The modified Cahn-Hilliard equation on general surfaces

Diblock copolymer melts are of great interest in industry today. Their ability to naturally self-assemble at a microscopic scale is a great asset in manufacturing complex materials such as plastics, textiles, and integrated circuits. The modified Cahn-Hilliard (mCH) equation studied in this thesis is a partial differential equation (PDE) based mathematical model for diblock copolymer self-assembly. This is a stiff, non-linear, fourth-order parabolic PDE that presents some challenges when numerically solving it on general surfaces. This thesis presents several methods employed in overcoming these difficulties and produces results that support the accuracy of these methods. Our software uses the Closest Point Method (CPM), a central feature of which is geometric flexibility. This allows us to compute on simple analytically defined shapes, such as the sphere, as well as on complex shapes defined by triangulation with no modification to the code.