PHYSICAL REVIEW A VOLUME 45, NUMBER 10 15 MAY 1992

Temporally linear domain growth in the segregation of binary fluids

Sanjay Puri* and Burkhard DunwegInstiiui fur Physik, Johannes Gutenberg-Universitat Mainz, Postfaeh 3980, Staudinger Weg 7, D 650-0 Mainz, Germany

(Received 1 July 1991)

We report the numerical observation of temporally linear domain growth in a phenomenological

model of segregating fluids. Our observation is facilitated by the use of computationally efficient cell

dynamical system models.

PACS number(s): 64.75.+g, 05.70.Fh

Much recent attention has focused on the dynamics ofsegregation of binary mixtures quenched below the mixingtemperature, e.g., spinodal decomposition [Il. Speci-fically, it is now well established that the segregatingdomains are characterized by a unique, time-dependentlength scale L(t)-t~ (where t is time and p is called thegrowth exponent). When hydrodynamic effects are notrelevant (e.g., binary alloys), it is known experimentally[2] and numerically [3] that p has an asymptotic value ofI /3. When hydrodynamic effects are relevant (e.g. ,

binary fluids), it is theoretically argued [4] that p=lasymptotically, and this temporally linear domain growthhas also been observed experimentally [5]. The theoreti-cal arguments leave many open questions such as the pre-cise time regime and the volume fraction where this tem-porally linear growth law should hold. Numerical workclarifying such questions would be very desirable. Todate, however, there have been no numerical confirma-

tions of this growth law. In this paper, we report the nu-merical observation of temporally linear domain growth ina phenomenological model of segregating binary fluids.Our simulation is facilitated by the use of computationallyefficient cell dynamical system (CDS) models, which havequite successfully elaborated the nature of spinodaldecomposition in the case without hydrodynamics [6],andin a variety of other problems involving reaction-diffusionequations [7].

Our phenomenological model for the dynamics of segre-gating fluids is a variant of the so-called model H [8],which describes the dynamics of binary fluids and has hadconsiderable success in correctly predicting the dynamicalcritical exponents of binary fluids. As in model H, ourmodel consists of a scalar density (order parameter) cou-pled to the hydrodynamic velocity field. With appropriaterescaling [9], our equations have the dimensionless form(in three dimensions)

By(x, r) = —V [y(x, t) —y(x, t) +V y(x, t)] —aV [y(x, t)J(x,r)],

BJ;(x,r )=riV'J;(x, r )+o g VVk Ji, (x,t)+ ay(x, t)V;[y(x, t) —y(x, t )'+V'y(x, r)] .

k I

In (]), y(x, i) and J;(x,t) (i =1,2, 3) are respectively theorder parameter and the dimensionless velocity field as afunction of dimensionless space x [=(xi,x2, x3)] and timeI The resc.aled parameters in (I) are the coupling con-stant a (a =0 corresponds to the usual case of binary al-loys) and the transport coefficients ri and c7 (which are theviscosities). Before we proceed, some comments on theform of (I) are in order. First, we consider only the deter-ministic case as we are interested only in the asymptoticbehavior, which is not affected by the presence of thermalnoise [IO]. In the late stages of growth, the presence ofnoise terms only alters the interfacial profile betweendomains and this is an irrelevant factor for sufficientlylarge domain sizes, i.e., at asymptotic times. Second, (I)would reduce to the standard form of model H [8] if weimpose the additional constraint V. J(x,t) =0. However,as was pointed out by Farrell and Valls [9], this extra con-straint causes numerical complications and we do not im-pose it.

Farrell and Valls [9] have numerically studied a more

I

complicated version of (I) (with thermal noise and non-linear convective terms) in two dimensions. In the latestages of their simulation, they find a domain growth lawL(I)-t ' . However, it is well known that continuumhydrodynamics is not well defined in two dimensions andit is not clear how this would affect numerical results.Furthermore, the simulation of Farrell and Valls [9] isslowed considerably by the presence of the additionalterms. We believe that (I ) constitutes the minimal modelfor studying the late stages of spinodal decomposition in

Auids and use it to derive a computationally efficient CDSmodel, which we use in our simulations.

We do not go into the details of CDS modeling here asthese are well documented in the literature [6]. Essential-ly, this procedure can be understood as an unconventionalmethod of discretizing the corresponding partial diff'er-

ential equations, which enables the use of rather coarsemesh sizes without any loss of numerical stability. TheCDS model (in three dimensions) we derive from (I) hasthe form

R6977 @1992The American Physical Society

R6978 SAN JAY PURI AND BURKHARD DUN%EG

Itr(n, t+ I) =I'(n, t) ——,' Ag[A tanh(Itr(n, t)) —y(n, t)+(D/6)ADItr(n, t)] —aVD. [Itt(n, t)J(n, t)],

3

J;(n, t+1)= J;(n, t)+rthDJ; (n, t)+o' g Vo,;VD I, JI, (n, t) (2)

+attI(n, t )VD;[A t anh( tIr(n, t)) —Itr(n, t)+ (D/6)boy(n, t)],

where y(n, t) is the order parameter at the discrete latticesite n on a simple cubic lattice at discrete time t; and A,D, a, g, and a are phenomenological parameters. The pa-rameters A and D are respectively measures of the quench

depth and the diffusion. The other parameters are analo-

gous to the corresponding parameters in the continuummodel (1). In (2), Ao is the isotropically discretized La-placian operator whose action on a function f(n) is

defined as t)af(n) =—6[((f(n))) —f(n)], where the angularbrackets refer to the average of f(n) on the sites neigh-

boring n Th.is average includes the nearest-, next-nearest,and next-next-nearest neighbors in the relative ratio 6:3:2.Also, in (2), VD is the symmetrically discretized gradientoperator. The values of the parameters are dictated bythe requirements that the scheme be stable and that theresults be reasonable [6]. We choose A =1.5 and D =0.5,which are good values for simulating the case without hy-

drodynamics [6]. We can associate mesh sizes with thesevalues of A and D by comparing the CDS scheme with theusual Euler discretization scheme for partial differentialequations. The corresponding values are t)t=0.5 and

hx =2.45. It is worth noting that these mesh sizes are toolarge for a stable simulation using an Euler discretizationof the given equations [3]. In (1), we set t) =1 and cr=2,following Farrell and Valls [9]. The phenomenologicalparameters I) and a in (2) are then fixed as rt=rtI5, t/(Ax) =0.08 and rr =rjht/(hx) =0.17. This should not

be interpreted as a rigorous prescription for fixing param-eters but rather as a general rule to associate parametervalues in a CDS model with those in the correspondingpartial differential equation model. Finally, we discussthe most important parameter in (2), viz. a, which fixes

the strength of the coupling between the order parameterand the velocity field. We have studied domain growth asmodeled by (2) for a range of values of a. Stronger values

of a accelerate the onset of the asymptotic time regime in

which L(t )- t, whereas smaller values of a delay the on-

set of the asymptotic regime and give an extended periodof slower growth. In the limit a 0, we recover theLifshitz-Slyozov growth law L(t)-t 't . In this paper, we

present results for a=0.41, which corresponds (in theprescription defined above) to a=2 in (1). Detailed re-

sults for different values of a will be published elsewhere[11].

We have implemented (2) with the above parametervalues on an N lattice with periodic boundary conditions.The results presented here are for the case N=80. Weused a Siemens-Fujitsu VP100 vector processor and a sin-

gle update of 80 lattice took approximately 1 CPUsec.All results presented here are for the case of a criticalquench, where there are equal concentrations of bothcomponents in the mixture. Results for off-criticalquenches will be presented later [11]. The time-

dependent structure factor for the order parameter is

I

defined as

S(k, t) =(Itr(k, t) Itr( —k, t)), (3)

gkS(k, t)(k)(t) =

S k, t

In (4), the sum over the k values is cut off atk „. „=trJ3/2. We have confirmed that the structure fac-tor has decayed sufficiently up to this wave vector so that

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—1 0 1 2 3

k/&k &

FIG. I. Plot of S(k, t)&k&' vs k/&k) for structure-factor datafrom a simulation of (2) on an 803 lattice with periodic bound-

ary conditions. Data were obtained as averages over 40 indepen-dent runs from initial conditions corresponding to a homogene-ous state. The structure factors are for update times 2000,4000, and 6000 (marked by the symbols indicated).

where It(rk, t) is the Fourier transform of the order param-eter field at wave vector k and the angular brackets referto an average over the ensemble of initial conditions. Inthe discrete case, we calculate (using the NAGLIB rou-tine C06FJF) Itt(k, t) as the discrete Fourier transform ofthe order parameter field Itt(n, t). The wave vectors k liein the first Brillouin zone of the lattice, viz. k=2tr(n„,n~, n, )/N where n„, n~, and n, are integers between andincluding N/2 and —N/2 —1. The structure factor is cal-culated as an average over 40 runs from different initialconditions, each of which consists of the order parameterand velocity field uniformly and randomly distributedabout a zero background with amplitudes 0.05 and 0.1,respectively. The structure factor is normalized asgkS(k, t)/N =1. The vector function S(k,t) is thenspherically averaged to give the scalar function S(k,t).The time-dependent characteristic length scale L(t) isdefined as the reciprocal of the first moment of the scalarstructure factor S(k,t), i.e., L(t) =[(k)(t)] ', where

TEMPORALLY LINEAR DOMAIN GROWTH IN THE. . . R6979

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FIG. 2. Plot of S(k, t)(k) vs k/lkl for structure factor datafrom a CDS simulation of model B [viz. (2) with a =01, imple-mented on an 80 lattice with periodic boundary conditions.The data were obtained by the same procedure as that describedin the text for Fig. l. The structure factors are for update times2000, 4000, and 6000 (marked by the symbols indicated).

the sum is a good approximation of the infinite integral.For purposes of comparison, we have also performed a

CDS simulation for the case without hydrodynamics (theso-called model B) on an 803 lattice. Our CDS model forthe case without hydrodynamics is simply (2) with a=0.We use the same values for A and D as in the case withhydrodynamics and the procedure whereby we obtainedthe structure factor and the characteristic length scale isthe same as that just described.

Figure I shows the scaled structure factor $(k, t)x [(k)(t)] plotted as a function of the scaled wave vectork/(k)(t) for data from update times 2000, 4000, and6000. The excellent data collapse indicates that dynarni-cal scaling [12] is valid and that the domain growth ischaracterized by a unique length scale. The form of theuniversal function in Fig. 1 is similar in shape to the casewithout hydrodynamics (shown in Fig. 2, for comparison)with the only major difference being that the peak of thescaled structure factor for model B is considerably higher.For earlier times, the form of the universal function forthe case with hydrodynamics is identica1 to that for thecase without hydrodynamics. Notice that there are no ad-justable parameters in our definition of the structure fac-tor.

Figure 3 shows the characteristic domain size L(t) as afunction of the update time r for our model (2) (markedby circles) and the case without hydrodynamics (markedby squares) on a double logarithmic scale. After an initialtransient regime (which can be extended by using a small-er value of a), the domain size for the hydrodynamic casegrows linearly in time, viz. , L(r) r At tim—es s. omewhatbeyond those shown in Fig. 3, freezing sets in because ofthe finite size of the system. To ensure that our data arenot affected by finite-size effects, we have also performed(less thoroughly) simulations on lattices of size 64 and100 . The results are the same as those presented herewith the only difference being that the onset of freezing is

-1.03.0

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6.0Int

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FIG. 3. Plot of the characteristic length scale L(t) as a func-tion of update time I for the case with hydrodynamics (marked

by circles) and the case without hydrodynamics (marked bysquares) on a double-logarithmic scale. The error bars on thedata points are smaller than the symbol sizes. The solid lineshave slopes of 3 and l, as marked.

S.P. is grateful to the Deutsche Froschungsgernein-schaft (DFG) for supporting his stay at Mainz under Son-derforschungsbereich 262. Both authors are grateful toK. Binder for many useful discussions and remarks andalso a critical reading of this manuscript. The necessarycomputer time was provided by the RHR at Kaiserslau-tern.

delayed in the larger systems. Domain growth for thecase without hydrodynamics is seen to obey the usualLifshitz-Slozov growth law L(t)-r '~ .

To summarize, we have reported the numerical obser-vation of a temporally linear domain growth in a phenom-enological model of segregating fluids. Our observation isfacilitated by (i) the use of the simplest possible(minimal) model, and (ii) the use of a CDS model to ac-celerate the onset of the asymptotic regime. The onset ofthis asymptotically linear growth is determined by thestrength of the coupling parameter a. For smaller valuesof a, we obtain a Lifshitz-Slyozov growth law for the earlyand intermediate times and this crosses over to the tem-porally linear growth reported here. We will provide de-tailed results in an extended publication, where we willpresent results for the decay of the current-current corre-lation function and also a comparison of our results withexperiments on binary fluids.

Note added. After the submission of this manuscript,we have become aware of cornplernentary works by Kogaand Kawasaki [13] and Shinozaki and Oono [14]. Kogaand Kawasaki [13] report a linear growth law in a binaryfluid system, where the hydrodynamic interactions are de-scribed by the Oseen tensor. Shinozaki and Oono [14]study spinodal decomposition in a Hele-Shaw cell.

R6980 SANJAY PURI AND BURKHARD DUNWEG

Permanent address: School of Physical Sciences,Jawaharlal Nehru University, New Delhi 110067, India.

[I] For reviews, see J. D. Gunton, M. San Miguel, and P. S.Sahni, in Phase Transitions and Critical Phenomena,edited by C. Domb and J. L. Lebowitz (Academic, New

York, 1983), Vol. 8, p. 267; K. Binder, in Phase Transformations of Materials (Materials Science and Technolo

gy), edited by P. Haasen (Springer-Verlag, Berlin, 1990),Vol. 5, p. 405.

[2] See, for example, A. F. Craievich, J. M. Sanchez, and C.E. Williams, Phys. Rev. B 34, 2762 (1986); B. D. Gaulin,S. Spooner, and Y. Morii, Phys. Rev. Lett. 59, 668(1987).

[3] Y. Oono and S. Puri, Phys. Rev. Lett. 5$, 836 (1987); S.Puri and Y. Oono, Phys. Rev. A 3$, 1542 (1988); T. M.Rogers, K. R. Elder, and R. C. Desai, Phys. Rev. B 37,9638 (1988).

[4] E. D. Siggia, Phys. Rev. A 20, 595 (1979); see also K.Kawasaki and T. Ohta, Prog. Theor. Phys. 59, 362(1978).

[5] See, for example, N. Wong and C. Knobler, J. Chem.

Phys. 69, 725 (1978); also Phys. Rev. A 24, 3205 (1981);Y. C. Chou and W. I. Goldburg, ibid 2.0, 2105 (1979);23, 858 (1981);D. Roux, J. Phys. (Paris) 47, 733 (1986).

[6] Y. Oono and S. Puri, Phys. Rev. A 3$, 434 (1988); alsothe first two papers of Ref. [3].

[7] N. Parekh and S. Puri, J. Phys. A 23, LI085 (1990); M.Mondello and N. Goldenfeld, Phys. Rev. A 42, 5865(1990).

[8] K. Kawasaki, Ann. Phys. (N. Y.) 6l, I (1970); P. C.Hohenberg and B. I. Halperin, Rev. Mod. Phys. 49, 435(1977).

[9] J. E. Farrell and O. T. Valls, Phys. Rev. B 40, 7027(1989).

[10]S. Puri and Y. Oono, J. Phys. A 2l, L755 (1988).[I I] S. Puri and B. Dunweg (unpublished).[12] K. Binder and D. Stauffer, Phys. Rev. Lett. 33, 1006

(1974); K. Binder and D. Stauffer, Z. Phys. B 24, 407(1976).

[13]T. Koga and K. Kawasaki, Phys. Rev. A 44, R817 (1991).[14] A. Shinozaki and Y. Oono, Phys. Rev. A 45, 2161 (1992).