Scripta METALLURGICA Vol. 29, pp. 1285-1290, 1993 Pergamon Press Ltd. et MATERIALIA Printed in the U.S.A. All rights reserved

A L A T T I C E M O D E L OF S O L I D I F I C A T I O N

Y. S. Yang, N. Bhdm, T. B. Abbott and J. F. McCarthy BHP Research Melbourne Laboratories

245 Wellington Road, Mulgrave, Victoria 3170, Australia

(Received March 2, 1993) (Revised August 16, 1993)

I n t r o d u c t i o n

Over the past decade, considerable progress has been made in the mathematical modelling of macroscopic phenomena in solidification processes using conventional finite difference/finite element computer simulations. However, few studies encompass micrustrncture formation. The ability of conventional simulations to resolve phenomena on the microscopic scale is limited both by the system size and by the continuum equations on which they are based [1]. Recently, an alternative approach using rule-based Monte Carlo models has been developed to explicitly model micrustructure -- i.e. columnar and equiaxed grain growth [2,31, and the dendritic growth of pure materials [4]. The model described in this paper advances on previous work by treating the case of solutal dendrite growth (both columnar and equiaxed) in alloy solidification, It is a statistical mechanical model of interacting classical particles, incorporating the effects of phase change, solute and thermal diffusion, latent heat evolution and nucleation.

T h e m o d e l

The model consists of a mixture of several types of classical particles, occupying the sites of a regular lattice. For example, in two dimensions, the solid,cation of an alloy can be simulated using two types of particles, call them A and B, with A particles representing volume elements which are depleted in solute relative to the mean concentration, while B particles represent solute-enriched volume dements. The concentration of A-particles is PA and the concentration of B-particles is Ps such that PA + Ps = 1 (see figure 1).

Figure 1: Schematic represen- tation of the binary mixture.

: Solid A particle.

~ : Solid B particle.

( ~ : Liquid A particle.

( ~ : Liquid B particle.

1285 0956-716X/93 $6.00 + .00

Copyright (c) 1993 Pergamon Press Ltd.

1286 LATTICE MODEL V o l . 2 9 , No. 10

The probability PLs(cO that a liquid particle a (a = A, B) bonds to a specified nearest-neighbour solid particle is defined by

i I ((x) when PLS(+ : (o){ - rl/[rL(+- r,(+] }' when

when

T < Ts(a )

TsCc,) < T < TL@

T >_ TL(~)

(1)

where T s and T L are solidus and liquidus temperatures, respectively, and PI is a constant (see figure 2 a). The limits for the phase change probability curve have been chosen to ensure that solidification occurs over a range of temperatures between the liquidus and solidus temperature, in accordance with the equilibrium phase diagram. The particular form of the curve has been chosen in an ad hoc manner and would have to be refined by calibration with experiment if the simulations were required to be accurate in detail.

P,(a~

TdoO T~(a)

P2(a =•• T

Ts(a) Tda)

Figure 2: (a) Probability that a liquid particle is bonded to a specified nearest-neighbour solid particle. (b) Probability that a liquid particle is detached from a specified neare~t-neighbour solid particle, a = A, B.

A liquid particle solidifies by bonding to a neighbouring solid particle. Hence, the probability p, for solidi- fication of a liquid particle with no nearest-neighbour solid particles is given by

p ° = 1 - ( 1 - P L s ) " " - ( 2 )

Equation (2) indicates that a liquid particle will not change to solid unless it is in contact with a solid particle. It also indicates that it is easier for a liquid particle to change into solid when it has more nearest-neighbour solid particles. This tends to make the solid clusters compact. Liquid to solid transitions will not happen when the temperature is higher than T n.

The probability PaL(a) that a solid particle ot (or ---- A, B) detaches from a specified nearest-neighbour solid particle is defined by

I 0 } when T < Ts(CX )

PSL(a) = P2(a){ [ T - Ts(a)]/[TL(~x) - Ts(a)] when Ts(cO < T < TL(a ) 2

P2(~) when T > T~(c~)

(3)

p~ = (Ps : ) "" (4)

where P2 is a constant (see figure 2 b). A solid particle changes to liquid when it detaches from all of its nearest-neighbour solid particles and when it is in contact with a liquid particle. Mathematically, the melting probability p~ of a solid particle a with n° nearest-neighbour solid particles is given by

Vol. 29, No. i0 LATTICE MODEL 1287

when at least one of its nearest-neighbours is a liquid particle. Equation (4) indicates that it is easier for a solid particle to change to liquid when it has fewer solid nearest-neighbours. This tends to remelt the ramified clusters. Solid to liquid transitions will not happen when the temperature is lower than T s.

In real materials, the solid state is more "ordered" than the liquid state. In general, a liquid particle experiences an energy barrier when changing to solid state due to the structural change. This factor depends on the type of particle. It is assumed that

P, (~,) = s(c , )P, P2(a) = P ,~ = A , B (s)

where S(a) is the '~tructural factor" to account for the energy barrier when a particle a changes from liquid state to solid state. S is left as an adjustable parameter to fit the experimental observations for different materials and under different conditions.

During the solidification process, both the entropy and the interaction energy will change. However, they are equivalent in that both contribute to the latent heat. In the current model, the change in entropy is converted into an equivalent amount of change in the particle interactions. It is assumed that each nearest-neighbour pair has an attractive interaction energy given by EaB where a, fl -- (solid A, solid B,liquid A, llquid B}. Heat capacities of the particles are given by C~. Values for E and C are determined from the material properties.

When the state of a particle changes, there will be an associated change in the interaction energy. It is assumed that when the interaction energy of a bond is changed, temperatures of the sites connected by the bond will change to conserve the total energy. For instance, when the energy of a bond connecting sites a and b is changed by AE, the temperatures of the sites will be changed by --~-AE/C, and -~AE/C~,, where Co and C b are the heat capacities of the particles a and b, respectively. In the current model, the latent heat in the solid-liquid transition is a result of the interaction energy difference between the liquid and the solid state.

Physically, heat conduction is the result of inelastic collisions at the atomic scale. For computational simplicity, heat conduction is simulated using the complete inelastic collision of nearest-neighbour pairs. For instance, for a randomly chosen nearest-neighbour pair of particles a and b with initial temperatures To and T b and heat capacities C~ and Cb, respectively, their temperature after the collision will be

T = (COT,, +CbTb)/(C a +C~,) (6)

When particles are in the liquid state, they diffuse according to the Boltzmann distribution. In simulations, particle diffusion is accomplished by the flipping of nearest-neighbeur pairs. Denoting E n as the total interaction energy of the system when the particle configuration is H, and E n, as the total interaction energy of the system when the particle configuration is II', where H' is obtained from H by flipping one of its nearest-neighbour pairs, the probabilities Pn and Pn, that the particle configurations is H and H' are expressed as the following

Pn = exp ( -En /T ) en, = exp ( -En , /T) (7) exp( -En/T) +exp ( -En , /T ) ' e x p ( - E n / T ) + e x p ( - E n , / T )

where T is the average temperature of the nearest-neighbour pair. In the equations, it is assumed that the particles are at local equilibrium. The numerators are the well-known Boltzmann factors and the denominators are normalization factors. Equation (7) can be re-written as

Pn = ( l + exp [ ( En - En, ) /T] } -I, Pn, = 1-Pn. (8)

As far as the energy difference E n - E n, is concerned, when a nearest-neighbour pair flips, only those nearest- neighbour interactions which connect to either site of the pair need be considered.

It was mentioned previously that, the model does not produce spontaneous nucleation. The nucleation rate is sensitive to many parameters and it is commonly treated independently. Following the work of Zhu and Smith [4l, the probability for nucleation to occur in a lattice cell containing a liquid particle is calculated using the relation

P,, = N(a)S(a) [1 - T/TL(a)] 2 (9)

where N(a) is a parameter which may be adjusted to fit experimental observations.

1288 LATTICE MODEL Vol. 29, No. I0

Initialization: E, C, w, h, pA, temperatures, etc. | #

Diffusion: w¢ w h times I t

I Phase change: wp w h times I

I Heat conduction: wh w h times I t

I Nucleation: w. w h times I f

I - Move down: speed I t

[. Display, save, etc. I t

Figure 3: Flowchart of the model. tu, h: width and height of the lat- tice. ~ua: weight for diffusion, w~: weight for p h u e change, tub: weight for heat conduction. ~u,~: weight for nucleation, speed is a program parameter which simulates the cast- ing speed for a continuous casting. Its value is amumed to be zero in the following numerical examples.

In practice, a uniform random number r, such tha t 0 < r < 1, is generated and compared with the nucleation probability P,, calculated for a ceil containir~g a liquid particle. If r < P~, then a liquid-solid phase change occurs. Nucleation can then occur through the liquid-solid phase change of a particle which is not in contact with a solid particle. Therefore, nucleation in the bulk of the liquid can occur. The lower the temperature is below TL(a), the higher is the nucleation rate.

The mechanisms discussed above are integrated in a computer program as shown in the flow chart in figure 3.

Numerical results for ~-Fe

In this section, connections of the model parameters with material properties and experimental conditions are discussed. Using ~f-Fe as an example, some numerical results on square lattice are presented.

Tr. and T s in equations (1) and (3) are determined by material properties. For 7-Fe, the melting temperature is T l - 1763K, and the solid-liquid range is A T 1 ---- 72K [5]. For "y-Fe containing approximately 0.6% of carbon in weight, A-particles(i.e. volume elements depleted in carbon) have higher transition temperatures than B- particles (i.e. volume elements enriched in carbon). It follows tha t

T L (A) = 1763K, T s (A) = 1727K, TI; (B) = 1727K, T s (B) = 1691K. (10)

For simplicity, it is assumed that the unit of energy is the Boltzmann constant k so tha t energy is expressed in terms of a temperature scale and the heat capacities of particles become pure numbers. Elements of the heat capacity vector can be expressed as

C a = c,w/lOOOp=R (11)

where a = {solid A,solid B, liquid A, liquid B} for the binary system, c= and p~ are the specific heat per unit volume and the density respectively, w is the atomic weight and R is the Molar gas constant.

For 7-Fe, c~ = 5.74 × 10SJ/mSK, c, = 5.73 x 10e J/ruSK, p~ -- 7.0 x 10Skg/m s, p, = 7.25 x 10Skg/m z [5] where l refers to the liquid state and s refers to the solid state. Using the atomic weight w = 55.8 for Fe and the Molar gas constant R -= 8.314J/Kmol [6], the dimensionless specific heat can be calculated as

C = (CSoli d A, CSolid B, CLiquid A, CLiquld B) = (5.3, 5.3, 5.5, 5.5) (12)

Elements of the interaction matr ix can be estimated using the value of the latent heat for solid-liquid transit ion Ah I and the value of the latent heat for liquid-gas transition AA 9. For Fe, Ah~ = 349.56k J/ tool and A h I = 13.81k J/ tool [6]. Assuming that the interaction in the gaseous s tate is zero, the interaction in liquid steel is calculated as

- - 2 A h J z R = 0.5 X 349.56 x 103/8.314 ---- - 21022K (13)

Vol. 29, No. 1O LATTICE MODEL 1289

and the interaction in solid steel is calculated as

-2 × (Ah, + A h t ) / z R -~ 0.5 × (349.56 + 13.81) × 103/8.314 = -21853K (14)

where the factor of 2 comes from the fact that each bond is shared by two sites and z = 4 is the coordination number of the square lattice in figure 1. The actual interaction matrix is set up from the physical considerations that the solid-liquid interactions should not be weaker than the liquid-liquid interactions but should not be stronger than the solid-solid interactions. The croes-type (A-B) interactions should not be stronger than the same-type (A-A or B-B) interactions. Assuming that the solid structure is of A-type, a solid particle should have a weaker interaction with a liquid B particle than with a liquid A particle. Also, E is a symmetric matrix.

In the simulations for "y-Fe, the following matrix is used

E =

[ -21853 -21853 -21438 -21022 | E.oIB..olA[ E.ol A, .ol A E.olE.ol B,-ol I ~ A ' ~l B E.olE.olB,|iqAA, |iq A E.ol A, liq BB = | --21853 --21853 --21438 --21022 |

S'°IB'iiq B ~--21438 --21438 --21022 --21022 1 Eliq A, ~I A Ellq A, sol B Ellq A, liq A Eilq A, |iq \ Eilq B, ~l A Eliq B, soIB Eliq B, liq A Euq B, llq B \--21022 --21022 --21022 --21022/

(15)

For the nucleation rate, the following values are used

w~ =0.0003, N(A) = 1, N(B) = 1. (16)

For the first example shown in figure 4, the relative weights of w v for phase change, w d for particle diffusion and w h for heat conduction, and the structural factors S(A) and S(B) take the following values

wp----0.1, w d=0 .7 , w h = 3 . 0 , S(A)=I .O, S(B)=O.O1. (17)

Figure 4: Solidification of the -y-Fe on square lattice. The initial temper- ature of the interior region is T O ---- 1793°K. The boundary temperature T b = 400°K. The weight for heat conduction is w h ---- 3.0. (a) After 900 iterations. (b) After 1800 iterations.

F i g u r e 5: Sol idif icat ion of the ~- Fe on square lat t ice . T h e initial t e m p e r a t u r e of the interior region is T O = 1703°K. T h e boundary t e m p e r a t u r e T b = 1400°K. The weight for heat conduct ion ~v~ = 15.0.

1290 LATTICE MODEL Vol. 29, No. i0

In the figure 4, different grey-scales represent solid A, solid B, liquid A and liquid B particles respectively. The lattice size is 250 x 300. Initially, A and B particles are distributed randomly with PA = PS -- 0.5. Particles on the left, right and bottom boundaries are in solid state and all the other particles are in liquid state initially. The left, right and bottom boundaries are kept at a constant temperature of 400°K and the top boundary is kept adiabatic. The initial temperature of other particles is 1793°K. Due to the low boundary temperature, the solid-liquid interface is considerably under-cooled. The high temperature gradient generated ensures columnar growth from the surface to the center line. There is a build up of solute enriched liquid at the centerline which simulates the central macro-segregatiun. For the second example shown in figure 5, the left, right and bottom boundary temperature is kept at 1550°K and the weight for heat conduction is to~ -- 15. All the other parameters are the same as the first example, Due to the lower thermal gradients, some grains have nucleated in the center, resulting in a columnaz to equiaxed transition.

The morphology of the solid, in each case, is irregtflar and highly branched with solid A tending to occur more within the core of the branches. The liquid composition varies with distance away from the solid interface. In the bulk of the liquid A and B particles are uniformly mixed, while close to the solid interface the liquid becomes depleted in A particles (see figure 4 a).

Discussion

A significant feature of the model presented in this paper is the incorporation of solute partitioning at the dendritic scale. The irregular branched morphology of the solid forms in a manner analogous to dendritic morphologies that arise from constitutional undercooling in real materials. Because of the different structure factors for A and B, liquid A becomes solid more readily, giving rise to a layer of A-depleted liquid adjacent to the solid. Irregularities on the solid interface will tend to protrude through this layer into relatively A-enriched liquid and so will grow preferentially.

Two numerical examples, using material properties for 7-Fe, are given to show the effect of the boundary temperature and the thermal conductivity upon the final solidification structure. The numerical simulations accord qualitatively with experimental observations. However, they are limited by being two dimensional. Also, it is not clear how the model parameters (e.g. the relative weights for heat conduction, particle diffusion, nucleation rate, phase change rate) are quantitatively related to the corresponding experimental parameters (e.g. thermal conductivity, thermal diffusivity, particle diffusivity). Further work is needed before the model can be quantitatively compared with experiment.

The model needs to be improved in several aspects. To approximate the solidification of steel with a binary mixture is too simplistic. Inclusion of more types of particles, or the use of a continuous carbon distribution, will be incorporated in future work. Also, work is in progress to incorporate growth anisotropy which plays a critical role in determining the fine structure of dendrites. Other features which could be incorporated are the effects of density differences in the liquid, and solidification shrinkage. This would allow a simulation of thermosolutal convection, shrinkage driven macrosegregation, and settling of equiaxed crystals.

Conc.lusions

• A model of solidification has been developed from a statistical mechanics basis that incorporates nucleation, thermal and solute di~usion, and latent heat of the phase change. The model can simulate the dendritic growth of columnar and equiaxed crystals and the segregation of solutes. The model has been applied to the solidification of~/steel and simulates the effects on the solidification pattern of nucleation conditions, thermal conductivity, thermal capacity, solute diiTusivity, and mould temperature.

• Some limitations in the model were noted. Techniques will be developed to further advance the simulations.

References

2 J.A. Spittle and S. G. R. Brown, Acta Metall. 37 1803 (1987) P. Zhu and R. W. Smith, Act~ Mete//. Mater. 40 683 (1992) S. G. R. Brown and J. A. Spittle, Scripta Metallurgica et MateriaIia 27 1599 (1992) W. Kurz and D. J. Fisher, Fundamenta/s of Solidifw~atfon (Trans Tech, 1989) H. L. Anderson, A Physicist's Desk Reference (AIP, 1989)