D:\PUBL MD & coworkers\pow³oki & secondary\164 GB_DIFTRANS1.WPD The Non-Reduced Solution of the Fischer Model, Grain Boundary Diffusion in 3

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1 July 7, 1998 D:\PUBL MD & coworkers\pow³oki & secondary\164 GB_DIFTRANS1.WPD The Non-Reduced Solution of the Fischer Model, Grain Boundary Diffusion in 3 M. Danielewski*, W. Krzy añski**, B. Figura*, P. Kita* and G. Pêcak* *University of Mining and Metallurgy, al. Mickiewicza 30, Cracow, Poland **Jagiellonian University, ul. Reymonta 3, Cracow, Poland KEYWORDS: stationary grain boundaries, diffusion, diffusion profiles, computer simulation, silver, partial differential equations, generalized solution. PACS: Lj; v; h; Lw; Jt. The studies of the grain boundary diffusion have been summarized in the handbook by Kaur et al. 1 and in several reviews, e.g., Bernardini and Gas 2. Grain boundary diffusion is generally analyzed depending on: 1) the ratio of the self and grain boundary diffusivities, 2) the ratio of the grain and the grain boundary thicknesses and 3) the duration of the experiment, i.e., depending on the "kinetic regime. Measurements are often carried out in the so called "type B" kinetic regime 1, 3. Some experimental studies have been carried on different kinetic regimes, namely the "type C" 1, 4 and the "type B1" 4, 5. Between the limits of type C and B kinetics 4 and their solutions remains the relative large field where the presently known solutions of the diffusion equation (2nd Fick's law) cannot be used 2. Thus, the more general classification of the diffusion regimes 5 includes the B1 regime which exists between C and B regimes. The interpretation of the experimental results that do not belong to B and C regimes, with the use of well known classical solutions need to be carefully justified and may often lead to systematic errors. To model the more realistic (more general) situation we analyzed diffusion in a bicrystal formed by two different crystalline phases (grains) having cubic geometry and different transport properties. The contact layer between the two media, the grain boundary, shows different diffusivity also. Through the upper surface of such bicrystal, the flow of mass occurs and the local concentration of mass on this surface is a known function. The initial distribution of the diffusing element is an arbitrary known function as well. We assume that through the other elements of the external boundary of the bicrystal the transport of mass does not occur. Finally we assume the Fick ian diffusion and mass conservation at all interfaces. We have formulated the variational form of such problem. We present an exact analytical formula, method of its solution, an effective software that allows to find the distribution of the diffusing element and practical computation of grain boundary diffusion.

2 INTRODUCTION We consider a bicrystal (domain ), formed by two media (e.g., different crystalline phases, grains) having cubic geometry and different transport properties, i.e., diffusivities D 1 and D 3, Fig. 1. Figure 1 Schema of the two cubic crystals with a contact layer (grain boundary). Three dimensional Fischer model. The media have one common interface (a boundary layer). This contact layer between the two media is called a grain boundary and is treated as a new phase having a markedly different property, in particular different transport properties -grain boundary diffusion coefficient D g = D 2. Assume that through the upper surface of such the domain, the flow of mass occurs. The local concentration (density) of mass on this surface is known and can be in a known function of time. The initial distribution of the diffusing element is an arbitrary known function as well. We assume that through the other elements of the external boundary of the domain the transport of mass does not occur. Finally we assume the Fick ian diffusion and mass conservation at all interfaces. Such a problem of grain boundary diffusion in 3 is analogous to the fundamental Fischer model 6 in 2 In the coordinate system shown in Fig. 1 the first and third medium occupy the domain and domain Their contact volume (grain boundary) occupies. The whole system forms the cubicoid domain. Assume that through the upper surface of the cubicoid domain denoted as: the flow of mass occurs. The concentration (density) of mass on

3 I is known and can be in general a function of time c b (t, x, y, z), where and denotes the observation time. We will examine the distribution of the element diffusing in the domain and, in particular the time evolution of its density in. On the top surface, I, of the domain we assume that the concentration of the diffusing element is given by: (1) We assume that through the other elements of the external boundary of the domain (at the other positions of the external boundary of the domain ) the transport of mass (i.e., diffusion) does not occur. Consequently, we postulate that gradient of diffusing element vanishes: (2) where denotes the external surface of, except its top ( I ) and denotes the whole boundary of. The vector is normal to and directed outward with respect to. Inside each domain, diffusion is governed by the second Fick's law: (3) where i > 0 denotes the diffusion coefficient of the diffusing element in the i-th domain. We postulate also that density of diffusing element,, is continuos at all interfaces. We assume continuity of the diffusion flux crossing the boundary of the domains, i.e., continuity of the normal parts of the diffusion fluxes crossing two interfaces in : (4) where is the outward normal vector to I. At the beginning of the process (t = 0 ), the distribution of the diffusing element is a known function: The grain boundary diffusion problem is to find a nonnegative function (5), which satisfies conditions (1) - (5), when the numbers t*, 1, 2, 3 > 0 and the functions are known. Solution of the problem. A solution of the problem (1) - (5) is derived in a weak sense 7, 8, i.e. as some continuos curve such that for every 0 < t < t* it is: 1 o ; 2 o, where is the trace operator 9, 10 ; 3 o where is the restriction operator 9, 10 ;

4 4 o ; 5 o, where the normal derivatives are understood in the distributional sense 9, 10. For almost every is weakly differentiable at t and 6 o and 7 o. Additionally we limit the rate of the diffusion process. We require that for every 0 < t < t* 8 o, where denotes 9 the Sobolev norm in the space H 1 ( ) Finally, we assume that and are given functions. A weak solution will be constructed as a Fourier series with respect to some orthonormal basis in L 2 ( ). The separation of variables leads to three, one-dimensional eigenvalue problems for the variables y, z and x. Let > 0. Consider the problem (6) (7) The resulting system of solution is (8) where From the general theory 8, 11 it follows that q 0, q 1,... is an orthonormal basis in. (9) Let > 0, consider the problem (10) (11)

5 The resulting system of solutions is (12) where (13) The general theory 8, 11 implies that r 0, r 1 `,... form an orthonormal basis in L 2 (-c, c). Let > 0 be given. For > 0 consider an eigenvalue problem (14) (15) (16) (17) (18) (19) (20) One can show that > min{ 1, 2, 3 }. Assume that 1 = 3 < 2. If 1 < / < 2, then solutions of the problem (14) - (20) are given by (21) where and. If / = 2, then (22) where. If 2 < /, then

6 (23) where and. The formulae (14) - (20) can be written as a homogenous system of six linear equations with six unknowns 1, 1,..., 3, 3. If the matrix for this system is ( ), then necessary condition for a nonzero solution is (24) Let k, l = 0, 1,... be fixed and = k + l. For such let (25) denote all the solutions of equation (24). One can check that the rank of M( k, l, m ) is 5, m = 0, 1,... Thus, for every k, l, m there is exactly one function p k,l,m (x) satisfying (14) - (20) such that (26) The functions p k, l, 0, p k, l, 1,... form an orthonormal basis 8, 11 in L 2 (- a, a). With the use of the functions p k, l, m (x), q k (y), r l (z), k, l, m = 0, 1,... we construct an orthonormal basis in L 2 ( ) One can check that where and s = s. (27) (28) If the function is constant (i.e., ) and c b satisfies some additional conditions then the weak solution of the problem (1) - (5) can be constructed. Theorem. Let and. Assume that there exists such that 1 o 2 o 3 o 4 o Then there exists a unique weak solution of the grain boundary diffusion problem. This solution is given by the formula (29)

7 (convergence in L 2 ( )), where I is the surface element on and is the scalar product in L 2 ( ). If additionally and, then the weak solution is differentiable almost everywhere in the classical sense and for every 0 < t < t* (30) A significant example of the process that follows the assumptions of the above Theorem is a case when: c b constant, then. COMPUTATIONS The exact analytical formula allows to obtain effective software and to find the distribution of the element diffusing in the domain and, in particular the time evolution of its density in. The results of simulation of such process in Ag bicrystal and in nanocrystalline material and in three-dimensions are shown in Figs A very large difference between lattice, D S = D 1 = D 3, and grain boundary diffusivity, D GB = D 2, for Ag at 400 C was found. There is negligible lattice diffusion in Ag at 400 C, yet grain boundary diffusion is substantial. In Figs. 2-5 we list the values of D S and D GB in Ag at two temperatures for comparison 12. The software used to compute evolution of densities was written in Borland C++ v.3.1. The demo version is available from the authors. All computations were carried out using IBM-compatible PC Pentium 200 MHz. The computations of evolution of in the system shown in Fig. 1 required ~20 minutes. All calculations were done upon assuming bicrystal geometry shown in Fig. 1. The following dimensions were applied: a = 25 nm, b = 25 nm and c = 200 nm, grain boundary thickness = 0.5 nm., initial density equals zero, concentration at the top surface c b = n cm - 3. An error in calculating the coefficients S in formulae (29) was The variable number of coefficients was used ( ). Number of iterations in order to find solution was 10. In order to show the diffusion process which affects concentrations at the boundaries, in Figs. 2-5 we show the calculated concentration profiles at 400 o C and 800 o C.

8 Figure 2 Concentration profiles, c(t i, x, 0, 0), in the cubicoid bicrystal. Figure 3 Cocentration profiles, c (t i, x, 0, 0), in the cubicoid bicrystal in the vicinity of grain boundary, region shown in Fig. 2.

9 Figure 4 Cocentration profiles, c (t i, x, 0, 0), in the cubicoid Ag bicrystal. Figure 5 Cocentration profiles, c (t i, x, 0, 0), in the cubicoid bicrystal in the vicinity of grain boundary, region shown in Fig. 2..

10 Summary The solution of the grain boundary diffusion problem describing this process (Fischer model in 3 ) in the bounded bicrystal are shown. The generalized model allows to obtain the quantitative data on the processes within a grain boundary zone and the bulk material. The numerical computations of diffusion in silver bicrystal show the evolution of the density profile and changes of densities at the boundaries. The computation suggests that the postulated boundary conditions correctly describe the transport process in the system. The generalized Fischer model of grain boundary diffusion allows to describe the diffusion processes in course grained as well as nanocrystals. The modern mathematics now-days allows for the modeling, practical calculation and the better understanding of the real transport problems. ACKNOWLEDGMENTS This work has been supported in 1998 by the Polish State Committee for Scientific Research under Grant AGH No REFERENCES 1.Kaur, I., Gust, W. and Kozma, L., Handbook of Grain and Interphase Boundary Diffusion Data, Ziegler Press, Stuttgart, Bernardini, J. and Gas, P., 1997, Defect and Diffusion Forum, , Harrison, L. G., 1961, Trans. Faraday Soc., 57, Kaur, I., Mishin, Y. and Gust, W., Fundamentals of Grain and Interphase Boundary Diffusion, J. Wiley & Sons, London, Mishin, Yu. M., and Razumovskii, IM., 1992, Acta Metall. Mater., 40, Fischer, J. C., 1951, J. Appl. Phys. 22, Friedman, A., Partial Differential Equations of Parabolic Type, Prentice-Hall, Englewood Cliffs, Renardy, M. and Rogers, R.C., An Introduction to Partial Differentail Equations, Springer- Verlag, New York, Adams, R., Sobolew Spaces, Academic Press, New York, Gilbarg, D. and Trudinger, N.S., Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, Courant,R., and Hilbert, Methods of Mathematical Physics, vol I, Wiley-Interscience, New York, Tu, K-N, Mayer, J. W. and Feldman, L. C., Electronic Thin Film Science, Macmillan Publ. Comp., New York, 1992.