  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.
