Kinetic rate laws as derived from order parameter theory I: Theoretical concepts

Abstract

A theoretical concept is outlined, which links the kinetics of structural transformations with thermodynamic theories of structural phase transitions. Starting from Landau theory and Markovian processes, the general rate laws for crystals with long correlation lengths are derived. The rate laws in Ginzburg-Landau theory are

$$1{\text{n }}\Delta Q - 1{\text{n }}f\left( Q \right) \propto - \frac{t}{\tau }{\text{ for }}T \ll T_c {\text{ and }}T \gg T_c$$

and Q ²∝ for T ≈ T _c . The physical meaning of the time constant τ and the correction term f(Q) are explained. Fluctuations of the order parameter lead to damping behaviour with explicit dependence on the wavelength of the fluctuation wave and modulation-dependent variations of the lattice strain.

Lattice relaxations and activation processes are discussed. Typical rate laws are found to follow

$$\begin{gathered} \ln \Delta Q = rln\Delta t, \hfill \\ ln\frac{Q}{Q} + \frac{{1\varepsilon }}{{2k_B T}}\left( {Q^2 - Q_0^2 } \right) = \frac{{\Delta t}}{{\tau *}} \hfill \\ \end{gathered} $$

which leads for short time intervals to a linear rate law

$$\Delta Q \propto \Delta t$$

It is shown that linear terms in the Landau potential are equivalent to a logarithmic decay of the excess entropy ΔS ∝ ln Δt which is also expected to be the dominant rate law in field-induced pseudo-spin glasses:

$$\Delta Q \propto 1{\text{n }}\Delta t{\text{ and }}1{\text{n}}\left( {\Delta {\text{Q}} \cdot \Delta {\text{t}}} \right) = A{\text{ }}\Delta t + B$$

Fluctuations lead to spatially heterogeneous distributions of the order parameter. A two phase field is found in this case where the nucleation energy is overcome by fluctuation processes.

Random fields, arising, for example, from lattice imperfections, lead also to spacially inhomogeneous material. The dominant microstructure is the lattice modulation mostly in the form of a cross hatched pattern (tweed) but also in the form of incommensurate modulations.