Multiscale Modeling of Theta ' Precipitation in Al-Cu Binary Alloys

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2976 V. Vaithyanathan et al. / Acta Materialia 52 (2004) 2973–2987 2.4. Phase-field model In the phase-field methodology, a precipitate microstructure is described by a set of continuum order parameter fields characterizing the difference in composition and structure between the precipitate and matrix [11]. In a two-dimensional representation of h 0 precipitates in a binary Al–Cu alloy, the precipitate microstructure is described by one composition field and two orientation-related long range order parameter fields. The chemical free energy of the microstructure, including the bulk and interfacial energy, is then expressed as Z F bulk þ F int ¼ f ðc; g 1 ; g 2 ; g 3 Þ þ a 2 ðrcÞ2 V þ 1 X b 2 ij ðpÞr i g p r j g p !dV ; ð4Þ p where a and b ij ðpÞ are gradient energy coefficients for composition and order parameters, respectively. b ij ðpÞ is retained as a second rank tensor to incorporate the (tetragonal) anisotropy in interfacial energy. The anisotropy in interfacial energy of the precipitate is dependent on the orientational variant, g, of the precipitate (see Fig. 3). Since the phase-field model used in this study is a dislocation-free model, semi-coherent interfaces are treated effectively as coherent interfaces with a larger interfacial energy. b ij ð1Þ ¼ b 11ð1Þ 0 ; 0 b 22 ð1Þ b ij ð2Þ ¼ b 11ð2Þ 0 0 b 22 ð2Þ c sc (η 1 = 0, η 2 =1 or -1) ¼ b 22ð1Þ 0 : 0 b 11 ð1Þ c (η 1 =1 or -1, η 2 =0) ð5Þ Fig. 3. Schematic of the h 0 precipitates in 2D showing the two variants along with their equilibrium order parameters. The interfaces of the variants are marked as coherent (c) and semi-coherent (sc) to show the symmetry and the interface orientation dependence on the variants. sc The bulk free energy describing the phase-separation and order-disorder transformation involved during precipitation of the h 0 phase is approximated using the following polynomial in c and g: f ðc; g 1 ; g 2 ; g 3 Þ ¼ A 1 ðc C 1 Þ 2 þ A 2 ðc C 2 Þðg 2 1 þ g2 2 þ g2 3 Þ þ A 41 ðg 4 1 þ g4 2 þ g4 3 ÞþA 42ðg 2 1 g2 2 þ g2 2 g2 3 þ g2 1 g2 3 Þ þ A 61 ðg 6 1 þ g6 2 þ g6 3 Þ þ A 62 fg 4 1 ðg2 2 þ g2 3 Þþg4 2 ðg2 3 þ g2 1 Þþg4 3 ðg2 1 þ g2 2 Þg þ A 63 ðg 2 1 g2 2 g2 3 Þ; ð6Þ where C 1 and C 2 are constants close to the equilibrium compositions of solid solution matrix and h 0 precipitates, respectively. The coefficients A 41 and A 61 are constrained to have negative and positive values, respectively, to describe a first order phase transition [20]. The elastic energy contribution to the total free energy of the system is obtained analytically using the method of Khachaturyan [21,22]. The stress-free misfit strain, 0 ijðrÞ, between precipitates and matrix, is expressed as a function of the structural order parameters, 0 ij ðrÞ ¼X 0 ij ðpÞg2 p ðrÞ; ð7Þ p where 0 ijðpÞ represents the stress-free strain of the pth variant. With the homogeneous modulus approximation (matrix and precipitates have similar elastic constants), the elastic energy in terms of order parameters can be expressed as [23] E el ¼ V 2 k X ijkl ij kl V k ijkl ij 0 kl ðpÞg2 p ðrÞ þ V 2 k X X ijkl 0 ij ðpÞ0 kl ðqÞg2 p ðrÞg2 q ðrÞ 1 X 2 p p q X Z q p d 3 g ð2pÞ 3 B pqðnÞfg 2 p ðrÞg g fg2 q ðrÞg g ; ð8Þ where k ijkl is the elastic stiffness tensor. B pq ðnÞ ¼ n i r ij ðpÞX jk ðnÞr kl ðqÞn l and X jk ðnÞ is the inverse matrix of X 1 jk ðnÞ ¼ n ik ijkl n l . n ¼ g=jgj is the unit vector in reciprocal space, fg 2 q ðrÞg g is the Fourier transform of the square of order parameter [g 2 q ðrÞ] and fg2 p ðrÞg g is the complex conjugate of fg 2 p ðrÞg g . ij ¼ 0 ij þ a ij is the sum of homogeneous strain caused by transformation and external constraints or applied stress, respectively. In the absence of applied strain, the fourth term in the elastic energy (Eq. (8)) is the dominant term controlling the precipitate morphology. The temporal evolution of the microstructure in the phase-field model is obtained by numerically solving the Cahn–Hilliard and Allen–Cahn equations [24] ocðr; tÞ ot ¼ Mr 2 dF dcðr; tÞ ; ð9Þ
V. Vaithyanathan et al. / Acta Materialia 52 (2004) 2973–2987 2977 og i ðr; tÞ ¼ ot dF Lð^/ p Þ ; i ¼ 1; 2; 3; ð10Þ dg i ðr; tÞ where M is the solute mobility and Lð^/ p Þ¼LAð^/ p Þ is the orientation-dependent interfacial kinetic parameter. L is the interfacial kinetic coefficient. Interface orientation is defined by the unit normal to the precipitate interface, ^/ p ð¼ ~rg p =j~rg p jÞ. The anisotropy in interfacial kinetics can be incorporated as a function of the interface normal (^/ p ). The temporal equations (Eqs. (9) and (10)) in dimensionless form can be reduced to oc ot ¼ M r 2 of nr 2 c ; ð11Þ oc og p ot ¼ Lð^/ p Þ L " of og p # w ii ðpÞr 2 i g p þ dE el ; ð12Þ dg p t ¼ LjDf jt; r ¼ r=l; ð13Þ M ¼ M Ll 2 ; n ¼ a jDf jl 2 ; f f ðc; gÞ ¼ ; E el jDf j ¼ E el jDf j ; w iiðpÞ ¼ b iiðpÞ jDf jl ; 2 ð14Þ where the quantities with asterisk ( ) represent the dimensionless equivalent of the corresponding dimensional values. l represents the grid spacing (Dx) or the characteristic length scale and Df represents the characteristic free energy (usually the maximum driving force for phase transformation from the constructed bulk free energy). The temporal equations are solved numerically using the semi-implicit Fourier-Spectral method [25]. 3. Results: first-principles calculations 3.1. Bulk chemical free energy 3.1.1. Solid solution phase The enthalpy and free energy (in meV/atom) as a function of composition and temperature, obtained from the combined first-principles/MSCE/Monte Carlo approach, are shown in Figs. 4(a) and (b), respectively. We note that the temperature dependence of enthalpy in Fig. 4(a) is due to a clustering-type short range order (SRO) in the Monte Carlo simulations of the Al–Cu alloy. For a more detailed discussion of the predicted and experimentally measured SRO in Al–Cu, see [26]. We also note that our calculated free energy for the solid solution phase includes configurational but not vibrational entropic contributions. 3.1.2. h 0 precipitate phase The MSCE Hamiltonian obtained for the solid solution free energy calculations based on first-principles ∆H ss (mev/atom) (a) ∆F ss (mev/atom) (b) sc. ∆F ss (mev/atom) (c) 0 -2 -4 -6 -8 -10 -12 -14 10 0 -10 -20 -30 5 4 3 2 1 0 0 0.02 0.04 0.06 0.08 0.1 573 K 673 K 723 K 773 K 873 K X Cu -40 0 0.02 0.04 0.06 0.08 0.1 X Cu -1 0 0.005 0.01 0.015 0.02 0.025 0.03 X Cu 573 K 673 K 723 K 773 K 873 K 573 K 673 K 723 K 773 K 873 K Fig. 4. (a) Enthalpy, (b) free energy and (c) scaled free energy of the Al–Cu solid solution as a function of solute composition and temperature, calculated using the first-principles (FLAPW-LDA) MSCE combined with thermodynamic integration. The scaled free energy is obtained by adding a linear term in composition ½ 3X Cu DF h 0 ðT ÞŠ, such that the rescaled h 0 free energy at X Cu ¼ 1=3 is zero. This linear scaling of the free energy does not affect the determination of equilibrium composition through the common tangent construction, but merely aids in visualization.
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Multiscale Modeling of Theta ' Precipitation in Al-Cu Binary Alloys

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