Kinetic rate laws as derived from order parameter theory II: Interpretation of experimental data by laplace-transformation, the relaxation spectrum, and kinetic gradient coupling between two order parameters

Abstract

A unifying theory of kinetic rate laws, based on order parameter theory, is presented. The time evolution of the average order parameter is described by $$\langle Q\rangle \propto \smallint P(x)e^{^{^{^{^{^{^{ - xt} } } } } } } dx = L(P)$$ where t is the time, x is the effective inverse susceptibility, and L indicates the Laplace transformation. The probability function P(x) can be determined from experimental data by inverse Laplace transformation. Five models are presented: (a) Polynomial distributions of P(x) lead to Taylor expansions of 〈Q〉 as $$\langle Q\rangle = \frac{{\rho _1 }}{t} + \frac{{\rho _2 }}{{t^2 }} + ...$$ (b) Gaussian distributions (e.g. due to defects) lead to a rate law $$\langle Q\rangle = e^{ - x_0 t} e^{^{^{^{^{\frac{1}{2}\Gamma t^2 } } } } } erfc\left( {\sqrt {\frac{\Gamma }{2}} t} \right)$$ where x0 is the most probable inverse time constant, Γ is the Gaussian line width and erfc is the complement error integral. (c) Maxwell distributions of P are equivalent to the rate law 〈Q〉∝e−k√t. (d) Pseudo spin glasses possess a logarithmic rate law 〈Q〉∝lnt. (e) Power laws with P(x)=xa lead to a rate law: ln〈Q〉=-(α + 1) ln t.