Symmetries and modelling functions for diffusion processes

arXiv:0808.2177v2 [physics.data-an] 1 Jun 2009 Symmetries and modelling functions for diffusion processes A.G. Nikitina, S.V. Spichaka, Yu. S. Vedulab and A.G. Naumovetsb aInstitute of Mathematics of National Academy of Sciences of Ukraine, 3 Tereshchenkivs’ka Street, Kyiv-4, Ukraine, 01601 e-mail: nikitin@imath.kiev.ua; bInstitute of Physics of National Academy of Sciences of Ukraine, 46 Prospect Nauki, Kyiv-28, Ukraine, 03028 e-mail: naumov@iop.kiev.ua Short title: Symmetries and modelling functions PACS numbers: 68.43.Jk - Diffusion of adsorbates; 66.30.Dn - Theory of diffu- sion and ionic conduction in solids. Abstract A constructive approach to theory of diffusion processes is proposed, which is based on application of both the symmetry analysis and method of mod- elling functions. An algorithm for construction of the modelling functions is suggested. This algorithm is based on the error functions expansion (ERFEX) of experimental concentration profiles. The high-accuracy analytical descrip- tion of the profiles provided by ERFEX approximation allows a convenient extraction of the concentration dependence of diffusivity from experimental data and prediction of the diffusion process. Our analysis is exemplified by its employment to experimental results obtained for surface diffusion of lithium on the molybdenum (112) surface pre-covered with dysprosium. The ERFEX approximation can be directly extended to many other diffusion systems. 1 Introduction Experimental and theoretical studies of diffusion processes are of a great importance for various branches of physics, biology, chemistry and other natural sciences. In ad- dition, such studies have important applications in medicine and many technological processes. A special interest is exited by surface diffusion processes which appear in many physical and chemical systems. In particular, they are used in various kinds of nanotechnologies. The theory of diffusion processes started in 1855 when Fick derived his classical diffusion equation [1] ∂θ ∂t − ∂ ∂xa  D ∂θ ∂xa  = 0, (1) which still is a corner stone of the diffusion theory. In equation (1) D is a diffusion coefficient, in general case depending on species concentration θ, and xa with a = 1, 2, 3 are spatial variables (summation over the repeated indices a is imposed). Being supplemented by an appropriate initial data, equation (1) serves as a background for description of such diffusion processes which are characterized by diffusion flows linear in concentration gradients and not depending explicitly on space and time variables. Two standard problems of a diffusion theory are: 1) To describe time evolution of the diffusion process, and 2) To specify the dependence of the diffusion coefficient on concentrations of dif- fusing species. Of course, these problems are closely related, since if we know how the diffusion coefficient depends on concentration θ, then the time evolution of the corresponding diffusion process can be found using the Fick equation (1) and the related initial data. On the other hand, if we know θ as a function of time variable t and spatial variables xa, then we can find D solving the inverse diffusion problem using again equation (1). Both mentioned problems are very complicated and in general need rather sophis- ticated techniques. Even if we know the diffusion coefficient as an explicit function of concentration, then generally speaking it is possible to find only an approximate (nu- merical) solution of the first problem if at all. The second problem has a much more complex character, but in the case of a sharp step-like initial θ profile it is possible to use the Boltzmann-Matano (BM) approach [2] and reconstruct the concentration dependence D(θ) of the diffusion coefficient. This approach enables one to make a numerical calculation of the diffusion coefficient, but its accuracy is not very high, especially for small and large concentrations θ. Experimental data and numerical solutions are very important for description of a diffusion process, but to formulate its theory it is desirable to create some analytical expressions for studied values. Unfortunately, there are only few known exactly solv- able realistic diffusion problems, the most famous of them is probably the Barenblat one [3]. Thus it is a common practice to use rather rough analytic presentations of D(θ) to make a qualitative analysis of diffusion process (see, e.g., [4]). 1 In the present paper we propose a new method for description of time evolution of a diffusion process and calculation of the diffusion coefficient. The distinct feature of our approach is that we find both functions θ = θ(t, x) and D = D(θ) in an explicit form, i.e., solve both problems 1) and 2) analytically. To achieve this goal we start with experimental data for a particular diffusion system and make the error func- tions expansion (ERFEX) of concentration profiles. Analytic description of diffusion processes is very convenient for their qualitative analysis. Moreover, our description appe

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