Closed, Two Dimensional Surface Dynamics

We present dynamic equations for two dimensional closed surfaces and analytically solve it for some simplified cases. We derive final equations for surface normal motions by two different ways. The solution of the equations of motions in normal direction indicates that any closed, two dimensional, homogeneous surface with time invariable surface energy density adopts constant mean curvature shape when it comes in equilibrium with environment. In addition, we show that the shape equation is an approximate solution to our equation of motion in the normal direction and is valid for stationary or near to stationary shapes. As an example, we apply the formalism to analyze equilibrium shapes of micelles and explain why they adopt spherical, lamellar, and cylindrical shapes. Theoretical calculation for micellar optimal radius is in good agreement with all atom simulations and experiments.