Nonaffine displacements in crystalline solids in the harmonic limit

Abstract

A systematic coarse graining of microscopic atomic displacements generates a local elastic deformation tensor D as well as a positive definite scalar χ measuring nonaffinity, i.e., the extent to which the displacements are not representable as affine deformations of a reference crystal. We perform an exact calculation of the statistics of χ and D and their spatial correlations for solids at low temperatures, within a harmonic approximation and in one and two dimensions. We obtain the joint distribution P(χ,D) and the two-point spatial correlation functions for χ and D. We show that nonaffine and affine deformations are coupled even in a harmonic solid, with a strength that depends on the size of the coarse-graining volume Ω and dimensionality. As a corollary to our work, we identify the field h(χ) conjugate to χ and show that this field may be tuned to produce a transition to a state where the ensemble average <χ> and the correlation length of χ diverge. Our work should be useful as a template for understanding nonaffine displacements in realistic systems with or without disorder and as a means for developing computational tools for studying the effects of nonaffine displacements in melting, plastic flow, and the glass transition.

Non-affine lattice dynamics of defective fcc crystals

“…In recent work 18,19 a similar connection has been observed in a system where non-affine displacements are induced via thermal fluctuations instead of structural disorder, which complements our findings. The analytical and numerical results 20 show that if the non-affine displacements are enhanced via an external field there is an accumulation of low-frequency modes in the density of states, also linking non-affine fluctuations to the boson peak.…”

Equilibrium and dynamic pleating of a crystalline bonded network

“…The minimisation procedure amounts to projecting [10] ∆ i onto a non-affine subspace defined by a projection operator P such that χ(R i ) = ∆ T P∆ where ∆ is the column vector constructed out of the ∆ i . In the projector P = I−R(R T R) −1 R T , the N d×d 2 elements of R are given by R jα,γγ = δ αγ R jγ (here, the central particle i is taken to be at the origin).…”

“…The non-affine field does not affect the statistics of the affine part of the displacements to linear order in h X . Space-time correlation functions of both affine and non-affine variables can also be obtained using a similar procedure [10,11]. In order to selectively excite lattice distortions that enhance non-affine displacements, we introduce an extended microscopic Hamiltonian [11] involving the thermodynamic conjugate variables h X and X, with X = N −1 N i χ(R i ).…”

“…In earlier work, it was demonstrated that local displacements of particles in a crystal away from their ideal positions may be decomposed into affine and non-affine components [10][11][12]. External stress couples to the affine part of the displacements.…”

Shear shocks in fragile networks

“…This method, which we tentatively label shear front rheology (SFR), is of general applicability, but it may be particularly appropriate to investigate critical systems whose dynamics is undergoing a dramatic slow-down, such as biopolymer networks or gels with a sufficiently small shear modulus and sound speed, so that transients become the main observables. Our findings demonstrate that the mechanical response near the critical point is inherently nonaffine, non-quasistatic, and nonlinear.First, the approach toward the critical point is accompanied microscopically by an increasingly heterogeneous displacement field that is nonaffine with respect to the uniform shear applied at the boundary (6,(15)(16)(17)(18)(19). We show that the increasingly nonaffine response, as Δz is lowered toward the rigidity transition, is reflected in SFR by the diverging widths of the shear fronts.…”