Formation of Random Solid Solution in Multicomponent Alloys: from Hume-Rothery Rules to Entropic Stabilization

Abstract

As proposed in the 1920s, the empirical Hume-Rothery rules have been guiding the alloy design for random solid solutions. In contrast, the recent proposal by Yeh et al. (Adv Eng Mater 6(5):299–303, 2004) suggested that formation of random solid solution could be attributed to the maximized configurational entropy of mixing of multicomponent alloys, also known as high entropy alloys. By taking into account the non-ideal mixing of atoms with inter-atomic correlations (correlated mixing), here we suggest that the idea of entropic stabilization could be theoretically connected with the Hume-Rothery rules. The non-ideal formulation of the configurational entropy of mixing of a multicomponent alloy rationalizes the recent data obtained from experiments and simulations, which show the temperature dependence of the configurational entropy of mixing, the metastability of random solid solution observed at a low temperature, and the coupled effect of atomic size and chemical bond misfit on the stability of random solid solution. Finally, we discuss the measures that one can take to maximize the configurational entropy of a multicomponent alloy by considering the possible correlations among their constituent elements.

Appendix

For a microcanonic ensemble, let us assume the occurrence probability of the individual microstate being equal. Furthermore, the number (Ω) of the microstate corresponding to the equilibrium macrostate is overwhelmingly large compared to others, which can be expressed as $\ln \varOmega = \ln N! + \sum\limits_{i} {n_{i} \ln g_{i} } - \sum\limits_{i} {\ln n_{i} !}$. For a large N, we can take Stirling’s approximation $\ln N! = N\ln N - N$ and, therefore, $\ln \varOmega = N\ln N + \sum\limits_{i} {n_{i} \ln \frac{{g_{i} }}{{n_{i} }}}$. Maximizing ln Ω subjected to the two constraints dE = ∑  _i ɛ _i dn _i  = 0 and dN = ∑  _i dn _i  = 0 leads to

$$d\left( {\ln \varOmega } \right) - \alpha dN - \beta dE = \sum\limits_{i} {\left( {\ln \frac{{g_{i} }}{{n_{i} }} - \alpha - \beta \varepsilon_{i} } \right)} dn_{i} = 0$$

(8)

where E is the potential energy; α and β are the Lagrangian multipliers. For any arbitrary change of dn _i , (8) holds only if $\ln \frac{{g_{i} }}{{n_{i} }} - \alpha - \beta \varepsilon_{i} = 0$. This leads to the Maxwell–Boltzmann distribution $n_{i} = g_{i} e^{{ - \alpha - \beta \varepsilon_{i} }}$. Define the partition function $Z = \sum\limits_{i} {g_{i} e^{{ - \beta \varepsilon_{i} }} }$, the particle number N of the system can be expressed as $N = \sum\limits_{i} {g_{i} e^{{ - \alpha - \beta \varepsilon_{i} }} = e^{ - \alpha } Z}$ or e ^−α = N/Z. As a result, the expression of ln Ω can be simplified to ln Ω = N ln Z + βE. Furthermore, for a canonic ensemble (constant temperature and constant volume), it can be shown that β = 1/k _B T[35] and thus $\ln \varOmega = N\ln Z + \frac{E}{{k_{B} T}}$. Taking a small fluctuation in the configurational energy E, we have $d\left( {\ln \varOmega } \right) = \frac{dE}{{k_{B} T}}$. According to the first law of thermodynamics, dE = δQ. Thus $d\left( {\ln \varOmega } \right) = \frac{\delta Q}{{k_{B} T}} = \frac{dS}{{k_{B} }}$. Finally, we obtain:

$$S = k_{B} \ln \varOmega$$

(9)

Here, it is worth noting that Ω in the Boltzmann entropy formula corresponds to the fixed potential energy E. Any disturbance in the potential energy E will cause a change in the Ω and thus the configurational entropy S.