Abstract

Binary phase polymer solution is interesting in that they expresses double-well local energy behavior, which means phase separation is preferred when condition is right. It is a feature that has been used to fabricate functional polymeric materials such as PDLC films for electro-optical devices (e.g. flat-panel displays and switchable windows). A uniformly mixed solution may be in one of three state: unstable, stable, or metastable. If the solution is unstable, then phase separation is spontaneous and proceeds by spinodal decomposition. If the solution is metastable, then the solution must overcome certain activation barrier for phase separation to proceed spontaneously. The activation barrier is usually the thermal noise or the fluctuation created by some external influence. This mechanism is called nucleation-and-growth. Manipulating morphology of phase separation has been of some great research interest because of its practical use. While spinodal decomposition has been well-studied, there are several other methods to further control morphology. For this thesis, the following methods are considered: double quench, anisotropic quenching with varying temperature or polymerization, surface-directed wetting, and concentration gradient. The methods are carried out within metastable or unstable regions or both. To numerically model, Cahn-Hilliard theory and FloryHuggins’ theory are used. This thesis is to also demonstrate that, present numerical method is very efficient and can work on complex geometry.

Highlights

  • Multiphase polymer solution has been known to express different peculiar behaviors when submitted to certain conditions

  • If phase separation from uniform state is spontaneous, i.e. it occurs without any external influence, it is said to be in unstable region

  • In order to initiate phase separation, several methods are used: 1. Double quench: Solution is first quenched into unstable region temperature or polymerization is changed so that the solution is in metastable region

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Summary

Introduction

Multiphase polymer solution has been known to express different peculiar behaviors when submitted to certain conditions. The stage the solution is in at specific time frames is identified and simulation is not always started from uniform state (e.g. Second quench for double quench), nonlinearity of Cahn-Hilliard equation must be accounted. Tim Davis [5] has done outstanding researches in large sparse matrices that arise frequently in science, engineering, and mathematics His algorithm is featured in many applications including Google Map. With the advance of powerful technology and extremely efficient algorithms, simulation time has been reduced from days and weeks into less than an hour for very similar model and method, and performed simulations on a standard home PC rather than high-end scientific computer with ever higher resolution.

Chapter 2 Literature Review
Chapter 3 Theoretical background
Flory-Huggins theory
Cahn-Hilliard theory and the dynamics of binary polymer solution
Double quench
Gradient of temperature, polymerization, and concentration
Wetting
Some notable theories
Chapter 4 Mathematical modeling
Galerkin finite element method (GFEM)
Governing equations, assumptions, and non-dimensionlizations
Implementation
On algorithm featuring parallel computation and SuiteSparse
Fourier transform and Inverse Fourier transform
Results and Discussions
Concentration gradient
Unstable first quench and unstable second quench
Unstable first quench, metastable second quench
Temperature gradient
Polymerization gradient
Biaxial gradient
Spinodal decomposition on the complex domain with curved boundaries
Conclusion
Solution to linearized Cahn-Hilliard and determining characteristic frequency in all dimensions
Flory-Huggins equation
Theory of Cahn-Hilliard
Dynamics of polymer mixture
Change of energy in binary polymer mixture
Basis functions in 1-d
Bicubic Hermitian basis functions in 2-d
Hermitian basis functions with serendipity in 3-d
A.10 Gaussian quadrature and isoparametric mapping in 1-d
A.11 Gaussian quadrature in 2-d and 3-d
A.12 Isoparametric mapping of global basis functions from local basis functions
A.13 Domain with curved boundaries and irregularly shaped mesh

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