Microscopic Approach to the Nonlinear Elasticity of Compressed Emulsions

Using confocal microscopy, we measure the packing geometry and interdroplet forces as a function of the osmotic pressure in a 3D emulsion system. We assume a harmonic interaction potential over a wide range of volume fractions and attribute the observed nonlinear elastic response of the pressure with density to the first corrections to the scaling laws of the microstructure away from the critical point. The bulk modulus depends on the excess contacts created under compression, which leads to the correction exponent $\alpha =1.5$. Microscopically, the nonlinearities manifest themselves as a narrowing of the distribution of the pressure per particle as a function of the global pressure.

Figure 1

Confocal 3D images for emulsions (a) creamed under gravity and (b) compressed at $P=2.2\mathrm{kPa}$ by uniaxial centrifugation. (c) A zoom-in of two reconstructed droplets labeled $\alpha$ and $\beta$ shows their geometric overlap area $A^{\alpha \beta }$, which is used to estimate the interdroplet force.

Figure 2

Scaling laws for the excess average number of contacts $\delta z$ with the changes in (a) global density $\delta \varphi =(\varphi -{\varphi }_{\mathrm{c}})$ and (b) the applied pressure $\delta P=(P-P_c)$. A parametric plot of $\delta P$ versus $\delta \varphi$ in (c) implies the nonlinear scaling law, in agreement with previous bulk measurements. The power-law fits are shown on log-log scales in the insets. (d) Probability density distributions of the pressure per particle $P_{\mathrm{L}}$ rescaled by the mean as a function of the applied pressure. The inset shows the corresponding coefficients of variation for all measured distributions. The pressure fluctuations are best fit with exponential tails at low pressures and a Gaussian distribution at high pressure.

Figure 3

Probability density distributions of local parameters (a) $z_{\mathrm{L}}$ and (b) ${\varphi }_{\mathrm{L}}$ are rescaled by the mean for three different global pressure values. The insets show the corresponding coefficients of variation for all measured distributions. The data are well fit with the granocentric model.