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

Herbert Kroll
Rate equations for non-convergent order-disorder processes; a review and application to orthopyroxene
European Journal of Mineralogy (February 2003) 15 (1): 7-19


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, Q, with time, t, dQ/dt = -k/RT[delta G (super ord) /delta Q] (sub p,T) (k = rate constant). (1) Equation (1) is valid for quasi-static processes in the linear regime between driving force delta G (super ord) /delta Q, and ordering rate, dQ/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 (super 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 (sub D) , dQ/dt = k[1/2(lnK (sub D) -lnK (super E) (sub D) )] (E = equilibrium). (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, dQ/dt = k (sub dis) [-A . (Q-Q (super E) )-B . (Q-Q (super E) ) (super 2) ] (A, B = constants) (3) and dQ/dt = k (sub dis) [-X (sub Fe) (super M2,E) X (sub Mg) (super M1,E) (1-exp(Delta G (sub exch) /RT))] (4) where X (super E) denotes equilibrium site occupancies and Delta G (sub 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), dQ/dt = k (sub dis) [X (sub Fe) (super M2,E) X (sub Mg) (super M1,E) (lnK (sub D) -lnK (sub D) (super E) )], (5) whereas (3) can be simplified by neglecting the quadratic term so that dQ/dt = k (sub dis) [-A(Q-QE)] (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 Q(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.

ISSN: 0935-1221
EISSN: 1617-4011
Serial Title: European Journal of Mineralogy
Serial Volume: 15
Serial Issue: 1
Title: Rate equations for non-convergent order-disorder processes; a review and application to orthopyroxene
Author(s): Kroll, Herbert
Affiliation: Westfalische Wilhelms-Universitat Munster, Institut fur Mineralogie, Munster, Federal Republic of Germany
Pages: 7-19
Published: 200302
Text Language: English
Publisher: Schweizerbart'sche Verlagsbuchhandlung (Naegele u. Obermiller), Stuttgart, Federal Republic of Germany
References: 39
Accession Number: 2003-025992
Categories: Mineralogy of silicates
Document Type: Serial
Bibliographic Level: Analytic
Illustration Description: illus. incl. 3 tables
Country of Publication: Germany
Secondary Affiliation: GeoRef, Copyright 2020, American Geosciences Institute.
Update Code: 200309
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