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: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 x0 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)=xa lead to a rate law: ln<Q>=-(α + 1) ln t. 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 x0 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)=xa 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.
- Publication:
-
Physics and Chemistry of Minerals
- Pub Date:
- November 1988
- DOI:
- 10.1007/BF00203197
- Bibcode:
- 1988PCM....16..140S
- Keywords:
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- Taylor Expansion;
- Unify Theory;
- Laplace Transformation;
- Maxwell Distribution;
- Polynomial Distribution