A. R. Allnatt

A self-consistent theory of matter transport in a random lattice gas and some simulation results

Abstract Linear response formulae for the phenomenological atomic transport coefficients LXY of non-equilibrium thermodynamics are analysed for the randomlattice-gas model in which all configurations have the same energy. A new self-consistent decoupling of the associated hierarchy of kinetic equations for time-dependent correlation functions is employed. For a binary system with an atom jump-rate ratio of ten to one for the two components the agreement with Monte Carlo simulations is generally good for both tracer and non-tracer coefficients and is more complete than for earlier theories. New Monte Carlo simulation results for non-tracer coefficients in a simple cubic binary random lattice gas are presented.

Computer simulation of phenomenological coefficients for atom transport in a random alloy

Abstract Linear‐response theory provides general formulae for the phenomenological coefficients for atom transport defined by non-equilibrium thermodynamics. These formulae can be written in forms which are simple generalizations of the Einstein formula for tracer diffusion; they can therefore be calculated in a similar manner to tracer diffusion coefficients by Monte Carlo simulation of a system at thermodynamic equilibrium. The method is applied to calculate the phenomenological coefficients LAA, LBB , and LAB =LBA for a random f .c.c. alloy of components A and B with a very small concentration of vacancies. The diffusion coefficients of A and B are simply related to these coefficients. Compared with the simulation calculation of tracer correlation factors, larger numbers of vacancy jumps are needed to obtain acceptable statistics, particularly for the cross-coefficient LAB- The results are in agreement with the theory of Manning (1971) within the accuracy attained.

Exact linear relations between the phenomenological coefficients for matter transport in a random alloy

Abstract The general linear response expressions for isothermal matter transport simplify for the random alloy model with the vacancy mechanism because the jump rates are independent of the local configuration of atoms on the lattice. It is shown that in consequence there are n exact relations between the non-equilibrium thermodynamic phenomenological coefficients for isothermal transport in a random alloy of n atomic components in addition to the Onsager reciprocal relations. These new relations are satisfied by earlier approximate theories. They are advantageous in Monte Carlo simulations of the model.

Computer simulation study of the Manning relations and related approximations in a strictly regular solution model

Abstract The phenomenological coefficients and the tracer diffusion coefficients for a strictly regular solution model of a binary f.c.c. alloy with a very small vacancy content have been calculated by Monte Carlo simulations for four temperatures above the critical solution temperature over the complete concentration range. The results have been used to test the Manning relations, which express the phenomenological coefficients in terms of the tracer diffusion coefficients, and some related approximations. The Manning relations are not adequate to calculate intrinsic diffusion coefficients over the complete composition range.

Approximate expressions for the phenomenological coefficients of a binary alloy with short range order

Approximate expressions are derived for the phenomenological coefficients for vacancy-mechanism matter transport in a nearest-neighbour interaction lattice gas model of a concentrated binary alloy using the Kikuchi-Sato jump frequency model. These expressions are constructed from the leading coefficients (moments) in the Taylor series expansions in powers of time of the time correlation functions working within the Mori continued fraction representation. Numerical results are compared with earlier Monte Carlo simulations and with the predictions of the path probability method. Comparison with new Monte Carlo results shows that the approximate time correlation functions are good at short times but are too small at relatively long times.

Random walk analysis of the collective cosine formulation of correlation effects in atom transport in a concentrated alloy

Abstract The phenomenological coefficients Lij, which are defined in non-equilibrium thermodynamics and which characterize atom transport near thermodynamic equilibrium, can be expanded in terms of so-called collective cosines. A typical such quantity is (cos θji (n)) which is defined as the average of the cosine of the angle between the direction of an initial jump of an atom of species j and a final jump of an atom of species i when there are exactly n jumps of atoms of species i following the initial j atom jump and n = 1,2,3,…New exact relations between the four quantities (cos θji (n)), for i,j = A,B and arbitrary n are derived for a binary random alloy of two atomic species (A,B) with transport by a very small concentration of vacancies. A new simplified formula for the off-diagonal coefficient Lij,j ≠ i, is also derived in terms of these collective cosines. The approximate calculation of the collective cosines by enumeration of random walks is examined; results for n = 1,2 and 3 are in very good ag...

Collective and tracer diffusion kinetics in the ternary random alloy

In this study, collective and tracer diffusion kinetics is addressed for the ternary random alloy. A formal solution from the self-consistent theory of Moleko et al (Moleko L K, Allnatt A R and Allnatt E L 1989 Phil. Mag. A 59 141) is derived for collective diffusion and compared with the corresponding solution for the binary random alloy. Tracer diffusion in the ternary alloy is treated from the perspective of a special case of the quaternary random alloy. Results from Monte Carlo calculations for tracer and collective correlation factors (for the bcc ternary random alloy) are found to be in excellent agreement with this self-consistent theory but in only semi-quantitative agreement with the earlier theory of Manning (Manning J R 1971 Phys. Rev. B 4 1111).

Self-consistent kinetic theory and the Manning theory of matter transport in a random alloy

Abstract A method for the self-consistent decoupling of many-body kinetic equations was applied by Holdsworth and Elliott (1986) to the study of the incoherent scattering function and tracer diffusion coefficients arising from vacancy transport in a multicomponent random lattice gas. The principle of this method is applied here to calculate from linear response formulae the phenomenological atomic transport coefficients Lxy as defined in non-equilibrium thermodynamics. The results at very low vacancy concentrations are the same as those of the Manning (1971) theory of the random alloy but results at higher vacancy constants cannot be obtained without some additional assumption.

Atomic transport in a random alloy at arbitrary vacancy concentration

Abstract For a random alloy with a very small vacancy concentration cv the theory of Manning (1971, Phys. Rev. B, 4, 1111) provides accurate predictions of all of the phenomenological atomic transport coefficients Lxy of non-equilibrium thermodynamics. A natural extension of these results to arbitrary vacancy content is sought. An earlier decoupling scheme for the hierarchy of time-correlation functions encountered in the analysis of the linear-response expression for Lxy is reformulated and extended to arbitrary cv. The final results generalize all Mannings results in a very simple manner and reduce to them for cv ←0. The predicted tracer correlation factors at large cv- are in poorer agreement with simulation results than those for cv←0. These results suggest that a simple and natural generalization of the complete Manning structure of relationships between the various transport coefficients of comparable accuracy to the original theory for cv←0 may not be readily found.

Diffusion kinetics in dilute binary alloys with the h.c.p. crystal structure

In this paper, an extended version of the matrix method is derived in order to address diffusion kinetics for the full anisotropic three-dimensional h.c.p. structure. It is shown that the diffusion anisotropy can be properly addressed with a model of 13 atom-vacancy frequencies which is an extended version of the well-known 5-frequency model for the f.c.c. lattice. Both tracer and phenomenological diffusion coefficients are calculated using this new approach. Extended Monte Carlo simulations are performed in order to cross-check some of the results of the matrix method. Applications of the proposed model to experimental diffusion data are discussed.