Jean-Marc Joubert
email: joubert@icmpe.cnrs.fr

Jean-Claude Crivello

G Deffrennes

Modification of Lu's (2005) high pressure model for improved high pressure/high temperature extrapolations. Part I: modeling of platinum at high pressure/high temperature

Keywords: Calphad, model, EOS, DFT, platinum, high pressure

    Modification of Lu's (2005) high pressure model for improved high pressure/high temperature extrapolations. Part I: modeling of platinum at high pressure/high temperature

Introduction

Modeling of both thermodynamic properties and phase diagrams at high pressure is very important not only for geological applications. It is also needed for the description of materials under thermomechanical stress, impact or explosion, for the prediction of molar volume... High pressure properties are obviously also strongly related to the basic physics of the materials. The models used must combine the possibility to reproduce well the data, the ease of use and accurate extrapolations. There exist different high pressure models in the literature that can be used in the frame of the Calphad methodology: Mie-Grüneisen (1912), Murnaghan (1944), Birch-Murnaghan (1947), Refs. [START_REF] Jacobs | A realistic equation of state for solids. The high pressure and high temperature thermodynamic properties of MgO[END_REF][START_REF] Lu | Implementation of a new model for pressure dependence of condensed phases in Thermo-Calc[END_REF][START_REF] Stixrude | Thermodynamics of mantle materials -I. Physical properties[END_REF][START_REF] Jacobs | The derivation of thermo-physical properties and phase equilibria of silicate materials from lattice vibrations: application to convection in the Earth's mantle[END_REF][START_REF] Brosh | Application of Calphad to high pressures[END_REF][START_REF] Brosh | Explicit Gibbs free energy equation of states for solids[END_REF][START_REF] Lu | A CALPHAD Helmholtz energy approach to calculate thermodynamic and thermophysical properties of fcc Cu[END_REF][START_REF] Jacobs | Thermodynamic properties and equation of state of fcc aluminum and bcc iron, derived from a lattice vibrational method[END_REF]. However, there is no consensus about which is preferable.

Among them, the model by Lu et al. [START_REF] Lu | Implementation of a new model for pressure dependence of condensed phases in Thermo-Calc[END_REF] has had indisputable achievements. It is particularly interesting because it has been implemented in the Thermo-Calc software, it is relatively easy to handle, it can treat any phase, solid or liquid, stable or metastable and it follows the Calphad formalism in its extrapolation possibilities to multi-component systems. However, it has been criticized for leading to sometimes erroneous extrapolations at high pressure and/or temperature (negative entropy and specific heat). We will describe the limitations of this model in more detail and their origin will be investigated. Solutions will be proposed in order to avoid the problems of extrapolation. It will be finally demonstrated that it may be used safely with the proposed modifications and that, in most cases, it should be preferred as being simpler, easier to handle and because it is better adapted to the frame of the Calphad method, to the description of complete systems and to the extrapolation to multicomponent systems.

As an application, the example of platinum at extreme pressure and temperature will be detailed in the present paper. A thermodynamic assessment of the available experimental data together with calculated data obtained in this work will be described. In the second part of this study [START_REF] Joubert | Modification of Lu's (2005) high pressure model for improved high pressure/high temperature extrapolations. Part II: modeling of osmiumplatinum system at high pressure/high temperature[END_REF], this model will be applied to the complete description of Os-Pt binary system .

Description of Lu's model

In high pressure models, the Gibbs energy is described with an additional contribution taking into account the temperature and pressure dependence of the volume: [START_REF] Jacobs | A realistic equation of state for solids. The high pressure and high temperature thermodynamic properties of MgO[END_REF] In Lu's model [START_REF] Lu | Implementation of a new model for pressure dependence of condensed phases in Thermo-Calc[END_REF], the volume is described as a function of pressure using the Grover equation, itself derived from the Murnaghan model (see the original paper for the complete description and derivation of the equations). Finally, the volume is calculated as a function of temperature and pressure in Equation 9 of Lu's paper adapted here with other notations: [START_REF] Lu | Implementation of a new model for pressure dependence of condensed phases in Thermo-Calc[END_REF] where is the exponential integral function and its inverse function. The model is fully implemented in Thermo-Calc software using the dedicated functions.

-V 0 : a constant equal to the molar volume at a reference temperature (here 0 K) and 1 bar -V A (T): integrated volume thermal expansion such as (3) If described as:

(4) then the average linear thermal expansion is:

(5) -V C (T): pressure dependence of the bulk modulus (B) -V K (T): isothermal compressibility at 1 bar Among these functions, V A , V K and V C are temperature dependent, though the use of temperature dependence for V C is not recommended. They can be optimized using the following data which are generally available from experiments:

-molar volume as function of temperature at ambient pressure makes it possible to determine V 0 and V A -thermal expansivity as a function of temperature is related to V A -molar volume as function of pressure at room temperature is related to V 0 , V K and V C -bulk modulus as a function of temperature is related to V K

The data used can be not only experimental but also calculated either from first principles calculations (V m = f(p)) or by phonon calculations (V m , α, B = f(T)), for example, in the quasiharmonic approximation.

Problems with Lu's (2005) model

Reported problems

Soon criticisms about this model appeared in the literature. The main criticism is that the model yields nonphysical extrapolated values of C p and entropy. As an example, in Fig. 1, the C p of bcc Fe is calculated from the assessment of Lu et al. [START_REF] Lu | Implementation of a new model for pressure dependence of condensed phases in Thermo-Calc[END_REF]. The figure shows that negative C p are obtained at high temperature at pressures above 100 GPa. Brosh et al. [START_REF] Brosh | Application of Calphad to high pressures[END_REF] analyzed the problem as being due to an 'incompatibility' between the SGTE description of the C p and the Mie-Grüneisen equation. By second derivation of Eq. ( 1), one may obtain the C p at high pressure. [START_REF] Brosh | Explicit Gibbs free energy equation of states for solids[END_REF] The C p is therefore the sum of the low pressure C p coming, for example, from the SGTE description and a contribution given by the model at high pressure.

Brosh et al. assume that the contribution of the second term at high temperature (above the ambient pressure melting point) is necessarily negative. Therefore, according to these authors, in order to avoid that the overall C p becomes negative at high temperature when pressure (and therefore melting point) is raised, then the first term should increase strongly as a function of temperature above the ambient pressure melting point of the element. This disagrees with the socalled SGTE approximation [START_REF] Guillermet | Critical evaluation of the thermodynamic properties of molybdenum[END_REF][START_REF] Andersson | A new method describing lattice stabilities[END_REF] tending to have constant C p at high temperature above the melting point of the elements as shown in Fig. 1. This is what the authors call an 'incompatibility' between the SGTE database and high pressure models.

The validity of the assumption of Brosh et al. is not obvious because it can be imagined as well that the high pressure contribution to the C p should not be negative and that the Mie-Grüneisen model itself badly reproduces the features at high pressure. So, it is not a question of incompatibility, it is rather a question of defining what should be the C p above the melting at low pressure and what should be the contribution to the C p at high pressure. It is not proven that this second contribution should be high and negative.

One can see, for example, that negative C p can be obtained with the model of Lu at 200 GPa at temperatures below the ambient pressure melting point of bcc Fe. The problem can therefore not be caused by the term in Eq. ( 6) since, at this temperature there is no other option than sticking to the consistently evaluated C p of the bcc phase. If one wants to avoid negative C p , then one should work on the contribution to the C p of the high pressure model itself.

Analysis of the problem

In order to discover the origin of the problem, we conducted several simulations using Lu's model. Calculated data will be reported for C p but note that the same tendency is also observed for entropy (i.e. when negative C p is calculated, negative entropy is also calculated). The effects of the compressibility and the thermal expansion are examined separately. The simulations for a phase without thermal expansion show that the compressibility itself is not a problem. Indeed, the C p calculated up to 1000 GPa, whatever the compressibility, is the same as the low pressure C p . This is also in contradiction with the analysis of Brosh et al. [START_REF] Brosh | Application of Calphad to high pressures[END_REF] i.e. the Gibbs energy contribution at high pressure calculated from the model does not bring any negative contribution to the C p and there is therefore no need for the low pressure C p to increase above the melting point.

Let's now examine the effect of thermal expansion for an uncompressible phase. The results are presented in Fig. 2. Negative C p is definitely observed at high pressure and temperature. This can be understood by writing the following equations from Eq. 1 for an uncompressible phase in which V does not depend on p:

(7) (8)
From this equation, there is actually a negative contribution. It is not directly caused by the thermal expansion (proportional to ) but rather by the increase of thermal expansion as a function of temperature (proportional to

). This can be checked in Fig. 2 where a constant thermal expansion is used without causing a significant C p decrease at high pressure.

Eq. 8 has been derived for the specific case of an uncompressible phase for the sake of simplicity. A more general expression can be obtained, without making any kind of assumption, by using the Maxwell equation (as quoted in Ref. [START_REF] Guillermet | Critical evaluation of the thermodynamic properties of molybdenum[END_REF]): [START_REF] Joubert | Modification of Lu's (2005) high pressure model for improved high pressure/high temperature extrapolations. Part II: modeling of osmiumplatinum system at high pressure/high temperature[END_REF] It indicates as well the C p decreases when pressure is raised if is positive.

Practically, at ambient pressure, is always positive and may be very large for several elements as obtained experimentally from ambient pressure measurements of volume or thermal expansion as a function of temperature (see in Ref. [START_REF] Lu | Theoretical modeling of molar volume and thermal expansion[END_REF] the assessment of the thermal expansion of many different elements). Whatever model reproducing the available experimental data should therefore reproduce this negative contribution to the C p at low pressure. Note that this is not model dependent, this contribution being inherent to the data. The problem is coming from this negative term. If kept constant as a function of pressure, it will have a dramatic influence on the C p when calculated at both high pressure and temperature. It is therefore compulsory that it decreases with pressure and this has not necessarily been considered in the models.

First, thermal expansion or high temperature volume data are rarely known at high pressure, so a fit of the pressure dependence of thermal expansion coefficients cannot be performed. For few systems in which thermal expansion has been measured at high pressure, as expected, the slope of thermal expansion as a function of temperature and even thermal expansion itself is observed to decrease (see e.g. MgO and MgSiO 4 in data reproduced in Refs. [1] [4]).

Second, such a fit should be allowed by the model. This is indeed the main problem with Lu's model: the thermal expansion controlled by the parameter V A (Eq. 4) is not made pressure dependent while it should. The uncontrolled behavior of at high pressure, especially if thermal expansion increases as a function of temperature, even more if the second derivative of thermal expansion is positive and high (see e.g. the example of Mo in Ref. [START_REF] Lu | Theoretical modeling of molar volume and thermal expansion[END_REF]), is responsible for the predicted negative C p when high pressure and high temperature are combined.

Possible solution

It is always possible to fit the volume as a linear function of temperature (no curvature in the volume curve or constant thermal expansion). As can be inferred from Eq.9, it would allow to maintain a constant C p as a function of pressure. This would however be a crude approximation resulting in a poor description of the data and therefore of the thermodynamic properties. Another option is to keep a good description of the low pressure thermal expansion data but make the thermal expansion constant in the high pressure regime. This is possible by using an exponential term intended to damp the terms in T 2 and T 3 in Eq. 4 at high pressure.

A value of the cut-off pressure p cut =10 9 Pa gave satisfactory results. Additionally, it is possible to make the thermal expansion decrease at high pressure by applying a similar exponential term on the T term as well. This second cut-off pressure (p cut' typically around 10 11 Pa) can be adjusted from the data when existing (for example volume at both high pressure and temperature) or set arbitrarily.

We propose such an equation instead of Eq. 4: [START_REF] Guillermet | Critical evaluation of the thermodynamic properties of molybdenum[END_REF] At low pressure, this results in a linear thermal expansion equal to: [START_REF] Andersson | A new method describing lattice stabilities[END_REF] identical with the original model.

The intermediate pressure limit, typically above p=p cut is: [START_REF] Lu | Theoretical modeling of molar volume and thermal expansion[END_REF] While, at high pressures (p>>p cut' ), the thermal expansion converges to zero.

It is also found that cross correlations between thermal expansion and compressibility parameters have a contribution to the C p , although to a lesser extent than the coefficients of thermal expansion. This may be seen from the thermodynamic identity: [START_REF] Arblaster | Crystallographic properties of platinum[END_REF] Therefore, it may be recommended to damp also, with the same exponential term, the temperature dependence of V K : [START_REF] Arblaster | Crystallographic properties of platinum[END_REF] Eq. 2 remains valid provided the term is used. Note that it represents V(T) only at low pressure but no longer at high pressure since V A is now pressure dependent.

Application

We present here an application of the modified model to the description of platinum. This system has been chosen because a lot of data is available up to very high pressure. It is also important because platinum may be used as a pressure standard in high pressure diffraction experiments. Both fcc and liquid phases will be described allowing a calculation of the pressure-temperature phase diagram.

Selection of data

fcc phase

The molar volume and thermal expansion data have been thoroughly reviewed by Arblaster in different publications [START_REF] Arblaster | Crystallographic properties of platinum[END_REF][START_REF] Arblaster | Crystallographic properties of platinum[END_REF]. These assessed data will be used as experimental data for the present assessment. It offers the advantage of a proper correction of the different temperature scales used in the older literature, a proper conversion from kX to Å when appropriate, a careful selection of the most accurate data and an assessment along the complete temperature range. Anyhow, there is no significant discrepancy in the literature, except perhaps the data of Refs. [START_REF] Mehmood | Thermophysical properties of platinum-copper alloys[END_REF][START_REF] Paradis | Thermophysical properties of platinum group metals in their liquid undercooled and superheated phases[END_REF][START_REF] Klotz | Thermophysical properties of platinum-rich alloys measured by sessile drop experiments[END_REF]. These data have been obtained in the frame of a study of the liquid volume with an optical measurement which is probably less reliable than high temperature diffraction results. Several chosen significant data are reported in Table 1 and will be plotted for comparison.

Compressibility data at room temperature are available from different authors [START_REF] Dewaele | Equations of state of six metals above 94 GPa[END_REF][START_REF] Fei | A critical evaluation of pressure scales at high temperatures by in situ X-ray diffraction measurements[END_REF][START_REF] Zha | P-V-T equation of state of platinum to 80 GPa and 1900 K from internal resistive heating/X-ray diffraction measurements[END_REF][START_REF] Anzellini | In situ characterization of the high pressure-high temperature melting curve of platinum[END_REF]. In general, the agreement is very good between the different datasets. We note the availability, for this system, of data at both high pressure and temperature [START_REF] Fei | A critical evaluation of pressure scales at high temperatures by in situ X-ray diffraction measurements[END_REF][START_REF] Zha | P-V-T equation of state of platinum to 80 GPa and 1900 K from internal resistive heating/X-ray diffraction measurements[END_REF][START_REF] Matsui | The temperature-pressure-volume equation of state of platinum[END_REF] which is very important in the frame of this work. More recent data by Huang et al. [START_REF] Huang | In situ synchrotron X-ray diffraction with laser-heated diamond anvil cells study of Pt up to 95 GPa and 3150 K[END_REF] have been found to be not only inconsistent with the other authors but also self-inconsistent. They were not considered in the present work.

Shock wave data are available in the form of Hugoniot plots and are essentially compatible with each other [START_REF] Mcqueen | The equation of state of solids from shock wave studies[END_REF][START_REF] Morgan | The equation of state of platinum to 680 GPa[END_REF][START_REF] Marsh | LASL shock Hugoniot data[END_REF][START_REF] Holmes | The equation of state of platinum to 660 GPa (6.6 Mbar)[END_REF]. In the three Refs. [START_REF] Morgan | The equation of state of platinum to 680 GPa[END_REF], [START_REF] Mcqueen | The equation of state of solids from shock wave studies[END_REF] (converted in Ref. [START_REF] Holmes | The equation of state of platinum to 660 GPa (6.6 Mbar)[END_REF]) and [START_REF] Holmes | The equation of state of platinum to 660 GPa (6.6 Mbar)[END_REF], they have been converted into isotherms at room temperature and compare very well with cold compressibility curves. These converted data will be used in the optimization. The advantage of shock wave data is the possibility to extend the description up to extremely high pressure not reached with conventional compressibility measurements. The adiabatic bulk modulus as a function of temperature measured by sound velocity [START_REF] Collard | High-temperature elastic constants of platinum single crystals[END_REF] has been converted into isothermal bulk modulus.

Density Functional Theory data (DFT) is also available for Pt. Both Refs. [START_REF] Menéndez-Proupin | Ab initio caculations of elastic properties of compressed Pt[END_REF][START_REF] Sun | Lattice dynamics and thermal equation of state of platinum[END_REF] use Local Density Approximation (LDA) functional in quasiharmonic phonon calculations. As usual for DFT data, it is difficult to reproduce exactly the experimental volume which causes problems if one wants to use the data in the optimization. An empirical correction is needed not only at low temperature (to compare with the cold compression curve) but also at high temperature to account for anharmonic phonon-phonon and electron-phonon contributions.

New DFT data have been obtained in the frame of the present work. Details of the calculations (classical DFT and phonon calculations in quasiharmonic approximation) can be found in our previous papers [START_REF] Bourgeois | Thermodynamic modeling of the Ni-H system[END_REF]. Calculations of the molar volume, thermal expansion and bulk modulus at zero pressure and volume as a function of pressure at 0 K have been obtained. In order to have the best agreement between the equilibrium volume and the experimental one, the three functionals PBE, PBEsol and LDA were tested [START_REF] Perdew | Generalized gradient approximation made simple[END_REF][START_REF] Perdew | Restoring the density-gradient expansion for exchange in solids and surfaces[END_REF]. Compared to experiments, PBEsol gives far better agreement on the volume than the other functionals. As a classical scheme, PBE overestimates while LDA underestimate the volume (see the comparison in Supplementary Data). With PBEsol, the volume at low temperature and the bulk modulus is perfectly reproduced. No significant difference is observed between the DFT volume and that obtained from the phonon calculation (9.042 x 10 -6 vs 9.062 x 10 -6 m 3 /mol). The thermal expansion is slightly overestimated (it is a bit better with LDA) which makes the agreement with the experimental volume and bulk modulus less good at high temperature. Overall, with PBEsol the comparison with experimental data is very good but not enough to be used in the frame of the thermodynamic optimization. DFT data is also not really needed when exists trustful experimental data. It will be shown on the graphs for comparison only.

More interesting is the derivation of both volume and thermal expansion at both high pressure and temperature. For this, the following procedure was adopted. Energy-volume relationships were obtained at different temperatures up to 3000 K from phonon calculations. The calculated data was fitted to a 4 th degree polynomial. The pressure as a function of volume obtained by derivation is then described as a 3 rd degree polynomial. The equation is solved in order to obtain the volume as a function of pressure at different temperatures. A numerical derivation as a function of temperature can then be made to obtain the thermal expansion at different pressures.

liquid phase

Molar volume as a function of temperature is available from different authors below (undercooled liquid) and above the melting point. One must stress the extreme difficulty of such measurements. Three groups of datasets can be distinguished. Stankus and Khairulin [START_REF] Stankus | Measurement of the thermal properties of platinum in the temperature interval 293-2300 K by the method of penetrating radiation[END_REF] and Mehmood et al. [START_REF] Mehmood | Thermophysical properties of platinum-copper alloys[END_REF] have higher values, Gathers et al. [START_REF] Gathers | Thermodynamic characterization of liquid metals at high temperature by isobaric expansion measurements[END_REF][START_REF] Gathers | Thermodynamic characterization of liquid metals at high temperature by isobaric expansion measurements[END_REF], Hixson and Winkler [START_REF] Hixson | Thermophysical properties of liquid platinum[END_REF], Ishikawa et al. [START_REF] Ishikawa | Non-contact thermophysical property measurements of liquid and supercooled platinum[END_REF] and Paradis et al. [START_REF] Paradis | Thermophysical properties of platinum group metals in their liquid undercooled and superheated phases[END_REF] have intermediate values and, finally, Klotz and Plevachuk [START_REF] Klotz | Thermophysical properties of platinum-rich alloys measured by sessile drop experiments[END_REF] have the lower values. Arblaster [START_REF] Arblaster | Selected values for the densities and molar volumes for the platinum group metals and of the initial melting curves of iridium, rhodium and ruthenium[END_REF] stated that the intermediate values for the volume change at the melting point should be preferred because they better agree with the measurement of the melting point as a function of pressure (the two depend on each other through the Clausius-Clapeyron relationship). The data of Ishikawa et al. and Paradis et al. are particularly valuable because they extend well below the melting point. In addition, the measurement of the size of a levitated droplet is more accurate than the measurement of a wire diameter used by other authors [START_REF] Mehmood | Thermophysical properties of platinum-copper alloys[END_REF][START_REF] Gathers | Thermodynamic characterization of liquid metals at high temperature by isobaric expansion measurements[END_REF][START_REF] Gathers | Thermodynamic characterization of liquid metals at high temperature by isobaric expansion measurements[END_REF][START_REF] Hixson | Thermophysical properties of liquid platinum[END_REF]. Among the two datasets, the one of Paradis et al. should be given a larger weight due to a better accuracy (J.F. Paradis, personal communication). The extension to extremely high temperature in the data of Gathers et al. [START_REF] Gathers | Thermodynamic characterization of liquid metals at high temperature by isobaric expansion measurements[END_REF][START_REF] Gathers | Thermodynamic characterization of liquid metals at high temperature by isobaric expansion measurements[END_REF] and Hixson and Winkler [START_REF] Hixson | Thermophysical properties of liquid platinum[END_REF] has been considered but the disagreement is significant.

The melting curve was studied by different authors. The most complete experimental works are those of Errandonea [START_REF] Errandonea | High-pressure melting curves of the transition metals Cu, Ni, Pd, and Pt[END_REF] for pressures up to 28 GPa and Anzellini et al. [START_REF] Anzellini | In situ characterization of the high pressure-high temperature melting curve of platinum[END_REF] for pressures up to 84 GPa. They agree very well with previous experimental work by Mitra et al. [41] and Vereshchagin and Fateeva [START_REF] Vereshchagin | Melting curves of graphite, tungsten, and platinum up to 60 kbar[END_REF] and theoretical predictions by the so-called Z method [START_REF] Anzellini | In situ characterization of the high pressure-high temperature melting curve of platinum[END_REF][START_REF] Belonoshko | High-pressure melting curve of platinum from ab initio Z method[END_REF][START_REF] Burakovsky | Z methodologies for phase diagram studies: platinum and tantalum as examples[END_REF] or by molecular dynamics [START_REF] Wang | Melting properties of Pt and its transport coefficients in liquid states under high pressure[END_REF][START_REF] Zhang | Melting curves of Cu, Pt, Pd, and Au under high pressures[END_REF]. The lower values of the melting point of Ref. [START_REF] Kavner | High-pressure melting curve of platinum[END_REF] and [START_REF] Patel | High pressure melting curve of platinum up to 35 GPa[END_REF] were not considered. Note that Ref. [START_REF] Burakovsky | Z methodologies for phase diagram studies: platinum and tantalum as examples[END_REF] predicts a randomly disordered hexagonal close-packed phase above 50 GPa in equilibrium with the liquid. This has been confirmed neither experimentally nor by other calculations and will not be considered in the present assessment. Finally, the bulk modulus has been derived from sound velocity measurements as a function of temperature at extremely high temperature [START_REF] Hixson | Thermophysical properties of liquid platinum[END_REF].

Previous assessments of the system

Several equations of state for fcc platinum are available in the literature [18-20, 22, 25, 27, 30, 49-51]. No model has been proposed for the liquid and no Calphad-type assessment is available in the literature except in the following reference, in Chinese, presenting the modeling of the fcc phase (no liquid) using the Debye-Grüneisen model [START_REF] Xu | Study of thermodynamic and thermophysical properties of fcc Pt through CALPHAD approach[END_REF]. It is also interesting to mention the description by Karbasi et al. [START_REF] Karbasi | The thermodynamics of several elements at high pressure[END_REF] using the model of Brosh et al. [START_REF] Brosh | Application of Calphad to high pressures[END_REF].

Results

All the parameters were optimized using the Parrot module of Thermo-Calc. The fcc phase was optimized first. A good agreement was obtained with our model using a limited number of parameters. The cold compression curve allows to optimize V 0 , bulk modulus and pressure dependence of the bulk modulus while thermal expansion terms were refined from the low pressure volume and thermal expansion function of temperature. For the compressibility, in the low pressure range (0-90 GPa), all the SRXD data are in excellent agreement and were used simultaneously. For higher pressure, only shock wave data exists. The optimization of the data of McQueen et al. [START_REF] Mcqueen | The equation of state of solids from shock wave studies[END_REF] were preferred because they better match the SXRD data. It is an intermediate between the data of Morgan [START_REF] Morgan | The equation of state of platinum to 680 GPa[END_REF] and Holmes et al. [START_REF] Holmes | The equation of state of platinum to 660 GPa (6.6 Mbar)[END_REF] extending to the highest pressures.

Then, the data combining high pressure and temperature were optimized. As mentioned before in the theoretical development of the model, in addition to the cut-off pressure for the temperature dependent terms for V K and terms in T 2 and T 3 for V A (p cut =10 9 Pa), it was found necessary to introduce a second cut-off pressure (p cut' ) for the term in T for V A corresponding to the constant thermal expansion. This cut-off pressure could be refined using the experimental data and our own DFT data at high temperature and a very good agreement is obtained with this single parameter both on the presentation of volume as a function of temperature at high pressure, and volume as a function of pressure at high temperature. This not only proves that thermal expansion should decrease as a function of temperature but also confirm the relevance of our exponential term to achieve this. Hugoniot data cannot be used directly in the optimization. They can, however, be calculated after the model is established. The comparison shows a relative agreement that can be explained by an underestimation of the thermal expansion at extremely high pressure. A better agreement of the Hugoniot was obtained by Karbasi et al. [START_REF] Karbasi | The thermodynamics of several elements at high pressure[END_REF] using the model of Brosh et al. [START_REF] Brosh | Application of Calphad to high pressures[END_REF].

Figs. 3 to 8 present the results with a comparison with experimental data. In Fig. 5, the low temperature range is evidently not described by our simple model of thermal expansion.

The liquid was then modeled with a limited set of parameters and parameters of the fcc phase fixed. V 0 and thermal expansion were refined from the volume as a function of temperature. Care was taken in order to have the volume of the liquid always higher than that of the solid. The volume of Gathers et al. (at 0.3 GPa) [START_REF] Gathers | Thermodynamic characterization of liquid metals at high temperature by isobaric expansion measurements[END_REF][START_REF] Gathers | Thermodynamic characterization of liquid metals at high temperature by isobaric expansion measurements[END_REF] is larger than that of Hixson and Winkler [START_REF] Hixson | Thermophysical properties of liquid platinum[END_REF] which should evidently not be the case. On the other hand, none of these two datasets is compatible with that of Paradis et al. [START_REF] Paradis | Thermophysical properties of platinum group metals in their liquid undercooled and superheated phases[END_REF] close to the melting point. A compromise was obtained between the data of Paradis et al. that were preferred because it is more accurate and that of Hixson and Winkler around 4000 K which allows also to reproduce the data of Gathers et al. around 7000 K. The melting temperature as a function of pressure was successfully described at the same time of the bulk modulus function of temperature. This confirms the good estimation of the volume change at the melting point and its temperature and pressure variation. The liquid has a lower bulk modulus than the solid but becomes stiffer with the pressure. The results are presented in Figs. 9 to 11 and the volume function of pressure for which no data is available is shown in Fig. 4a.

Above 150 GPa melting can no longer be described. This is because the fcc phase becomes stable over the liquid at very high temperature, even at low pressure, due to unconsidered variation of the C p of the fcc phase at high temperature as illustrated in Fig. 12. Except this anomaly caused by the SGTE description of the fcc phase that will be discussed later, no strange behavior is observed up to extreme pressures demonstrating the efficiency of our correction to Lu's model. Table 2 presents the refined optimized parameters for the two phases.

Discussion

Other models

As observed in the introduction, many models are available in the literature. We will limit our comparison to the most recent ones [START_REF] Brosh | Application of Calphad to high pressures[END_REF][START_REF] Brosh | Explicit Gibbs free energy equation of states for solids[END_REF][START_REF] Lu | A CALPHAD Helmholtz energy approach to calculate thermodynamic and thermophysical properties of fcc Cu[END_REF][START_REF] Jacobs | An alternative use of Kieffers's lattice dynamics model using vibrational density of states for constructing thermodynamic databases[END_REF]. These models are based on the quasi-harmonic approximation, the Debye-Grüneisen model, or the modeling of phonon frequencies, respectively. They offer very sound and complete approaches encompassing a closer relation to the physics and a superior use of DFT data and phonon calculations.

However, as a consequence of their link to the DFT or phonon calculation, most of them are limited to the only phases that can be calculated by these methods i.e. crystalline and stable phases. Unstable structures that have, for example, imaginary phonon frequencies or the liquid phase cannot be described using these models. Additionally, they do not use the SGTE description of the pure elements which is a severe limitation if one wants to implement pressure models in already assessed systems using the SGTE description.

These two features limit severely their applicability as regards the extension to the calculation of complete phase diagrams (including liquid) and the extension to multicomponent systems involving metastable (unstable) end-members as usual for the Calphad technique.

As an exception, Brosh's (2007) model [START_REF] Brosh | Application of Calphad to high pressures[END_REF] has been designed to follow SGTE at low pressure and converges to the quasi-harmonic model at high pressure. It is therefore physically correct and has had undeniable achievements including for platinum, as we already mentioned [START_REF] Karbasi | The thermodynamics of several elements at high pressure[END_REF], and even with the modeling of binary systems (Al-Si [START_REF] Brosh | Application of Calphad to high pressures[END_REF], Fe-C [START_REF] Fei | Experimental study and thermodynamic calculations of phase relations in the Fe-C system at high pressure[END_REF]).

But, all the models, including Brosh's, because of their complexity and the high number of parameters they use, are not adapted to systems for which only few data exist, or sometimes no data at all as it may be the case for metastable phases. Anticipating and fixing values, in such cases, may be difficult since the meaning and the effective influence of the model parameters on the volume and its derivative is difficult to comprehend. They may also neither be programmed nor used with the optimization modules of thermodynamic software packages because of their complexity. As an example Karbasi et al. [START_REF] Karbasi | The thermodynamics of several elements at high pressure[END_REF] write that Cr and Ni could not be treated with Brosh's model due to the scarcity of data.

Advantages of Lu's model

In comparison, the advantages of Lu's (2005) model are numerous. First, it is simple to use and the meaning and the influence of each parameter on the physical quantities like the volume or its pressure and temperature derivatives is easy to understand. As a consequence, it is uncomplicated to limit the number of parameters, or fix some, depending on the amount of data. It is fully compatible with SGTE. But it does not prevent the use of DFT and phonon calculation, as shown here, and as will be shown in the second part of this study [START_REF] Joubert | Modification of Lu's (2005) high pressure model for improved high pressure/high temperature extrapolations. Part II: modeling of osmiumplatinum system at high pressure/high temperature[END_REF].

It is fully adapted to the Calphad methodology. It is phenomenological like the Calphad method is. This includes the extrapolation possibilities to describe complete or non-complete solid solutions [START_REF] Huang | Modeling of the molar volume of the solution phases in the Al-Cu-Mg system[END_REF]. It may be used for stoichiometric or non-stoichiometric compounds within the sublattice model, like for example to describe interstitial solid solutions [START_REF] Bourgeois | Thermodynamic modeling of the Ni-H system[END_REF]. It can be used for any kind of phase, crystalline or not (solid or liquid), stable or not (not stable end-members, pure elements in nonstable structures). Of course, assumptions should be made to describe metastable phases when no data exist. This will be shown in Part II of this paper. The functions are compatible with the Calphad mixing formalism, they can be extended as a function of composition using excess terms similar to other (energetic) mixing parameters. This means that complete phase diagrams can be described and that predictions made in the frame of classical Calphad extrapolations can be made by combining systems. Another advantage is that it is also fully implemented in Thermo-Calc so that there is no need to develop an additional code for its application.

Lu's phenomenological model is therefore probably better than other more physical model for the construction of Calphad databases. The only drawback of the model was the wrong extrapolation of entropy and C p at high temperature and pressure. This paper has shown that this problem was essentially due to the fact that thermal expansion (and to a lesser extent temperature dependence of the compressibility) was left uncontrolled at high pressure. We have also shown that the problem can be easily solved.

We obtained a complete description of the Pt system including the liquid phase with a very good description of the available data. The model is simple and easily transferrable to any other system. It can also be extrapolated to higher order systems as will be shown for Os-Pt system in the second part of this study [START_REF] Joubert | Modification of Lu's (2005) high pressure model for improved high pressure/high temperature extrapolations. Part II: modeling of osmiumplatinum system at high pressure/high temperature[END_REF].

Problem of high temperature heat capacity from SGTE

More than Lu's model, the SGTE database should be criticized. It is not adapted to the use of high pressure models. This is for two different reasons. First, the lattice stabilities are not adapted to a raise of the melting point. For example, if liquid is suspended (or made less stable by the effect of pressure), fcc Os is stable against hcp phase at 5200 K without any reason. Second, the extrapolation of the C p of the solid above the melting point is wrongly done for many elements. In principle, the socalled SGTE approximation [START_REF] Guillermet | Critical evaluation of the thermodynamic properties of molybdenum[END_REF][START_REF] Andersson | A new method describing lattice stabilities[END_REF] involving a convergence of the C p of the solid phase toward that of the liquid should be used. An examination of the PURE database [START_REF] Sundman | SGTE Pure Element Database, version 5.1[END_REF] shows that, surprisingly, it is not the case for many elements. This was shown here for Pt. But this is also the case for Os and many other more common elements (Ta, Ti…). As a consequence, in our case, the uncontrolled C p of fcc Pt at high temperature causes the stabilization of this phase over the liquid at 11000 K and ambient pressure. Such high temperatures are rarely used in normal thermodynamic assessments, but they are common in the high pressure context as shown in Figs. 8 and11.

At least the SGTE approximation should be used systematically. But, better, a revision of the PURE database is strongly recommended that would encompass the evaluation of sensible values for the solid above the ambient pressure melting point because they no longer characterize a metastable phase when pressure is raised. A very efficient way of doing this and solving this long standing problem (see e.g. the discussions on the C p of Al above its melting point [START_REF] Bigdeli | An insight into using DFT data for Calphad modeling of solid phases in the third generation of Calphad databaes, a case study for Al[END_REF][START_REF] Schick | A modified Neumann-Kopp treatment of the heat capacity of stoichiometric phases for use in computational thermodynamics[END_REF][START_REF] Sundman | A method for handling the extrapolation of solid crystalline phases to temperature far above their melting point[END_REF]) would be the possibility to actually use high pressure in order to stabilize the solid phase and measure the C p above the normal melting point. This is probably difficult experimentally and, as far as we are aware, such measurements do not exist for metallic materials. This should be investigated, given the importance of such determination. A possibility would be to use shock wave measurements and Hugoniot curves for such a purpose. Another option would be to impose a high temperature limit for the solid (but why not also for the liquid) equal to 3R. This is the expected limit at high pressure. So when the melting point raises with pressure the C p of the stable phase would naturally converge towards 3R. Finally, on may mention that our model can be used easily with 3 rd generation databases.

Remaining issues

The presently modified model conveniently removes the negative contribution to the C p at high pressures and temperatures that was present in Lu's model. It allows a consistent description of the thermal expansion as a function of temperature when pressure is increased with a decrease of the slope around and a decrease of the thermal expansion itself around . This allows a good description of p-V curves at high temperature (Fig. 7). However, several anomalies remain. For example, a small increase of the bulk modulus as a function of temperature can be seen at p=10 11 Pa in Fig. 6. The slope decrease of the thermal expansion function of temperature at high pressure is not necessarily well controlled. One may see in Fig. 5 that thermal expansion is constant at 50 GPa while DFT calculations in the present and in other works [START_REF] Sun | Lattice dynamics and thermal equation of state of platinum[END_REF][START_REF] Hammerschmidt | Including the effects of pressure and stress in thermodynamic functions[END_REF] show a remaining slope at this pressure. This may have an effect on the heat capacity. Indeed, at very high pressure, C p is expected to converge towards C V as seen in Ref. [START_REF] Hammerschmidt | Including the effects of pressure and stress in thermodynamic functions[END_REF]. This can anyway not be the case in our model since, by construction, C p converges towards the ambient pressure SGTE value. The C p at high pressure is therefore essentially unchanged from the ambient pressure value which is at odds with the predictions of physically-derived models [START_REF] Hammerschmidt | Including the effects of pressure and stress in thermodynamic functions[END_REF][START_REF] Kangarlou | Thermodynamic properties of copper in a wide range of pressure and temperature within the quasi-harmonic approximation[END_REF][START_REF] Tirone | On the use of thermal equations of stateand the extrapolation at high temperature and pressure for geophysical and petrological applications[END_REF]. Also, a minimum of the C p is observed around 1 GPa which is an anomaly. Again, and as detailed in the previous paragraph, more attention should be given to the extrapolation of this parameter in the metastable range. It seems that Brosh's model [START_REF] Brosh | Application of Calphad to high pressures[END_REF] has more realistic behavior. But, these anomalies, because they relate to second derivatives, have a second order effects and therefore no significant consequences on volume and phase diagram descriptions. Evidently, extrapolations at extremely high temperatures and pressures should always be taken with care. But we believe that these anomalies have small effects compared to the experimental uncertainties on which the model is optimized.

Conclusion

A modification has been proposed to Lu's (2005) model to suppress the anomalous extrapolation of specific heat and entropy at high temperatures and pressures. With this modification, we have shown that such anomalies are not present in a typical system like platinum which has been described completely including the liquid phase. A model is always a tradeoff between accuracy, predictability and usability. Modified Lu's model is indeed so easy to handle compared to many other models that it should be considered again as a powerful tool in the Calphad approach, especially in the frame of assessment of multicomponent systems, including liquid, metastable end-members, and compounds for the development of complete and useful databases. The question of the validity of the SGTE database has also been raised. We will show an application to a binary system (Os-Pt) in the second part of this study [START_REF] Joubert | Modification of Lu's (2005) high pressure model for improved high pressure/high temperature extrapolations. Part II: modeling of osmiumplatinum system at high pressure/high temperature[END_REF]. [START_REF] Edwards | High temperature structure and thermal expansion of some metals as determined by X-ray diffraction data. I. Platinum, Tantalum, Niobium, and Molybdenum[END_REF] [51EDW], [START_REF] Stankus | Measurement of the thermal properties of platinum in the temperature interval 293-2300 K by the method of penetrating radiation[END_REF] [92STA], [START_REF] Arblaster | Crystallographic properties of platinum[END_REF] [97ARB], [START_REF] Matsui | The temperature-pressure-volume equation of state of platinum[END_REF] [09MAT] (at 24 and 40 GPa), [START_REF] Mehmood | Thermophysical properties of platinum-copper alloys[END_REF] [12MEH] and [START_REF] Klotz | Thermophysical properties of platinum-rich alloys measured by sessile drop experiments[END_REF] ). The experimental data are from Refs. [START_REF] Paradis | Thermophysical properties of platinum group metals in their liquid undercooled and superheated phases[END_REF] [14PAR], [START_REF] Gathers | Thermodynamic characterization of liquid metals at high temperature by isobaric expansion measurements[END_REF] [79GAT] (at 3.10 8 Pa), [START_REF] Hixson | Thermophysical properties of liquid platinum[END_REF] [93HIX], [START_REF] Stankus | Measurement of the thermal properties of platinum in the temperature interval 293-2300 K by the method of penetrating radiation[END_REF] [92STA], [START_REF] Ishikawa | Non-contact thermophysical property measurements of liquid and supercooled platinum[END_REF] [06ISH], [START_REF] Mehmood | Thermophysical properties of platinum-copper alloys[END_REF] [12MEH] and [START_REF] Klotz | Thermophysical properties of platinum-rich alloys measured by sessile drop experiments[END_REF] [16KLO]. [67MIT], [START_REF] Kavner | High-pressure melting curve of platinum[END_REF] [98KAV], [START_REF] Belonoshko | High-pressure melting curve of platinum from ab initio Z method[END_REF] [12BEL], [START_REF] Errandonea | High-pressure melting curves of the transition metals Cu, Ni, Pd, and Pt[END_REF] [13ERR], [START_REF] Burakovsky | Z methodologies for phase diagram studies: platinum and tantalum as examples[END_REF] [14BUR], [START_REF] Wang | Melting properties of Pt and its transport coefficients in liquid states under high pressure[END_REF] [16WAN], [START_REF] Zhang | Melting curves of Cu, Pt, Pd, and Au under high pressures[END_REF] [16ZHA], [START_REF] Patel | High pressure melting curve of platinum up to 35 GPa[END_REF] [18PAT] and [21] [19ANZ]. The data from Ref. [START_REF] Burakovsky | Z methodologies for phase diagram studies: platinum and tantalum as examples[END_REF] [14BUR] above 40 GPa represent equilibrium between hcp and fcc or liquid and are not to be considered.

Figure 1 :Figure 2 :Figure 3 :

 123 Figures:

Figure 6 :Figure 7 :Figure 8 :Figure 9 :

 6789 Figure 6: Bulk modulus of fcc Pt as a function of temperature at different pressures. The experimental data is from [28] [92COL] and has been transformed from adiabatic to isothermal bulk modulus. Calculated data are obtained in this work from PBEsol DFT and phonon calculation at zero pressure.

Figure 10 :Figure 11 :

 1011 Figure 10: Bulk modulus of liquid Pt (fcc phase is shown for comparison) at p=10 5 Pa. Experimental data is from Ref. [37] [93HIX].

Figure 12 :

 12 Figure 12: C p of fcc and liquid Pt at different pressures.

Figure 1 :Figure 2 :Figure 3 :Figure 4 :

 1234 Figure 1: Molar volume of fcc Pt as a function of temperature calculated with three different functionals.

Table 1 :

 1 summary of relevant data for the optimization of platinum (HT: High Temperature, HP: High Pressure, (S)XRD: (Synchrotron) X-Ray Diffraction, DAC: Diamond Anvil Cell).

	Phase	Data type	Range	Reference	Technique
	fcc	V-T	300-2000 K	[62]	HT XRD
		at p=10 5 Pa
			300-2040 K	[34]	penetrating gamma
					radiation
			0-2041 K	[13]	assessment of
					experimental data
			1740-2042 K	[15]	optical
					measurement
			1790 -2030 K	[16]	optical
					measurement
			1788-2033 K	[17]	optical
					measurement
		α-T	246-843 K	[63]	interferometry
		at p=10 5 Pa
			10-1800 K	[64]	assessment of
					experimental data
			0-2041 K	[13]	assessment of
					experimental data
			293-2041 K	[14]	assessment of
					experimental data
		V-p	0-268 GPa	[24]	shock wave
		at 300 K
			0-680 GPa	[25]	shock wave
			(293 K)
			0-536 GPa	[27]	shock wave
			0-89 GPa	[18]	HP SXRD
			3-22 GPa	[19]	HP SXRD
			3-78 GPa	[20]	HP SXRD
			0-80 GPa	[21]	HP SXRD
		B-T	300-1400 K	[28]	Sound velocity
		V-T-p	300-1873 K	[19]	HT HP SXRD
			3-28 GPa

  [16KLO]. Calculated data are obtained in this work from PBEsol DFT and phonon calculation at zero pressure and at different pressures (see text). Linear thermal expansion of fcc Pt as a function of temperature for different pressures. The experimental data are from Refs.[START_REF] Edsinger | Thermal expansion of platinum and platinumrhodium alloys[END_REF] [86EDS],[START_REF] Kirby | Platinum-A thermal expansion reference material[END_REF] [91KIR],[START_REF] Arblaster | Crystallographic properties of platinum[END_REF] [97ARB] and[START_REF] Arblaster | Crystallographic properties of platinum[END_REF] [06ARB]. Calculated data are obtained in this work from PBEsol DFT and phonon calculation at zero pressure and at different pressures (see text).

	Linear ) -1 V m thermal (m 3 /mol) (K expansion	6.0x10 -6 7.0x10 -6 8.0x10 -6 2.5x10 -5 9.0x10 -6 5.0x10 -6 1.0x10 -5 1.5x10 -5 2.0x10 -5	Shock wave [70McQ] [89HOL] p=100 GPa p=50 GPa [74MOR] (293 K) Calculated This work (0 K) a pressure calculation Calculated p=0 Pa HP SXRD Experimental [86EDS] [04DEW] [04FEI] [08ZHA] [19ANZ] [91KIR] p=1 GPa [97ARB] [06ARB] p=10 5 Pa
				fcc		p=150 GPa
		5.0x10 -6 0.0		liquid		p=200 GPa	p=1000 GPa
			0.0 0	500	2.0x10 11 1000	4.0x10 11 1500 2000	6.0x10 11 2500	3000
						p (Pa) T (K)
	0.0 Figure 4: Molar volume of fcc Pt as a function of pressure at 300 K (full scale with the liquid phase (a) 2.0x10 10 4.0x10 10 6.0x10 10 8.0x10 10 1.0x10 11 7.5x10 -6 8.0x10 -6 8.5x10 -6 9.0x10 -6 b Shock wave [70McQ] [74MOR] (293 K) [89HOL] Calculated This work (0 K) HP SXRD [04DEW] [04FEI] [08ZHA] [19ANZ] V m (m 3 /mol) p (Pa) and lower pressure region (b)). The experimental data are from Refs. [24] [70McQ], [25] [74MOR], 500 1000 1500 2000 2500 1x10 11 2x10 11 3x10 11 4x10 11 5x10 11 6x10 11 7x10 11 Figure 5: 0 8x10 11
	[27] [89HOL], [18] [04DEW], [19] [04FEI], [20] [08ZHA] and [21] [19ANZ]. Calculated data are
	obtained in this work from PBEsol calculation.

Acknowledgements

DFT calculations were performed using HPC resources from GENCI CINES (Grant A0080906175).

Supplementary materials to:

Modification of Lu's (2005)


Baptiste Blaineau

Sébastien Dutour
email: sebastien.dutour@laplace.univ-tlse.fr

Thierry Callegari

Pascal Lavieille

Marc Miscevic

Stéphane Blanco

Benoît Schlegel

Yves Bertin

Adel M Benselama

Benoit Schlegel

Experimental investigation of a dielectric liquid-vapor interface between two vertical planar electrodes: Influence of the DC electric field and temperature

Keywords: Liquid-vapor interface, Electrohydrodynamics, Two-phase cooling systems

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Heat pipes, Capillary Pumped Loops (CPL) and Loop Heat Pipes (LHP) are passive two-phase systems which are involved in electronics cooling. The vaporization of the working fluid in a porous wick or sub-millimeter grooves results in a natural fluid circulation between the heat exchangers. These devices are highly efficient and reliable. However, the maximum heat load transferred by the cooling system is mainly restricted by the maximum pumping capability of the capillary structure which is given by the maximum pressure difference generated at the vaporisation interface (the capillary limit). In order to improve the performance of such cooling systems, it is now considered to assist them with an active device supplying power to the fluid. One option is to use an electric field and the electrohydrodynamic (EHD) forces induced in the fluid. Actually, within this approach, the power consumption is very low compared to mechanical pumping and no moving pieces are involved reducing the risk of failure.

The use of an electric field in order to enhance the performance of the capillary systems has been considered since the seventies and was firstly motivated by space applications [START_REF] Jones | Electrohydrodynamic heat pipes[END_REF], [START_REF] Jones | Electrohydrodynamic heat pipe experiments[END_REF], [START_REF] Loehrke | Performance characteristics of several ehd heat pipe designs[END_REF]. The approach considered that the capillary structure of heat pipes might be augmented or replaced by electrodes and EHD pumping. Equivalent thermal performance of conventional axial-groove heat pipes was achieved but questions regarding the long-term stability of dielectric fluids in the presence of corona and intermittent discharges drastically restrained their development.

Later on, Babin et al. [START_REF] Babin | Experimental investigation of an ion-drag pump-assisted capillary loop[END_REF] coupled an ion-drag pump with a small CPL filled with R11 fluid. The approach was slightly different than previously because the electrodes could be placed in the liquid line of the capillary pump. Here, unipolar charges were injected by a set of needles to overcome the pressure drop in the loop. The comparison of performance in a horizontal orientation with and without the ion-drag pump showed an enhancement of 20% to 60% of the heat transport capacity. Once the conduction regime based on the dissociation of the liquid molecules into ions has been clarified [START_REF] Atten | Electrohydrodynamically induced dielectric liquid flow through pure conduction in point/plane geometry[END_REF], the investigation became more active as a longer stability of the liquid properties can be expected. A determinant effort was the fact of J. Yagoobi and his co-workers who demonstrated a 100% enhancement in the transport capacity of a monogroove heat pipe [START_REF] Bryan | Heat transport enhancement of monogroove heat pipe with electrohydrodynamic pumping[END_REF], [START_REF] Jeong | Experimental study of electrohydrodynamis pumping through conductino phenomenon[END_REF]. They also investigated two-phase cooling systems based on the pumping of millimetric liquid films [START_REF] Pearson | Experimental study of linear and radial two-phase heat transport devices driven by electrohydrodynamic conduction pumping[END_REF], [START_REF] Patel | Combined Dielectrophoretic and Electrohydrodynamic Conduction Pumping for Enhancement of Liquid Film Flow Boiling[END_REF]. An important demonstration for the potential of conduction pumping at a larger scale was due to Jeong et al. [START_REF] Jeong | Thermal control utilizing an electrohydrodynamic conduction pump in a two-phase loop with high heat flux source[END_REF], [START_REF] Jeong | Performance characteristics of electrohydrodnamic conduction pump in two-phase loops[END_REF] who tested a ten-meters long two-phase loop filled with R134a. They found that for an applied voltage of 15 kV and a sink temperature of 20 • C, the generated pressure head was approximately 9.6 kPa and the transported heat load reached 130 W.

A somewhat different approach was provided by Mo et al. [START_REF] Mo | Startup time reduction in an electrohydrodynamically enhanced capillary pumped loop[END_REF], [START_REF] Mo | Capillary pumpedloop thermal performance improvement with electrohydrodynamic technique[END_REF] who investigated a larger CPL (maximum allowable heat-transport capacity of 1 500 W) filled with R134a and involving a EHD-assisted cylindrical evaporator. Here, a spring electrode was inserted under the wick within the evaporator core while the casing was grounded in order to influence the liquid-vapor separation inside the evaporator and to assist the pumping menisci in the wick. The results showed that up to three times heat-transfer coefficient enhancement could be obtained. They also revealed that the deprime condition of a capillary pump can be prevented in the presence of an electric field. Harvel et al. [START_REF] Harvel | Electrohydrodynamically enhanced capillary evaporator[END_REF] confirmed a significant decrease of the partial dry out mechanism of the evaporator using a coaxial electrode directly located in the evaporator core.

Concurrently to pumping, EHD has been widely used with the objective of the two-phase heat transfer enhancement. Up to 20 times of augmentation of the average heat transfer coefficient was reported in the existing literature as compiled by Laohalertdecha et al. [START_REF] Laohalertdecha | A review of electrohydrodynamic enhancement of heat transfer[END_REF]. A recent example of application to a gravity heat pipe filled with HFE-7000 and using AC electric field in the evaporator is proposed by Smith et al. [START_REF] Smith | Electrohydrodynamic augmentation of a reflux thermosyphon[END_REF]. A cylindrical electrode was inserted along the thermosyphon axis whilst the casing was grounded. They found an enhancement up to 40% of the evaporator heat transfer coefficient for heat flux lower than 8 kW/m 2 due to the the thinning of the liquid film in contact with the evaporator wall. But, the electric field also led to a partial dry-out of the evaporator for greater heat flux.

These numerous demonstrations of a significant improvement in the performance of two-phase cooling systems contrast with the fact that, to our knowledge, none of these devices has been implemented in thermal management applications. This is primarily due to some technical issues: (i) charge injection at the surface electrode leads to a fast alteration of both electrodes and fluid properties so it must be avoided; as this regime is highly affected by the wall shape and its surface state, the design of the electrodes and their integration in the cooling system remain extremely sensitive, (ii) the implementation of EHD devices with high voltage is difficult with some widely used fluids which are highly conductive such as water and ammonia. In this case, an electrode coating with a dielectric layer is necessary to limit the current; defaults in this coating can breakdown at relatively low voltages limiting the operating range. Another limitation is the high complexity of the phenomena involving different electrostatic effects, fluid flow, interface dynamics and instabilities, temperature gradients and heat flux. This is indeed demanding for theoretical investigations and numerical efforts with the objective of the cooling system dimensioning. In addition, there is still a lack of understanding of some observations especially in the interfacial region. For instance, the origin of the saturation of the contact angle in electrowetting on dielectric remains an open question [START_REF] Bazant | Electrotunable wetting, and micro-and nanofluidics: general discussion[END_REF].

Facing this, there is a serious need of investigations that report carefully designed experiments in simple geometric configurations with the objectives to analyze the influence of temperature and heat transfer as encountered in operating conditions of the cooling systems. Thus, we propose to study the behavior of a two-phase dielectric fluid confined between two vertical and parallel electrodes. This configuration, which actually corresponds to the original work of Pellat [START_REF] Pellat | Mesure de la force agissant sur les diélectriques liquides non électrisés placés dans un champ électrique[END_REF], has many advantages: it is an experiment commonly used to understand different well controlled problems and we thus benefit of some studies clarifying separately the electrical conduction in the liquid [START_REF] Atten | Electrohydrodynamically induced dielectric liquid flow through pure conduction in point/plane geometry[END_REF], the electrowetting phenomenon at the wall [START_REF] Welters | Fast electrically switchable capillary effects[END_REF], [START_REF] Jones | Frequency-dependent electromechanics of aqueous liquids :electrowetting and dielectrophoresis[END_REF] and some effects of surface tension [START_REF] Jones | Capillarity and dielectrophoresis of liquid deuterium[END_REF]. Then, it offers a simple way to control the heating of the fluid from above the interface or directly through the liquid zone. In measuring the liquid height-of-rise, it is possible to determine the global force acting on the liquid column. Additionally, in observing the interface shape and dynamics, a more local information can be achieved and this is determinant for heat transfer specially close to the wall. The objective is firstly to study the interface behavior when a DC electric field strength is increased and, in a second hand, to determine the influence of the temperature based on some isothermal observations performed at different temperatures.

Background

The electrical force acting on a fluid is usually formulated in terms of the Korteweg-Helmoltz body force density:

f = q E - E 2 2 ∇ε + 1 2 ∇ ρ ∂ε ∂ρ T E 2 (1)
where q is the volume density of free charge, E the electric field, ρ the mass density and ε is the permittivity which is assumed to be a function of the mass density and temperature. The first term is the Coulomb's force. In a dielectric fluid, free charges can have two different sources. Firstly, they can be created by dissociation/recombination of the fluid molecules into ions with an opposite charge. Secondly, at larger electric field, charges could come from injection at the electrode-fluid interface where an electron is exchanged between the electrode metal and a molecule of the liquid.

The two last terms of the equation (1) which are respectively the dielectrophoretic (DEP) force and the electrostriction force are due to the polarization of the fluid with the electric field. In a two-phase system, the polarization forces act principally in the liquid-vapor interfacial zone where there is an important variation of the permittivity with the fluid mass density. Moreover, the literature converges to the suggestion that electrostriction has no observable influence on the incompressible electrohydrostatics [START_REF] Jones | Frequency-dependent electromechanics of aqueous liquids :electrowetting and dielectrophoresis[END_REF].

Experimental set-up and fluids properties

Experimental set-up

The test cell is based on the Pellat's original experiment: two vertical planar electrodes are partially immersed in the liquid (figure 1). The electrodes are made of a 1 mm-thick plates of glass coated with indium tin oxide (ITO). The conductive layer has a height of 40 mm and a width of 30 mm. The spacing between the plates D can be set to 2.03 mm and 3.02 mm using glass rods which are placed horizontally at the bottom and at the top of the electrodes so that the liquid is free to fall and flow on the side. The experiment takes place within an enclosure made of a double casing glass cylinder in which a transparent coolant circulates at controlled temperature (figure 2). On one end, a disk made of sapphire is placed in order to perform the interface visualizations and also IR measurements. The test cell can be placed with the electrodes pair either parallel or perpendicular to the viewport providing respectively a side-view or an end-on view of the liquid height-of-rise.

The experimental procedure consisted in increasing the voltage by steps of 0.25 kV of the high voltage (HV) electrode while the other electrode was grounded. A simultaneous high-speed video camera capture was used to determine the mean liquid height-of-rise h and the end-of-view interface shape and dynamics. As the electrodes formed an opened channel, the liquid fell out by the side and a lateral motion was observable. Nevertheless, a large unperturbed zone existed in the center of the cell where the interface was stable. The observations focussed on this region. The image resolution was 35 pixels/mm and the recording was set at 4 000 frames/s. The maximum voltage of each of the series was achieved either when the liquid reached the top of the electrodes or just before the breakdown limit in the covering vapor was reached. The maximum temperature for each fluid was fixed by the maximum pressure allowed by the enclosure (2 bar). Actually, following the temperature modification, the pressure changed from 0.24 bar at 2 • C to 2.2 bar at 60 • C for the HFE-7000 and from 0.10 bar at 2 • C to 1.8 bar at 80 • C for the HFE-7100.

For a given fluid sample and a given pair of electrodes, the repeatability of the height-of-rise measurements was established on a test during which the steps amplitude (increasing or decreasing steps, small and large amplitudes), the time duration before another step were changed. We found that the deviations were always lower than ±2 mm. The experiments were performed both at D = 2.03 mm and at D = 3.02 mm. No significative influence of the inter-electrodes spacing were found on the results even though the deviations were more noticeable for HFE-7100.

Liquid phase properties

The working HFE fluids are dielectric fluids with low global warming potential that have been developed for thermal management applications. The purity was 99.5%. In order to remove non-condensable gases, the fluids were preliminarily boiled in a separated device while high vacuum was reached in the test cell before filling. Then, the fluid saturation was controlled in situ by comparing the saturation temperature obtained from the pressure measurement with the fluid temperature. Some additional physical properties at 25 In order to investigate qualitatively the electrical conduction in both liquids, some dedicated experiments were made with the inter-electrodes spacing entirely filled with liquid. Figure 3 shows the current density in both liquids as a function of the voltage and the mean electrical field at different temperatures. The quasi-ohmic regime was observable below 2 kV. Then, for a given temperature, the curves deviated from a linear voltage dependence which is typical to the dissociation/recombination regime and (i) at high temperature, the current became almost constant and the saturation regime was reached; (ii) at low temperature, the current still increased with the voltage but with a different rate. This latter behavior is representative of the field-enhanced ionic dissociation regime. As predicted by Onsager [START_REF] Onsager | Deviation from ohm's law in weak electrolytes[END_REF], the dissociation rate is greater at low temperature for a given electrical strength.

In the ohmic regime, the ratio current density to the electric field j/E gives an order of magnitude of the electrical conduction in the device. For the HFE-7100, this ratio j/E ≈ 0.93 10 -8 S /m is consistent both with the electrical conductivity provided in table 1 and with the measurements performed by [START_REF] Nassar | Experimental investigation of the variation of HFE electric properties with temperature[END_REF]. For HFE-7000, j/E ≈ 1.65 10 -8 S /m which is two orders of magnitude lower than the 3M value. Surprisingly, no reference measurements are available in the literature for this fluid. However, based on the results obtained for HFE-7100, we could reasonably suggest that the electrical conductivity provided by 3M for the HFE-7000 may not be accurate. Obviously, this statement has to be confirmed by some specific measurements.

Results and discussion

The DEP force and the liquid height-of-rise

Figure 4 shows the liquid column height-of-rise h versus the electric field for both fluids and for temperatures ranging from 0 • C to 80 • C. A secondary axis was built by multiplying the haxis by a constant liquid density (see table 1) with the objective to give an estimation of the hydrostatic pressure. Actually, this value deviated from 4% (underestimation) at the minimum temperature to 10% (overestimation) at the maximum temperature. For HFE-7000, the deviation was about ±4%.

The electric field dependency of h is a square law and the data are consistent with the dielectric height-of-rise h DEP . This suggests that the DEP force is dominant as h DEP is defined by [START_REF] Jones | Frequency-dependent electromechanics of aqueous liquids :electrowetting and dielectrophoresis[END_REF] in a perfectly insulating fluid taking into account the DEP force only:

h DEP = 1 2 ε 0 (ε r -1) E 2 ρ l g (2)
where the electric field is assumed to be E = V/D along a surface enclosing the interface entirely. The variation with temperature of the liquid dielectric constant ε r was estimated using the temperature dependent expressions for the mass density provided by Rausch et al. [START_REF] Rausch | Density, Surface Tension, and Kinematic Viscosity of Hydrofluoroethers HFE-7000, HFE-7100, HFE-7200, HFE-7300, and HFE-7500[END_REF] and the Clausius-Mossotti relation:

ε r -1 ε r + 2 1 ρ l = const ( 3
)
where the constant was determined using the data given in table 1. Based on this approach, the liquid dielectric constant decreased with temperature in a quasi-linear trend. This is consistent with [START_REF] Nassar | Experimental investigation of the variation of HFE electric properties with temperature[END_REF] for HFE-7100 even though their results were slightly lower (8% to 10%). The effect of temperature is shown in more details in Figure 5 for several electric field strengths. The breakdown limit curve gives the maximum field strength for which a height-of-rise could be maintained without the vapor became conductive. For both fluids, h was not really affected by the temperature in the range of investigation. The larger discrepancies (about 4 mm) with the dielectric height-of-rise were observed for HFE-7100, (i) when E ≤ 3.1 kV/mm within an intermediate temperature range from 30 • C to 60 • C and (ii), when the field was greater than 4.2 kV/mm. The latter case corresponded to the more intense fluid circulation through the electrodes spacing and then the interface rise may be slightly limited by the pressure drop in the liquid. In the first case, the electrical conduction in the liquid could be a reason, as it is discussed in the next section. However, except for these few values, the data compared quite satisfactorily with the dielectric model prediction and then, according to equations ( 2) and (3), the rather weak effect of temperature could be correlated to the decrease of the dielectric constant.

Finally, the main influence of the temperature was given by the extension of the vapor breakdown domain. Actually, a drastic limitation of the operating range was observed with the decrease of the temperature. The most concerned fluid was HFE-7100. For instance, the hydrostatic pressure was limited at 100 Pa (h ≈ 5 mm) at 0 • C while it could exceed 500 Pa (h ≈ 30 mm) at 40 • C. The breakdown limit is mainly influenced by the vapor pressure: the lower the pressure, the lower the voltage threshold (except at extremely low pressure where the breakdown field strength increase again). This behavior was consistent with the observations as the lowest vapor pressure was achieved for HFE-7100.

Effect of the electrical conduction in the liquid

Figure 6 shows the influence of the electric field on the shape of the interface for a temperature of 60 • C and for HFE-7100. When no voltage was applied, the end-on view exhibited an interface profile that was symmetric and curved by capillary effect. As the electric field was increased, the meniscus first remained curved but became more and more asymmetrical. The liquid accumulated on the HV electrode and the meniscus rose for almost 2 mm higher than in the opposite side at E = 1.39 kV/mm. Then, it started to become smoother and smoother and finally the interface was maintained totally flat beyond E = 2.79 kV/mm. A similar behavior was observed for HFE-7000 even though the asymmetry was less pronounced. As it is nearly close to zero, the contact angle is not affected by the EHD forces so that the interface profile behavior results from gravitational, surface tension and electrical forces. The relative importance of each term can be measured using dimensionless numbers such as the Bond number Bo and the dielectric Bond number Bo E defined by:

Bo E = 1 2 (ε l -ε 0 ) E 2 2 γ D = (ε l -ε 0 ) E 2 D 4 γ (4)
The Bo number gives the relative importance of gravity and surface tension: the capillary force is dominant when Bo ≪ 1. In a similar way, the Bo E number compares the DEP force with surface tension. The ratio Bo E /Bo corresponds to the electrogravitational number and it is also useful as it measures the relative importance of DEP and gravitational forces.

These numbers are given on figures 6. As expected, the Bond number which was greater but close to 1 highlights that the surface tension had a significant but not a dominant influence on the meniscus at the initial state. When the voltage increase, all the forces were first quite comparable until the DEP force became dominant compared to the surface tension force when the electric field exceeded E ≈ 2.13 kV/mm. The interface profiles of both fluids were still affected by gravity up to E ≈ 2.77 kV/mm. But beyond this value, both the electric bond number and the electrogravitational number Bo E /Bo were greater than 10: the electric field strength made the DEP force fully dominant. The flat interface was thus produced by this force which erased the curving effect of surface tension even though the contact angle remains close to zero. However, following this reasoning, a flatter and flatter interface would be observable with the increase of the electric field and then, it is not possible to appreciate why the meniscus was still curved and tilted at intermediate voltage.

Free charges and the corresponding Coulomb force in the liquid could be an additional contribution influencing the interface profile. Actually, the weak electrical conduction in the liquid may generate a horizontal pressure gradient in the liquid which could consequently affect the interface profile. The bulk distribution of the dissociated ions can be identified based on the dimensionless number C 0 which is the ratio between the transit time t 0 = D/(2 bE) and the charge relaxation time τ = ε l /(n eq e 2 b):

C 0 = t 0 τ = n eq e D ε l E (5)
where n eq is the charge density of ions at dissociation/recombination equilibrium when no electric field is applied. For C 0 ≫ 1 i.e. at low voltage for a given fluid, the quasi-ohmic conduction regime is established. The charges form two thin heterocharge layers located close to each electrode. The effect on liquid pressure is then limited as the liquid bulk largely remains electro-neutral. As the voltage increases, the layers gradually spread in the liquid. When C 0 ≤ 1, the net charge (positive + negative charges) profile fills the interelectrodes spacing entirely and, as a consequence, the liquid pressure gradient is fully affected by the Coulomb term. The equilibrium densities were determined for both fluids using the voltage/current characteristic curves determined experimentally. Based on a one-dimensional model for the conduction in a dielectric liquid between parallel infinite electrodes [START_REF] Thomson | Conduction of electricity through gases[END_REF], it can be demonstrated that the intersection point between the ohmic regime and the saturation asymptote on figure 3 is actually defined such as C 0 = 1. As a consequence, we have:

n eq = ε l U s e D 2 (6)
where U s is the voltage at the intersection point. We found that n eq ≈ 0.9 10 17 m -3 for HFE-7000 and n eq ≈ 1.2 10 17 m -3 for HFE-7100. Therefore, C 0 was quite similar for both fluids in our experiments. For D = 3.02 mm, it linearly decreased from C 0 ≈ 1 at E = 0.69 kV/mm to C 0 ≈ 0.2 at E = 3.48 kV/mm. For D = 2.03 mm, C 0 was 2/3 times smaller. Then, the conduction mainly corresponded to regimes in which free charges may impact the horizontal pressure profile in the liquid bulk entirely. Following the same 1D approach used by Atten and Yagoobi [START_REF] Atten | Electrohydrodynamically induced dielectric liquid flow through pure conduction in point/plane geometry[END_REF] for the conduction regime, it could be demonstrated (see Appendix A) that, for the saturation regime, the horizontal pressure difference in the liquid between the electrodes and the

• for the positive net charge layer close to the ground electrode,

∆p G sat ≈ n 2 eq e 2 ε l D 2 2 b - b + (8)
This shows that (i) without a significant field-enhanced dissociation, the horizontal pressure difference increases when the voltage increases and finally reaches a maximum value with no electric field dependency at the saturation regime, (ii) the asymmetry of the pressure profile between the electrodes is introduced by the ratio of the charge mobilities and actually, these relations verify that the pressure profile is symmetrical when b -= b + as ∆p HV sat = ∆p G sat and L = D/2 . A dimensionless number Bo C,sat is introduced in order to measure the relative influence of this horizontal pressure gradient on the interface profile. The capillary pressure is still used as the reference scale according to the relation (4):

Bo C,sat = n 2 eq e 2 ε l D 2 2 2 γ D = n 2 eq e 2 D 3 4 γ ε l (9)
Based on the equilibrium densities previously estimated, we found that Bo C,sat ≈ 2 for the HFE-7100 at 60 • C. This shows that the influence of the horizontal pressure gradient was comparable with surface tension and gravity at the initial state and with DEP force at low voltage. Then, the increase of the electric field made the DEP force dominant as the ratio Bo E /Bo C,sat became greater than 10 when E ≥ 2.9 kV/mm. This is consistent with the fact that we observed a non-flat interface until this electric field strength. On top of that, a greater liquid heightof-rise in contact with the HV electrode suggested that for both fluids, the mobility of negative charges is somewhat lower than those of the positive ones. The influence of the space charge on the pressure gradient is increased by this effect according to equation [START_REF] Jeong | Experimental study of electrohydrodynamis pumping through conductino phenomenon[END_REF].

Figure 7 shows the influence of the temperature on the meniscus shape for HFE-7100. No significant influence of the temperature was perceptible at the greatest electric field as the DEP force was fully dominant on the temperature range. At lower electric field strengths, as the Bond number was weakly affected, the analysis must rely on the temperature influence, (i) on the conduction structure in the liquid and (ii) on the relative importance of the DEP versus the electrical conduction.

Regarding the first point, we previously observed (figure 3) that the saturation regime was established for temperatures greater than 40 • C. Below this temperature, the charges distribution at E = 1.06 kV/mm was close to those of the quasi-ohmic regime and then, the horizontal pressure drop in the liquid [START_REF] Atten | Electrohydrodynamically induced dielectric liquid flow through pure conduction in point/plane geometry[END_REF] was lower than ∆p sat . However, it could be significantly affected by field-enhanced dissociation at greater field and at low temperature. This is consistent with the observations where the interface profile was mostly affected by the electrical conduction for T ≥ 30 • C. At greater temperature, the relationship between ∆p sat and the temperature is then decisive.

We have seen that the dielectric constant was slightly affected by the temperature so that it can be assumed that ∆p sat mainly depends on the temperature variation of n eq . Actually, this could be estimated by the relationship between the charge mobility and the temperature considering that n eq = σ/(b + + b -). [START_REF] Nassar | Experimental investigation of the variation of HFE electric properties with temperature[END_REF] found that σ is a decreasing function of temperature within the operating range. Moreover, according to Walden's rule, the charge mobility depends on the inverse of the liquid viscosity which is divided by twice between 0 • C and 80 • C for the HFE-7100; so, when the temperature rises, the mobility of the ions could double and n eq could significantly decrease with temperature. As a consequence, when the saturation regime is established, the effect of the conduction should decrease with an increase of temperature. On top of that, the relative importance of the DEP versus the electrical conduction should increase. This was actually noticeable for both low field strengths. At E = 1.06 kV/mm, the effect of the DEP force was not really significant and the sole influence of the temperature on conduction could explain the slightly diminution of the asymmetry of the profile with increasing temperature. At E = 2.09 kV/mm, the DEP force was almost 4 times greater so that the increase of the relative importance of the DEP with the temperature probably led to a flatter interface with temperature.

A charged interface behavior at the wall

Figure 8 shows a capture of one of some liquid jets regularly arising at high electric field in the interfacial region where the liquid accumulated along the HV electrode. The process started from the occurrence and the instability of a liquid film (0 to 7 ms). A deformation of this film into a micro-cone led to a horizontal cylindrical jet of liquid (11 ms) spreading along the electric field line. This jet was maintained between both electrodes until 15.5 ms and then returned to HV electrode until a new jet appeared. Sometimes, the jets broke in a Rayleigh-like instability to form some droplets.

The electrojet process is actually well-known. A review and detailed descriptions of the free surface instabilities in a electric field is given by Zhakin [START_REF] Zhakin | Electrohydrodynamics of charged surfaces[END_REF]. It is shown that this behavior typically concerns the electrodynamics of a charged surface in which both bulk and surface conductivities need to be considered (non-equipotential surface). As this process could sig-nificantly affect the structure of the interface, the occurrence of these instabilities may play a major role in deteriorating the control of the cooling system performance.

Conclusion

The effect of a DC electric field on a liquid-vapor interface was investigated at different temperatures using a Pellat-like test cell filled with HFE-7000 and HFE-7100 dielectric fluids.

The experimental results confirmed first that the DEP force controls the mean position of the meniscus as it compared quite favorably with the dielectric height-of-rise model. Moreover, the visualizations of the meniscus revealed that the DEP force could become locally so dominant compared to surface tension force so that the interface could be almost totally flattened when the electric field is increased.

These visualizations also revealed that the electrical conduction in the liquid could play a role in affecting, at the second order, the pressure profile in the liquid bulk. The modifications of the liquid horizontal pressure by the dissociation of ions have been determined on the basis of a 1D model of conduction for the saturation regime as most of our experiments were given by C 0 ≤ 1. We demonstrated that the asymmetry of the profile could be due to a difference between the mobilities of positive and negative charges. An additional effect of the free charges was identified with the emergence of some interface instabilities leading to the formation of micro-jets close to the triple line. In our experiments, the electric field was mainly parallel to the interface and this phenomenon did not significantly perturb the interface position and its control. However, this may be an issue to consider for DC voltage and for a different electrodes configuration.

The effect of temperature of the interface behavior was quite weak and it was given by the temperature dependence of both the dielectric constant and ions mobilities when the electrical conduction was significant in the liquid. However, the temperature introduces a major drawback for cooling applications as the interface control was mainly limited by the temperature dependence of the vapor dielectric strength which is highly sensitive to vapor pressure. As a consequence, a drastic limitation of the maximum electric field that the fluid can withstand before breakdown existed at low temperature and, finally, if the opportunity of using the EHD forces to control a liquid-vapor interface was confirmed, the temperature dependence for the vapor dielectric strength must be carefully considered in determining the working fluids which are suitable for EHD enhanced heat transfer applications.

• close to the HV electrode,

Figure 1 :

 1 Figure 1: Test cell configuration and schematic end-of-view of the meniscus.

Figure 2 :

 2 Figure 2: Test cell in its enclosure and view points of the liquid height-of-rise (SV: side-view; EOV: end-on view).

  for HFE-7100.

Figure 3 :

 3 Figure 3: Current density versus the voltage and the electric field at different temperatures.

  for HFE-7100.

Figure 4 :

 4 Figure 4: Liquid height-of-rise versus the electric field for (a) HFE-7000 and (b) HFE-7100 at different temperatures. DEP labelled curves correspond to equation (2) predictions.

Figure 5 :

 5 Figure 5: Liquid height-of-rise versus the temperature for (a) HFE-7000 and (b) HFE-7100 and for different electric field strengths.

Figure 6 :

 6 Figure 6: Influence of the electric field on the shape of the interface for a temperature of 60 • C and for HFE-7100; high voltage was applied on the right-hand side electrode.

Figure 7 :

 7 Figure 7: Interface profile on the operating range of temperature for HFE-7100; the HV electrode was on the right-hand side; the red line delimits the range in which the interface profile was significantly affected by the conduction in the liquid.

Figure 8 :

 8 Figure 8: Formation and dynamics of a micro-jet close to the HV electrode wall at high electric field strength in HFE-7100.

6 )Figure A. 9 :

 69 Figure A.9: Coulomb body force and pressure profiles in liquid at saturation regime when the mobility of positive ions is greater than those of the negative ions.

table 1 .

 1

	HFE-7000 HFE-7100

• C are given in

Table 1 :

 1 HFE liquid properties at 25

• C from 3M data-sheets.

Preprint (final version, doi: 10.1016/j.expthermflusci.2019.03.017) submitted to Journal of Experimental Thermal and Fluid Science July 12, 2021

Acknowledgment

Financial support from the French Research Agency, for EDYPHICE project undergrant number ANR-14-CE05-0031, is gratefully acknowledged.

Nomenclature

Symbol, definition, SI units b

Ionic mobility, m 2 /(V s) c

Neutral species density, /m Appendix A. Effect of pure conduction on the pressure in a liquid dielectric between two infinite parallel plane electrodes when C 0 ≪ 1

Thomson and Thomson [START_REF] Thomson | Conduction of electricity through gases[END_REF] proposed a model for pure conduction in a dielectric medium in which (i) the processus of charge creation is only due to dissociation/recombination (ii) the charge diffusion is not significant compared to ions drift and (iii) no medium motion takes place. These developments have been reproduced for C 0 ≫ 1 in Atten and Yagoobi [START_REF] Atten | Electrohydrodynamically induced dielectric liquid flow through pure conduction in point/plane geometry[END_REF]. In the C 0 ≪ 1 case, it was shown that an approximate solution is given by neglecting the recombination in the bulk entirely. Assuming a constant dissociation rate and satisfying the boundary conditions at the the electrodes i.e. n + (0) = 0 (the HV electrode is located in x = 0) and n -(D) = 0, it was demonstrated that the Coulomb body force and thus the horizontal pressure gradient in the liquid is a linear function of space:

where n + is the charge density of positive ions, n -is the charge density of negative ions, E is the local field strength, K d the rate of dissociation, c eq is the constant concentration of neutral species, b + and b -are respectively the charge mobilities of positive and negative ions. The horizontal pressure in the liquid is then parabolic and the maximum pressure difference is established between the wall and the vertical plane such as the net charge q = (n -n -) e = 0 (see figure A.9). Using the previous equation, the position (from the HV electrode) x = L of this plane is given by:

Integrating the equation A.1 between x = L and the high voltage position x, the pressure difference between both positions is:

In a similar way, the pressure difference between the grounded electrode and x = L is:

At equilibrium (when no electric field is applied), we have n eq = n + eq = n - eq and K d,eq c eq -K r n 2 eq = 0 in which the recombination constant is given by Langevin relation K r = (b + + b -) e/ε l . Assuming that the rate of dissociation is not significantly affected by the electric field i.e. K d ≈ K d,eq , we finally obtained: