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 |
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| Authors: | E Salje B Wrack |
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| Institution: | 1. Department of Earth Sciences, Downing Street, CB23EQ, Cambridge, UK
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| 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: - Polynomial distributions of P(x) lead to Taylor expansions of 〈Q〉 as $$\langle Q\rangle = \frac{{\rho _1 }}{t} + \frac{{\rho _2 }}{{t^2 }} + ...$$
- 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 x 0 is the most probable inverse time constant, Γ is the Gaussian line width and erfc is the complement error integral.
- Maxwell distributions of P are equivalent to the rate law 〈Q〉∝e?k√t .
- Pseudo spin glasses possess a logarithmic rate law 〈Q〉∝lnt.
- Power laws with P(x)=x a lead to a rate law: ln〈Q〉=-(α + 1) ln t.
The power spectra of Q are shown for Gaussian distributions and pseudo spin glasses. The mechanism of kinetic gradient coupling between two order parameters is evaluated. |
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