Issue 
Int. J. Simul. Multisci. Des. Optim.
Volume 5, 2014



Article Number  A23  
Number of page(s)  9  
DOI  https://doi.org/10.1051/smdo/2014003  
Published online  08 July 2014 
Article
Numerical simulation of vertical Bridgman solidification of CdZnTe
^{1}
LabEM (LR11ES34), University of Sousse, ESSTH Sousse, rue Lamine Abbassi, 4011
Hammam Sousse, Tunisia
^{2}
CNRSSPCTS/CEC, 12 rue Atlantis, 87068
Limoges, France
^{3}
Université de Lorraine, Lab. Energétique de Longwy – LERMAB/FJV, Longwy, France
^{*} email: hanene.jamai@etu.unilim.fr
Received:
8
February
2014
Accepted:
17
April
2014
The solidification of the singular CdZnTe in axisymetric Vertical Bridgman (VB) cavity is investigated in the current study. The numerical approach is developed by using COMSOL Multiphysics software, which is based on finite elements method (FEM) with Arbitrary LagrangianEuleurian (ALE) formulation allowing treating the moving interface. Numerical aspects of different parameters that affect the solidification interface and present an important factor during the crystal growth’s process are presented and analyzed. Especial attention will be paid to investigate the thermophysical properties such as thermal diffusivity, heat capacity. Thermal conductivity and the effect of thermal condition specially the effect of cold temperature T_{c} which affects significantly the concavity and the convexity of the interface for specific ranges in the solidification of CdZnTe phenomena, but it is of great importance the crystal growth experts are not it given importance. It has been shown that this properties, the effect of T_{c}, and the variation of interface velocity affect the solid/liquid interface shape and the crystal growth process.
Key words: CdZnTe solidification / Axisymetric Bridgman Vertical (BV) / ALE formulation
© H. Jamai et al., Published by EDP Sciences, 2014
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1 Introduction
The vertical Bridgman method for crystal growth from the melt was firstly developed in 1925. Nowadays, it is the most useful technique for the growth of crystals. This technique is explained in Figure 1. Its perfections to control temperature gradients and to provide provides pure crystals of large sizes by allowing testing the boundary conditions of mass and heat transfer. Hence, such technique is widely utilized in order to produce II–VI compound semiconductor crystals. Their industrial applications are mostly concentrated in detectors and solar cell production fields. Among these semiconductors, the Cadmium zinc telluride (CdZnTe) is considered as one of the most important and promising applied materials for roomtemperature γ and Xray spectroscopy. Despite the fact that great efforts have been made to produce high quality of CdZnTe dominated by the Bridgman method, its production still suffers from small device size and low quality.
Figure 1. System. (a) Vertical Bridgman, (b) design domain, and (c) isotherms profile. 
The solidliquid interface shape is specially determined by heat transfer in melt crystal growth process. The search focus in heat transfer during Bridgman growth are presented in references [1–5], these others studied the factors affecting the shape of the liquidsolid interface, such as the such as the thermal conductivities of the crystal, melt, and ampoule, simple furnace characteristics and latent heat generation. In particular, Adornato and Brown [6] examined the effect of furnace profiles and ampoule materials on interface shape and satisfactorily compared their results to experiments of Wang et al. [7]. Jasinski and Witt [8, 9] clarified notable insights on how interface shape may be controlled in Bridgman processes and considered the growth of CdTe as an example system.
In the aim of better understanding the solidification phenomenon and solving the physical problem during the crystal growth, researchers have chosen to test them by implementing a number of different numerical methods that can predict both the shape and position of the phase change front that are compatible with the governing conservation equations in each phase [10]. For instance, one of the most used methods is the front tracking approach which solves weak formulations of these equations by interface tracking by remeshing both states at every temporal step [11]. However, it is known that such method is not appropriate to study solidification process of alloys that undergo a frequent “mushy region” instead of sharp interface. Furthermore, another used method is the enthalpy method which is an approach that fixes the energy equation by using coordinates transformation. On another hand, the interface position is recovered a posteriori [11] and an extension of such a front fixing approach has been proposed in literature in order to introduce and predict the convection phenomenon within the two mushy region phases [12–14]. El Ganaoui et al. [15–17] improved this method by implementing a timedependent enthalpyporosity formulation for directional solidification, in which the energy equation is introduced by using the enthalpy variable computing implicitly the thermal Stefan condition at the interface with an implicit procedure.
The current work uses COMSOL Multiphysics software, to simulate solidification in Bridgman Vertical (BV) configuration during CdZnTe, which is an II–VI compound crystal growth. To the best knowledge of the authors, it seems that thermophysical properties such as thermal diffusivity, heat capacity, and thermal conductivity have not been well investigated. In fact, these thermophysical properties have important factors in the Stefan equation and therefore have an effect on the solidification, interface of CdZnTe. It is worth mentioning that both governing parameters the temperature of the solid domain is not well studied in literature and the growth rate phenomena still need better understanding and we propose to address such issue in the present work.
From this point of view, the main idea of the current investigation is to control the shape interface, in solidification process. In particular, we will focus on the various parameters that affect the concavity and convexity of the interface that are also an important factor in crystal growth, namely, the thermal conductivity, temperature of the solid domain, and growth rate.
2 Materials and methods
2.1 Physical properties of the materials
The physical properties of CdZnTe are listed in Table 1 [18, 19]. It should be noted that these properties depend on the given temperature, and they are in accordance with the thermal boundary conditions, for each domains.
Physical properties of CdZnTe.
2.2 Physics model
The Bridgman crystal system is schematically displayed in Figure 1a. The furnace uses a tapered heating element to establish a linear temperature profile Figure 1b. The ampoule is axially positioned at the center of the growth furnace. During the process, the cylindrical ampoule containing the material in the solid and liquid phase is fixed at the constant rate V_{t} = 0. Because of the symmetry of the system, the right half of Figure 1c is taken as the calculation domain only supposed.
There is only heat conduction between the melt and crystal.
We can assume that this problem can be considered as a two dimensional axissymmetry one, the temperature field, the Stefan equations are expressed in the cylindrical coordinates system.
The energy equations for the melt and the crystal given by:(1) (2)
The shape of the solidliquid interface is a compulsory controlling factor that allows producing a single crystal in vertical Bridgman enclosure. This interface shape is predicted by heat transfer and primarily by the direction of the heat flux near the interface.
This interface is governed by the Stefan’s equation expressed as:(3)
The aforementioned parameters in the above equations K_{s}, K_{l}, ρ, Cp_{l}, and Cp_{s} are the thermal conductivities, the density, specifics heats, in solid (denoted by s) and liquid (denoted by l), respectively. Here, ∆H expresses the latent heat and V_{int} denoted the velocity at the interface.
The boundaries conditions corresponding to the temperature field are as follows: the left and right boundaries are kept adiabatic while the upper, the lower, and the interface position are maintained at T_{h} = 1400 K, T_{c} = 1200, and T_{m} = 1364 K, respectively.
3 Numerical tools
For solving the mathematical model industrial software has been used, namely COMSOL which essentially integrated MATLAB which includes mathematics and computation; data acquisition; modeling, algorithm development; simulation, and prototyping; data analysis, exploration, and visualization. On another hand, the MATLAB language is particularly well suited to designing predictive mathematical models and developing specific algorithms [20, 21]. Where, these algorithms contain operations for linear algebra, partial differential, matrix manipulation, or differential equation solving, data analysis and statistics.
First results concerned the validation of the computational approach versus the mathematical model. For that a test validation has been defined and conducted. It refers to available experimental results with considering shaped solid/liquid interface.
3.1 Test code validation
The validation herein concerns the physical problem of CdZnTe solidification in Vertical Bridgman (BV). In fact, the choice of this configuration is based on the experimental data [22–24] for CdZnTe crystal growth: R = 0.0375 m, L = 0.20 m, the temperature of the Bridgman furnace the upper of them being at a higher temperature than the melting point while the lower one is below the melting point. Thus, the system is cooled from below, and the solidification process begins from the bottom of the ampoule.
We noticed the same shape interface between our numerical computations (Figure 2b) and experimental ones (Figure 2a) in which the authors [18] presented the half of longitudinal section of grown ingot of CdZnTe. In fact, they mentioned, that the interface shape at the end of growth is concave toward the crystal, with interface bending of 3 mm Figure 2a. The same results are shown numerically Figure 2b and a good agreement is observed with those obtained by Sen et al. [18]. In fact, the same results are revealed by the current simulation in which the shape interface is concaved and the progress rate is found to be around 3 mm in 9 h.
Figure 2. Comparison between experimental data obtained by Sen et al. [18] (a) and our numerical results (b) in terms of the interface growth of CdZnTe. 
4 Results and discussion
Our results concerned the solidification process of CdZnTe in the Vertical Bridgman (VB) configuration. In fact, we will present the different parameters which affect the solidification of CdZnTe, with the aid of numerical simulations.
As aforementioned, one of the main ideas of our investigation is to present a general study of different parameters that affect the quality of the single CdZnTe crystal. Especially, we will give a great importance to study the thermophysical properties of CdZnTe in each domain (solid or liquid) such as the thermal diffusivity α, the specific heat Cp (at constant pressure), and the thermal conductivity, that all depend on the temperature and which are measured for both solid and liquid CdZnTe. In addition, thermal diffusivity is an important factor in the current investigation. In fact, not only it offers a convenient and accurate method of determining the thermal conductivity (k), but also it is a crucial parameter in the conduction equation which governs heat transfer during crystal growth [18]. In addition, these parameters have a great impact on the Stefan equation that interferes in the solidification interface process.
In fact, as shown in Figures 3a and 3b illustrating the profile of thermal diffusivity in both solid and liquid domains, we note that this diffusivity increases for temperature in the solid domain while it decreases with temperature in the liquid state. Indeed, one can note that the diffusivity goes through a sharp minimum in the region of melting point; the value in the liquid is higher than that in the solid just before the onset of melting. The same results are showed by Sen et al. [18], who treated the influence of thermal diffusivity for CdTe and Cd0.96Zn0.04Te. However, their results are significantly different from those reported by Marina et al. [22] who claimed that thermal diffusivity increased sharply above the liquid temperature for Hg_{1 – x } Cd_{ x }Te.
Figure 3. Thermal diffusivity, Specific heat and Thermal conductivity of CdZnTe relatively to the solid domain (a) and the liquid one (b), respectively for fixed parameters T _{c} = 1200 K, T _{m} = 1364 K, and T _{h} = 1400 K. 
In the same context, we study the shape of the heat capacity shown in Figures 3c and 3d in both the solid and liquid domain. Besides, as shown in Figures 3a, the heat capacity is observed to be progressively augmented in the solid domain until reaching a maximum of temperature, precisely at the value T _{c} = 1260 K. Consequently, by approaching the melting temperature, the profile of thermal capacity undergoes a decreasing trend until reaching a constant value in the liquid domain Figure 3b.
Similarly, we propose now to study the thermal conductivity and its influence on the interface shape and position as determined, in this case. In fact, we observed that the profile of thermal conductivity Figures 3e and 3f is similar to that relative to thermal diffusivity in both solid and liquid domain as revealed in Figures 3a and 3b. This makes good sense because thermal conductivity is proportional to thermal diffusivity which expression is defined by α = k/ρCp. In fact, it was found that the profile of solid thermal conductivity K _{s} is more enhanced compared to K _{l} in liquid state.
Figure 4a shows the temperature according to the solidification of CdZnTe along the both liquidsolid state plotted as function of axial coordinate, where z = −0.01 m denotes the bottom tip of solid domain, and z = 0.20 m corresponds to both liquid and liquid states. This thermal profile is based on two linear segments designed to yield a gradient in the melt and a steeper cooling rate in the crystal [25]. And, the visualization of interface evolution denoted by the Figure 4b, proved that is increased with increasing time.
Figure 4. Vizualisation of temperature field, and the spatial and temporal variations of the interface evolution in both solid and liquid domains. 
Figure 5 shows the interface temperature parameter as function of time. In fact, as note that after the initial transient, the temperature presented a peak which decreases steadily until a mild increase near the end of the run. The decreasing peak temperature coincides with decreasing melt height and volume as growth occurs. It is worth mentioning that similar results have been revealed by Cerny et al. [26], which mentioned that the time development of interface temperatures reveals well the establishment of the pseudosteady regime of crystal growth after 15 h, and by Zang et al. [27] in the problem of solidification of CdZnTe crystal growth from the melt.
Figure 5. Temporal variation relative to the temperature at the interface for T _{h} = 1400 K, T _{c} = 1200 K, and T _{m} = 1364 K. 
Thereafter, we will present the effect of the thermal conductivities ratio defined by K in the shape interface. We found that it is of great importance, because it changes not only the sign of curvature and consequently the concavity and the convexity of the interface shape, but also the curvature magnitude. Therefore, it was argued that for T _{h} = 1400 K, T _{c} = 1200 K and T _{m} = 1364 K, the parameter K ensures the achievement of a convex interface when it is less than 1.1 and over which the interface becomes concave Figure 6. Therefore, it is to conclude that the conductivity ratio is of great importance in the CdZnTe solidification because it helps controlling the process of crystal growth.
Figure 6. Influence of thermal conductivity ratio K = K _{l}/K _{s} on the interface shape for T _{c} = 1200 K, T _{h} = 1400 K, and T _{m} = 1364 K. 
Among the problems encountered in numerical simulation is the interface computation. These problems are generally called Stefan problems (Eq. (3)) that connects the axial temperature gradient in the liquid and the solid domain by the velocity growth. In fact, we devote this section to study the evolution of velocity interface for different values taken in the solidification of CdZnTe. The main issue in this section is to give a better visualization of the velocity growth and surface evolution which will allow to better understand the phenomenon of growth, especially the evolution and of the interface shape Figure 7 of great interest in this portion is to vary the velocity interface while keeping the speed of the bulb at zero, in other words the bulb is held fixed. The velocity profile depicted in Figure 7 demonstrates that the velocity value determined by the Stefan equation is strengthened compared to that computed during the growth of CdZnTe especially in the solidification of the material. This value is seen to be nearly 10^{−5} m/s given by COMSOL after resolution and compilation of the Stefan equation while respecting the requirement for thermal boundary conditions s in both liquid and solid areas.
Figure 7. Temporal and spatial variations of isovalues relative to the interface velocity. 
It was also noted that the decrease of the interface velocity from 10^{−5} to 10^{−10} m/s has a great effect on the evolution and the interface shape which becomes progressively weakened in Figure 8. Therefore one can conclude that the velocity variation is an important factor in the industrialization of crystal growth.
Figure 8. Effect of the interface velocity on the interface shape for T _{c} = 1200 K, T _{h} = 1400 K, and T _{m} = 1364 K. 
In this part we attempt to focus on predicting the effect of the variation of cold temperature T _{c} on the isotherms in the liquid state which they are affected by the advancement of the shape interface and consequently the concavity and the convexity magnitude. In fact, as we mentioned before in the introduction, although this factor is also important in the understanding of crystal growth industrialization, it has not been well investigated until now. As revealed from Figure 9, we can mention that the variation of T _{c} affects the concavity and the convexity of the interface. In fact, for T _{c} ∈ [1200 K − 1264 K], the interface remains concave while it becomes convex for T _{c} ∈ [1265 − 1354 K] especially, when the cold temperature reaches the value T _{c} = 1265 k according to the fixed values T _{h} = 1400 K, T _{m} = 1364 K, and R = 0.037. However, by increasing T _{c} over the value 1354 K, the problem is seen to diverge because the temperature approaches the specific value of T _{m}.
Figure 9. Isotherm patterns with respect to different values of the cold temperature T _{c} for T _{h} = 1400 K and T _{m} = 1364 K. 
We can finally conclude that the variation of T _{c} is of great importance such as the thermal conductivities ratios and the velocity interface variation, and this parameter can help to solve some problems encountered during the growth of CdZnTe, especially the problem of the concavity and convexity of the interface shape which affects the quality of crystal growth.
Do not forget also that the purpose of any study is to solve the problems not only in the semiconductor factory, but also to obtain crystals of good qualities. Investigations must help to choose the value of the cold temperature which allow operators to obtain a concave or convex interface
In the same as mentioned in Figure 10 the variation of axial coordinate with radial coordinate for different values of cold temperature which reveals the same interpretation proved in Figure 9.
Figure 10. Effect of the cold temperature on the interface shape at fixed parameter values T _{c} = 1200 K, T _{h} = 1400 K, and T _{m} = 1364 K. 
Our numerical findings may contribute in controlling the parameters for the growth process, and it represents a strong support to resolve difficulties for experimentalists.
5 Conclusion
In the current investigation, solidification of the singular CdZnTe in axisymetric Vertical Bridgman (VB) cavity is studied. All numerical computations are obtained using COMSOL Multiphysics software, which is based on finite elements method (FEM) with ALE (Arbitrary LagrangianEuleurian) formulation allowing treating the interface displacement. The visualization of temperature fields in terms of isotherms and spatial and temporal profiles and according to the solidification of CdZnTe is presented and discussed with the help of COMSOL to provide a better understanding of phase change. Especial attention has been attributed to study the thermophysical properties of CdZnTe in each domain (solid or liquid) such as the thermal diffusivity α, the specific heat Cp (at constant pressure), and the thermal conductivity, which are measured for both solid and liquid CdZnTe. In this context, it is found that isosurfaces of thermal diffusivity in both solid and liquid domains show that diffusivity increases for temperature in the solid domain while it is lowered with temperature in the liquid state. Furthermore, the diffusivity goes through a sharp minimum in the region of melting point and the value in the liquid is higher than that in the solid just before the onset of melting. In addition, the heat capacity is observed to progressively augment in the solid domain until reaching a maximum of temperature, at exactly the value T _{c} = 1260 K. On another hand, it is found that the variation of T _{c} affects the concavity and the convexity of the interface. In fact, it is seen that for specific interval of T _{c} the interface remains concave while it becomes convex for another one. Concerning the effect of the interface velocity variation, it is found that the decrease of the interface velocity from 10^{−5} to 10^{−10} m/s has a great influence on the evolution and the interface shape which becomes progressively weakened. In this vein, analysis of the results shows that the velocity variation is seen to be an important factor in the industrialization of crystal growth.
Acknowledgments
The fist author thanks Profs. R. Bennacer (ENS Cachan) and A. Aydi (university of sfax) for fruitful discussion.
References
 Chang CE, Wilcox WR. 1974. Control of interface shape in the vertical BridgmanStockbarger technique. Journal of Crystal Growth, 21, 135–140. [CrossRef] [Google Scholar]
 Sen S, Wilcox WR. 1975. Influence of crucible on interface shape, position and sensitivity in the vertical BridgmanStockbarger technique. Journal of Crystal Growth, 28, 26–40. [Google Scholar]
 Fu TW, Wilcox WR. 1980. Influence of insulation on stability of interface shape and position in the vertical BridgmanStockbarger technique. Journal of Crystal Growth, 48, 416–424. [CrossRef] [Google Scholar]
 Naumann RJ. 1982. An analytical approach to the thermal modeling of Bridgmantype crystal growth. II. Twodimensional analysis. Journal of Crystal Growth, 58, 569–584. [CrossRef] [Google Scholar]
 Naumann RJ, Lehoczky SL. 1983. Effect of variable thermal conductivity on isotherms in Bridgman growth. Journal of Crystal Growth, 61, 707–710. [CrossRef] [Google Scholar]
 Adornato P, Brown RA. 1987. Convection and segregation in directional solidification of dilute and nondilute binary alloys: effects of ampoule and furnace design. Journal of Crystal Growth, 80, 155–190. [CrossRef] [Google Scholar]
 Wang CA, Witt AF, Carruthers JR. 1984. Analysis of crystal growth characteristics in a conventional vertical Bridgman configuration. Journal of Crystal Growth, 66, 299–308. [CrossRef] [Google Scholar]
 Jasinski T, Witt AF. 1985. On control of the crystal melt interface shape during growth in a vertical Bridgman configuration. Journal of Crystal Growth, 71, 295–304. [CrossRef] [Google Scholar]
 Jasinski T, Witt AF, U.S. Patent: Apparatus for Growing Crystals, 1986. [Google Scholar]
 El Ganaoui M, Lamazouade A, Bontoux P. 2002. Computational solution for fluid flow under solid/liquid phase change conditions. Computers & Fluids, 31, 539–556. [CrossRef] [Google Scholar]
 Crank IJ. 1984. Free and moving boundary problems. Clarendon Press: Oxford, UK. [Google Scholar]
 Voller R, Prakash C. 1987. A fixed grid numerical modelling methodology for convection diffusion mushy region phase change problems. International Journal of Heat and Mass Transfer, 30(8), 1709–1719. [Google Scholar]
 Bennon WD, Incropera FP. 1987. A continuum model for momentum, heat and species transport in binary solidliquid phase change systems: 1. Model formulation. International Journal of Heat and Mass Transfer, 30(10), 2161–2170. [CrossRef] [Google Scholar]
 Voller R, Cross M. 1980. Accurate solutions of moving boundary problems using the enthalpy method. International Journal of Heat and Mass Transfer, 24, 545–556. [CrossRef] [Google Scholar]
 El Ganaoui M. Modélisation numérique de la convection thermique instationnaire en présence d’un front de solidification déformable. Thèse de l’Université d’AixMarseille, Octobre, 1997. [Google Scholar]
 El Ganaoui M, Mazhorova OS, Bontoux P. 2000. Computer simulation of pure and alloys melt growth. Microgravity Quart Rev, 7(4), 171–178. [Google Scholar]
 Elganaoui M, Bontoux P, Morvan D. 1999, Localisation d’un front de solidification en interaction avec un bain fondu instationnaire, Paris: CR Acad Sciences, Série II b, t. 327, p. 41–48. [Google Scholar]
 Sen S, Konkel WH, Tighe SJ, Bland LG, Sharma SR, Taylor RE. 1988. Crystal growth of largearea singlecrystal CdTe and CdZnTe by the computercontrolled vertical modifiedBridgman process. Journal of Crystal Growth, 86, 111. [CrossRef] [Google Scholar]
 Bruder M, Figgemeier H, Schmitt R, Maier H. 1993. Mat. Res. Eng. B, 16, 40. [CrossRef] [Google Scholar]
 Kuppurao S, Brandon S, Derby JJ. 1995. Journal of Crystal Growth, 155, 93. [CrossRef] [Google Scholar]
 MATLAB. The Language of Technical Computing – Programming Version 7, MA, USA: The Mathworks, Inc, 2005 [Google Scholar]
 Marchenko MP, Golyshev VD, Bykova SV. 2007. Investigation of Cd_{1x}Zn_{x}Te composition inhomogeneity at crystal growth by AHPmethod. Journal of Crystal Growth, 303, 193–198. [CrossRef] [Google Scholar]
 Huang ZH, Conway PP, Thomson RC, Dinsdale AT, Robinson JAJ. 2008. A computational interface for thermodynamic calculations software MTDATA. Computer Coupling of Phase Diagrams and Thermochemistry, 32, 129–134. [CrossRef] [Google Scholar]
 Bruder M, Figgemeier H, Schmitt R, Maier H. 1993. Mat. Res. Eng. B, 16, 40. [CrossRef] [Google Scholar]
 Fang HS, Wang S, Zhou L, Zhou NG, Lin MH. 2012. Influence of furnace design on thermal stress during directional solidification of multicristalline silicon. Journal of Crystal Growth, 346, 5–11. [CrossRef] [Google Scholar]
 Cerny C, Kalbac A, Prikryl P. 2000. Computational modeling of CdZnTe crystal growth from the melt. Computational Materials Science, 1734–1760. [Google Scholar]
 Zhang N, Yeckel A, Derby JJ. 2012. Maintaining convex interface shapes during electrodynamic gradient freeze growth of cadmium zinc telluride using a dynamic, bellcurve furnace profile. Journal of Crystal Growth, 355, 113–121. [CrossRef] [Google Scholar]
Cite this article as: Jamai H, Pateyron B, Sammouda H & El Ganaoui M: Numerical simulation of vertical Bridgman solidification of CdZnTe. Int. J. Simul. Multisci. Des. Optim., 2014, 5, A23.
All Tables
All Figures
Figure 1. System. (a) Vertical Bridgman, (b) design domain, and (c) isotherms profile. 

In the text 
Figure 2. Comparison between experimental data obtained by Sen et al. [18] (a) and our numerical results (b) in terms of the interface growth of CdZnTe. 

In the text 
Figure 3. Thermal diffusivity, Specific heat and Thermal conductivity of CdZnTe relatively to the solid domain (a) and the liquid one (b), respectively for fixed parameters T _{c} = 1200 K, T _{m} = 1364 K, and T _{h} = 1400 K. 

In the text 
Figure 4. Vizualisation of temperature field, and the spatial and temporal variations of the interface evolution in both solid and liquid domains. 

In the text 
Figure 5. Temporal variation relative to the temperature at the interface for T _{h} = 1400 K, T _{c} = 1200 K, and T _{m} = 1364 K. 

In the text 
Figure 6. Influence of thermal conductivity ratio K = K _{l}/K _{s} on the interface shape for T _{c} = 1200 K, T _{h} = 1400 K, and T _{m} = 1364 K. 

In the text 
Figure 7. Temporal and spatial variations of isovalues relative to the interface velocity. 

In the text 
Figure 8. Effect of the interface velocity on the interface shape for T _{c} = 1200 K, T _{h} = 1400 K, and T _{m} = 1364 K. 

In the text 
Figure 9. Isotherm patterns with respect to different values of the cold temperature T _{c} for T _{h} = 1400 K and T _{m} = 1364 K. 

In the text 
Figure 10. Effect of the cold temperature on the interface shape at fixed parameter values T _{c} = 1200 K, T _{h} = 1400 K, and T _{m} = 1364 K. 

In the text 
Current usage metrics show cumulative count of Article Views (fulltext article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.
Data correspond to usage on the plateform after 2015. The current usage metrics is available 4896 hours after online publication and is updated daily on week days.
Initial download of the metrics may take a while.