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

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. We study this process in detail for a simple triangular, harmonic lattice in two dimensions. Coarse-graining local strains obtained from particle configurations in a Monte Carlo simulation generates nontrivial, nonlocal strain correlations (susceptibilities). These 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 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 $\left(\chi \right)$ made up of nonaffine displacements. Several properties of $\chi$ 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 $\left(\chi \right)$ is studied. We find that for any inverse temperature $\beta$, spring constant $f$, density $\rho$ and coarse-graining length $\Lambda$ the probability distribution can be obtained from a master curve of the scaling variable $\mathcal{X}=\chi \beta f/\rho {\Lambda }^2$.

Figure 1

(Color online) The analytic form of the strain-strain correlation functions. For each function a surface plot and next to it a density plot are shown. In the density plot the maxima are white, minima are black. The set of parameters used for plotting was obtained from a Monte Carlo simulation of a harmonic triangular lattice in the $NVT$ ensemble with periodic boundary conditions: $a_1=97.4$, $a_2=48.1$, $a_3=198.2$, $c_1=54.3$, $c_2=37.4$, $c_3=118.2$, $c_1^{\prime }=-86.1$, $c_2^{\prime }=-1.3$, and $c_3^{\prime }=-18.8$. (a) $G_{11}\left(\vec{k}\right)$, (b) $G_{22}\left(\vec{k}\right)$, (c) $G_{33}\left(\vec{k}\right)$, and (d) $G_{2\theta 2\theta }\left(\vec{k}\right)$.

Figure 2

(Color online) Cuts of the analytic strain-strain correlation functions along specific directions in Fourier space. The set of parameters used for plotting is: $a_1=100$, $a_2=50$, $a_3=200$ and $c_1=54.3$, $c_2=37.4$, $c_3=118.2$, $c_1^{\prime }=-86.1$, $c_2^{\prime }=-1.3$, and $c_3^{\prime }=-18.8$. The correlations lengths were obtained from a Monte Carlo simulation of a harmonic triangular lattice in the $NVT$ ensemble with periodic boundary conditions. ${\tilde{G}}_{11}\left(\vec{k}\right)$ is shown along the direction $k_y=2k_x$, ${\tilde{G}}_{22}\left(\vec{k}\right)$ along the diagonal $k_x=k_y$, ${\tilde{G}}_{33}\left(\vec{k}\right)$ along the $k_y$ axis and ${\tilde{G}}_{2\theta 2\theta }\left(\vec{k}\right)$ along the diagonal $k_x=k_y$ and along the $k_y$-axis.

Figure 3

(Color online) Strain-strain correlation functions of a harmonic triangular lattice at zero hydrostatic pressure as obtained from Monte Carlo simulations in the $NpT$ ensemble with periodic boundary conditions. Results for a system with $N=3120$ particles and a spring constant $\beta a^{2}f=200/\sqrt{3}$ are shown. For each function a surface plot and next to it a density plot are displayed. In the density plot the maxima are white, minima are black.

Figure 4

(Color online) Cuts of the strain-strain correlation functions in the $NpT$ ensemble with periodic boundary conditions at zero hydrostatic pressure along specific directions in Fourier space. (a) ${\tilde{G}}_{11}\left(\vec{k}\right)$ along $k_y=2k_x$, (b) ${\tilde{G}}_{22}\left(\vec{k}\right)$ along $k_x=k_y$, (c) ${\tilde{G}}_{33}\left(\vec{k}\right)$ and (d) ${\tilde{G}}_{2\theta 2\theta }\left(\vec{k}\right)$ along the $k_y$ axis. For a comparison the results for different system sizes are displayed. The horizontal broken lines mark the value expected from theory for the correlation functions at $\vec{k}=\vec{0}$.

Figure 5

(Color online) Strain-strain correlation functions of a harmonic triangular lattice as obtained from Monte Carlo simulations in the $NVT$ ensemble with periodic boundary conditions. Results for a system with $N=3120$ particles and a $\beta a^{2}f=200/\sqrt{3}$ are shown. For each function a surface plot and next to it a density plot are displayed. In the density plot the maxima are white, minima are black.

Figure 6

(Color online) Cuts of the strain-strain correlation functions in the $NVT$ ensemble with periodic boundary conditions along specific directions in Fourier space. (a) ${\tilde{G}}_{11}\left(\vec{k}\right)$ along $k_y=2k_x$, (b) ${\tilde{G}}_{22}\left(\vec{k}\right)$ along $k_x=k_y$, (c) ${\tilde{G}}_{33}\left(\vec{k}\right)$ and (d) ${\tilde{G}}_{2\theta 2\theta }\left(\vec{k}\right)$ along the $k_y$ axis. For a comparison the results for different system sizes are displayed. The horizontal broken lines mark the value expected from theory for the correlation functions at $\vec{k}=\vec{0}$.

Figure 7

(Color online) Cuts of the strain-strain correlation functions in the $NpT$ ensemble with periodic boundary conditions along specific directions in Fourier space. Shown are the correlation functions obtained in a system without (crosses) external pressure and a system with $\beta a^{2}p=20/\sqrt{3}$ (circles). Both systems have $N=3120$ and a spring constant $\beta a^{2}f=200/\sqrt{3}$. (a) ${\tilde{G}}_{11}\left(\vec{k}\right)$ along $k_y=2k_x$, (b) ${\tilde{G}}_{22}\left(\vec{k}\right)$ along $k_x=k_y$, (c) ${\tilde{G}}_{33}\left(\vec{k}\right)$, and (d) ${\tilde{G}}_{2\theta 2\theta }\left(\vec{k}\right)$ along the $k_y$ axis.

Figure 8

(Color online) A schematic drawing of the displacement field $\vec{u}\left(\vec{r}\right)$ and below the corresponding strain fields as they result from a given perturbation at the origin: (a) a dilatation and (b) a rotation a the disk at the origin. If the analyzed volume $V_B$ (broken red line) does not coincide with the volume of the complete system $V$ (black line), the energy needed for the displacements in the embedding medium $\left(V-V_B\right)$ must be taken into account in the interpretation of the correlation functions.

Figure 9

(Color online) Compliances $S_{ii}$ as they are obtained from the sum rule as a function of the ratio of analyzed volume $V_B=L_B^2$ to the complete simulation volume $V=L^2$. Shown are the results of Monte Carlo simulations of a harmonic triangular lattice with $N=5822$ particles and spring constant $\beta a^{2}f=200/\sqrt{3}$. Lines are a fit to the data with the formulas given in the text, while the dotted lines are a guide for the eyes. ( a) $NpT$ ensemble with periodic boundary conditions. (b) $NVT$ ensemble with periodic boundary conditions. (c) $NVT$ ensemble with open boundary conditions. Here dashed lines show the value for $L_B/L=1$ as a comparison.

Figure 10

(Color online) Plots of the probability distribution of the nonaffinity parameter $\chi$ from simulations in the $NpT$ ensemble with periodic boundary conditions and $N=3120$ particles. (a) Plot of the probability distribution of $P\left(\chi \right)$ vs $\chi$ for various coarse-graining length $\Lambda$. (b) The probability distribution of $\chi /{\Lambda }^2$ shows a data collapse for $\Lambda \ge 2.2$. (c) Data for various spring constants $f$ collapse onto each other. The prediction from a simple cell model (black line) is shown for comparison. (d) Data from simulations at various hydrostatic pressure $p$ scale with the resulting average density $⟨\varrho ⟩$.

Figure 11

(Color online) Example configuration (a) consisting of a central particle 0 and neighboring particles $i=1,6$ which can be decomposed into a purely affine deviatoric distortion (b) together with a nonaffine displacement $\vec{s}$ of the central particle 0 (c).

Figure 12

(Color online) (a) The autocorrelation function of the nonaffinity parameter $\chi$ for the harmonic triangular lattice with spring constant $\beta a^{2}f=200/\sqrt{3}$ in the $NpT$ ensemble with periodic boundary conditions along the $x$ and the $y$ axes. The red line is a fit to an exponential form. (b) A surface plot of its Fourier transform ${\tilde{G}}_{\chi \chi }\left(\vec{k}\right)$.