T Sayari

T Honorio

S Mohaine

F Robert

J.-L Adia

M Lion

P Gotteland

C Clergue

L 'aloia

F Benboudjema

CONCRETE DRYING KINETICS: DEVELOPMENT OF AN ACCELERATED DRYING PROTOCOL IN FIRE TESTINGS

Keywords: concrete drying kinetics, water content, concrete spalling fire, microstructure

This work proposes an accelerated drying protocol that allows reaching a hydric and mechanical state representative of normal service conditions while optimizing the storage time of the specimen before the fire resistance test. The temperature and relative humidity conditions are chosen in a range where microstructure changes are insignificant. An extensive experimental campaign on four concrete formulations is performed, including drying experiments (ambient and accelerated) and fire tests with different durations depending on the type of storage (normalized vs. accelerated drying). Then, the accelerated protocol is proposed based on the modeling of drying at moderate temperature. According to the predictive study, the target moisture content could be achieved under several drying conditions, showing that ad hoc solutions can be proposed. The reproducibility of the accelerated drying protocol is essential to ensure widespread application in research and industrial laboratories. This article focuses on the assessment of the water status.

INTRODUCTION

The fire behavior of concrete is a critical aspect of civil engineering works since spalling, resulting from fire exposition, can lead to severe durability and safety problems. Fire characterization tests are therefore required to study and prevent spalling. The spalling of concrete is a complex phenomenon involving several mechanisms, such as water vapor pressure (liquid water can also be thermal pressurized), thermomechanical stresses, and deformation incompatibility between cement paste and aggregates [START_REF] Hertz | Heat-induced explosion of dense concretes[END_REF]- [START_REF] Mindeguia | Experimental discussion on the mechanisms behind the fire spalling of concrete[END_REF]. Furthermore, it is strongly dependent on the hydric state and material permeability [START_REF] Maier | On the effect of pore-space properties and water saturation on explosive spalling of fire-loaded concrete[END_REF]. This means that if the concrete water content is high the day of the fire test and if it has low transfer properties, its resistance to spalling would be minor. The fire resistance test standard NF EN 1363-1 stipulates that at the testing time, the condition of the specimen should be representative of its normal service condition (mechanical strength and water content), considered to be obtained at equilibrium after storage in an ambient atmosphere. A conditioning period of at least three months is preconized. Drying processes play, therefore, a crucial role in the characterization of concrete susceptibility to spalling and should be studied carefully.

Concrete drying is a complex phenomenon involving several mechanisms, i.e., molecular diffusion, Knudsen diffusion, surface diffusion, and permeation. The hydric and hygric gradient between the material and its external environment generates the movement of: liquid water, water vapor, and dry air. Several research works focus on the description of the drying mechanisms [START_REF] Xi | Moisture diffusion in cementitious materials Moisture capacity and diffusivity[END_REF]- [START_REF] Carette | Identifying the mechanisms of concrete drying: An experimental-numerical approach[END_REF] and show that concrete drying occurs in two main phases. The first phase is fast and lasts only a few hours to a few days. The second phase is much slower and evolves as a square root of time. This last phase is even slower in massive structures. So, the preconized conditioning period may underestimate the fire resistance due to higher spalling susceptibility associated with high water content. This shortcoming is also reported by [START_REF] Lenglet | 2nd international RILEM Workshop on concrete spalling due to fire exposure[END_REF], who observed that a longer normalized conditioning time reduces the susceptibility of concrete to spalling. This study aims at optimizing the fire test protocol by reducing the conditioning period via the definition of a hydric state representative of a structure with a few years of exploitation. The approach adopted is based on the combination of an experimental and numerical study of drying in concrete. The experimental campaign allows defining the material parameters (porosity, density, etc.), characterizing the drying kinetics (desorption isotherm test, mass loss monitoring, water content measurement, etc.), and validating the numerical predictions of the water profiles in concrete structures.

MATERIAL AND METHODS

Materials

To extend the reproducibility of the accelerated drying protocol and investigate its effectiveness on a wide range of concrete, four mix designs covering two prominent families of concrete were selected for this study. The table below shows the main characteristics of each mix design. The average compressive strength fcm is measured at 28 days on 12 specimens per concrete. The porosity is measured at 28 days also according to the AFPC-AFREM recommendations. PERFDUB38 mix contains 8,4 % of silica fume and PERFDUB41mix contains 25 % of metakaolin.

Concrete drying simulation

Drying at an ambient temperature

We briefly recall the nonlinear diffusion equation describing concrete drying. The formulation is based on the assumption of mass conservation and allows simulating the movement of water in its liquid and vapor forms in cementitious materials using the Darcy and Fick laws [START_REF] Mainguy | Modèles de diffusion non linéaire en milieux poreux. Applications a la dissolution et au séchage des matériaux cimentaires[END_REF]. The liquid phase is assumed to be incompressible, so the density constant. The gaseous phase decomposes into dry air and water vapor. An isothermal temperature at 20 °C is considered [START_REF] Thiery | Modélisation du séchage des bétons[END_REF], [START_REF] Mainguy | Role of Air Pressure in Drying of Weakly Permeable Materials[END_REF]. The Van Genuchten model is used to simulate the desorption isotherm [START_REF] Van Genuchten | A Closed-form Equation for Predicting the Hydraulic Conductivity of Unsaturated Soils[END_REF]. Under these assumptions, the balance equation used to simulate the concrete drying at ambient temperature reads: where ρl liquid density, ρv vapor density, Φ porosity, pc capillary pressure, Sl liquid water saturation degree, μl dynamic liquid viscosity, Dv diffusion coefficient, Kl apparent permeability, Mw molar density, R gas constant, T the temperature and t time.

(

) 1
Boundary conditions are Dirichlet-type ones. [START_REF] Carette | Identifying the mechanisms of concrete drying: An experimental-numerical approach[END_REF], [START_REF] Bakhshi | Moisture loss characteristics of cement-based materials under early-age drying and shrinkage conditions[END_REF], [START_REF] Zhang | Effects of model boundary conditions on simulated drying kinetics and inversely determined liquid water permeability for cement-based materials[END_REF] show that this condition is sufficient to simulate drying in cementitious materials.

Drying at a moderate temperature

The choice of temperature is limited to moderate temperatures (up to 60 °C) to assume that no significant microstructural changes occur safely. For this study, the temperature is taken equal to 40 °C ± 2°C. The temperature effect on the material parameters (porosity, density, intrinsic permeability, etc.) is considered negligible [START_REF] Cen | Eurocode 2: calcul des structures en béton -Partie 1-2: Règles générales -Calcul du comportement au feu[END_REF]. We explicitly account for the temperature-dependence of the water viscosity [START_REF]Water -Density Viscosity Specific Weight | Engineering Reference and Online Tools[END_REF] (cf. equation ( 2)), the saturation vapor pressure [START_REF] Bary | Simulations of the thermo-hydro-mechanical behaviour of an annular reinforced concrete structure heated up to 200°C[END_REF] (cf. equation ( 3)), the intrinsic diffusivity (cf. equation ( 5)) [START_REF] Vries | On the value of the diffusion coefficient of water vapour in air[END_REF] and the desorption isotherm [START_REF] Chhun | Modélisation du comportement thermo-hydro-chemo-mécanique des enceintes de confinement nucléaire en béton armé-précontraint[END_REF] (cf. equation ( 4)):

𝜇 𝑙 = 𝐴 𝜇 10 𝐵 𝜇 𝑇-𝐶 𝜇
where T temperature, Aμ=2,414e -5 Pa, Bμ=247,8 K, and Cμ=140 K.

(2)

𝑝 𝑣𝑠 = 𝑝 𝑎𝑡𝑚 . 𝑒 40500 𝑅 ( T-373 373𝑇 )
where R=8,314 J mol -1 K -1 and patm= 101325 Pa.

(

) 3
𝑆 𝑙 = [(( 𝑝 𝑐 𝑎 𝑚𝑢 (𝑇) ) 𝑏 𝑚𝑢 𝑏 𝑚𝑢 +1 + 1) - 1 𝑏 𝑚𝑢 ] , 𝑎 𝑚𝑢 (𝑇) = 𝑎 𝑚𝑢0 𝑒 (- 𝑇-𝑇 0 𝑇 𝑘𝑣𝑔𝑛 -𝑇 0 )
where T0 reference temperature, amu0, bmu and Tkvgn material parameters.

(4)

𝐷 𝑣0 = 2,17.10 -5 𝑝 𝑎𝑡𝑚 𝑝 𝑔 ( 𝑇 𝑇 0 ) 1,88
pg gas pressure taken equal to patm.

(

) 5
Drying kinetics differ from one concrete to another depending on its porosity, pore size distribution, permeability, etc. Thierry [START_REF] Thiery | Modélisation de la carbonatation atmosphérique des matériaux cimentaires : prise en compte des effets cinétiques et des modifications microstructurales et hydriques[END_REF] shows that ordinary concretes drying is driven mainly by the diffusion of water vapor, whereas in high-performance concrete concretes, the drying is driven primarily by the permeation of liquid water. Nonetheless, the temperature does not act the same way on the diffusion as on the permeation. The problem is even more complicated since the weight of each mechanism in the drying process is unknown. The acceleration factor obtained on VERCORS concrete with the adopted approach is compared to other methods in literature where an equivalent transfer coefficient is considered [START_REF] Granger | Comportement différé du béton dans les enceintes de centrales nucléaires : analyse et modélisation[END_REF]- [START_REF] Adia | Concrete Drying Modelling in a Variable Temperature Environment[END_REF]. Figure 1 shows an acceleration factor of about 3.5 for diffusion and about 1.5 for permeation, going from 20°C to 40°C. To avoid this problem, an equivalent transfer coefficient will be used in the drying simulation, as shown in equation [START_REF] Xi | Moisture diffusion in cementitious materials Moisture capacity and diffusivity[END_REF]. The hydration and the wall effects are not considered in this model. [START_REF] Granger | Comportement différé du béton dans les enceintes de centrales nucléaires : analyse et modélisation[END_REF], [START_REF] Cheikh Sleiman | Influence of common simplifications on the drying of cement-based materials up to moderate temperatures[END_REF].

𝜌 𝑙 [𝛷 (1 - 𝜌 𝑣 𝜌 𝑙 )] 𝜕𝑆 𝑙 𝜕𝑡 = div [ 𝜕𝑝 𝑐 𝜕𝑆 𝑙 (𝐾 é𝑞 )𝑔𝑟𝑎𝑑(𝑆 𝑙 )] (6)

Experimental program

Drying kinetics characterization

The experimental characterization of the drying process consisted firstly of measurements of mass loss, porosity, and desorption isotherms for the numerical identification of the concrete transfer parameters.

Mass loss experiments were conducted on cylinders Ø 11x10 cm to accelerate the equilibrium achievement while respecting the Representative Elementary Volume (generally superior or equal to 2.5 times the diameter of the largest aggregate). Thinner specimens (a few mm thick) were used to measure the desorption isotherm. Secondly, water content measures were carried out at different conditioning durations to validate the numerical prediction. The tests were launched in the two conditioning modes: -23 °C, 50 % RH which corresponds to the normalized conditioning, and -40 °C, 65 % RH which corresponds to the accelerated drying (defined numerically for each concrete). The reference geometry corresponds to the geometry of the test bodies of the fire test, i.e., slabs of dimensions 1.7 x 1 x 0.3 m 3 . for technical, operational reasons and to simplify the numerical study and the of water profile evolution is carried out on cylinders Ø 15x30 cm which dry unidirectionally in height. This drying mode represents an element located in the middle of the slab (Figure 2).

Fire testing

Table 2 summarizes all the fire tests performed so far and provides the different parameters of the tests for each family of concrete (fire curve, loading, geometry of the specimens, and duration and conditioning mode). The tests carried out at three months of standardized conditioning correspond to the usual tests on concrete structures. An examination under the ISO 834 fire curve is conducted on PERFDUB concrete proof bodies subjected to the accelerated drying protocol. This accelerated drying is representative of a 3-month drying period under standard conditions according to the first approach of the numerical study. The objective of this first test is to evaluate the sensitivity to spalling in the drying modes (normalized vs. representative accelerated ones). One slab per concrete in an unloaded configuration is tested in each condition.

RESULTS AND DISCUSSION

Prediction of concrete drying under normalized conditioning

In the first part, the following figures illustrate the numerical results after identification of material parameters for VERCORS concrete compared to the experimental results. Figure 3 compares the theoretical/experimental results of the mass loss of a cylinder Ø 15x30 cm of VERCORS concrete drying unidirectionally. The results are consistent given the uncertainty associated with this measurement (± 0.1 % due to material variability).

Figure 4 shows the theoretical/experimental results of the average water content per 2.5 cm in VERCORS concrete. With only the first section, the theoretical and empirical results are very close considering the standard deviation on the experimental measurement. For the first slice, which corresponds to the skin area, the drying is underestimated, which may suggest the impact of wall effects (i.e., the impact of the formwork, microcracking, hydration evolution, etc.). The wall effects are not considered in the drying model at this study stage.

Development of an accelerated drying protocol

Numerical prediction and validation

Theoretically, to reach a similar water profile as a 23 °C, 50 % RH curing after 90 days, it is sufficient to cure for 15 days at 40 °C, 65 % RH with a 7-day conservation period in sealing condition before accelerated curing. This period is necessary to allow the hydration to progress. The allowed deviation is chosen to be equal to ± 0,4 % in absolute value for the water content. It is set to be equal to the maximum uncertainty value in the experimental measurements after a literature review. Experimental results are used in the numerical identification of drying parameters, and their uncertainty will propagate to the numerical results. In this case, it is not possible to set a more restrictive permissible deviation. The accelerated drying protocol is tested in the short term on PERFDUB38 concrete. The experimental results show good consistency between the moisture content measured in the two drying modes. The deviation is minimal and remains below the permissible deviation. This first test shows that it is possible to achieve a given moisture profile target by modifying the drying conditions and duration. This first short-term test of the accelerated protocol provides more confidence for the future and a test targeting a water profile at a few years of accelerated drying.

Effect of accelerated drying on microstructure

Carbonation and microcracking are the two main possible microstructural changes that may occur because of the drying acceleration at moderate temperature.

Carbonation is maximal for a relative humidity range of about [40 % -70 %]. The effect of temperature is more controversial in the literature [START_REF] Mori | Carbonation of blast-furnace slag cement[END_REF]- [START_REF] Papadakis | Fundamental modeling and experimental investigation of concrete carbonation[END_REF], and the comparison of studies is not evident since the materials and test conditions are different. Papadakis and al. [START_REF] Drouet | Carbonation of hardened cement pastes: Influence of temperature[END_REF] have recently shown, through accelerated carbonation tests at temperatures ranging from 20 °C to 80°C, that carbonation increase linearly with temperature in concretes formulated with CEM I. However, the increase in the depth of carbonation is not significant. Therefore, the effect of the accelerated drying protocol (40 °C, 65 % RH) on the increase of the carbonation risk remains low. The evaluation of carbonation at 23 °C, 50 % RH and at 40 °C, 65 % RH confirms this hypothesis. The experiment consists of spraying Phenolphthalein on the Ø 11x10 cm cylinders having dried for 13 months uni-directionally. At 23 °C, 50 % RH, the carbonate thickness observed in PERFDUB concretes from 0 to 1 mm, from 0 to 3 mm in VERCORS concrete and from 2 to 5 mm in CETU concrete. At 40 °C, 65 % RH, the carbonate thickness observed in PERFDUB concretes varies from 0 to 2 mm. It varies in the same way in VERCORS concrete and from 2 to 3 mm in CETU concrete (Figure 6). Microcracking is observed with a Keyence optical microscope on Ø 11x10 cm cylinders dried at 40 °C, 65 % RH, and cylinders of the same geometry and size dried at 23 °C, 50 % RH. In both conditioning modes, the drying lasted 13 months and occurred in all directions. The maximum cracking rate and maximum openings are observed in the CETU concrete. Minor cracking is observed in VERCORS concrete. The PERFDUB concrete shows almost no cracking. This observation is valid for both drying modes. Comparing the cracking rate observed for the two drying modes in each concrete, the microcracking is almost identical for PERFDUB concretes, and more important in the case of drying at 23 °C for both VERCORS and CETU concretes. In this case, the effect of the deformations due to the thermal gradients passing from 23 °C to 40 °C might be negligible. The impact of the water gradient will be investigated to affirm whether drying at 23 °C, 50 % RH, induces more microcracking than drying at 40 °C, 65 % RH or not. For the moment, the result obtained is encouraging in the sense that the microcracking rate is not higher in the case of the accelerated drying.

Spalling sensibility: effect of the concrete conditioning

The effect of the conditioning mode on spalling due to fire has been tested on both PERFDUB mixes ( §0). Table 4 shows the comparison of spalling sensibility, water content, and compressive strength between the normalised conditioning and the accelerated drying in the PERFDUB38 concrete mix.

Results show good representativeness of the accelerated protocol compared to the normalized conditioning of 3 months. In the case of a drying time of 3 months, only the first centimeters of the specimen have dried, the core remains almost saturated. Therefore, for a water status representative of a short duration standardized drying (≤ 3 months), the 40 °C, 65 % RH accelerated drying protocol is validated. Furthermore, this accelerated drying protocol is also validated for a representative skin water status regardless of the drying duration. The same findings are observed on PERDUB41 concrete mix.

CONCLUSION AND PERSPECTIVES

In this work, the experimental study allowed us to identify the concrete drying parameters and to validate the theoretical predictions. The drying prediction allowed us to evaluate the hydric profile evolution in the reference structure in short and long term and to define the target profile and then the optimal drying conditions to reach it in a few months only. The main conclusions are:

-The modeling of the drying phenomenon at different thermal and hydric solicitations shows that temperature plays an essential role in the acceleration of concrete drying. The relative humidity controls the water content on the surface of the test bodies but can delay in counterpart the drying if its value is high. -According to the numerical results so far, drying during 15 days at 40°C, 65% RH after preservation in sealed condition for 7 days allows to reach the same water profile as for drying 90 days at 23 °C, 50% RH.

-Theoretically, several options of accelerated drying protocol are possible. A target water profile could be reached by different temperature and relative humidity combinations. In our study, the temperature is limited to 40°C in order to not to alter significantly the material microstructure. -The first test of the accelerated drying protocol shows good consistency between theoretical predictions and experimental results of concrete water content. It also indicates that the accelerated drying did not modify the fire behavior. Perspectives of this study include optimizing the numerical study by separating the two main drying mechanisms: and optimizing and refining the accelerated drying protocol by type of concrete. Also, we envision validating the accelerated drying protocol in the long term by assuming a target hydric state in a structure with a few years of operation. Applications of this study are broad and can go beyond fire behavior because of the strong dependence on the water profile of, for instance, delayed behavior and pathologies in concrete.

Figure 1 -

 1 Figure 1 -Diffusion and permeation acceleration factor as a function of temperature. Comparison with factors obtained by other approaches in literature[START_REF] Granger | Comportement différé du béton dans les enceintes de centrales nucléaires : analyse et modélisation[END_REF],[START_REF] Cheikh Sleiman | Influence of common simplifications on the drying of cement-based materials up to moderate temperatures[END_REF].

Figure 2 -

 2 Figure 2 -The reference geometry and the drying configuration of the representative specimens.

Figure 3 -

 3 Figure 3 -Theoretical and experimental mass loss of a cylinder Ø 15x30 cm of VERCORS concrete drying unidirectionally.

Figure 4 -

 4 Figure 4 -Theoretical and experimental water content per cylinder Ø 15x30 cm slice of uni-directionally drying VERCORS concrete.

Figure 5 -

 5 Figure 5 -Predicted water profile of a uni-directionally drying PERFDUB38 concrete after 90 days of normalized drying compared to the profile after 15 days of accelerated drying. On the right, the average water content per 2,5 cm from the drying surface to the core of Ø 15x30 cm cylinder.

Figure 6 -

 6 Figure 6 -Evaluation of carbonation by colored indicator (Phenolphthalein) in specimens of CETU concrete after unidirectional drying for 13 months at 23°C, 50% RH (left), and at 40 °C, 65 % RH (right).

Table 1 -

 1 The main characteristics of the studied concrete formulations.

	Concrete formula	Source	fcm [MPa]	Porosity [%] *
	PERFDUB38	French national project PERFDUB [12]	86 ± 2,3	12,3 ± 0,2
	PERFDUB41	French national project PERFDUB [12]	93 ± 3,3	11,7 ± 0,2
	VERCORS	EDF [13]	45 ± 2,0	15,4 ± 0,3
	CETU (containing polypropylene fibers)	CETU	44 ± 3,3	21,1 ± 0,9
	* Corresponds to the standard deviation.

Table 2 -

 2 Assessment of the fire tests carried out to date

	Concrete	Fire curves	Loading [MPa]	Specimens	Conditioning period	Conditioning mode
		ISO 834	0 and 4	Two slabs (1,7 x 1 x 0,3 m 3 )	90 days	Normalized conditioning
	PERFDUB
		ISO 834	0	Two slabs (1,7 x 1 x 0,3 m 3 )	15 days	Accelerated drying
	Vercors and CETU	HCM	0 and 4	Two slabs (1,7 x 1 x 0,3 m 3 )	90 days	Normalized conditioning

Table 3 -

 3 Selection of microscopic cracking images (3300 x 2500 µm) on the PERFDUB38 and CETU concretes in the two drying modes (23 °C, 50 % RH) and (40 °C, 65 % RH).

	Concrete	23 °C, 50 % RH	40 °C, 65 % RH
	PERFDUB38
	CETU

Table 4 -

 4 The impact of the storage conditions on the spalling susceptibility PERFDUB38 concrete mix.

		Storage	90 days under 23 °C, 50 % RH	15 days under 40 °C, 65 % RH
		fcm [MPa]	91	88
		Water content
		Max depth [mm]	21	25
		Mean depth [mm]	12	12
	Spalling	topography of the
		spalling


Takwa Sayari
email: takwa.sayari@ens-paris-saclay.fr

Tulio Honorio

Farid Benboudjema

Rita Tabchoury

Jean-Luc Adia

Christian Clergue

Numerical and experimental investigation of wall effect in concrete

Keywords: durability, wall effect, concrete drying, aggregates distribution, transport properties, mechanical properties, Grand Canonical Monte Carlo (GCMC) simulations, micromechanics

This study presents a new numerical approach using tools developed to perform a molecular simulation to investigate the wall effect on aggregates and mortar distribution in concrete. Aggregates are represented by spheres interacting via a generalized truncated Lennard-Jones potential. This approach allows obtaining the particle profiles according to the reference frame of interest (e.g., the confined directions). Then the particle-based distributions are transformed into continuum profiles of volume fractions using a convolution. Based on volume fraction profiles, transport or mechanical properties are estimated by the Mori-Tanaka scheme from classical homogenization. Results are compared to experimental work. The numerical method could be generalized and used for other applications in different fields. In civil engineering, perspectives include using the aggregate distribution to conduct a finer analysis, and the results would be extremely relevant for the prediction of the water content profile and the evolution of pathologies such as carbonation, corrosion, ISR, etc.

Introduction

One of the main advantages of using concrete is that the material in its fresh state can be poured into formworks and assume the most variable shapes. Nevertheless, the casting is not without effect on concrete. The formwork perturbates the arrangement of aggregates; this effect is known as 'the wall effect' and it is generally neglected in both experiments and numerical calculations. So far, there is no validated method to quantify the gradation of transport and mechanical properties induced by the wall effect. The manifestation of the wall effects can be decomposed on the contributions of (i) the effects of aggregate distribution from the surface; and (ii) the interactions with the environment such as drying (which may induce cracks because of drying shrinkage gradient and strain incompatibility between cement paste and aggregates), hydration slowing down (due to rapid water evaporation on exposed surfaces), and carbonation in long-term. Altogether, these aspects lead to changes in the properties of a superficial layer, and these changes can be relevant to some applications of cement-based materials and concrete structures potentially affecting their performances [START_REF] Kreijger | The skin of concrete composition and properties[END_REF], [START_REF] Bissonnette | Concrete Surface Engineering[END_REF]. The skin of the concrete is in direct contact with aggressive agents such as chlorides and carbon dioxide [START_REF] Bentz | Influence of aggregate characteristics on concrete performance[END_REF]. If it has weaker properties than the bulk area of the material, this implies an increase in the risk of the appearance of pathologies or the acceleration of its appearance. In reinforced concrete, the concrete cover (generally 3 cm thick on average) is critical to prevent serious durability problem such as corrosion. A better understanding of the factors affecting the properties of this zone might contribute to improve durability prediction and strategies of design for durability.

Experimental studies have identified property gradation due the wall effects in cement-based materials [START_REF] Kreijger | The skin of concrete composition and properties[END_REF], [START_REF] Bissonnette | Concrete Surface Engineering[END_REF], [START_REF] Liu | Study on microstructure and composition distribution of concrete surface zone based on fractal theory and XCT technology[END_REF]. From the early 1960s, the importance of considering the effect of the (granular) inclusion of concrete was highlighted as a factor influencing the compressive strength of concrete [START_REF] De Larrard | Structures granulaires[END_REF]. [START_REF] Liu | Study on microstructure and composition distribution of concrete surface zone based on fractal theory and XCT technology[END_REF] showed that the surface of the concrete is mainly composed of cement paste, and the coarse aggregates start to appear as one moves away from the walls. Numerical approaches have also been adopted to quantify how formworks alters aggregate/cement paste distribution adjacent to the walls. Simulation of hard spheres [START_REF] Zheng | Simulation of Two-Dimensional Aggregate Distribution with Wall Effect[END_REF] and ellipsoidal [START_REF] Xu | Effects of particle size distribution, shape and volume fraction of aggregates on the wall effect of concrete via random sequential packing of polydispersed ellipsoidal particles[END_REF] particles allowed the quantification of aggregates and cement paste volume fraction profiles in simplified system. To date, no modelling work have studied how such microstructure gradation affects property on the cover zone. The goal of this work is, on the one hand is to estimate the aggregates volume fraction and concrete properties profiles and to quantify numerically the gradation induced by the formwork. On the second hand, the experimental work shows whether the skin effect modifies or not the overall behavior of the concrete (about transfer and mechanical properties).

MATERIAL AND METHODS

Materials

VeRCoRs REF concrete is used in this study for both numerical and experimental work. This concrete formula is developed at EDF's laboratories for the project aiming to investigate the ageing of containment vessels. VeRCoRs concrete is an ordinary concrete C35/40 and 0.52 water-cement ratio. Table 1 shows the composition of VeRCoRs concrete. Experiments are conducted on prismatic 7x7x28 cm3 specimens which are widely used in civil engineering laboratory testing. The specimens are manufactured and then stored under sealed conditions for 60 days. Under this condition, drying (and the resulting differential cement hydration progress) is limited but the appearance of microcracks induced by desiccation shrinkage might occur. and second to enable the hydration reaction to progress. Otherwise, these parameters would interfere the formwork effect.

Experiments

Mass loss is monitored on 9 specimens while drying at 23°C, 50%HR over one surface (upper, lower, and left or right surface, 3 specimens per surface state). The studied drying configurations follow the scheme depicted in Fig. 1. The aim of this experiment is to investigate the impact of concrete surface state on transport properties such as permeability. The second experiment consists of conducting 3-point bending test considering the same previous configurations of specimen's surface. The idea is to investigate the effect of concrete skin state on its mechanical properties such as Young modulus and tensile strength. Another experimental investigation of the wall effect consists of analysing the microcracking rate in molded vs non-molded concrete surfaces using a microscopic observation on a 7x7x28 cm specimen.

Simulation granular packing under confined geometries

We propose a new approach to simulate aggregate distribution in confined geometries based on a tool developed to perform molecular simulations (using LAMMPS) [START_REF] Plimpton | Fast Parallel Algorithms for Short-Range Molecular Dynamics[END_REF]- [START_REF] Plimpton | Effect of interatomic potential on simulated grain-boundary and bulk diffusion: A molecular-dynamics study[END_REF]. The idea is to represent the aggregates by polydisperse spheres, adopt interactions approximating hard-sphere potential, and perform Monte Carlo simulations to fill the confined geometry. Post-processing allows computing aggregates profiles that will be used in the estimation of property gradation.

Pseudo Hard-Sphere potential

The hard-sphere (HS) pair potential reads:

𝑈 𝐻𝑆 (𝑟 𝑖𝑗 ) = { ∞, 𝑟 𝑖𝑗 < 𝜎 𝑖𝑗 0, 𝑟 𝑖𝑗 ≥ 𝜎 𝑖𝑗 (1)
where 𝑟 𝑖𝑗 is the distance between the centers of particles 𝑖 and 𝑗, and

𝜎 𝑖𝑗 = 𝜎 𝑖 +𝜎 𝑗 2
is the average of the diameters 𝜎 𝑖 and 𝜎 𝑗 of particles 𝑖 and 𝑗, representing their center distance when they are in contact. The discontinuous and non-bounded nature of this potential make difficult its numerical implementation. One solution is the use of a pseudo-HS potential that is continuous. The HS potential can be approximated by a shifted and truncated generalized Lennard-Jones (or Mie) potential [START_REF] Jover | Pseudo hard-sphere potential for use in continuous molecular-dynamics simulation of spherical and chain molecules[END_REF]:

𝑈 𝐻𝑆 𝑃 (𝑟 𝑖𝑗 ) = { 𝜆 𝑟 𝜆 𝑟 -𝜆 𝑎 ( 𝜆 𝑟 𝜆 𝑎 ) 𝜆 𝑎 𝜆 𝑟 -𝜆 𝑎 𝜖 𝑖𝑗 [( 𝜎 𝑖𝑗 𝑟 𝑖𝑗 ) 𝜆 𝑟 -( 𝜎 𝑖𝑗 𝑟 𝑖𝑗 ) 𝜆 𝑎 ] + 𝜖 𝑖𝑗 , 𝑟 𝑖𝑗 < 𝑟 𝑐𝑢𝑡𝑜𝑓𝑓 0, 𝑟 𝑖𝑗 ≥ 𝑟 𝑐𝑢𝑡𝑜𝑓𝑓 (2)
where 𝜆 𝑟 and 𝜆 𝑎 are the exponents of the repulsive and attractive terms of the Mie potential, respectively; 𝜖 is the potential well depth; and 𝑟 𝑐𝑢𝑡𝑜𝑓𝑓 = (

𝜆 𝑟 𝜆 𝑎 ) 𝜆 𝑎
𝜆 𝑟 -𝜆 𝑎 𝜎 𝑖𝑗 is the pair-wise cutoff distance. Therefore, only the repulsive part of the potential is effective, and the shift makes the potential smooth (class 𝐶 1 ) for all 𝑟 𝑖𝑗 . Jover et al. [START_REF] Jover | Pseudo hard-sphere potential for use in continuous molecular-dynamics simulation of spherical and chain molecules[END_REF] shows that using 𝜆 𝑟 = 50 and 𝜆 𝑎 = 49 lead to a good reproduction of the HS behavior. For polydisperse systems, the Lorentz-Berthelot mixing rule 𝜖 𝑖𝑗 = √ 𝜖 𝑖𝑖 𝜖 𝑗𝑗 and 𝜎 𝑖𝑗 = (𝜎 𝑖𝑖 + 𝜎 𝑗𝑗 )/2 can be applied to describe the interactions among the different classes of particle size.

The confining walls interaction are described by the usual 12-6 Lennard-Jones potential but also in a truncated form and with a shift:

𝑈 𝐿𝐽 𝑊 (𝑟 𝑖𝑤 ) = { 4𝜖 𝑤 [( 𝜎 𝑖 𝑟 𝑖𝑤 ) 12 -( 𝜎 𝑖 𝑟 𝑖𝑤 ) 6 ] + 𝜖 𝑤 , 𝑟 𝑖𝑗 < 𝑟 𝑐𝑢𝑡𝑜𝑓𝑓 0, 𝑟 𝑖𝑗 ≥ 𝑟 𝑐𝑢𝑡𝑜𝑓𝑓 (3)
where the distance between the center of particle 𝑖 and the wall, and 𝜖 𝑤 is the energy well depth associated with wall interactions. The LJ wall is placed at the edge of the simulation box, and the cutoff in this case is 𝑟 𝑐𝑢𝑡𝑜𝑓𝑓 = 𝜎 𝑖 2 ⁄ . With this assumption, the wall function as a hard wall. Fig. 2 shows the LJ potential (usual 12-6 and truncated LJ).

GCMC simulations of concrete granular packing

Grand Canonical Monte Carlo (GCMC) simulations are adapted to model system with varying number of particles and is used here to simulate the granular packing in a confined system. In a GCMC simulation a particle insertion is accepted following the probability [START_REF] Frenkel | Understanding Molecular Simulation -2nd Edition[END_REF].

acc(N→N+1)=min (1, V Λ 3 (N+1) exp [- ∆U-μ k b T ]) (4)
where 𝑉 is the volume, 𝛬 is the thermal De Broglie wavelength, ∆𝑈 = 𝑈(𝑁) -𝑈(𝑁 + 1) is the variation in potential energy due to the trial insertion, 𝜇 is the chemical potential, 𝑘 𝑏 is the Boltzmann constant, and 𝑇 is the temperature. Note that in our simulations, the temperature and chemical potential do not necessarily need to have a physical meaning and can be used to enforce particle insertion and to promote a better packing if combine with dynamics. Removals can be prevented by choosing a large value of 𝜇, which favors insertion of particles.

Aggregate volume fraction profiles

The volume fraction of the aggregates can be computed directly from the particles using the convolution [START_REF] Nygård | Anisotropic Pair Correlations and Structure Factors of Confined Hard-Sphere Fluids: An Experimental and Theoretical Study[END_REF]:

ϕ i s (x)=N i (x)⨂P sph (x,R i )= ∑ N i (x j ) P sph (x j -x,R i ) j ( 5
)
𝑃 𝑠𝑝ℎ (𝑥, 𝑅) is the projection of the sphere with the center at the point o i and R is the radius of the sphere.

𝑃 𝑠𝑝ℎ (𝑟, 𝑅) = { 0, 𝑟 < -𝑅 𝜋(𝑅 2 -𝑟 2 ), 𝑟 < -𝑅 0, 𝑟 ≥ 𝑅 (6)

Estimating property gradation using micromechanics

Representative elementary volume (REV) and microstructure gradation Homogenization schemes can be used to estimate the effective properties of heterogenous materials in which a representative elementary volume (REV) can be defined. In a REV, all the heterogeneities (different phases present) and their spatial variability (in term of spatial distribution and orientation) should be statistically represented. A REV cannot be defined in systems with microstructure gradation, but a local approximation can be done. For the zone perpendicular to the microstructure gradation, the criteria to define a REV are respected. The local behaviour per slice can be approximated as the behaviour of a 3D medium with similar microstructure as the slice. In this framework, the usual homogenization schemes can be applied locally to obtain the evolution of properties as a function of the microstructure gradation. Mori-Tanaka homogenization scheme [START_REF] Mori | Average stress in matrix and average elastic energy of materials with misfitting inclusions[END_REF] is used to compute properties profile. The concrete in this case is considered a two-phase material, where mortar constitute the matrix and aggregates the inclusionary phase (cf. Fig. 3). More details about the used homogenization method are described in the next section.

Estimates of elastic properties

For an (N+1)-phase heterogeneous material with a matrix/inclusion morphology in which N isotropic spherical inclusions are randomly placed in a host medium (percolating matrix phase), the Mori-Tanaka estimate of the effective bulk and shear moduli are, respectively [START_REF] Mori | Average stress in matrix and average elastic energy of materials with misfitting inclusions[END_REF]:

K MT -K 0 K MT + 4 3 G 0 = ∑ f i K i -K 0 K i + 4 3 G 0 N i=1 (7) G MT -G 0 G MT +H 0 = ∑ f i G i -G 0 G i +H 0 N i=1 (8)
with

H 0 = G 0 ( 3 2 K 0 + 4 3 G 0 ) K 0 +2G 0 (9)
where 𝑓 𝑖 is the volume fraction of the phase 𝑖. In this expression, the subscript 0 stands for the matrix phase and the subscripts 𝑖 ∈ [1, 𝑁] for the inclusions.

The local effective young modulus 𝐸 𝑀𝑇 and Poisson ratio 𝜈 𝑀𝑇 can be computed from the bulk and shear moduli using

E MT =9K MT G MT /(3K MT +G MT ) ( 10
)
ν MT =(3K MT -2G MT )/[2(3K MT +G MT )] (11)
Estimates of conductivity-like properties By mathematical analogy, the upscaling of the thermal conductivity, electrical conductivity, dielectric permittivity, and diffusivity are equivalent [START_REF] Torquato | Random Heterogeneous Materials: Microstructure and Macroscopic Properties[END_REF]. The expression of Mori-Tanaka (or Maxwell-Garnet) scheme for conductivitylike properties for an (N+1)-phase heterogeneous material with a matrix/inclusion morphology in which 𝑁 isotropic spherical inclusions are randomly placed in a host medium (percolating matrix phase) is given by [START_REF] Mori | Average stress in matrix and average elastic energy of materials with misfitting inclusions[END_REF], [START_REF] Hashin | A variational approach to the theory of the elastic behaviour of multiphase materials[END_REF]:

λ MT -λ 0 λ MT +2λ 0 = ∑ f i λ i -λ 0 λ i +2λ 0 N i=1 ( 12
)
where again the subscript 0 stands for the matrix phase and the subscripts 𝑖 ∈ [1, 𝑁] for the inclusions. Table 2 reports the properties of each phase. Bulk and shear moduli are taken from [START_REF] Bentz | Influence of aggregate characteristics on concrete performance[END_REF]. The permeability of aggregates is very low compared to mortars permeability. In this study, aggregates permeability is considered null.

Mass loss monitoring

Figure 4 reports the mean mass loss measurements on 3 prismatic specimens for each drying configuration. Mass loss is very similar at the beginning of the drying process. Then it becomes higher for the non-molded surface drying. Samples drying over the bottom and the side surfaces (which were in contact with the formwork) show a similar drying behavior. The reported difference can be explained by the roughness of the unmolded surface and/or by the rate of micro-cracking compared to a molded surface.

3-point bending test

Microcracking rate in molded vs non-molded concrete surfaces

Microscopic visualization of molded and non-molded surfaces of 7x7x28cm concrete specimens shows a higher rate and more opened microcracks. Fig. 5 shows an illustration of the visualized microcracks. Numerical RESULTS: Polydisperse systems

1D 'confinement'

A calculation of the granular distribution in the concrete is carried out once the method is validated for a monodisperse distribution case. The application consists in simulating the granular distribution of a concrete formula starting from the granulometric curves. The simulated concrete is EDF's VeRCoRs concrete with two aggregate classes G4/11 and G8/16 as shown in Fig. 6. We consider 11 different diameters with a total volume fraction of 40% and a maximum diameter of Dmax = 25 mm. 1D and 3D confinement are studied. Confinement along the x-direction. Y independent configurations are tested to evaluate the filling variability, the effect of the confined width, and then the of the maximum aggregate diameter (by comparing the results obtained with a calculation based on a different aggregate distribution). The second part consists of performing a calculation considering a confinement configuration on a 7x7x28 cm VeRCoRs concrete specimen in a configuration similar to the one considered in the experimental part of this work., i.e. formwork on all surfaces + a free non-molded surface.

Fig. 6. Granulometric curves of VeRCoRs concrete aggregates

The configurational packing variability is evaluated by conducting 5 independent simulations in which the seed for the Monte Carlo simulation is changed. Fig. 7 reports the mean value and standard deviation of the aggregates volume fraction, and the estimated profile of the effective properties. The resulting variability is low on the aggregates volume fraction (maximum standard deviation is equal to 0,015) and very much lower on the effective properties. To evaluate the thickness of the zone perturbed by the wall, in Fig. 8 we compute the profile of the aggregate volume fraction over the first 50 mm of the confined direction for various confinement widths: 3Dmax, 4Dmax, 5Dmax, and 10Dmax. In the most confined system (3Dmax) the effects of the wall opposed walls superposes leading to an augmentation of the average volume fraction of aggregates even in the sample core. Confinement width does not have a major effect on the thickness disturbed by the wall: in all cases the distance perturbed by the wall is roughly Dmax /2. This observation corroborates previous studies indicating that the thickness disturbed by the formwork depends directly on the maximum aggregate diameter [START_REF] Granger | Comportement différé du béton dans les enceintes de centrales nucléaires : analyse et modélisation[END_REF]. We verify the validity of this observation for other aggregates granulometries in Fig. 9, in which aggregates used in other concrete mix designs are considered: F1 VeRCoRs concrete with Dmax = 25 mm, F2 CETU concrete with Dmax = 25 mm and a different aggregates distribution, and F3 the B11 concrete with Dmax = 12.5 mm (the total volume fraction is the same and is equal to 40%). The formulation with the lower Dmax exhibited a smaller thickness perturbed by the wall as expected. Thus, the disturbed thickness depends strongly the maximum aggregate diameter. It is quite inferior to half the maximum aggregate diameter. [START_REF] Granger | Comportement différé du béton dans les enceintes de centrales nucléaires : analyse et modélisation[END_REF] shows the gradation in the wall zone where the distribution of cement pastes and aggregates would be constant from a distance R (greater than the maximum aggregate diameter) away from the wall as reported in Fig. 10. The estimated permeability profile is compared with the results obtained by Kreijger about the estimation of the concrete properties over the first 5 mm. The results are comparable in mean value as reported in Fig. 11. However, the modulus is lower on the first millimeter according to the study [START_REF] Kreijger | The skin of concrete composition and properties[END_REF].

3D 'confinement'

We also perform simulation with 3D confinement using the same dimensions 7x7x28 cm3 of the specimen evaluated in the experimental part of this work. The results are presented in Fig. 12 and Fig. 13. Depending on the width according to x, the thickness disturbed by the formwork at x = 0 and x = 7 cm is equal to about 8 mm, which is 1/3 of the maximum aggregate diameter in this case. Between the two walls, there is some fluctuation around the homogeneous value of about 5 % on the volume fraction profile. Regarding the profile of the estimated effective properties, from the surface of the material to 5 mm in depth, the permeability on the surface is twice that of the bulk zone. Therefore, drying and transfer in general are faster in the disturbed zone than in the bulk zone. The Young's modulus in the disturbed zone loses 10% on average compared to its value in the bulk. Since the elastic constants correlate with the mechanical strength, it is expected that the skin zone is more prone to cracking than the bulk zone.

In the y-direction, y = 0 cm corresponds to the free unmolded surface; y = 7cm corresponds to a molded surface. The volume fraction profile of the aggregates shows that there are aggregates that stand out at the free surface. The same observations are made about the disturbed thickness and the interference on the estimated effective properties These results highlight the effect of the formwork on the concrete, which is expressed by a higher mortar content in a zone whose thickness depends on the maximum diameter of the aggregates.

Conclusion and perspectives

We combine experiments and simulation strategies to quantify the wall effects and the repercussion in the properties of the skin of concrete. We proposed a new simulation strategy based on tool developed to performed molecular simulations. This strategy enables generating simplified microstructures in confined geometries (1-, 2or 3D) for polydisperse particles and the computation of aggregates volume fraction profiles. These profiles can be then used as input for multiscale approaches to estimate property profiles.

In agreement with previous studies, we observe that the disturbance induced by the formwork only depends on the maximum size of the aggregates and its thickness is slightly lower than the diameter of the largest aggregate. Properties profiles obtained from the modelling approach show good agreement with the experimental results of the literature [START_REF] Kreijger | The skin of concrete composition and properties[END_REF], despite the simplifying assumption adopted (regarding sphericity of aggregates).

Fig. 1 .

 1 Fig. 1. The different drying configurations (1 = free non-molded surface, 2 = side surfaces in contact with the formwork and 3 = bottom surface in contact with the formwork).

Fig. 2 .

 2 Fig. 2. Usual 12-6 Lennard-Jones potential vs truncated LJ potential.

Fig. 3 .

 3 Fig. 3. Matrix and inclusionary phases of Mori-Tanaka homogenization used to estimate properties gradation.

Fig. 4 .

 4 Fig. 4. Mass loss in 7x7x28 cm specimens one-sided drying in different surface configurations: bottom surface in contact with the formwork, side surfaces in contact with the formwork and free non-molded surface.

Fig. 5 .

 5 Fig. 5. Concrete surface visualization using optical microscope.

Fig. 7 .

 7 Fig.7. Aggregates packing-related variability respectively on volume fraction profile fg, on Young modulus E, and on permeability k.

Fig. 8 .

 8 Fig. 8. Evolution of the aggregate volume fraction for different values of the confined width x over the first 50 mmvariable simulation box size dx, dy=dz=20xDmax.

Fig. 9 .

 9 Fig. 9. Aggregate volume fraction in 3 concrete formulas: F1 VeRCoRs concrete with Dmax = 25 mm, F2 CETU concrete with Dmax = 25 mm and a different aggregates distribution, and F3 the B11 concrete with Dmax = 12,5 mm (the total volume fraction is the same and is equal to 40%).

Fig. 10 .

 10 Fig. 10. Distribution of aggregates in a concrete element [20]. D is the maximum aggregate size.

Fig. 11 .

 11 Fig. 11. Comparison of Young's modulus profile estimated over the first 5 mm with the profile obtained by Kreijger in 1984 [1].

Fig. 12 .

 12 Fig.12. Evolution of the aggregate volume fraction, permeability and Young modulus in 7x7x28cm specimen according to x.

Fig. 13 .

 13 Fig.13. Evolution of the aggregate volume fraction, permeability and Young modulus in 7x7x28cm specimen according to y (y=0 presents the non-molded side of the specimen).

Table 1 .

 1 Composition of VeRCoRs concrete.

	______________________________________
	Components	Mass ________	Volume fraction ___________
	(kg.m -3 ) ______________________________________ (%)
	Cement	316.2	13.7
	Sand 0/4	820.1	35.4
	Gravel 4/11	439.7	18.9
	Gravel 8/16	543.5	23.4
	Water	192.9	8.3
	Superplasticizer ______________________________________ 2.6 0.1

Table 2 .

 2 Properties of phases.

	_______________________________________________
	Properties _____________________________________________________ Aggregates Cement matrix
	Bulk modulus (GPa)	67	13
	Shear modulus (GPa)	30	9
	Permeability (-) ________________________________________________ 0 1
	3	Experimental results

Table 3

 3

reports the results of the 3-point bending test. Young modulus in the first test configuration is lower than test conducted on molded surfaces as bearing. The roughness of the non-molded surface is the main reason for the decrease in the Young modulus value. Results in the two other test configurations are almost similar.

Table 3 .

 3 Young modulus measured on 7x7x28cm specimens considering different bearing surface condition (molded and non-molded).

	___________________________________________
	Condition of the	Specimens	Young
	bearing surface ____________________________________________ modulus (GPa)
	Free non-molded	1	22.78
	surface	2	23.87
	3 ____________________________________________ 22.65
	Left or right molded	4	32.04
	surface	5	31.99
	6 ____________________________________________ 31.91
	Bottom molded	7	32.34
	surface	8	32.34
	9 ______________________________________ 32.23

The experimental work shows that the effect of the formwork and the condition of the concrete surface have a very low impact on the overall behavior and on the global properties of the material. The effect of the formwork is counterbalanced by the effect of micro-cracking and the roughness of the non-formed surface. In view of this study, the wall effect on concrete drying process will be evaluated by exploiting the profiles of the granular distribution and the estimated effective properties i.e., the permeability profile. The overall effect on durability could be thus predicted.


Itaï Ben

ON THE EXPRESSIVE POWER OF QUANTIFIERS IN CONTINUOUS LOGIC

Keywords: 2020 Mathematics Subject Classification. 03C66 Continuous logic, quantifier, Vietoris topology

In this short note we compare the expressive power of real-valued continuous logic (or just continuous logic, in recent literature) with that of compact-valued continuous logic, proposed by Chang and Keisler. We conclude that the two logics have the same expressive power, and moreover, that this remains true if we replace the plethora of potential quantifiers of compact-valued logic with a single "primordial" one.

This note summarises remarks the author has made on various occasions, and to various people, regarding the similarities and differences between two variants of "continuous logic" that exist in the literature, and how to reconcile them: the one proposed by Chang and Keisler [START_REF] Chung | Continuous model theory[END_REF], which we shall call compact-valued logic, and that proposed in [START_REF] Itaï | Continuous first order logic and local stability[END_REF], which we call here real-valued. Despite striking similarities, the latter was conceived in ignorance of the former (at least as far as the author of the present note is concerned), essentially through reverse-engineering the idea of compact type spaces with open continuous variable restriction maps between them. Let us also refer to the True/False logic one learns in kindergarten as two-valued.

For most of this note we are going to consider "bare bones" versions of either logic, in a purely relational language, without a distinguished symbol for equality or distance. The latter means that a structure may contain two points that are formally distinct and yet indistinguishable by the logic, even with parameters. For our purposes this will be of no importance whatsoever (see a further remark to this effect at the end). Relation symbols applied to variable symbols (in the absence of function symbols) make atomic formulas, and these are combined using continuous connectives and appropriate first order quantifiers to generate all the formulas of the logic.

In compact-valued logic, the truth values of a formula ϕ belong to a fixed compact value space (this note being written in France, compact spaces are Hausdorff). In particular, the logic associates to each relation symbol P a compact value space X, and then the atomic formula P(x) is X-valued. If (ϕ i : i < α) is a sequence of, respectively, X i -valued formulas, and θ : ∏ X i → Y is continuous, then we view θ as a connective that combines the given formulas into a Y-valued formula θ • (ϕ i : i < α).

In real-valued logic, formulas take values in compact intervals of the reals, and connectives are continuous functions θ : R α → R (necessarily bounded on each product of compact intervals). In some texts, one restricts truth values to [0, 1], but the result is the same.

Remark. With these definitions, real-valued formulas are closed under limits of uniformly Cauchy sequences. Indeed, for any such a sequence (ϕ n ) there exists a continuous θ : R N → R such that θ • (ϕ n ) is the limit. Moreover, θ can be chosen in a manner that depends only on the convergence rate of (ϕ n ), so, possibly passing to a sub-sequence, a single such θ suffices (see the forced limit construction in [START_REF] Itaï | Continuous first order logic and local stability[END_REF]). By the Stone-Weierstraß Theorem, closing under this special θ plus, say, constants, addition and multiplication, is equivalent to closing under all possible continuous connectives.

Clearly, atomic real-valued formulas are also atomic compact-valued ones, and real-valued continuous connectives, once restricted to an appropriate product of compact intervals, are also compact-valued ones.

In the opposite direction, consider a compact-valued language L. For each X-valued relation symbol P of L, choose an embedding X ⊆ [0, 1] α , and let (P i : i < α) be new [0, 1]-valued relation symbols of the same arity. Let L R be the real-valued language consisting of all such symbols. An L-structure M will be tacitly identified with an L R -structure M R defined so P(a) = P i (a

) : i < α ∈ X ⊆ [0, 1] α .
Say that an X-valued L-formula ϕ(x) is coded in L R if for every continuous function θ : X → R, there exists an L R -formula ϕ θ (x) that agrees with θ • ϕ on all L-structures. An equivalent condition is that the same hold for a family Ξ of functions ξ : X → R that separates points in X. Indeed, under this assumption, the induced map X → R Ξ is continuous and injective, and therefore a topological embedding. By Tietze's Extension theorem, any continuous θ : X → R extends continuously to R Ξ , and then ϕ θ = θ • (ϕ ξ : ξ ∈ Ξ) will do. By construction, every atomic L-formula is coded in L R . By the same Tietze argument as above, any continuous combination of coded formulas is again coded. It follows that every quantifier-free X-valued L-formula is coded in L R . In other words, at the quantifier-free level, the expressive power of real-valued and compactvalued logic is the same.

For real-valued logic we proposed sup and inf as quantifiers, which requires some justification. A first argument is that, a priori, these may be presented as the real-valued analogues of the classical quantifiers ∀ and ∃. A second argument, a posteriori, is that these quantifiers "deliver the goods", in the sense that real-valued logic does behave as a generalisation of two-valued logic. In particular, when stated appropriately, Łoś's Theorem and the Compactness Theorem hold in real-valued logic, as do many other results. The discussion which follows will provide, in a sense, a third argument, which may be qualified as ab initio: the choice of sup and inf as quantifiers is correct because they have the same expressive power as the primordial quantifier, from which any (admissible) quantifier may be constructed.

In a general compact-valued logic there are no obvious analogues for ∀ and ∃. Chang and Keisler proposed that quantifiers should be specified at the same time as other symbols of the language. A compact-valued quantifier Q, in this sense, is a function that takes a collection of truth values in some value space X (e.g., all values of ϕ(a, b) in a structure M, as a varies over M) and returns a truth value in some value space Y (the value of (Qx)ϕ(x, b) in M). Notice that Qx can only be applied to a formula ϕ(x, y) whose value space is X, and then the value space of (Qx)ϕ(x, y) is Y. In order to be admissible as a quantifier, Q is also required to be continuous in a sense that we shall make precise below. With such continuous quantifiers, compact-valued logic is well-behaved (in particular, Łoś and Compactness hold).

Definition. Let X be a Hausdorff topological space, and let K(X) denote the space of all non-empty compact subsets of X. For an open set U ⊆ X, let

O 1 U = K ∈ K(X) : K ⊆ U , O 2 U = K ∈ K(X) : K ∩ U = ∅ . We equip K(X)

A basic open set of the Vietoris topology is of the form

O 1 U ∩ i<n O 2 V i .
It is Hausdorff, and if X is compact, then so is K(X). See also Kechris [Kec95, Chapter I.4.F], where K(X) also contains the isolated point ∅.

We can now state the continuity requirement for a compact-valued quantifier: it should be of the form

Q(A) = θ Q (A), where θ Q : K(X) → Y is continuous (and ∅ = A ⊆ X, so A ∈ K(X)).
Example. If I ⊆ R is a compact interval, then the upper bound function sup : K(I) → I is continuous in the Vietoris topology, and similarly for inf. Therefore, the real-valued quantifiers are also compact-valued quantifiers in the sense of Chang and Keisler.

Let us simplify things and define a variant of compact-valued logic in which the aforementioned functions θ Q : K(X) → Y are exactly all possible identity maps. Stated somewhat differently, we allow a unique quantifier Q, which may be applied to any formula, with any value space. If ϕ(x, y) is an X-valued formula, then (Qx)ϕ(x, y) is K(X)-valued. If M is a structure and b ∈ M, then, with truth values calculated in M:

(Qx)ϕ(x, b) = ϕ(a, b) : a ∈ M ∈ K(X).
Let us call this Q the primordial quantifier. For any other continuous quantifier Q ′ , in the sense of Chang and Kiesler, we may express (Q ′ x)ϕ(x, y) as a composition θ Q ′ • (Qx)ϕ(x, y), now viewing θ Q ′ as a mere continuous connective.

Let us compare he expressive power of quantifiers in real-valued and in compact-valued logic. We have already observed that the real-valued quantifiers sup and inf can be expressed as a composition of continuous functions with the primordial quantifier. For the opposite direction, let us observe first that K is a functor. Indeed, if θ : X → Y is continuous, and K ⊆ X is compact, then θ(K) ⊆ Y is compact as well, defining a map K(θ) : K(X) → K(Y). In addition, assuming that U ⊆ Y is open and V ⊆ X is its inverse image by θ, then O 1

V and O 2 V are the inverse images by K(θ) of O 1 U and O 2 U , respectively, so K(θ) is continuous. In particular, if θ : X → [0, 1] is continuous, then so is the function sup •K(θ) : K(X) → [0, 1], which we shall denote by (sup θ). Assume that K, F ∈ K(X) are distinct, say x ∈ F K. By Urysohn's Lemma there exists a continuous θ : X → [0, 1] that vanishes on K and equals one at x. Then (sup θ) vanishes at K and equals one at F. It follows that the functions (sup θ) separate points in K(X).

Assume now that ϕ is an X-valued L-formula, in compact-valued logic, and that it is coded in L R . In other word, for every continuous θ : X → [0, 1], the L-formula θ • ϕ agrees with a real-valued L R -formula ϕ θ . Then (sup θ) • (Qx)ϕ agrees with sup x ϕ θ , and since the functions (sup θ) separate points, (Qx)ϕ is also coded in L R .

We conclude that real-valued continuous logic, equipped with the two quantifiers sup and inf (or just with one, from which the other can be recovered), has exactly the same expressive power as the compact-valued logic of Chang and Keisler, even when restricting the latter to the primordial quantifier alone.

In order to make this a little more precise, define a compact-valued theory T to be (the set of consequences of) a collection of closed conditions σ ∈ K, where σ is, say, an X σ -valued sentence, and K ⊆ X σ is closed. We could define a real-valued theory in the same manner, but we usually prefer to restrict to sets of conditions of the form σ = 0, which may then be identified with just a collection of sentences. With these definitions, the class of L R -structures that code L-structures is elementary, defined by a (universal) base theory T 0 . General classes of models of L-theories are in bijection with classes of models of L R -theories extending T 0 .

Remark. Two-valued logic usually assumes the presence of a distinguished relation symbol that interprets equality. When constructing real-valued logic, one argues that the "natural" real-valued analogue of equality is a symbol that interprets a (bounded, complete) distance function, with respect to which all other symbols are uniformly continuous. In addition to being natural, this is extremely useful for applications, making realvalued logic suitable for the model-theoretic treatment of metric structures. This potential was unfortunately missed by Chang and Keisler, who followed the tradition of two-valued logic, assuming an equality symbol.

In order to designate a real-valued binary relation symbol d as a distance symbol, we need to add to our base theory the universal axioms asserting that d interprets a pseudo-distance with respect to which every other symbol respects a fixed uniform continuity modulus (restricting the value space of d to {0, 1} would make it an equality symbol). If M is a model of these axioms then we may quotient out the zero-distance relation, and complete it (as a structure), without affecting the interpretation of the formulas.

Remark. Once we have a distance (or equality) symbol, we may introduce function symbols (without one, function symbols may misbehave). Adding a (uniformly continuous) function symbol f (x) is the same as adding a relation symbol P(x, y) representing d f (x), y , together with ∀∃ axioms that assert that P is indeed of that form.

  with the Vietoris Topology, namely the topology generated by all sets O 1 U and O 2 U , as U varies over all open subsets of X.

Author supported by ANR project AGRUME (ANR-17-CE40-0026). The author wishes to thank Todor TSAKNOV for having pushed him into writing this note, as well as for some mushy chocolate.


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The Initial Drift of a 2D Droplet at Zero Temperature Raphaël Cerf, Sana Louhichi To cite this version: Raphaël Cerf, Sana Louhichi. The Initial Drift of a 2D Droplet at Zero Temperature. Probability Theory and Related Fields, 2007, 137 (3-4), pp.379-428. 10.1007/s00440-006-0007-4. hal-03713003 ×