... of the form Q od 2 A R ( T − T c ) = ln ⁡ t − Δ H * R T + Δ S * R − ln ⁡ τ 0 . Values of A = −2.1 × 10 −5 , T c = 2215 °C, Δ H * = 124 ± ~15 kcal/mol, and τ 0 exp(−Δ S */R) = 9 × 10 −21 s were obtained. This type of rate law can be understood in terms of solutions to the Ginzburg-Landau rate equation...

... in the linear regime between driving force, ∂G ord /∂Q̄, and ordering rate, dQ̄/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...

...F igure 11. Thermodynamic models: thin solid line is a fit to our data using O’Neill and Navrotsky (1983) model, and the dashed line is the Landau fit. Thick solid line is the calculated non-equilibrium behavior using the Ginzburg-Landau rate law with the thermodynamic driving force given...

... in a significant temperature interval, or whether the data may consistently be described by a mean-field Landau model for all temperatures T < T C . Ginzburg et al. (1987) have shown that if the traditional way to fit a power law to the temperature dependence of the order parameter is followed...

... by O’Neill (1994) in terms of the Mueller rate law and by Harrison and Putnis (1999) in terms of the Ginzburg-Landau rate law. The Mueller rate law is a special case of the more general Ginzburg-Landau rate law, but it only applies to homogeneous ordering. However, the Ginzburg-Landau rate law is rather...

... exchange equilibrium was achieved. Calculations of K were performed using the Ginzburg-Landau equation, the kinetic model of Mueller, and the mathematical formulation proposed by Sha and Chappell (1996) to check the responses of various kinetic models for a Ca-rich pyroxene on the same data set...

... b/c ratio ( Ginzburg et al. 1987 , Redfern 1992 ). Figure 5. The dependence of the critical exponent β on the Landau coefficients. The critical exponents were calculated by least squares fitting of an equilibrium order parameter evolution from a Landau potential. The transition...

...Figure 6. ( a ) Timescales for Al/Mg-inversion on the merged layer (referred as M-layer: T Mg I − t + O Al I − t ) within the twinned region. The time taken for inversion is calculated by integrating the Ginzburg-Landau rate law at different temperatures under...

... behavior were analyzed using a Ginzburg-Landau model giving activation energies for Mg-Ni exchange between M1 and M2 for samples of composition Mg 1.6 Ni 0.4 SiO 4 and Mg 1.0 Ni 1.0 SiO 4 of 145 ± 5 and 160 ± 5 kJ/mol, respectively. The model also shows that the characteristic time scale for re...

... in a simple linear fashion with dopant concentration. This supposition can be rationalized by a Landau-Ginzburg excess Gibbs free-energy expansion ( Salje et al. 1991 ). A simple second order phase transition for phase A with the regular free energy expression (4) \batchmode \documentclass[fleqn,10pt...

.... As the temperature increases and approaches the transition point, the transitional areas between pericline twins becomes more and more diffuse. This is in complete agreement with the predictions of the continuum Landau-Ginzburg theory, according to which the width of a domain wall increases with temperature...

... and kinetics of Al/Si ordering in anorthite, I: Incommensurate structure and domain coarsening . American Mineralogist , 76 , 1110 – 1119 . Carpenter , M.A. ( 1991b ) Mechanisms and kinetics of Al/Si ordering in anorthite, II: Energetics and a Ginzburg-Landau rate law . American Mineralogist...

... stability of polytypes ( Sassi et al., 1994 ). Redfern et al. (1997) got insights into the mechanisms governing the cation order-disorder reactions in FeMn and MgMn olivines, at high-temperature conditions, and gave an interpretation based upon the Landau-Ginzburg expansion of the associated free...

... et al. 1994 ), and formation of microstructures during cation exchange processes can be rationalized, in a Landau-Ginzburg approach in which solids are treated as elastic continua , by taking into account local correlation of the order or disorder events. A theoretical interpretation of spatial...

... in anorthite: I. Incommensurate structure and domain coarsening . American Mineralogist , 76 , 1110 – 1119 . Carpenter , M.A. ( 1991b ) Mechanisms and kinetics of Al,Si ordering in anorthite: II. Energetics and a Ginzburg-Landau rate law . American Mineralogist , 76 , 1120 – 1133...

... towards the equilibrium order-disorder line. Using Ginzburg-Landau theory, which relates the driving force for ordering to the rate of change of order, one can obtain a kinetic description of the expected t-T- η pathway which relates to the thermodynamic description of the non-convergent disordering...

... and domain coarsening . American Mineralogist , 76 , 1110 – 1119 . Carpenter , M.A. ( 1991b ) Mechanisms and kinetics of Al-Si ordering in anorthite: II. Energetics and a Ginzburg-Landau rate law . American Mineralogist , 76 , 1120 – 1133 . Carpenter , M.A. ( 1992 ) Equilibrium...

... properties in a first approximation. Hooke’s law (a material equation connecting stresses with strains) is used with additional terms proportional to strain derivatives; the factors of proportionality are components of the gyration tensor b, the tensor of fifth rank, invariant with respect to a group...