Stochastic Eutectic Growth

Transcription

1 VOLUME 72, NUMBER 5 PHYSCAL REVEW LETTERS Stochastic Eutectic Growth 31 JANUARY 1994 K. R. Eler, Frangois Drolet, J. M. Kosterlitz 2 an Martin Granti Department of Physics, McGill University, Rutherfor Builing, S6 University Street, Montreal, Quebec, Canaa SA BT8 Department of Physics, Brou)n University, Provience, Rhoe slan 8912 (Receive 1 July 1993) A full phase fiel moel of eutectic growth is propose, which incorporates the generic features of a eutectic phase iagram an reuces to the sharp-interface moel in the appropriate limit. Large scale two-imensional simulations are presente for the isothermal soliification of a uniformly unercoole eutectic melt, in which the Avrami exponent is 3. The results of this stuy ientify three possible growth mechanisms: iffusion limite growth, lamellar growth, an spinoal ecomposition. PACS numbers: 64.7.Dv, 5.7.Fh, 82.2.Mj A eutectic is characterize by a point in the temperature concentration plane (T, C) known as the eutectic point at which a liqui coexists with two soli phases of ifferent concentration. sothermal soliification of a eutectic liqui is a complex process involving the nucleation an growth of two soli phases an can lea to a multitue of microstructures. The stuy of the kinetics of such pattern formation an omain growth has been greatly enhance by the concepts of universal ynamical scaling. t is now generally accepte [1,2] that systems with conserve an nonconserve orer parameters form separate universality classes efine in part by power law growth of the average omain size with ynamical exponents z of 1/3 an 1/2, respectively. Eutectic soliification involves the coupling of conserve concentration an nonconserve liqui/soli orer parameter fiels. To investigate this, a full phase fiel moel is introuce. A phase fiel moel is neee for this stuy since the stanar sharp-interface moel [3 6], use for irectional eutectic growth in which a liqui/soli interface is pulle through a temperature graient, cannot treat the stochastic nature of nucleation an is ifficult to implement for a collection of multivalue interfaces. To verify the moel an corresponing iscrete map, many known results for irectional eutectic soliification [3,5] are recovere. Finally large scale numerical simulations of isothermal eutectic soliification from an unercoole melt are presente. These inicate three possible growth mechanisms for the concentration fiel: iffusion limite growth (z = 1/2), lamellar growth (x = ), an spinoal ecomposition or Ostwal ripening (x = 1/3). The moel is presente in terms of a free energy E which is a functional of c (x C CE, the eviation from the eutectic concentration, an a liqui/soli orer parameter Q. To lowest orer, E can be written as nomenological constants. Soli (liqui) phases are represente by Q & (1[/ ( ). T is fixe externally, which is an excellent approximation for 2D films, metals, an metallois where concentration iffusion is orers of magnitue slower than that of T, so latent heat generate by the transition may be ignore. The ynamics is realize by Langevin equations, Oc4 ay/at = r~(sy/s-y) + q~, ac/at = r, V'(bZ/bc) + ~.. S S (2) (3) are mobilities, (g@(r,t)rig(, )) = 2'ykbTb(r)b(t), an ([7,(r, t)rk(, )) = 21',kbT~[7zb(r)b(t). The meanfiel phase iagram of this moel is illustrate in Fig. 1 for parameters (r, b, u), n, P) = (1, 1,,.15,.15). Also inclue in this iagram are the metastable extensions of the liquius an solius lines. When u) ( P(r/u)i/z an n(r/u)i/z (( 1, the eutectic point is given by X(c, cp) fr[/(cg)+kc[vrp[ /2+K, [V, c[/2[(2),.5 where f(c, tp) = rqz/2 + uq /4 + (nkt pc )@ + u/c /2+ bc /4, b,t = T T with T the melting temperature at P =, an the other parameters are phe- C FG. 1. Phase iagram of Eq. (1) for (r, b, u), n, P)= (1, 1,,.15,.15). Dashe (soli) lines: bounaries of metastable (stable) phases. 8 (C) is & (liqui ( 2[['/ ) phase

2 VOLUME 72, NUMBER 5 PHYSCAL REVE% LETTERS 31 JANUARY 1994 (DT~, c~) = ([2P(r/u) i!2 u!]z/8ba, ). Similar expressions can be erive for the coexistence lines an chemical potential y,. Phase fiel moeling of liqui/soli transitions has been escribe elsewhere [7]. The sharp-interface moel can be obtaine from Eqs. (2) an (3) using stanar techniques [8,9] in the appropriate limit. The two assumptions that must be mae are that the interface of with ( is in local thermoynamic equilibrium an that the curvature is small or r( &. Also, the interface must exist. Note that the phase fiel moel provies a phenomenological escription of eutectic growth even when growth cannot be escribe by an interface moel. The Gibbs-Thomson conition is 6c/c;, = +[o!!,+ (AT AT@)/m~;, ], where p = o/(c;, By/Bc), o = 2K, f u(bc"/bu), c;, is the miscibility gap, m is the slope of the liquius, c~ is a stationary planar interface at AT = AT@, an bc an b, T are evaluate at the interface. The plus an minus signs refer to the c ( an c ) phases. ntegration of Eq. (3) across a moving interface gives v (c ci)n y = D!Bbc/Bu~~ D, Bbc/ Bu] where the subscripts l an s refer to the liqui an soli sies of the interface an v is the velocity normal to the interface. The iffusion constants are given by D!, = ', (B f/bc ) ~!,, These bounary conitions an the fact that Eq. (3) reuces to a iffusion equation in the liqui an soli phases comprise the sharp-interface moel [3 6]. For computational efficiency, a simple iscrete map was constructe from Eqs. (2) an (3) using Euler's metho for the time erivative an a nearest neighbor approximation for the Laplacians. This was simulate on both square an hexagonal lattices with time step.5, mesh size 1.3, '@ = ', = r = b = K, = 1, an P = a. As a consequence of the universality of omain growth, the continuum moel, iscrete map, an sharp-interface moel are all equally vali phenomenological moels of eutectic growth, which has been exploite in many numerical stuies [2]. To support this, the map was use to recover many known results [3,5] on irectional eutectic growth. Directional eutectic growth is implemente numerically through b,t = G(y vt), where y is the pulling irection an v is the pulling velocity. n these simulations, (G, a, w, Ky) = (.1,.15,.1, 1). For a given v, steay state interface profiles were obtaine as a function of lamellar wavelength A an, as in other works [3,5], the minimum unercooling assumption was use to select A an the average interfacial unercooling LT. Details will be reporte in a future paper. A summary of this stuv is shown in Fig. 2 in which the following well-known [3,5, 1] relationships were obtaine: A oc v [Fig. 2(a)] an (ET~ AT~) oc v [Fig. 2(b)]. Discrete branches of solutions exist for ifferent initial conitions [5]. The recently iscovere [4,5, 11,12] tilt wave instability was stuie by fixing A an increasing v until the lamellae unergo a tilting transition. Figure 2(c) shows the tilt.2 Q5 4 p vx &3 3 8 ) 2 p 2 s i i ( V x &l 8 & ~.5 4 i i i r i «& i i i i i v&13 1 J 1 FG. 2. Simulation of irectional eutectic growth. v as function of (a) 1/A; (b) (DT ATz); (c) tilt angle (e). () Points are soli/soli interface position ten lattice spaces behin the liqui/soli front as a function of time. angle as a function of v, where 8 = ' correspons to lamellae that are perpenicular to the interface. All these results are consistent with earlier simulations [5]. Tilt waves can also be generate by ranom initial conitions at sufficiently large v [Fig. 2()] or by thermal Quctuations. Using the same iscrete map, a 2D numerical stuy of isothermal soliification of a uniformly unercoole eutectic liqui was performe on a hexagonal lattice with perioic bounary conitions an cp = c@ =. Nucleation an growth of soli roplets an the phase separation process are illustrate respectively in the 6rst an secon columns of Fig. 3. The parameters use are of size (K~, u!,a, AT) = (1/8,,.15,. 4) on a system 256 x 256 with thermal fluctuations of magnitue.29 in g. The initial unercoole liqui state is represente = 1+ ri an c(x, y) = rl, where ]r[ &.1 is ranom. The thir column of Fig. 3 correspons to a secon type of simulation in which a small soli roplet is incorporate in the initial conitions but without thermal fluctuations. The system size was 512 x 512 an the parameters were (K@,tu, a, AT) = (1,,.15,.2). n these an other simulations, the lamellar wavelength A* selecte ecrease with increasing interface velocity v which is proportional to 6T, although the minimum unercooling assumption cannot be mae as 6T is fixe. The statistics of orering for the 6rst set of simulations was analyze by monitoring the soli volume fraction X(t) an the spherically average structure factors of both fiels: S~(k, t) :(]g(k, t)[z) an S,(k, t)

3 h ~.h i VOLUME 72, NUMBER 5 PH YSCAL REVEW LETTERS 31 JANUARY 1994 a) l g) e) -O 15 x 1 A M.4. 3L h,.2 2 t 4 FG. 3. Grey scale plots of orering fiels. n (a) to (c) black, soli (@ ) ); light grey, liqui (g ( ) at times t = 1, 16, an 315. () to (f) are counterparts of (a) to (c) for concentration fiel. White: c =.6; black: c =.6. (g) to (i) show concentration fiel at t = 25, 375, an 5. See text for parameters an initial configurations. (c(k, t)2). The results were average over 33 inepenent runs. X(t) can be fitte by a Kolmogorov [13] form with an Avrami exponent of 3: X(t) = 1 e where is the nucleation rate an ts the waiting time. The fit an sample structure factors for both fiels are shown in Fig. 4. S@(k,t) shows poor quantitative agreement with Sekimoto's expression [14]. The peak position k~, height S~ = S,(k~, t)/(c ), an with tv of S,(k, t) are isplaye in Fig. 5. Before interpreting these results, it is useful to consier several simple examples. At late times, when X(t) = 1, Eq. (3) reuces to the Cahn-Hilliar-Cook [15] moel of spinoal ecomposition. n this limit, k /tv an S k are constant an k oc t * with x = 1/3. 2x/k is interprete as the average omain size. n contrast, if the omain size A' is fixe, then k is also an S is proportional to the average size of the growing roplets. Since omains grow at constant v an S~ oc /tu, /tv oc t~. This is seen in the last column of Fig. 3 an will be terme lamellar growth. f a collection of uncorrelate soli roplets each consisting of a set of stripes or lamellae in c of average size 2x/k is consiere, the structure factor takes the form S,(k, t)j(cz) = k "(k /ur) f([k k ]/tv). n principle m an k are both time-epenent quantities. During the early stages of growth, both (S~tU/k~) / an 1/k increase as t~/~. n this time regime, the initial lamellar wavelength is smaller than A' an is increasing at the expense of the surrouning liqui matrix. This FG. 4. Open squares: soli volume fraction X(t); the soli line is a fit to the Kolmogorov [13]form given in the text. Top left inset: Sample structure factor for Q fiel at t = 25. Bottom right inset: Sample structure factor for c fiel also at 5 = 25. iffusion limite process leas to an exponent of 1/2. As the roplets coalesce, spinoal ecomposition takes over, which will eventually give an exponent of 1/3. The smaller exponent seen in Fig. 5 (z 1/4) is simply a precursor to the asymptotic exponent of 1/3. Many numerical stuies [2] have shown that the ynamic exponent in spinoal ecomposition starts at a value smaller CY 2 ~ ~ ~ ~ ~ 1 ~ ~ ~ ~ ' o o[k /w] ~ [S W/k ]~' ~ ~ 7.5 tn(t) FG. 5. Open circles, soli circles, an open triangles correspon to n(q) = jn(1/k ~ ) +.55, 1n([S tu/k ] ) + 1.7, an 1n(k /m), respectively

4 VOLUME 72, NUMBER 5 PHYSCAL REVEW LETTERS 31 JANUARY 1994 than 1/3. These simulations may also be hampere by an initial state that contains lamellar structures. The ynamics of k /tv provies aitional insight into the orering. Most striking are the three istinct plateaus seen in Fig. 5. Figures 3() an 3(e) inicate that the 6rst an secon plateaus correspon to the growth of the initial rop of precipitate an the next shell or layer of the opposite phase, respectively. The with to ecreases as the secon shell grows since it is highly correlate with the initial precipitate. The last plateau correspons to spinoal ecomposition occurring when the sample is approximately 97 soliifie. To summarize, three istinct growth mechanisms have been ientifie. nitially, iffusion limite growth occurs as shown in Figs. 3() an 3(e) leaing to a growth exponent of 1/2. f the soli rops are allowe to grow large without coalescence, as in Figs. 3(g) to 3(i), lamellar orering occurs in which k~ oc to an tv oc t ~. Finally when the soli rops coalesce spinoal ecomposition takes over as in Fig. 3(f). For this moel, spinoal ecomposition, or Ostwal ripening for off-eutectic concentrations, will be the infinite time mechanism. However, which growth mechanisms can be observe epens on the quench. Preliminary experimental results on the eutectic crystalization of amorphous metallic glasses have been reporte by Fischer et al. [16]. n these experiments, both the small an large angle time resolve x-ray scattering patterns are measure, which respectively measure correlations in the electron ensity (c) an the crystal structure (g). Thus the scaling exponents shoul be irectly measurable. However, irect comparisons are complicate by ifferences in the average ensity of the amorphous an crystal phases an lattice mismatches at grain bounaries. We are presently incorporating some of these features to better escribe such experiments. This work is supporte by the Natural Sciences an Engineering Research Council of Canaa an le Fons pour la Formation es Chercheurs et 1'Aie a la Recherche e la Province e Quebec. J.M.K. was partially supporte by NSF Grant No. DMR We thank Dr. Mark Sutton, Dr. Henry Fisher, Dr. lan Graham, an Dr. Mohame Laraji for useful iscussions. [1] J. D. Gunton, M. San Miguel, an P. S. Sahni, in Phase Ybunsitions an C~~ Phenomena, eite by C. Domb an J. L. Lebowits (Acaemic, Lonon, 1983), Vol. 8. [2] Y. Oono an S. Puri, Phys. Rev. Lett. 58, 836 (1987); S. Puri an Y. Oono, Phys. Rev. A $8, 1542 (1988); T. M. Rogers, K. R. EMer, an R. C. Desai, Phys. Rev. 8 3V', 9638 (1988). [3] K. A. Jackson an J. D. Hunt, Trans. Metall. Soc. AME 2$8, 843 (1966); 288, 1129 (1966). [4] Alain Karma, Phys. Rsv. Lett. 59, 71 (1987) [5] K. Kassnsr an C. Misbah, Phys. Rev. Lett. 85, 1458 (199); 88, 445 (1991); Phys. Rev. A 44, 6513 (1991); 44, 6533 (1991). [6] J. S. Langsr, Phys. Rev. Lett. 44, 123 (198); V. Datye an J. S. Langer, Phys. Rev. B 24, 4155 (1981). [7] A. A. Wheeler, G. B. McGaen, an W. J. Boettinger, Phys. Rev. E 47, 1893 (1993); A. A. Wheeler, B. T. Murray, an R. J. Suu'. far, Physica (Amsteram) 88D, 243 (1993). [8] B. Grossmann, H. Guo, an M. Grant, Phys. Rev. A 4$, 1727 (1991); J. B. Colliins an H. Levine, Phys. Rev. B $1, 6119 (1985). [9] K. R. Eler, J. Vinals, an M. Grant, Phys. Rev. Lett. 68, 324 (1992); Phys. Rev. A 48, 7618 (1992). [1] n Ref. [5] eviations from A oc v an DT ATE oc v' were observe at small v an explaine by corrections to the exponent 1/2. We also see eviations from 1/2 but prefer to explain them by corrections to scaling, A oc v '~ + b. The ata of Ref. [5] are also consistent with this form. [ll] G. Faivre, S. De Cheveigne, C. Guthmann, an P. Kurowski, Europhys. Lett. 9, 779 (1989). [12] B. Caroli, C. Caroli, an S. Faivrs, J. Phys. (France) 2, 281 (1992). [13] A. N. Kolmogorov, Bull. Aca. Sci. USSR, Phys. Ser. 3, 335 (1938). [14] K. Sekimoto, Physica (Amsteram) 1$5A, 328 (1986). [15] J. W. Cahn an J. E. Hilliar, J. Chem. Phys. 28, 258 (1958); J. W. Cahn, Acta Metall. 14, 1685 (1966); Trans. Metall. Soc. AME?42, 166 (1968); H. E. Cook, Acta Metall. 18, 297 (197). [16] H. E. Fischer, S. Brauer, J. O. Strom-Olsen, M. Sutton, an G. B. Stephenson, in nterface Dynamics an Growth, eite by K.S. Liang, Materials Research Society Symposium Proceeings No. 23? (MRS, Pittsburgh, 1992). 68

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