Solidification Morphology and Bifurcation Predictions with the Maximum Entropy Production Rate Model

The use of the principle of maximum entropy generation per unit volume is a new approach in materials science that has implications for understanding the morphological evolution during solid–liquid interface growth, including bifurcations with or without diffuseness. A review based on a pre-publication arXiv preprint is first presented. A detailed comparison with experimental observations indicates that the Maximum Entropy Production Rate-density model (MEPR) can correctly predict bifurcations for dilute alloys during solidification. The model predicts a critical diffuseness of the interface at which a plane-front or any other form of diffuse interface will become unstable. A further confidence test for the model is offered in this article by comparing the predicted liquid diffusion coefficients to those obtained experimentally. A comparison of the experimentally determined solute diffusion constant in dilute binary Pb–Sn alloys with those predicted by the various solidification instability models (1953–2011) is additionally discussed. A good predictability is noted for the MEPR model when the interface diffuseness is small. In comparison, the more traditional interface break-down models have low predictiveness.

The maximum entropy production rate-density, MEPR, or the Maximum Entropy Production Principle, MEPP, are the acronyms used in the literature [2,6,9,10] for analyses that employ the entropy rate maximization principle. We have chosen to use MEPR [2] in this article to emphasize the importance of the "rate" in the acronym. A MEPR hypothesis is tested in this article for initiating interface diffuseness

The MEPR Model
During the one dimensional solidification of a pure metal or a binary molten alloy, which is at freezing temperature under a fixed temperature gradient and with constant interface velocity there is a loss of work potential from the dissipation of kinetic energy, giving rise to entropy generation rate density . ϕ max (J m −3 K −1 s −1 ) is given by [1] in the region of the diffuse interface with dimensions ζ (m). The subscript max. indicates that that the maximum value of this new entropy generation rate is given by: . ϕ max = ∆ρ k V 3 2 ζ 2 G SLI (1) where ∆ρ k (kgm −3 ) is the overall density shrinkage given by (ρ l ∆ρ/ρ s ), ∆ρ (kgm −3 ) is the density change from liquid to solid (ρ s − ρ l ); ρ s (kgm −3 ) and ρ l (kgm −3 ) are the densities of the fully solid and fully liquid zones, respectively. The symbol G SLI (Km −1 ) is the temperature gradient across the solid-liquid interface, including the diffuse interface. This gradient is difficult to measure experimentally, so it is commonly approximated as the average between the rigorous-solid and rigorous-liquid regions. In this article, G SLI is assumed to be approximately equal to the temperature gradient in the fully liquid zone, G L (Km −1 ), (i.e., G SLI ≈ G L ). At the solid-liquid interface region, during directional solidification of a binary material, the existence of diffuseness or a non-planar morphology (such as cellular) can produce new entropy. Following Sekhar [2], the entropy rate maximization in this region when compared between various morphological pathways can be thought to be somewhat analogous to the free energy selections between various phases. A cellular structure produces entropy of a positive value that increases with velocity, as does the planar diffuse structure [1,2], but at different rates. The postulate of MEPR applied to solidification morphology is that the highest entropy-rate-producing configuration is the most stable. During directional solidification (one dimensional growth in casting, as is done for turbine blades or jewelry manufacture), the first transition from a stationary planar interface is the evolution in the interface region from an atomically sharp to a diffuse interface between the rigorous solid and the rigorous liquid [2]. For an alloy, further topographical variations become possible as the entropy generation rate per unit volume reaches a peak, beyond which a cellular or other non-planar structure (e.g., cells or dendrites) can overtake the planar entropy production rate at any given composition of the alloy [1,2]. Detailed calculations for developing Equation (1) for the diffuseness dimension and the instability criterion are shown in [1]-only a brief review is provided below. The maximum entropy generation rate per unit volume (or the entropy rate density) [1,2] is related to . s LG (J m −3 K −1 s −1 ), which is the entropy transfer rate from the solute gradient in the liquid and . s E (J m −3 K −1 s −1 ) (i.e., the main component of the entropy generation rate that describes the entropy generated due to exchange of matter and heat in the SLI), expressed as: . s E and . s LG are given by V∆h sl G SLI /T 2 M and ∆T O V 2 R g ln(1/k)/D L 4 m L , respectively [1]. The term ∆h sl (Jm −3 ) is the heat of fusion, which is an approximation for ∆h m [2]; ∆h m (Jmol −1 ) is the heat of fusion with defects; m L (Km 3 mole −1 ) is the slope of the liquidus line at the solid-liquid boundary for a binary material; k (dimensionless) is the partition coefficient that can be obtained from the binary phase diagram; D L (m 2 s −1 ) coefficient of solute diffusion in the alloy. Here, ∆T O (K) is the solidification temperature range, which is expressed as (T l − T s ) or (C O (1 − k) m L /k), where T l and T s are the liquidus and solidus temperatures, respectively, and can be obtained from the phase diagram. The conditions given by Equations (3a) and (3b) below for a maximum or minimum defines a possible onset of a bifurcation condition (morphological instability). Note that arguments to indicate the maximization condition are provided in [2,6,9,10]. Note that is negative when inferring a maximization condition. Although T si and T li are unknown based on binary alloy materials, the thickness ζ of the diffuse interface (m) can be approximated for dilute solutions by assuming that T si ≈ T m and T li ≈ T m , and following the procedures developed in [1], the standard solute balance at steady state growth along with Equation (3a) above can used to yield: Here, C O (wt % or mole m −3 ) is the solute concentration in the alloy. Similarly, Equation (2) yields, Now by using (3b) and (5), ζ can be written as: Here, k eff is the effective partition coefficient for a diffuse interface. The equation is valid for extremely dilute alloys. Changing the formulation of Equation (6) by placing back into Equation (5) now also gives the driving force diffuseness for a binary alloy material as: , where d is the interplanar lattice spacing normal to the growth direction. However, note that the exact bifurcation may occur at any velocity and temperature gradient greater than that set by Equation (5) (i.e., Equation (5) only sets one boundary condition). By analyzing the entropy generation density for a wavy interface [1], one can also infer that the interface will break down between This is the MEPR condition for describing the breakdown limits. Because the condition is based on the comparison of the entropy rate maximization, it may also be recast in terms of the cooling rate (VG SLI ) C .
Note that Figure 1 establishes a relationship between the diffuseness and the break down variable. Figure 1 shows the plot of the total diffuseness as a function of (V/G sli ) c . The figure plots the set of measurable breakdown parameters. For any alloy this would be a straight line as per the MEPR model. However, we note that that the band is the similar across various alloys thus highlighting the previously unanticipated relationship between interface diffuseness and the solidification parameters. This implies that the results shown below in Figures 2 and 3, namely the maximums in the entropy generation rate, are anticipated by the experiments. Additionally, the slopes are different for faceted materials when compared to the non-facet situation, possibly indicating features of diffuseness not fully captured by the MEPR model. The plot shows measured experimental conditions at breakdown in the abscissa and calculated interface diffuseness on the ordinate. If the total interface diffuseness is greater than one or two atomic layers, then there is a possibility of non-facet morphology at breakdown, otherwise it should be facet morphology [1]. The relationship between total diffuseness and the ratio of the velocity/temperature gradient (V)C/(GSLI)C should yield a straight line irrespective of material parameters for any growth direction (or crystal plane spacing normal to a growth direction) in the MEPR model. The values V and GSLI are experimentally measured numbers at breakdown, and ηT is calculated from the model [1]. Note that succinonitrile (SCN) alloys are non-faceted by an additional thermal and possibly rotational diffuseness at the melting temperature, which makes the SCN material transformation always appear with a non-faceted morphology particularly when observed at optical level magnifications. Experimentally, the materials shown below the dashed line (log10 ηT = 2) are recorded to be macroscopically faceted [1]. For the zone for facet materials, a different slope than in the non-facet region may represent different mechanisms for growth (e.g., nucleationdominated or dislocation-dominated) [22,32,43].
Equation (6) can be related to the processing parameters for constrained or unconstrained solidification namely, (V/G SLI ) or the cooling rate ( VG SLI ) respectively to yield the following entropy rate based criteria, Where c refers to critical and N is defined below Equation 6 (please also see nomenclature). Figures 2 and 3 show the plot of the entropy rate density as a function of the alloy parameters (diffuseness) and the processing and for the MEPR model. The first and second derivatives w.r.t. to V at constant ζ and GSLI indicate that the entropy generation rate will increase linearly with velocity unless solute partitioning into the liquid is allowed. When solute partitioning is possible, the entropy rate generation term indicates a maximum when plotted as a function of velocity ( Figure 2). If no other interface configuration is feasible (those that display a higher entropy rate generation, such as a seaweed, jagged or fine tip interface), the interface will remain planar during growth. Note that φ̇m ax cannot be less than zero (second law of thermodynamics). This implies that regardless of the sign of GSLI, the critical φ̇m ax can only have a lower value of zero for a planar interface. Thus, a nonplanar shape can always overtake a plane front morphology for a negative temperature gradient, or in other words a negative temperature gradient will always imply a breakdown into cells or other The plot shows measured experimental conditions at breakdown in the abscissa and calculated interface diffuseness on the ordinate. If the total interface diffuseness is greater than one or two atomic layers, then there is a possibility of non-facet morphology at breakdown, otherwise it should be facet morphology [1]. The relationship between total diffuseness and the ratio of the velocity/temperature gradient (V) C /(G SLI ) C should yield a straight line irrespective of material parameters for any growth direction (or crystal plane spacing normal to a growth direction) in the MEPR model. The values V and G SLI are experimentally measured numbers at breakdown, and η T is calculated from the model [1]. Note that succinonitrile (SCN) alloys are non-faceted by an additional thermal and possibly rotational diffuseness at the melting temperature, which makes the SCN material transformation always appear with a non-faceted morphology particularly when observed at optical level magnifications. Experimentally, the materials shown below the dashed line (log 10 η T = 2) are recorded to be macroscopically faceted [1]. For the zone for facet materials, a different slope than in the non-facet region may represent different mechanisms for growth (e.g., nucleation-dominated or dislocation-dominated) [22,32,43].
Entropy 2020, 22, x 6 of 13 patterns (unless a high-velocity-plane front transition occurs [2]). Additionally, because cellular shapes with a diffuse interface are seemingly restricted by the bounds of entropy from the diffuseness of alternate shapes, additional configurational entropy production rate increases for complex features (e.g., dendrites) are feasible and so will always emerge as an alternative structure unless a very wide diffuse interface topographies are possible with no partitioning. This is a possible explanation for why well-defined cellular features are not commonly noted in microstructures, such as atomized powders that solidify with a negative temperature gradient. Figures 2 and 3 show the entropy generation rate density as a function of various solidification features and collapsed parameters that are known to influence instability of a particular topography or morphology. Note the definitions of B, M, and N from Equations (4)-(6) and the nomenclature. When B becomes greater than or equal to M, then N is either zero or negative; consequently, the interface diffuseness becomes undefined. The maximum entropy generation rate density increases with the corresponding increase in diffuse interface thickness and falls only when the parameter B approaches 0.5 M. The growth of the interface can be steady when N is greater than one. When the temperature gradient is zero, the diffuse interface thickness becomes undefined, thus allowing keff to take on a high value closer to one. When B is equal to M, then N is zero, and ζ and φ̇m ax are both undefined. From the transition instability criterion defined above, the peak for φ̇m ax against velocity occurs when M/B (dimensionless) is equal to 2 (i.e., M/N 0.5 is equal to ( 2 ∆h sl ∆ρ k T m 2 ). Figure 2 in [1] shows the plot of the entropy generation rate as a function of the diffuseness. When M > B, then the number of pseudo-atomic layers present within the diffuse interface region is easily related to the driving force diffuseness in an almost linear manner [1]. Note that the deviation from linearity sets in at a lower V/GSLI as the concentration increases. At the condition where M ≥ N > 1, noted in Figure 2, a steady slope is observed, where the V/GSLI ratio shows a strong effect on the number of pseudo-atomic-spacings [1]. As the condition for 1 > N > 0 is encountered, even a small change in the V/GSLI ratio can lead to a rapid change in the number of pseudo-atomic spacings at the interface. Figure 3 shows the parabolic-like profile of the entropy generation rate density as a function of V/Gsli. Both Figures 2 and 3 indicate that a peak is noted in the entropy generation rate density for a planar interface, essentially giving other entropy producing morphologies a possibility to dominate over the plane front structure (whether diffuse or not). An example is shown in Figure 3 of how the entropy rate for a cellular pattern or a dendritic morphology may indicate transitions to those shapes. Note that an implication of the results in Figure 1 is that ΔTSLI approaches ΔTO, but a morphological transition prevents the full attainment for this separation for the plane front (i.e., if the high velocity plane front condition is not encountered) [2,13,[38][39][40].   ϕ max for the diffuse plane front reaches its highest value at the peak of the curve (i.e., when M becomes equal to 2B).  The φ̇m ax increases with decreasing solute concentration and reaches a maximum value. At extremely low solute concentration, the binary material behaves similarly to a pure material (linear dark line) and φ̇m ax increases indefinitely with V/GSLI, like a pure metal [1]. Note the two thin schematic lines, one for cells (that begins at the origin) and the other for some form of dendrites, are also shown to indicate how a bifurcation transition may be reached, and further how dendrites can overtake cellular morphologies (see reference [2] for more details on types of dendrites). Note that in Equation (8) a similar graphical relationship for the entropy rate density generation density is seen when the abscissa is the cooling rate (V*GSLI) [1]. For unconstrained dendrites [18] the cooling rate is a preferred grouping of processing variables to indicate particularly the fineness of the secondary dendrites with increased cooling rate [32,33,43,44].

The CUT Model
The first interface breakdown model was proposed qualitatively by Rutter and Chalmers [22], and then quantitatively described by Tiller, Rutter, Jackson, and Chalmers [19]. This model describes the interface instability (from planar to non-planar) as being enabled by a region of constitutionally undercooled liquid that forms ahead of the solid-liquid interface during growth from solute partitioning. For a binary alloy, the CUT criterion for instability is written as: where GL (K m −1 ) is the temperature gradient in the liquid, DL (m 2 s −1 ) is the solute diffusion coefficient in the liquid, and ΔTO (K) is the equilibrium solidification range (Tl−TS) for a liquid at composition CO (mole m −3 ). Also, Tl (K) and TS (K) are the equilibrium liquidus and solidus temperatures captured in equilibrium phase diagrams.

The LST Model
In 1964, Mullins and Sekerka [31] proposed the linear stability model (LST) that considered the stability of a planar interface to a perturbation of an infinitesimal amplitude. In this stability model, the interface is unstable if any wavelength of a sinusoidal perturbation grows, and conversely the interface is stable if none of the perturbations can grow (regardless of their wavelength and surface energy) can grow. This LST criterion gives the instability criterion for a binary material as: ϕ max increases with decreasing solute concentration and reaches a maximum value. At extremely low solute concentration, the binary material behaves similarly to a pure material (linear dark line) and . ϕ max increases indefinitely with V/G SLI , like a pure metal [1]. Note the two thin schematic lines, one for cells (that begins at the origin) and the other for some form of dendrites, are also shown to indicate how a bifurcation transition may be reached, and further how dendrites can overtake cellular morphologies (see reference [2] for more details on types of dendrites). Note that in Equation (8) a similar graphical relationship for the entropy rate density generation density is seen when the abscissa is the cooling rate (V * G SLI ) [1]. For unconstrained dendrites [18] the cooling rate is a preferred grouping of processing variables to indicate particularly the fineness of the secondary dendrites with increased cooling rate [32,33,43,44]. Equation (6) can be related to the processing parameters for constrained or unconstrained solidification namely, (V/G SLI ) or the cooling rate (VG SLI ) respectively to yield the following entropy rate based criteria, where c refers to critical and N is defined below Equation (6) (please also see nomenclature). Figures 2 and 3 show the plot of the entropy rate density as a function of the alloy parameters (diffuseness) and the processing and for the MEPR model. The first and second derivatives w.r.t. to V at constant ζ and G SLI indicate that the entropy generation rate will increase linearly with velocity unless solute partitioning into the liquid is allowed. When solute partitioning is possible, the entropy rate generation term indicates a maximum when plotted as a function of velocity ( Figure 2). If no other interface configuration is feasible (those that display a higher entropy rate generation, such as a seaweed, jagged or fine tip interface), the interface will remain planar during growth. Note that . ϕ max cannot be less than zero (second law of thermodynamics). This implies that regardless of the sign of G SLI , the critical . ϕ max can only have a lower value of zero for a planar interface. Thus, a non-planar shape can always overtake a plane front morphology for a negative temperature gradient, or in other words a negative temperature gradient will always imply a breakdown into cells or other patterns (unless a high-velocity-plane front transition occurs [2]). Additionally, because cellular shapes with a diffuse interface are seemingly restricted by the bounds of entropy from the diffuseness of alternate shapes, additional configurational entropy production rate increases for complex features (e.g., dendrites) are feasible and so will always emerge as an alternative structure unless a very wide diffuse interface topographies are possible with no partitioning. This is a possible explanation for why well-defined cellular features are not commonly noted in microstructures, such as atomized powders that solidify with a negative temperature gradient. Figures 2 and 3 show the entropy generation rate density as a function of various solidification features and collapsed parameters that are known to influence instability of a particular topography or morphology. Note the definitions of B, M, and N from Equations (4)-(6) and the nomenclature. When B becomes greater than or equal to M, then N is either zero or negative; consequently, the interface diffuseness becomes undefined. The maximum entropy generation rate density increases with the corresponding increase in diffuse interface thickness and falls only when the parameter B approaches 0.5 M. The growth of the interface can be steady when N is greater than one. When the temperature gradient is zero, the diffuse interface thickness becomes undefined, thus allowing k eff to take on a high value closer to one. When B is equal to M, then N is zero, and ζ and . ϕ max are both undefined. From the transition instability criterion defined above, the peak for  Figure 2 in [1] shows the plot of the entropy generation rate as a function of the diffuseness. When M > B, then the number of pseudo-atomic layers present within the diffuse interface region is easily related to the driving force diffuseness in an almost linear manner [1]. Note that the deviation from linearity sets in at a lower V/G SLI as the concentration increases. At the condition where M ≥ N > 1, noted in Figure 2, a steady slope is observed, where the V/G SLI ratio shows a strong effect on the number of pseudo-atomic-spacings [1]. As the condition for 1 > N > 0 is encountered, even a small change in the V/G SLI ratio can lead to a rapid change in the number of pseudo-atomic spacings at the interface. Figure 3 shows the parabolic-like profile of the entropy generation rate density as a function of V/G sli . Both Figures 2 and 3 indicate that a peak is noted in the entropy generation rate density for a planar interface, essentially giving other entropy producing morphologies a possibility to dominate over the plane front structure (whether diffuse or not). An example is shown in Figure 3 of how the entropy rate for a cellular pattern or a dendritic morphology may indicate transitions to those shapes. Note that an implication of the results in Figure 1 is that ∆T SLI approaches ∆T O , but a morphological transition prevents the full attainment for this separation for the plane front (i.e., if the high velocity plane front condition is not encountered) [2,13,[38][39][40].

The CUT Model
The first interface breakdown model was proposed qualitatively by Rutter and Chalmers [22], and then quantitatively described by Tiller, Rutter, Jackson, and Chalmers [19]. This model describes the interface instability (from planar to non-planar) as being enabled by a region of constitutionally undercooled liquid that forms ahead of the solid-liquid interface during growth from solute partitioning. For a binary alloy, the CUT criterion for instability is written as: where G L (Km −1 ) is the temperature gradient in the liquid, D L (m 2 s −1 ) is the solute diffusion coefficient in the liquid, and ∆T O (K) is the equilibrium solidification range (T l − T S ) for a liquid at composition C O (molem −3 ). Also, T l (K) and T S (K) are the equilibrium liquidus and solidus temperatures captured in equilibrium phase diagrams.

The LST Model
In 1964, Mullins and Sekerka [31] proposed the linear stability model (LST) that considered the stability of a planar interface to a perturbation of an infinitesimal amplitude. In this stability model, the interface is unstable if any wavelength of a sinusoidal perturbation grows, and conversely the interface is stable if none of the perturbations can grow (regardless of their wavelength and surface energy) can grow. This LST criterion gives the instability criterion for a binary material as: where S (no units) is the Mullins-Sekerka stability constant [31], which is equal to one for low velocities; K L and K S (J m −1 K −1 s −1 ) are the thermal conductivities for the fully solid and the fully liquid states, respectively. Note that the CUT model in Equation (10) and LST model in Equation (11) converge for the limit of K s ∼ = K L . A study by Burgeon et al. [38] with in situ interface imaging in microgravity conditions prevalent during the ordering of a cellular array structure concluded that the cause of interface dynamics and breakdown are more than just an account of the undercooled liquid ahead of the interface. An experimental study by Inatomi et al. [39] also cast doubt on whether an undercooled liquid or solute pile-up ahead of the interface is always present. They have argued persuasively that none of the theories for breakdown may be correct. For an interface's topographical instability in the case of facet prone materials, a strain accumulation model [34] has also been considered as describing the interface breakdown. However, Inatomi et al. [39] argue against a general strain model as the cause for the instability. In reference 1, the variables Z CUT and Z LST were developed as parameters that describe the deviations from the experimental values. For the conditions where the interface instability occurs at high velocities, especially for very low alloy composition materials or with very low temperature gradients [1], both the CUT and LST models lose even more predictive capability [1,18]. Additionally, it should be noted the CUT and LST models do not address the facet/non-facet transitions or diffuseness at a molecular level, which is easily treated by the MEPR model [1,2].

Comparison with Experiments for the Diffusion Coefficient Prediction
The experimentally reported values of D L from non-solidification experiments for Pb-Sn alloys at different concentrations are reported in [18] and are summarized in Table 1. The experimental D L values shown in Table 1 directly measured from non-solidification experiments are corrected by an Arrhenius-type correction for the liquidus temperature if the report is at a higher temperature than the liquidus [30][31][32][33][34][35][36][37][38][39][40]. However, note that these corrected numbers only impact the results in a minor way for the dilute alloy compositions considered. The results shown in Table 1 for the D L predictions for both CUT and LST show consistent and significant deviation from experimental measurements, as pointed out by De Cheveigne et al. [26] and Bensah et al. [18]. From Table 1, we note that the MEPR model shows stronger predictive capability of D L compared to CUT and LST for Pb-Sn alloys when compared with experimental values. However, even with the MEPR model, large deviations are noted for experimental conditions with a small temperature gradient. Pb-Sn alloys are known to have very wide diffuse interfaces [1,21,[35][36][37]. A lower G SLI dramatically influences the diffuse interface as noted above, and consequently the partition coefficient. The expectation that k eff approaches one with increased interface diffuseness is a reasonable assumption for dilute Pb-Sn and icosahedral alloys with a diffuse interface [32,33,[40][41][42][43][44][45][46][47][48]. Should k eff , therefore, change from 0.636 to 0.95 because of the low temperature gradient and diffuse interface, the D L value that is calculated is shown to become much lower to match the experimental data for even these low solidification temperature-gradient experiments. The corrected values are also shown in italics in Table 1 in the highlighted part of the table. Regardless, it should be noted that the assumed change in k eff to 0.95 is arbitrary and is only set to this number to illustrate the influence of the partition function on the calculated number. Table 1. A summary of results for D L in instability conditions for experimental breakdown compared with the value obtained from three instability models. Note that for MEPR, G SLI was assumed to be equal to G L for the calculations . The diffusion data and physical constant from [18] for Pb-Sn alloys is Ks = 33.6 (J/mKs), K L = 15.4 (J/mKs); the equilibrium partition coefficient is k = k eff = 0.636, Tm = 600.65 K, ∆h sl = 2.

Summary Discussions
Topographical and diffuse interface reconfigurations occur with a change in the solidification rate. In this article, we pursue the hypothesis that the interface configuration during solidification is determined by the maximum rate of entropy production in the region between a rigorous solid and rigorous liquid phase. We posit that when an interface begins to migrate, there are several stable configurations that are possible. These include atomically planar, diffuse-planar, facet non-planar, and cellular nonplanar configurations. The configuration and topographical condition that affords the maximum entropy production rate (MEPR) yields the most stable interface configuration. The principle of MEPR is applied to (1) describe atomically smooth and diffuse interfaces, (2) provide quantitative results for the diffuse interface thickness and the number of pseudo-atomic layers in the interface region, and (3) predict the transition from planar to a non-planar facet or non-facet cellular morphology as a function of solidification velocity or temperature gradient. The MEPR model provides for an assessment of the interface diffuseness at the breakdown condition. It also allows for the break down condition to be expressed in terms of the cooling rate and the entropy generation rate.
Numerous experimental investigations spanning sixty years have failed to comprehensively validate any of the existing solid-liquid interface (SLI) growth instability models. With the MEPR model for the first time, breakdown conditions are predicted with a fair degree of accuracy for several binary alloys, where no previous theoretical model had predictability. The model considers steady-state solidification at close-to-equilibrium and far-from-equilibrium conditions. For dilute Pb-Sn alloys, the MEPR model gives closer D L predictions compared to the predictions made by the more traditional CUT and LST models. Regardless of the success of the model to date, it should be noted that the model remains untested for alloys with a significant amount of solute content. . s gen total irreversible entropy generated rate density at an interface (Jm −3 K −1 ) . s LG entropy generation rate density by the solute gradient in a liquid (Jm −3 K −1 ) (S gen ) max maximum entropy generation due to lost work (JK −1 ) dS cv /dt total steady state entropy rate in a control volume (JK −1 s −1 ) ds cv /dt total steady state entropy rate density in a control volume (Jm