Abstract
We generalise the non-affine theory of viscoelasticity for use with large, well-sampled systems of arbitrary chemical complexity. Having in mind predictions of mechanical and vibrational properties of amorphous systems with atomistic resolution, we propose an extension of the Kernel Polynomial Method (KPM) for the computation of the vibrational density of states (VDOS) and the eigenmodes, including the $\Gamma$-correlator of the affine force-field, which is a key ingredient of lattice-dynamic calculations of viscoelasticity. We show that the results converge well to the solution obtained by direct diagonalization (DD) of the Hessian (dynamical) matrix. As is well known, the DD approach has prohibitively high computational requirements for systems with $N=10^4$ atoms or larger. Instead, the KPM approach developed here allows one to scale up lattice dynamic calculations of real materials up to $10^6$ atoms, with a hugely more favorable (linear) scaling of computation time and memory consumption with $N$.
Highlights
For the case of elasticity of centrosymmetric crystalline solids, Born and Huang developed a theory which can straightforwardly predict and compute the elastic moduli from the atomistic structure [1]
Having in mind predictions of mechanical and vibrational properties of amorphous systems with atomistic resolution, we propose an extension of the kernel polynomial method (KPM) for the computation of the vibrational density of states and the eigenmodes, including the correlator of the affine force field, which is a key ingredient of lattice-dynamic calculations of viscoelasticity
We show that the results converge well to the solution obtained by direct diagonalization (DD) of the Hessian matrix
Summary
For the case of elasticity of centrosymmetric crystalline solids, Born and Huang developed a theory which can straightforwardly predict and compute the elastic moduli from the atomistic structure [1]. Recently it was shown that so-called nonaffine corrections to the original Born and Huang approach offer a pathway for the prediction of glass viscoelasticity [2,3,4] These corrections account for additional relaxations of atomic positions in noncentrosymmetric cases and result in an overall softening of a material. A key component of lattice dynamics computations is the analysis of the spectral density of dynamical matrices and their eigenvector characteristics or eigenmode spectrum [4,9,18] For this relatively large number of atoms, direct diagonalization (DD) of the Hessian matrices ceases to be a viable method to obtain the eigenfrequencies and eigenmodes since, typically, DD becomes prohibitive for N 104 due mainly to memory requirements. The framework allows lattice dynamic calculations of viscoelastic moduli of amorphous solids to be performed on systems larger than N = 105. The proposed framework provides a working solution to the problem of bridging time and length scales in the molecular simulation of materials mechanics
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