Membranes self-assembled from rod-like molecules in aqueous suspension have attracted much attention recently. Depletion force is the key factor to make the molecules self-assemble. The membranes formed can appear in nematic and smectic-\textit{A} phase when the concentration of the molecules themselves and the concentration of the depletant in the solution are tuned. The latter is effectively varying the strength of the depletion force. Inside the smectic-\textit{A} phase, different shapes of the membranes are observed as flat disks, twisted ribbons and 'cookie disks'. The major goal of this work is to establish a theory to explain the appearance of these shapes, discuss their relative stabilities and study the phase transitions. In order to achieve this goal, the model free energy of the monolayer is written down by adding two parts together. The first of which is the de Gennes free energy describing the phase transition between nematic and smectic-\textit{A} phases and the cost of deformation in the director field. The second is the Helfrich free energy related to the curvature of the membranes. The model is applied to different shapes of the membranes and the corresponding Euler-Lagrange equations are solved either analytically or numerically. The numerical methods used include a partial differential equation solver in MATLAB and a Monte Carlo (MC) method. Good results are obtained about the relative stabilities of the flat disks, twisted ribbons and 'cookie disks'. First order phase transitions are found to lie between the flat disks and twisted ribbons when the molecules are chiral and also between the flat disks and 'cookie disks' when the molecules are of opposite chiralities are mixed, which effectively makes the membrane achiral. Our theoretical results qualitatively agree with the experimental observations.
Tu, Hao,
"Theoretical and Numerical Studies of Entropy-Driven Self-Assembled Smectic-\textit{A} Monolayer Membranes"
(2013).
Physics Theses and Dissertations.
Brown Digital Repository. Brown University Library.
https://doi.org/10.7301/Z0319T6C
