Jin Wang

Laurie Lacarriere
email: laurie.lacarriere@insa-toulouse.fr

Alain Sellier
email: alain.sellier@insa-toulouse.fr

Multicomponent modelling of cement paste dehydration under different heating rates

Keywords: Dehydration, multicomponent modelling, elevated temperature, cement paste, heating rate

The aim of this study is to establish a dehydration model that is able to calculate the water release of cement paste at elevated temperature. The main hydrates in ordinary Portland cement paste (C-S-H, portlandite, aluminates and sulfo-aluminates) were modelled separately and then combined to form the model of the cement paste. In this way, the model is able to predict the dehydration of cement pastes with different compositions without re-fitting. For each hydrate, the dehydration law takes account not only of the maximum quantity of water that can be released at each temperature but also of the kinetics to reach that equilibrium. The model is thus able to reproduce the paste dehydration for any heating rate. Thermogravimetric tests were carried out with different heating rates to validate

the model in comparison to experimental results from the literature. The results demonstrated the capability of the model in different situations.

Introduction

At elevated temperature, cement paste will undergo loss of chemically bound water of the hydrates. This dehydration is of interest when studying the durability and stability of concrete structures in severe environments such as elevated temperatures that can be encountered, for instance, in nuclear confinement structures when there is a severe accident, or in tunnels subjected to fire. Indeed, the paste modification induced by these elevated temperature leads to a loss of mechanical and transfer properties of the concrete. In most cases of structural applications, the dehydration at slightly elevated temperatures is modelled by empirical dehydration law that required to be identifed on a laboratory test performed on the paste used to formulate the concrete or by linear evolution according to temperature supposed to be independent of the cement nature [START_REF] Bary | A coupled thermo-hydro-mechanical-damage model for concrete subjected to moderate temperatures[END_REF][START_REF] Ranc | Comportement haute temprature du bton de structure: Impact de la fissuration sur les transferts hydriques[END_REF][START_REF] Gawin | An Overview of Modeling Cement Based Materials at Elevated Temperatures with Mechanics of Multi-Phase Porous Media[END_REF]. However it can be observed in tests with thermogravimetric analysis (TGA) that dehydration is a non-linear phenomenon that depends on the heating rate. These empirical laws, used, for instance, to reproduce the behaviour of a nuclear confinement vessel during a severe accident (inside temperature increasing to 200 • C in a matter of minutes), are identified from laboratory tests such as TGA, which are usually performed with a heating rate of 5 K/min. The heating rate used to identify the parameters of such a dehydration law is thus very different from the real heating kinetics (which is linked to the heating source on the surface but may be different at the core because it is also linked to the thermal conductivity) and will have an impact on the dehydration kinetics. Moreover, such empirical laws required a test on the same paste formulation than the one used in the structure and cannot be used as a prediction tool for other concretes. There is thus a need for a model able to predict separately the dehydration of each hydrate of the cement paste with specific kinetics laws adapted to the different mineralogy of hydrates (crystalline or amorphous). Based on previous experience of hydration modelling [START_REF] Buffo-Lacarrire | Multiphasic finite element modeling of concrete hydration[END_REF], this study deals with the formulation of such multicomponent dehydration model that could manage the bound water release of each hydrate using intrinsic laws based on mineralogy and stoichiometry considerations.

The effect of the heating rate on dehydration kinetics can be taken into account by introducing a kinetic parameter into the dehydration law, as proposed, for instance, by Zhang [START_REF] Zhang | Dehydration kinetics of Portland cement paste at high temperature[END_REF]. For a well crystallized phase such as calcium hydroxide (CH), all the chemically bound water is released as soon as the temperature that corresponds to the phase equilibrium is reached (with specific kinetics). The dehydration is thus complete even if the temperature is maintained at this value corresponding to the disequilibrium of the phase. In this case, a purely kinetic approach like the one proposed in [START_REF] Zhang | Dehydration kinetics of Portland cement paste at high temperature[END_REF] can be justified. However for amorphous structures like C-S-H (main phase in hydrated cement paste), the mass loss is not linked to a complete dissolution of the hydrate but to a progressive release of the interlayer water depending on the temperature. The most accessible water layers are thus released at lower temperatures than the most strongly bound water layers. In this case, the dehydration begins at moderate temperature but, if the temperature is maintained at a moderate value, only the most accessible water layers, the ones that are accessible with this energy, are released. The dehydration will thus stop when the water accessible at this temperature has been lost. However, when using a pure kinetic dehydration law, even if the temperature is maintained at a moderate level, the dehydration will continue until all the water has been released. This is due to the fact that progressive dehydration equilibrium with temperature is not considered.

Multicomponent dehydration model

The formulation of our model was inspired by the approach proposed by Ferraille [START_REF] Fresnet | Le role de l'eau dans le comportement haute temprature des btons[END_REF], which considered both kinetics and equilibrium. However, in order to obtain a model that could easily be used for different cements without any refitting to avoid new characterization tests, in our approach the kinetics of water mass loss due to dehydration is calculated from the mass loss kinetics for each separate hydrate j of the cement paste using eq. 1.

ẇdehyd = - j Q W j • ξj (1)
where:

• ξ j is the dehydration degree for hydrate j ;

• Q W j is the chemically bound water contained in the hydrate j before the dehydration.

This quantity of chemically bound water Q W j contained in the hydrate j at the beginning of dehydration (t 0 des ) can be calculated from the hydration degree of the anhydrous phase i for which the reaction produces or consumes hydrate j and from stoichiometric consideration [START_REF] Buffo-Lacarrire | Multiphasic finite element modeling of concrete hydration[END_REF][START_REF] Kolani | Hydration of slag-blended cements[END_REF]. For instance, for Portland cements, only clinker is considered, so i = {clinker}. In this case, and for cement with low sulfure content, the produced hydrates are CH, C-S-H and AFm, so the calculation of Q W would have to be performed for these 3 hydrates.

Q W j = t 0 dehyd 0 i H Hyd j • ∂Hyd j ∂α i • ∂α i ∂t • dt (2)
where:

• α i is the hydration degree of anhydrous phase i;

• H Hyd j is the mass of water in 1 mole of hydrate j;

• ∂Hyd j ∂α
is the molar quantity of hydrate j produced (> 0) or consumed (< 0) by the reaction of 1 unit of mass of the anhydrous phase i (calculated from stoichiometric considerations [START_REF] Kolani | Hydration of slag-blended cements[END_REF]).

The dehydration kinetics for each hydrate is determined at each time step by considering the difference between the equilibrium and actual degrees of dehydration and a characteristic time managing the kinetics (equation 3). The kinetics is thermoactivated in order to consider the faster dehydrations observed at high temperatures (for instance, as shown for portlandite in [START_REF] Zhang | Dehydration kinetics of Portland cement paste at high temperature[END_REF]).

ξj = H(T -T j o ) onset effect • ξ j eq (T ) -ξ j disequilibrium 1 τ j exp - E j a R 1 T - 1 T j o kinetic (3)
where:

• ξ j eq (T ) is the final dehydration degree at a given temperature; • H(T -T j o ) is the Heaviside function; • T j o is the temperature from which the hydrate j begins to dehydrate (onset temperature) . This temperature is also used as reference temperature for the thermoactivation of dehydration. Temperatures T and T j o must be given in [K]; • τ j is the characteristic time associated with hydrate j; • E j a is the activation energy associated with hydrate j. The equilibrium degree of dehydration (ξ j eq (T ) in equation 3) depends on the crystalline structure of the hydrate concerned. For a well crystallized hydrate (such as CH), it is equal to 1, characterizing the complete dissolution of the hydrate. In contrast, for amorphous hydrates (such as C-S-H), the equilibrium dehydration degree is a function depending on temperature and thus characterizing a dependence of interlayer water loss on the temperature.

The aim of this work is to determine the equilibrium degree (ξ j eq ) and the constants managing the kinetics (τ j ) and its dependence on temperature (E j a ) for each hydrate of a cement paste. These parameters are then considered as intrinsic because they depend only on the nature of the hydrate,and not on its quantity, for example.

Determination of parameters of dehydration law for each hydrate

In this section the parameters of the dehydration laws for each hydrate are determined by minimizing the difference between theoretical and experimental results issued from literature. For each hydrate, the theoretical dehydration curves are obtained integrating equation 3 using a finite difference method with explicit scheme in which the time step time step is controlled to limit the variation of ξj to 10 -4 .

Dehydration of CH

Due to its crystalline structure, the equilibrium dehydration degree of calcium hydroxide (ξ CH eq in eq. 3) is equal to 1. The progressive mass loss when temperature increases is only managed by the kinetic parameters, which have to be identified (τ CH and E CH a in eq. 3). Zelic [START_REF] Jelica Zelic | The First International Proficiency Testing Conference[END_REF] performed TGA tests on specimens of pure CH at different heating rates in order to highlight these kinetic phenomena. Those tests thus allowed the kinetic parameters of our model to be identified. The values found were τ CH = 17.4 h and E CH a = 158 kJ/mol. E a is considered equal to the average of the values that can be identified with the classical method for determination of activation energies Fig. 1: Dehydration of CH (experimental results from [START_REF] Jelica Zelic | The First International Proficiency Testing Conference[END_REF]) used for instance in [START_REF] Zhang | Dehydration kinetics of Portland cement paste at high temperature[END_REF]. The comparison between the model and Zelic's results is illustrated in Fig. 1. It must be noticed that only the kinetic parameters can be fitted on these results on pure CH. The onset temperature (from which the decomposition begins) observed in pure CH is indeed lower than the one observed for CH in cement paste (as shown by the authors [START_REF] Jelica Zelic | The First International Proficiency Testing Conference[END_REF]). The onset temperature for CH in cement paste (presented on table 1) is fixed to 375 • C according to the observation of [START_REF] Jelica Zelic | The First International Proficiency Testing Conference[END_REF].

Dehydration of tricalcium aluminate hexahydrate

Like CH, tricalcium aluminate hexahydrate (C 3 AH 6 , written TCA) also has a crystallized structure that gives it an equilibrium dehydration degree (ξ T CA eq in eq.

3) of 1. The progressive mass loss when temperature increases is managed only by the kinetic parameters that have to be identified (τ T CA and E T CA a in eq. 3). From Rivas-Mercury's TGA-DSC results [START_REF] Rivas-Mercury | Dehydration of ca3al2(sio4)4(3-y) (0< y< 0.176) studied by neutron thermodiffractometry[END_REF], it can be noted that tricalcium aluminate hexahydrate (TCA) loses most of its water above 200 • C . Horváth proposed that the dehydroxylation of TCA occurred in the 260-400 • C temperature range and that the activation energy of the reaction was 85.4 kJ/mol [START_REF] Horvath | Activation energies of the thermal decompositions of $\mathrmC 3ah 6$ AND $\mathrmC 3ad 6$ by the isothermal TG method[END_REF]. S.K. Das reported that the peak of the differential scanning calorimetry test (DSC) in the process of dehydration occurred at 310 • C and the activation energy was found to be 35.58 kJ/mol [START_REF] Das | Thermal analysis of hydrated calcium aluminates[END_REF]. Rivas-Mercury performed the TGA test and neutron thermodiffractometry and claims that the whole dehydration process consisted of two stages [START_REF] Rivas-Mercury | Dehydration of ca3al2(sio4)4(3-y) (0< y< 0.176) studied by neutron thermodiffractometry[END_REF]. Nevertheless, most of the mass loss (around 80%) took place in the 200-300 • C temperature range. It was mentioned that the TGA test might have been Results from Horváth [START_REF] Horvath | Activation energies of the thermal decompositions of $\mathrmC 3ah 6$ AND $\mathrmC 3ad 6$ by the isothermal TG method[END_REF] identify the activation energy of dehydration E T CA a = 85.4 kJ/mol. Using this value with Rivas-Mercurys results [START_REF] Rivas-Mercury | Dehydration of ca3al2(sio4)4(3-y) (0< y< 0.176) studied by neutron thermodiffractometry[END_REF] allows the characteristic time to be identified. The value adopted here is τ T CA = 3.3 h and a comparison of the model with Rivas-Mercury's results is illustrated in Fig. 2.

Dehydration of monosulfoaluminate

The water in the monosulfate (C 4 A SH 12 , written AFm) is usually considered to be made up of two types: the interlayer water and the bound water [START_REF] Dilnesa | Stability of Monosulfate in the Presence of Iron[END_REF][START_REF] Leisinger | Thermodynamic modeling of solid solutions between monosulfate and monochromate[END_REF][START_REF] Pan | Mao Ruoqing, Research of dehydrated calcium sulphoaluminate hydrates (afm) and its hydration[END_REF]. The water loss of AFm with temperature occurs through two different mechanisms. Leisinger [START_REF] Leisinger | Thermodynamic modeling of solid solutions between monosulfate and monochromate[END_REF] showed that the interlayer water was released progressively from around 40 • C . This can be explained, as for C-S-H, by the fact that, when the temperature increases, the associated energy allows a new water layer to be extracted. At higher temperature, when this interlayer water has all been released, AFm exhibits a crystalline structure which is shown by a peak in the DSC curve centred around 300 • C [START_REF] Leisinger | Thermodynamic modeling of solid solutions between monosulfate and monochromate[END_REF]. A quantitative analysis of Leinsingers results shows that the crystalline structure contains 6 moles of water. In the study exposed in the present paper, only the release of water due to temperature increase alone is considered. The interlayer water that can be released by dehydration corresponds to around 6 mol/mol of AFm [START_REF] Dilnesa | Stability of Monosulfate in the Presence of Iron[END_REF][START_REF] Leisinger | Thermodynamic modeling of solid solutions between monosulfate and monochromate[END_REF][START_REF] Pan | Mao Ruoqing, Research of dehydrated calcium sulphoaluminate hydrates (afm) and its hydration[END_REF].

The degree of dehydration of monosulfate can thus be calculated as a weighted combination of the dehydration degree corresponding to the crystalline part of AFm ( ξAFm cryst ) and that corresponding to the interlayer water which is not able to move by a decrease of relative humidity alone ( ξAFm layer ) (eq. 4). AFm contains 12 moles of water per mole of AFm before dehydration and, as the water content in each part of AFm is equal to 6 moles per mole of AFm, the weighted coefficient is equal to 6/12=0.5. ξAFm = 0.5 • ξAFm layer + 0.5

• ξAFm cryst (4)
The loss of interlayer water is modelled using an equilibrium degree of dehydration (ξ AFm layer eq in eq. 3) that depends on the temperature as proposed in equation 5 (to reproduce progressive loss of water). As this interlayer water does not correspond to a crystalline structure the dehydration process does not need the temperature to reach a specific value (corresponding to the crystalline structure destabilisation). The dehydration process is thus considered to begin from the reference temperature of 40 • C but with a kinetic depending on temperature (lower kinetic for low temperatures).

ξ AFm layer eq = 1 -exp -0.06 T -T 0 AFm layer 0.8 (5)
This equation is represented in Fig. 4 and is compared with the evolution of the equilibrium degrees of other hydrates. The parameters of this equation 5, together with the constants managing kinetics and its dependence on temperature (eq. 3), are identified using literature results for temperatures below 150 • C which is the onset temperature associated with the loss of the structural water (AFm cryst ). The values adopted for the interlayer water are τ AFm layer = 0.2 h and E AFm layer a = 10 kJ/mol and the comparison between the model and the experimental results is illustrated in Fig. 3 (temperature below 150 • C ).

As for other crystalline structures, the loss of structural water (decomposition of C 3 A SH 6 ) is associated with an equilibrium degree of dehydration (ξ AFm cryst eq in eq. 3) equal to 1 and to an onset temperature below which the crystalline structure is stable (equal to 150 • C for AFm). The constants managing the kinetics and its dependence on temperature (eq. 3) are identified using literature results above 150 • C . The values adopted are τ AFm cryst = 0.5 h and E AFm cryst a = 20 kJ/mol, and the comparison between the model and experimental results is illustrated in fig. 3.

Dehydration of C-S-H

Based on the work of Jennings [START_REF] Jennings | A model for the microstructure of calcium silicate hydrate in cement paste[END_REF], the bound water of C-S-H, which will be managed by the dehydration model, is considered to correspond to an H/S ratio of 1.4 whatever the C/S ratio. The complementary water is considered as free water and is managed by the hydric equilibrium and the water retention curves. It means that the differences in water content observed on C-S-H with different C/S ratios is only due to free water which can be removed without heating (for Fig. 3: Dehydration of AFm (experimental results from [START_REF] Pan | Mao Ruoqing, Research of dehydrated calcium sulphoaluminate hydrates (afm) and its hydration[END_REF][START_REF] Leisinger | Thermodynamic modeling of solid solutions between monosulfate and monochromate[END_REF][START_REF] Dilnesa | Stability of Monosulfate in the Presence of Iron[END_REF]) instance by decreasing the relative humidity). It is confirmed on fig. 4 where it can be seen that the water loss from 100 • C to 450 • C is similar for C-S-H with different C/S ratios.

Due to their amorphous structure with layers, the loss of bound water from C-S-H, even when equilibrium is reached, depends on the temperature which corresponds to the energy level of a specific part of the interlayer water. The equilibrium dehydration degree (ξ CSH eq in eq. 3) adopted in this model is thus, as for the interlayer water of AFm, an exponential function (eq. 6), even though the parameters of this function are expected to differ from that of AFm (interlayer water in C-S-H is bound by the hydrogen bond and, as in AFm, it is connected with Ca 2+ ). This equation is represented in Fig. 4, where it is compared with the evolution of the equilibrium degrees of other hydrates. As for the interlayer water in AFm, the bound water in C-S-H does not correspond to a crystalline structure, so the dehydration process does not need the temperature to reach a specific value to start. The dehydration process is thus considered to begin from the reference temperature of 40 • C but with a kinetic depending on temperature (very slow kinetic for temperature below 100 • C .

ξ CSH eq = 1 -exp -0.01 T -T 0 CSH 0.9 (6)
The parameters of this equation 6 as well as the constants managing kinetics and its dependence on temperature (eq. 3) were identified using literature results by Rodriguez [START_REF] Rodriguez | Thermal stability of C-S-H phases and applicability of Richardson and Groves' and Richardson C-(A)-S-H(I) models to synthetic C-S-H[END_REF] and Foley [START_REF] Foley | Synthesis and nano-mechanical characterization of calcium-silicate-hydrate C-S-H made with 1.5 C/S mixture[END_REF]. Below 100 • C and depending of the sample preparation, some free water (the one able to move with only a decrease of relative humidity and corresponding to H/S > 1.4) can be remaining in the samples. That's why, to identify the parameter of the dehydration law, only the loss of

Summary of the dehydration law parameters

The results of the identification are summarized for the different hydrates in Table 1, which gives the onset temperature, the characteristic time and the activation energy managing the kinetic part of the dehydration laws. Table 2 presents the functions of equilibrium degree for each hydrate, with the parameters identified using the literature results presented earlier. These functions are illustrated in Fig. 5 for temperatures between 50 • C and 700 • C İt can be observed that, in a cement paste, below 200 • C the loss of water will be due only to the dehydration of C-S-H and AFm.

Table 2: Equilibrium degree of dehydration for each hydrate

ξ j eq CH 1 TCA 1 AFm layer 1 -exp -0.06 (T -40 • C ) 0.8 AFmcryst 1 CxSH 1.4 1 -exp -0.01 (T -40 • C ) 0.9
4 Application to the simulation of experimental results on cement pastes and parametric study

In order to evaluate the ability of the model to reproduce (without any refitting) the dehydration of cement pastes with different heating rates, 2 TGA tests were performed on a paste cast with W/C=0.5, using CEM I 52.5 cement. The specimen had been cured in sealed cylindrical plastic tubes (φ30 mm × 60 mm) at 20 • C for 60 days. To remove the free water, the specimens were soaked in liquid nitrogen for 5 minutes immediately after being taken out of the tube, and then dried with lyophilization at -50 • C using the Labconco Free Zone 4.5 lyophilizer for 48h. After drying, the samples were ground until all of them passed the 80µm sieve before the TGA test. The TGA was run on the STA 449 F3 analyser manufactured by Netzsch. The temperature range for the test was 40-800 • C at heating rates of 5 K/min and 10 K/min. Two other cement pastes taken from the literature were tested. They were selected among several available data because, in these works, all the experimental conditions (cement composition, heating rate, pre-drying process, etc.) were given. Morandeau [START_REF] Morandeau | Investigation of the carbonation mechanism of ch and csh in terms of kinetics, microstructure changes and moisture properties[END_REF] tested an ordinary paste with a W/C ratio of 0.45 and Alarcon-Ruiz [START_REF] Alarcon-Ruiz | The use of thermal analysis in assessing the effect of temperature on a cement paste[END_REF] performed TGA on a paste with a low W/C ratio (0.33) and with a cement containing more Al 2 O 3 . The compositions of the cements used in the experiments are listed in Table 3. The heating rate was 10 K/min for Morandeau and 5 K/min for Alarcon.

The chemical compositions of the 3 cements tested are given in Table 3. As the model was developed as a multicomponent model, the dehydration of different cement pastes should be predicted without any refitting, with only the chemical composition of cement and water to cement ratio as data. Indeed, the water content per unit volume of cement pastes, before the beginging of the dehydration test, can be calculated using the stoichiometry for each hydrate and the hydrate mole number computed by the hydration model developed previously in our laboratory [START_REF] Buffo-Lacarrire | Multiphasic finite element modeling of concrete hydration[END_REF][START_REF] Kolani | Hydration of slag-blended cements[END_REF], only from the oxide content in cement and the water to cement ratio. This quantity of water content of each kind of hydrate (in kg/m 3 of paste) at the end of the curing period is given in Table 4 for the paste tested in our laboratory and for the 2 pastes drawn from the literature.

For the three studies (LMDC and [START_REF] Morandeau | Investigation of the carbonation mechanism of ch and csh in terms of kinetics, microstructure changes and moisture properties[END_REF][START_REF] Alarcon-Ruiz | The use of thermal analysis in assessing the effect of temperature on a cement paste[END_REF]) the pastes were dried before TGA in order to eliminate free capillary water, using techniques that also released part of the interlayer water of AFm (D-drying and 105 • C oven drying), this pre-release of 4 moles of H per mole of AFm thus had to be deducted from the water evaluated by hydration. The quantity of water lost due to pre-drying (that should be subtracted from the water content in each hydrate before applying the dehydration model) is also given in Table 4.

The results of the TGA tests are compared to the predictions of the dehydration model, with the parameters identified from the literature results for each hydrate and recalled in Table 1. The results are presented in terms of percentage of mass of bound water remaining in the paste (as a percentage). This remaining bound water (%m W bond ) is calculated from the mass of the specimen tested (m spe ) as given in equation 7 (supposing that the mass obtained at 700 • C corresponds to the mass of the solid to avoid taking into account the effect of decomposition of carbonates). In the tests performed at LMDC, as we want to decouple the effect of dehydration and carbonation, a specific care was taken to avoid any carbonation of the sample during the cure and preparation. No loss of mass corresponding to decomposition of carbonate (that should be identified in TGA-DSC test with the measurement of CO 2 release) was observed. In the two litterature tests used to evaluate the prediction capability of the model [START_REF] Morandeau | Investigation of the carbonation mechanism of ch and csh in terms of kinetics, microstructure changes and moisture properties[END_REF][START_REF] Alarcon-Ruiz | The use of thermal analysis in assessing the effect of temperature on a cement paste[END_REF], some carbonation was observed. In this first study the hydration model used to predict the initial state of the hydrated paste, before TGA test, is not coupled with a carbonation model. Therefore some differences are expected between the model prediction and the results.

%m W bond (T ) = m spe (T ) -m spe (700 • C) m spe (100 • C) -m spe (700 • C) (7)
The dehydration model was applied without any re-fitting. The model and test results obtained for 3 different pastes and 2 different heating rates are compared in Figures 6, 7, 8 and 9.

It can be seen that these tests are globally well reproduced by the model for both heating rates (comparison with LMDC results) and for the three different pastes (different W/C ratios and different ordinary cements). The slight difference between model prediction and experimental results for the tests presented in [START_REF] Morandeau | Investigation of the carbonation mechanism of ch and csh in terms of kinetics, microstructure changes and moisture properties[END_REF][START_REF] Alarcon-Ruiz | The use of thermal analysis in assessing the effect of temperature on a cement paste[END_REF] may be explained by the partial carbonation of the paste before the dehydration test (carbonation identified by the authors from the analysis of the TGA test). The carbonation of some hydrates in the specimen during the cure can lead to two sources of misprediction of the model. First, the initial quantity of bound water content at the beginning of the dehydration test can be slightly overestimated because the hydration model that calculated the composition of the hydrated paste does not take into account the conversion of some hydrates into carbonates (which releases some bound water). This could be improved by coupling the hydration and dehydration models with a carbonation model. Then, as the mass of the specimen at 700 • C includes the mass of carbonates, the calculation of the bound water as exposed in equation 7 is not exactly made on the same basis.

The model is then tested to illustrate the effect of much broader variation of heating rates such as large heating rates (50K/min) which can be quite representative of fire exposition near the exposed surface and a theoretical extremely slow heating rate which allows to represent the equilibrium degree of dehydration at each temperature (figure 10). It must me noticed that, as the identification of the Fig. 9: Comparison between the modeling results and the experiment in [START_REF] Morandeau | Investigation of the carbonation mechanism of ch and csh in terms of kinetics, microstructure changes and moisture properties[END_REF] Fig. 10: Parametric study for a cement paste subjected to different heating rates model's parameters was made for heating rate of 5 or 10K/min, the extrapolation of the model prediction to these much broader variation of heating rates remains theoretical and should be verified with experimental devices allowing such heating configurations. It can be noticed that for heating rates not very fast (5 and 10K/min), the bound water remaining around 400 • C corresponds to the one predicted if the equilibrium is reached, that is to say that C-S-H and aluminates lost quite all their bound water before CH decomposition. On the contrary for high heating rates some water remains in these phases before CH decomposition and the profile of the water loss is quite linear according to temperature between 150 and 550 • C .

The model is also tested to simulate the effect of a very different cement composition, to illustrate the possible applicability of the model to blended cements in the future. Figure 11 presents the results of dehydration for an ordinary Portland cement and for a cement with a low calcium content which induces that the hydrated paste contains almost no CH.

Conclusion

The multicomponent dehydration model for cement paste was established on the basis of the dehydration model for each hydrate in the cement paste. The hydrates taken into account in this study were C-S-H, CH, TCA and AFm. The model contains both equilibrium and kinetic parts, and can be applied on cement pastes having different compositions without the need to re-fit the parameters. Dehydration models for CH, AFm, TCA and C-S-H were identified from experiments on pure hydrates reported in the literature, in order to ensure that Fig. 11: Prediction of the dehydration of a portland cement and a "low calcium" cement for a heating rate of 5K/min the model was able to predict the dehydration of every Portland cement paste without refitting, based only on the composition of the paste (quantity of the different hydrates in the hydrated paste), which can be obtained using a previously developed hydration model. The model was compared with experiments run at different heating rates and with different compositions of cement. It showed its good ability to predict the dehydration process of Portland cement paste between 100 and 650 • C . A coupling with ageing model (carbonation for instance) can be envisioned in the future works before extending this multicomponent dehydration model to the prediction of the effect of dehydration on the mechanical properties.

6 Compliance with Ethical Standards:

Fig. 2 :

 2 Fig. 2: Dehydration of TCA (C 3 AH 6 ) (experimental results from [9])

Fig. 4 :

 4 Fig. 4: Identification of dehydration parameters for C x SH 1.4 (experimental mass loss after 100 • C calculated from the results of [16])

Fig. 5 :

 5 Fig. 5: Equilibrium degree of dehydration for the different hydrates

Fig. 6 :Fig. 7 :

 67 Fig. 6: Comparison with LMDC TGA test under 5K/min

Table 1 :

 1 Dehydration law's parameters for each hydrate

		T j o ( • C )	τ j (h)	E j a (kJ/mol)
	CH	375	17.4	158.53
	TCA	200	3.3	85.4
	AFm layer	40	0.2	10
	AFmcryst	150	0.5	20
	CxSH 1.4	40	1.	34

Table 3 :

 3 Composition of the cement (%w.t. for oxyde and mass ratio for W/C)

		LMDC	[18]	[19]
	SiO 2	20.6	22.12 19.45
	Al 2 O 3	4.5	3.69	4.8
	Fe 2 O 3	2.3	4.	1.75
	CaO	63.2	65.03	59
	SO 3	3.3	2.94	2.95
	W/C	0.5	0.45	0.33

Table 4 :

 4 Water content of each hydrate in cement paste (kg per 1 m 3 cement paste)

		LMDC	[18]	[19]
	CH	54.43	55.1	60.76
	AFm	86.65	82.2	97.64
	TCA	18.15	27.39	28.0
	C 1.7 SH 1.4	90.16	103.07 107.30
	loss by pre-drying	28.88	27.4	32.50

Funding: This study was funded by the Chinese Scholarship Council (CSC) through Jin Wang's thesis grant and by the French National Research Agency (ANR-PIA) under the MACENA research program 11-RSNR-0012 (Control of nuclear vessel in accident conditions).


X Artru
email: x.artru@ipnl.in2p3.fr

J M Richard

J Soffer

POSITIVITY DOMAINS FOR PAIRS OR TRIPLES OF SPIN OBSERVABLES 1

Positivity restrains the allowed domains for pairs or triples of spin observables in polarised reactions. Various domain shapes in 1 2 + 1 2 → 1 2 + 1 2 reactions are displayed. Some methods to determine these domains are mentioned and a new one based on the anticommutation between two observables is presented.

The spin observables

We consider the polarised 2 × 2 reaction

A + B → C + D , (1.1)
where A, B, C and D are spin one-half particles. An example is pp → ΛΛ .

(1.

2)

The fully polarised differential cross section of (1.1) can be expressed as

dσ dΩ = I 0 F S A , S B , ŠC , ŠD , (1.3)
where F contains the spin dependence. S A and S B are the polarisation vectors of the initial particles (|S| ≤ 1). ŠC and ŠD are pure polarisations (| Š| = 1) accepted by an ideal spin-filtering detector. They must be distinguished from the emitted polarisations S C and S D of the final particles. The latter ones depend on the polarisations of the incoming particles, e.g., S C = ∇Š C F S A , S B , ŠC , ŠD = 0 / F S A , S B , ŠC = 0, ŠD = 0 (1.4)

F is given in terms of the Cartesian reaction parameters [1] by

F S A , S B , ŠC , ŠD = C λµντ S λ A S µ B Šν C Šτ D . (1.5)
In the right-hand side the S 's are promoted to four-vectors with S 0 = 1. The indices λ, µ, ν, τ , run from 0 to 3, whereas latin indices i, j, k, l, take the values 1, 2, 3, or x, y, z. A summation is understood over each repeated index. S x , S y , S z are measured in a triad of unit vectors {x, ŷ, ẑ} which may differ from one particle to the other. A standard choice is to take ẑ along the particle momentum and ŷ common to all particles and normal to the scattering plane. For example, C 0000 ≡ 1, C xy00 ≡ A xy is an initial double-spin asymmetry, C 000y is the spontaneous polarisation of particle D along ŷ, C 0y0y ≡ D yy is a spin transmission coefficient from B to D and C 00xy ≡ C xy is a final spin correlation.

The Cartesian reaction parameters are given by

C λµντ = Tr{ M [σ λ (A) ⊗ σ µ (B)] M † [σ ν (C) ⊗ σ τ (D)] } / Tr{ M M † } , (1.6)
which will be symbolically abbreviated as a sort of expectation value:

(λµ|ντ ) ≡ C λµντ = σ λ (A) σ µ (B) σ ν (C) σ τ (D) , (1.7)
with σ 0 = ½ ≡ 1 0 0 1 .

The positivity constraints

The cross section (1.3) is positive for arbitrary independent polarisations of the external particles, that is to say

F S A , S B , ŠC , ŠD ≤ 1 for S A , S B , ŠC , ŠD ∈ unit ball |S| ≤ 1 . (2.8)
However there are positivity conditions which are stronger than (2.8). The full positivity condition can be obtained from the positivity of the cross section matrix R defined by

c, d|M|a, b a ′ , b ′ |M † |c ′ , d ′ = a ′ , b ′ ; c ′ , d ′ |R|a, b ; c, d = a ′ , b ′ ; c, d| R|a, b ; c ′ , d ′ (2.9)
in terms of the helicity or transversity amplitudes c, d|M|a, b . R is the partial transpose R, the transposition R → R bearing on the final particles 2 . All spin observables of reaction (1.1) can be encoded in R or R. The diagonal elements of R or R are the fully polarised cross sections when the particles are in the basic spin states. By construction, R (but not necessarily R) is semi-positive definite, that is to say Ψ|R|Ψ ≥ 0 for any Ψ. Equations (1.3), (1.5) and (1.6) can be rewritten as:

dσ dΩ (ρ A , ρ B , ρC , ρD ) = Tr{ R [ρ A ⊗ ρ B ⊗ ρC ⊗ ρD ] } , C λµντ = Tr{ R [σ λ (A) ⊗ σ µ (B) ⊗ σ ν (C) ⊗ σ τ (D)]} / Tr R , = Tr{R σ λ (A) ⊗ σ µ (B) ⊗ σ t ν (C) ⊗ σ t τ (D) } / Tr R , (2.10)
2 Alternatively, keeping the same R, one may define R as the full transpose of that given by (2.9). Then the partial transposition between R and R would bear on the initial particles. This choice was done in Ref. [START_REF] Artru | Proc. Dubna Spin Workshop[END_REF], where R is called "grand density matrix".

with ρ = 1 2 (½ + S • σ), ρ = 1 2 (½ + Š • σ).
The last two equations of (2.10) can be inverted

as R1 ≡ (2 4 / Tr R) R = C λµντ σ λ (A) ⊗ σ µ (B) ⊗ σ ν (C) ⊗ σ τ (D) , or R 1 ≡ (2 4 / Tr R) R = C λµντ σ λ (A) ⊗ σ µ (B) ⊗ σ t ν (C) ⊗ σ t τ (D) .
(2.11)

The matrix R1 is normalised to have the same trace as the unit matrix and is directly obtained from F replacing the S µ 's by σ µ 's.

Various domains for pairs of observables

For one observable, for example

O = C 0µ0ν ≡ σ µ (B) σ ν (D) we have the trivial positivity condition O ∈ [-1, +1]. For a pair {O 1 , O 2 } of such observables we have therefore {O 1 , O 2 } ∈ [-1, +1] 2 .
However, in many cases the allowed domain is more restricted than the square. An empirical but systematic method [2,3] to find the domain simply consists of generating random, fictitious helicity or transversity amplitudes, computing the observables and plotting the results the one against the other. Once the contours revealed, it is an algebraic exercise to demonstrate rigorously the corresponding inequalities. Table 1 summarises the shapes of the domains for the sixteen independent observables of the reaction (1.2). These domains are either the full square [-1, +1] 2 or the unit disk or a triangle.

Table 1: Domain allowed for pairs of observables: the entire square ( 2 ), the unit disk ( ), the triangle

|2O 1 | ≤ O 2 + 1 ( ∇ ), or |2O 2 | ≤ O 1 + 1 ∆
, where O 1 is horizontal and

O 2 vertical.
The symbol ⊗ indicates that the pair of observables is constrained in the unit disk, but the corresponding operators do not anticommute.

⊗ ⊗ P n ∆ 2 2 2 2 2 2 2 2 A n 2 2 ∇ 2 2 2 2 ∇ C nn 2 2 2 2 2 D nn 2 2 2 2 K nn ⊗ ⊗ 2 2 C ml 2 2 ⊗ 2 D mm 2 2 2 2 2 C mm 2 2 2 2 2 C ll ⊗ 2 2 ⊗ D ml 2 K mm 2 ⊗ K ml 2 2 2 C nlm 2 2 C nml ⊗ ⊗ C nmm 2 ⊗ C mnl ⊗ C mln 2 C mnm

Anticommutation method

Disk-shaped domains are, in many cases, straightforward results of anticommutation of the observables of the pair. From the last equation of (2.10), one can consider the observables as expectation values of operators. Since each σ 2 µ is equal to the identity, we have O 2 = ½.

Furthermore two such operators O and O ′ differing by at least one index (λ, µ, ν or τ ) either commute or anticommute.

For pairs of anticommuting observables, disk domains result from the following theorem:

If O 2 = O ′2 = ½ and O and O ′ are anticommuting, then O 2 + O ′ 2 ≤ 1 . (3.12) Proof: set x = O 2 + O ′ 2 , O = ax, O ′ = bx.
Then a 2 +b 2 = 1 and aO+bO ′ = x.

From O 2 = O ′2 = ½ and OO ′ + O ′ O =

Various domains for triples of observables

The empirical and anticommutation methods generalise straightforwardly to triple of observables. Figure 1 shows the boundary of the domains that we have identified for the observables of the reaction (1.2): the unit sphere, a pyramid, an upside-down tent, a cone, a cylinder, the intersection of two orthogonal cylinders or a double cone which is slightly smaller than this intersection, a combination of the disk, square and triangle projections delimiting a volume similar to a "coffee filter", the intersection of three orthogonal cylinders (larger than the unit sphere!), a tetrahedron, the intersection of two cylinders and two planes, an octahedron, or figures deduced by mirror symmetry.

Can the domain of a triple be the whole cube? Suppose now that for instance 3 observables O 1 , O 2 , and O 3 , each of which has +1 and -1 as extreme eigenvalues, are commuting and that no symmetry relates a pair of them to a non-commuting pair. Does it means that their joint positivity domain D{O 1 , O 2 , O 3 } is the whole cube? A partial negative answer is the following: If the reaction depend on N independent amplitudes, D{O 1 , O 2 , O 3 } can reach at most N corners of the cube [4]. The domains shown in Figure 1 are those of the reaction (1.2), which has N = 6 and indeed none of them reaches more than 6 corners, this number being obtained for the domain (i). More generally, if N < 8, all triple observables are restricted in domains smaller than the cube.

Outlook

We have seen that the positivity restricts the pairs or triples of observables to subdomains of the square or the cube, some of which having non-trivial shapes. Here we have presented only two methods for determining these domains. Other methods use the Cauchy-Schwarz Figure 1: Some allowed domains encountered in simulating randomly three observables: the unit sphere (a), the intersection of three orthogonal cylinders of unit radius (b), the intersection of two cylinders (c), or a slightly smaller double cone (d), a cylinder (e), a cone (f), a pyramid (g), a tetrahedron (h), an octahedron (i), a "coffee filter" (j), an inverted tent (k), and the intersection of two cylinders and a dihedral (l). For clarity, part of the limiting surface is sometimes removed. Some figures transformed by parity with respect to the centre of the cube or by interchange of the axes are also obtained.

inequality or the positivity of the subdeterminants of R whose diagonal elements are on the diagonal of R. For exclusive reactions, R is of rank one, therefore all diagonal 2 × 2 subdeterminants vanish. This links the observables by a large number of quadratic identities, from which inequalities can be obtained straightforwardly. We must tell, however, that inequalities expressing the positivity of R define joint domains for many observables and it is sometimes a straightforward but lengthy task to obtain the projected domain for two or three observables.

We thank M. Elchikh and O.V. Teryaev for help, useful discussions and comments.

  0 one gets (aO + bO ′ ) 2 = ½ which means that aO + bO ′ has eigenvalues ±1. Its expectation value x has to be within these eigenvalues, therefore x 2 ≤ 1.Note that a disk can occur even if the observables commute, for instance if, due to some symmetry, O 2 has the same expectation value as another operator O ′ 2 which anticommutes with O 1 and O 3 . Examples of this situation are indicated by crossed circles of Table1.

Talk given by Xavier Artru at "DSPIN-07", XII Workshop on High-Energy Spin Physics, Dubna, Sept.3-7,

2007, to appear in the Proceedings