Computer Simulations Combining Finite Difference and Finite Element ...

Computer Simulations Combining Finite Difference and Finite Element Methods: Solute Drag on Migrating Grain Boundaries in Three-Dimension M. C. Gao1, a, J. Gruber1,b, A D Rollett1,c and A. P. Kuprat2,d 1

Carnegie Mellon University, Pittsburgh, PA 15213, USA.

2

Pacific Northwest National Laboratory, Richland, WA 99352, USA

a

[email protected], [email protected], [email protected], d [email protected]

Keywords: Grain Growth; Recrystallization; Solute Drag; Finite Difference; Finite Element Method

Abstract. The current study aims to improve our fundamental understanding of solute segregation and solute drag on migrating grain boundaries (GB) in three dimensions. Computer simulation combines finite difference and finite element methods. An exemplary case study is reported, in which a spherical grain is embedded inside a cubic grain and shrinks as a result of motion by curvature, as a preliminary to modeling grain growth in single phase materials. The results agree qualitatively with literature studies in 1-D. Introduction Grain boundaries (GB) are planar defects where two grains of different orientation meet. GB structures are generally complex although there exist some special GBs whose structures are relatively simple such as symmetrical twin boundaries. The atomic misfit at GB causes an excess free energy of the system called GB energy. Consequently, solute or impurities will tend to diffuse into and then segregate at GB in order to lower the total system free energy. Segregation is much more pronounced for species that occupy the interstitial sites in the bulk than those that prefer substitutional sites because strain energy associated with the former is typically much more significant. Typical segregation mechanisms in metals or alloys are a combination of several driving forces including the partial elastic strain energy release, decrease in the grain boundary energy [1, 2]. For a comprehensive review of GB structure and segregation, see Refs. [1-4]. The presence of segregated solutes or impurities at GB can substantially slow down the kinetics of grain growth and recrystallization [3]. This is because segregated solutes tend to catch up with a migrating GB as if there is a force, opposite to the GB motion, exerted on the GB, which partially consumes the driving force for grain growth or recrystallization. (In addition, it is known that solute segregation tends to diminish the anisotropy of GB energy distribution in alloys.) Review papers that address GB structures, anisotropy in energy and mobility, and GB motion can be found in Refs. [2, 3, 5-7]. The interaction between solute and migrating GBs has been studied by a number of researchers since the 1950’s [3, 8-37]. There are two basic approaches to the problem: one is the energy dissipation approach used by Hillert and coworkers [11, 12, 17, 21, 22, 33, 35] and the other is the drag force approach originally proposed by Lucke and Cahn [8-10, 20]. Both approaches should generate the same results if both are physically correct. Hillert argues that both approaches become equivalent only if one correlates the solute-GB interaction potential proposed in Ref. [9] with thermodynamic quantities (i.e. ∆µ B − ∆µ A ) [33]. Note that only the triangle potential was used in Cahn’s analysis for an ideal bulk solution with a perfectly flat GB [9] and others [24] for simplicity, mainly because the exact mathematical form of such solute-GB interaction is not known.

Theoretical studies on solute drag effect have been largely confined to one dimension [8-11, 24, 31], and they all are limited to a perfectly flat planar GB migrating at a constant velocity i.e. a steady state. This scenario is very rare in real materials even though it is not clear that there is a steady state in grain growth. Clearly, 1-D analysis and simulation of solute drag is severely limited especially when compared to experiments since real samples are all in 3D and each grain interacts with its neighbors. Therefore, there emerges a need for theoretical studies of microstructure in 3D and accordingly solute-GB interaction in 3D. In this study, we employ gradient-weighted moving finite element code (GRAIN3D) [38, 39] to simulate evolution in 3D. Solute diffusion is simulated using the finite difference method on a regular grid of size 100x100x100. The solute-GB interaction is implemented by modifying the diffusion equation by considering solute-GB interaction potential using Cahn’s approach [9]; the GB position is taken from GRAIN3D directly. To begin, one simple microstructure is considered: a shrinking spherical grain embedded inside a cubic grain driven by curvature, to mimic grain growth in single-phase materials. No assumption of steady state grain boundary migration was made in the current simulations. The Model Ideal thermodynamic solution. Since the segregation effect is solely due to the interaction between solute atoms and grain boundaries, it would be reasonable to introduce an interaction parameter, E , as proposed initially in Refs. [8-10]. Given that the solute-grain boundary interaction decays with increasing distance from the GB, it is reasonable to introduce an interaction width w , only within which solute-GB interaction is important (but neglected beyond this zone). For solutes that are located outside w , E is set to zero. The chemical potential of solute µ at GB is assumed to be:

µ = °µ + RT ln C + E

(1) where R =gas constant, T =temperature, C =solute concentration, µ is the chemical potential at the reference state. The flux is then calculated from: r {D}C∇µ = −{D}⎡∇C + C ∇E ⎤ J =− ⎥⎦ ⎢⎣ RT RT (2) o

where {D} is the diffusivity tensor of the impurity component. For simplicity, we assume an isotropic diffusivity with a constant value, Do , such that the solute composition evolution can be calculated based on mass balance: r ∂C C ⎤ ⎡ = −∇ ⋅ J = D∇ ⋅ ⎢∇C + ∇E ⎥ ∂t RT ⎦ ⎣ (3) Drag force. The interaction potential E is assumed to have a parabolic dependence on distance: 2

⎛d ⎞ E ( x, y , z ) = E o − E o ⎜ ⎟ ⎝ w⎠ where d is the distance between a solute and the GB plane. Eq. 7 applies only when d ≤ a o ; E ( x, y , z ) = 0 when d > a o . w is the interaction zone width: w = n dx 2 + dy 2 + dz 2

(4)

(5) where dx , dy and dz is the mesh increment along x, y and z direction, respectively, to represent solute diffusion field. n is a scalar constant. The drag force experienced by a GB due to the presence of a chemical composition gradient is assumed to obey a simple linear relationship to a first order approximation:

r r r (6) F drag = φ ⋅ n.∑ (ni ⋅ ∇C GB ) i r where n is the GB normal vector, φ is an scalar coefficient. The summation is performed for all the

solutes within the interaction zone width for each GB plane on both sides. ∇C

GB

denotes the

composition gradient at the center of GB plane, and it is evaluated on the discrete grid as: ∇C

GB ( i , j , k )

=

C (i, j, k + 1) − C (i, j, k − 1) C (i, j + 1, k ) − C (i, j − 1, k ) C (i + 1, j, k ) − C (i − 1, j , k ) zˆ yˆ + xˆ + 2 ⋅ dx 2 ⋅ dy 2 ⋅ dz

(7)

Simulation Results and Discussions The simulation is performed with GRAIN3D, which is designed to simulate microstructure evolution driven by curvature. The details of the code are described elsewhere [38, 39]. Briefly, GRAIN3D approximates the interfaces in a grain boundary network as a mesh of triangular elements [38-40]. Nodal velocities are calculated by minimizing a functional that depends on the local geometry of the mesh and the (anisotropic) properties of the grain boundaries. Since the focus of present study is on the solute drag effect, the GB energy and GB mobility are treated as isotropic and constant. The effect of anisotropy of GB energy and mobility on grain growth has been reported earlier by Demirel et al., and by Gruber et al. [40]. Fig. 1 illustrates the compositional profile across a GB for the case in which the GB attracts solute (i.e. the solute-GB interaction potential is negative inside the GB, which causes solute segregation at the GB). The boundary moves to the right. Note that the composition profile is asymmetric with respect to the GB plane, and the solute is depleted much more significantly ahead of the GB.

Fig. 1 Composition profile along a line across the center of a spherical grain embedded in a cubic grain. Effect of interaction width (see Eq. 5) is also shown. The initial condition is that the solute concentration is 10 at.% throughout the both grains including the GB region. The solute-GB interaction energy is set at -1000 J/mol (i.e. a weak GB segregation) to gain computational efficiency. The size of the simulation domain is a 1x1x1 box. The segregation level is higher with zero velocity. The effect of interaction width is also shown (n is the integer in Eq. 5). Fig. 2 shows the effect of the scalar coefficient of the drag force (see Eq. 6) on the grain boundary migration kinetics, which are quantified by measuring the rate of decrease in total GB area. A positive coefficient contributes to solute drag and a negative coefficient acts as a driving

force for grain growth thus accelerates grain growth. Increasing the value of the coefficient (in the positive regime) slows down grain growth kinetics further. When the coefficient reaches 100, grain growth becomes very sluggish. When the drag force balances the driving force, grain growth will eventually halt.

Fig. 2 Effect of the scalar coefficient (see Eq. 6) on grain growth kinetics of the shrinking sphere.

Fig. 3 shows the effect of solute diffusivity on grain growth kinetics when holding all other parameters fixed. We examined a set of values of diffusivities, namely 0, 0.01, 0.1 and 1.0. Diffusivity equal to zero of course corresponds to the case in which grain growth is free of solute drag, which corresponds to the physical situation of boundary migration at temperatures low enough to suppress solute diffusion, e.g. recrystallization near room temperature in high purity aluminum. It is seen that increasing diffusivity in the low value regime increases the solute drag effect thus slowing down GB motion and the grain growth kinetics. This is true because too-slowdiffusion solute atoms would find themselves unable to catch up with the migrating GB thus lowering the segregation level and the drag force. Note that when the diffusivity exceeds some threshold, then the solute drag effect decreases; this is because the solute diffuses so rapidly that the solute drifts along with the migrating boundary, the equilibrium segregation profile is preserved (symmetric) and therefore drag is minimized.

Fig. 3 Effect of solute diffusivity on grain growth kinetics of the shrinking sphere. Note that an intermediate value of solute diffusivity (D=0.1 in this case) causes the strongest solute drag effect.

The temperature effect is shown in Fig. 4. Lowering the temperature monotonically decreases the grain growth kinetics due to the increase in drag force. This is physically reasonable because the

equilibrium segregation concentration increases exponentially (up to saturation) with decreasing temperatures to a first approximation [2].

Fig. 4 Effect of temperature on grain growth kinetics of the shrinking sphere.

Summary The current study uses simulation to examine the solute drag phenomenon in 3D for the first time. A simple microstructure was examined and a set of parameters relevant to solute-GB interactions are examined and all the results are physically reasonable. Therefore, this study provides a solid base for performing 3D simulations of grain growth of a large-scale microstructure with a statistically meaningful number of grains that will be performed in the near future. Acknowledgements The authors would like to acknowledge financial support from the Computational Materials Science Network, a program of the Office of Science, US Department of Energy. MCG would like to thank Profs. P. Wynblatt and S. Ta'asan for valuable discussions. References [1] [2] [3] [4] [5] [6] [7]

[8] [9] [10] [11] [12] [13]

D. McLean: Grain Boundaries in Metals (Clarendon Press, Oxford 1957). P. Lejcek and S. Hofmann: Crit. Rev. Sol. State. Mat. Sci. Vol. 20 (1995), p. 1. G. Gottstein and L.S. Shvindlerman: Grain Boundary Migration in Metals: Thermodynamics, Kinetics, Applications (CRC Press, Boca Raton, FL 1999). A.P. Sutton and R.W. Balluffi: Interfaces in Crystalline Materials (Clarendon Press, Oxford 1995). H.V. Atkinson: Acta Metall. Vol. 36 (1988), p. 469. G.S. Rohrer: Annual Rev. Mat. Res. Vol. 35 (2005), p. 99. R.D. Doherty, D.A. Hughes, F.J. Humphreys, J.J. Jonas, D.J. Jensen, M.E. Kassner, W.E. King, T.R. McNelley, H.J. McQueen and A.D. Rollett: Mater. Sci. Eng. A Vol. 238 (1997), p. 219. K. Lücke and K. Detert: Acta metall. Vol. 5 (1957), p. 628. J.W. Cahn: Acta metall. Vol. 10 (1962), p. 789. K. Lücke and H.P. Stüwe: Acta Metall. Vol. 19 (1971), p. 1087. M. Hillert and B. Sundman: Acta Metall. Vol. 24 (1976), p. 731. M. Hillert and B. Sundman: Acta Metall. Vol. 25 (1977), p. 11. Z.K. Liu, J. Agren and M. Suehiro: Mater. Sci. Eng. A Vol. 247 (1998), p. 222.

[14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40]

M. Suehiro: Isij Int. Vol. 38 (1998), p. 547. M. Suehiro, Z. K. Liu and J. Agren: Metall. Mater. Trans. A 1998, 29, 1029. D. Fan, S.P. Chen and L.Q. Chen: J. Mater. Res. Vol. 14 (1999), p. 1113. M. Hillert: Acta Mater. Vol. 47 (1999), p. 4481. R. Le Gall and J.J. Jonas: Acta Mater. Vol. 47 (1999), p. 4365. A. Michels, C.E. Krill, H. Ehrhardt, R. Birringer and D.T. Wu: Acta Mater. Vol. 47 (1999), p. 2143. J.W. Cahn and A. Novick-Cohen: Acta Mater. Vol. 48 (2000), p. 3425. M. Hillert and M. Schalin: Acta Mater. Vol. 48 (2000), p. 461. M. Hillert, J. Odqvist and J. Agren: Scripta Mater. Vol. 45 (2001), p. 221. M.I. Mendelev and D.J. Srolovitz: Phil. Mag. A Vol. 81 (2001), p. 2243. M.I. Mendelev and D.J. Srolovitz: Acta Mater. Vol. 49 (2001), p. 589. A. Vilenkin: Interface Sci. Vol. 9 (2001), p. 323. P.R. Cha, S.G. Kim, D.H. Yeon and J.K. Yoon: Acta Mater. Vol. 50 (2002), p. 3817. M. Jahazi and J.J. Jonas: Mater. Sci. Eng. Vol. 335 (2002), p. 49. R. Kirchheim: Acta Mater. Vol. 50 (2002), p.413. M.I. Mendelev and D.J. Srolovitz: Model. Sim. Mat. Sci. Eng. Vol. R79 (2002), p. 10. Y.J.M. Brechet and G.R. Purdy: Acta Mater. Vol. 51 (2003), p. 5587. N. Ma, S.A. Dregia and Y. Wang: Acta Mater. Vol. 51 (2003), p. 3687. N. Maruyama, G.D. W. Smith and A. Cerezo: Mater. Sci. Eng. Vol. 353 (2003), p. 126. M. Hillert: Acta Mater. Vol. 52 (2004), p. 5289. W.A. Soer, J.T.M. De Hosson, A.M. Minor, J.W. Morris and E.A. Stach: Acta Mater. Vol. 52 (2004), p. 5783. M. Hillert: Z. Metallkd. Vol. 96 (2005), p. 104. M.I. Mendelev, D.J. Srolovitz, G.J. Ackland and S. Han: J. Mater. Res. Vol. 20 (2005), p. 208. A.L. Korzhenevskii, R. Bausch, R. Schmitz: Acta Mater. Vol. 54 (2006), p. 1595. A.P. Kuprat: SIAM J. Sci. Comput. Vol. 22 (2000), p. 535. A. Kuprat, D. George, G. Straub and M.C. Demirel: Comput. Mater. Sci. Vol. 28 (2003), p. 199. J. Gruber, D.C. George, A.P. Kuprat, G.S. Rohrer and A.D. Rollett: Scripta Mater. Vol. 53 (2005), p. 351.