On the Effect of Atoms in Solid Solution on Grain Growth Kinetics

EMMANUEL HERSENT 0 KNUT MARTHINSEN 0 ERIK NES 0 0 EMMANUEL HERSENT, Research Engineer, formerly with the Department of Materials Science and Engineering, Norwegian University of Science and Technology (NTNU) , 7491 Trondheim, Norway , is now with Gr anges Technology AB , 612 33 Finsp ang, Sweden . KNUT MARTHINSEN, Professor, Deputy Head of Department , and ERIK NES, Professor Emeritus, are with the Department of Materials Science and Engineering, Norwegian University of Science and Technology (NTNU). Contact The discrepancy between the classical grain growth law in high purity metals (grain size D / t1=2) and experimental measurements has long been a subject of debate. It is generally believed that a time growth exponent less than 1/2 is due to small amounts of impurity atoms in solid solution even in high purity metals. The present authors have recently developed a new approach to solute drag based on solute pinning of grain boundaries, which turns out to be mathematically simpler than the classic theory for solute drag. This new approach has been combined with a simple parametric law for the growth of the mean grain size to simulate the growth kinetics in dilute solid solution metals. Experimental grain growth curves in the cases of aluminum, iron, and lead containing small amounts of impurities have been well accounted for. - where D and D0 are the instantaneous and the initial average grain sizes, respectively; t is the isothermal annealing time; and K a parameter depending on material and temperature, and the exponent n, which also varies with material and temperature, is usually referred to as the grain growth exponent. It follows from the classical treatment by Burke and Turnbull[2] that n = 2. However, in high purity metals and solid solution alloys, n is usually found to be larger than 2 as illustrated by the results shown in Figures 1 and 2, taken from Gordon and El-Bassyouni[3] and Hu[4] on high purity aluminum and zone-refined iron, respectively. In the work by Beck et al.[5] on aluminum n-values in the range from 3 to 18 were reported. For a review, see Humphreys and Hatherly.[6] No generally accepted theory has been developed which accounts for n being larger than 2, but a common belief is that non-linear dependence of velocity on driving pressure due to solute effects is the main cause for this phenomenon. The classical solute-drag theory established by Cahn[7] and Lu cke and St uwe[8,9] is of little help in this context (this theory will be referred in following as the CLS theory). In the extremes, i.e., in situations of breakaway or loaded boundaries, it follows that n should be equal to 2. Only in the unstable situation in-between these extremes, deviation in n-values from 2 can be expected. The solute-drag theory, however, has not been developed to a level which makes it possible to predict the relevant n-values in this region of solute-boundary instability. However, the new statistical solute-pinning approach to the effect of solute atoms on boundary migration recently developed by the present authors,[10] and extended to the effect of multiple components highly diluted in solid solution,[11] is both physically and mathematically more transparent, which opens for a theoretical analysis of grain growth kinetics in dilute solid solutions. MODELING GRAIN GROWTH KINETICS The simulation of the grain microstructure evolution by phase-field methods looks very promising.[1216] Each grain is characterized by a continuous-phase variable whose value is ranging from 0 outside the grain to 1 inside and whose evolution is described by the physical phenomena to be simulated. It has been shown recently by Gro nhagen et al.[14] and Kim et al.[15] that solute drag could be taken into account in a manner consistent with the CLS theory by introducing a concentration dependence in the double-well potential in the Gibbs energy expression. Solute drag in non-steady-state conditions (the CLS theory assumes stationary conditions) and abnormal grain growth induced by solute drag have Annealing time t (hours) Fig. 1Grain growth evolution with time in a zone-refined aluminum which has been added 4 ppm Cu (cf. Table I; data redrawn from Ref. [3]). The exponent n averages about 3.4 for the different curves, except for the 590 K (317 C) curve for which the exponent n equals 5.8. n=2.0 n=1.9 n=3.2 n=3.8 Fig. 2Grain growth in zone-refined iron during isothermal annealing (data redrawn from Ref. [4]). For each temperature, it has been indicated the value of the exponent n which corresponds to the best fit of the law Dn = Kt to the experimental data (values taken from Fig. 2 in Ref. [4]). been studied, respectively, in Reference 16 and in Reference 15 with the help of this approach. However, this procedure is computer intensive and as an alternative, it is suggested here to treat grain growth in a simple parametric way, the objective being to define a computational procedure for predicting the time dependence of the average grain size during annealing of a polycrystalline solute containing metal at a constant temperature. It is simply assumed that the growth rate of the average grain size, D, during isothermal annealing at a temperature T will follow a relationship of the form where a is a geometric factor depending upon the shape of the grain and vbc; T; P is the steady-state migration rate of a grain boundary acted upon by a pressure P in a material containing a solute concentration c. In simplistic terms, a can be set equal to 2 as grain boundaries move apart by vb dt. However, the exact value of a is of no consequence since in the present modeling work, a becomes a trivial fitting parameter. In a grain growth situation, the driving pressure is assumed to be well represented by P 2cGB=r; where cGB is the specific grain boundary energy and r is the grain radius (r D=2). An expression for the boundary migration rate can be found in the solutepinning analysis referred to above,[10] and a brief summary of some salient ideas in this approach seems necessary in this context. Further, this summary will also include the equations required in the simulations below. A. The Solute-Pinning Approach The migration rate of a high-angle grain boundary in a pure metal at a temperature, T, is commonly given by an expression in the following form[17]: In this equation, m is the boundary mobility, Cp is a constant, b is a typical inter-atomic spacing (Burgers vector), mD is the Debye frequency, k is Boltzmanns constant, and UbSB is an activation energy associated with boundary migration. This activation energy is typically found to have a value half that of self-diffusion. If solute atoms are added to the metal, then the situation will change. The present treatment assumes a boundary region potential for the boundary-solute interaction as schematically outlined in Figure 3(a) i.e., Ux U0 for xt 2k xt xt 2k and U(x) = 0 for all other values of x, where x(t) is the instantaneous positi (...truncated)