Phase Field Models with Non-Conserving Kinetics

Abstract

In this chapter, we present a mathematical analysis of the dynamics of phase transitions with a non-conserved order parameter e (Model A phase transitions). In this connection, we refer to Section 4.4, where a mathematical model for such phase transitions has been developed that explains the possible occurrence of an accompanying hysteresis in the framework of Landau-Ginzburg theory. The approach of Section 4.4 led to

$$ \frac{{\partial e}}{{\partial t}} = K(e,T)\;\left( {div(\frac{{\gamma (e,T)\nabla e}}{T}) - \frac{1}{{2T}}\;\frac{{\partial \gamma }}{{\partial e}}(e,T)\;|\nabla e{|^2} - \frac{1}{T}\;\frac{{\partial F}}{{\partial e}}(e,T)} \right) $$

((0.1))

as the kinetic equation governing the evolution of the order parameter e, while the balance of internal energy was given by

$$ \frac{\partial }{{\partial t}}\left( {F(e,T) - T\frac{{\partial F}}{{\partial T}}(e,T) + \frac{1}{2}(\gamma (e,T) - T\frac{{\partial \gamma }}{{\partial T}}(e,T))|\nabla e{|^2}} \right) + div\left( {k(e,T)\nabla (\frac{1}{T})} \right) = g $$

((0.2))

.