Mathematical modelling of complex dynamics

Abstract

Soft materials have a wide range of applications, which include the production of masks for nano–lithography, the separation of membranes with nano–pores, and the preparation of nano–size structures for electronic devices. Self–organization in soft matter is a primary mechanism for the formation of structure. Block copolymers are long chain molecules composed of several different polymer blocks covalently bonded into a single macromolecule, which belong to an important class of soft materials which can self–assemble into different nano–structures due to their natural ability to microphase separate. Experimental and theoretical studies of block copolymers are quite challenging and, without computer simulations, it is difficult and problematic to analyse modern experiments. The Cell Dynamics Simulation (CDS) technique is a fast and accurate computational technique, which has been used to investigate block copolymers.
The stability has been analysed by making use of different discrete Laplacian operators using well–chosen time steps in CDS. This analysis offers stability conditions for phase–field, based on the Cahn–Hilliard Cook (CHC) equations of which CDS is the finite difference approximation. To overcome grid related artefacts (discretization errors) in the computational grid, the study has been done for employing an isotropic Laplacian operator in the CDS framework. Several 2D and 3D discrete Laplacians have been quantitatively compared for their isotropy. The novel 2D 9–point BV(D2Q9) isotropic stencil operators have been derived from the B.A.C. van Vlimmeren method and their isotropy measure has been determined optimally better than other exiting 2D 9–point discrete Laplacian operators. Overall, the stencils in 9–point family Laplacians in 2D and the 19–point stencil operators in 3D have been found to be optimal in terms of isotropy and time step stability.
Considerable implementation of Laplacians with good isotropy has played an important role in achieving a proper structure factor in modelling methods of block copolymers.
The novel models have been developed by implementing CDS via more stable implicit methods, including backward Euler, Crank–Nicolson (CN) and Alternating Direction Implicit (ADI) methods. The CN scheme were implemented for both one order and two order parameter systems in CDS and successful results were obtained compared to forward Euler method. Due to the implementation of implicit methods, the CDS has achieved second–order accuracy both in time and space and it has become stronger, robust and more stable technique for simulation of the phase–separation phenomena in soft materials.