Dynamic scaling theory for a tethered membrane in solution.

We present the dynamic scaling behavior for the specific viscosity and diffusion coefficient of a single membrane and membranes with nonzero concentration in solution. Starting from the membrane free energies, we derive their Langevin equations. The corresponding Kirkwood diffusion equation, describing the time evolution in configuration space, contains two kinds of time scales that are separated by the external dimension 4/(2-D) where D is the dimension of the internal space. These time scale separation behaviors depend strongly on the hydrodynamic screening effect. For a single membrane solution, we resolve the dynamic scaling exponents for the diffusion coefficient and intrinsic viscosity by the dimension reduction method. For a concentrated membrane solution, the effective excluded volume strength and draining parameter are introduced. The effective medium argument is applied to obtain a concentration dependent power law form for the specific viscosity and diffusion coefficient, whose results contribute to a fundamental understanding of membrane in solution and of hydrodynamic screening and excluded volume effects in many different solvents.