Phase Behavior Of Rods In Cylinder And Slit Confinement

In this paper,we consider the rods in cylinder and parallel walls.We use selfconsistent field theory to simulate the phase behaviors of rods.Our question is a four-dimensional question,which includes 1 dimension in space,2 dimension of orientation and 1 dimension of molecule length.Different methods are adopted for cylinder and slit during numerical simulation.We use Runge-Kutta method and upwind method to cylinder and adopt forward Euler method and upwind method for slit in order to solve the diffusion equation.In numerical integration,Simpson integral formula is used in cylinder and Trapezoidal formula is used in slit.In order to find the Saddle point in equilibrium,we chose Picard mixing method as an iterative method to update filed function of the two medium.Due to the existence of cylinder and plate walls,rigid molecules will be more arranged near walls.In order to better perform the distribution of molecules,we use nonuniform grid in spatial direction and use uniform grid in other directions.Three phase states were obtained from the numerical solution by self-consistent filed theory: uniaxial,biaxial and condense.This paper displays three states at first,and then,presents two phase diagrams of rod molecules,one in cylinder and another in slit.And on the basis of the three phase states,we discuss how do biaxial state comes in to condense state in cylinder.As grid number and the degree of uniformity of the division of spatial both effect the numerical solution,we give a few examples in cylinder to analysis which point number and is suitable for fixed parameters.At last,we analysis the change of error and energy goes during the iteration of rod molecules in cylinder.