Comparison of the Helmholtz, Gibbs, and collective-modes methods to obtain nonaffine elastic constants

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Abstract

We review and compare the Born-Huang and the Lemaitre-Maloney theories that lead to analytical expressions for elastic constants, accounting for affine and nonaffine deformations in a lattice (or in a disordered solid). The Born-Huang method is based on Helmholtz free energy while the Lemaitre-Maloney formalism focus on the Gibbs ensemble with the focus on local force. Although starting from different perspectives, in the linear elastic limit, and in equilibrium, material elastic constants must be the same in all these methods. This is explicitly verified on examples of linear chains, and the numerical simulation of a non-centrosymmetric crystal.

Introduction

It is manifest that elastic materials experience internal resistance to the deformation caused by external forces. They tend to return to original sizes and shapes when the external influence is eliminated. The elasticity of materials is generally described by a stress-strain curve, which exhibits a characteristic linear region for sufficiently small deformations. This linear regime is vital for, e.g. elastic waves, and most elastic theories are established in this linear regime. In a one-dimensional rod, the simplest linear relation between stress and strain is known as Hooke’s law; in three dimensions, the general proportionality between stress and strain is a 4th-rank tensor of stiffness coefficient (Landau and Lifshitz, 1960).

At zero temperature, once the relative initial positions of atoms are known, it is then a simple task to add all contributing interactions to elastic constants for homogeneous (affine) deformations. The resultant elastic constants are often called affine. When the two assumptions, zero temperature and homogeneous displacements, are not valid, one needs to develop a more complicated theory of nonaffinity (local, inhomogeneous). Early works focus on thermal effects on elasticity in crystals (Hoover, Holt, Squire, 1969, Squire, Holt, Hoover, 1969). In recent decades, athermal systems, like granular materials or foams, raise a lot of attention, investigating corrections to the affine elasticity (Lacasse, Grest, Levine, Mason, Weitz, 1996, Langer, Liu, 1997, Radjai, Roux, 2002, Tanguy, Wittmer, Leonforte, Barrat, 2002, Wittmer, Tanguy, Barrat, Lewis, 2002). In other words, even at zero temperature, particles (atoms) do not always follow homogeneous displacement fields. They instead attempt to minimise the potential energy of the system, and in some cases, this requires additional local nonaffine displacements, no matter how small deformation the system is strained to. The nonaffine correction to the elastic constants can be prominent, which has been found in simulation of a non-centrosymmetry lattice (Cui et al., 2019b). The formal expressions for the nonaffine corrections were systematically developed by Lemaitre and Maloney (LM) via studying the Gibbs ensemble with the local force acting on each particle in the system (Lemaitre and Maloney, 2006). Through performing normal mode decomposition, their analysis relates nonaffine corrections to the correlator of a fluctuating force field, which can be extended to the viscoelastic dynamical response of the system.

Prior to LM, the linear elastic constants were studied in detail in the work of Born and Huang (BH). The most familiar BH results are for the basic affine elastic constants, although they have actually discussed the nonffine deformation case in great detail (but have not derived complete analytical expressions for nonaffine corrections) (Born and Huang, 1954). However, reviewing the BH theory, and comparing it with LM formalism, we find that they address the elasticity problem from two complementary angles: LM approach works by identifying the local nonaffine forces, essentially working in the Gibbs ensemble framework, while BH arguments are based on optimization of local nonaffine displacements (i.e. in the conjugate Helmholtz ensemble). To test the comparison between these two approaches, we also consider the vibrational lattice waves in the lattice, which represent collective motions: in the long-wavelength limit these waves provide an additional path to material elastic constants, which must match both of LM and BH results.

This paper is organised as follows: Section 2 reviews the three approaches to elastic constants, including their interpretations of both affine and nonaffine contributions. Section 3 begins with clarifying the link between Gibbs and Helmholtz frameworks, with supporting examples of 1D linear lattices and a 3D non-centrosymmetric crystal, where we compare in detail the different ways of calculating elastic constants. Finally, in Section 4, we draw our conclusions and suggest an insight of practical applications of these methods.

Section snippets

BH: Elastic constants for non-ionic crystals

In non-ionic crystals, only short-range pairwise interaction need to be considered. The long-range Coulombic forces that usually cause a notional divergence are ignored here, although there exist ways to tackle the issue of divergence (see Section 3.2 below). To make it convenient for calculation, we assume the pair interacting potential depends on the square of interparticle distance near its equilibrium. We also require the system we are studying in the paper remains at zero temperature

Nonaffine elasticity

In general, it is cumbersome to apply the BH method directly. One has to express potential energy in terms of Helmholtz displacements, which consist of affine and nonaffine displacements. The affine displacements are related to the external strain, whereas the nonaffine displacements must be solved via Eq. (6). However, we note that, in the BH method, objects like{μνξι};{IJμν};{Iνξμ},which appeared, e.g., in the energy density Eq. (5), are mathematically equivalent to the affine elastic

Conclusion

In conclusion, having reviewed different approaches to calculating linear elastic constants, we find that in the BH framework, nonaffine elasticity is essentially due to equilibration of the local additional (Helmholtz) displacements particles experience within the unit cell. In contrast, in the LM formalism, the field of local (Gibbs) forces arising from the breaking of inversion symmetry are instead the cause of nonaffinity. The two methods are equivalent, in the sense that the change of

Declaration of Competing Interest

None.

Acknowledgements

This work was supported by the CSC-Cambridge Scholarship. Discussions with A. Zaccone are gratefully acknowledged.

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