Coarse-graining microscopic strains in a harmonic, two-dimensional solid: Elasticity, nonlocal susceptibilities, and nonaffine noise

Abstract

In soft matter systems the local displacement field can be accessed directly by video microscopy enabling one to compute local strain fields and hence the elastic moduli in these systems using a coarse-graining procedure. Here, we study this process in detail for a simple triangular lattice of particles connected by harmonic springs in two-dimensions. Coarse-graining local strains obtained from particle configurations in a Monte Carlo simulation generates non-trivial, non-local strain correlations (susceptibilities), which may be understood within a generalized, Landau type elastic Hamiltonian containing up to quartic terms in strain gradients (K. Franzrahe et. al., Phys. Rev. E 78, 026106 (2008)). In order to demonstrate the versatility of the analysis of these correlations and to make our calculations directly relevant for experiments on colloidal solids, we systematically study in detail various parameters such as the choice of statistical ensemble, presence of external pressure and boundary conditions. Crucially, we show that special care needs to be taken for an accurate application of our results to actual experiments, where the analyzed area is embedded within a larger system, to which it is mechanically coupled. Apart from the smooth, affine strain fields, the coarse-graining procedure also gives rise to a noise field (χ) made up of non-affine displacements. Several properties of χ may be rationalized for the harmonic solid using a simple "cell model" calculation. Furthermore the scaling behavior of the probability distribution of the noise field (χ) is studied. We find that for any inverse temperature β, spring constant f , density ρ and coarsegraining length Λ the probability distribution can be obtained from a master curve of the scaling variable X = χβf /ρΛ 2 .

Nonaffine displacements in crystalline solids in the harmonic limit

“…We note that the large distance (small q) anisotropies of e 2 u (q) and e 2 s (q) are also consistent with a mean field, continuum theory, calculation shown in detail in Ref. 22. Interestingly, that calculation restricted attention to fluctuating strain fields that satisfy force balance, so one concludes that it is indeed these configurations that dominate the scaling for large distances.…”

“…The q space structure of the uniaxial and the shear correlation functions as implied by (56), Figs. 11(b) and (c), and further supported by continuum theory [22], shows that these correlation functions have singularities at q = 0, with the second terms in (54) and (57) vanishing along specific directions (q x = 0, q y = 0 in e u and q x = ±q y in e s ). These singularities lead to slow (∼ 1/R 2 ) decay of the correlation functions in real space (see Fig.…”

“…. = σ 12 = 0 that do not contribute to (22) as they correspond to purely affine distortions within Ω. The eigenvectors associated with the 8 non-affine displacements are shown in Fig.3.…”

Coarse-grained density and compressibility of nonideal crystals: General theory and an application to cluster crystals

“…Other symmetries with known solutions are hexagonal 42 and pentagonal 43 . The corresponding problem in two dimensions can be found in [44]. We consider a three dimensional sphere with volume V B embedded in a spherical matrix V of the same isotropic material.…”

Strain fluctuations and elastic moduli in disordered solids

“…One common approach focuses on thermally induced microscopic strain fluctuations [4,8,12,13]. In these studies, the fluctuations of a locally-defined strain field are aggregated over time to arrive at a distribution of strains at each point.…”