Nonlocal elastic compliance for soft solids: Theory, simulations, and experiments

Abstract

The nonlocal elastic response function is crucial for understanding many properties of soft solids. This may be obtained by measuring strain-strain autocorrelation functions. We use computer simulations as well as video microscopy data of superparamagnetic colloids to obtain these correlations for two-dimensional triangular solids. Elastic constants and elastic correlation lengths are extracted by analyzing the correlation functions. We show that to explain our observations displacement fluctuations in a soft solid need to contain affine (strain) as well as nonaffine components.Elasticity theory [1,2] is based on the assumption that the displacement field u(r) is smooth and continuous for all po sitions r within a volume V. It is expected that this assump tion breaks down [3] below a limiting length g el . How small is g el in practice, and what are the consequences of the break down? What implications does this have on the mechanical response of the material? While for most conventional engi neering problems such issues are practically irrelevant, they do become important when investigating microelastic and microrheological properties of soft solids [6]. We answer these questions in the context of the nonlocal mechanical response of a soft solid in two dimensions (2D). Nonlocal elastic effects are important whenever strain gradients are large, e.g., near crack tips, grain boundaries, and interfaces [4]. They should also be important if the linear dimension of the solid is not much larger than g el . Both of these criteria are satisfied in colloidal solids under usual experimental condi tions [5].Recently, there has been a lot of interest in obtaining the elastic moduli of soft solids [6,7] from fluctuations of par ticle coordinates [8][9][10][11][12]. These methods have several advan tages. First, no specialized equipment is necessary other than standard video microscopy [7]. Second, no external forces need to be applied which may change the very properties that are being measured. This is especially convenient when try ing to obtain elastic constants near the dislocation unbinding transition [1] where the solid is sensitive to external pertur bations [11]. Lastly, these techniques are general enough so that they may be applied to a wide variety of systems [13]. Here we extend this procedure a step further to study nonlo cal [4] elasticity for solids in two dimensions far from the melting transition.The nonlocal elastic response function, or compliance X ij (r , r') (i = x , y), is defined as the strain e ij (r') produced at position r' due to a stress C ij (r) at r. This is given by X ij = (k B T) −1 G ij , where, for a homogeneous solid, G ij (r') = (e i (0)e ( j r')> is the strain-strain correlation function. The (¯> denote a thermal average (and one over the choice of origin) and k B T is the Boltzmann constant times the tempera ture [1]. The linear combinations of components of the strain tensor e y , x ( u derivatives of the displacement field =1,2,3 i ( i e relevant to a 2D solid are ) defined as ) calcu...

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

“…A local dilatation as perturbation results in a superposition of eigenmodes with four-as well as eight-fold degeneracy. This leads to the eight-fold rotational symmetry visible for the strain correlation functionG 11 . In contrast to this the two possible shear perturbations will excite elastic waves that are superpositions of exclusively eigenmodes with four-fold degeneracy.…”

“…fromG 11 ( k = 0) the coefficients c 1 and c ′ 1 can also be obtained by fitting one of the correlation functions along a cut in Fourier space. So in principle all 9 parameters of the free energy functional can be determined from an analysis of the strain-strain correlation functions.…”

“…As the considerations in section III D will show, it will be helpful to define the strain variable e 2θ = 2θ and analyze its two-point correlation function in embedded systems, such as the colloidal crystal discussed in [11]. Switching to a discretized notation, appropriate for comparison with our simulations, so that e 1 ( x) → e 1, m and x → R m , where m (a tuple of integer lattice indices) specifies the position on a coarse-graining, square mesh the free energy can be rewritten as,…”

“…Firstly, we present in great detail the analytic background of the coarse-graining procedure for obtaining elastic moduli and non-local susceptibilities (or strain-strain correlation functions) used in our earlier work [11]. In order to demonstrate the versatility inherent to the analysis of these non-local elastic susceptibilities, we present systematic studies of a simple two dimensional lattice of particles connected by harmonic springs in the current paper.…”

“…Recently, this procedure was further extended in Ref. [11] to obtain even the non-local elastic susceptibilities.…”