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# Melting Entropy and Specific Heat Capacity of Binary Eutectic Alloys without Solid State Solubility

Pierre Guiraldenq

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Pierre Guiraldenq. Melting Entropy and Specific Heat Capacity of Binary Eutectic Alloys without Solid State Solubility. 2025. hal-05374198

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Distributed under a Creative Commons Attribution 4.0 International License
MELTING ENTROPY AND SPECIFIC HEAT CAPACITY OF BINARY EUTECTIC ALLOYS

WITHOUT SOLID STATE SOLUBILITY

Pierre GUIRALDENQ*

*Emeritus Professor, Ecole Centrale de Lyon, Ecully, France, e-mail : pierre.guiraldeng@orange.fr

Conflicts of interest: the author declares no conflict of interest

OVERVIEW

- Summary

- Recall of previous results

- The heat capacity of special binary eutectic alloys

- Different new concepts applied to the subject

- Conclusion

- References

- Acknowledgements

SUMMARY

An unusual class of binary alloys has been analyzed taking simple considerations about the thermal
melting process with the entropy. This study concerns only the eutectic binary systems without
reciprocal solid-state solubility. The results obtained, with an extension of the Einstein theory for the
heat capacity, are presented. Different mechanisms are suggested in relation with the increasing
entropy during the solid → liquid transformation. Theoretical and experimental investigations are
proposed for potential applications.

KEYWORDS

Binary eutectic alloys, melting entropy, specific heat capacity, fractal diffusion, fractal temperature, order-disorder transition.
RECALL OF PREVIOUS RESULTS

In two preceding papers (1) (2), we have detailed some properties of singular binary eutectic alloys, without or very low solid state solubility at the melting temperature T~E~: Particularly, it has been found an entropy gain corresponding to the process solid → liquid. This increasing melting entropy has been connected with the solid state diffusion of each element and the thermos topology of each binary system (the family of rare earth alloys is excluded because too many allotropic transitions are present).

We have obtained for a lot of this family a linear relationship between the entropy gain and the
thermal parameter T~E~/T_m obtained with the two elements (T~E~: eutectic temperature, T_m1, T_m2: melting
temperatures of the two components at the atm. pressure) with a slope which is like the higher heat
capacity obtained with the Einstein theory of solids (3kN) (by extension of the Dulong and Petit Law),
and reviewed later by other authors, like in (3) (Figure 1).

A structural mechanism at a nanoscale level has been proposed taking firstly into account the interfaces properties between the two phases and the evolution of the transient structure to a full disordered liquid state at T~E~.

Other physical properties in relation to the bulk self-diffusion, like the frequency factor, have also been studied to achieve the mechanism which would be like a damped oscillation process starting from every solid interface between the two basic components.

In this paper, we give new details and extension in relation to the heat capacity of each component
of these alloys and some ideas connected with recent advanced models.

THE HEAT CAPACITY OF SPECIAL BINARY EUTECTIC ALLOYS

Taking into account the values of the specific heat capacity of each element (4), we have analyzed
the ratio between the heat capacity C<sub>p</sub> (given at room temperature by the Tables) and the constant
3kN (or 3R) seen before, for which the maximum value is 1. Two approaches have been then
followed:

In a first time, at a macroscopic level with the classical thermodynamic properties, we have computed the C<sub>p</sub>/3KN ratio with the two concentrations C<sub>1</sub> and C<sub>2</sub> (at. °/°) given at TE for each equilibrium diagram (the references are given by the classical books cited in (1)).

In a second case, the C<sub>p</sub>/3KN ratio, look at a nano scale level for the interfaces between the 2
elements, where exist, before any local inter diffusion, the same atomic concentrations (C<sub>1</sub> = C<sub>2</sub> = 50
at. °/°).

The Figure 2 gives the results obtained with the two configurations and the parameter (T<sub>E</sub>/T<sub>m1</sub>+T<sub>E</sub>/T<sub>m2</sub>) which characterizes the place of TE versus the melting point of each element.
We observe that all the alloys have approximately the same value $C_p/3KN$ ratio around the line corresponding to $C_p/3KN = 1$, without a great difference between the binary alloys with a high entropy gain and the second group with low entropy excess values (given in the Figure 1).

But, looking at these results in detail, we note some examples drawn aside with low or higher values ($C_p/3KN > 1$ or $C_p/3KN < 1$).

For example: Si-Au with $C_p/3KN = 0.94 - 0.90$, B-Pt with $0.80 - 0.74$, Be-Si with $0.80 - 0.70$ and other cases, like As-Pb, Al-Sn, Nb-Th, Ga-Sn, with values $C_p/3KN$ a little more than 1.

This means that other physical factors are to be considered, like singular diffusion processes near the primary solid interfaces, at temperatures near the melting point $T_E$, according a fractal process to give an amorphous phase, as proposed in the preceding papers.

## DIFFERENT NEW CONCEPTS APPLIED TO THE SUBJECT

First remark concerning the diffusion processes at TE:

We recall the preceding results (ref. 1 and 2), in which it has been proposed to take into account the first contribution of interfaces (or grain boundaries) diffusion, given a long time ago for all crystal structures with the same liquid self-diffusion at the melting point (6) (7) (8) in ref. (1). So, the binary interfaces in the present binary duplex structures should be connected with a transient transformation to the liquid state beginning by an amorphous phase (Fig 3 - page 11 in (2)). The diffusion mechanisms may concern a new local medium with a high anisotropy and several space dimension levels (2D – 3D) to obtain the final liquid state.

Another question concerns the diffusion from these interfaces (long time ago after the work of Leonard Brillson (5) for electronic devices) for which we have proposed a progressive evolution of the solid structure by damping oscillations to the liquid phase at $T_E$ (Fig. 3 and 4 in ref. (2)).

Just at the beginning of the transformation near $T_E$, we can also consider the possible place of the bulk self-diffusion coefficients of each element at $T_E$, without taking into account the interfacial zones with possible coincidence sites:

For example, with the best experimental results given by the literature (ref. 7 and 8 in (2), we can compare the relative importance of the bulk self-coefficients just at $T_E$ ($D_1/D_2$) when is obtained an increasing entropy, like with the systems: Si-Au, Si-Ag, Ge-Au, Ge-Ag.

In spite of large extrapolations with the Arrhenius plots to calculate the $D_1/D_2$ ratio of the four preceding systems, we have obtained:

<table><thead><tr><th>System</th><th>Si-Au</th><th>Si-Ag</th><th>Ge-Au</th><th>Ge-Ag</th></tr></thead><tbody><tr><td>D<sub>1</sub>/D<sub>2</sub></td><td>10<sup>-17</sup></td><td>10<sup>-10</sup></td><td>10<sup>-8</sup></td><td>10<sup>-5</sup></td></tr><tr><td>at T<sub>E</sub> (K)</td><td>643</td><td>1103</td><td>629</td><td>924</td></tr></tbody></table>

Concurrently, it has been observed in (2) a equivalent decreasing evolution of the ratio between the vibration frequencies of the two elements 1 and 2 at $T_E$ (10<sup>+4</sup>, 10<sup>+3</sup>, 10<sup>+2</sup>, 10<sup>+1</sup>) obtained with the
experimental values given by the Arrhenius relations. Such unbalance observed for D₁/D₂ and the frequencies ratio can contribute to a higher disorder of each couple, taking place in the disturbed interfacial zones near T_E.

Second Remark concerning new concepts:

Another approach of the problem is to consider the contribution of the temperature with fractal properties, concept proposed by A.K. Golmankhaneh and co-authors (6) (7). So, new ideas can emerge with the fractal heat capacity derived from the Einstein solid model, given by the relation:

$$C_p = 3kN / \Gamma ( 1 + \mu )$$

- μ: being the fractal dimension of the temperature completing the structural evolution at T_E between the two initial solid phases and the final alloyed liquid phase,

- the numerical values of Γ are in basic mathematical books

So, we can consider an instability of T_E because the Gibbs' Rule cannot be applied exactly according to the well classical Gibbs law for an eutectic binary system (at T_E, the degree of freedom for the alloy C_E is zero) as described a long time ago by Rhines (8). The presence of a new transient structure at T_E, during the transformation of the two primary solid phases, includes a modification of the result with a new degree of freedom (at atm. pressure).

So, fractal temperature near T_E and fractal diffusion near the interfaces should be the middle point for the solid → liquid transformation of these "special binary" eutectic systems.

## CONCLUSION

Many studies about eutectic alloys have covered the research a long time ago and this domain has received important industrial applications, fine solid state analysis and recently a complete synthesis about nano advanced and multi component eutectic alloys, as described in (9).

The contribution of this study was, at the beginning, a research about the most simple binary eutectic alloys without solid state solubility near the melting temperature T_E. For this family, we have discovered interesting cases, characterized by an entropy excess obtained at the solid - liquid transition, described in (1) and (2).

New results concerning the specific heat capacity have been observed with the generalized Einstein model and the level obtained for the C_p/3kN ratio seems to complete recent theoretical works to be applied to the transition at T_E:

Fractal diffusion during the emerging amorphous zones joined to fluctuant and fractal temperatures at the final melting stage.

In a next future, these eutectic materials may have some thermal applications or other properties with the group of high entropy alloys (mixture of couples like Si-Au, Si-Be, Si-Ag, Si-Al, Ge-Al, Ge-Au, Ge-Ag, etc.) but it remains to confirm, in a first time, the nanostructures present at the interfaces before the liquid phase.
REFERENCES

(1) P. Guiraldenq: A new approach to the solid - liquid transition for simple binary eutectic systems in relation to the melting entropy and the periodic table. HAL 2023 Id: hal-04348252.

(2) P. Guiraldenq: New comments on the melting entropy of binary eutectic alloys without solid state solubility. HAL 2023 Id: hal-04032265v3.

(3) M. Beekman, I. Fernsler, T.D. Gutierrez: Clarification on Einstein's model for heat capacity of a solid 2021 https://arxiv.org/pdf/2102.04258

(4) Molar thermal capacities of elements in the periodic table Wikipedia.org 2025, and Engineers EDGE: "specific heat capacity of the elements". Table Chart 2020-2025 LLC

(5) L. Brillson: The structure and properties of Metal-Semiconductor Interfaces - Surface Science Reports November 1982 p. 123-326.

(6) A. K. Golmankhaneh: Statistical mechanisms involving Fractal temperature Fractal Fract.2019, 3, 20; doi: 10.3390:FractalFract3020020

(7) A.K. Golmankhaneh, C. Cattoni, R. Pesechnik, S. Firuichi, P.E.T. Jergensen: Fractal calculus of variations for problems with constraints. *Modern Physics Letters A* (40) (7-8) 2025 march 14 Art. 2550001

(8) F. N. Rhines: Phase Diagrams in Metallurgy – Their development and applications McGraw-Hill Book Company N.Y. 1956

(9) B. Chanda, G. Potnis, P.P. Jana, J. Das: A review on nano-/ultrafine advanced eutectic alloys. J. of Alloys and Compounds 2020 827 Art. 154226

ACKNOWLEDMENTS

The author thanks very much his colleagues Dr Michelle Salvia - Gobin, Associate Professor Emeritus and Bernard Beaugiraud, CNRS Research Engineer at Laboratoire de Tribologie et Dynamique des Systemes (LTDS), Centrale Lyon, for their aid and discussions. Many thanks also to the "Michel Serres" Library for the team efficiency.
<figure><img src="image_3.png" /><figcaption>Figure 1: Recall about the increasing entropy obtained at T<sub>E</sub> as a function of the factor (T<sub>E</sub>/T<sub>m1</sub>+T<sub>E</sub>/T<sub>m2</sub>) for binary eutectic alloys without solid state solubility (1) (2).</figcaption></figure>

T<sub>E</sub>: Eutectic Temperature; T<sub>m1</sub> and T<sub>m2</sub>: Melting Temperatures of 1 and 2 components.
<figure><img src="image_4.png" /><figcaption>Figure 2: Evolution of the Cp/3kN ratio around the value "1" for the binary eutectic alloys without solid state solubility, computed at two levels with the factor ($T_E/T_{m1}+T_E/T_{m2}$):</figcaption></figure>

<footnote>①: macroscopic level and ②: nanoscopic level</footnote>