Abstract
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.
Highlights
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 mechanical properties of emulsions are controlled by two seemingly irreconcilable energy scales: Dilute emulsions like cream and vinaigrette are fluids with osmotic moduli proportional to the ratio of thermal energy, kBT, to droplet volume, 4πR3/3, while compressed emulsions like mayonnaise are jammed solids composed of droplets that are pressed together into amorphous, elastic packings with elastic moduli proportional to the ratio of interfacial tension, σ, to droplet size, R [1, 2]
The magnitudes of the shear modulus, G, and γy are determined by the strength of the contacts between abutting droplets, which decrease with decreasing osmotic pressure, Π: Reducing Π makes the solid softer and more fragile, and this direct link between Π and G makes it possible to vary the shear modulus of a compressed emulsion over several orders of magnitude [1,2,3,4,5,6]
Summary
The elastic moduli of the sintered bead pack and PTFE block and the size of their scattering structures are too large for the amplitude of their thermally induced vibrations to exceed picometer scales. For λ = 532 nm and γ ≈ 2, this condition limits resolvable displacements to 50 nm The value of this RMSD is connected to the shear modulus of the emulsion by the equipartition relation of elastic energy (2d), which is the long lag-time limiting form of the the Generalized. Measurements performed on aging samples are not stationary, ; the average described in (2a) should not be done over the waiting time, t, but over independent speckles. With τ = 104, and use (2a-c) to compute ∆r(τ = 104) , we can equate the the shear modulus of the reference sample with that of the aged sample by choosing γ = 2.27 This value of γ is within the commonly observed range for this parameter; but, given the uncertainty of the measurement, we choose to fix γ ≡ 2 to compute the shear moduli of the quiescent emulsion. The difference between shear moduli computed with either value of γ is less than 30%, and rescales G(Π) by a constant factor
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