Enthalpy-entropy compensation of atomic diffusion originates from softening of low frequency phonons

Simon Gelin,Alexandre Champagne-Ruel,Normand Mousseau

doi:10.1038/s41467-020-17812-2

Abstract

Experimental data accumulated over more than 120 years show not only that diffusion coefficients of impurities ordinarily obey the Arrhenius law in crystalline solids, but also that diffusion pre-exponential factors measured in a same solid increase exponentially with activation energies. This so-called compensation effect has been argued to result from a universal positive linear relationship between entropic contributions and energy barriers to diffusion. However, no physical model of entropy has ever been successfully tested against experimental compensation data. Here, we solve this decades-old problem by demonstrating that atomistically computed harmonic vibrational entropic contributions account for most of compensation effects in silicon and aluminum. We then show that, on average, variations of atomic interactions along diffusion reaction paths simultaneously soften low frequency phonons and stiffen high frequency ones; because relative frequency variations are larger in the lower region of the spectrum, softening generally prevails over stiffening and entropy ubiquitously increases with energy.

Highlights

Experimental data accumulated over more than 120 years show that diffusion coefficients of impurities ordinarily obey the Arrhenius law in crystalline solids, and that diffusion pre-exponential factors measured in a same solid increase exponentially with activation energies
Our numerical study starts with the observation that all Arrhenius parameters of diffusion processes reported in Fig. 1a fall on a single straight line, extending over more than three electronvolts in activation energies and seven orders of magnitude in pre-exponential factors
While there is a large dispersion on an event per-event basis, we find clear trends when averaging over these large data sets: prefactors increase in CuZr and amorphous solids: silicon (a-Si), according to the compensation law, and decrease in Ni80P20 and LJ, showing an anti-compensation that has already been observed in a non-physical LJ glass[21]

Summary

Introduction

Experimental data accumulated over more than 120 years show that diffusion coefficients of impurities ordinarily obey the Arrhenius law in crystalline solids, and that diffusion pre-exponential factors measured in a same solid increase exponentially with activation energies. The two most popular explanations for the compensation of energy barriers by entropy are based on phenomenological models of migration entropy: Zener’s model, drawing on reaction rate theory, ascribes compensation to a loosening of crystalline lattices’ elastic moduli at transition states (TSs)[16], while the multiexcitation entropy model explains the increase of entropic contributions as resulting from the increasing number of ways phonons can assemble to overcome higher energy barriers[9] Because they resort to qualitative descriptions of atomic diffusion, both models introduce arbitrary parameters that make them untestable, so that it remains unknown whether any of them identifies the correct physical origin of compensation. Density functional theory data in aluminium—obtained with the local density approximation (LDA) or the generalized gradient approximation (GGA)— obey compensation, with a compensation factor of 3.8 eV−1

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Conclusion