Strain fields in repulsive colloidal crystals

The concept of a local linear elastic strain field is commonly used in the metallurgical research community to approximate the collective effect of atomic displacements around crystalline defects. Here we show that the elastic strain field approximation is a useful tool in colloidal systems. For colloidal crystals with repulsive particle interaction potentials, given similar mechanical properties, sharper potentials lead to (1) free energies of deformation dominated by entropy, (2) lower variance in strain field fluctuations, (3) increased tension-compression asymmetry near dislocation core regions, and (4) smaller windows of applicability of the linear elastic approximation. We show that the window of linear behavior for entropic colloidal crystals is broadened for pressures at which the interparticle separation sufficiently exceeds the range of steep repulsive interactions.

Figure 1

The isotropic pair potentials used in this study, shown in units of system thermal energy. Darker blue curves are potentials with larger hardness values (labeled on curves). The Lennard-Jones potential (shifted in energy by $\varepsilon$) is also shown for comparison.

Figure 2

Pressures at which the moduli of SWCA systems most closely approach those of an LJ system at zero pressure. (a) Plots of the absolute value of the fractional difference in strain free energy between the SWCA and LJ system. Here $F_{\mathrm{strain}}^{\mathrm{ref}}$ is the strain free energy of the LJ system against which the matching is being evaluated. Error bars are calculated via bootstrapping of moduli measurements, and represent one standard deviation. (b) Absolute pressure vs. hardness. Tick marks indicate the uncertainty of $P_{\mathrm{match}}$, one standard deviation of the minima from the fit in (a), and the color map is the fit value from (a), truncated at 10% maximum difference.

Figure 3

Energy and entropy of deformation. (a) Internal (potential and kinetic) energy of deformation of potentials in this hardness family. Solid lines are linear fits to data collected from systems homogeneously deformed along the $\left[1\overline{1}0\right]$ direction. (b) Ratio of internal energy change to free energy change (evaluated at 1% strain). Free energy is calculated from the sampled elastic moduli and the applied homogeneous strain. Error bars of one standard deviation are much smaller than shown data points.

Figure 4

The standard deviation of per-particle strain energy values (normalized to a reference strain) as a function of the number of decorrelated samples considered. For each sample number, bootstrapping is used to obtain the standard deviation. The normalization strain is chosen to be the per-particle strain energy of a 1% volumetric expansion in LJ, which estimates the strain energy of a dilatory point defect such as a vacancy. With increasing $h$, fewer samples are required to obtain the same standard deviation values.

Figure 5

Strain fields around partial dislocation pairs. This view is along the length of the dislocation line, with the Burgers vector pointing from left to right in the plane of the image. The three strain components with the largest magnitudes are shown. The other three strain components contain small magnitudes because they involve the $\left[11\overline{2}\right]$ direction, along which there is little change in the displacement field around a straight dislocation line. See Ref. [33] for these components. At low hardness, and in particular for LJ, we find a close match between the analytically generated strain distribution and the sampled one. As the hardness of the potential is increased, an asymmetry develops between the compressive and tensile regions of ${\eta }_{xx}$ and ${\eta }_{zz}$.

Figure 6

Stress-strain relationship for different values of $h$ under a strain ramp. The region of linear behavior shrinks and the asymmetry of the stress curve increases as the pair potential hardness is increased. All potentials are first equilibrated under appropriate $P_{\mathrm{match}}$ and then a uniaxial strain is applied.

Figure 7

(a) Stress-strain curves for a system with $h=0.95$, found by equilibrating the system under a hydrostatic pressure and then applying a uniaxial strain. As the pressure is reduced, the linearity of the stress-strain relationship improves. However, at pressures other than $P_{\mathrm{match}}$, the solids cannot be said to have similar mechanical properties to the other systems shown in this study. (b) Strain to contact for systems in (a). Strain to contact is the strain needed geometrically to make particle centers sit a distance of $r_{\mathrm{cut}}$ from each other. Error bars represent one standard deviation (due to uncertainty in measurement of the lattice parameter). The region of linear behavior for a hard repulsive material must be a fraction of the strain-to-particle contact. Together these data show that hard potentials behave most elastically linear when the rattle volume of a particle at its lattice site is large. The time averaged effect of many hard-particle collisions produces an effective harmonic potential.