Brice Saint-Michel

Marc Georgelin

Sylvain Deville

Alain Pocheau
email: alain.pocheau@univ-amu.fr

Wall friction and Janssen effect in the solidification of suspensions

We address the mechanical effect of rigid boundaries on freezing suspensions. For this we perform the directional solidification of monodispersed suspensions in thin samples and we document the thickness h of the dense particle layer that builds up at the solidification front. We evidence a change of regime in the evolution of h with the solidification velocity V with, at large velocity, an inverse proportionality and, at low velocity, a much weaker trend. By modelling the force balance in the critical state for particle trapping and the dissipation phenomena in the whole layer, we link the former evolution to viscous dissipation and the latter evolution to solid friction at the rigid sample plates. Solid friction is shown to induce an analog of the Janssen effect on the whole layer. We determine its dependence on the friction coefficient between particles and plates, on the Janssen's redirection coefficient in the particle layer, and on the sample depth. Fits of the resulting relationship to data confirm its relevance at all sample depths and provide quantitative determinations of the main parameters, especially the Janssen's characteristic length and the transition thickness h between the above regimes. Altogether, this study thus clarifies the mechanical implication of boundaries on freezing suspensions and, on a general viewpoint, provides a bridge between the issues of freezing suspensions and of granular materials.

Introduction

The solidification of a suspension (i.e. of a two phase mixture involving solid particles dispersed in a fluid) arises in a number of situations either natural as the freezing of soils [1][2][3][4][5][6][7][8] or man-made as in the food industry [START_REF] Velez-Ruiz | Handbook of food preservation[END_REF] , the casting of particle rich alloys [START_REF] Stefanescu | [END_REF][11][12] or the making of bio-inspired composite materials 13 . Its physics involves phenomena referring either to solidification, to suspension, or to the interaction between a solidification front and the suspension particles. The latter phenomenon gives rise, by van der Waals or electrostatic interactions, to a thermomolecular force between front and particles that is usually repulsive. It then results in the formation of a dense layer of particles that is pushed by the advancing solidification front and in which phenomena pertaining to the physics of suspensions may occur. However, the specific effects induced by an assembly of particles have been poorly addressed to date in the context of suspension freezing. Here, we wish to study them by addressing the effects of boundaries on the compacted particle layer formed ahead of the front.

Boundaries will be provided by the sample plates that sandwich the suspension. Attention will be focused on the steady thickness reached by the compacted particle layer when particles succeed in being trapped by the front. In particular, its evolution with respect to the solidification velocity and the sample depth will be documented at several sample depths. Two regimes will be found : one, dominated by viscous friction of the fluid flowing across the dense layer to feed solidification ; the other, dominated by solid friction between particles and plates.

Whereas the viscous regime has been recently studied in detail first at a single 14 , then at multiple sample depths 15 , the solid friction regime has not, to the best of our knowledge, been documented yet. It will be the objective of this study to clarify it.

The combined mechanical model of particle layer and particle trapping proposed in ref. 14,15 will be modified to include Coulomb frictional contacts at the sample plates and the resulting stress redistribution in the particle layer. The new model will enable to recover the main features of both regimes as well as the transition between them. In particular, a cumulative effect of friction will become prominent in the solid friction regime, when the compacted layer exceeds some characteristic size. This effect is similar to the Janssen effect in granular materials following which the apparent weight of granular columns confined in silos is bounded at a value corresponding to a characteristic height λ , thanks to cumulative friction effects at the silo's walls and to stress redistribution in the granular column [16][17][18][START_REF] Andreotti | Granular media[END_REF] . The length λ , called the Janssen's length, will state here the transition between the above regimes in terms of layer thickness. Its variation with the sample depth e will be evidenced and found to agree with the modelling. This will allow us to clarify its link with mechanical coefficients such as the friction coefficient and the Janssen's redirection coefficient. In addition, an indirect measurement of the thermomolecular repulsive force between the solidification front and a nearby particle will be performed from the fits of data to the relationship derived from modelling.

Altogether, this study thus clarifies the effect of solid boundaries on solidifying suspensions and provides a bridge between the physics of granular suspensions and that of freezing suspensions.

Experiment

The experiment aims at performing the solidification of suspensions in a controlled way so as to allow quantitative analyses in link with appropriate modelling. For this, care is taken to manage the main parameters of solidification (temperature gradient, growth velocity), of suspensions (monodispersity, particle volume fraction), and of sample geometry (parallelepiped, depth) and to obtain relevant real-time visualization of the freezing suspension.

Setup

The experimental setup has been adapted from a previous one dedicated to the solidification of binary solutions [START_REF] Georgelin | [END_REF]21 . It aims at solidifying a thin sample at a prescribed velocity in a fixed thermal gradient while allowing visualization of the vicinity of the solidification front [Fig. 1(a)].

The sample consists of glass plates separated by calibrated spacers that delimit a parallelepiped space in which the suspension is introduced by capillarity [Fig. 1(b)]. Its 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 at least) free of lateral boundary effects and compatible with solidification over hours. Observations will always refer to it. The sample depth can be changed by varying the spacer thickness. In practice, six depths were studied, 16, 30, 50, 75, 100 and 125µm, each in specific samples but with otherwise the same setup and the same suspensions. Samples were sealed with cyanoacrylate and epoxy glue.

The suspensions, manufactured by Magsphere Inc., were stable over months. They contain plain polystyrene (PS) spheres of density 1.05 and volume fraction φ 0 = 10% or 20% ±0.5%. Their size distribution, measured by a Coulter counter by the manufacturer, leads a mean diameter of 3.0 µm with a relative standard deviation as low as 4%, thus resulting in quasi mono-dispersed suspension. The solvent was water with a small amount of surfactant and less than 0.09% of sodium azide. When solidified alone, it led a planar front to destabilize at a large growth velocity of several µm•s -1 , thus indicating a low concentration of additive. Its dynamical viscosity µ was thus taken as that of water : µ = 1.8 × 10 -3 Pa.s.

The sample was pushed on a linear track (THK) by a microstepper motor (ESCAP) driving a recirculating ball screw (Transroll). The elementary displacement on a microstep was, with 6400 microsteps by turn and a 5 mm screw pitch, as small as 0.8µm. At the end of each microstep, the motor uses an electronic damping to slow down its rotation and thus prevent vibrations. The available velocity range extends from 0.07 to 50 µm.s -1 with relative modulations less than 3%.

Heaters and coolers sandwich the sample so as to induce a controlled thermal gradient in it. They are made of copper blocks either heated by resistive sheets (Minco) or cooled by Peltier devices (Melcor) and are separated by a 10 mm gap. They are electronically regulated at temperatures of ±20 • C following which the melting isotherm takes place in the middle of the gap. This facilitates visualization in the optical window that stands in between the thermal blocks (Fig. 1-a) and minimize the dependence of the thermal gradient on the sample velocity V [START_REF] Georgelin | [END_REF]22 . Extra heat released by the Peltier devices or conducted from the heaters to the lateral sides of the setup were extracted by an external circulation of cryogenic fluid at -30 • C. Insulating polystyrene walls were finally placed all around the whole setup so as to provide a closed dry atmosphere and help avoiding condensation and ice formation.

Visualization was achieved by forming an image of the solidification front on a CCD camera. Care has ben taken to avoid thermal perturbation by placing the optical devices far from the front. For this, a home-made optical setup with a large focal length of 50 mm has been preferred to a microscope. A photographic lens was then placed at a distance of about a focal length from the solidification front so as to provide an enlarged image on a camera placed about a meter apart. Despite its simplicity, an excellent image sharpness could be obtained since the low inclination of the optical rays enables the Gauss approximation and stigmatism to be satisfactorily fulfilled.

Samples were observed either by transmission or by reflection. In the former case, light crossed the sample ; in the latter case, it was reflected by the particles. In both cases, the image intensity was linked to the particle volume fraction φ : at large φ , low intensity in transmission but high intensity in reflection, and conversely at small φ . Both methods reveal in figure 2 the dense particle layer that forms ahead of the front as dark (transmission) or bright (reflection). At the scale of the layer, its transition with both the suspension and the solid phase is sharp, both by reflection and transmission. Some modulations appear on the transverse direction parallel to the solidification interface but do not yield instabilities in the velocity regime studied here. In practice, in the following, we mostly employed the reflection method.

Interestingly, Figure 2 reveals that the extent of the compacted layer is the same whatever the optical method. As the reflection method probes the vicinity of the plates up to one or two particle diameters and the transmission method the whole layer from one plate to the other, the fact that the same image of the particle layer is displayed except for the symmetry bright/dark indicates that the layer thickness remains constant from one plate to the other. In particular, if the compacted layer had been significantly curved in between the plates, the image by transmission would have displayed a continuous transition between the grey suspension and the dark zone, in the domain where the compacted layer would have partially filled the space between the plates.

The arrangement of particles adjacent to a plate is revealed in figure 3 by confocal microscopy. Particles appear to be close packed within a plane parallel to the plate. They show an ordering over a characteristic distance of few particles and, in the bottom of the figure, over a much larger range. This possible long-range order corroborates the quasi mono-dispersed nature of the suspension and confirms their diameter value. These long-range ordered patches correspond to the bright patches observed by light reflection in figure 4. This particle ordering is presumably triggered by the impenetrable plate which forces adjacent particles to lie in a plane. It is reminiscent of the ordering induced by planar boundaries in monodisperse granular materials [23][24][25] or of the particle crystallization evidenced near the walls in monodisperse mixtures in channel flows 25 , Couette flows 26 , or by shaking 27 . It is expected to fade away with the distance to the plate and to disappear beyond a few particle diameters from the plate to leave place for a random arrangement.

As the sample is horizontal and the solidification velocities low, particles have enough time to sediment on the bot-tom plate before encountering the advancing compacted particle layer. As they remained dispersed, they then redistribute in between the sample plates and fill the compacted layer in the whole sample depth at the same rate as if the suspension had remained homogeneous. As the essential physical mechanisms will concern the particle layer and the solidification front, they are thus unaffected by this transient sedimentation. Its sole observable effect then stands in a thin redistribution zone ahead of the compacted layer which rounds its transition with the suspension over a distance that we estimate to a sample depth. As the layer thickness h was measured after thresholding, this affected its measurement accuracy. Some additional phenomena induced by particles or solidification may appear intermittently and induce fluctuations. They refer to the compaction of some particle patches that makes them enter the front as a whole, to the stress propagation in the particle layer which is known to involve intrinsic fluctuations 28 , to grain boundaries at the solidification front that favor particle trapping ... They are responsible for small undulations of the particle layer (Figs. 2 and4) and thus to fluctuations of h. Their relative amplitude has been measured to be 15% at most and 6% in average and may be taken as an estimation of the width of error bars.

In the remainder of the paper, we assign the x-axis, y-axis and z-axis to the directions of the solidification front, the sample depth and the thermal gradient respectively [Fig. 1(b)]. Samples are thus pushed in the z direction along which the dense particle layer develops.

Particle layer

At the beginning of solidification, the solidification front faces a homogeneous suspension with a particle volume fraction φ 0 . However, in the velocity range of the experiment, particles are first repelled by the solidification front. They then accumulate ahead of it in a dense layer that is visible in figure 4 as a bright zone induced by light reflection. Meanwhile, the thickness of the layer grows at a constant rate that we have quantitatively studied in the same setup at the largest sample depth e = 125µm 14 . By applying particle conservation, this provided the average particle volume fraction φ l of the layer, φ l = 0.634 ± 0.007, equal to the random close packing value φ rcp = 0.634. This is thus consistent with a random accumulation of particles in the layer.

At some thickness h, the dense layer stops growing and particles appear in the solid phase. Despite repelling forces ex-

φ 0 = 20% , e = 16µm, V = 1 µm•s -1 .
The image is obtained by reflection. Its top corresponds to the liquid phase and its bottom to the frozen phase. Both appear grey as their particle volume fraction, φ 0 , is low. In between them, the bright zone reveals a noticeable increase of particle density which corresponds to the compacted particle layer. It involves bright spots which correspond to crystallized patches. The steady state thickness reached by the particle layer beyond its build-up is labelled h.

erted by the solidification front, particles then succeed in entering the ice phase at a rate equal to their incoming rate in the layer. This steady value h of the layer thickness is thus linked to the repelling/trapping transition of particles by the front. Interestingly, it thus corresponds to a macroscopic variable linked to a micronic or submicronic phenonemon. In this regard, it may enable us to get insight into a small scale issue by a large scale measurement. We shall thus focus our attention on it from now on. It will appear to depend on the particle volume fraction φ 0 , the growth velocity V and the sample depth e.

Following the build-up of the dense particle layer, the particle volume fraction has become inhomogeneous with a step from φ 0 to φ rcp at the frontier between the suspension and the dense layer and the reverse step at the solidification front. When the layer thickness is steady, mass conservation of fluid and particles implies a related step in their respective velocities with respect to the solidification front, v f = v f e z for the fluid and v p = v p e z for the particles (Fig. 5). In particular, whereas both velocities are opposite to the growth velocity in both the suspension and the solid phase,

v f = v p = -V , they amount to v f = -V (1 -φ 0 )/(1 -φ l ) and v p = -V φ 0 /φ l
in the layer. As they differ, a viscous dissipation is generated. It depends on the volume flux of fluid U = Ue z with respect to the particle matrix,

U = (1 -φ l )(v f -v p ), i.e. : U = -V (φ l -φ 0 ) φ l (1)
As φ l > φ 0 , U is negative, so that the viscous force exerted by the fluid on the particle matrix is directed towards the front. To avoid sign confusion, its magnitude will be hereafter labeled

|U|.

The velocity U, called the Darcy velocity, is thus more relevant to the physical process at work in the layer than the growth velocity V. In particular, using it instead of V to report the evolution of h leads to a collapse of data on master curves (see for instance Fig. 6) instead of a slight shift depending on φ 0 otherwise 14 . We shall thus adopt it in the rest of the paper. This will reduce the dependence of h on U and e only. The relative fluctuations of U will result from those of V and of φ 0 and amount to 4%.

Fig. 5 Sketch of a cross-section of the suspension in the z direction of the thermal gradient. It displays the suspension, the compacted layer of particles and the frozen phase. As particles share the same radius, the apparent variation of their radii is an artifact of the section, the particles being randomly distributed in space. The mean velocities with respect to the solidification front of the liquid fluid, the particles and the solid phase are denoted v f , v p and v s respectively. In the suspension and in the solid phase, they all equate to the opposite -V of the solidification velocity. However, in the compacted layer of particles, v f and v p differ following the rise of particle volume fraction φ .

Particle layer thickness

We report the evolution of the steady layer thickness h with respect to both the magnitude |U| of the Darcy velocity and the sample depth e.

Evolution with the Darcy velocity U at a given e

We focus our attention here on e = 125µm which is representative of the evolution at any sample depth e. These two trends signal the occurrence of two different physical regimes. At large velocities, the trend h ∝ 1/|U| will be related in section 4 to a dominant viscous dissipation in the particle layer. At low velocities, the much weaker variation of h with |U| will be related in the same section to a dominant dissipation provided by solid friction between the particles and the sample plates.

Evolution with the Darcy velocity U at various e

When decreasing e from 125µm to 16µm, figure 7 shows similar trends of h with respect to |U|. In particular, at all sample depths e, one recovers at large velocities a linear increase of h with 1/|U| followed, at low velocities, by a concave evolution towards a much slower raise. However, beyond these qualitative similarities, the graphs show quantitative differences. In particular, the transition between the two different trends is encountered at lower h and lower 1/|U| as e decreases. Similarly, both the slope of the linear trend h ∝ 1/|U| displayed at low 1/|U| and the value of h reached at 1/|U| = 15 s.µm -1 decrease with decreasing e.

The remainder of the paper will be devoted first to understand each regime and their transition, then to model the evolution of the layer thickness on the whole velocity range and, finally, to determine the relevant physical variables and their evolution with e. This will yield in figure 7 the non-linear fits displayed by full lines and a location of the transition at the dashed lines h = λ with a transition thickness λ (e) dependent on e. As these fits satisfactorily recover data in figure 7, we gather them in figure 8 to get an overview of the evolution of the whole data. It shows both the similarities between the evolutions of h(1/|U|) at different e and their quantitative differences.

Modelling

The compacted layer ceases to grow when particles begin to be trapped by the solidification front. The steady thickness that it then reaches is thus linked to the repelling/trapping transition of particles at the front. We develop below a mechanical model of this transition by considering the forces that act on a particle nearing the front. In this context, the critical state involved at the repelling/trapping transition will correspond to a net force balance on a particle entering the front. As some dissipative forces will appear to depend on h and U, this balance will provide a link between these variables that will be confronted to our data.

Forces on particles

We consider a particle adjacent to a solidification front. It is surrounded by the front, the suspension fluid and the particle matrix, all of them being suitable to exert a force on it (Fig. 9). We review these different kinds of forces below : i) Force exerted by the solidification front : thermomolecular force F T It is a short range force that results from van der Waals and electrostatic interactions between the particle and the front. Its magnitude and intensity varies with materials. However, it stands here as a repelling force.

On an elementary particle surface, it yields a normal force whose intensity quickly decreases with its distance to the front, as its inverse cube for non-retarded van der Waals interactions [START_REF] Israelachvili | Intermolecular and Surface Forces[END_REF][START_REF] Wilen | [END_REF] and as an exponential for electrostatic interactions [START_REF] Israelachvili | Intermolecular and Surface Forces[END_REF][START_REF] Wilen | [END_REF][31] . On a spherical particle, the net force is, symmetry, parallel to the thermal gradient direction e z (Fig. 9). Its intensity depends on the distance between the particle base and the front but, as the present particles stand in the critical state of the repelling/trapping transition, we shall assume that this distance and thus the force intensity F T on a particle is a constant.

ii) Force exerted by the fluid : viscous force F L It results from the pressure and viscous stresses exerted by the fluid on the surface of an entering particle. It should be emphasized that the thermomolecular force induces an additional pressure between particles and front which, for thermodynamic reasons, maintains a liquid phase between them, whatever the smallness of their distance 4,32 . Accordingly, there thus always exists a liquid film, so-called premelted film, in between an entering particle and the front. As it is submicronic, it gives rise by far to the largest dissipation force, F L , that we index by "L" since it corresponds to a lubrication force.

This force mainly comes from the depression induced in the premelted film. It is thus a trapping force that pushes particles towards the front. By reason of symmetry, it is parallel to the thermal gradient direction e z . As it is induced by creeping flows, its intensity is linearly related to the flow magnitude and thus, to the Darcy velocity U : F L =f L |U|, the prefactor f L depending on the geometry of the premelted film (see ref. [33][34][35] for details) and F L denoting the z-component of the force. In practice, we shall see below that this prefactor is simply related to the critical trapping velocity U c at which a single particle succeeds to be trapped.

iii) Force exerted by the particle matrix : contact force F p It is transmitted to a particle nearing the front by contacts between particles along the compacted layer. It results from the accumulation, along the whole particle matrix, of the stresses induced by the fluid and the sample plates and is thus expected to increase with the layer thickness h. The former stresses refer to the pressure and viscous stresses induced by the fluid flowing across the whole particle matrix. The latter stress refers to the solid friction exerted by the sample plates on the particles following the drift of the particle layer pushed by the growing solidification front.

The fluid force is directed in the direction of the Darcy flow U. The solid friction force opposes the motion of the particle layer with respect to the plates. It is thus opposite to the growth direction V. Both forces are thus directed towards the front. So is their resultant force F p on an entering particle, which is thus a trapping force. As all forces on an entering particle are directed on the zdirection, only their magnitude will matter in the sequel. For convenience, we shall then normalize their z-component by the particle cross-section πd 2 /4 to deal with effective mean pressures : the thermomolecular pressure PT , the lubrication pressure PL and the particle matrix pressure Pp . The repelling pressure PT will then be positive and the two latter trapping pressures, negative.

Trapping state

As the particle layer stands in a critical state for trapping, the trapping pressures PL and Pp have grown up to just reach a force balance on particles nearing the front. In term of mean pressure, this expresses as : Pp + PL + PT = 0.

Here, PT is expected to be constant, PL is proportional to |U|, PL = -g|U|, and Pp should depend on the layer thickness h. This offers the opportunity to reduce the number of pressure variables by considering the critical Darcy velocity U c at which the lubrication force is sufficient to induce particle trapping on a single isolated particle. No particle layer can then build up since all particles coming on the front are trapped without delay : h = 0 and Pp = 0. In this single particle case, the force balance thus reduces to PL + PT = 0 with PL = -gU c and g = 4 f L /πd 2 . This allows us to express the prefactor g with respect to PT and U c , g = PT /U c . In the compacted layer case, this yields at any |U| ≤ U c , PL = -PT |U|/U c , which eventually leads to a relationship between Pp and PT , parametrized by |U| and U c :

Pp + PT (1 -|U|/U c ) = 0 (2)
To close the model it now remains to express the mean pressure Pp with respect to h and |U|. For this, we address below the various stresses involved in the suspension.

Suspension stresses

The mechanical description of a suspension involves the force experienced by a unit surface of the suspension and the resulting stresses. As this surface involves both the liquid phase and the particles, part of the force is carried by the contacts between particles while the remaining part is carried by the fluid. This allows to decompose the suspension stress σ s into a particle stress σ p and a fluid stress σ f : σ s = σ p + σ f .

Here the suspension stands in a dense state in which the fluid shear stress is usually neglected in comparison to the frictional particle stress [START_REF] Andreotti | Granular media[END_REF] . In addition, as the compacted particle layer is pushed by the solidification front at a uniform velocity V, it stands in the front frame in a steady state which can be modeled as a homogeneous Darcy flow. At the mesoscopic level, this means that the fluid phase experiences no shear stress, so that σ f reduces to the pressure part : σ f i j = -pδ i j where p denotes the fluid pressure. Steadiness also implies that the suspension stress σ s yields no force on suspension elements : ∇σ s = 0. Altogether, these statements yield the following mechanical equation for the particle phase :

∇σ p = -∇σ f = ∇p (3)
On the other hand, the mechanical equation for the fluid

8 | 1-14
phase yields the Darcy equation:

∇p = -µ U k = µ |U| k e z (4)
where the right hand side expresses the volumic viscous friction force in the present porous medium, k denoting its permeability. In particular, in the present monodisperse suspension, the Kozeny-Carman relation provides the link between the permeability k, the particle volume fraction φ l of the layer and the particle diameter d :

k(φ l , d) = d 2 180 (1 -φ l ) 3 φ 2 l ( 5
)
Relation (4) shows that a pressure drop ∆p = p(0)p(h) = -µ|U|h/k sets in the particle layer, between the frontier with the suspension (z = h) and the solidification front (z = 0). It is responsible on one hand for the cryosuction of liquid 32 that makes the fluid flow towards the front to feed solidification and, on the other hand, for a part of the trapping force exerted by the particle matrix on an entering particle.

It now remains to solve for the particle stress σ p by adding boundary conditions to relation (3).

Boundary conditions

We seek to express the boundary conditions imposed to the particle stress σ p at the sample plates by solid friction. For this, we place the origin of the frame (x, y, z) on the front, at the same distance from the plates and from their lateral boundaries. Accordingly, the front thus lies in the plane (x, y, 0) and the plates in the planes (x, ±e/2, z).

Following Coulomb's law, solid friction implies a link between the normal stress σ p yy and the tangential stress σ p yz at the plates :

σ p yz (x, ±e/2, z) = µ W σ p yy (x, ±e/2, z) (6)
where µ W denotes the friction coefficient. On the other hand, granular materials are known to redistribute stresses on perpendicular directions so that :

σ p yy = K σ p zz (7)
K designing the Janssen's redirection constant 16,17 . Altogether, relations ( 6) and ( 7) thus yield :

σ p yz (x, ±e/2, z) = µ W K σ p zz (x, ±e/2, z) (8)

Janssen effect

Considering the z-component of relation ( 3) yields the force balance :

∂ x σ p xz + ∂ y σ p yz + ∂ z σ p zz =

d p dz

It is convenient to integrate it from y = -e/2 to y = +e/2 to make connection with the boundary conditions at the sample plates. Denoting with a tilde the mean variables along the sample depth, one obtains :

∂ x σ p xz + ∂ z σ p zz + ∆σ p yz e = d p dz ( 9
)
where ∆σ p yz = σ p yz (x, e/2, z)σ p yz (x, -e/2, z)]. The symmetry σ p yz (x, e/2, z) = -σ p yz (x, -e/2, z) and the boundary conditions (8) give :

∆σ p yz = 2 µ W K σ p zz (x, e/2, z)
We may now invoke the following assumptions that are widely used in granular materials [START_REF] Andreotti | Granular media[END_REF]36 :

• a uniform normal stress σ p zz (x, y, z) in the y direction : σ p zz (x, e/2, z) = σ p zz (x, z). This assumption, commonly used in granular materials 18,[START_REF] Andreotti | Granular media[END_REF]36,37 , follows from the propagation of stresses inherent to these materials 28 and the large uniformity of pressure in piles formed from a uniform rain of grains 38 . In addition, it is supported here by the observations of a flat solidification front, a flat compacted layer and the same layer thickness by both reflection and transmission (Fig. 2). Following them, the layer thickness h keeps the same value when going from one plate to the other. As the particle matrix pressure Pp depends on h and is linked to the stress σ p zz (see section 4.6), the homogeneity of h in between the plates supports that of the particle stress σ p zz (x, y, z) in the y direction. We finally note that the above considerations refer to stresses, independently of the particle volume fraction. In particular, the fact that stresses propagate and redistribute in the layer 28 enables them to homogenize whatever the distribution of particle volume fraction.

• constant mean stresses σ p i j on the x direction : ∂ x σ p i j = 0. This follows from the Hele-Shaw geometry of the samples for which the sample width l is quite large compared to the sample depth e : l e. Then, the effect of boundaries on the x direction may be overlooked in most of the suspension so that its mechanical variables may be considered as independent of x in most of the domain. This in particular results in :

• a constant mean tangential stress σ p xz on the x direction :

∂ x σ p xz = 0.
These assumptions enable us to reduce the stress equation ( 9) to :

∂ z σ p zz + 2 µ W K e σ p zz = d p dz (10)
1-14 | 9

to which we add the boundary condition σ p zz (h) = 0 at the frontier z = h between the boundary layer and the suspension bulk. It follows from the fact that particles are not yet into contact there, so that σ p = 0.

As φ may be considered as independent of z in the compacted particle layer, so must be d p/dz following (4) ( 5) and (1). This enables to easily integrate (10) to obtain :

σ p zz (z) = -λ d p dz [exp( h -z λ ) -1] with : λ = e 2µ W K (11)
At the solidification front, z = 0, this yields the normal stress exerted by the particle matrix :

σ p zz (0) = -λ d p dz [exp(h/λ ) -1] (12)
This exponential response with respect to the height h of the layer is reminiscent of the Janssen effect in granular materials with gravity replaced here by the pressure gradient force -∇p 16,17,[START_REF] Andreotti | Granular media[END_REF] . In addition, the characteristic length λ of the exponential trend takes the same expression as in the Janssen effect, except that it corresponds here to an amplification length instead of a relaxation length. This difference traces back to the fact that friction forces are directed here in the same direction as the leading force, the pressure gradient force -∇p. Their action thus enhances the effect of the leading force, leading to an exponential amplification instead of an exponential relaxation otherwise. The analogous situation in granular materials would correspond to particles pushed against gravity 37 .

Particle layer thickness

To determine the particle layer thickness h, we need to apply the pressure balance (2) on a particle entering the front. For this, we wish to link the particle matrix pressure Pp on this particle to the mean normal stress σ p zz (0) exerted by the particle matrix at the solidification front.

Normal stresses σ p zz (x, y, z) correspond to the z-component of the mean force conveyed by particle contacts on a unit surface of the suspension normal to the z-direction. However, in average, a part ϕ only of this surface is occupied by particles. At the level of a particle, this means that the mean force σ p zz (x, y, z) exerted on a unit surface is distributed over particles occupying an average surface ϕ. On each particle, it then corresponds to a mean pressure Pp (x, y, z) = σ p zz (x, y, z)/ϕ. To determine the particle surfacic fraction ϕ, let us notice that it does not depend on z since the compacted layer is homogeneous. Considering a volume dV = Sdz, the averaged number of particles in it then writes dN = (ϕS)dz. However, it also amounts to dN = φ dV by definition of the particle volume fraction. Accordingly, surfacic and volumic particle fraction appear to be equal : ϕ = φ .

In particular, at the level z = 0 of the solidification front, the mean pressure on a particle exerted by the particle matrix reads Pp = σ p zz (x, y, 0)/φ l and for uniform normal stresses, Pp = σ p zz (0)/φ l . Then, following (2) and ( 12), one obtains :

h = λ ln[1 + χ λ ( 1 |U| - 1 U c )] (13)
where

χ = φ l |U| d p/dz PT ( 14
)
is, from ( 4) and ( 5), independent of |U|.

Janssen effect

We now confront the modelling (13) of the layer thickness h to our experimental data, first to validate the model and the relevance of the Janssen effect to freezing suspension and, second, to extract information on the Janssen's length λ and the mean thermomolecular pressure PT .

For this, we consider the best fit of relation (13) to the data obtained, at each sample depth e, at various Darcy velocities U (Fig. 7). As the particle diameter d and the fluid viscosity are known, this fit involves three parameters : (λ , χ,U c ). Among them, the critical velocity U c involves specific features that need to be discussed.

The critical velocity U c refers to a force equilibrium on an isolated particle entering the front, i.e. for h = 0. The forces involved are then the trapping lubrication force at the base of the particle, proportional to U, and the repelling thermomolecular force exerted by the front. As this trapping issue is disconnected from the presence of other particles, it is expected to be independent of e. Microscopic models of trapping [33][34][35] then provide a determination of U c that yields U c = 12µm.s -1 14 in the present context.

However, here, the fits provide U c = 20µm.s -1 for e = 16µm and for the other sample depths, either irrelevant negative values or too large positive values of about 30, 80 or 130µm.s -1 . This means that, above e = 16µm, the critical velocity is not a sensitive parameter of the fit. It is then largely dependent on data fluctuations so that its best fit value is not physically relevant. As the critical velocity should be a constant of the study, it is then relevant to fix it to a definitive value on the whole data set to prevent its spurious variation with e to affect the fitting of the two other parameters. In view of the expected theoretical value U c = 12µm.s -1 and of the observation of vanishing layer thickness close U = 15µm.s -1 , we adopt this latter value for the whole data set. We nevertheless checked that U c was not essential for the remaining fit parameters λ and PT so that close values were obtained for them by fixing U c either to infinity or to 20µm.s -1 .

| 1-14

The data fits performed at each sample depth e with fixed parameter U c = 15µm.s -1 and free parameters λ and χ are displayed in figure 7 and summarized in figure 8. They all agree with the data and provide values of λ (e) and PT (e) that we comment below.

Influence length λ (e)

As shown in figure 10-a, the influence length λ appears to be proportional to the sample depth e, λ = 10.0 e, in agreement with (11). This proportionality confirms the relevance of the Janssen effect to the particle layer. The measured coefficient, 10.0, corresponds to a product µ W K = 0.05 that we confront below to available data.

The value of the friction coefficient µ W depends on materials, here polystyrene for the particles and glass for the sample plates. It is about 0.35 for polystyrene-steel and 0.5 for polystyrene-polystyrene 39 . For polystyrene-glass, although we are not aware of direct measurements of µ W for these materials, we may assume its order of magnitude to be 0.5. On the other hand, particles are immersed here in the liquid phase of the suspension. This makes them be surrounded by water films which are known to decrease friction coefficients by a factor at least four between hydrophilic (glass) and hydrophobic (polystyrene) surfaces 40 . Finally, confocal microscopy evidences an ordering of particles at the sample plates (see Fig. 3) which is known to reduce the friction coefficient by a factor of about 5/6 25 . Altogether, one may then expect the effective friction coefficient of particles on the sample plates to be about 0.5 × (1/4) × (5/6) ≈ 0.1.

The value of the Janssen's redirection constant K for granular materials stands between that of solid, K = 0, and of fluids, K = 1. It amounts to 0.58 for a compact triangular stack [START_REF] Durand | Powders, and Grains: An Introduction to the Physics of Granular Materials[END_REF] and is usually taken as about 0.5 for particles assemblies.

Altogether, we thus obtain an order of magnitude for the product µ W K of 0.05, in agreement with the estimation from fit measurements. We may compare this value to a measure of µ W K obtained from the Janssen effect in the case of glass beads constrained by a polystyrene wall : µ W K ≈ 0.37 37 . Although the present situation is symmetric regarding materials with beads made of polystyrene and walls of glass, the product µ W K should be similar except that particles are immersed in water here. As this reduces the friction coefficient by a factor four at least 40 , one may finally estimate µ W K to about 0.09 which is of the order the above estimates.

Thermomolecular pressure PT (e)

The fit parameter χ allows from (4) ( 5) and ( 14) an indirect determination of the mean thermomolecular pressure PT over entering particles. As this pressure corresponds to a local submicronic interaction between a particle and the solidification front, it is expected to be independent of large scale features, in particular the sample depth e. However, figure 10-b surprisingly displays values that vary nearly over a factor four on our sample depth range.

To address the relevance of the value of PT obtained from the fit of the non-linear relation (13) over the whole data range, we restrict attention to the domain h < λ which contains most of our data. Using (4) ( 5) and ( 14), it then a priori refers, at first order in h/λ , to the linear and λ -independent relationship :

h = φ l k(φ l , d) µ PT ( 1 |U| - 1 U c ) + o(h) (15)
The largest data density and the reduction to a single fit parameter PT is then expected to provide a more relevant determination that we label Pl T . However, figure 11-a still shows a large variation whereas figure 11-b reveals that both determinations PT and Pl T are largely correlated since they exhibit a ratio of two : Pl T (e) ≈ PT (e)/2. This confirms that the variation with the sample depth of the determination of PT is independent of the fit method and of the fit domain. On the other hand, the ratio between the two evaluations is striking since, following the expansion (15), one might have expected equality instead of a doubled value. To explain it, we question the relevance of this expansion regarding the fit domain h < λ of figure 7. In particular, it appears that, to provide noticeable values of h, most of our data belong to the range 0.3 < h/λ < 1 instead of 0 < h/λ < 1. Accordingly, the relevant tangential approximation to relation (13) in our data range should be applied around the center of the actual fit domain, h/λ = 0.65, instead of h/λ = 0, yielding :

h = 0.17λ + φ l k(φ l , d) µ PT 1.9 ( 1 |U| - 1 U c ) + o(h -0.65λ ) (16)
In comparison to relation (15), the concavity of relation ( 13) thus yields the slope of h(1/|U|) to refer to PT divided by 1.9. This means that the values Pl T determined from (15) were underestimated by a factor 1.9, in agreement with figure 11-b.

Accordingly, although the range h < λ is more suitable for fitting due its largest data density, the resulting values of the thermomolecular pressure will have to be doubled to recover the relevant values.

It nevertheless remains that the thermomolecular pressures determined this way unexpectedly depend on the sample depth e. We stress that this issue is decoupled from the relevance of relation (13) and thus of the Janssen effect, and only concerns the interpretation of the parameter χ in terms of thermomolecular pressure PT . We address it in a forthcoming paper 15 by questioning the effect of the flatness of the sample plates on the particle volume fraction in their vicinity. Although confocal microscopy reveals a particle ordering at the plates (see figure 3 and ref. 14 ), the dynamics of the compacted layer build-up shows that its mean particle volume fraction is close to the random close packing value 14 . This suggests that the long-range ordered close packing only holds close to the boundaries, whereas particles in the bulk indeed arrange randomly. This ordering echoes the particle ordering observed on few layers close to planar boundaries in monodisperse granular materials [23][24][25] . It implies inhomogeneous volume fractions on the sample depth axis with implications on the permeabil-ity k from (5) and thus on the viscous stresses undergone by the particle matrix. Invoking the redistribution of stresses in the particle packing enables us to reevaluate the proportionality between χ and PT and then to reinterpret the variations of slopes near the origin in figures 7 and 8 as a sole effect of an evolution of volume fraction φ l (y) over about 8 particle diameters and at a single thermomolecular pressure value 15 .

12 | 1-14

Conclusion

For growth velocities below the critical trapping velocity of isolated particles, the solidification of suspensions spontaneously yields an heterogeneous situation in which a dense compacted layer of particles builds up ahead of the solidification front. As freezing proceeds, this dense layer is pushed by the advancing front and drifts onto the sample boundaries. Meanwhile, it reaches a steady thickness h that we have measured at various Darcy velocities U and sample depths e. At given e, h shows two regimes with an evolution linear in 1/U at large U followed, at lower U, by a much weaker raise. To understand this evolution, we have linked the steady state of h to a critical trapping state at the front and modeled it by a force balance. One of these forces, conveyed by the contact between particles, accumulates all along the particle layer the dissipative effects provided by viscous friction in the layer bulk and by solid friction at the sample plates. Expressing the force balance then yields a relationship h(U, e) whose fit to data provides a fine agreement. This leads the two regimes to be associated to a specific dominant dissipation, viscous friction at large velocities and solid friction at low velocities.

In the solid friction regime, at low velocities, both the logarithmic trend of the curve h(1/U) at each sample depths e and the proportionality of its influence length with e are analogous to the features brought about by the Janssen effect in granular materials. However, here, gravity is replaced by pressure gradient and friction forces enhance the leading force instead of being opposed to it. Accordingly, in freezing suspensions, the way friction forces are mobilized appears similar to that involved in granular material. However, instead of reducing the apparent weight of particle assemblies, the Janssen effect increases here the trapping force applied by the particle matrix on the particles that near the front. It then yields a macroscopic implication by largely reducing the layer thickness required to make particles enter the advancing solidification front at low growth velocities.

Altogether, these results clarify the formation and the main features of the dense particle layer formed ahead of solidification fronts in freezing suspensions. In particular, they identify the respective role of the two dissipative phenomena at work in the particle layer, viscous friction and solid friction, and evidence the occurrence of a Janssen effect induced by the latter. Beyond the model experiment in parallel sample plates performed here, the present insights in the behavior of dense particle layers could be useful in more natural or complex situations involving largely disperse suspensions with large particles playing the role of boundaries for the thinnest. Finally, on a more general viewpoint, the evidence of a Janssen effect in this study provides a useful bridge between the issues of freezing suspensions and those of granular materials.

Fig. 1 (

 1 Fig. 1 (Color online) Sketch of the experimental setup. (a) A thin sample is pushed in between heaters and coolers by a micro-stepper motor coupled to a linear track. Real-time non-invasive visualization of the solidification interface is provided by an optical access through which an objective allows the formation of a magnified image on a camera. (b) Samples containing the suspension are made of two glass plates separated by calibrated spacers which enable the sample depth e to be varied. Suspensions are filled in by capillarity into a large domain, 100 mm long and 45 mm wide, that is sealed before solidification. In a freezing suspension, a compacted particle layer of thickness h stands ahead of the solidification front. The axes x, y and z refer to the directions of the solidification front, the sample depth and the thermal gradient respectively.

Fig. 2

 2 Fig. 2 Images of the a compacted layer by reflection (left) and transmission (right) : φ 0 = 20% , e = 16µm ; image width 640 µm.They involve a top zone (the liquid phase), a bottom zone (the frozen phase) and an intermediate zone (the compacted particle layer). Bright zones refer to large φ by reflection (left) and to low φ by transmission (right). The change of optical methods thus reverses the zone intensity : the top and bottom zones remain grey but the intermediate zone turns from bright to dark. We notice that, regardless of the optical method, the compacted layer displays the same extent. This indicates that it keeps the same thickness from one plate to the other.

Fig. 3 (

 3 Fig.3(Color online) Image of the compacted layer of particles at a sample plate by confocal microscopy . The particle diameter is 1 µm. The image shows close packed particles filling a plane adjacent to the plate. A particle ordering spanning at least few particles is noticeable all over the image. It even extends to a large-scale hexagonal order at its bottom.

Fig. 4

 4 Fig.4Compact layer of particles formed ahead of the solidification front : φ 0 = 20% , e = 16µm, V = 1 µm•s -1 . The image is obtained by reflection. Its top corresponds to the liquid phase and its bottom to the frozen phase. Both appear grey as their particle volume fraction, φ 0 , is low. In between them, the bright zone reveals a noticeable increase of particle density which corresponds to the compacted particle layer. It involves bright spots which correspond to crystallized patches. The steady state thickness reached by the particle layer beyond its build-up is labelled h.

  Figure 6-a shows a decrease of h with the Darcy velocity U. However, a finer analysis of this evolution in figures 6-b and 6-c reveals two different trends : First, the graph of the product h|U| with respect to |U| shows in figure 6-b that, above about |U| = 2µm.s -1 , h is inversely proportional to |U|. This corresponds in figure 6-c to the linear trend displayed at low 1/|U| below 0.5 s.µm -1 . In contrast, at small velocities, figure 6-b shows that, below about |U| = 2µm.s -1 , h decreases quicker than |U| -1 , as |U| -2 at least. This is confirmed by the concavity of the graph of figure 6-c above 1/|U| ≈ 1µm.s -1 , which reveals a much weaker growth of h with 1/|U| at very low velocity.

Fig. 6 (

 6 Fig. 6 (Color online) Evolution of the layer thickness h with the Darcy velocity |U| at sample depth e = 125µm. (a) h(|U|) : h shows a continuous decrease with |U| ; (b) The plot of h|U| as a function of |U| shows that, for |U| larger than about 2µm.s -1 , h is inversely proportional to |U|. The line provides the data average of h|U| for |U| > 2µm.s -1 . ; (c) The plot of h as a function of the inverse Darcy velocity 1/|U| shows a linear trend up to about 0.5 s.µm -1 , followed by a concave trend. The curve corresponds to a fit of data to relation (13).

Fig. 7

 7 Fig. 7 Evolution of the thickness h of the compacted layer with the inverse of the Darcy velocity |U| for various sample depths e. The full lines correspond to the best fits of data to the relation (13) with the critical velocity U c fixed at U c = 15µm.s -1 . The dashed line shows the Janssen's length λ = 10 e that is identified in figure 10-a from the fit results. This length provides the transition between a linear trend with dominant viscous dissipation and a concave trend with a dominant solid friction. Sample depth : (a) e = 125µm. (b) e = 100µm. (c) e = 75µm. (d) e = 50µm. (e) e = 30µm. (f) e = 16µm. The particle diameter is d = 3µm. Fluctuations of h and U are indicated as error bars, most of which being smaller than the data symbols. They have not been taken into account for fitting data.

1- 14 | 7

 147

Fig. 8 (

 8 Fig. 8 (Color online) Summary of the fits of data to the relation (13) with U c fixed at 15µm.s -1 . At a given At given Darcy velocity, the compacted layer thickness h grows with the sample depth e. All show a similar type of concavity, with parameters dependent on e.

Fig. 9 (

 9 Fig.9(Color online) Sketch of the forces acting on a particle entering the frozen phase: a thermomolecular repelling force F T exerted by the solidification front (black arrow); a lubrication force F L induced by the liquid flowing in the thin premelted film in between the particle and the front (red arrow); a force F p exerted by the particle matrix on the particle, as a result of the viscous friction and the pressure drop induced by the liquid flow through the whole layer (green dotted arrow). The liquid flow is represented by blue curved arrows. The direction e z , normal to the solidification front, is parallel to the thermal gradient. Particles have the same diameter and are in contact with each other. However, their random three-dimensional distribution results in different apparent cross-sections and in an apparent lack of contacts between them.

Fig. 10 Fig. 11 (

 1011 Fig. 10 Fit parameters λ and PT at various sample depths e. They have been obtained from the non-linear evolution of the graphs h(1/|U|) of figure 7. (a) Janssen's length λ at various sample depth e. The full line shows the best linear fit of data : λ = 10.0 e. (b) Thermomolecular pressure PT at various sample depth e.

| 1-14

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.

Conflicts of interest

There are no conflicts of interest to declare.


Marie-Pierre Comard

Rachel Calvet

John A Dodds

Henri Balard

C oupling of inverse gas chromatography at infinite dilution (IGC-ID) with a controlled modification of a solid surface a a , a b *

Keywords: Inverse gas chromatography, Steric hindrance, Surface properties, Talc

published or not. The documents may come

Coupling of inverse gas chromatography at infinite dilution (IGC-ID) with a controlled modification of a solid surface

. Introduction

access to apparent thermodynamic data because probes mainly interact with the high-energy adsorp-Inverse gas chromatography at infinite dilution tion sites [2]. We try to take advantage of this (IGC-ID) is a useful method for exploring the limitation by coupling IGC-ID with a progressive thermodynamic surface properties of a homogeneous absorption impregnation of a suitable polymer of the solid at the molecular scale. The complexity of the sites having the highest energy. A polymer is used to interactions between probes and the surface of a progressively saturate the more interactive sites by solid depends on the amount of probe injected.

preferential adsorption, thus causing the probes to Further theoretical details can be found elsewhere access the less energetic sites remaining free. So, by [1]. In IGC-ID, very small amounts of probes are progressively increasing the amount of polymer injected in such a way that the interactions between added to the solid, the probes will access different probe molecules can be considered to be negligible energy sites on the surface of the solid. We have and only the interactions between the surface of the applied this method to talc which is a phyllosilicate solid and an isolated probe molecule are important.

with a heterogeneous lamellar structure, having two However, a limitation of IGC-ID appears in the case kinds of surface: of a heterogeneous surface where IGC-ID only gives

• weak energetic basal surfaces with basic Si-O-Si groups, and • more energetic lateral surfaces with acidic Si-OH *Corresponding author. Tel.: 133-5-6349-3064; fax: 133-5groups and residual magnesium cations.

6349-3025.

E-mail address: calvet@enstimac.fr (R. Calvet).

The intention is to use a basic polymer to interact preferentially with the lateral acidic surfaces of the beaker and mixed with a given volume of the talc. When sufficient polymer is adsorbed on the polymer solution until the mixture became homogesurface to saturate the lateral surfaces, the probes neous (|5 min). The polymer solutions cover ranges, will then access the less energetic sites on the basal respectively, from 0.01 to 30, 0.01 to 80, and 0.07 to surfaces. Therefore the surface energy of the basal 32 mg for the PEG, PS and PEVE per gram of talc. surfaces, inaccessible to the probe with conventional After evaporation of the solvent in a vacuum oven, IGC-ID, are then determined.

the column was filled with the polymer impregnated Impregnation of the surface of a solid with a talc. These columns were maintained under a nitropolymer also allows the characterisation of solid-gen atmosphere to prevent polymer oxidation. polymer interactions. Here we study effects induced by three types of basic polymers: linear, branched and cyclic which have different degrees of surface steric hindrance, i.e. polyethyleneglycol 20 000 3 . Results (PEG), polyethylvinylether 3800 (PEVE), and polystyrene 230 000 (PS).

In the first part of the study, the evolution of both

Stainless steel chromatographic columns with an

The impregnation ratio has no real meaning from a inside diameter of 5.3 mm and a length of 10 cm, molecular point of view, and the coverage ratios of 2 closed at the output end with a stainless steel frit of the monomer unit expressed in MU / nm are calcu-0.5 mm porosity, filled with |2 g of talc were used in lated using the following formula: the tests.

Two Hewlett-Packard 6890 gas chromatographs were used, each fitted with two flame ionisation detectors (FID). The carrier gas was helium with a flow of |30 ml / min. The injector and detector were heated to 150 8C. All the columns were conditioned at 120 8C overnight and the analyses made at 100 8C. The second part of this paper examines the and the amount of polymer deposited on the surface influence of the steric hindrance of different polyof the solid corresponds to a mono-layer of monomer mers on the evolution of the thermodynamic surface units. It may be assumed that the whole surface of properties of the impregnated talc. In particular the the solid is covered with a mono-layer of PEG effect of PEG, which has a linear structure, is monomer units. compared to the effects shown by two other basic

In domain D, multi-layers of PEG monomer units polymers: a branched polymer (PEVE) and a cyclic d 2 are formed. g becomes equal to 25 mJ / m (value s polymer (PS) (Fig. 2). already obtained on silica impregnated with a high ratio of PEG) which corresponds to the interactions of the probes with a thick layer of PEG. I increases m and exceeds the value of 1, which reflects the 4 . Discussion absorption of the probe in a thick film of polymer. The impregnation by PEG of the surface of the 4 .1. Impregnation of the surface of a reference talc seems to be sensitive not only to the surface talc with PEG morphology, but also to the chemical functions. PEG is able to insert partially into the slot like sites in the The impregnation of the surface of a reference talc lateral surfaces or in the structure defects, such as with an increasing amount of PEG allows following grain joints between the talc crystallites. In addition, s m impregnation of talc may be compared with the The differences observed between the polymers influence of two other basic polymers: a branched are attributed to their steric hindrance even though polymer (PEVE) and a cyclic polymer (PS).

the molecular masses of the three polymers used are very different. In a previous study [6], polymer 4 .2. Impregnation of the surface of a reference impregnations were made with PEG with a range of talc with PEVE and PS molecular masses (2000, 20 000, 35 000, 100 000).

The molecular mass of the polymer was found to d 2

The plateau value of g at 160 mJ / m , determined have no influence on the evolution of the surface s with low impregnation ratios of PEG, is not observed properties of talc as a function of the impregnation with PEVE. This is also the case for the plateau at ratio. In addition, a pyrogenic silica was impregnated 2 d 45 mJ / m , attributed to the coverage of the basal with PEG, PEVE and PS and the g curves were s surfaces, which does not occur with PEVE. Despite found to present the same form with one break point the fact that the PEVE is a basic polymer like PEG, on each curve. It may be concluded that the molecuthese two polymers differ notably by their stereo-lar masses of these different polymers do not inchemistry. PEG has a linear structure and PEVE has fluence the results. some branches from the linear chain and asymmetric carbons. Because of its structure and its steric hindrance, PEVE can not fit a rough surface as easily R eferences as can linear PEG. Balard, Macromol. Symp. 169 (2001) 19.

[5] M.E. Schrader, S. Yariv, J. Colloid Interface Sci. 136 (1990) chemistry of the macromolecular chain and the low 85.

potential interaction with the lateral surfaces can the polymer first covers the basal surfaces, which are

  d the dispersive component of the surface energy, g , s and the nanomorphological index, I [3], were m 2 . Experimental studied as a function of the amount of PEG added to d the talc (Fig. 1). The g is a measurement of the s The talc used in the experiments was provided by dispersive London forces between the surface of the the Luzenac Europe Company and had an average solid and linear alkanes injected as probes. The I m particle size of 10 mm, as measured by a Mi-reflects size exclusion effects, or a measurement of cromeritics Sedigraph 5100. The PEG was supplied the surface of the solid that can be reached by a by Fluka and the two other polymers, PEVE and PS, branched alkane. It is determined by the injection of by Aldrich. High purity dichloromethane obtained a branched or cyclic probes. In the present work, the from Fisher Scientific International Company was I was determined using cyclooctane as injected m used for the preparation of the polymer solutions. probe.

  Fig. 1. Evolution of the dispersive component of the surface d

  d the evolution of g and I . The results, which have this basic polymer interacts preferentially with the s m acidic sites on the lateral surfaces, causing the probes not visible in IGC-ID, rather than the lateral surfaces d to access the less energetic free sites on basal which alone contribute to the high value of g . After s surfaces of the talc, inaccessible by conventional 20 MU, the amount of PS is so great that it almost d IGC-ID. covers all the surface of micro-crystallites and g s The influence of the chemical structure of the decreases to reach the energy of a talc surface d polymer on the evolution of g and I with the entirely covered by PS.

d

  The curve of g as a function of the impregnation [1] M.N. Belgacem, A. Gandini, Surf. Sci. Ser. 80 (1999) 41. s [2] H. Balard, E. Brendle, E. Papirer, in: K. Mittal (Ed.), ratio of PS presents a completely different shape Acid-Base Interactions: Relevance to Adhesion Science and from the two others. This polymer is unable to hide Technology, VSP, Utrecht, The Netherlands, 2000, p. 316. the high-energy sites until a high degree of ad-[3] H. Balard, E. Papirer, Progr. Org. Coating 22 (1993) 1. d sorption coverage is achieved. The high value of g [4] M.P. Comard, R. Calvet, S. Del-Confetto, J.A. Dodds, H. s remains constant until 20 MU of PS. The stereo-

  . Comard, R. Calvet, J.A. Dodds, H. Balard, Recents explain the behaviour of the PS. It is probable that `´´Ṕrogres Genie Procedes 15 (2001) 197.


M.-P Comard

R Calvet
email: calvet@enstimac.fr

H Balard

J A Dodds

Influence of the geological history, particle size and carbonate content on the surface properties of talc as determined by inverse gas chromatography at infinite dilution

Keywords: Inverse gas chromatography, Talc, Carbonate ratio, Surface properties

published or not. The documents may come

Influence of the geological history, particle size and carbonate content on the surface properties of talc as determined by inverse gas chromatography at infinite dilution

Mp Comard, Rachel Calvet, H Balard, John A.

Introduction

Inverse gas chromatography at infinite dilution, i.e. IGC-ID, is a convenient method for the examination of the surface properties of a finely divided solid [1][2][START_REF] Papirer | Adsorption on New and Modified Inorganic Sorbents[END_REF]. It consists in injecting clearly identified molecules, called probes, into a column made up of the solid under test. The injected probes interact with the surface of the solid as they are moved through the column by a carrier gas. The complexity of these interactions depends on the amount of probe injected. Two IGC techniques may be distinguished: infinite dilution or finite solute concentration. This study has been carried out at infinite dilution (IGC-ID) where very small amounts of probes are injected in such a way that the interactions between the probe molecules themselves can be considered to be negligible and only the interactions between an isolated probe and the surface of the solid exist. By IGC-ID, one can access the apparent dispersive component of the free surface energy γ d s (injection of linear alkanes). However, IGC-ID gives only information on absolute thermodynamic parameters on perfectly homogeneous surface. On a heterogeneous surface, like talc, the sites of higher energy contribute mainly to the chromatography process. In the present work, we try to take advantage of this limitation by coupling the IGC-ID with a controlled modification of talc surface with PEG 20M. The idea is to progressively kill the more interactive sites by preferential adsorption so that the progressively increasing surface coverage ratio by PEG macromolecules forces probes to interact with the less and less energetic free sites.

Talc (Mg 3 Si 4 O 10 (OH) 2 ) is a phyllosilicate with a lamellar structure. It consists of octahedral brucite sheets sandwiched between two tetrahedral silica sheets, to form talc layers superimposed indefinitely. The crystal is composed of two kinds of surfaces:

• Weakly energetic basal surfaces with basic and hydrophobic Si-O-Si groups.

• More energetic lateral surfaces with acid and hydrophilic Si-OH groups and residual magnesium cations.

The structure of talc directs the choice of the macromolecules used for the impregnation. It can be expected that a basic polymer such as polyethyleneglycol 20M (PEG 20M) will interact preferentially with the acid lateral surfaces. Then after saturating the acid sites with PEG, one can access the less energetic sites of the basal surfaces.

The industrial use of talc depends on the surface properties and on the ratio of basal to lateral surfaces. Different techniques allow access to this data such as immersion calorimetry [START_REF] Michot | [END_REF], gas adsorption (nitrogen, argon, water vapor . . . ) [5,6] and the measurement of contact angles [7]. IGC-ID is an interesting supplement to these techniques because it gives a vision of the surface of the solid at the scale of the injected probe.

The aim of the study is to observe the variation of the surface properties of talc as a function of several parameters that play a very important role in many industrial applications, that says, the particle morphology of talc (microcrystalline, lamellar or very lamellar structure), the carbonate content and the particle size.

For this purpose, the change of γ d s has been studied according to the polymer coverage ratio. The dispersive component of the surface energy γ d s is obtained by analyzing the retention times of a series of linear alkanes injected as probes and gives a measurement of the dispersive London forces. Further theoretical details can be found elsewhere [8].

Materials and methods

Materials

The talcs used in this investigation were supplied by the Luzenac Company and their characteristics are given in Table 1. The mean particle size, d 50 , is measured with a Micromeritics Sedigraph 5100. It is observed that for series M talcs, no correlation exist between the d 50 and the specific surface area because their porosity. The BET specific 2. The PEG was obtained from Fluka, and was purified by dissolution in chloroform and precipitation in heptane.

Experimental IGC conditions

Two Hewlett Packard 6890 gas chromatographs each fitted with two flame ionisation detectors (FID) are used. The carrier gas is helium. The injector and detector are heated to 150 • C. All the columns were conditioned at 120 • C overnight and the analyses made at 100 • C. Net retention times are calculated by subtracting the retention time of a non retained molecule, methane, from the retention time of the probe.

Preparation of the samples: controlled impregnation of talc

A solution of polymer with a known concentration is prepared in dichloromethane. A given volume of this solution is diluted in a beaker with 5 ml of the same solvent. The mass of talc required to fill the column (around 2 g), is put in the same beaker and stirred with the polymer solution. The mixture is then put in a vacuum oven to evaporate the solvent and deposit the polymer on the surface of the talc.

The chromatography column is afterwards filled with the impregnated talc.

The EG monomer unit coverage ratios expressed in MU/nm 2 were calculated using the following formula:

n MU = 6023τ M MU S
where τ (%), coverage ratio expressed as a percentage; n MU , PEG monomer unit number per nm 2 of talc; M MU , molecular weight of the EG monomer; unit, 44 g/MU; S, specific area of powder in m 2 /g.

Results

Preliminary study

As a first step, a reference talc denoted T00 was examined. The changes in the dispersive component of the surface energy (γ d s ) with increasing PEG coverage ratios has been described and commented in a previous paper [9]. To summarize, four domains can be distinguished: This plot allows an assessment of the ratio of lateral to basal surfaces of the talc (estimated to 7.5/80 for the T00) [START_REF] Comard | Caractérisation de surfaces solides hétérogènes: etude du couplage de la chromatographie gazeuse inverse à dilution infinie avec la modification de surface par imprégnation contrôlée par des macromolécules[END_REF].

Influence of the particle morphology

Talc samples of different geographical origins exhibit different morphologies. Changes due to the geological history of the talc are examined in this study on the basis of three different talcs coming from three different countries: France (T00), Italy (L) and Spain (M2). The first has a "normal" lamellar structure, the second a very pronounced lamellar structure and the third is a microcrystalline talc. The results obtained are presented in Fig. 1.

The main differences observed between the three talcs are the height of the first plateau at low polymer coverage ratios. The microcrystalline talc (M2) presents the highest γ d s on this plateau. It is difficult to interpret this result because these talcs have not only different lamellarities but also different chemical structures (different chlorite ratios for example). So, it's difficult to attribute the γ d s variations to only one parameter. It is possible that the microcrystalline talc has many structural defects into which the linear alkanes will insert giving high retention times. Thus these defects could be responsible for the high value of γ d s on the first plateau. The higher the lamellarity of the talc, the lower is the number of defects and therefore the lower is the value of γ d s at low polymer coverage ratios. The grinding processes are also different between the three talcs, T00 is obtained by pendular mill, L by air jet mill and M2 by pendular mill and hammer mill. So, an other observation is that the low γ d s value (130-140 mJ/m 2 ) at the low polymer ratios is obtained with the talc (L) obtained by air jet mill.

Influence of the particle size for microcrystalline talcs

In order to study the influence of the particle size, a microcrystalline talc was ground to obtain three samples with different particle sizes: 1.6 m (M3), 4.5 m (M2), and 11 m (M1). The results obtained are presented in Fig. 2. These three talcs are obtained with different grinding processes, M3 by pendular mill and air jet mill, M2 by pendular mill and hammer mill and M1 by pendular mill. With air jet mill, we observe a low γ d s value on the first plateau at the low polymer ratios, 160 mJ/m 2 for M3 compared to 190 mJ/m 2 for M2 and M1. At the polymer ratios higher to 1 MU/nm 2 , the change in γ d s with the coverage ratio is sensitive to the mean particle diameter. When the particle size decreases, the decrease of the γ d s between 2 and 10 MU/nm 2 is shifted towards the low coverage ratio. Bigger particles need higher coverage ratios of PEG to cover the slot and wedge-like sites.

Influence of the particle size for lamellar talcs

The influence of the particle size was also studied on three lamellar talcs (Fig. 3), L1: 3.85 m, L2: 2.75 m and L3: 0.8 m. These three talcs are obtained with air jet mill. They remain truly lamellar after fine grinding, we have the evidence by SEM pictures (Fig. 4). They all present a low γ d s value (130 mJ/m 2 ) on the first plateau. The air jet mill process seems to be responsible of this behavior, as we saw it on talcs L and M3, also obtained by air jet mill. If we compared the decrease of the γ d s , the talcs L1 and L2 present the same behavior as microcrystalline talcs. A shift of the curve towards the low coverage ratio is observed between 1 and 3 MU/nm 2 when the particle size decreases. This occurs in a smaller range of PEG coverage than for the microcrystalline talcs, which can be explained by:

• Firstly, noting that the range of particle size for the lamellar talcs is smaller: from 1 to 4 m than that for the microcrystalline talcs from 1 to 11 m. • Secondly, the structural defects are fewer in lamellar talcs than in a microcrystalline talcs and because the grinding of more macroscopic crystal leads to the formation of few defaults.

Influence of the carbonate content

As impurities are always present in talc, another parameter examined is the carbonate content. Fig. 5 allows comparison of the evolution of the γ d s with the polymer coverage ratios for different samples with three different carbonate contents (M5: 1.5%, M4: 4.5%, and M2: 10%). These three talcs are obtained by the same grinding process: pendular mill and hammer mill. A difference can be seen in the higher part of the first plateau and in the slope of the first decrease. At the present stage of our study, it is difficult to give a final clear-cut interpretation of these results, and to say if the carbonate content has or has not a real influence on the γ d s .

Conclusion

In conclusion, coupling IGC-ID and controlled surface modification through polymer adsorption is a powerful method for obtaining information on the surface heterogeneity of finely divided solids. This paper shows that the surface properties of talc are influenced by particle size and grinding process rather than the geological origin. All the talcs obtained by air jet mill have the same behavior at the low polymer ratios. It also seems that the carbonate content has no real influence on these properties but further investigations are necessary to complete the results and clarify the interpretations.

•

  A first plateau (γ d s = 160-170 mJ/m 2 ) corresponding to the coverage of the slot like high energetic sites present on the lateral surfaces of the talc. • A rapid decrease of γ d s attributed to the saturation of the wedge-like sites situated between the talc crystallites. • A second plateau (γ d s = 45 mJ/m 2 ) equal to that measured by contact angle measurements corresponding to the progressive coverage of the basal surfaces. • A final plateau (γ d s = 28 mJ/m 2 ) attributed to a thick layer of PEG.

Fig. 1 .

 1 Fig. 1. Influence of the particle morphology on the variation of the dispersive component of the surface energy, γ d s , of three impregnated talc samples with the PEG 20M coverage ratio.

Fig. 2 .

 2 Fig. 2. Influence of the particle size on the variation of the dispersive component of the surface energy, γ d s , of impregnated microcrystalline talc samples as a function of the PEG 20M coverage ratio for three samples of talc having different mean particle sizes.

Fig. 3 .

 3 Fig. 3. Influence of the particle size on the variation of the dispersive component of the surface energy, γ d s , of impregnated lamellar talc samples as a function of the PEG 20M coverage ratio for three samples of talc having different particle mean diameters.

Fig. 4 .Fig. 5 .

 45 Fig. 4. SEM pictures of lamellar talcs (L1, L2 or L3).

Table 1

 1 Characteristics of the talc samples provided by Luzenac Europe Company

	Sample a	Geographical origin	Talc-chlorite (%)	Lamellarity	S BET (m 2 /g)	d 50 (m)
	T00	France	92-7	Lamellar	3	10.6
	M1	Spain	86.5-<0.5	Microcrystalline	8	11
	M2	Spain (10% of carbonates)	86.5-<0.5	Microcrystalline	11.5	4.5
	M3	Spain	86.5-<0.5	Microcrystalline	13	1.6
	M4	Spain (4.5% of carbonates)	96-0	Microcrystalline	10	4.5
	M5	Spain (1.5% of carbonates)	97-1.5	Microcrystalline	10	4.5
	L	Italy	95-2	Very lamellar	14.5	1.2
	L3	Italy	95-2	Very lamellar	20.5	0.8
	L2	Italy	95-2	Very lamellar	9.5	2.75
	L1	Italy	95-2	Very lamellar	7.5	3.85

The mean particle size, d 50 , is measured with a Micromeritics Sedigraph 5100. The BET specific surfaces, S BET , are obtained by argon adsorption. a L, lamellar; M, microcrystalline.

Table 2

 2 Milling processes of the talc samples

	Samples	Milling processes
	T00	Pendular mill
	M1	Pendular mill
	M2 M3 M4 M5 L	Pendular mill + hammer mill Pendular mill + air jet mill Pendular mill + hammer mill Pendular mill + hammer mill Air jet mill
	L3	Dry mill
	L2	Dry mill
	L1	Dry mill
	surface, S BET , is obtained by argon adsorption with a Mi-
	cromeritic ASAP 2010. The milling processes of these talcs
	are given in Table