Apparent and effective viscosity of 2D suspensions

Many lipid and surfactant monolayers exhibit coexistence of discrete condensed domains in a continuous 2D liquid. These domains are typically more resistant to deformation than the surrounding liquid, and therefore alter the effective rheology of the monolayer. Such a system is well approximated by a 2D liquid containing rigid inclusions, with the added caveat that the inclusions grow at the expense of the continuous phase upon compression. We look at the dilatational rheology of such a monolayer. When the 2D liquid is surface inviscid, we use a simple linear model for the exchange flux between the condensed and liquid phases to show that the monolayer behaves like a Maxwell fluid. Notably, our model suggests a rate-dependent elastic modulus during phase coexistence and an apparent surface dilatational viscosity, both in agreement with recent experiments. On the other hand, when the continuous phase is inherently surface viscous, the presence of rigid domains introduces an additional source of viscous dissipation. We calculate the hydrodynamic stress and effective surface viscosity in this case, and illustrate its implications in experimental systems.