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

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.

Figure 1

(Color online) Analytic strain-strain correlation functions for a triangular lattice with parameters obtained from fits to simulations: subfigures show real space density plots of the strain-strain correlation functions over a box of size $L=34.35a$: (a) ${(L∕a)}^2{G}_{11}$ (gray scale black to white $[-0.003,0.015]$), (b) ${(L∕a)}^2{G}_{22}$ $\left([-0.1,0.1]\right)$, and (c) ${(L∕a)}^2{G}_{33}$ $\left([-0.03,0.03]\right)$. (d) Cuts of ${\tilde{G}}_{jj}$ along various directions in $k$ space: $k_x=k_y$ for $j=2$, $k_x=0$ for $j=3$, and $k_y=2k_x$ for $j=1$. Note the discontinuity in ${\tilde{G}}_{11}$ at $k=0$.

Figure 2

(Color online) Simulation of a harmonic triangular lattice. Subfigures show real space density plots of the strain-strain correlation functions over a box of size $L=34.35a$. (a) ${(L∕a)}^2{G}_{11}$ (gray scale black to white $[-0.003,0.015]$), (b) ${(L∕a)}^2{G}_{22}$ $\left([-0.1,0.1]\right)$, and (c) ${(L∕a)}^2{G}_{33}$ $\left([-0.03,0.03]\right)$. (d) Simulation data (symbols) and fits (lines) of ${\tilde{G}}_{jj}$ along various directions in $k$ space: $k_x=k_y$ for $j=2$, $k_x=0$ for $j=3$, and $k_y=2k_x$ for $j=1$.

Figure 3

(Color online) Colloidal crystal at $\Gamma =150$ [19]. The strain correlation functions were obtained from 3600 independent configurations for a defect-free square region $(L=12.881a)$. (a)–(c) show real space strain-strain correlation functions ${(L∕a)}^2{G}_{11}$ (gray scale, black to white $[-0.0015,0.0015]$), ${(L∕a)}^2{G}_{22}$ $\left([-0.05,0.075]\right)$, and ${(L∕a)}^2{G}_{2\theta 2\theta }$ $\left([-0.02,0.04]\right)$. (d) Data (symbols) and fits (lines) of ${\tilde{G}}_{jj}$ along various directions in $k$ space: $k_x=k_y$ for $j=2$, $k_x=0$ for $j=3$ and $j=2\theta$, and $k_y=2k_x$ for $j=1$.

Figure 4

(Color online) (a) Correlation function of the nonaffine parameter $G_{D^2}(x,y)$ for the colloidal crystal at $\Gamma =150$ [19]. (b) Correlation function $G_{D^2}(x,y)$ for the harmonic system $(f=1200∕\sqrt{3})$. (c) Probability distribution $P\left(D^2\right)$ for different spring constants $f$ and the colloidal solid.