A Monte Carlo simulation of solidification structure formation under melt shearing

Transcription

1 Materials Science and Engineering A365 (2004) A Monte Carlo simulation of solidification structure formation under melt shearing A. Das,Z.Fan Brunel Centre for Advanced Solidification Technology, Brunel University, Uxbridge UB8 3PH, UK Abstract Semi-solid metal (SSM) processing of alloys under applied shear has a promising future for near net-shape production of components. Although non-dendritic growth of solid is critical for semi-solid metal processing, a comprehensive understanding of morphological evolution under melt shearing is lacking. Through Monte Carlo simulation a growth-based evolution of solidification structure under different melt shearing conditions is proposed in the present work. It has been demonstrated that the nature of the diffusion boundary layer determines the growth morphology of the solid under shear. In effect, the nature of fluid flow determines the morphology of the solid by influencing the geometry of the boundary layer. A generalized explanation of morphological evolution under melt shearing is presented that qualitatively corroborates the experimental observations Elsevier B.V. All rights reserved. Keywords: Semisolid metal processing; Solidification; Microstructure; Monte Carlo simulation 1. Introduction Following the original discovery by Spencer [1] that under shear partially solidified alloys show the fluidity of a machine oil and exhibit thixotropic behavior, semi-solid metal (SSM) processing has become one of the most promising technological developments for near net-shape production of components. In SSM processing alloys are processed within the freezing range, usually under the action of shear through mechanical or electromagnetic means, and exhibit non-dendritic morphological evolution of the solid that gives rise to the observed rheological properties [2 4]. Furthermore, suppression of dendritic growth also reduced porosity and segregation in SSM-processed components. The intensity of stirring and the time spent in the semisolid state is shown to influence the morphological evolution of the particles as equiaxed dendrites, rosettes or dense spheroids with entrapped liquid [2]. Ji et al. have introduced high intensity melt shearing (previously unattainable) through a twin-screw rheomoulding process and demonstrated that near mono-size distribution of fine spherical particles evolve in the semisolid slurry [5]. Corresponding author. Tel.: x2337; fax: address: amitabha.das@brunel.ac.uk (A. Das). In spite of the importance of non-dendritic morphological evolution during SSM processing, theoretical understanding of the solidification process in stirred melt has not progressed much. The theories proposed so far appear to be inconsistent with each other and do not offer a generalized explanation for the multitude of growth patterns observed under different stirring conditions. In an attempt to understand non-dendritic structural evolution, Vogel and Cantor concluded from a stability analysis that shearing destabilizes the solid liquid interface and promotes dendritic growth [6] much in contrast to the experimental observation. Accordingly, Doherty et al. have proposed a dendrite fragmentation mechanism due to the mechanical action of shear in the melt to account for the experimental observation [7]. The dendrite fragmentation theory, the most explicit available explanation for SSM processed microstructure, has been questioned by Hellawell who argued that shear forces are confined within the elastic limit, and therefore, dendrite fragmentation by a mechanical force is unlikely to occur [8]. Moreover, from three-dimensional microstructural observation in sheared melts, it has been suggested that the dendritic fragmentation mechanism might not be as influential as believed [9]. On the other hand, continuous nucleation due to periodic passage through different temperature zones as described by Hellawell [8] is at least unlikely in the twin-screw rheomoulding process of Ji et al. [5]. Extremely /$ see front matter 2003 Elsevier B.V. All rights reserved. doi: /j.msea

2 A. Das, Z. Fan / Materials Science and Engineering A365 (2004) intense convection following a complex fluid flow pattern between the fine intermeshed twin screws is able to homogenize instantaneously the smaller amount of entrapped liquid with respect to temperature. Furthermore, no explanation for observed rosette type morphology under low shear rate have been presented. Through a coupled free boundary and cellular automaton model Mullis proposed that dendrite bending could give rise to rosette formation without invoking mechanical effects [10]. Molenaar et al. suggested that rosette type morphology evolves due to cellular growth of solidification structures under shear but ignores the formation of spherical particles observed at high shear rate [11]. The present work, therefore, attempts a generalized explanation on the evolution of different solidification structures under different melt stirring conditions through a Monte Carlo simulation. A complex non-linear flow behavior and turbulence of the stirred liquid is difficult to model, and thus, a simplified multiparticle diffusion limited aggregation approach [12] is adopted. The model accounts for diffusion and fluid flow by incorporating random or directional motion of the liquid atoms, atom attachment at the solid liquid interface, and incorporates a surface rearrangement process to minimize the solid liquid interfacial energy under the influence of capillary. The main objective of the work is to form a qualitative basis for structural evolution based on the growth behavior of the solid and the accurate description of the movement of liquid atoms is not essential. Different flow patterns were generated and the effects of fluid flow characteristics on the solidification structures were investigated in order to formulate a general insight into the morphological evolution during SSM processing. 2. The model A two-dimensional square lattice is filled up sequentially with an A or B atom according to the average concentration of the binary alloy. In the simulations presented in this paper a lattice has been considered. As nucleation is not addressed in the model, solidification is seeded from two rows of solid atom at the top horizontal edge of the lattices or from an isolated solid atom at the center of the lattice. As the lattice is completely filled and devoid of vacancies, an atomic exchange mechanism is used to address diffusive or convective motion in the liquid. A random lattice position is chosen at first; if a liquid atom with at least one nearest or next-nearest solid neighbor is found, solidification is attempted as described later. Otherwise, the liquid atom is allowed to exchange position with any of its four neighbors with equal probability (pure diffusive motion) or with a higher probability of exchanging position in a pre-determined direction (convective motion). In case of convective motion, allowing the liquid atom to exchange position (with another liquid atom) more than one lattice spacing away can increase the intensity of convection. Each atom in the liquid experiences on an average one attempted exchange in a single (arbitrary) time step, which is also denoted as a Monte Carlo step. However, as the sites for such potential jumps are chosen at random, within a particular time step some atoms may attempt more than one exchange whereas some atoms may not attempt any jump at all. A periodic boundary condition is employed at the free edges of the lattice so that a liquid atom escaping the lattice at one edge is reintroduced at the opposite edge at exactly the same height. When solidification is simulated from the top horizontal edge, the bottom horizontal edge is considered impermeable so that no liquid atom can escape from this edge. Periodic boundary conditions are employed only at the vertical edges of the lattice under this condition. Solidification probability of an atom is calculated considering the total Gibbs energy change ( G T ) incorporating the changes in volume and surface energy and is given by (say, for a solidifying A atom), G T = µ A + (n A A + φn n A A )E A A N V + (n A B + φn n A B )E A B + (n After + φn n After )E (1) where, µ is the change in chemical potential, N V the Avogadro number, n A A and n A B the numbers of nearest A A and A B bonds formed and n After represents the total number of unsatisfied nearest neighbor bonds created at the solid liquid interface upon solidification of the atom. Superscript n to these terms refers to the numbers of corresponding next-nearest neighbor bonds. E A A and E A B are the energy values associated with the nearest neighbor A A and A B solid bonds, and φ is the weighing factor for the corresponding energy terms for the next-nearest neighbor solid bonds. E is the energy of an unsatisfied bond at the solid liquid interface which, for simplicity, is taken equal for solid A or B atoms at the interface. A similar expression can be derived for a solidifying B atom. The probability of solidification of the liquid atom, W ( sticking coefficient ), is defined as, W = 1 τ s exp( G T /kt) 1 + exp( G T /kt) where k is the Boltzmann constant and τ s is some arbitrary factor, usually taken as unity. An atom is adsorbed at the surface if a pseudo-random number smaller than W is generated. The possibility for this atom to return back to the liquid is considered and if remelting does not occur then the atom is considered to have solidified. After each successful solidification attempt a surface rearrangement trial is performed by allowing the solidified atom to travel to an energetically more favorable position at the solid liquid interface within a square area of edge length (2L+1) x with the solidified atom at the center. With L as an integer and x as the lattice parameter, the surface diffusion length (SDL) is defined as L x. A surface rearrangement can be viewed upon as a melting event at the position considered and a resolidification event at a new (2)

3 332 A. Das, Z. Fan / Materials Science and Engineering A365 (2004) position of the solid liquid interface and essentially takes into account the effect of capillary by keeping the curvature information at the interface. The transition probability (W SR ) for surface rearrangement to all probable sites are calculated following a procedure similar to the case of solidification and the atom is moved to the most probable position only if W SR is larger than 0.5. If many of these positions are equally probable then one of them is chosen at random. 3. Results and discussion 3.1. Solidification microstructures formed under different shearing intensities A twin-screw rheomoulding machine is utilized to investigate the nature of microstructure solidifying under different melt stirring conditions. The quenched microstructures from a model Sn 15 wt.% Pb alloy are shown in Fig. 1. Without the application of shear the quenched specimen shows well developed dendrites dispersed throughout the microstructure as shown in Fig. 1a. At low and intermediate shear rates coarsened solidification structures resembling rosettes are observed as shown in Fig. 1b under a shear rate of 41 s 1 in the twin-screw machine. Very fine white particles observed in the microstructure are solidification products of the remnant liquid formed during the quenching process. Under a very high shear rate (3625 s 1 ) a distribution of fine spherical particles is observed in the microstructure (Fig. 1c). From the experimental observation it is obvious that the intensity of shear in the twin-screw machine governs the solidification structure. The fluid flow characteristic inside the machine is quite complex and at high shear rate the flow can mainly be described by high intensity turbulence while at a low shear rate the flow is, perhaps, predominantly laminar in nature [5]. Therefore, it appears that the change in the fluid flow characteristics resulting from different shearing intensities is associated with the corresponding change in the microstructure observed. Accordingly, the Monte Carlo simulation is utilized to evaluate the growth pattern of the solid under different fluid flow characteristics Effect of fluid flow on the morphology of solid growing from a fixed substrate In the present simulation different types of fluid flow are characterized by the direction of gross motion of the liquid atoms relative to the growth direction of the solid rather than trying to simulate an exact laminar or turbulent type flow in the bulk of the liquid that is difficult to implement. This is a reasonable simplification as the growth of the solid is controlled only by the nature of the diffusion boundary layer existing around the liquid-solid interface and the interaction of liquid atoms with it. The exact nature of atomic movement throughout the bulk of the liq- Fig. 1. Quenched microstructures of Sn 15 wt.% Pb alloy during continuous cooling showing (a) dendrites under normal solidification without shearing, quenched from 205 C, (b) rosettes formed under low shear rate (41 s 1 ) quenched from 197 C, and (c) spherical particles formed under high shear rate (3625 s 1 ) quenched from 197 C. uid (outside diffusion boundary) has insignificant influence on the solidification process. Therefore, in the present simulation a distinction between laminar and turbulent flow can be made on the basis of liquid penetration or the lack of

4 A. Das, Z. Fan / Materials Science and Engineering A365 (2004) it inside the diffusion boundary layer, and accordingly, a gross directional motion can be set in the liquid as discussed below. A pure shear flow or laminar flow can be visualized as one where mass transport is diffusive inside the diffusion boundary layer (around the particle) and convective outside. Increased intensity of convection is expected to decrease the boundary layer thickness but unlikely to assist forced liquid penetration into the interdendritic region of dendrites growing from a substrate. The condition can be best approximated allowing a gross directional motion in the liquid perpendicular to the growth direction of the solid. A turbulent flow, on the other hand, can be visualized as an ensemble of eddies that can destabilize the diffusion boundary layer promoting liquid penetration into the interdendritic regions of the growing solid. Accordingly, a gross liquid motion parallel to the solid growth direction represents the best approximation to study the influence of turbulent flow on the diffusion boundary layer. Solidification structures simulated for a binary alloy (atomic fraction of A, C 0 = 0.7) under different flow conditions are shown in Fig. 2. The specific A solid atom is represented by a black pixel while solvent (A) and solute (B) atoms in the liquid are represented by white and gray pixels, respectively, in the simulated microstructures. Grayscale intensity represents the local solute concentration in the liquid. It may be noted that an exact correspondence with the composition of Pb Sn alloy used in the experiments is not necessary as the primary objective is to understand the evolution of solidification structure rather than the equilibrium volume fraction and composition of solid. Any alloy composition will produce similar morphological development of solidification structures in the simulations (albeit in different time intervals) and C 0 = 0.7 has been chosen as an optimum value for reduced computational time and clearly identifiable rejected solute profile in the melt (grayscale intensity in figures). As the present model does not address nucleation, solidification on a fixed substrate (say, mould wall) is seeded from two rows of solid atoms at the top edge of the lattice. Values of the simulation parameters are reported in the caption of the figures. Morphological development under pure diffusive motion, where the liquid atoms have equal probability of exchanging position with any of its neighbor, is shown in Fig. 2a. As expected in a pure diffusive growth, the structure is dendritic with well-developed primary and secondary arms growing along the 1, 1 direction in the lattice for energetic reasons. When the liquid atoms are assigned a directional motion (as shown by the black arrowhead) predominantly perpendicular to the growth direction of the solid, a clear destabilization of the solid liquid interface occurs and dendritic solidification is favored (Fig. 2b) as evidenced from the reduced thickness of the initial solid layer prior to the initiation of dendritic growth in Fig. 2b compared to Fig. 2a. This result is in essential agreement with the theoretical stability Fig. 2. Effect of fluid flow on the morphology of solidification structures growing directionally from a fixed substrate. The arrows in the figures show direction of fluid flow. Growth under (a) diffusive mass transfer, (b) fluid flow perpendicular to the growth direction of solid, and (c) fluid flow parallel to the direction of growth of solid. Parameters used for the simulation are C 0 = 0.7, µ A /N V kt = 10, µ B = 5N V kt, E = 2kT, E A A = 2kT, E A B = 0.1kT, φ = 0.1, SDL = 3. analysis of Vogel and Cantor [6]. On the other hand, when the motion of the liquid atoms is predominantly parallel to the growth direction of the solid, a compact solidification structure results with less or no liquid entrapment and a flat solid liquid interface as shown in Fig. 2c.

5 334 A. Das, Z. Fan / Materials Science and Engineering A365 (2004) The morphological evolution can be explained as follows. A lateral motion in the liquid (from left to right) in Fig. 2b in essence establishes a laminar type flow in the liquid over a random diffusive motion where no interdendritic liquid penetration occurs but mass transport in the rest of the liquid takes place by convection. Growth is favored at the tips of the protruding instabilities growing in the liquid where the supply of solvent atoms is relatively abundant due to convective mixing in the liquid (compared to purely diffusive conditions). On the other hand, solute rejected in the interdendritic regions (lagging behind the growth front) cannot be transported away and strong growth retardation occurs in these locations. Therefore, a forced laminar motion as in the cases of Fig. 2b eventually promotes dendritic growth for solid growing from a fixed substrate. A vertical motion of the liquid atoms in Fig. 2c, on the other hand, represents a turbulent flow where mass transport in the whole of the liquid takes place by convection and constant liquid penetration into the interdendritic regions occurs. As the rejected solute is quickly transported away from the interdendritic region, at every point of the growing solid liquid interface the melt concentration is relatively uniform thereby preventing perturbation formation and promoting growth of compact structures Effect of fluid flow on the morphology of particles suspended in the melt So far the simulations have been restricted to solid growing from a fixed substrate under forced fluid motion. In the actual SSM process, however, growing solid particles will be removed from the mould wall under the action of the fluid flow and grow as isolated particles suspended in the melt. Moreover, it has been predicted earlier that the isolated and suspended particles will rotate at a speed of 2 γ in a simple shear flow with a shear rate of γ. As rotation of the particle itself is difficult to implement in the simulation, it is kept stationary and the liquid is allowed to change direction of flow in every time (Monte Carlo) step. This, in effect, is equivalent to simulating the effect of particle rotation in a shear flow. Fig. 3a shows equiaxed dendritic growth of a solid particle suspended in the melt under pure diffusive liquid motion with well-developed primary, secondary and tertiary arms. Parameters used for the simulation are reported in the caption of the figure. On implementing rotation and allowing a directional motion in the liquid with sufficient intensity (shown by black arrowhead) the dendritic structure is replaced by a coarse rosette type pattern where the growth of tertiary and secondary arms are suppressed as shown in Fig. 3b. When the liquid atoms are assigned gross directional motion such that the liquid atoms impinge upon the growing solid particle from all directions, a compact solidification morphology evolved (Fig. 3c). The growth of dendritic morphology under pure diffusive liquid motion and compact morphology under liquid impingement on the solid Fig. 3. Growth morphology of an isolated particle suspended in the melt under (a) purely diffusive mass transfer, (b) unidirectional fluid flow, and (c) liquid impingement from all directions. Parameters used for the simulation are C 0 = 0.7, µ A /N V kt = 10, µ B = 5N V kt, E = 2kT, E A A = 2kT, E A B = 0.1kT, φ = 0.1, SDL = 3. (turbulent motion) can be explained on the basis of the interaction of the liquid motion with the interdendritic region as suggested in the earlier section. However, the present simulation shows that unlike directional growth of solid from a

6 A. Das, Z. Fan / Materials Science and Engineering A365 (2004) substrate where a laminar flow promotes dendritic growth, the rotation of isolated particles under laminar type flow stabilizes the solid liquid interface to some extent and promotes rosette morphology. It appears that rotation of the particle ensures a periodic change in the liquid flow characteristics around every region of the growing particle. Consequently, every part of the solid liquid interface alternately experiences a lateral flow promoting dendritic growth (as in the case of growth from fixed substrate) and an impinging flow penetrating the interdendritic regions promoting coarsening resulting in rosette morphology. On the basis of the experimental observations and Monte Carlo simulation results, a growth based morphological evolution of solid under melt stirring can now be presented. Morphological development would depend on the geometry of the diffusion zone (diffusion boundary layer) around the growing particle, and therefore, on the effect of the nature of fluid flow on the diffusion boundary layer. Without the application of shear solute transfer takes place by diffusion through the entire volume of liquid (infinite diffusion boundary layer) and the growth structure is purely dendritic as observed in normal casting process (also see Figs. 2a and 3a). At low and intermediate shear rate a laminar flow is established in the liquid. Under such condition a finite boundary layer exists around the growing solid beyond which the melt becomes homogenized due to the imposed convection. Solute transport inside the finite boundary layer is still diffusive and it has been shown that dendritic growth from a fixed substrate is enhanced under laminar flow (see Fig. 2b). This is in agreement with the earlier prediction by Doherty et al. [7] that convection destabilizes the solid liquid growth front except that the present analysis predicts destabilization under laminar flow only for directional growth of solid from a substrate. Dendritic growth is never observed in SSM processed materials and the present simulation predicts that particles detached from the substrate due to shear will be rotating in a laminar flow experiencing periodic destabilizing and stabilizing effects on the solid liquid interface promoting rosette type morphology (see Fig. 3b) as observed under low intensity shearing experiments (Fig. 1b). At a high shear rate the flow characteristics will change to turbulence and the diffusion boundary layer around the solid will be completely destabilized due to liquid penetration into the interdendritic region. Under such condition solid will grow with a compact morphology as solute will, transported away from the interdendritic region discarding any possibility of constitutional undercooling (Figs. 2c and 3c) and explains the occurrence of compact spherical particles under very high intensity shear in the experiments (Fig. 1c). 4. Conclusions A Monte Carlo simulation that accounts for diffusive as well as forced motion of the liquid atoms, attachment kinetics at the solid liquid interface, and a surface rearrangement process under the effect of capillary is utilized to understand microstructure formation under melt shearing. Based on the simulation results a growth based morphological evolution influenced by the nature of fluid flow is presented that qualitatively corroborate experimental observations. A laminar type fluid flow appears to promote dendritic morphology for solid growing directionally from a fixed substrate but promotes rosette type growth morphology for isolated particles suspended in the melt under laminar flow. A turbulent type fluid flow that penetrates into the interdendritic region seems to prevent dendritic growth and stabilize the solid liquid interface such that compact solidification morphology develops both for solid growing from a substrate and for individual particles suspended in the melt. References [1] D.B. Spencer, R. Mehrabian, M.C. Flemings, Metall. Trans. 3 (1972) [2] M.C. Flemings, Metall. Trans. 22A (1991) 957. [3] D.H. Kirkwood, Inter. Mater. Rev. 39 (1994) 173. [4] Z. Fan, Inter. Mater. Rev. 47 (2002) 49. [5] S. Ji, Z. Fan, M.J. Bevis, Mater. Sci. Eng. A299 (2001) 210. [6] A. Vogel, B. Cantor, J. Cryst. Growth 37 (1977) 309. [7] R.D. Doherty, H.-I. Lee, E.A. Feest, Mater. Sci. Eng. 65 (1984) 181. [8] A. Hellawell, in: D.H. Kirkwood, P. Kapranos (Eds.), Proceedings of the 4th International Conference on Semi-Solid Processing of Alloys and Composites, The University of Sheffield, Sheffield, UK, 1996, p. 60. [9] B. Niroumand, K. Xia, Mater. Sci. Eng. A283 (2000) 70. [10] A.M. Mullis, Acta Mater. 47 (1999) [11] J.M.M. Molenaar, L. Katgerman, W.H. Kool, R.J. Smeulders, J. Mater. Sci. 21 (1986) 389. [12] A. Das, E.J. Mittemeijer, Philos. Magn. A 81 (2001) 2725.