Phase field modelling of microstructural evolution using the Cahn-Hilliard equation: A report to accompany CH-muSE

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1 Phase field odelling of icrostructural evolution using the Cahn-Hilliard equation: A reort to accoany CH-uSE 1 The Cahn-Hilliard equation Let us consider a binary alloy of average coosition c 0 occuying the (2D) xy-lane. Let the alloy consist of two hases and, that is, it is ket at a teerature (say T ) that corresonds to the two hase region in the hase diagra (see Figure 1). We assue that the teerature reains a constant, i.e., our resent forulation is an isotheral one. Let the coosition at any oint r in the xy-lane at tie t be denoted by c(r, t). Let the icrostructure of the syste be coletely described by the coosition field. Let us choose a doain bounded by of such a syste. Further, we assue the coosition field to be eriodic on, i.e., is the reresentative area for the 2-diensional syste and reeats infinitely to fill the entire xy-lane (see Figure 2). This assution of eriodicity hels us avoid accounting for the surfaces in our calculations. Given an initial coosition rofile, say c(r,0), the coosition rofile at any future tie t can be obtained by solving the following (Cahn-Hilliard) non-linear diffusion equation (with eriodic boundary conditions): t = M µ, (1) where, M is the obility, c is the (scaled) coosition t is the tie, and µ is the cheical otential, given by µ = δf ch δc, (2) 1

2 Teerature T + c c 0 c Coosition Figure 1: Scheatic hase diagra. 2

3 Figure 2: Periodic doain. where δ denotes the variational derivative with resect to coosition, and δc F is the cheical free energy F ch = N AV [f 0 (c) + κ( c) 2 ]d, (3) where, N AV is the Avogadro nuber, κ is the gradient energy coefficient, and f 0 (c) is the bulk free energy density, and is given by f 0 (c) = Ac 2 (1 c) 2, (4) where A is a ositive constant indicating the energy barrier between the two equilibriu hases and (see Figure 3). Using the exressions (3) and (4) in the definition of the cheical otential, we obtain where, µ = h 2κ 2 c, (5) h = f 0 = 4Ac(1 c)(1 2c). (6) We assue the obility M and the gradient energy κ to be (scalar) constants: this aounts to assuing the interfacial energies and the diffusivities to be isotroic. 3

4 bulk free energy density coosition Figure 3: Bulk free energy density as a function of coosition for A=1. Note that the energy barrier is one-sixteenth of A, and the equilibriu coositions of the and hases are scaled to be zero and unity resectively Using the Equation (5) above, and the fact that M is a constant, we obtain the Cahn-Hilliard equation as follows: t = M 2 (h 2κ 2 c). (7) 2 Nuerical ileentation and bencharking 2.1 Ileentation details Non-diensionalisation All the araeters used in our siulations are non-diensional. The nondiensionalisation was carried out using the characteristic length L, energy E, and tie T, as follows: L = ( κ A)1 2, (8) 4

5 E = AL 3, (9) T = L 2 (c e c e ) 2 ME, (10) where c e and c e are the equilibriu reciitate and atrix coositions, resectively. We note that this choice of non-diensionalisation renders the non-diensional κ, A, and M unity. Thus, using these non-diensionalisation we obtain the following Cahn- Hilliard equation (fro Equation (7)): Fourier transfor and discretization t = 2 (h 2 2 c). (11) Let us consider the equation (11) above and take the (satial-) Fourier transfor on both sides of the equations: {c} g t = g 2 ({h} g + 2g 2 {c} g ), (12) where, { } g reresents the (satial-) Fourier transfor of the quantity { }. The sei-ilicit discretization of the above equation is then obtained as follows: c(g, t + t) c(g, t) t = g 2 {h} g 2g 4 c(g, t + t) (13) c(g, t + t) = c(g, t) g2 {h} g t tg 4, (14) where t is the tie ste for the nuerical integration. Thus the roble of icrostructural evolution reduces to nuerically solving the discretized equation above Algorith for icrostructural evolution The following is the algorith for icrostructural evolution: 5

6 1. Given a coosition rofile at tie t, we calculate the h and its Fourier transfor as well as the Fourier transfor of c. 2. Using {h} g and {c} g in equation (14), we calculate the coosition rofile at soe future tie t + t. 3. The inverse Fourier transfor of c(t + t) gives the coosition rofile at tie t + t. 4. We reeat stes 1-3 to arch in tie for the given nuber of tie stes. The (discrete) Fourier transfors needed for our calculations have been carried out using FFTW develoed by Frigo and Johnson [1]. 3 Acknowledgeents We would like to thank T.A. Abinandanan, H. Raanarayan, R. Sankarasubraanian, S. Chatterjee, Saswata Bhattacharyya, R. Mukherjee, and generations of ME students for discussions regarding the forulation and ileentation of the Cahn-Hilliard and Allen-Cahn equations. References [1] M Frigo and SG Johnson (2005), The design and ileentation of FFTW3, Proceedings of the IEEE, 93, 2, (See also htt:// 6

7 List of Figures 1 Scheatic hase diagra Periodic doain Bulk free energy density as a function of coosition for A=1. Note that the energy barrier is one-sixteenth of A, and the equilibriu coositions of the and hases are scaled to be zero and unity resectively