponding to the minima reached by the heat flux because a looping with variable amplitude appears at each run. This loop is due to the very important latent heat coming from water to ice transformation. We also use a thermocouple located at the surface of the copper finger using during the freezing experiment. When the radiography of the initial zone is acquired, the freezing temperature corresponds to the temperature when the first ice crystals are observed.

X-ray radiography and tomography

A monochromatic high coherent X-rays beam with an energy of 20.5 keV is sent through the sample. A CDD camera with 2048 x2048 sensitive elements is placed at 20 mm behind the sample. The advancement of the freezing front is followed by radiography with a spatial resolution of 2.8 µm. A sequence of pictures is acquired without moving the sample to follow the advancement of the freezing front, so that we can accurately measure the freezing front velocity. On radiography, the grey level is proportional to the transmission of x-rays, so low values of grey levels correspond to a zone with low absorption of X-rays, that is, ice in our case. High grey levels correspond to the particle-rich phase.

Tomography consists in taking radiographies at different viewing angles on 180°. To ensure a good observation, microstructure must not evolve during acquisition. If we would perform a tomography during the freezing, as the scan takes about ten minutes, the freezing front would move and the microstructure would evolve. We therefore observe the microstructure when the sample is completely frozen. The frozen sample is maintained at a constant temperature during the scan. The spatial resolution is 1.4 µm. After a computer treatment, a 3D map of the absorption coefficient is obtained. Contrary to radiography, the grey level is proportional to the absorption of X-rays.,Low grey levels correspond to the phase with the higher atomic number, the particle-riche phase in our case.

Image analysis

The height of transitional zone is measured from tomography acquisition. A sequence of 1600 projections of 2048 x 2048 pixels is obtained. Such large files are difficult to handle. We resize the picture to obtain a square of 400 x 400 px with a resolution of 4.7 µm. Then each picture in grey level is transformed in binary pictures by using a segmentation criterion, provided as plug-in in the software Fiji 13 . A scan of 1600 binarised pictures is obtained. By image analysis, we calculate the fraction of ice and the fraction of the particle-rich phase. By reporting the variation of the rich particles phase fraction as a function of position z in sample (z is the freezing direction), we can measure the height of the transitional zone. It is characterized by a rapid decrease of the rich particles phase fraction, followed by an increase [START_REF] Deville | In Situ X-Ray Radiography and Tomography Observations of the Solidification of Aqueous Alumina Particle Suspensions. Part I: Initial Instants[END_REF] . This sudden variation originates from the extinction of R-crystals, which concentrate particles in the z direction, while Z-crystals concentrate them mostly in the xy plane, roughly perpendicular to the temperature gradient.

Results

The freezing temperatures were found in the range -10 and -20 °C, for a cooling rate included between 2 and 5.3 °C.min -1 regardless of the composition and the technique of measurement (figure 1). It appears that the counter-ion Na + used in dispersant seems to be the only parameters affecting the freezing temperatures; lower freezing temperatures are systematically obtained with this ion. In order to decouple the influence of the dispersant and the influence of the particles, we measured, by cold-DSC only, the freezing temperature of aqueous solutions containing the quantity of dispersant (or chlorhydric acid) in similar quantities the alumina suspensions contain (figure 2). Surprisingly, distilled water freezes between -20 and -25°C, far from the equilibrium temperature of 0 °C, and the addition of dispersant or acid increases the freezing temperature. The freezing temperature measured for distilled water can be explained by the specific freezing conditions in the cold-DSC experiments. A very small amount of ultrapure water is put in a clean capsule, so that little nucleation sites are available. Water can be supercooled before the liquid/solid transition occurs. Surfactants (dispersants) and particles incorporated in water act as nucleation sites [START_REF] Deville | Influence of Particle Size on Ice Nucleation and Growth During the Ice-Templating Process[END_REF] . The freezing temperature of water, measured in such conditions, cannot be compared to the freezing temperature of the suspensions. At the freezing temperature, the ice nucleus overcome their critical size and can grow into ice crystals 14 . Solidification can proceed.

By tracking the position of the freezing front by X-ray radiography (figure 3), we obtain the instantaneous freezing front velocity, and report its evolution according to the time for different suspensions, in terms of solid loading, nature of dispersant, and quantity of dispersant or chlorhydric acid. All the points are on the same trajectory, but do not necessarily have the same initial and final interface velocity.

Careful observations of the radiographs allow us to rationalize preliminary observations previously reported [START_REF] Deville | In Situ X-Ray Radiography and Tomography Observations of the Solidification of Aqueous Alumina Particle Suspensions. Part I: Initial Instants[END_REF] . When freezing begins, large ice crystals, with a very dendritic morphology, grow very quickly into the suspension. Their growth velocity is five to fifteen times greater than that of the freezing front appearing later (figure 4). When they stop growing, they partially melt back, with the melting starting from the top.

Reconstruction of the tomography data show that the microstructure and magnitude of the transitional zone depends on the composition of the suspension. For the suspension containing 2wt%D[NH4 + ], we clearly observe the end of transitional zone, characterized by the apparition of a continuous lamellar morphology (figure 5b). A perpendicular cross section shows a typical microstructure previously observed, of R-crystals and Z-crystals (figure 6a). This microstructure is less well defined for the suspension with 0.2%D[NH4 + ], but the location of the end of the transitional zone can still be identified. Suspensions containing 0.2 or 2wt% D[Na + ] (figure 5c,d) have a transitional zone larger than that of suspensions containing D[NH4 + ] (figure 5a,b). A perpendicular cross section shows that the transitional zone is just constituted by cellular ice-crystals. No lamellar ice-crystals are observed (figure 6b).

Discussion

To understand the role of the large dendritic ice crystals, their dimensions, assessed on the radiography, can be compared with the frozen microstructure of the corresponding zone (figure 7.a, b). We observe that the location where these large ice crystals stop coincides with the location of the end of the transition zone. When the transition zone extend beyond the observation window, in the case of suspensions containing D[Na + ], the top of the large dendritic crystals is not observable on the radiographs. It is therefore safe to assume that the growth and dimensions of these crystals corresponds to the formation and extent of the transitional zone, and that these large crystals correspond to the previously identified R-crystals. The large dendritic ice crystals should be the first formed nuclei reaching their critical size. These nuclei are formed in a highly supercooled suspension. The location where their growth come to a halt should thus corresponds to the location where the suspension is at its equilibrium temperature. Similar conclusions were recently reached by Spannuth & al. [START_REF] Spannuth | Particle-scale structure in frozen colloidal suspensions from small-angle x-ray scattering[END_REF] , using a very different approach. They observed that the transitional zone (referred as stage I in their paper) takes end when the sample is warmed to the melting temperature. We can also assume that the top of the large dendritic ice crystals melt back because of the important latent heat released during their growth. The extinction of the R-crystals marks the end of the transitional zone [START_REF] Deville | In Situ X-Ray Radiography and Tomography Observations of the Solidification of Aqueous Alumina Particle Suspensions. Part I: Initial Instants[END_REF] . We enter in the steady state of the freezing regime. Finally the dendritic aspect of the R-crystals can be explained by their very high growth velocity, cellular and dendritic morphology being favoured by high interface velocity.

We observe a different freezing behaviour for the suspensions containing D[Na + ]. The presence of Na + , introduced as a counter ion in the dispersant, decreases the freezing temperature. The ammonium polyacrylate is known to exhibit an antifreeze effect [START_REF] Funakoshi | Thermal hysteresis induced by ammonium polyacrylate as antifreeze polymer[END_REF] . As the sodium polymethacrylate has a similar molecular configuration, we suppose it can exhibit a similar antifreeze activity. This activity is reinforced by the structuring nature of the sodium cation and its capacity to rearrange molecular water [START_REF] Jolivet | De la solution à l'oxyde[END_REF] and might thus explain the decrease of the freezing temperature. By decreasing the freezing temperature, the supercooling effect increases. As the first ice crystals appearing do so in a supercooled suspension, they grow very fast to quickly reach their equilibrium position, where the freezing temperature and the temperature in the suspension are equal. The greater the degree of supercooling, the greater the freezing front velocity. We know that the freezing front velocity determines the ice crystal morphology, which is lamellar at intermediate velocity and cellular when velocity increases (ref: Qian L, Ahmed A, Foster A, Rannard SP, Cooper AI, Zhang H. Systematic tuning of pore morphologies and pore volumes in macroporous materials by freezing. Journal Of Materials Chemistry 2009;19:5212.). If we report the evolution of the time it takes for the freezing front to propagate through the observation window on the radiography (transit time) as a function of the freezing temperature, we can observe that the greater the degree of supercooling (the lower the freezing temperature), the greater the freezing front velocity (low transit time) (figure 8). The cellular microstructure (figure 6b) in case of D[Na + ] might thus be explained by an increase of the freezing front velocity, due to an important supercooling effect.

We can adjust, to some extent, the dimensions of the transitional zone with initial degree of supercooling (figure 9), which increases when the freezing temperature of alumina suspensions decreases. The closer the freezing temperature from equilibrium temperature, the lower the initial degree of supercooling. The position where the suspension is at equilibrium temperature will be closer to the base of the sample and Rcrystals will stop growing earlier, resulting in a shorter transitional zone. With these observations, we show that is possible to adjust the height of the transitional zone through the control of the initial supercooling, which is dependent on the composition of the suspension, but also the nucleation conditions, controlled by, among other parameters, the particle size and the shape of the cooling ramp applied. By changing the counter ion of the dispersant used from NH4 + to Na + , the freezing temperature can be decreased and consequently the height of transitional zone can be increased.

Conclusions

The current results allow us to rationalize all previous observations of the transitory stage occurring during the solidification of colloidal suspension. The cooling conditions and suspensions composition and characteristics will determine the degree of supercooling reached when the first ice nuclei reach their critical size. The transitory stage is determined by this supercooling, and can be controlled to some extent by adjusting the freezing temperature of the suspension or controlling the nucleation conditions. The later can be achieved by the incorporation of ice nucleation sites, or cooling conditions less favourable to supercooling. These parameters can be used to control the graded structure of the transitional zone, undesirable for applications of porous materials obtained by ice-templating.

Alumina content

Fig. 2 . 1 Fig. 3 .

 213 Fig.2. Freezing temperature obtained by cold-DSC, of aqueous solutions (without alumina particles) containing the same dispersant or acid in similar quantity that alumina suspensions contain. The domain where the distilled water freezes is indicated by a dashed rectangle. The cooling rate is 2°C.min -1

Fig 4 .

 4 Fig 4. Binary image obtained by subtraction of three consecutive radiographies. The white arrow designates the large ice crystals with a dendritic morphology, the black arrow indicates the freezing front. Scale bar: 200 µm.

Fig 5 .

 5 Fig 5. Cross sections of reconstructed tomography, parallel to the freezing direction, showing the frozen microstructure of the transitional zone for the alumina suspensions containing 0.2wt%D[NH4 + ] (a), 2wt%D[NH4 + ] (b), 0.2wt%D[Na + ] (c), 2wt%D[Na + ] (d). The end of the transitional zone can only be observed for the both first suspensions. Scale bars = 200 µm.

Fig 6 .Fig. 7 .

 67 Fig 6. Cross sections of reconstructed tomography, perpendicular to the freezing direction, showing the frozen microstructure of the suspension 2wt% D[NH4 + ] (a) and 0.2wt%D[Na + ] in the transitional zone. The first microstructure is a mix between R and Z-crystals (a) whereas with D[Na + ] no Z-crystals are observed, only cellular R-crystals with a cellular morphology appear. Scale bar = 200 µm

Fig 8 .

 8 Fig 8. Freezing front transit time for different suspension as a function of the freezing point measured with a thermocouple. The lower values correspond to suspensions prepared with D[Na + ]. The greater the supercooling, the higher the freezing front velocity, which is inversely proportional to the freezing front transit time.

Fig. 9 .

 9 Fig.9. Height of transitional zone versus freezing temperature. The lower the freezing temperature, the higher the transitional zone.

Table 1 .

 1 Composition of the ice-templated alumina suspensions

		Nature of additive	Quantity of additive
	32vol%	D[NH4 + ]	0.2-0.4-0.7-1-2wt%
	50 vol%	D[NH4 + ]	0.2-2 wt%
	32 vol%	D[Na + ]	0.2-2wt%
	32 vol%	d[NH4 + ]	0.2-1wt%
	32 vol%	HCl	3.1 10 -2 -6 10 -2 mol.L -1

Fig.

1

. Freezing temperature of alumina suspensions, measured by cold DSC (cross points) and by a thermocouple during ERSF experiments (rhomb points). The freezing temperature is measured for different compositions, the dispersant used is reported on the abscissa and all alumina suspensions contain 32vol% of particles except when it is indicated.

Acknowledgments

Financial support was provided by the National Research Agency (ANR), project NA-CRE in the non-thematic BLANC programme, reference BLAN07-2_192446. Beamline access was provided by the ERSF, under the proposal MA997. Acknowledgements are due, as usual, to local staff of the beamline: Elodie Boller, Paul Tafforeau and Jean-Paul Valade for the technical and scientific support on the beamline ID-19 at ERSF.


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

Eric Maire

Audrey Lasalle

Agnès Bogner

Catherine Gauthier

Jérôme Leloup

Christian Guizard

Part I: Initial instants

Keywords: alumina, interfaces, porous materials, suspensions, X-ray computed tomography

est destinée au dépôt

In situ X-ray radiography and tomography observations

of the solidification of aqueous alumina particle suspensions.

Introduction

The solidification of suspensions of particles is of critical importance in various and sometimes extremely different fields, such as the preservation of biological materials [START_REF] Ishiguro | Mechanical interactions between ice crystals and red blood cells during directional solidification[END_REF] (cells, food), the processing of composites materials 2 (metals with inclusions, ceramic materials), the freezing of soils [START_REF] Viklander | Laboratory study of stone heave in till exposed to freezing and thawing[END_REF] or the purification of water from pollutants [START_REF] Gay | Forced migration of nonsoluble and soluble metallic pollutants ahead of a liquid-solid interface during unidirectional freezing of dilute clayey suspensions[END_REF] . Recent developments in the so-called freeze-casting processing routes, in particular for ceramic materials [START_REF] Deville | Freeze-Casting of Porous Ceramics: A Review of Current Achievements and Issues[END_REF] but also for metals [START_REF] Chino | Directionally freeze-cast titanium foam with aligned, elongated pores[END_REF] , have drawn attention on the behaviour of such suspensions undergoing directional solidification.

What is understood so far of the freeze-casting process has been analysed mostly from post-mortem observations [START_REF] Deville | Ice-templated porous alumina structures[END_REF] : cross-section imaging of ceramic scaffolds obtained after complete solidification of the starting suspension, followed by sublimation and densification of the green body by sintering at high-temperature. Two major limitations have hindered further progress in the comprehension of the solidification process. First, the observations are two-dimensional, hence giving only a partial view of the structure or the arrangement of particles. Second, only a static picture of the particles arrangement after solidification is obtained, and this arrangement is also likely to be modified by the sublimation and sintering steps with mass transport occurring at hightemperature. Hence both three-dimensional and dynamic observations of the solidification stage are highly desirable.

In addition, the knowledge regarding the behaviour of the solidification of colloidal suspension has been largely extrapolated from the behaviour of suspension of single and larger particles [START_REF] Lipp | On the engulfment of spherical particles by a moving solid liquid interface[END_REF][START_REF] Cisse | A study of the trapping and rejection of insoluble particles during the freezing of water[END_REF][START_REF] Korber | Interaction of particles and a moving ice-liquid interface[END_REF][START_REF] Asthana | Engulfment of foreign particles by a freezing interface[END_REF] , because these can be more easily observed (using conventional optical microscopy techniques for example). The behaviour of suspensions of large particles differs largely from that of colloidal suspensions, at least regarding two aspects: the segregation of particles, which does not occur with colloidal particles where Brownian motion is dominating, and the interaction between the particles and between the particle and the solid/liquid interface. Any conclusion drawn from such experiments with large particles should be extrapolated to colloidal particles with great care.

In situ observation is therefore highly desirable to investigate several unknown aspects of the solidification process, including the morphological evolution of crystal shapes that are growing below the liquid/solid interface during the first steps of solidification and the particles redistribution occurring in front of this interface during crystal growth. Finally, in situ observation allows systems with more simple formulation to be used: when processing porous ceramic samples using the solidification of a colloidal suspension, an organic binder must be added to the formulation, so that the particles arrangement does not collapse during sublimation. However, this binder might affect several aspects of the process, such as the particle's interactions and surface energy, the viscosity of the suspension, the freezing temperature of the media, the diffusion of the species ahead of the interface and so forth. The presence of the binder is not necessary when performing in situ experiments, so that the system is closer to the ideal model systems (particles suspended in water) investigated theoretically [START_REF] Peppin | Solidification of colloidal suspensions[END_REF][START_REF] Peppin | Morphological instability in freezing colloidal suspensions[END_REF] . X-ray radiography and tomography experiments provide several advantages in this context [START_REF] Maire | On the application of X-ray microtomography in the field of materials science[END_REF][START_REF] Salvo | X-ray micro-tomography an attractive characterisation technique in materials science[END_REF] : transparent materials (like those used in [START_REF] Sekhar | Solidification microstructure evolution in the presence of inert particles[END_REF] ) are not required since the imaging is based on absorption contrast; a high spatial resolution (submicronic) can be obtained, so that individual particles can be resolved, and a three-dimensional reconstruction of the particle's arrangement and the solvent crystals is possible, providing more information's than 2D observations from cross-sections.

The interest of these techniques is illustrated here through the in-situ observation of directional solidification of colloidal alumina suspensions. In this first part of the two companion papers, we focus on the initial instants of solidification. The companion paper [START_REF] Deville | In Situ X-Ray Radiography and Tomography Observations of the Solidification of Aqueous Alumina Particles Suspensions. Part II: Steady state[END_REF] is focussed on the steady state behaviour.

Experimental methods

Slurries were prepared by mixing distilled water with a small amount (1 wt% of the powder) of ammonium polymethacrylate anionic dispersant (Darvan C, R. T. Vanderbilt Co., Norwalk, CT) and the alumina powder (TM-DAR from Taimei, Japan (D50=0.2µm, SSA 14.5m²/g). The particle content in the suspension was 28 vol.%. The powder content should be chosen below the breakthrough concentration, which lies around 56-58 vol.%. At the breakthrough concentration [START_REF] Shanti | Particle Redistribution During Dendritic Solidification of Particle Suspensions[END_REF] , the capillary pressure exceeds the osmotic pressure and the solid/liquid interface break through the spaces between the concentrated particles, so that no particle redistribution can occur anymore. If the particle content in the suspension is too close to the breakthrough concentration, particle redistribution is very limited and the growth behaviour of crystals can be modified accordingly [START_REF] Deville | Ice-templated porous alumina structures[END_REF] .

Slurries were ball-milled for 20 hrs with alumina balls and de-aired by stirring in a vacuum desiccator, until complete removal of air bubbles (typically 30 min).

Solidification experiments were performed by pouring the suspensions into of tube like shape (which allows a constant thickness to be crossed by the Xrays beam, i.e; with no significant contribution to X-rays data, in addition to a low absorption coefficient) placed at the surface of a cold finger, in a cell cooled using liquid nitrogen (Figure 1), available at the ESRF beamline. Freezing kinetics were controlled by a heater placed at the bottom of the cold finger (copper rod) and a thermocouple also placed at the bottom. The experimental setup allows only linear ramp of temperature during cooling. A cooling ramp of 5°C/min was used in the experiments, similar to the conditions of previous experiments [START_REF] Deville | Ice-templated porous alumina structures[END_REF][START_REF] Deville | Freezing as a path to build complex composites[END_REF][START_REF] Deville | Freeze casting of hydroxyapatite scaffolds for bone tissue regeneration[END_REF] .

The experimental part of this study focuses on the use of two and threedimensional non-destructive techniques, i.e. high-resolution X-ray absorption radiography and tomography, to directly image nucleation and growth of the crystals in the suspension and the corresponding particles redistribution. Tomography combines information from a large number of X-ray radiographs taken with different viewing angles of the sample. The technique includes a computed step, i.e., a recalculation step during which a 3D map of the local absorption coefficients in each elementary volume of the sample is retrieved from the set of absorption radiographs. In this study, the HST program available at the ESRF was used [START_REF]HST Program[END_REF] . The reconstructed map gives an indirect image of the 3D microstructure.

Although dramatic improvement in the acquisition time of tomography images have been made in the last decade, acquiring a full 180° scan cannot be obtained in less than approximately 30s, and such time span depends on the desired resolution; higher resolution acquisitions require 5 to 10 min. To obtain a reconstruction of good quality, the interface must remain at a constant location during acquisition. Hence, in situ tomography observations can only be obtained for very low interface velocities (<1 µm/s). Only one scan of in situ tomography of the moving interface during solidification was performed, the results of which are reported in the companion paper (part II: Steady State). Because of the typical solidification front velocities in these experiments, a different approach was adopted. Dynamics of solidification was followed by X-ray radiography, with an acquisition frequency of 3 Hz, which allows to precisely follow the interface evolution in two dimensions. The frozen structures after complete solidification were characterized in three dimensions afterwards using a low speed highresolution tomography acquisition, and the results were then combined with the data obtained by radiography.

In the present study, the samples were scanned using a high-resolution X-ray tomograph located at the ESRF (beam line ID 19) in Grenoble (France). X-ray tomography was performed at a voxel size of (1.4 μm) [START_REF] Viklander | Laboratory study of stone heave in till exposed to freezing and thawing[END_REF] . The energy was set to 20.5 keV. The distance between the sample and the detector was 20 mm. Because of the extremely high coherence of the X-ray beam on this beam line, absorption is not the only source of contrast in the obtained radiographs and phase contrast is also present, but in a small amount. A set of 1200 projections was taken within 180°. The detector was a CCD camera with 2048 × 2048 sensitive elements coupled with an X-ray-sensitive laser screen. To improve the acquisition frequency, the radiographs were acquired in a binning configuration (i.e. with a 1024x1024 pixels resolution, half that of tomography acquisitions). The corresponding spatial resolutions of the X-ray absorption radiography and tomography results presented here are given in table 1.

Results

Observation artefacts and defects

Several precautions must be taken to obtain high quality, artefacts-free images. First, any condensation from the atmosphere at the surface of the mold must be avoided. Such condensation would initiate growth of ice crystals at the surface of the mold, which would be detrimental for the observations. A special cooling cell was therefore developed (see again figure 1), including a double circulation of nitrogen at different temperatures, to avoid condensation at the surface of the mold, while ensuring that solidification remains directional. Second, the prolonged exposition of the suspension under the beam induces the formation of bubbles, which are rapidly growing with time. Under typical experimental conditions, bubbles start to appear after 5 to 10 min of irradiation and rapidly grow. To avoid their presence during the image acquisition, every experiment was performed twice: once with a constant exposition under the beam, which allows the precise determination of the observation time frame (which depends on the experimental conditions). A second go was performed, switching on the X-ray beam (and hence the radiography acquisition) only a few seconds before the useful observation time frame. This experimental strategy is time consuming but ensures bubbles-free observations. The frozen structures are not affected by the beam during the tomography acquisitions, no particles movement is possible.

Finally, the effect of the size of the mold in which suspensions were poured was investigated. The signal is collected on a 2048 pixels wide detector. The mold inner diameter must fit into this dimension to get an optimal reconstruction of the data; therefore, the spatial resolution is dependent on the size of the sample to be imaged. The diameter of the mold must therefore be adjusted as a function of the desired spatial resolution. It has been verified in this study that similar results (in terms of solidified structure) were obtained with mold diameters in the range 0.6 -6 mm. Side effects were never observed, even for the lowest diameter. For normal-resolution observations, a tube diameter of 3.2mm was used. For high-resolution observations (shown in the companion paper), a mold diameter of 0.6 mm was used.

Initial instants of solidification

The initial instants of solidification are shown in the pictures sequence in figure 2, obtained by radiography. The acquisition frequency is still too low to observe precisely the nucleation stage. In the time span (0.3s) between the last frame with no crystals and the next one, crystals have already grown around 250 µm in this example. The minimum growth velocity during these first instants can accordingly be estimated to be 750 µm/s, and is probably faster than that. The interface velocity can be measured from the pictures sequence. To limit the error range of the measurements, four sets of measurements were done, at various locations along the x-axis defined in figure 2. The results are plotted in figure 3. The crystals are first growing at a very high rate (>750 µm/s) that dramatically slows down within the first 4 seconds. An almost steady state is then observed, with a rate around 50 µm/s. Such profile of the interface velocity is not surprising considering the cooling conditions. A linear cooling ramp being applied, an inverse exponential variation of the velocity should be expected, as observed.

Secondary crystals, highly dendritic, can also be noticed in the initial stages, ahead of the main interface, in particular in figure 2b and2c. These crystals seem to be growing only during the first nucleation and growth instants, then come to a halt and, before the main interface catches up, seem to melt back. Two explanations can be considered: these crystals can be either particularly well-oriented crystals (crystallographically speaking), growing initially very fast, or they can be crystals growing at the surface of the mold. In this latter case, the thermal conditions might be somewhat different from the ones at the copper surface, and very likely, nucleation might be easier. Their growth rapidly stops, the thermal conductivity of the polypropylene mold being low. The radiography pictures providing only a twodimensional information, a definitive conclusion cannot be drawn from the current observations.

Using the tomography experiment performed on the frozen body, the solidified structure corresponding to the initial instants observed in figure 2 can be reconstructed. The corresponding cross-sections in the xz plane (parallel to the growth direction) and xy plane (perpendicular to the growth direction) are shown, respectively, in figure 4a and figures 4b-e. The z location of the xy cross-sections are indicated in figure 4a. The bottom part of figure 4a reveals the copper surface on which nucleation occurs. The ice crystals appear in dark in the pictures, while the lighter phase corresponds to the particle rich phase (concentrated entrapped particles between the ice crystals). A morphological transition is clearly visible, with an evolution of the crystals shape and arrangement from the copper surface to point c. In the initial stages (figure 4d and4e), a dual microstructure is observed, with both vertical lamellar crystals (tagged 1) and almost horizontal lamellar crystals (tagged 2). The structure cannot be resolved within the first 80 µm, the spatial resolution achieved by tomography being too large compared to the typical size of the microstructure in this portion of the sample. A lamellar structure is observed in point c and from then on remains constant.

The attenuation contrast between the ice crystals and the concentrated particles being strong enough, a quantitative analysis of the phase fraction can be performed on the xy cross-sections, to investigate the evolution of this parameter along the growth direction. The results are plotted in figure 5. The plotted particle rich phase fraction corresponds to the fraction of the phase composed of concentrated particles and the ice surrounding these particles (hence, if all particles are entrapped with no redistribution taking place, the particle rich phase fraction defined here would be equal to 1), as opposed to the phase composed of pure ice crystals (particle-free phase) † . Five distinct areas can be identified on the graph. A first region A where the tomography resolution was not good enough to get reliable images. In region B, the particle rich phase fraction goes progressively down to a minimum. A second rapid increase of the particle rich phase fraction occurs in region C. The fraction is then progressively decreasing in region D, until becoming constant in region E, reaching a constant value around 0.49.

Discussion

The existence of a morphological transition of the interface in the initial stages of solidification under similar conditions has been proposed previously [START_REF] Deville | Ice-templated porous alumina structures[END_REF] , to explain the evolution of the structure close to the cooling surface as revealed by SEM observations. However, the observations were only two-dimensional. From the qualitative and quantitative results presented here, the scenario can be greatly clarified.

It was previously suspected that the interface was initially flat and then undergoing a destabilization by a Mullins-Sekerka type mechanism [START_REF] Mullins | Stability of a planar interface during solidification of dilute binary alloy[END_REF] , leading to a progressive evolution of the interface morphology from flat to cellular and finally lamellar. It is clear from the observations presented here that this is not the case. Instead, two types of crystals are present from the beginning: a first population of lamellar crystals with their main axis oriented along the z-axis (cooling direction, and hereafter called z-crystals), and a second population of crystals, more or less lamellar, called r-crystals (r for random, Figure 6), but lying more or less in the xy-plane (radial direction). The lenticular or lamellar shape of these crystals can be explained through the strong growth anisotropy of the ice crystals in the conditions of temperature and pressure used here. Growth along the c axis of hexagonal ice is theoretically two orders of magnitude slower than growth along the a and b axis, explaining their lamellar morphology. Hence, both types of crystals exhibit the same crystallography, but present a different orientation to the temperature gradient. The nucleation stage occurring at the surface of the copper rod should be spatially homogeneous (no epitaxy relationships between the copper surface and the ice); crystals with all crystallographic orientations are initially nucleating and growing, explaining the two populations of crystals observed. Although we cannot conclude completely regarding the presence of both types of crystals in the 0-80 µm zone, it seems reasonable to assume that they are both nucleating at the surface. It is also worth mentioning that the relative growth kinetics of the two types of crystals cannot yet be discriminated with radiography experiments. They can possibly be different, with one type of crystals growing faster than the other one, although this is still speculation at this time. † The actual particle fraction in the phase composed of the concentrated particles and the surrounding ice is theoretically close to 0.55-0.58 [START_REF] Shanti | Particle Redistribution During Dendritic Solidification of Particle Suspensions[END_REF] , depending on particle size.

The evolution of the ice crystals and particles redistribution is then dependent on several parameters, namely:

-The growth kinetics of the various types of ice crystals. z-crystals are growing much faster in the z direction than r-crystals.

-The ability and efficiency of these crystals to repel the particles and therefore induce particles redistribution in the suspension (figure 7). The particles redistribution efficiency of a given crystals interface section can be defined as the product of the interface surface of the considered portion by the interface velocity of this same portion. Particles are repelled in a direction tangential to the interface displacement direction. Only the tip of the z-crystals is redistributing particles in the z-direction in figure 7a, hence a low redistribution efficiency is found in the z-direction, while the opposite situation is encountered in figure 7b, with redistribution mostly occurring in the z direction. Any tilt of the crystals will result in an intermediate situation.

-The differences in residual stress distribution. The r-crystals arrangement is connected with lower mechanical constraint perpendicular to the crystal layers, as the strain induced by crystal growth can be accommodated by a vertical displacement of the suspension above the crystals (see Fig. 7). Therefore, the r-crystals arrangement would be a favoured growth direction for ice crystals, causing a higher redistribution of the second phase particles.

-The thermal conditions. In these experiments, solidification is directional; the suspension is continuously cooled from the bottom (starting from room temperature), so that a temperature gradient along the z-axis is progressively appearing. In addition, the time dependence of temperature profile is influenced by the changing thermal conductivity due to the rearrangement of the particle distribution, as discussed hereafter.

-The suspension properties which depend on particles concentration. In particular, it has been previously shown that particles can only be concentrated up to a certain breakthrough point [START_REF] Shanti | Particle Redistribution During Dendritic Solidification of Particle Suspensions[END_REF] . Particles will be pushed along the interface until the capillary drag force pushing the particles is countered by the force resulting from the osmotic pressure of the suspension [START_REF] Korber | Interaction of particles and a moving ice-liquid interface[END_REF][START_REF] Asthana | Engulfment of foreign particles by a freezing interface[END_REF][START_REF] Rempel | The interaction between a particle and an advancing solidification front[END_REF][START_REF] Azouni | Thermophysical properties effects on segregation during solidification[END_REF] . The resulting critical particles packing fraction (0.55-0.58) was close to the maximum packing, which is itself dependent on the particles characteristics (size distribution, roughness, etc…). This aspect is particularly important for low interface velocities [START_REF] Peppin | Experimental verification of morphological instability in freezing aqueous colloidal suspensions[END_REF] .

-The efficiency of particles packing between the growing crystals. When the particles dimensions and the crystals spacing are of the same order of magnitude, the optimal particles packing (defined in the previous point) will be more difficult to achieve (figure 8). We can define the true particles fraction in the particle rich phase by the following equation:

Cpt=Cpmax.Peff (equation 1)

where Cpt is the true particles fraction, Cpmax is the maximum particles fraction obtained at the breakthrough (when the osmotic force takes over the capillary force), and Peff is the packing efficiency. Peff is a coefficient that is always smaller than 1, a value of 1 corresponding to the situation where no packing faults are present, i.e. a situation of colloidal crystal. The lower limit of Peff is dictated by geometrical considerations (likely percolation arguments) and might be estimated at a value around 0.3 based on the percolation theory for spherical particles (Cpt=0.17). When Peff=1, then the value of Cpt corresponds to that obtained through the models mentioned earlier, and is around 0.55-0.58. The nominal particles fraction in the suspension is obtained by the product of the measured particle rich phase fraction (corresponding to the concentrated particles entrapped in ice) by Cpt. We can verify here that with a measured value of Cpt close to 0.48 at the end of the initial stage (Figure 5) and a nominal particle fraction of 0.28 (initial formulation of the suspension), we obtain a fraction of particles at breakthrough of 0.58, in excellent agreement with the model exposed previously.

Based on these parameters and the experimental qualitative and quantitative observations, the following scenario for the dynamical selection of interface pattern can be proposed:

-Zones A (defined as the portion of the sample where the tomography resolution was not good enough to image the crystals and particles distribution). When the temperature at the copper surface reaches the nucleation temperature, homogeneous (spatially speaking) nucleation of ice crystals occurs. Crystals with random crystallographic orientation are nucleating and growing very fast, so that all particles are initially entrapped. As the crystals start to grow along the z-axis, the interface velocity is diminishing, and particles redistribution can progressively take place, as particles start being repelled by the interface. All crystals with their c axis almost perpendicular to the temperature gradient align rapidly along the temperature gradient direction, so that their growth kinetics anisotropy corresponds to the favourable conditions dictated by the temperature gradient. This rapidly results in a situation where two populations of crystals are present: z-crystals and r-crystals. Since nucleation is spatially homogeneous at the surface of the cold finger, these populations are spatially homogeneously distributed, as seen in figure 9. The most favourable crystallographic orientation, as described previously, is also the one present in majority.

-Zone B. As the crystals grow larger and larger, particles' packing between the crystals becomes more and more efficient. The entrapped particles fraction therefore progressively diminishes. Crystals with all orientations grow along the z-axis. As the effect of the temperature gradient becomes more important, the growth kinetics along the z-axis slows down. r-crystals grow, repelling particles in the z-direction. The z-crystals do not affect the particles distribution in the z-direction, as they repel particles in the x and y directions (Figure 7). The coexistence of both types of crystals and their progressive growth are observed in figure 4d and4e. The relative fraction (in the place) of z-crystals and rcrystals is plotted in Figure 5. The measured particle rich phase fraction corresponds to an average value of the two populations of crystals (Figure 10). The fraction of z-crystals progressively but slowly increases, corresponding to region B.

-Zone C. The r-crystals stop growing (or turn into z-crystals), as seen in figure 5, a much more favourable configuration under these directional solidification conditions. This evolution can be considered as a morphological instability of the crystallization front. The influence of r-crystals on the particle rich phase fraction progressively diminishes to zero in region C; the particle rich phase fraction increases to its corresponding value for z-crystals only (figure 10). The extinction of r-crystals and preferential growth of z-crystals can be understood by two phenomena: the increasing residual stress by the anisotropic stress free strain caused by the ice crystals, and the enhanced anisotropic thermal gradient due to the higher thermal conductivity in the particle phase. Since the thermal conductivity of ice (1.6-2.4 W/mK, depending of temperature) and alumina particles (40 W/mK) largely differs (almost two orders of magnitude), we can estimate the apparent thermal conductivity in the z-crystals (assuming that thermal resistance of particle rich phase and ice act in parallel, the apparent thermal conductivity is close to that of alumina) and r-crystals domains (thermal resistance of particle rich phase and ice associated in series, the apparent thermal conductivity is assumed to be that of ice). Thus, the average thermal conductivity in z-crystals domain is larger than that in r-crystals domain (figure 11). After a certain time, the temperature ahead of the z-crystals domains is therefore lower than ahead of the r-crystals domain. Particles concentrate above the stopping r-crystals, as shown in figure 7b and figure 12a-c, likely contributing to the short and sharp increase of the particle rich phase fraction.

-Zone D. Only z-crystals grow in the suspension, as seen in figure 4b-c; the concentrated particles above the recently stopped r-crystals still occupy a fraction of the xy-plane. It takes some time for the z-crystals or for the r-crystals turning into z-crystals to occupy that space (figure 12 d-g), so that these concentrated particles zones are progressively incorporated in the frozen structure. Particles redistribution is dictated by the orientation of the z-crystals to the growth direction. A progressive tilt of the crystals can be observed in figure 4a, corresponding to the alignment of the z-axis crystals to the direction of the temperature gradient. The particle rich phase fraction is dictated by the efficiency of particles packing, as discussed previously, but this parameter is now dependent only on one type of crystals (z-crystals) (Figure 10).

-Zone E. Particles redistribution occurs in the xy plane. The particles fraction Cpt is therefore constant, and is only dependent on the particle fraction initially present in the suspension. These aspects are discussed in more details in the companion paper (part II: Steady state). Spikes in the particle fraction and interface velocity in figure 5 can be noted, close to z=2100 µm. These spikes are significant (they do not fall within the measurement error range), and correspond indeed to interface instabilities, but this will be discussed in more details in a separate paper.

Several consequences arise from this analysis. First, the magnitude of particle redistribution locally depends on the orientation of the ice crystals to the growth direction. This has been proven with the different behaviour associated to the z-and r-crystals. Therefore, any modification of the crystals morphology should be locally reflected to some extent into the particle fraction distribution in the solidified body, and hence in the structure of the materials processed through this technique. This also implies that removing this initial transition zone, which could be detrimental for functional properties, will be difficult to achieve. It would imply favouring the z-crystals from the beginning. This could be done by avoiding nucleation of the r-crystals, which could be achieved through a carefully selected templating of the cold surface with a substance exhibiting a crystal lattice correspondence (epitaxy) with the a-axis of the hexagonal ice structure. An alternative solution could be to suppress the strong anisotropy of growth kinetics with additives blocking certain growth directions, as encountered in nature [START_REF] Raymond | Inhibition of Growth of Nonbasal Planes in Ice by Fish Antifreezes[END_REF] . Using a different material, or with different thermal properties, the dimensions of the initial transition zone should be affected accordingly.

The second important result here is the identification of several parameters affecting the efficiency of particle packing Peff. The coefficient Peff is not only dependent on the size relationships between the particles and the available space for packing (i.e. the interlamellar spacing), but also on the orientation of the growing crystals. The global value of Peff, which dictates the evolution of the measured particle fraction during the initial stages, can be separated here into two contributions: Peff(r-crystals) and Peff(z crystals). More generally, Peff, is probably dependent on the angle q between the local interface displacement direction (of the portion of the interface actually repelling the particles) and the temperature gradient. Peff can therefore be defined as:

( ) ( ) å = = 90 0 q q q eff eff P f P (equation 2)
where f(q) is the fraction of crystals with the q orientation, and Peff(q) is the corresponding packing efficiency. In addition, the results presented here (in particular Figure 5) reveal that Peff(r-crystals) >Peff(z crystals). This is justified by two facts: the orientation of the r-crystals might be more favourable for proper packing, and, more importantly, their size is larger than z-crystals, as seen in figure 6; less packing defects can therefore be expected, in agreement with the results shown in figure 5. When rcrystals stop growing, only the z-crystals are remaining, and the particle fraction is increasing to its new equilibrium value, dictated by Peff(z-crystals). The scenario exposed here could explain the previously mentioned influence of the particles dimension over the dimensions of this initial transition zone [START_REF] Deville | Freeze casting of hydroxyapatite scaffolds for bone tissue regeneration[END_REF] . Larger particles lead to transition zone spreading over longer distances. Since the size of the transition is largely dependent on the particle redistribution behaviour, the redistribution is expected to be more difficult with larger particles, as it takes more time for Peff to reach its maximum value (i.e. for a given interlamellar spacing, particles packing is more efficient with smaller particles).

The third consequence of the presence of lamellar crystals from the beginning is the possibility to use templated surfaces to induce ordered nucleation at the surface, leading to lamellar structures with long range order in the xy plane.

Conclusions

Based on the experimental in situ investigations by X-ray radiography and tomography of the controlled solidification of concentrated alumina particles suspensions, the following conclusions can be drawn, regarding the initial stages of solidification and the dynamical selection of interface pattern:

-Nucleation of ice crystals is occurring homogeneously (spatially speaking) at the cold surface, leading to two types of lamellar crystals, z-crystals and r-crystals, both corresponding to crystals growing perpendicular to the c-axis of the hexagonal ice structure, but with a different orientation in regard to the temperature gradient.

-Particle redistribution in general and in particular during these first stages is different for the different types of crystals growing into the suspensions. zcrystals growing vertically are redistributing particles in the xy plane (perpendicular to the solidification direction), while r-crystals are repelling particles in xz and yz planes (parallel to the solidification direction). The relative importance and evolution of both populations of crystals is reflected in variations of the fraction of particles entrapped in the solidified structure. Particles' packing between the growing crystals is becoming progressively more efficient up to a distance where the r-crystals stop growing, leaving only z-crystals growing into the suspension.

-The efficiency of particles packing, defined as the ratio of the maximum particles concentration achievable and the actual particle fraction obtained, is dependent on the ratio of particle size and intercrystals (or lamellar) spacing, the orientation relationships between the ice crystals and the temperature gradient and the relative amount of residual anisotropic stress-free strain caused by the ice crystals. These cross-sections can be used to measure the relative fraction of z-and r-crystals (see figure 5).

Figure 10: Schematic plot of particle rich phase fraction vs. distance from the cold surface, expected for the r-crystals (lower curve), z-crystals (upper curve) and their mixture (intermediate curve). r-crystals exhibit a better packing efficiency of particles in the particle rich phase, hence a lower particle rich phase fraction. When both types of crystals coexist (region B), an average particle rich phase fraction is obtained. When r-crystals stop growing (region C), the particle rich phase fraction is increasing to its corresponding value for z-crystals only. An equilibrium situation, where the particle rich phase fraction is equal to Feq, is reached in any case; the value of Feq is given by the particles concentration at breakthrough and the nominal particle fraction in the suspension. It takes a longer time for the particle rich phase fraction to reach its equilibrium value in the case of z-crystals.

Figure 1 :

 1 Figure 1: Cooling cell available at the ID19 beamline at the ESRF for the in-situ experiments, to perform directional solidification while avoiding condensation at the surface of the cell.

Figure 2 :

 2 Figure 2: Solidification sequence in the initial instants, particle size 0.3 µm. Time picture was taken: a=0s, b=1s, c=2.1s, d=3.6s, e=9.7s, f=16.3s. Scale bar: 500 µm. Picture a is the last frame taken before nucleation and growth begin. The copper surface is visible in black at the bottom of the pictures. The white arrows indicate fast growing crystals.

Figure 3 :

 3 Figure 3: interface velocity vs. time in the first frozen zone, measured from the radiography images shown in figure 2, particle size 0.3 µm. The velocities values comprised four sets of measurements, performed at different location of the sample along the x-axis of the radiographs (see Fig. 2 for axis definition).

Figure 4 :

 4 Figure 4: Tomography reconstructed slice showing the structure in the first frozen zone, vertical (left, perpendicular to solidification front) and horizontal (right, parallel to solidification front) cross-sections, particle size 0.3 µm. Scale bars: a : 250 µm, b-e : 150 µm. The location of the b-e cross-sections is shown in the micrograph a. The dashed line in a indicates the location of the copper surface, where nucleation occurs. Legend of panel e: zone 1: r-crystals, zone 2: z-crystals.

Figure 5 :

 5 Figure 5: Particle rich phase fraction (▲) and fraction of z-crystals (•) and r-crystals (¨) (F(z-crystals)+F(r-crystals)=1, since only these two types of crystals are present) vs. distance from cold surface in the first frozen zone, measured from radiography and tomography data, particle size 0.3 µm. Dashed areas in region B indicate the measurement range of error, which increases as the distance from the cold surface decreases. The resolution was not good enough to obtain data in the region 0-500 µm (zone A). A similar particle fraction variation profile was obtained for all experiments, with particles fraction at the beginning of zone B up to 0.7-0.8, depending on particle size.

Figure 6 :

 6 Figure 6: z-crystals (left) and r-crystals (right). Three-dimensional reconstruction from the tomography data, bounding box dimensions: 160x160x160 µm 3 . The r-crystals have a larger interspacing.

Figure 7 :

 7 Figure 7: Schematic of particles redistribution as a function of the crystals orientation. (a) z-crystals (q=90°), where particles redistribution occurs in the xy plane and (b) rcrystals (q=0°), where particles redistribution occurs in the xz and yz planes. Dashed area: crystals at time t0. Continuous line: crystals at time t0, dashed line: crystals at time t0+ e. Particles are repelled in a direction tangential to the interface displacement direction. Only the tip of the z-crystals in redistributing particles in the z-direction in figure a, while the opposite situation is encountered in figure b.

Figure 8 :

 8 Figure 8: Schematic of the efficiency of particles packing for different particle size. With large particles (a), obtaining the maximum packing fraction is more difficult to achieve than with small particles (b).

Figure 9 :

 9 Figure 9: Cross-section in the xy-plane from tomography data showing the distribution of crystals in the initial stage of solidification. The majority of the crystals are z-crystals (lamellar shape in this cross-section) with their c axis in the xy plane. Scale bar 500 µm.These cross-sections can be used to measure the relative fraction of z-and r-crystals (see figure5).

Figure 11 :

 11 Figure 11: Thermal conduction and resistance for the r-crystals domain and z-crystals domain. The thermal conductivity of the z-crystals domain is larger; the temperature ahead of these domains should therefore be lower.

Figure 12 :

 12 Figure 12: Cross-sections in the xy-plane from tomography data showing the end of transition zone. Detail showing the crystals evolution, with z-crystals progressively replacing r-crystals. Scale bar 100 mm. Z position: a:z0, b: z0+39mm, c: z0+60mm, d: z0+80mm, e: z0+111mm, f: z0+129mm, g: z0+151mm. Pictures a to c are in zone C, d to g in zone D (reference zone defined in figure 5).

Table 1 :

 1 ). Spatial resolution (voxel or pixel size) obtained for radiography and tomography data. High resolution observations are shown in the companion paper (part II)[START_REF] Deville | In Situ X-Ray Radiography and Tomography Observations of the Solidification of Aqueous Alumina Particles Suspensions. Part II: Steady state[END_REF] .

		Radiography	Tomography
	Normal	2.8 µm/pixel	1.4 µm /voxel
	resolution
	High resolution	0.56 µm /pixel	0.28 µm /voxel

Acknowledgements

We acknowledge the European Synchrotron Radiation Facility for provision of synchrotron radiation beam time and we would like to thank Elodie Boller and Jean-Paul Valade for their irreplaceable assistance in using beamline ID19. Financial support was provided by the National Research Agency (ANR), project NACRE in the nonthematic BLANC programme.