In this thesis Monte Carlo techniques were applied to diffusional problems in material physics. The field of interest was substitutional diffusion in binary alloys. Diffusion is a process covering several time and length scales. On the microscopic scale of several Ångström and several femto-seconds the funda-mental diffusion event is described by atomic site exchanges. Whereas on the macroscopic level (sample sizes of cm and diffusion times of days or even weeks) diffusion manifests itself in the smearing of continuous concentration profiles. Due to the difference in the involved scales different concepts have been developed to give an appropriate description of the diffusional process on the scale investigated. Bridging the gap between macroscopic and micro-scopic description, i.e. calculation of the macromicro-scopic quantities describing diffusion out of the atomic details of the elementary jump processes, is the field of statistical mechanics. But because of the complexity of the involved systems exact solutions are possible only for few, simple systems. To gain results for models closer to physical reality the use of approximations in the calculations is inevitable. Since the impacts of these approximations is not always well understood it was the aim of this thesis, to compare the pre-dictions of different macroscopic theories to the results of atomistic Monte Carlo simulations for a simple model alloy. Starting from an atomistic com-puter experiment – where all atomic details of the diffusional process were known – continuum mechanical averages were calculated and compared to the predictions of the theoretical approaches by Darken, Manning, Svoboda and Moleko, which show different levels of approximation. It was shown that Manning’s and Svoboda’s concept, although derived on different paths, are equivalent. Determining the tracer diffusion coefficients and Onsager’s coeffi-cients in two independent simulation runs provided the basis for a comparison
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44 5. Conclusion and Outlook - Diffusion
of the different models. It was shown that Darken’s concept gives only a poor description of the diffusional processes, whereas Manning’s/Svoboda’s model describes the situation better. The most accurate description is given by the model of Moleko, with the drawback that Moleko’s model is especially designed for the random alloy and can not readily be applied to more gen-eral systems. Although Manning’s model was originally also developed for the random alloy, it was shown by Lidiard that Manning’s results are valid beyond the random alloy. On the other hand Svoboda’s concept is entirely based on a macroscopic description, the only input are diffusion coefficients, and is therefore applicable to all model systems (if diffusion coefficients can be defined). But the version of Svoboda’s model described in this paper has another big drawback. In this thesis it was shown that it is not possible to describe the random alloy exactly by Svoboda’s description. This is due to the fact that the dissipation matrix was assumed diagonal. Our results show that an exact description of diffusional problems demands to include off-diagonal terms in the dissipation matrix. Therefore the drawback of Svo-boda’s theory – the incomplete description of diffusional processes with a diagonal dissipation matrix – is at the same time its big advantage: it is possible to generalise Svoboda’s concept beyond the concept of Manning to account for all diffusional effects.
Additionally to the determination of Onsager’s coefficients interdiffusion problems were studied. The starting configuration of a classical diffusion couple was prepared and the resulting concentration profiles were monitored.
The obtained profiles, which showed excellent scaling behaviour, were anal-ysed with a Sauer, Freise, den Broeder analysis which gave the interdiffusion coefficient as a function of composition. This was then compared to the expressions given by Darken and Svoboda. Furthermore using the tracer diffusion coefficients obtained from a Monte Carlo experiment concentration profiles were also obtained by continuum mechanical calculations. Both, the atomic and vacancy concentration profiles, were compared to the Monte Carlo results. The reduced profiles show very similar behaviour. Small devi-ations are most probably due to the approximation of a diagonal dissipation matrix in Svoboda’s approach.
Several experiments can be thought of to continue this work. One exten-sion is the generalisation to alloys not forming an ideal solution, most striking ordering solids. These alloys can be most effectively studied by implement-ing atomic interactions in the model, e.g. denotimplement-ing each configuration pair hiji) an interaction energy ij. Describing the configuration of the lattice with spin variables results in an Ising-like interaction Hamiltonian that now governs the thermodynamic behaviour – especially the jump frequencies – of the system. This is the natural extension of the random alloy model,
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the exchange frequencies of the atomic species not being constant any more but dependent on the local environment of atom and vacancy. Within this model the attraction or repulsion of atoms and vacancies, respectively, is eas-ily tuneable. Apart from effects arising from ordering tendencies of the alloy, interesting new features can also be expected by introducing some attraction by the vacancy and one atomic species.
Another route to extend the presented work is the investigation of multi-component alloys beyond the binary alloy. For more than two atomic com-ponents the results of Manning (relating jump frequencies and Onsager’s coefficients) and Moleko are not equivalent any more. An interesting ques-tion to answer is, if the presented descripques-tions stay accurate for a ternary or higher alloy or if they fail with the increasing complexity of the system.
46 5. Conclusion and Outlook - Diffusion