Dendritic solidification of binary alloys

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1 University of Ljubljana Faculty of Mathematics and Physics Department of Physics Seminar I b - 3rd year, 2nd cycle Dendritic solidification of binary alloys Author: Tadej Dobravec Mentor: prof. dr. Božidar Šarler Ljubljana, May 2015 Abstract The purpose of this seminar is to discuss the thermodynamic and kinetic background for simulation of dendritic growth in binary alloys. The equilibrium and non-equilibrium effects on phase diagram are considered. Theory of homogeneous nucleation is presented. Interface balances are considered by taking into account the curvature effect. Simulation of dendritic growth of Al-7Si alloy is presented, based on quadtree-adaptive cellular automaton algorithm and explicit finite volume method.

2 Contents 1 Introduction 1 2 Thermodynamics of binary systems A single phase solution Ideal and regular solution model Equilibrium of two phases Departure from equilibrium Curvature contribution Nucleation and growth of a new phase 5 4 Numerical modeling of dendritic solidification Al-Si phase diagram Mesoscale model for dendritic growth Conclusions 10 References 11 1 Introduction Dendritic morphology is probably the most common and the most observed and studied phenomenon in solidification of metalic alloys [1, 2]. By understanding the physical phenomena, key to solidification, one can predict microstructure of casts, which is closely associated with the mechanical properties of final product [3]. In the seminar thermodynamics of solidification processes is considered, introducing the important concepts of free energy, chemical potential and equilibrium - these concepts are important for understanding equilibrium binary phase diagrams [4]. Departure from equilibrium induced by interface energy between two various phases is also considered along with nucleation and growth of a new phase, e.g., nucleation of a solid phase in undercooled liquid. A numerical model for dendritic solidification, based on cellular automaton method is described in the final chapter [5]. Derivations in the seminar are based on book Solidification [4] and can also be found in other similar books, e.g., Fundamentals of Solidification [6] and Principles of Solidification [7]. 2 Thermodynamics of binary systems 2.1 A single phase solution A binary system containing n A moles of component A and n B moles of component B (atoms or molecules) is considered (figure 1(a)). Since molecules or atoms can interact, Gibbs free energy is written as G = G(T, p, n A, n B ); the total derivative of Gibbs free energy yields dg(p, T, n A, n B ) = V (p, T, n A, n B )dp S(p, T, n A, n B )dt + µ A (p, T, X B )dn A + µ B (p, T, X B )dn B, (1) where p, T, V, S, X B, µ A and µ B are pressure, temperature, volume, entropy, mole fraction of component B (X A + X B = 1) and chemical potential of component A and B, respectively. Gibbs free energy is homogeneous function of n J, i.e., its magnitude is directly proportional to the amount of its constituents. At constant temperature and pressure G is written as G = n A µ A + n B µ B. (2) Equating total derivative of Eq. (2) to Eq. (1) at constant T and p yields the Gibbs-Duhem equation Dividing Eq. (2) by the total number of moles n = n A + n B yields n A dµ A + n B dµ B = 0. (3) G m = X A µ A + X B µ B, (4) where G m is the molar Gibbs free energy. Differentiation of Eq. (4) with respect to X B yields G m X B = µ A µ B, (5) 1

3 where Eq. (3) was used. Combining Eqs. (4) and (5) yields ( ) G µ A (T, p, X B ) = G m m (T, p, X B ) X B, (6) X B T,p ( ) G µ B (T, p, X B ) = G m m (T, p, X B ) + (1 X B ). (7) X B Eqs. (6) and (7) define tangent rule construction for calculation of chemical potentials of solution having a composition X B (Figure 1). T,p (a) Figure 1: Schematics of the mixing of n A moles of A atoms/molecules with n B moles of B atoms/molecules (a) and molar free energy as a function of the mole fraction of element B, showing the tangent rule construction to compute µ A and µ B. [4] 2.2 Ideal and regular solution model Consider to have one mole solution with X A moles of A and X B moles of B. The formation of totally disordered solution (figure 1(a)) will create on average 0.5(X A N 0 )(X A N b ) bonds (A A) 0.5[(X A N 0 )(X B N b ) + (X B N 0 )(X A N b )] bonds (A B) 0.5(X B N 0 )(X B N b ) bonds (B B) N 0 is Avogadro s number and N b is the coordination number or number of the atomic bonds per atom. The molar internal energy of such solution can be written as E m i = 0.5N 0 N b (X 2 Aɛ AA + X 2 Bɛ BB + 2X A X B ɛ AB ), (8) where ɛ IJ is bond energy between components I and J. Eq. (8) can also be written as E m i = 0.5N 0 N b (X 2 Aɛ AA + X 2 Bɛ BB + X A X B ɛ AA + X A X B ɛ BB ) N 0 N b X A X B (2ɛ AB ɛ AA ɛ BB ). By using X A +X B = 1 and introducing E m ia = 0.5N 0N b ɛ AA, E m ib = 0.5N 0N b ɛ BB and Ω m = N 0 N b (ɛ AB (ɛ AA + ɛ BB )/2), the molar internal energy can be rewritten to E m i = X A E m ia + X B E m ib + Ω m X A X B. (9) The last term in Eq. (9) is called the enthalpy of mixing H mix. The relative affinity of atoms A and B is defined through the bonding energy ɛ AB = ɛ AB 0.5(ɛ AA + ɛ BB ). ɛ AB = 0 corresponds to the case where all bonds in our solution are equivalent, i.e., ɛ AA = ɛ BB = ɛ AB, and consequently H mix = 0. The tendency when ɛ AB > 0 is to have demixing between A and B atoms, since the bond energy between A and B atoms is greater than the average A-A and B-B bonding energy. ɛ AB < 0 corresponds to the case when system can lower its energy by forming A-B bonds and consequently an ordered phase will have tendency to form. The free energy of the solution is written as G m = X A G m A + X m B G m B + Ω m X A X B T S m mix, (10) 2

4 where G m A and Gm B are the molar free energies of the pure A and B respectively, and Sm mix the entropy of mixing, defined as Smix m = k B ln (N A + N B )! (N A )!(N B )!, where k B is Boltzmann s constant, and N A and N B numbers of A-atoms and B-atoms in our solution (N A +N B = N 0 ). By using Stirling formula ln x! = x ln x x for x 1, Eq. (10) can be rewritten as G m = X A G m A + X B G m B + Ω m X A X B + RT (X A ln X A + X B ln X B ), where R = k B N 0. Figure 2 shows the molar Gibbs free energy for Ω m = 0, known as ideal solution G m ideal (curved dashed-line on all three sub-figures). Figure 2(a) shows molar Gibbs free energy for Ω m < 0 and figure 2 for Ω m > 0 (regular solutions). G m for Ω m < 0 is always deeper than G m ideal and overall concave. When Ωm > 0, G m is higher than G m ideal and overall concave only for high temperatures - at low temperatures it is possible for Gm to become convex, leading to phase separation. (a) Figure 2: Molar Gibbs free energy for regular solution with Ω m < 0 (a) and with Ω m > 0. [4] 2.3 Equilibrium of two phases Consider a two phase system containing solid and liquid phase with molar Gibbs free energies G m s and G m l. At fixed pressure and temperature, the molar free energies for such a system are shown in Figure 3(a); curves intersect at X B0, but that does not mean that system is in solid phase for X B < X B0 and in liquid phase for X B > X B0. Use of tangent rule construction (figure 3(a)) yields µ As µ Al and µ Bs µ Bl - system can decrease energy by transferring solute atoms B from the solid to the liquid and solvent atoms A from liquid to the solid. This can be written as dg = (µ As µ Al )dn A + (µ Bl µ Bs )dn B < 0. (11) From Eq. (11) the equilibrium condition for co-existence of two phases is obtained µ As = µ Al ; µ Bs = µ Bl. (12) Figure 3 shows equilibrium condition from Eq. (12) - the dashed line is known as common tangent. For X B < X Bs system is in solid and for X B > X Bl in liquid phase. When X Bs < X B < X Bl both phases coexist with composition X Bs for a solid and X Bl for a liquid phase (the ratio X Bs /X Bl of the solute B in solvent A is called the partition coefficient, k m 0 ). In other words - the overall energy can be minimized for intermediate fractions X Bs < X B < X Bl by splitting the solution into a mixture of liquid and solid phase, and mixing them in proportion to generate overall system fraction [8]. If upper analysis is performed over a range of temperatures, phase diagram with so-called solidus T sol (X B ) and liquidus T liq (X B ) curves can be constructed, as shown in Figure 4(a). System is in liquid phase if T > T liq (X B ) and in solid phase if T < T sol (X B ); for T sol (X B ) < T < T liq (X B ) both phases coexist. 3

The global equilibrium applies to the entire system where no gradients of the temperature, pressure and the chemical potentials of the components are present.

5 (a) Figure 3: Molar Gibbs free energies of two phases, solid and liquid, as a function of the composition X B. [4] 2.4 Departure from equilibrium In previous section systems, for which the global equilibrium was assumed, were considered. The global equilibrium applies to the entire system where no gradients of the temperature, pressure and the chemical potentials of the components are present. When gradients in the various phases appear, equations from previous section can still be applied at the interface between the two phases. This is called assumption of local equilibrium the phase diagrams, e.g., Fig. 4(a) can still be used to determine values at the interface. In other situations, where the departure from equilibrium is not due to gradients, one can no longer use values from the phase diagrams. There are three main sources of (true) departure from equilibrium: surface energy of a curved interface, attachment kinetics, trapping of solute elements. Only curvature contribution to departure from equilibrium is considered. It has to be accounted for in every microscopic numerical model for simulation of solidification process and plays an important role in development of microstructure. (a) Figure 4: Simple phase diagram (a) and excess Gibbs free energy of atoms at the interface and interfacial energy. [4] Curvature contribution The Gibbs free energy of atoms or molecules at the interface between, e.g., solid-liquid interface differs from one in the solid or liquid. The integral of excess free energy over thickness of the interface, multiplied by the 4

6 molar volume, yields solid-liquid interfacial energy γ sl (Figure 4). By assuming that γ sl is isotropic, the total Gibbs free energy can be written as G = G m s n s + G m l n l + A sl γ sl, (13) where n s and n l are the number of moles in the solid in liquid and A sl is the interfacial area. The total derivative of Eq. (13), assuming that the number of moles n = n s + n l is fixed, yields equilibrium condition where κ is the mean curvature of the solid defined as G m l = G m s + 2V m κγ sl, κ = 1 A sl = 1 2 V s 2 ( ), R 1 R 2 where R 1 and R 2 are the principal curvatures [4]. The decrease of melting point, associated with curvature, is known as curvature undercooling and can be calculated as T R = T f T R f = 2Γ sl κ, where T f is melting temperature of planar interface, T R f melting temperature of curved interface and Γ sl is Gibbs-Thompson coefficient, defined as Γ sl = γ slv m S m f = γ sl ρ s s f = γ slt f ρ s L f, (14) where V m, S m f, ρ s, s f, T f and L f are molar volume, the entropy of fusion (defined as S m f = L f /T f ), the solid phase density, the specific entropy of fusion, the equilibrium melting temperature and the enthalpy of fusion, respectively. Figure 5 shows how positively curved solid-liquid interface shifts Gibbs free energy curves and solidus and liquidus lines in phase diagram. Figure 5: Equilibrium of solid and liquid phases for binary alloy including the curvature contribution. [4] 3 Nucleation and growth of a new phase If phase transformation reduces the overall energy, then some of the energy, given up on transformation, is available for creating the free surface. This is the driving force for nucleation and growth of the new phase [8]. The simplest case, where the new phase grows from an old phase with no heterogeneities, e.g., precipitates, voids, vacancies, grain boundaries or dislocations, is considered. Such nucleation is known as homogeneous nucleation. The driving force in this case is the difference in free energy between the liquid system containing the solid particle of volume V s and surface area A sl and completely liquid system. The driving force G is calculated by using Eq. (13) as G m s G m l G = V s V m + A sl γ sl. (15) It can be shown that for small undercooling T, G m s G m l Sf m T ; Eq. (14) yields Sm f /V m = ρ s s f - by inserting these two relations and appropriate values of A sl and V s for sphere of radius R in Eq. (15), G can be written as G = 4 3 πr3 ρ s f T + 4πR 2 γ sl (16) 5

7 Figure 6 shows variation of free energy from Eq. (16) with radius R for pure Al, when T > 0. The curve has maxima G homo n at critical radius R c. For R < R c, a nucleus will shrink and for R > R c a nucleus will grow. Values R c and G homo n can easily be calculated by finding the maxima in the curve R c = 2γ sl ρ s f T = 2Γ sl T, Ghomo n = 4πγ slr 2 c 3 = 16π 3 γ 3 sl (ρ s f ) 2 T 2. As T 0, R c and G homo n, so nucleation becomes impossible. Maxima G homo n is actually the barrier to nucleation - a particle of size R c must spontaneously arise by atomic vibrations and diffusion, otherwise a growth of the new phase is not possible. Consider freezing of purified and filtered water - the freezing occurs a lot sooner if some particles are presented. A presence of particles or favourable surface reduces the energy barrier G n and makes nucleation easier. Such cases represent heterogeneous nucleation [8]. Figure 6: Surface, bulk and total free energy of a spherical solid as a function of its radius for fixed undercooling T = 5K. [4] 4 Numerical modeling of dendritic solidification The complex dendrite morphology has significant effects on the mechanical and material properties of cast alloys. Efficient and accurate numerical models help us understand microstructure evolution during solidification in detail and allow metal production industries to optimize the cast alloy properties [9]. Different numerical models have been developed to simulate dendritic growth, such as front tracking [10, 11], level set [12, 13], phase-field [14, 15] and cellular automata [1, 3] methods. In this section, numerical model, based on cellular automata method [2] is described and simulation of dendritic growth of Al-7Si alloy is presented. 4.1 Al-Si phase diagram Figure 7(a) shows binary phase diagram for Al-Si alloy; α and β denote solid phases of Al and Si, respectively. One can observe regions where only liquid phase l exists, only solid phase α exists, liquid l and solid phase α coexist, liquid l and solid phase β coexist, both solid phases α and β coexist (growth in that region is also known as eutectic solidification). Temperature T = 577 C is know as eutectic temperature T eut and weight percent C = 12.6% as eutectic concentration C eut. Physical constants (table 1) for hypoeutectic alloy containing 7% silicon, i. e., Al-7Si alloy (green line in figure 7(a);), are used for testing the developed numerical model. 6

8 Property L [J/kg] ρ [kg/m 3 ] λ [W/m/K] c p [J/kg/K] T L [ C] C 0 [wt.%] Value Property C eut [wt.%] k [ ] D L [m 2 s 1 ] D S [m 2 s 1 ] m L [ C% 1 ] Γ [K m] Value Table 1: Thermophysical properties of Al-7Si alloy. [2] (a) Figure 7: Binary phase diagram of Al-Si alloy (a) and two-dimensional domain for dendritic solidification. [2, 16] 4.2 Mesoscale model for dendritic growth We simulate dendritic growth in two dimensional domain Ω, i. e., growth of a solid phase in undercooled liquid is simulated, as shown in figure 7. Heat transfer equation (ρh) = (λ T ), (17) t where ρ, h and λ stand for density, specific enthalpy and thermal conductivity, respectively, and solute diffusion equation C L C S = (D L C L ), = (D S C S ), (18) t t where C L and C S are concentrations (weight percent Si) and D L and D S diffusion coefficients in liquid and solid phase, respectively, must be solved in Ω. Approximation h = c p T + f L L, is used for specific enthalpy, where c p, f L and L are specific heat, liquid fraction and latent heat of fusion, respectively; constant ρ, λ and c p are used for solid and liquid phase. Initial conditions for temperature and concentration are known (T 0, C 0 ), along with boundary conditions on the surface Ω; Robin boundary conditions are used for solving Eq. (17) λ T ( ) m = χ T T B m, Ω Ω where m, χ and T B are outward unit normal on the surface Ω, surface heat transfer coefficient and temperature at the boundary, respectively; values χ = 1000 W/m 2 /K and T B = T 0 are used in simulation. For solving Eq. (18), Neumann boundary conditions are used C i m = f, Ω where f is prescribed solute flux and i stands for liquid or solid. f is set to 0 in simulation. 7

Interface velocity v of growing solid phase can be calculated through solute conservation at solid/liquid interface: ( ) 1 C S v = D S C L (1 k) n D L S n, (19) L C L where CL, k and n are interface

9 Interface velocity v of growing solid phase can be calculated through solute conservation at solid/liquid interface: ( ) 1 C S v = D S C L (1 k) n D L S n, (19) L C L where CL, k and n are interface concentration in the liquid phase, partition coefficient and solid/liquid interface normal vector, respectively. Interface concentration in the liquid phase is calculated as C L = C 0 + (T T L + Γ κf(ϕ, θ)) m L, where C 0, T L, T, Γ, κ, f and m L are initial concentration, corresponding liquidus temperature from phase diagram, interface temperature, Gibbs-Thompson coefficient, average interface curvature, the anisotropy of the surface tension and liquidus slope from phase diagram, respectively. Average interface curvature is calculated with counting-cell technique [2]. A four folded symmetry is assumed; in such case the anisotropy of the surface tension is calculated as [4] f(ϕ, θ) = 1 + δ cos(4(ϕ θ)), where θ is the growth angle, δ degree of anisotropy and ϕ angle, calculated as v x ϕ = arccos, vx 2 + vy 2 where v x and v y are x and y components of interface velocity v from Eq. (19). δ is set to 0.04 in the simulation. Square domain Ω with side length l is divided into cells with quadtree algorithm [17], as shown in figure 8. Temperature T, concentration in solid and in liquid phase C S and C L, solid fraction f S (f S + f L = 1) and state of a cell (liquid,interface or solid) are defined in each cell. The Explicit Finite Volume Method [18] is used to calculate the temperature and concentration in each cell as described by Eqs. (17) and (18). Solute transfer equation is solved on adaptive grid, where grid has the biggest accuracy on solid/liquid interface. The length of cells a varies from a min to a max (figure 8). As the typical length scale for thermal diffusion is much larger than for solute diffusion, heat transfer equation is calculated on a regular, time independent mesh with larger finite volumes (squares with side length a temp > a). In table 2 parameters, used in current simulation can be found. Simulation parameter l [µm] a temp [µm] a min [µm] a max [µm] T 0 [ C] C 0 [wt.%] θ[ ] Value Table 2: Simulation parameters. (a) Figure 8: Adaptive grid based on quadtree algorithm in the beginning (t = s) and in the end (t = s) of the simulation. At the beginning of the simulation states of all cells are set to liquid (f S = 0) - solid phase is nucleated by changing state of the cell in the middle of the domain from liquid to interface. Knowing the interface velocity from Eq. (19), the solid fraction increment in an interface cell is calculated as [19]: δf S = δt a ( v x + v y v x v y δt a 8 ),

$Once solid fraction of interface cell become unity (f S = 1), the cell changes its state from interface to solid.$

10 where a is the size of the cell and δt the time step, calculated as δt = 1 5 min ( amin v max, a2 min D L, a2 min D S, a ) temp. λρc p Since the grid has the biggest accuracy on solid/liquid interface, length of interface cells is always equal to a min. Once solid fraction of interface cell become unity (f S = 1), the cell changes its state from interface to solid. In parallel, a random number rand is generated; if rand is smaller than capture probability p c, defined as p c = tan 2 θ + 1, the solid cell nucleates its liquid neighbours, i. e., their states are changed from liquid to interface. Figure 9 shows results of current simulation, i. e., concentration C = f L C L + f S C S in domain Ω at six different times - typical dendrite morphology is obtained. (a) (c) (d) (e) (f) Figure 9: Time evolution of concentration during dendritic solidification of Al-7Si alloy. 9

11 5 Conclusions Equilibrium binary phase diagram derivation and treatment of departure from equilibrium is represented in the seminar. Nucleation and growth of spherical nuclei is considered and explained why growth of a new phase reduces the system energy and is therefore thermodynamically favorable. A simple dendritic growth model based on the heat and solute transport is represented, where growth velocity of the solid phase is calculated through the solute conservation at the solid/liquid interface. The verification of the cellular automaton model will be performed by comparing results to analytical model [4], (numerical) phase field model [9] and experimental results. The main disadvantage of current model is mesh dependence, i. e., the dendrites can only grow at two orientations: θ = 0 and θ = 45. There are two reasons, why mesh dependence occurs: first is the inherent configuration of the orthogonal mesh, and second is the growth and capture rules of cellular automaton [1]. We have two main goals for the future research: finding appropriate algorithm for reducing dendrites orientation dependence, e.g., decentered square algorithm [1] or grid rotating technique [3] and extend current algorithm to simulate eutectic solidification. 10

12 References [1] R. Chen, Q. Xu, B. Liu, Journal of Materials Science & Technology, Vol. 30, No. 12, 1311 (2014). [2] L. Nastac, Modeling and Simulation of Microstructure Evolution in Solidifying Alloys (Kluwer Academic Publishers, Massachusetts, 2004). [3] L. Ying, X. Qingyan, L. Baicheng, Tsinghua science and technology, Vol. 11, No. 5, 495 (2006). [4] J. A. Dantzig, M. Rappaz, Solidification (EPFL Press, Switzerland, 2009). [5] K. G. F. Janssens, D. Raabe, E. Koezeschnik, M. A. Miodownik, B. Nestler, Computational Materials Engineering (Elsevier Academic Press, USA, 2007). [6] W. Kurz, D. J. Fisher, Fundamentals of Solidification (CRC Press, UK, 1998). [7] M. E. Glicksman, Principles of Solidification (Springer, USA, 2011). [8] (4/21/2015). [9] M. A. Zaeem, H. Yin, S. D. Felicelli, Applied Mathematical Modelling, Vol. 37, No. 5, 3495 (2013). [10] N. Al-Rawahi, G. Tryggvason, J. Comput. Phys., Vol. 194, No. 2, 677 (2004). [11] P. Zhao, J.C. Heinrich, J. Comput. Phys., Vol. 173, No. 2, 765 (2001). [12] L. Tan, N. Zabaras, J. Comput. Phys., Vol. 211, No. 1, 36 (2006). [13] K. Wang, A. Chang, L. Kale, J.A. Dantzig, J. Parallel Distrib. Comput., Vol. 66, No. 11, 1379 (2006). [14] R. Kobayashi, Phys. D.,Vol. 63, No. 3-4, 410 (1993). [15] J.A. Warren, W.J. Boettinger, Acta Metall., Vol. 43, No. 2, 689 (1995). [16] G. K. Sigworth, International Journal of Metalcasting, Vol. 8, No. 1, 7 (2014). [17] Finkel, R. A., J. L. Bentley, Acta Informatica, Vol. 4, No. 1, 1 (1974). [18] H. K. Versteeg, W. Malalasekera, An Introduction to Computational Fluid Dynamics, The Finite Volume Method, 2nd Edition (Pearson Education Limited, England, 2007). [19] H. K. D. H. Bhadeshia, Mathematical Modeling of Weld Phenomena 3 (Institute of Materials, London, 1997). 11

Equilibria in Materials

009 fall Advanced Physical Metallurgy Phase Equilibria in Materials 10. 13. 009 Eun Soo Park Office: 33-316 Telephone: 880-71 Email: espark@snu.ac.kr Office hours: by an appointment 1 Contents for previous