A Fabbri

O Coussy

T Fen-Chong

P J M Monteiro

ARE DEICING SALTS NECESSARY TO PROMOTE SCALING IN CONCRETE?

Keywords: Concrete, cryosuction, durability, frost, poromechanics, porous media, thermodynamics, spalling

The main purpose of the present study is to investigate the role of the material parameters such as permeability, thermal diffusivity, and pore size distribution on the mechanical behavior of cementitious structures submitted to frost action, such as surface scaling. An experimental device, in which a cement paste specimen is exposed to freezing-thawing cycles under a thermal gradient, has been developed. The experimental results show that under high thermal gradient (up to 1.5 o C/mm), skin damage can occur without a saline layer in contact with the frozen surface. This can be explained and quantified in the framework of poromechanics. The model is based on the coupling between liquid -ice crystal thermodynamic equilibrium, liquid water transport, thermal conduction and elastic properties of the different phases that form the porous material. It eventually predicts that a less permeable sample is more susceptible to be damaged by surface defacement, which explains the observed experimental result.

INTRODUCTION

When concrete structures are exposed to freezing and thawing cycles, two types of damage can occur: internal frost damage and surface scaling [START_REF] Pigeon | Microstructure et résistance au gel des ciments et bétons[END_REF]. The former takes place and generates micro-damage within the whole medium. Theoretical study of concrete 1 Université Paris-Est, Institut Navier, LMSGC 2 allée Kepler, 77420 Champs-sur-Marne, France. 2 new address: Laboratoire de Géologie, UMR 8538 (Ecole Normale Supérieure, CNRS), Paris, France 3 Department of Civil and Environmental Engineering, University of California at Berkeley, CA, USA. 1 behavior at low temperature at the material scale started with the work of Powers in 1949 [START_REF] Powers | The air requirement of frost-resistant concrete[END_REF][START_REF] Powers | Theory of volume changes in hardened portland-cement paste during freezing[END_REF]. In this seminal work, he attributed the expansion of concrete to the hydraulic pressure originated by the expulsion of liquid water from the freezing pores due to the liquid-to-ice volumetric expansion. Early attempts at evaluating frost performance of concrete were based on the comparison between the tensile stress induced by the hydraulic pore pressure and the tensile strength of the material. However, the picture was not so simple as demonstrated by the fact that expansion is observed in cement paste saturated with benzene, whose density increases with solidification [START_REF] Beaudoin | The mechanism of frost damage in hardened cement paste[END_REF]. Nowadays, it has been recognized that the mechanical response of a saturated or partially saturated porous material at freezing temperatures is the result of the volumetric increase of water during its solidification, the transport of unfrozen liquid water through the porous network and the thermo-mechanical properties of all the phases of the composite material. Physico-mechanics based models have been developed to capture and quantify all these phenomena both at the pore scale [START_REF] Vignes | A model for the freezing of water in a dispersed medium[END_REF][START_REF] Coussy | Crystallization, pore relaxation and microcryosuction in cohesive porous materials[END_REF] and at the material scale [START_REF] Coussy | Poromechanics of freezing materials[END_REF][START_REF] Zuber | Predicting the volume instability of hydrated cement systems upon freezing using poro-mechanics and local phase equilibra[END_REF][START_REF] Bazant | Mathematical-model for freezethaw durability of concrete[END_REF]).

On the other hand, scaling is the result of a local flaking or peeling of the concrete surface. Generally, it starts as localized small patches which later on often merge and extend to larger areas. Moderate and severe scaling expose the coarse aggregate from the concrete surface and may involve losses up to 3 to 10 mm of the surface, which is harmful as it reduces the cover of the steel reinforcement. The discovery of scaling in the early 1950s prompted a series of experimental studies [START_REF] Verbeck | Studies of salt scaling of concrete[END_REF][START_REF] Sellevold | Frost/salt-testing of concrete: Effect of test parameters and concrete moisture history[END_REF][START_REF] Pigeon | Frost resistant concrete[END_REF][START_REF] Valenza | Mechanisms of salt scaling[END_REF][START_REF] Penttala | Surface and internal deterioration due to saline and non-saline freezethaw loads[END_REF]). The results indicate that scaling is largely enhanced by the presence of deicing salts and that any surface desaturation prevents the specimen from scaling. Despite the numerous studies, the mechanisms responsible for surface damage, and especially the action of deicing salts, have not been clearly identified.

Indeed, while according to many studies scaling only happens when a saline layer is in contact with the surface submitted to frost action [START_REF] Sellevold | Frost/salt-testing of concrete: Effect of test parameters and concrete moisture history[END_REF][START_REF] Valenza | Mechanisms of salt scaling[END_REF], in others [START_REF] Verbeck | Studies of salt scaling of concrete[END_REF][START_REF] Penttala | Surface and internal deterioration due to saline and non-saline freezethaw loads[END_REF], scaling can happen even when the saline surface is not present, albeit not often. This work investigates the role of the material parameters such as permeability, thermal diffusivity, pore radii distribution and connectivity (which determines the ice content as a function of temperature at the scale of the representative volume element) on the mechanical behavior of cementitious materials submitted to frost action. Let us emphasize that its goal is not to predict in-situ scaling but to understand how these material parameters may influence the mechanical response of a cementitious structure submitted to frost action and to determine if the action of deicing salts is the only phenomenon that produces surface deterioration.

It is first shown experimentally that scaling can occur without the presence of deicing salts. This experimental result is then explained by a poromechanical-based approach at a scale where the importance of the permeability and the amount of ice formed on frost durability is indubitable.

EXPERIMENTAL EVIDENCE OF SCALING WITHOUT SALTS

Samples of hardened cement paste, with 0.4 water-cement ratio (W/C) by mass, were prepared in a 5-liter mortar mixer, and cast in 150 mm high cubic moulds. Ordinary Portland Cement similar to ASTM Type I and distilled water were used. One day after casting, the specimens were removed from their mould and stored in moist condition (relative humidity = 95% ± 5%) for 6 months, when they were cored and cut into 20mm-thick slices with 40mm diameter and remained in water until tested. Some specimens (index d) were dried in an oven at 55 o C then saturated with degassed distilled water at 3 kPa air pressure before tested.

The tested sample was inserted between two hollow pistons each filled with a fluid from a cryostat. Their temperature was controlled by a PT100 sensor. The piston in contact with the bottom side of the specimen was held at a constant temperature of 10 o C. The other one was subjected to 56 cycles ranging from 0. 1, the temperature rate was 10 o C/hour. Freezing was stopped at -20 o C and the sample was thawed to 0.1 o C. At the end of freezing, the temperature was held constant for one hour and at the end of thawing the temperature was held constant for two hours. Let us recall that the purpose of this paper is not to simulate and/or to predict in-situ frost scaling, but to investigate if scaling can occur without deicing salt. In this context, in order to enhance the effect of structural gradients, a large thermal gradient through the sample was intentionally imposed.

1 o C ± 0.1 o C to -20 o C ±0.1 o C. As sketched in figure
The thermal insulation of the lateral surfaces of the specimen was achieved by an expanded polystyrene ring. In order to avoid surface desaturation during freezing-thawing cycles, each specimen was wrapped by a moisture resistant Parafilm sheet. Thus, the surface submitted to frost action is not in contact with a frost layer (i.e. a water or brine layer).

A picture of the experimental device is reported in figure 2.

After each fourteen cycles, the specimen was weighed in order to verify that no water supply nor evaporation have occurred during the test. Then, the Parafilm sheet was removed and scales were collected, dried at 55 o C during 4 days and weighed.

Table 1 shows the evolution of the mass of scales collected per unit of surface. As it can be seen, no scaling occurs on previously dried P4-E-3d and P4-E-4d samples. Indeed, these two specimens were totally disintegrated by internal cracking during the first fourteen cycles (see figure 3). On the other hand, a significant scaling occurred on P4-E-1 and P4-E-2 samples (figure 4). This result clearly indicates that scaling can occur without a frozen brine layer in contact with the surface submitted to frost action. The lack of scaling without free liquid on the surface commonly observed [START_REF] Verbeck | Studies of salt scaling of concrete[END_REF]) may be due to a surface desaturation during the test, which is not possible in this study thanks to the presence of the parafilm sheet.

In all cases, the solicitation was the same and the differences only relied on the cure of the specimen (pre-dried or not). The next step is then to identify the material characteristics which are significantly changed by a drying-resaturation process and to investigate their actions on frost-thaw behavior.

CHARACTERIZATION OF MATERIAL PROPERTIES

It is established that drying strongly affects the intrinsic permeability (κ 0 ) of concrete [START_REF] Hearn | Self-sealing property of concrete -experimental evidence[END_REF] and the amount of ice formed during freezing [START_REF] Fabbri | Dielectric capacity, liquid water content, and pore structure of thawing-freezing materials[END_REF][START_REF] Kaufmann | Experimental identification of damage mechanisms in cementitious porous materials on phase transition of pore solution under frost deicing salt attack[END_REF]). The permeability of the pre-dried sample is evaluated through the Katz-Thompson-Garboczi relation based on Mercury Injection Experiments [START_REF] Katz | Quantitative prediction of permeability in porous rock[END_REF][START_REF] Garboczi | Modelling the microstructure and transport properties of concrete[END_REF] (see appendix 1). This is not possible for virgin specimens because MIP measurements can only be made on dried samples. In consequence, their permeability is estimated from the predried one according to the study performed by [START_REF] Hearn | Self-sealing property of concrete -experimental evidence[END_REF], where 2 orders of magnitude are observed between the permeability of a virgin and a predried sample. This can be explained by the formation of micro-cracks produced by oven-drying that noticeably increases connectivity and size of capillary pores and thus makes the liquid transport easier [START_REF] Shafiq | Effects of initial curing condition on the fluid transport properties in OPC and fly ash bended cement concrete[END_REF]. Results lead to κ 0 = 4.3 × 10 -20 m 2 for pre-dried samples and κ 0 = 4.3 × 10 -22 m 2 for virgin ones. The porosity (φ 0 ) was determined through the mass loss between the saturated (m sat ) and the oven-dried at 55 o C (m dry ) states: φ 0 = (m satm dry )/ρ 0 l /V where ρ 0 l is the mass density of liquid water and V is the sample volume. This leads to φ 0 = 0.29 for pre-dried samples and φ 0 = 0.28 for virgin ones. The dependency of the volumetric ratio of unfrozen water (S l ) as a function of temperature (θ) was determined by a capacitive sensor apparatus developed at the Navier Institute. The full description of the experimental set-up and its calibration are reported in [START_REF] Fen-Chong | Determination of liquid water content and dielectric constant in porous media by the capacitive method[END_REF]) and [START_REF] Fabbri | Dielectric capacity, liquid water content, and pore structure of thawing-freezing materials[END_REF]. As shown in figures on heating, where the slope changes significantly. However, the amount of ice formed in pre-dried hardened cement pastes appears to be significantly higher than in virgin ones.

Assuming that the in-pore ice formation results from the propagation of ice crystals through the connected porous network [START_REF] Scherer | Freezing gels[END_REF], the mechanical (Young-Laplace's law) and chemical equilibria between ice crystal and surrounding water provide a relation between the water to ice transition temperature θ l→c and the ice crystal mean curvature c * [START_REF] Brun | A new method for the simultaneous determination of the size and the shape of pores : The thermoporometry[END_REF]:

θ l→c ≈ - γc * Σ f (1)
where γ is the liquid-ice surface tension while Σ f stands for the entropy of fusion.

As c * increases with decreasing pore radius in which the water is confined, pores with larger radius will freeze at higher temperature. From studies made with low temperature calorimetry [START_REF] Sellevold | Low temperature calorimetry as a pore structure probe[END_REF] and nuclear magnetic resonance [START_REF] Jehng | Pore structure of hydrating cement paste by magnetic resonance relaxation analysing and freezing[END_REF], a maximum of three distinct types of pores, associated with the three freezing peaks, was observed in a freezing test: large capillaries (r c > 50 nm,

θ l→c >-1 o C), small capillaries (r c = 2 nm to 5 nm, θ l→c -25 o C
) and open gel pores (r c < 1 nm, θ l→c <-40 o C). In our experiment, the lowest temperature is -30 o C, so only the water confined in large and small capillaries can freeze. This is why only two freezing peaks are presented in the figures 5 and 6. As a consequence, the larger amount of ice formed in P4-E-3d and 4d cement pastes samples can be explained by the damage caused by micro-cracks produced by oven-drying that reduces the proportion of open gel pores and raises the proportion of small and large capillary pores.

The characteristics of the hardened cement paste samples are summarized on table 2.

At this point, strong differences on permeability and amount of ice formed between the two kinds of specimens have been quantified. Now, in order to evaluate and predict the consequence of these differences on the frost-thaw behavior, theoretical modelling is necessary.

POROMECHANICS OF A PARTIALLY FROZEN CONTINUUM UNDER A THERMAL GRADIENT

The poromechanical model built in this study assumes that the thermodynamic equilibrium between water in liquid form (index l) and its ice crystal (index c) is reached at any time. This requires the equality of their specific chemical potentials:

µ l = µ c (2)
The natural arguments of the chemical potential µ j per mass unit of phase j = l or c are the absolute temperature T and the current pressure p J . The differentiation of (2) and standard thermodynamics methods give:

1 ρ l dp l -s l dT = 1 ρ c dp c -s c dT (3)
where ρ j and s j are the density and entropy per unit of mass of the phase j.

Hereafter the atmospheric pressure (conveniently set equal to zero in all that follows) and the corresponding freezing point T 0 = 273.15 K are adopted as the common reference for pressure and temperature. In this study, small transformations and small displacements are considered. Let K j and α j be the bulk modulus and the volumetric thermal coefficient of the phase j. As shown by (Lide 2001), the variation of K j and α c with temperature can be neglected while the variation of α l one can not. The linear form α l = α 0 l + ∆α l θ for the thermal dilatation of supercooled water obtained from the linearization of [START_REF] Speedy | Thermodynamic properties of supercooled water at 1 atm[END_REF] data in the [-20 o C, 0 o C] temperature range is adopted. Then, under the hypothesis of small density variations, the density of liquid water and ice crystal are:

ρ l = ρ 0 l 1 + p l K l -α 0 l + ∆α l (T -T 0 ) 2 (T -T 0 ) (4a) ρ c = ρ 0 c 1 + p c K c -α 0 c (T -T 0 ) (4b)
where ρ 0 l , ρ 0 c , α 0 l and α 0 c stand for the reference state value of ρ l , ρ c , α l and α c .

Assuming that the entropy s j of the phase j per unit of porous media volume does not depend on pressure, standard thermodynamics leads to:

s j = s 0 j + c 0 p,j ln T T 0 (5)
where s 0 j and c 0 p,j are the bulk specific entropy and the specific heat capacity, both constant and associated with the reference state while T 0 is the freezing temperature of bulk liquid water.

Let us define the capillary pressure as p cap = p cp l . The integration of (3) from the reference state, while using (4-5), the hypothesis of small density variations and neglecting second order terms, but taking into account variations of temperature from 253.15 to 273.15 K leads to (see [START_REF] Fabbri | Physico-mécanique des matériaux cimentaires soumis au gel-dégel[END_REF]) for more details):

p cap = Σ f (T 0 -T ) -C f (T -T 0 ) 2 2 T 0 (6)
where

Σ f = ρ 0 c (s 0 l -s 0 c ) and C f = ρ 0 c c 0 p,l -c 0 p,c
are respectively the entropy of fusion and the heat capacity difference between water and ice per unit of volume.

Unsaturated poroelasticity

Let us consider the initial state to be [σ](x, 0) = 0, [ε](x, 0) = 0, p l (x, 0) = p c (x, 0) = 0, T (x, 0) = T 0 , Σ(x, 0) = Σ 0 and ϕ l (x, 0) = ϕ c (x, 0) = 0 where [σ], [ε] and Σ are respectively the stress tensor, the strain tensor and the entropy of the empty porous medium (solid matrix and empty porous network) while ϕ j is the deformation of the porosity occupied by phase j. The relationship between the related porosity of the phase j (noted φ j ), the saturation ratio of the phase j and ϕ j is given by [START_REF] Coussy | Poromechanics of freezing materials[END_REF]:

φ j = ϕ j + φ 0 S j (7)
The lowest temperature reached in the scaling experiments is -20 o C. As indicated in figures 5-6 and in table 2, less than 60% of water is solidified in the tested samples. The interfacial liquid-ice energy remains small enough to be negligible in the expression of the pressure actually transmitted to the solid matrix by the crystal and liquid phases [START_REF] Fabbri | Physico-mécanique des matériaux cimentaires soumis au gel-dégel[END_REF]. In this case the complete set of constitutive equations of the empty porous medium [START_REF] Coussy | Poromechanics[END_REF][START_REF] Coussy | Poromechanics of freezing materials[END_REF] are then found to simplify to [START_REF] Fabbri | Physico-mécanique des matériaux cimentaires soumis au gel-dégel[END_REF][START_REF] Coussy | Unsaturated poroelasticity for crystallization in pores[END_REF]:

[σ] = K - 2 3 G [1] + 2G[ε] - j=l,c (b j p j ) + αK(T -T 0 ) [1] (8a) ϕ j = b j + 1 N jl p l + 1 N jc p c -α φj (T -T 0 ); j = l or c (8b) Σ -Σ 0 = αK - j=l, c (α φj dp j ) + C m ln T T 0 (8c)
where = tr ([ε]) is the volumetric dilatation; K, G, C m and α are respectively the bulk modulus, the shear modulus, the heat capacity, and the thermal volumetric dilatation coefficient of the empty porous medium; b j and N ij are the generalized Biot coefficients and the generalized Biot coupling moduli while α φj is the coefficient related to the thermal dilatation of the pore volume occupied by phase j. They depend upon the elastic properties of the solid matrix and the saturations of liquid and ice according to:

1 N jj + 1 N lc = b j -φ 0 S j K m ; b c + b l = 1 - K K m and α φ j = α (b j -φ 0 S j ) where j = l or c (9)
with K m the bulk modulus of the solid matrix.

The generalized Biot coefficients bj can be estimated from the Biot coefficient b = 1 -K/K m by use of the so-called iso-deformation hypothesis expressing that there is no morphological difference between the forming ice crystal and the liquid water domains (they are both bounded by pores having the same shape). This yields [START_REF] Chateau | Micromechanics of saturated and unsaturated porous media[END_REF]:

b j = S j b; j = c, l (10)
Let us assume an one-dimensional problem without any external mechanical load. The momentum balance (div(σ ij ) = 0) and the geometric compatibility of the strain tensor

([ε] = [grad(u)] + [grad(u)] T )
, where u is the displacement vector, require σ ij, i =j and σ xx to be null everywhere, while only the ε xx term of the strain tensor ε ij is non zero:

σ ij, i =j = σ xx = 0; ε xx = ∂u x ∂x ; ε ij = 0 for ij = xx (11)
Then, the substitution ( 6), ( 9) and ( 11) in the constitutive equation (8a) allows for the computation of volumetric strain:

= b p l + b c C f T -T 0 -T ln T T 0 -Σ f (T -T 0 ) + αK(T -T 0 ) K + 4G/3 (12)
Due to the temperature gradient across the specimen, the pressure field is not uniform and a liquid flow is created. Assuming that it is governed by Darcy's law, we write:

ω = -ρ 0 l κ(S l ) η l (T, p l ) grad(p l ) ( 13
)
where ω is the relative flow vector of mass fluid, while κ(S l ) and η l (T, p l ) are respectively the permeability of the porous medium and the viscosity of liquid water. The expression of the permeability has been extensively studied in case of drying, as reviewed in [START_REF] Baroghel-Bouny | Assessment of transport properties of cementitious materials: a major challenge as regards durability?[END_REF]. In case of freezing/thawing, the NMR-based paper [START_REF] Kleinberg | NMR measurements of permafrost: unfrozen water assay, pore-scale distribution of ice, and hydraulic permeability of sediments[END_REF] concludes that ice forms from the pore centres. Since no liquid flow can occur through ice crystal, such a morphology can be represented by the usual self-consistent differential scheme yielding [START_REF] Dormieux | Introduction la micromécanique des milieux poreux[END_REF]:

κ = κ 0 (S l ) 3/2 (14)
where κ 0 is the constant intrinsic permeability, while the term (S l ) 3/2 stands for the relative permeability accounting for the change in porous volume due to the solidification of liquid water.

Because the flow of ice is significantly slower than the usual cooling timescales, it will be neglected. In this case, the overall mass conservation of water (m = m l + m c ), in both liquid and crystal forms, requires:

∂m ∂t = -div(ω) (15)
Using ( 7), the global mass of liquid and crystal per unit of porous medium volume can be expressed as:

m j = ρ j φ j = ρ j (φ 0 S j + ϕ j ) ; j = l or c (16)
Finally, due to the small density difference between liquid and crystal phases of water (|ρ 0 c /ρ 0 l -1| ≈ 0.09 1), the combination of ( 4), ( 8b), (9-10), ( 12), ( 13), ( 15) and ( 16)

under the assumption of small perturbation allows us to express the overall mass conservation according to the only unknown variables p l and T :

div ρ 0 l κ(S l ) η l (T, p l ) grad(p l ) = ∂ ∂t (A p l ) + ṙ∆ρ + ṙT + ṙcryo (17)
with:

A = φ 0 S c 1 K 0 c - 1 K 0 l + b 2 K + 4G/3 + b -φ 0 K m + φ 0 K l (18a) ṙ∆ρ = φ 0 ρ 0 c ρ 0 l -1 ∂S c ∂t (18b) ṙT = φ 0 [α -α * l + S c (α * l -α * c )] -αb 4G/3 K + 4G/3 ∂T ∂t (18c) ṙpcap = φ 0 S c 1 K m - 1 K 0 c -b c b K + 4G/3 + 1 K m ∂p cap ∂t (18d)
The source term of liquid pressure in ( 17) is made of three distinct contributions. The first contribution, ṙ∆ρ , is a positive term originating from the expansion occurred when liquid water freezes. It is the only contribution considered in Power's hydraulic theory [START_REF] Powers | The air requirement of frost-resistant concrete[END_REF]. The term ṙT accounts for the pressure in the liquid caused by the relative thermal contraction between the solid matrix and the in-pore constituent. And finally, ṙpcap accounts for the pressure in the liquid caused by flows at the microscopic scale. The physical understanding of the driving force of these flows, which drives liquid water to the already frozen sites in order to meet the liquid-crystal equilibrium condition (6), was previously discussed in [START_REF] Vignes | A model for the freezing of water in a dispersed medium[END_REF][START_REF] Vignes-Adler | On the origin of the water aspiration in a freezing dispersed medium[END_REF].

Equation of heat transfer

The second law of thermodynamic applied to the porous medium for a reversible evolution leads to:

T ∂Σ s ∂t + m l ∂s l ∂t + m c ∂s c ∂t = T (s l -s c ) ∂m c ∂t -div(q) (19)
where Σ s is the entropy of the skeleton, which, in the present study, is assumed to be equal to the entropy of the empty porous medium Σ. The term (s ls c ) ∂mc ∂t is the heat source due to the latent heat of liquid to ice phase change. The heat flow (q) is unidirectional and the medium is assumed to be isotropic. Applying Fourier's Law:

q = -λ(S l ) ∂T ∂x x ( 20
)
where λ(S l ) is the isotropic thermal conductivity. Assuming that ice crystal is growing from the center of the pore in which it is confined, λ can be estimated using the (n+1)-phases multi-scale scheme developed by [START_REF] Hervé | Thermal and thermoelastic behaviour of multiply coated inclusionreinforced composites[END_REF] for isotropic composite media. Thus:

λ(S l , T ) = λ m (T ) 1 + φ 0 1-φ 0 3 + λm(T ) λ i (S l ,T )-λm(T ) with λ i (S l , T ) = λ l (T ) 1 + 1 -S l S l 3 + λ l (T ) λc(T )-λ l (T ) (21)
where λ s , λ l and λ c are the isotropic thermal conductivity of matrix, liquid water and ice crystal respectively.

Moreover, assuming that the entropy of the skeleton Σ s does not depend on pressure and strain, and under the small transformation hypothesis, (8c) and ( 16) gives the following approximation for ∂Σs ∂t and ∂mc ∂t :

∂Σ s ∂t = C m ∂T ∂t (22a) ∂m c ∂t = -ρ 0 c φ 0 ∂S l ∂t (22b)
Let C pl = ρ 0 l c 0 p,l be the heat capacity of water per volume unit. The substitution of ( 20) and ( 22) in ( 19) leads to:

div λ ∂T ∂x x = ∂T ∂t (C m + φ 0 (C pl -S c C f ) ) -T φ 0 Σ f + C f ln T T 0 ∂S c ∂T (23)
where

-T φ 0 Σ f + C f ln T T 0 ∂Sc
∂T stands for the latent heat of solidification (resp. fusion) of water and C m + φ 0 (C pl -S c C f ) for the average heat capacity of the porous medium.

NUMERICAL APPLICATION

The specimen is modelled as a one-dimensional structure made up of an isotropic medium, of length L and lateral surface S, ideally insulated on its lateral surfaces. The Cartesian coordinate system (O, x) is used, with O the node of the surface which is submitted to frost action and x following the symmetry axis from the top to the bottom of the specimen. In this case, at the macroscopic scale the flow of heat and liquid only occurs in the x direction.

As a consequence, the elementary volume is given by dΩ = Sdx.

At the initial condition, the temperature of the sample is T (x, t = 0) = T 0 = 273.15 K with no water overpressure (p l (x, t = 0) = 0 MPa) where T stands for absolute temperature in Kelvin (T [K] = θ [ o C] + 273.15) and t stands for time in second. At t = 0, the x = 0 surface is submitted to a progressive decrease of temperature while the x = L surface is increased to T b = 283.15 K:

T (0, t) = T s (t); T (L, t) = T b (24)
where T s (t) follows the same time-evolution as the experimental protocol, thus:

T s (t) =
In this study, no external loading is applied to the structure, which leads to:

[σ] • n L = 0
where n L is the outward unit vector of the bottom surface.

The parafilm sheet used to wrap the specimen should not be sufficient to prevent the in-pore water to be expelled from the x = L surface. Consequently, the most realistic liquid pressure boundary condition is:

p l (L, t) = 0 (28)
Similarly, we may assume that the water which solidifies at the x = 0 surface is stress-free.

Then, the following boundary condition should be taken:

p c (0, t) = 0 (29)
Alternatively, we may suppose that the water which solidifies at the x = 0 surface during the first freezing-thawing cycles will play the role of an impermeable frost layer. In this case, no flow (ω) through the frozen surface has to be prescribed as boundary condition:

ω • n 0 = 0 for x = 0 (30)
where n 0 is the outward unit vector perpendicular to the frozen surface.

In the following, the influences of each of ( 29) and ( 30) boundary conditions on frost behavior is studied.

At last, the poromechanical-based approach of the problem can be summarized by equations (17-23) where p l and T stand for the main unknown variables. This system is solved using the Newton-Raphson method on a structure discretised according to the finite volume method implicit scheme [START_REF] Eymard | The finite volume method[END_REF]. Mathematical functions used to implement the S l (θ) experimental curves on the model are reported on appendix 2 and the thermophysical properties values used for the calculation are reported on appendix 3.

The predicted liquid and ice pressure profiles at the end of the cooling stage, for the predried and virgin hardened cement pastes and the ( 29) and ( 30) boundary conditions, are on figure 7.

DISCUSSION

Whatever the permeability and the boundary condition used, the maximum ice pressure (round to 25.5 MPa and 41.0 MPa for the less permeable cement paste and round to 8.5

MPa and 32.0 MPa for the more permeable one) is close to or higher than the usual tensile elastic limit of an Ordinary Portland Cement Paste (f t ), which is between 5 and 10 MPa (Taylor, 1997). In case of the p c (0, t) = 0 boundary condition, the ice pressure peak of the virgin sample is localized at 2.7 mm beneath the surface, which is in appropriateness with the depth of scaling commonly observed under lower thermal gradients. On the contrary the ice pressure peak of the pre-dried sample is broad and localized in the sample core (8 mm beneath the surface). In case of the ω(0, t) • n 0 = 0 boundary condition, the profiles shape are also noticeably different. Indeed, for the predried sample, the p c (x) curve is quite homogeneous all over the specimen, while for the virgin specimen a great peak of overpressure is observed near the frozen surface.

However, the knowledge of the ice pressure is not sufficient to estimate the susceptibility of the solid matrix to be crushed. To do so, we rather use the stress in the solid matrix, σ m,xx .

As reported in [START_REF] Chateau | Micromechanics of saturated and unsaturated porous media[END_REF], neglecting the influence of the interfacial tension between liquid water and pore walls, it can be linked to the porous medium stress and the pressures of liquid and crystal by the relation (1

-φ 0 ) σ m,xx = σ xx + φ l p l + φ c p c .
Then, assuming that the thermophysical properties of the solid matrix are constant over the temperature cycle, σ m,xx , can be expressed through ( 6) and ( 11) as:

σ m,xx = φ 1 -φ 0 p l - φ c 1 -φ 0 Σ f (T -T 0 ) + C f (T -T 0 ) 2 2 T 0 (31)
The σ m,xx profiles are in figure 8.

For the pre-dried specimen, whatever the boundary condition used, no peak of tensile stress is predicted near the surface submitted to frost action. Moreover, only the ω(0, t)•n 0 = 0 boundary condition results in a significant risk of frost damage through the structure (σ m,xx is higher than 5 MPa until 8 mm depth). Then, the hypothesis of the existence of an impermeable frozen film on the skin surface leads to a better suitability between experimental results and theoretical calculations. For virgin samples, both boundary conditions give a matrix stress close to or higher than f t only within the first millimeters depth (until 3mm).

This fact, enhanced by a poor surface quality of the material and an increasing number of freezing-thawing cycles, can explain the observed damage on the skin surface.

The explanation of the behavior difference between the two types of samples is pointed up by the p l (x) profiles. Indeed, liquid over-pressure appears to be easily dissipated through the porous network for a pre-dried sample than in a virgin one, where liquid over-pressurization peaks are observed. This is why, while the amount of ice formed in pre-dried samples is significantly higher than in virgin ones (at -20 o C, S c is about 0.1 for a virgin sample and S c is about 0.5 for a predried one), the maximum ice pressure of the virgin sample is higher: a higher permeability allows a more efficient in-pore overpressure relaxation.

Actually, in case of too low permeable cement pastes, the liquid flow expulsion is not important enough to relax the whole local overpressure caused by thermal stresses, cryogenic aspiration and liquid-ice difference of density. Thus, this pore overpressure, which produces a local tensile stress on the matrix, remains localized near the x = 0 surface where the ice formation is the most important. This fact is illustrated on figure 9 where a decrease of the maximum tensile stress on the matrix is predicted during the 1 hour temperature step at -20 o C. This decrease is clearly due to the pore relaxation due to slow water transport from over-pressurized zone to the bottom and/or the top of the specimen.

This explanation leads to the conclusion that scaling is enhanced by the inability of the frozen porous network to relax local overpressure from the skin surface. This is consistent with the experimental data from [START_REF] Baroghel-Bouny | Vieillissement des bétons en milieu naturel : une expérimentation pour le XXI e siècle. IIIpropriétés de durabilité des bétons mesurées sur éprouvettes conserveés en laboratoire[END_REF] where a concrete with a compressive strength equal to 50 MPa exhibits a better frost durability than a less permeable but higher resistance concrete (with a compressive strength equal to 75 MPa). In addition, the common lack of efficiency of air-entraining agent in a high resistant concrete (LCPC 2003) can then be attributed to the difficulty of the liquid water to migrate from the porous network to air voids in case of too low permeable material.

To sum up, similarly to the experimental observations, the poromechanical model rather predicts an internal frost damage for pre-dried samples and scaling for virgin samples even though they are submitted to the same solicitation. The observed difference between the two mechanisms seems to be caused by the contrast between permeability and amount of ice formed. Indeed, in the case of initial and boundary conditions which force the top surface temperature to be lower than the bottom one, water will first freeze near the skin surface.

If the amount of ice formed is high enough and the permeability is too small to relax pore overpressure, scaling will occur. Thus, it appears to be an "internal frost"-like damage localized at the x = 0 surface.

According to a recent work performed by [START_REF] Valenza | Mechanisms of salt scaling[END_REF], the increase of scaling due to the presence of salts is a consequence of mechanical interaction between the skin surface and the frozen brine layer. This effect can be taken into account in the present model by changing the boundary conditions at the top surface (x = 0). For this case, all components of the strain tensor have to be taken into account. It leads to lengthy three-dimensional calculations but the global analysis remains the same.

CONCLUSION

An experimental device, in which hardened cement paste specimens are submitted to freezing-thawing cycles under a thermal gradient, has been developed. This study reveals occurrence of scaling without a brine layer in contact with the frozen surface. To explain this behavior, a poroelastic model has been worked out. The model predicts that scaling can occur without salts due to the localization of pore overpressure near the skin surface, caused by the low permeability of cementitious material. Finally, a link is established between the frost behavior at the structural scale and the two key material parameters toward frost durability: the permeability and the amount of ice formed as a function of temperature.

The permeability influences tensile matrix stress localization while the amount of formed ice rather acts on its global amplitude.

Appendix 1: Permeability of the saturated porous specimen

The permeability is estimated using a mercury injection experiment and the Katz-Thompson relation [START_REF] Katz | Quantitative prediction of permeability in porous rock[END_REF]:

κ 0 = 1 226 l 2 c C C 0 (32)
where l c is the threshold pore diameter in mercury injection experiment (MIE), C the sample conductivity and C 0 the conductivity of the water in the pore space. The relative conductivity C/C 0 is also estimated by a mercury injection experiment through relationship reported in [START_REF] Garboczi | Modelling the microstructure and transport properties of concrete[END_REF], based on a 3-D electrical analogy of the cement paste and assuming that the relative conductivity of capillary water is 1 while all the other constituents ones are null:

C C 0 = 0.001 + 0.07 φ 2 MIE + 1.8 [[φ MIE -φ c ]] (φ MIE -φ c ) 2 (33)
where φ MIE is the porosity estimated by a mercury injection experiment while φ c is the capillary porosity under which the pore space becomes non percolated and is commonly estimated to be approximately 0.18 [START_REF] Powers | Capillary continuity or discontinuity in cement pastes[END_REF][START_REF] Garboczi | Modelling the microstructure and transport properties of concrete[END_REF]] is the Heaviside function. As reported in [START_REF] Garboczi | Modelling the microstructure and transport properties of concrete[END_REF][START_REF] Cui | Permeability and pore structure of OPC cement paste[END_REF], a good agreement is found between (33) and experimental data for ordinary cement pastes by many authors.

Appendix 2: Fit of the freezing-thawing curves

The freezing-thawing curves shapes, quite similar to water retention capillary curves ones, are well described by the following "Van-Genuchten"-like (Van Genuchten 1980) sigmoidal functions:

S l,f rost (θ) = k i=1 φ i 1 + n i θ θ * i 1 1-n i n i + 1 - k i=1 φ i (34a) S l,thaw (θ) = φ T + (1 -φ T ) 1 + n θ θ * 1 1-n -n (34b)
where S l,f rost (θ) and S l,thaw (θ) are the liquid saturation ratio on freezing and thawing, while k stands for the number of freezing peaks, θ * i for the temperature at which each of them happens, and θ * for the temperature at which the thawing peak happens. These coefficients can be evaluated from the analysis of the S l (θ) derivative curve.

Let θ th be the temperature at which thawing begins. Because the liquid saturation ratio is a continuous function, S l,f rost (θ th ) and S l,thaw (θ th ) are equal. As a consequence, the coefficient φ T must be:

φ T = S l,f rost (θ th ) -f n,θ * (θ th ) 1 -f n,θ * (θ th ) (35) with f n,θ * (θ) = 1 + n θ θ * 1 1-n -n .
Finally coefficients φ i , n i and n, which are the amplitude and the shape factor of the i th freezing sigmoidal function and the shape factor of the thawing curve, need to be fitted from experimental data. It must also be emphasized that θ * and n appear to be independent of θ th on the two kind of tested cement pastes (virgin or pre-dried). Thus, the fit of only one thawing curve is sufficient to estimate the amount of ice within the porous network whatever the lower temperature reached during the freezing stage. Figures 5 and6 show the comparison between the experimental data and the fitted curves using equation ( 34

5

  and 6, in the [-30 o C; 0 o C] temperature range, both S l (θ) curves varies linearly, except for two particular temperatures on cooling, around -5 o C and -20 o C, and one, around -5 o C

FIG. 7 .

 7 FIG. 1. Freezing-thawing cycle imposed at the sample top surface.

Appendix 3: Thermophysical properties

Numerical application is made with ρ 0 l = 9.97 × 10 -3 MN/m 3 , ρ 0 c = 9.17 × 10 -3 MN/m 3 , α 0 l = -68.7×10 -6 K -1 and ∆α l = 24.732×10 -6 K -2 , α 0 c = 160×10 -6 K -1 , K l = 1970 MPa, [START_REF] Brun | A new method for the simultaneous determination of the size and the shape of pores : The thermoporometry[END_REF],

C m = 1, 36 MPa/K (de Schutter and Taerwe 1995), α = α m = 30×10 -6 K -1 [START_REF] Ulm | The "chunnel" fire. ii: Analysis of concrete damage[END_REF] , K = 14100 MPa, K m = 31800 MPa, b = 0.55 [START_REF] Ulm | Is concrete a poromechanics material ? -a multiscale investigation of poroelastic properties[END_REF],

mN/m [START_REF] Zuber | Predicting the volume instability of hydrated cement systems upon freezing using poro-mechanics and local phase equilibra[END_REF], λ m = 1.9 W/(mK) (deduced from the experimental values of saturated and a dried cement pastes conductivity given by [START_REF] Kim | An experimental study on thermal conductivity of concrete[END_REF]).

The evolution of the viscosity of supercooled water with temperature and liquid pressure by numerous empirical relationships. In this study, one of the most recent proposed by [START_REF] Harris | Temperature and volume dependence of the viscosity of water and heavy water at low temperatures[END_REF]) is chosen (p l in MPa and T en K):

with a 1 = 0.411011 mPa s, a 2 = 0.00854113 mPa s/MPa, a 3 = -2.67340 mPa s K/MPa, a 4 = -1.35750 mPa s/MPa 2 , a 5 = 0.00525404 mPa K/MPa 2 , a 6 = -5.15042 mPa s/MPa 3 , a 7 = -2350.32 mPa s K and a 8 = 654401 mPa s K 2 .

Let us emphasize that the cement paste did not have air-entraining agent so there was no relief on the hydraulic pressure by the air voids.

List of


Antonin Fabbri

Teddy Fen-Chong

Aza Azouni

Jean-François Thimus

Investigation of water to ice phase change in porous media by ultrasonic and dielectric measurements

Keywords: Freeze-thaw, Pore size distribution, Hysteresis, Cement, Clay, Thermodynamics

The main objective of this paper is to study the evolution of the ice content of porous media submitted to sub-zero temperatures by dielectric and ultrasonic measurements. Dielectric measurements are made by a capacitive sensor-based apparatus. The amount of ice formed within the tested sample is estimated from the global dielectric constants of the sample and of all the phases that form the tested composite material. On the other hand, ultrasonic measurements are based on the evolution of the ultrasonic wave velocity through the tested sample during a freezing-thawing cycle. These two methods lead to very close results and appear to be cheaper alternatives to low temperature calorimetry. The ice content curves are analyzed with the help of thermoporometry concepts in order to characterize the pore size distribution. Results appear to be complementary to mercury intrusion porosimetry ones. Moreover the commonly observed hysteresis of the ice content during a freezing-thawing cycle is investigated with respect to material microstructure.

INTRODUCTION

Damage induced by frost action upon concrete structures is a source of main concern in cold climates [START_REF] Pigeon | Microstructure et résistance au gel des ciments et bétons[END_REF][START_REF] Dash | The premelting of ice and its environmental consequences[END_REF]. The mechanical response of a porous mate-1 rial is credited to result from the combination of the liquid-solid expansion, the transport of unfrozen liquid water through the porous network, and the presence of air voids (Wang et al. 1996). Theoretical approaches have been recently developed to understand and quantify the phenomena at the pore scale [START_REF] Coussy | Crystallization, pore relaxation and microcryosuction in cohesive porous materials[END_REF], at the material scale [START_REF] Coussy | Poromechanics of freezing materials[END_REF] and at the structure scale [START_REF] Fabbri | Are deicing salts necessary to promote scaling in concrete?[END_REF]. Whatever the approach carried out, the volumetric ratio of frozen water (commonly called the ice saturation ratio, S c ) versus temperature turns out to be the key curve governing the mechanical behavior of porous materials upon frost action. Freezing of cement based materials is usually studied with low temperature calorimetry [START_REF] Bager | Ice formation in hardened cement paste -part II : Drying and resaturation on room temperature cured pastes[END_REF]. However this device only allows to test very small samples (characteristic length round the millimeter) which are often crushed.

Hence tests on non-crushed heterogeneous materials like mortar and concrete (aggregate characteristic length around the centimeter) should be difficult to set up. For such materials requiring larger samples, dielectric [START_REF] Fabbri | Dielectric capacity, liquid water content, and pore structure of thawing-freezing materials[END_REF]) and ultrasonic [START_REF] Thimus | Ultrasonic measurements in frozen soils to -120 o C: results and interpretation using different theoretical models[END_REF]) methods are more appropriate while being less expensive. This paper investigates these two methods and compares the results they provide on the ice / water transition in cohesive porous materials such as cement pastes. Then the hysteresis on ice content commonly observed during a freezing-thawing cycle is investigated in relation with the material microstructure.

BASIC MEASUREMENT PRINCIPLE

Ultrasonic method

Ultrasonic methods rely on the difference between the longitudinal compression wave velocities in liquid water (V l = 1428 m/s at 0 o C) and ice crystal (V c = 3844 m/s at 0 o C) to evaluate the amount of unfrozen water in a saturated partially frozen porous medium.

After a review of some classical schemes used in ultrasonic experiments [START_REF] Thimus | Determination of unfrozen water content of an overconsolidated clay down to -160 o c by sonic approachescomparison with classical methods[END_REF][START_REF] Thimus | Ultrasonic measurements in frozen soils to -120 o C: results and interpretation using different theoretical models[END_REF], the link between the wave velocity through a partially frozen porous sample (V ) and the volumetric ratio of ice (S c ), is established by the Timur model (Timur 1968):

1 V = φ 0 (1 -S c ) V l + φ 0 S c V c + 1 -φ 0 V s (1)
where V s is the longitudinal compression wave velocity in the solid matrix and φ 0 the porosity of the porous medium.

The relation (1) leads to the following expression in which S c is directly linked to V :

S c = a V + b (2)
where

a = V c V l /(φ 0 (V l -V c )) and b = -V c V l /(φ 0 (V l -V c )) ((1 -φ 0 )/V s + φ 0 /V l ) .
However a great dispersion on V l and V c measurements on the same kind of sample has been observed. In fact these two parameters seem to strongly depend on material cure, age and sampling. In consequence, in this study, the coefficients a and b are fitted from the totally frozen (S l = 0) and unfrozen (S l = 1) states so that:

a = V (S c = 0)V (S c = 1) V (S c = 0) -V (S c = 1) (3a) b = V (S c = 1) V (S c = 1) -V (S c = 0) = - a V (S c = 0) (3b)

Capacitive method

The capacitive method is based on the contrast between water (ε l about 80 at 20 o C) and ice (ε c about 3 between -40 o C and 0 o C) dielectric constants in the radio-frequency range [START_REF] Fen-Chong | Determination of liquid water content and dielectric constant in porous media by the capacitive method[END_REF][START_REF] Fen-Chong | Freezing and thawing porous media : Experimental study with a dielectric capacitive method[END_REF][START_REF] Fabbri | Dielectric capacity, liquid water content, and pore structure of thawing-freezing materials[END_REF]. In consequence, any solidification of the porous network liquid will induce a notable decrease of the global dielectric constant of the sample (ε). To get the freezing curve, a relation between ε and the ice saturation ratio S c must be established. To do so, after a review of some classical mixture schemes (Zakri et al. 1998;[START_REF] Fen-Chong | Determination of liquid water content and dielectric constant in porous media by the capacitive method[END_REF]), the Lichtenecker model was chosen. This model comes from the Effective Medium Theory in which the self-consistent scheme is adopted. This finally gives [START_REF] Fabbri | Dielectric capacity, liquid water content, and pore structure of thawing-freezing materials[END_REF])):

ε = ε 1-φ 0 m (1 -S c )ε φ 0 l + S c ε φ 0 c (4)
where ε m is the dielectric constant of the solid matrix. Whereas ε c is well known and remains constant during a freezing-thawing cycle, ε l varies with the temperature. In addition, the dielectric constant of the hydration water [START_REF] Mercury | Thermodynamics of ice polymorphs and ice-like water in hydrates and hydroxydes[END_REF]) that is part of the mineral matrix, changes with temperature, which significantly affects ε m .

Therefore, a semi-empirical method has been developped [START_REF] Fabbri | Dielectric capacity, liquid water content, and pore structure of thawing-freezing materials[END_REF]. It is based on the experimental determination of the dielectric constant of the solid matrix ε m in a dried sample (no capillary water) and that of the liquid water ε l,s for the same sample but in fully saturated conditions. The dielectric constant of the unfrozen water has been experimentally

found to be obtained by extrapolating the affine ε l,s (T ) relation. By doing so, one gets:

ε m (T ) = 1-φ 0 ε dried (T ) ε φ 0 a ( 5
)
where ε a = 1 is the dielectric constant of air, and

ε l (T ) = φ 0 ε(S c = 0, T ) ε 1-φ 0 m (T ) (6)
Using ( 5) and ( 6), ( 4) can be eventually inverted in the form:

S c = A(T ) ε + B(T ) (7)
where

1/A = ε 1-φ 0 m ε φ 0 c -ε φ 0 l and B = -ε φ 0 l / ε φ 0 c -ε φ 0 l .

MATERIALS AND RELATED PROPERTIES

In this study we used the following materials. sheet stacking and its mineralogy is alike that given in [START_REF] Romero | Water permeability, water retention and microstructure of unsaturated compacted boom clay[END_REF]. From the analysis of its microstructure by SEM [START_REF] Dehandschutter | Microfabric of fractured boom clay at depth: a case study of brittle-ductile transitional clay behaviour[END_REF]) and Mercury Intrusion experiment (figure 1), Boom Clay appears to be a lightly consolidated medium with an important open porosity and composed of particles lined up with the stratigraphy. Three porosity peaks are identified on the pore diameter distribution : 150 nm, 2 µm and 100 µm.

The 300 µm peak is attributed to the presence of cracks on the sample surface while the 2 µm and 150 nm-peaks are attributed to the inter-aggregate pore modes [START_REF] Romero | Water permeability, water retention and microstructure of unsaturated compacted boom clay[END_REF]. Cylindrical samples of 60-mm mean diameter and 25-mm mean thickness were used.

Hardened cement paste (HCP): Cement paste with water-cement ratio of 0.5 are prepared with a 5-liter mortar mixer and cast in cylinder moulds with a 40-mm in diameter and 100-mm high. Ordinary Portland Cement CEM I CPA 52.5 N CP2 and distilled water are used. The porous distribution got from Mercury Intrusion Experiments is reported on figure 1 where the two observed peaks correspond to capillary porosity (100-nm in diameter) and inter-hydrates porosity ( 4-nm in diameter) (Baroghel-Bouny and Chaussadent 1993).

One day after casting, samples are removed from their mould and stored in moist condition (relative humidity rh = 95%±5%) during 6 months. Then they are cut in about 25-mm slices before being put in water until testing. Samples were saturated with degassed distilled water at 3 kPa air pressure and tested.

Table 1 gives the tested materials main characteristics which are their global water porosity (φ 0 ), initial water content (ω), mean pore diameters (D p ), density of the dried material (γ d ), ultrasonic coefficients a and b (see ( 2)) and dielectric constants defined in ( 5) and ( 6).

The fitted values from ultrasonic and capacitive experiments of a, b, ε m , and ε l empirically take account of the mineralogy, microstructure of the porous media, as well as the interfacial polarization effects at stake in clays and cement materials. Indeed one can see on figure 2 that the dielectric constant of the in-pore water in SGB is alike that of bulk water [START_REF] Ellison | Water: a dielectric reference[END_REF][START_REF] Kaatze | The dielectric properties of water in its different states of interaction[END_REF][START_REF] Fabbri | Physico-mécanique des matériaux cimentaires soumis au gel/dégel[END_REF]) whereas this is not the case for HCP and more importantly for BC. At last figure 3 indicates that the dielectric constant of the mineral solid in HCP is much higher than those in SGB and BC, which results from the presence of water molecules inside the C-S-H gel phase of cementitious materials.

EXPERIMENTS AND RESULTS

For each type of material the ice saturation ratio was evaluated at least four times: twice with the ultrasonic apparatus and twice with the capacitive method. The measurement sensitivity on S c is estimated to be around 0.01 for the two apparatuses [START_REF] Thimus | Ultrasonic measurements in frozen soils to -120 o C: results and interpretation using different theoretical models[END_REF][START_REF] Fabbri | Dielectric capacity, liquid water content, and pore structure of thawing-freezing materials[END_REF].

For ultrasonic experiments two samples are used. The first one is inserted into two transducers of 50-mm in diameter (an emitter and a receiver). Electric impulses (100 Volts pulse amplitude and 3 ps pulse width at a rate of 40 pulses per second) were generated by a PUNDIT unit (portable ultrasonic non-destructive digital indicating tester) and transformed into ultrasonic longitudinal compression waves in the emitter. After propagation through the sample, the signals were recorded by the receiver and treated by a numerical oscilloscope and a PC at a sampling rate of 2 MHz. The wave is supposed to be received by the receptor when the recorded amplitude reached 10% of the first peak value. Then its velocity is calculated by the ratio between the sample thickness and the delay of the wave through the sample. At the interface between the transducers and the sample, grease was used as a coupling agent to improve wave transmission [START_REF] Couvreur | Te properties of coupling agents in improving ultrasonic transmission[END_REF]. The second sample is drilled to differences. But, this is not the case of thawing where the heating rate is significantly higher during ultrasonic measurements. Consequently, the comparison between the two methods will be made only through the freezing curves.

Ice saturation results from the ultrasonic experiments are presented in figures 6A, 7A and 8A, while those from the capacitive method are shown in figures 6B, 7B and 8B. The curves obtained from the ultrasonic apparatus and the capacitive one appear to be consistent: For Sintered glass beads, the whole porous network freezes and thaws in a similar way at 0 o C. This behavior is not surprising since the sintered glass beads consist of a material with a monodisperse micrometric porosity. For Boom Clay, an important freezing-thawing peak is difference remains lower than 0.06 which is of order of magnitude of the accuracy of the measurement. Moreover, in the case of hardened cement pastes, the capacitive and the ultrasonic methods provide respectively at -30 o C S c equal to 0.22 and 0.16, which is in good agreement with the value of S c = 0.22 obtained at the same temperature by low temperature calorimetry [START_REF] Badger | Ice formation in hardened cement paste -part Iroom temperature cured pastes with variable moisture content[END_REF]. In conclusion, the two apparatus presented in this study can predict with a good accuracy the amount of the ice formed within a consolidated porous medium. Consequently they represent cheaper alternatives to the low temperature calorimetry.

ICE SATURATION CURVE AND POROUS NETWORK CHARACTERIZATION

As observed in the previous experimental results, water does not solidify in the whole porous volume at the same temperature: freezing occurs progressively as the temperature is reduced. Once formed somewhere, ice cannot propragate instantaneously through the whole material because of the Thomson-Gibbs law and its porous microstructure features (such as the pore sizes distribution -from nm to micrometers at least -and geometry). In the following, we present how the experimentally-determined function S c (T ), giving the amount of ice versus the temperature, can be used with the Thomson-Gibbs law to obtain such microstructural information.

Thomson-Gibbs law

The Thomson-Gibbs law [START_REF] Brun | A new method for the simultaneous determination of the size and the shape of pores : The thermoporometry[END_REF][START_REF] Scherer | Freezing gels[END_REF][START_REF] Fabbri | Dielectric capacity, liquid water content, and pore structure of thawing-freezing materials[END_REF]) comes from the solid-liquid equilibrium (described by the chemical potentials) and from the mechanical equilibrium of a curved interface (Young-Laplace law), which is presented below.

Let µ J be the chemical potential per mass unit of phase J = l (for liquid water) or c (for ice crystal) and T (T [K] = θ [ 0 C] + 273.15) the absolute temperature. The thermodynamic equilibrium between water in liquid form (index l) and its ice crystal (index c) requires the equality of their specific chemical potentials, reading:

µ c (T, σ c ) = µ l (T, p l ) (8)
where σ c and p l are the stress tensor and the pressure of the solid ice and the liquid water, respectively. As indicated in [START_REF] Scherer | Freezing gels[END_REF][START_REF] Coussy | Unsaturated poroelasticity for crystallization in pores[END_REF] the shear contribution in ice can be neglected so that (8) reduces to:

µ c (T, p c ) = µ l (T, p l ) (9)
where p c is the pressure in the ice crystal. Hereafter the atmospheric pressure (set equal to zero such that p J refers to relative pressure) is adopted as the common reference pressure and the corresponding freezing point T 0 as the reference temperature (273.15K for water).

From the differentiation of ( 9), the Gibbs-Duhem relation dµ J = 1 ρ J dp Js J dT furnishes:

1 ρ c dp c -s c dT = 1 ρ l dp l -s l dT (10)
where ρ J is the mass density which is assumed to be constant (ρ c 0.9167 g/cm 3 and ρ l 0.9998 g/cm 3 (Petrenko and Whitworth 1999)), and s J are the entropy per mass unit of phase J. Since the strain latent heat (that is the heat exchanged by the considered system with the outside in an evolution when both temperature and pressure are held constant) and the variation of heat capacity with temperature are negligible, the specific entropy is written as:

s J = s 0 J + c p,J ln T T 0 (11)
with s 0 J the bulk specific entropy and c p,J the specific heat capacity, constant and associated with the reference state. Defining the capillary pressure as p cap = p cp l , integrating (10) from the reference state, and using (11), we derive:

p cap = S f (T 0 -T ) + C f T -T 0 + T ln T 0 T = g(T ) (12)
where The mechanical equilibrium of the ice/water interface (Young-Laplace law) gives:

S f = ρ c (s l -s 0 c )
p cap = γκ ( 13
)
where γ 36 + 0.25 (T -T 0 ) (Zuber and Marchand 2004) is the water / ice interface energy and κ = dA lc /dV c = 1/r 1 + 1/r 2 the interface curvature where A lc is the water/ice interface area, V c the ice crystal volume and r J=1,2 are the principal curvature radii of the ice / water interface. Combination of ( 12) and ( 13) gives the Thomson-Gibbs law.

Thermoporometry and results

From now, the liquid-ice interface is supposed to be spherical. Thus, we classically have:

κ = 2 r ( 14
)
where r is the radius of the ice crystal. Assuming that the ice crystal is surrounded by a homogeneous layer of unfrozen water, r is related to the pore radius r p and the layer thickness e that remains constant and equal to 9 Å, by the relation [START_REF] Brun | A new method for the simultaneous determination of the size and the shape of pores : The thermoporometry[END_REF][START_REF] Dash | The premelting of ice and its environmental consequences[END_REF]:

r = r p -e. ( 15
)
The freezing and the melting temperatures T * f and T * m , respectively, for an ice crystal just filling the cylindrical pore and in equilibrium with liquid water in adjacent pores, are related to the corresponding pore radius r f p and r m p , respectively, from ( 12) to ( 15):

T * i = g -1 2 γ r i p -e ; i = f or m (16)
Based on relation ( 16), the measurement of the successive freezing or melting temperatures provides a means of assessing the pore radii distribution of a porous material. S c is linked to the volume of ice crystal, V c, and the volume of the sample, V s , by the relation

S c = V c /(φ 0 V s ).
Assuming spherical pore shape, V c is linked to the volume of the unfrozen pores, V uf , by the relation:

dV uf (T ) = - r p r p -e 3 dV c (T ) (17)
Thus, a S c -T curves can be translated to a V ufr p curve, which leads to the following expression for the logarithmic pore size distribution curve ϕ(r p ):

ϕ(r p ) = - φ 0 γ d r p r p -e 3 ∂S c ∂ log(r p ) (18)
As shown in figure 9, whatever the experimental device used, the ϕr p curve of boom clay samples obtained from a freezing and a thawing experiment are consistent and exhibit two porosity peaks: around 2 nm and around 10 nm. The existence of these two peaks, associated to intra-aggregate porosity, is in good agreement with literature data [START_REF] Romero | Water permeability, water retention and microstructure of unsaturated compacted boom clay[END_REF].

Thermoporometry results of hardened cement paste samples are shown in figure 10.

Freezing tests lead to a large peak around 1-2 nm and a smaller one around 50 nm. Thawing tests lead to similar results except that the peaks occur at slightly larger pores radius than for freezing.

Comparisons of the ϕr p curves from capacitive measurement, from ultrasonic measurement and from porosimetry intrusion test are shown in figure 11 for Boom clay samples and in figure 12 for hardened cement paste. This comparison was only made for the ϕr p curves given by a freezing experiment as the kinetics of thawing for the two methods could not be the same.

For hardened cement paste samples, ultrasonic and capacitive results appear to be close and seem to be complementary with the mercury intrusion porosimetry tests: the 100 nm one is not observed on thermoporometry curves while only the beginning of the 2 nm peak is observed on mercury intrusion porosimetry curves. Thus thermoporometry appears to be more suitable for scanning pore radii less than 100 nm while mercury intrusion porosimetry test is more suitable for scanning pore radii higher than 10 nm.

The same conclusion can be made for Boom clay measurements where ultrasonic and capacitive curves are almost the same: only the 2 nm and 10 nm peaks are observed. Conversely, the larger pores are only observed by mercury intrusion porosimetry.

The freezing-thawing temperature was linked to the radius of the pore in which the water or the ice are confined. The pore size distributions obtained from capacitive and ultrasonic measurements are in good agreement with data in the literature and complementary to mercury intrusion porosimetry tests. The next step is to understand the physical meaning of the liquid to ice phase change hysteresis and to understand how to link it to the porous network characteristics.

Physical meaning of the water to ice phase change hysteresis

It is well-known [START_REF] Scherer | Freezing gels[END_REF][START_REF] Béjaoui | Mécanismes de formation de la glace au sein des pâtes de ciment et des bétons[END_REF]) that a liquid water filled pore of radius r p does not generally freeze at T * f (r p ) due to supercooling (metastable liquid phase). Instead in-pore ice formation results from ice percolation through smaller (nanoscopic) channel-like pores connected to already frozen pores [START_REF] Scherer | Freezing gels[END_REF] or from nucleation process which can only occur at a temperature lower than T f (r p ) [START_REF] Corr | A microscopic study of ice formation and propagation in concrete[END_REF]. Consequently, the liquid water filling a pore with radius r p is likely to freeze (r p > r * p ). If it is connected with almost one frozen channel, its water will freeze, quasi-instantaneously, by propagation. However if it is only connected with ice crystal through a smaller channel (which can be called "percolation channel", r c < r * p ) which is not frozen (figure 13), solidification does not occur. It will happen when the temperature decrease will be large enough to allow propagation of ice through the smaller channel. As nucleation can only occur at temperatures below the equilibrium one, the apparent radius calculated from the measured temperature (r f ) will be smaller than the real one (r p ), so :

r c ≤ r f ≤ r p (19)
This point can be illustrated by figures 14 and 15 where the temperature evolution of the sample core follows the imposed temperature except around -5 o C where a temperature jump is observed. This jump is commonly attributed to the inner massive freezing due to the nucleation process that generates solidification of all the supercooled water contained in the biggest connected pores. After that, no temperature jump is observed until the end of the freezing process. After this inner nucleation process, the ice crystal will progressively grow by propagation through smaller and smaller pores as the temperature decreases.

On the other hand, during heating, thawing appears necessarily at T * m (r p ). This is attributed to existence a thin water film between the crystal and the pore wall which prevents from metastability [START_REF] Dash | The premelting of ice and its environmental consequences[END_REF]. Hence, the thawing process is governed by the radius of the pore body. The hysteresis between the freezing pore radius and the thawing pore radius can be explained by a traditional ink-bottle scheme in which r c plays the role of the neck pore radius [START_REF] Dullien | Pourous Media -Fluid Transport and Pore Structure -Second Edition[END_REF]. A polydisperse porous network composed by "big" spherical-like pores connected through small capillaries will engender a great hysteresis while a cyclindrical-like or a thin sheet stacking porous structure should show a reversible behavior between freezing-and thawing.

As shown in figure 2b from [START_REF] Romero | Water permeability, water retention and microstructure of unsaturated compacted boom clay[END_REF], the Boom clay microstructure exhibits spherical-like inter-aggregate porosity and intra-porosity of a thin sheet stacking structure.

However, as it is shown in figure 11, in the tested material, inter-aggregate pores volume appears to be significantly lower than the intra-aggregate ones. In consequence, the lack of hysteresis observed on the Boom clay experiments is consistent with the global thin sheet stacking of the material porous structure.

On the other hand, the hysteresis observed in hardened cement paste samples tests are consistent with their global polydisperse microstructure composed by either spherical-like pores and cylindrical-like nanoscopic tubes [START_REF] Baroghel-Bouny | Caractérisation microstructurale et hydrique des pâtes de ciment et des bétons ordinaires et très hautes performances[END_REF].

In conclusion, the study of the freezing-thawing hysteresis can provide information on the pore shape: the more pronounced the hysteresis is, the higher the ratio r p /r c is. The importance of this ratio has been already emphasized by recent studies on frost damage [START_REF] Coussy | Unsaturated poroelasticity for crystallization in pores[END_REF] and salt crystallization [START_REF] Coussy | Deformation and stress from in-pore drying-induced crystallization of salt[END_REF].

CONCLUSION

Dielectric capacitive and ultrasonic apparatus were used to experimentally determine the thermodynamic state relation between the ice volume ratio S c and the temperature. These quantitative methods were successfully applied to samples of sintered glass beads, Boom clay and hardened cement paste. The results obtained by the two methods are found to be in good agreement with data in the literature for hardened cement paste.

Pore radii distribution was similar to the data in the literature and complementary to mercury intrusion porosimetry: capacitive and ultrasonic apparatus are more suitable for scanning pore radii less than 100 nm while the latter is more suitable for scanning pore radii higher than 10 nm. In addition, we succeeded to establish a link between the freezing-thawing hysteresis and the porous network structure.

Therefore we can conclude that the two methods presented here are appropriate to analyze the ice formation in porous samples and to provide insights of their pore structure.

BC

  insert a T-type thermocouple and to get the sample temperature history for a given imposed temperature time-evolution. The two samples are put in a 50 cm × 50 cm × 40 cm thermally insulated box connected to a liquid nitrogen reservoir. The temperature rate is 20 o C/hour. Freezing is stopped at -80 o C and thawing at 20 o C. A one hour temperature step is made at the end of freezing (figure 5). Such an important freezing rate is chosen in order to reach quite rapidly the totally frozen state of the tested sample. A schematic representation of the ultrasonic device is shown in figure 4a. For the capacitive experiments, the sample is inserted between two 60 mm diameter stainless steel electrodes connected to a 30 MHz-50 MHz oscillator electronic device. All of them form an oscillating electric circuit. Then, the dielectric constant of the sample (ε) which is linked to the resonant frequency of the oscillating circuit, is measured. The temperature is imposed by a Galden PFPE HT200 cryogenic fluid (from Solvay Solexis) which circulates from a computer-controlled Hueber cryostat and measured by a T-type thermocouple inserted into the sample core. Likewise, in the Ultrasonic experiments the temperature rate is 20 o C/hour. Freezing is stopped at -40 o C for the SGB and BC experiments and et -30 o C for the HCP experiments. Thawing is stopped at 0.1 o C. A two hours temperature step is made at the end of freezing (figure 5). The thermal insulation of the lateral surface is achieved by an expanded polystyrene annulus. In order to avoid surface desaturation during freezingthawing cycles, each sample is wrapped by a moisture resistant Parafilm sheet. A schematic representation of the capacitive device is shown in figure 4b. A comparison of the temperature developed in the core of the sample in the two methods is shown in figure 5. During the freezing stage, the temperature of the sample follows the same pattern in both experimental devices used. It is then possible to compare the formation of ice in a cooling sample by the two devices without any effects from cooling rate

  observed near -4 o C. Then saturation varies almost linearly with temperature until -80 o C. Moreover likewise the SBG experiments, freezing and thawing behavior are nearly similar. Finally the hardened cement paste experiments exhibit small ice formation until -30 o C, followed by a more important linearly variation of S c until -90 o C while the thawing peaks are respectively around -60 o C and 0 o C. It should be emphasized that, unlike in the SBG and BC experiments, a hysteresis loop is observed in the S c -T curve of the hardened cement paste. So that the link between the temperature and the saturation degree cannot reduce, in this case, to a one-to-one relation. The quantitative comparison between the ice saturation values obtained by ultrasonic measurements (S U c ) and capacitive measurements (S C c ) is given in table 2. As we can see results obtained from the two methods appear to be very close. Indeed, the S C c -S U c

  is the entropy of fusion per unit of ice crystal volume, C f = ρ c (c p,lc p,c ) the heat capacity difference between water and ice per unit of ice crystal volume. Function g(T) is the state function which links the temperature T and the capillary pressure p cap once the thermodynamic equilibrium is reached between the liquid water and the ice already formed[START_REF] Coussy | Poromechanics of freezing materials[END_REF].

  FIG. 1. Pore size distribution in boom clay and hardened cement paste, from mercury intrusion experiments.

  FIG.7. Variation of ice saturation during a freeze-thaw cycle of a specimen of boom clay, evaluated with the ultrasonic apparatus (A) and the capacitive apparatus (B).

TABLE 1 .

 1 Main characteristics of tested materials. The temperature T is in o C.

	HCP	SBG

PhD, Université Paris-Est, Institut Navier, LMSGC

allée Kepler -77420 Champs-sur-Marne, France.2 New Address: BRGM-GEO,

avenue Claude Guillemin, BP 6009 -45060 Orléans cedex 2, France.3 Research Fellow, Université Paris-Est, Institut Navier, LMSGC 2 allée Kepler -77420 Champs-sur-Marne, France.

4 CNRS Research Director, Université Paris-Est, Institut Navier, LMSGC 2 allée Kepler -77420 Champssur-Marne,

France. 5 Unité de Génie Civil et Environnemental, Université Catholique de Louvain, B-1348 Louvain-la-Neuve, Belgique.

ACKNOWLEDGEMENTS

We are indebted to Damien Delmoitié and Patrick Zélis for their contribution on dielectric and ultrasonic measurements. We are grateful to Philippe Vié for the fabrication of the sintered glass beads, Alex Bertholet for the set-up of the Ultrasonic device and Eric Dupuis for the development of the data acquisition system.

China University Technology Press, 611-616. Timur, A. (1968). "Velocity of compressional waves in porous media at permafrost temperatures. " Geophysics, 33, 584-595. Wang, K., Monteiro, P., Rubinsky, B., and Arav, A. (1996). "Microscopic study of ice propagation in concrete." ACI Materials Journal, 93, 370-377. Zakri, T., Laurent, J.-P., and Vauclin, M. (1998). "Theoretical evidence for Lichtenecker's mixture formulae based on the effective medium theory." Journal of Physics D: Applied Physics, 31, 1589-1594. Zuber, B. and Marchand, J. (2004)