VOLUME 74, NUMBER 11 PH YS ICAL REVIEW LETTERS 13 MARcH 1995

Reaction-Controlled Morphology of Phase-Separating Mixtures

Sharon C. Glotzer, ' Edmund A. Di Marzio, ' and M. Muthukumar'~'Polymers Division and Center for Theoretical and Computational Materials Science,

National Institute of Standards and Technology, Gaithersburg, Maryland 208992Polymer Science and Engineering Department, University of Massachusetts at Amherst, Amherst, Massachusetts 01003

(Received 17 December 1993)

The role of externally-controlled chemical reactions in the selection of patterns in phase-separatingmixtures is presented. Linearized theory and computer simulation show that the initial long-wavelengthinstability characteristic of spinodal decomposition is suppressed by chemical reactions, which restrictdomain growth to intermediate length scales even in the late stages of phase separation. Our findingssuggest that such reactions may provide a novel way to stabilize and tune the steady-state morphologyof phase-separating materials.

PACS numbers: 64.60.Cn, 47.54.+r, 61.41.+e, 64.75.+g

Pattern formation in reaction-diffusion systems occursthroughout nature. It is well known, for example, that spi-ral waves and other interesting steady-state patterns can begenerated by simple chemical reactions [1]. In contrast,transient patterns are formed during phase separation byspinodal decomposition in both small molecule and poly-mer mixtures [2,3J. These patterns, whose characteristiclength scale depends on the specificity of the componentsof the mixture, coarsen and disappear when macroscopicphase separation is achieved at asymptotically long times.It would be desirable to devise a mechanism by which thesephase-separating morphologies could be stabilized. In thisLetter, we argue that chemical reactions can be used to sta-bilize and tune the characteristic length scale of patternsarising in phase-separating materials. Unlike the usual sce-nario of spinodal decomposition, where concentration fluc-tuations of all length scales larger than a certain criticallength scale spontaneously grow with time, we show thatchemical reactions introduce two cutoff lengths, therebyrestricting the growth of fluctuations to a narrow range oflength scales. Pattern tunability is achieved through ap-propriate selection of the rate constants governing the ex-ternally controlled chemical reactions [4]. Interestingly,our simplest model describing this phenomenon results inan equation identical in form to an empirical equation usedto model microphase separation in block copolymer melts[5] and other systems [6] where short-range attractive andlong-range repulsive interactions compete. However, un-like the majority of these pattern-selecting systems, chemi-cal reactions offer a tremendous opportunity to controlthe final morphology of phase-separated materials, espe-cially polymers. Since the kinetics of spinodal decompo-sition in polymer mixtures and small molecule mixturesis similar in many respects [7], for simplicity we focushere on the effect of chemical reactions on small moleculesystems [8].

Consider a mixture of molecules of types A and8 which has been quenched to a thermodynamicallyunstable state, and which simultaneously undergoes thereaction [4]

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I,n~A + n~B n~C, (1)

I2

where I J and I 2 are the temperature-dependent forwardand backward reaction rates, respectively, and the n; arethe stoichiometric coefficients. The equations of motionfor the concentration P, (x, t) of component i are

BfJA —nAr14A" QB' + nAr24c' + hA,

BfJB nBrl fA @B + nBr21t c + hB (2)

A B. (3)I2

The equation of motion for this immiscible, chemicallyreactive system is [11]

'~ = Mv '"~' —r, p+ r, (1 —p),Bt 6$

where we have dropped the subscript "A" on the lo-cal concentration@ and assume incompressibility (@A +pB = 1). Quenching below the spinodal temperature willresult in demixing via spinodal decomposition and, simul-taneously, mixing via the reaction A=B.

The free energy functional F(P/I is typically writtenas the sum of the bulk free energy f(@), which has adouble-well structure below the critical point, and the

(4)

~Ac nA n.cBf

& ' Jc + ncrlyA @B ncr2$c + hc,

where J; = —g, M;, (6F/6$, ), F is the free energy func-tional appropriate to the mixture, and the h s are reac-tion terms arising from spatial inhomogeneities. Alternateapproaches to the coupling of diffusion and chemical reac-tions are possible [9]. In these equations, the local trans-port of heat and momentum, which in general couple tomass flow [10J, has been ignored. The essential physicsunderlying the stabilization and tunability of pattern for-mation in phase-separating materials can be illustrated byconsidering a simpler, two-component system undergoingthe following reaction:

VOLUME 74, NUMBER 11 PHYSICAL REVIEW LETTERS 13 MARcH 1995

usual square-gradient approximation to the interfacial freeenergy [2,3,12]. For small molecules mixtures, Eq. (4)can be written as [13]

2

Bt (BP )= AV'i —2~V'y —(I, + I,) y + r, , (5)

where A —= Mk&T and f(P) has been divided by k&T. Welinearize Eq. (5) about the initial average concentrationbefore the quench, @0, and replace P by Po + 8$, where6$ is a small perturbation about @o [2]. After Fouriertransforming, we obtain

" = [2~Ak'(k,' —k') —(I, + 1,)]a@„

+ [r, —(I., + 1.,)@,]a(k), (6)where (8 f/8 P )@„(0 in the two-phase region, k,. —=

(i d f/8P i p, /2')'t, and 6(k) = 0(1) when k @ 0 (k =0). For nonzero values of k, this equation is solved by asimple exponential function [14],

6$„(t)= 6@„(0)e"l"l', (7)with the growth rate

(k) = 2 Ak (k, —k ) —(I'i + I'2). (8)Figure 1 shows the growth factor cu(k) for spinodal de-composition both with and without chemistry. Withoutchemistry (I &

= I 2 = 0), the growth factor is the usualone predicted from Cahn's linear theory, with a cutoffat large wave vector k, . Thus concentration fluctuationswith k ~ k, decay and those with k ( k, grow, with themaximum growth rate occurring for k = k, /~2. How-ever, the simultaneous occurrence of the reaction A=8decreases the usual growth factor by an amount propor-tional to the sum of the forward and backward reactionrates I ~ and I 2. This shifts the small-wavelength cutoffto larger wavelengths, and introduces a large-wavelengthcutoff. Thus concentration fluctuations at large wave-lengths (small k) are suppressed by the reactions. Suchsuppression of long-wavelength fluctuations is a natural

SD——— SD+ CR

FIG. l. Early-time growth factor cu(k) vs wave vector k, bothwith (dotted line) and without (solid line) chemistry. In theabsence of chemical reactions, concentration fluctuations atall wave vectors k ~ k, grow. Chemical reactions introducecutoffs both at large k and small k, so that growth occurs onlyfor intermediate-wavelength fluctuations.

mechanism for pattern selection in a variety of systems[1,5,6], such as block copolymers. The mathematical ori-gin of the similarity between spinodal decomposition withchemical reactions and ordering of block copolymers liesin the fact that a term linear in @ in dynamics [Eq. (5)]can be absorbed into a redefined free energy functional asan additional nonlocal quadratic coupling of P's [5,15].

Our analysis shows that due to the reactions, only fluc-tuations at an intermediate length scale grow initially.However, solution of the full nonlinear equation is nec-essary to explore the later stages of phase separation whenthe nonlinearities are important [15]. We numerically in-tegrated Eq. (5) on a two-dimensional lattice with f(P),the bulk free energy, taken as that for an incompressible,small molecule mixture,

f(4) = 4»4 + (1 —4)»(I —0) + XA(I —@),kpT

(9)where the dimensionless interaction parameter g re-lates the interaction energies between the two speciesof molecules [12]. We take ~ = gA2, where A is theaverage range of the intermolecular interaction [12,16].Equation (5) can then be written asa@, t'

8t (1 —(6)= AV In~ —2gP —2~A V P

—(Ii+ I'.)A+ I'2 (10)

R(t) —t

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Our simulations were performed by discretizing Eq. (10)using a simple finite difference scheme in two dimensions.Computational details of the integration method will begiven elsewhere [7].

The critical point of the free energy in Eq. (9) is givenby @, = 1/2 and ~, = 2. The concentration at eachlattice site was initialized to P = 1/2 ~ 6@,where BP isa random number in the range [—0.0001, 0.0001]. Latticesof size 2562 and larger were then quenched to ~ = 4.0 forvarious choices of equal forward and backward reactionrates I i and I'2 (zero heat of reaction is implied). WhenI

&

= I 2 = 0, the system phase separates in the usual way[3,17]. This system is shown in Fig. 2 in the late stagesof phase separation after a time (a) r = 512 and (b) 7. =2048. Figures 3(a) and 3(b) show the same system as inFig. 2(b) at r = 2048, but with I i

= I q= 0.05 and 0.2,

respectively. Clearly, the steady-state, lamellar structureexhibited by the reactive systems in Figs. 3(a) and 3(b)is very different from the transient, interconnected, self-similar morphology of the nonreactive mixture in Fig. 2.

We measured the average domain size R(t) by calculat-ing [18] the inverse of the first moment of the structurefactor 5(k, t), for various choices of reaction rates. Foreach system, the reaction rates were chosen to be equal:I

&

= I 2=—I . In the absence of chemistry (1 = 0) the

system exhibits the expected Lifshitz-Slyozov [19]growthlaw at late times,

VOLUME 74, NUMBER 11 PH YS ICAL REVIEW LETTERS 13 MARcH 1995

FIG. 2. Concentration field for 2562 lattice at a time (a) r =512 and (b) r = 2048 following a quench of Eq. (10) to theunstable region, in the absence of chemical reactions (i.e. ,

I, = I 2= 0). A-rich regions are shown black and B-rich

regions are shown grey.

FIG. 3. Concentration field for 256 lattice at a time 7. = 2048following a quench of Eq. (10) to the unstable region, withreaction rates (a) I = 0.05 and (b) I = 0.20. A-rich regionsare shown black and 8-rich regions are shown grey. Furtherevolution of the system tends to align domains, but the steady-state domain width has already been selected.

where n = 0.32 ~ 0.02 [17]. However, for nonzero reac-tion rate, the domain growth saturates at a certain steady-state value RF. In the steady state, dimensional analysisof Eq. (10) shows that [t] =. [1/I ], so that the domainsize RJ; should obey the scaling law

RF —(1/I ) . (12)

The steady-state inverse domain size RF' is plotted double

logarithmically against the reaction rate I in Fig. 4.

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Indeed, we find that n appears to be approaching 1/3 forsmall reaction rates [20]. Thus, the simultaneous presenceof the chemical reaction A=8 selects intermediate lengthscales for growth, even in the late stages of spinodaldecomposition [21].

The suppression of long-wavelength fiuctuations by theinterplay between chemical reactions and thermodynamicinstability provides a ubiquitous mechanism for patternselection in nature. The underlying mechanism for pat-tern selection in typical reaction-diffusion systems arisesfrom a competition between diffusion and chemical re-

VOLUME 74, NUMBER 11 PHYSICAL REVIEW LETTERS 13 MARcH 1995

/JO

/

//

//

/'

4 IC

.00$

FIG. 4. Double logarithmic plot of equilibrium inverse aver-age domain size RF ' vs I'. The straight line has slope 1/3.

action [I]. The length scale characterizing the transientpatterns in spinodal decomposition of mixtures withoutchemical reactions is dictated by the competition betweenthe square-gradient interfacial term and the thermody-namic instability inherent to the system. When externally-controlled chemical reactions and spinodal decompositionoccur simultaneously, these two selection mechanismscombine to determine the length scale of the steady-statepattern. The two mechanisms can be tuned independentlyof one another, thereby allowing one to control the finalmicrophase-separated structure of the material [4]. Gener-alization of the approach developed here to polymeric sys-tems and more complicated chemical reactions promisesto be of significant technological importance.

S.C. G. would like to thank J. Cahn, A. Coniglio,J. Douglas, and M. Grant for helpful discussions, and theCenter for Computational Science at Boston Universityand the Supercomputing Center at the University of Mary-land for computational resources. M. M. acknowledgesthe National Science Foundation for support.

[1] M. C. Cross and P. C. Hohenberg, Rev. Mod. Phys.65, 851 (1993); S.A. Kauffman, The Origins of Order(Oxford University Press, New York, 1993).

[2] J.W. Cahn Acta Met. 9, 795 (1961).[3] K. Binder, in Material Science and Technology: Phase

Transformations in Materials, edited by P. Haasen (VCH,Weinham, 1990), Vol. 5, pp. 405 —471.

[4] We consider here open systems in which the reactionsare induced and controlled externally, e.g. , by irradia-tion. See, e.g., Q. Tran-Cong, T. Nagaki, T. Nakagawa,O. Yano, and T. Soen, Macromolecules 22, 2720 (1989);T. Tamai, A. Imagawa, and Q. Tran-Cong, Macro-molecules 27, 7486 (1994), and references therein.

[5]

[7]

[9]

[11]

[12]

[13]

[14]

[15]

[16]

[18]

[20]

[21]

Y. Oono and Y. Shiwa, Mod. Phys. Lett. B 1, 49 (1987);Y. Oono and M. Bahiana, Phys. Rev. Lett. 61, 1109(1988); A. Chakrabarti, R. Toral, and J. D. Gunton, Phys.Rev. Lett. 63, 2661 (1989); F. Liu and N. Goldenfeld,Phys. Rev. A 39, 4905 (1989).C. Sagui and R. C. Desai, Phys. Rev. Lett. 71, 3995(1993); Phys. Rev. E 49, 2225 (1994), and referencestherein; C. Roland and R. C. Desai, Phys. Rev. B 42,6658 (1990); L. Q. Chen and A. G. Khachaturyan, Phys.Rev. Lett. 70, 1477 (1993); S. A. Langer, R. E. Goldstein,and D. P. Jackson, Phys. Rev. A 46, 4894 (1992); A. J.Dickstein et al. , Science 261, 1012 (1993); E. H. Brandtand U. Essmann, Phys. Status Solidi (b) 144, 13 (1987).S.C. Glotzer, in Annual Reviews of ComputationalPhysics, edited by D. Stauffer (World Scientific, Singa-pore, to be published), Vol. 2, and references therein.M. Gitterman, J. Stat. Phys. 58, 707 (1990), and refer-ences therein; S. Puri and H. L. Frisch, J. Phys. A 27,6027 (1994).N. G. Van Kampen, Stochastic Processes in Physics andChemistry (North-Holland, Amsterdam, 1992).B.J. Berne and R. Pecora, Dynamic Light Scattering(J. Wiley and Sons, New York, 1976).This equation can also be derived directly from amaster equation based on the Ising model or lattice gasHamiltonian with reactions.G. Fredrickson, in Physics of Polymer Surfaces and Interfaces, edited by I. C. Sanchez (Butterworth-Heinemann,Boston, 1992).By rewriting the concentration field @ in terms ofthe order parameter P, and taking f(P) = rg2 + gP,Eq. (5) can be transformed to an equation relevant to thephenomenological study of patterns in diblock copolymerswithout any chemistry, where the origin of the "reaction"term is different. See Ref. [5].Note that for critical quenches (Po = 1/2) with equalforward and backward reaction rates, Eqs. (7) and (8) alsohold for k = 0.This equation has been solved analytically for theGinzburg-Landau free energy functional in the limit ofinfinite-component order parameter, in S.C. Glotzer andA. Coniglio, Phys. Rev. E 50, 4241 (1994).M. A. Kotnis and M. Muthukumar, Macromolecules 25,1716 (1992).T. M. Rogers, K. R. Elder, and R. C. Desai, Phys. Rev. B37, 9638 (1988), and references therein.S.C. Glotzer et al. , Phys. Rev. E 49, 247 (1994), andreferences therein.I.M. Lifshitz and V. V. Slyozov, J. Phys. Chem. Solids19, 35 (1961).Note that RF is expected to scale as a power law with Ionly when the crossover in domain growth occurs afterthe system has already entered the scaling regime whereR —tThese results have also been confirmed by recent MonteCarlo simulations of the Ising analog of Eq. (5), in whichKawasaki exchange dynamics was used to model thephase separation while simultaneous random spin Hipswere used to model the chemical reaction; S.C. Glotzer,D. Stauffer, and N. Jan, Phys. Rev. Lett. 72, 4109 (1994).

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