Kinetics order parameter models

By virtue of their simple stnicture, some properties of continuum models can be solved analytically in a mean field approxunation. The phase behaviour interfacial properties and the wetting properties have been explored. The effect of fluctuations is hrvestigated in Monte Carlo simulations as well as non-equilibrium phenomena (e.g., phase separation kinetics). Extensions of this one-order-parameter model are described in the review by Gompper and Schick [76]. A very interesting feature of tiiese models is that effective quantities of the interface—like the interfacial tension and the bending moduli—can be expressed as a fiinctional of the order parameter profiles across an interface [78]. These quantities can then be used as input for an even more coarse-grained description. 

Here we explain how the thermodynamic and kinetic anomalies can be explained in the same framework of our two-order-parameter model. 

In the past there have been attempts to treat the kinetics of glass formers in terms of single ordering parameter models. (13) For experiments which measure volume, it is convenient to define a new parameter called delta as the normalized departure from equilibrium. 

Three specific predictions of the order parameter model (either in its thermodynamic or in its kinetic form) have generated much controversy and confusion and the situation has been summarized (10). First, the model predicts two relationships between the pressure dependence of Tg and discontinuities in... 

Similar to generalized mass-action models, lin-log kinetics provide a concise description of biochemical networks and are amenable to an analytic solution, albeit without sacrificing the interpretability of parameters. Note that lin-log kinetics are already written in term of a reference state v° and S°. To obtain an approximate kinetic model, it is thus sometimes suggested to choose the reference elasticities according to simple heuristic principles [85, 89]. For example, Visser et al. [85] report acceptable result also for the power-law formalism when setting the elasticities (kinetic orders) equal to the stoichiometric coefficients and fitting the values for allosteric effectors to experimental data. 

We review Monte Carlo calculations of phase transitions and ordering behavior in lattice gas models of adsorbed layers on surfaces. The technical aspects of Monte Carlo methods are briefly summarized and results for a wide variety of models are described. Included are calculations of internal energies and order parameters for these models as a function of temperature and coverage along with adsorption isotherms and dynamic quantities such as self-diffusion constants. We also show results which are applicable to the interpretation of experimental data on physical systems such as H on Pd(lOO) and H on Fe(110). Other studies which are presented address fundamental theoretical questions about the nature of phase transitions in a two-dimensional geometry such as the existence of Kosterlitz-Thouless transitions or the nature of dynamic critical exponents. Lastly, we briefly mention multilayer adsorption and wetting phenomena and touch on the kinetics of domain growth at surfaces. 

If in addition to a thermodynamic driving force, a system has kinetic mechanisms available to produce a phase transformation (e.g., diffusion or atomic structural relaxation), the rate and characteristics of phase transformations can be modeled through combinations of their cause (thermodynamic driving forces) and their kinetic mechanisms. Analysis begins with identification of parameters (i.e., order parameters) that characterize the internal variations in state that accompany the transformation. For example, site fraction and magnetization can serve as order parameters for a ferromagnetic crystalline phase. 

The Allen-Cahn equation applies to the kinetics of a diffuse-interface model for a nonconserved order parameter—for example, the order-disorder parameter r](f,t)... 

Numerical models of conserved order-parameter evolution and of nonconserved order-parameter evolution produce simulations that capture many aspects of observed microstructural evolution. These equations, as derived from variational principles, constitute the phase-field method [9]. The phase-field method depends on models for the homogeneous free-energy density for one or more order parameters, kinetic assumptions for each order-parameter field (i.e., conserved order parameters leading to a Cahn-Hilliard kinetic equation), model parameters for the gradient-energy coefficients, subsidiary equations for any other fields such as heat flow, and trustworthy numerical implementation. 

The solid lines in Figs. 25 to 28 are the result of model predictions of metal deposition based on porphyrin reaction pathways. Curves are generated using intrinsic kinetic rate parameters and effective diffusion coefficients for the metal species on the order of 10 6 cm2/sec. These values are similar to diffusion coefficients measured in the independent studies referenced. 

Parameter Estimates, Standard Errors, Root Mean Square Errors, and Correlation Coefficients for First-Order Kinetic and Irreversible Models for Various Columns... 

The limitation of the five parameter model necessitated the development of a model that treats the overall reaction order as a separate, independent parameter. Through the relationship of the general equations for the first order (n = 1) kinetics ... 

We use here a simplified kinetic model of cracking reactions in order to illustrate the role of secondary cracking steps on product distribution. More detailed and rigorous models are available but the additional rigor is not essential to describe the concepts that we illustrate here. We assume that sites within transport-limited pellets (all kinetic-transport parameters as in Figs. 16, 17, and 19) catalyze the cracking of olefins with a probability given by... 

The homogeneous tar conversion is described by an empirical model. The results show that the simple one-tump, single first-order reaction model describes the depletion of the gravimetric tar satisfactory well. The comparison with literature studies shows a good agreement of the determined kinetic parameters. 

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