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Rate equations for non-convergent order-disorder processes - a review and application to orthopyroxene

Institut für Mineralogie, Westfälische Wilhelms-Universität Münster, Corrensstr. 24, D-48149 Münster, Germany

^* kroll{at}nwz.uni-muenster.de

Various rate equations for non-convergent order-disorder processes in minerals are currently in use in the literature. Their interrelations are investigated and their formal equivalences are worked out. Their performance in reproducing kinetic experiments and modelling ordering paths is compared using orthopyroxene as an example.

Macroscopic thermodynamic consideration of the entropy production in irreversible order-disorder processes yields for the variation of the order parameter, , with time, t,

(1)

Equation (1) is valid for quasi-static processes in the linear regime between driving force, G^ord/, and ordering rate, d/dt. Under these constraints the Ginzburg-Landau equation (Salje, 1988) which is based on microscopic arguments reduces to the same form.

Equation (1) can be elaborated along different routes. (1) The Gibbs energy due to ordering, G^ord, may be expressed in terms of classical or Landau formulations. From the reciprocal solution model a rather simple rate law is obtained in which the driving force relates to the difference between the logarithmic temporal and equilibrium distribution coefficients, K_D,

(2)

(2) Reaction rate theory provides another path to rate equations. The popular Mueller (1967, 1969) equation can be developed along two lines yielding formally different, but numerically identical results,

(3)

and

(4)

where ^E denotes equilibrium site occupancies and G_exch is the reaction Gibbs energy for order-disorder.

(3) Equations (3) and (4) yield new expressions when reformulated for processes close to equilibrium. From (4) one obtains a form that is similar to (2),

(5)

whereas (3) can be simplified by neglecting the quadratic term so that

(6)

This expression also results (a) when a second order Landau potential is inserted into (1), or (b) when the formalism of Sha & Chappell (1996a) for two-site multi-cation ordering is rewritten for the usual case of two-cation ordering. It is thus seen that rate equations are closely interrelated although they are derived from different starting points and appear in quite different formulations in the literature.

Inspection of the driving forces for ordering and disordering reveals that equations (2) and (5) predict disordering rates to be faster than ordering rates, whereas the opposite is predicted by equations (3) and (4), and no preference is expected from equation (6). Testing these predictions on literature data of kinetic experiments fails, however, because the data scatter is too large to allow distinction between the predictions.

Rate constants were calculated from equations (2)–(6) using published kinetic data (t). The composition and temperature dependences of the rate constants were fitted to an Arrhenius equation. The activation energy was found to vary quadratically with composition when fitting rate constants derived from the Mueller related equations (3)–(6), whereas a linear variation resulted for rate constants obtained from (2). From the quality of fit of the Arrhenius equations no decision could be made on the most appropriate rate equation. This failure may be related to both the close formal correspondences between the rate equations and a lack of consistency in the experimental rate constants.

However, when employing the rate equations in ordering path calculations to obtain cooling rates, differences turn up. They are largest for equations (2) and (3) amounting in the worst case to a factor of two in the rate of a slowly cooled compositionally intermediate orthopyroxene.

Key-words: rate equations, non-convergent ordering, orthopyroxene, cooling rate calculations.

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