Lindemann Unjamming of Emulsions

We study the bulk and shear elastic properties of barely-compressed, “athermal” emulsions and find that the rigidity of the jammed solid fails at remarkably large critical osmotic pressures. The minuscule yield strain and similarly small Brownian particle displacement of solid emulsions close to this transition suggests that this catastrophic failure corresponds to a plastic-entropic instability: the solid becomes too soft and weak to resist the thermal agitation of the droplets that compose it and fails. We propose a modified Lindemann stability criterion to describe this transition and derive a scaling law for the critical osmotic pressure that agrees quantitatively with experimental observations.

Figure 1

Diffusive light scattering from sedimented emulsion. (a) $7.2\mu \mathrm{m}$ diameter droplets are sealed in a thermostated glass tube, where they settle and consolidate for several months. The vertical position of the tube is adjusted so that the laser illuminates the sediment at a specific vertical distance below the top, $d$. A camera and optical fiber collect cross-polarized, backscattered light. (b) Scattered light intensity autocorrelations $g_2(\tau )-1$ measured at distances $d=0.5$ (circle dot), 1.3 (circled circle), 4.0 (circle), 4.9 (circle plus), and 8.5 cm (circleX) below the top of a sedimented emulsion held at $31.5°\mathrm{C}$ show clear separation between solidlike and fluidlike behaviors. (Inset) Droplets closer to the bottom of the sediment reach a stable plateau MSD, while droplets closer to the top are slowed by the crowding of their neighbors but continue to move.

Figure 2

Vertical depth and pressure dependence of sediment shear modulus. (a) Vertical depth dependence of shear moduli $G(d)$ inferred using DWS microrheology from the plateau values of $g_2(\tau )$ collected from a sediment held at $31.5°\mathrm{C}$. Colored symbols refer to moduli inferred from data presented in Fig. 1. The dashed line is a guide for the eye. (b) $G(d)$ measured for sediments held at $27°\mathrm{C}$ (square), $31.5°\mathrm{C}$ (circles), and $34.9°\mathrm{C}$ (filled triangle). (C) Osmotic pressure $\mathrm{\Pi }$ dependence of shear moduli shown in (b). Moduli measured at different temperatures collapse onto a single curve when $d$ is replaced by $\mathrm{\Pi }$, and show that $G(\mathrm{\Pi })\sim \mathrm{\Pi }$ when $\mathrm{\Pi }\gtrsim 2.5$ Pa, but tends towards zero below it. (Inset) Expanded range of pressures and shear moduli.

Figure 3

Oscillatory rheology of weakly jammed emulsion. Strain amplitude dependence of the elastic ($G^{\prime }$, filled circle) and viscous ($G^{\prime \prime }$, circle) shear moduli of a homogeneous emulsion measured in a mechanical rheometer at 0.005 Hz. The yield strain ${\gamma }_y$ of this soft and fragile solid can be estimated as $\approx 0.1\%$ by the intersection of the dashed lines fit to the linear elastic and shear thinning regimes.

Figure 4

Magnetic densitometry of a sedimented emulsion. The vertical depth dependence of droplet volume fraction $\varphi (d)$ of a sediment composed of oil droplets in ${\mathrm{D}}_2\mathrm{O}$ measured with a NMR spectrometer programmed to act as a high-resolution densitometer. (a) Simplified form of the spin- and gradient-echo pulse sequence used for proton imaging. (b) Osmotic pressure dependence of oil volume fraction $\varphi (\mathrm{\Pi })$ computed from $\varphi (d)$ (see text). The dashed red line is proportional to $(\mathrm{\Pi }R/{\varphi }_c\sigma )$. The dashed blue line is a linear fit to $\varphi (\mathrm{\Pi })$ for small pressures. The dashed black curve is a fit to Eq. (5). (Inset) Direct measurement of $\varphi (d)$.