  The time-dependent Ginzburg-Landau (TDGL) equation for a single component
non-conservative structural order parameter is used to study the
spatio-temporal evolution of a second phase in the vicinity of an edge
dislocation in an elastic crystalline solid. A symmetric Landau potential of
sixth-order is employed. Dislocation field and elasticity modify the
second-order and fourth-order coefficients of the Landau polynomial,
respectively, where the former makes the coefficient singular at the origin.
The TDGL equation is solved numerically using a finite volume method, where a
wide range of parameter sets is explored. Computations are made for
temperatures both above and below the transition temperature of a defect-free
crystal Tc0. In both cases, the effects of the elastic properties of the solid
and the strength of interaction between the order parameter and the elastic
displacement field are examined. If the system is quenched below Tc0, steady
state is first reached on the compressive side of the dislocation. On the
tensile side, the growth is held back. The effect of thermal noise term in the
TDGL equation is studied. We find that if the dislocation is introduced above
Tc0, thermal noise supports the nucleation of the second phase, and steady
state will be attained earlier than if the thermal noise were absent. For a
dislocation-free solid, we have compared our numerical computations for a
mean-field (spatially averaged) order parameter versus time with the late time
growth of the ensemble-averaged order parameter, calculated analytically, and
find that both results follow late time logistic curves.
