טכניון מכון טכנולוגי לישראל
הטכניון מכון טכנולוגי לישראל - בית הספר ללימודי מוסמכים  
M.Sc Thesis
M.Sc StudentKrush-Bram Marina
SubjectAn Integro-Differential Model for Phase Change with
Memory: The One Dimensional Theory
DepartmentDepartment of Applied Mathematics
Supervisor Professor Amy Novick-Cohen


Abstract

In the present work we analyzed a mathematical model describing solid-liquid phase transitions with memory, which extends the classical phase field equations for solid-liquid phase transitions to an integro-differential system for both the temperature field and the order parameter. The order parameter here is assumed to be a smooth function which is equal to -1 in the solid and +1 in the liquid and which varies rapidly in the neighborhood of interfaces.

We focused on an equation for the order parameter which can be obtained as a particular case when the influence of latent heat and entropy can be neglected. Under these conditions the equation for the order parameter constitutes a memory version of the Allen-Cahn equation.

We studied domain walls evolution in one dimension for three groups of memory kernels: (i) positive type with sufficiently rapid decay, (ii) Abel's type, (iii) Jeffreys' type. We obtained a closed system (to leading order) of integro-differential equations for the motion of the domain walls for each group of memory kernels.

We studied the behavior of an isolated domain for the first and third groups. For the first group finite time collapse was proven. Asymptotic solutions for the first and third groups were found. Numerical solutions for simple kernels from the first group were found. The asymptotic solutions were seen to agree well with the numerical solutions with 10-3 accuracy for all parameters values tested.