kinetics of ordering in fluctuation-driven first-order transitions: simulation and theory

PHYSICAL REVIEW E NOVEMBER 2000VOLUME 62, NUMBER 5

Kinetics of ordering in fluctuation-driven first-order transitions: Simulation and theory

N. A. GrossCollege of General Studies, Boston University, Boston, Massachusetts 02140

M. Ignatiev and Bulbul ChakrabortyMartin Fisher School of Physics, Brandeis University, Waltham, Massachusetts 02454

~Received 2 March 2000; revised manuscript received 8 June 2000!

Many systems involving competing interactions or interactions that compete with constraints are well de-scribed by a model first introduced by Brazovskii†Zh. Eksp. Teor. Fiz.68, 175 ~1975! @Sov. Phys. JETP41,85 ~1975!#‡. The hallmark of this model is that the fluctuation spectrum is isotropic and has a maximum at anonzero wave vector represented by the surface of ad-dimensional hypersphere. It was shown by Brazovskiithat the fluctuations change the free energy structure from af4 to a f6 form with the disordered statemetastable for all quench depths. The transition from the disordered phase to the periodic lamellar structurechanges from second order to first order and suggests that the dynamics is governed by nucleation. Usingnumerical simulations we have confirmed that the equilibrium free energy function is indeed of af6 form. Astudy of the dynamics, however, shows that, following a deep quench, the dynamics is described by unstablegrowth rather than nucleation. A dynamical calculation, based on a generalization of the Brazovskii calcula-tions, shows that the disordered state can remain unstable for a long time following the quench.

PACS number~s!: 05.10.Cc, 05.40.2a, 05.45.2a

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I. INTRODUCTION

Kinetics of growth can be broadly classified into two caegories: nucleation and spinodal decomposition~or continu-ous ordering!. The former applies to situations where thinitial state is metastable and the latter where the initial sis unstable. The distinction becomes unclear near the limmetastability and in systems where the concept of metability itself is ill defined. A class of systems where the denition of metastability becomes ambiguous is one whthere is a fluctuation-driven first-order phase transition.theoretical model describing these transitions was propoby Brazovskii in 1974@1#. The Brazovskii model has beeshown to apply to the nematic to smectic-C transition inliquid crystals@2#, to the onset of Rayleigh-Be´nard convec-tion @3,4#, and to microphase separation in symmetdiblock copolymers in the weak segregation limit@5,6#. Inthe diblock copolymer system, experiments have shownthe Brazovskii scenario can describe the nature of the fiorder transition@7#.

It was shown by Brazovskii, within a self-consistent Hatree approximation, that the fluctuations destroy the mefield instability and lead to a first-order phase transition@1,8#.Theories of nucleation and growth have been construcbased on the idea that the static, Brazovskii-renormalitheory can provide an effective potential for a stochasLangevin equation@8,9#. In this paper we examine the validity of this description by using numerical simulationsstudy the relaxational dynamics of a model described byBrazovskii Hamiltonian. These results are compared topredictions of ‘‘static’’ nucleation theories and to the predtions of a theory@10# based on the dynamical action formaism.

The rest of the paper is laid out as follows. The dynamiequation under consideration is introduced in the next stion along with an outline of the computational approach.

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the third section a review of the static results of Brazovsand the simulation results that confirms this picture is psented. A scaling presented by Hohenberg and Swift@9# isalso confirmed. A review of a dynamical perturbation theopresented in Ref.@10# is given in Sec. IV. This work uses thdynamical action formulation@11# to study the contributionsof perturbations around the classical path. The contributiconsidered are of the same order as those considered byzovskii in the static calculation. These results are compato simulations. A numerical analysis of the theory suggethat, after the system is quenched to low temperatures,disordered metastable well develops slowly. During this elution of the free energy the system has time to develolamellar structure. This picture is supported by our simution results in which the effective curvature of the disorderfree energy is measured. That curvature is shown to be ntive at early times after a quench and becomes positive oat late times. Thus the evolution of the order parametebetter described by continuous ordering rather than nuation. This is the main result of this paper. The last sectcontains suggestions for testing these results experimenas well as plans for future work.

II. EQUATION OF MOTION AND NUMERICALSIMULATIONS

Our starting point is the same as in Ref.@9# and describesa system with modelA dynamics @12# and a BrazovskiiHamiltonian ~cf. @13# for consideration of block copoly-mers!. The Brazovskii Hamiltonian is characterized byfluctuation spectrum whose maximum occurs at a nonzwave vectoruqu5q0 and can be represented by a hypesphere ind dimensions. The full form of the Hamiltonian i

H5E dxS 1

2f~¹21q0

2!2f11

2tf21

1

4!lf4D . ~1!

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PRE 62 6117KINETICS OF ORDERING IN FLUCTUATION-DRIVEN . . .

The dynamics is taken to be relaxational and so the equaof motion is given by the Langevin equation,

df

dt52M

dH

df1h, ~2!

whereM is the mobility ~which sets the time scale for thproblem! and h is the usual Gaussian noise@^h&50 and

^h(xW ,t)h(xW8,t8)&5d(xW2xW8)d(t2t8)].The stochastic Langevin equation derived from t

Hamiltonian differs from the usual Ginzburg-Landau dscription@12# in the appearance of an unusual gradient teThis dynamical equation is usually referred to as the SwHohenberg~S-H! equation and falls into the typeI s classifi-cation of Cross and Hohenberg@14#. In this classification thesystem is unstable to a static, spatially periodic structure

The complete equation of motion is

df

dt52M S q0

2¹2f1¹4f1~q041t!f1

l

6f3D1h. ~3!

In Eq. ~3!, the coupling constantl has been rescaled by thnoise strength. For systems where the noise strength is ssuch as Rayleigh-Be´nard convection@4#, the effective cou-pling constant is also small. In diblock copolymers, the nostrength is of the order ofkBT.

For numerical calculations, the Langevin equation mbe approximated by a discrete equation:

f~ t1Dt !5f~ t !2MDtS Lf1L2f1~r 1q04!f1

l

6f3D

1@MDt/~Dx!3#1/2h. ~4!

L here is the discrete Laplacian. This can take on sevforms, although for this work we use the simpleform, d2f/dx2'Lf5(@(1/Dx2)(x1Dx)1f(x2Dx)22f(x)#, where the sum is over the dimensions of the ltice. Other choices that include next nearest or more comcated neighborhoods are possible@15#. This is important ifisotropy is of concern; however, the effect is small and cbe ignored. The scaling of the noise in Eq.~4! reflects thefact that, as the cell size or time step becomes sma~larger!, the possible fluctuations become larger~smaller!.

For these simulations we setq051 and choose the latticspacing such that one lamellar spacing spans six lattice(Dx56/2p) For most of this work the overall lattice size603. The mobilityM is set to 1. In order for the simulationto be numerically stable the time scale must be smaller tsome stability time,Dt;1/Dx4. To satisfy this the time scaleis chosen to beDt50.01 and measurements are takenintervals of 100Dt or longer.

Previous theories of nucleation have analyzed the abequation under the approximation that the effects of flucttions can be incorporated into astatic renormalized free energy FR which replaces the bare free energyF in Eq. ~2!@8,9#. We will compare our numerical results to the Brzovskii predictions for the static renormalized parametand show that the static scenario is beautifully borne outsimulations in three dimensions. We will then analyzetime dependence of the structure factor as observed innumerical simulations and show that this time dependenc

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inconsistent with the dynamical picture based on a strenormalized free energy. A theory based on thedynamicalperturbative treatment of Eq.~3! can provide a qualitativelycorrect description of the numerical simulations.

III. STATICS

A. Brazovskii theory

Brazovskii’s treatment of the model within the Hartreapproximation can be restated in terms of an expansionthe thermodynamic potentialG(f), the generating functionafor the vertex functions@11#. This approach has been described in detail by Fredrickson and Binder@8#, and withinthe Hartree approximation leads to a renormalized mass t~r! in Eq. ~1!. The diagrams up to one loop are shown in F1, and the mass renormalization relation is given by

r 1~q22q02!25t1~q22q0

2!2

1l

2E dq

~2p!d@t1~q22q0

2!2#21. ~5!

This approximation is made self-consistent by replacthe bare parameter in the integrand with the renormaliparameter. Essentially the bare propagator on the loop in1 is replaced by a renormalized propagator. This leads toHartree result,

r 5t1l

2E dq

~2p!d@r 1~q22q0

2!2#215t1al/Ar . ~6!

a includes the geometric factors that depend on the dimsion of the system and in three dimensionsa51/4p2. Ac-cording to Brazovskii@1# the Hartree approximation is gooonly for l26!1.

The interesting point about Eq.~6! is that, for all positiveand negative values of the bare parametert, the renormal-ized parameterr is always positive. This implies that thdisordered phase is always metastable. Brazovskii went oshow that the bare coupling parameterl gets renormalized toa negativevalue leading to af6 theory and the possibility ofa first-order phase transition@8#.

B. Computational results

Numerical simulations can measure the static structfactor in the disordered phase which is predicted by the Bzovskii theory@1,8# to be

S~q!215r 1~q22q02!2, ~7!

FIG. 1. Diagrams used for static renormalization of the mparameter in the free energy. The Hartree approximation usesgrams only up to orderl and then replaces the bare parameterthe integrand with the renormalized one to solve the equationsconsistently.

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6118 PRE 62N. A. GROSS, M. IGNATIEV, AND BULBUL CHAKRABORTY

FIG. 2. The renormalized control parameterr~dimensionless! plotted against the bare controparametert ~dimensionless!. The different datasets are for different values ofl. The Gaussiancasel50, which was used to set the scale, is alplotted for comparison. The inset shows the rgion near the mean-field transition. Note that trenormalized parameter does not cross zero h

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where r is the renormalized control parameter@1,9#. Thus,S(q0)21 is just r. The renormalized mass can, therefore,measured by monitoring the peak of the structure factor.Hartree calculations predictr to be positive and asymptotically approaching zero ast→2`. Experiments in symmetric diblock copolymers@22# have verified that the behavioof S(q0) is consistent with the Brazovskii predictions andvery different from the mean-field prediction@S(q0)5t21#@7#.

Figure 2 shows results forr obtained from our simulationsfor different values of the coupling constantl. The Gaussiancase (l50) is also shown for comparison. The system wrun for 1000 time steps~each time step being 100Dt) and100 samples were taken. The data points plotted are theerage of the samples while the error bars represent the sqroot of the standard deviation. For large positive valuesthe bare control parametert, the fourth-order term becomeless important andr approaches the Gaussian value. Thisseen clearly for small values ofl. As t approaches zero, thmeasured values ofr deviate from the Gaussian case and spositive for all t. Some care was taken to normalize tvalues presented in Fig. 2. For the Gaussian case,S(q0)21

should be linearly related tot and the slope of the lineshould be 1. In calculatingS(q0) several normalizations arneeded, including the normalization due to the Fourier traform ~FT! and the normalization due to circular averaginWhile the FT normalization is just related to the system sithe normalization due to circular averaging is more comcated to calculate. Instead of calculating these normalizatdirectly, the slope of the raw result from the Gaussian cwas used to normalize all of the data. Another concern isfor an infinite continuous system this line should have axintercept at zero; however, for the finite systems on a gused in the simulations thex intercept is slightly negative. Toaccount for this, all of the data are shifted by an amoequal to that intercept. This is important because whescaling is applied the values of the results around zeromagnified. A small negative~positive! value becomes amuch larger negative~positive! value and so shifts fromnegative to positive will become important.

In Ref. @9# the authors show that, within the Hartree aproximation, ther (t) curves for differentl ’s can be de-scribed by a single functional form in terms of scaled vaablest* and r * :

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r * 5r * ~al!22/3,~8!

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As stated above, in three dimensionsa51/4p2. Figure 3shows plots ofr * versest* obtained from simulations forthree different values ofl. The curves are seen to scale quwell but the scaled curve falls significantly below the theretical prediction@1,8,9# shown as the thick black line. Ththeoretical prediction that the value ofr * ~and r ) never be-comes negative and is asymptotic to zero is, however, cleborne out by the simulations. The deviation from the theretical line could be a system size effect; however, resultslarger and smaller systems are consistent with the dshown in Fig. 3. For the Gaussian caser * 5t* and for largepositive values of the bare parameter both the theorysimulations approach this. Since the theoretical resultslarger than the simulations and both are above the Gausour simulations suggest that the contributions to the twpoint function from diagrams not included in the Hartrapproximation serve to lower the overall correction to mefield theory. Another property that the simulation data ehibit in Fig. 3 which is not predicted by the theory is that thcurves for different values ofl diverge from each other anegative values oft, i.e., the scaling is not perfect. Smallevalues ofl lie closer to the theory, as expected. It is alimportant to note that the smallest value ofl used wasl50.06 while the approximation used to derive the theoretiresults is valid only forl;1026; so the theoretical scalingpredictions seem to be far more robust than expected.

To further explore the nature of the phase transition in tmodel, we can compare scaled values ofS(q0) obtainedfrom hot ~random! and cold~purely modulated order! initialconfigurations. Figure 4 shows just such a plot. As inprevious plot, the theory is shown as a thick black line. Feach value ofl there are two lines presented: one fordisordered start and one for an ordered start. The averawas done as described above. For positive and small ntive values oft the results from the disordered start and tordered start are nearly the same. However, fort* belowsome ts* the two differ, with the ordered phase havinghigher peak. It should be pointed out that for these valuet* the disordered state is not in equilibrium~metastable orstable! and there is a slow but definite time evolution of th

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FIG. 3. Scaling ofr (t) using the form de-duced from Hartree theory@Eq. ~8!#. The datacollapse well except at the deepest quenches.scaling curve predicted by the Hartree appromation is shown for comparison. It is in fairlygood agreement with the data considering thatvalues ofl used in the simulations are four orders of magnitude larger than that for which thapproximation is expected to be valid. Bothr *andt* are dimensionless.

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structure factor over the time period in which the averaare taken. This time evolution will be discussed in modetail in Sec. IV. For this work the disordered start pointsincluded in the plot simply as a comparison to see whereordered phase melts. The value where the two data setverge can be considered an estimate for the limit of stab~spinodal! of the lamellar phase. As a consistency checkaverage value of the wave vector was measured. For vaof t abovets the wave vector is small and points in a radom direction, while for lower values oft the average wavevector is large and points along the direction that the syswas prepared in. For all three sets of data the lamellar sodal lies in the range of22.1,ts* ,21.9. This is consistenwith the prediction from Hartree theory that the first-ordtransition to the lamellar phase occurs at a lower value oft* ,t t* 522.74 @9#. That is to say, our measured value for tlamellar spinodal is above the predicted transition tempeture as it should be. For values oft* below ts* , however,systems prepared in the disordered phase always evolvward the ordered phase. It is possible that the lamellardered phase is enhanced by the boundary conditions~thelattice size is always chosen to be commensurate with

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lamellar wavelength!; however, it should be noted thalamellae form in many possible directions where the systsize is not commensurate with the lamellar wavelength. Dcussion in Sec. IV should shed some light on this issue.

IV. DYNAMICS

A. Relaxational dynamics based on static renormalizedparameters

Previous analyses of nucleation and metastability, incontext of fluctuation-driven phase transitions, have bebased on a Langevin equation with the force obtained frthe static, Hartree-renormalized free energy function@8,9#.Fredrickson and Binder@8# used this approach to computhe nucleation barrier and the completion rate of nucleatand growth in diblock copolymers. The interfacial tensiwas found to be small, leading to a small nucleation barand rapid nucleation for deep quenches. The shape ofnucleating droplet was more carefully analyzed by Hohberg and Swift@9# by taking into account spatial inhomogeneities in the effective free energy function. Their analyrelied on constructing a coarse-grained free energy fu

simulating aperatures

higher peak

FIG. 4. Temperature dependence of the average intensity at the peak of the structure factor~in arbitrary units!. To show the transition,results from two different starting configurations are compared. The shapes of the symbols denote different values ofl, whereas the shadingdistinguishes between the starting configurations. The closed symbols represent systems that were started in a disordered state,hot start, while the open symbols are for systems that were started with an ordered lamellar structure already present. At high temthe two starting conditions give nearly the same results, while at low temperatures the systems prepared in the ordered state haveintensities, implying that the systems prepared in the disordered states are in a metastable state or not in equilibrium~see text!.

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tional. The coarse graining was based on a momentum-srenormalization idea where fluctuations with momentaaway from the shell defined byuqu5q0 are successively integrated out. This analysis led to nonspherical dropletswas consistent with the picture of spinodal nucleation thad been obtained earlier@16#. In this picture, there is noessential difference between the kinetics of a fluctuatidriven first-order transition and a weak first-order transitioThe results of our simulations suggest a very different snario for the growth of the lamellar structures in a Brazovsmodel.

As pointed out in the work of Hohenberg and Swift@9#, acomplete theory of the dynamics of fluctuation-driven firorder transitions would have to be based on a coarse graiof the full dynamics as expressed by the original Langeequation with the bare Brazovskii Hamiltonian. In this papwe compare the results of our numerical simulations topredictions of a Hartree renormalization of the full dynamcal equation as expressed in Eq.~2!. Our emphasis has beeon understanding the nature of the dynamics of the mstable phase and we have not analyzed, in any detail,spatial structures associated with the growth process.

B. Perturbative analysis of the dynamics and simulation results

In order to systematically analyze the effects of fluctutions on the kinetics of growth of the lamellar phase, pertbative techniques analogous to the static Hartree approxtion have to be applied to the Langevin equation. Thismost conveniently done through the dynamical action fmalism@11#. The application of this method to the S-H eqution was outlined by Ignatievet al. @10#. In the dynamicalaction formalism, the average value of an operatorO(f)over the noise history is rewritten as a functional integra

^O$f~x,t !%&5E Df exp~2S@f#!, ~9!

whereS@f# is the dynamical action@11#.As in the static case, the calculation of dynamical cor

lation functions is most conveniently formulated through tconstruction of a generating function@11#. To establish theclosest analogy with the static calculations, it is usefulwork with the generator of vertex functionsGdyn@f#, andestablish a diagrammatic expansion that is the exact anof the diagrams retained by Brazovskii in the static calcution. The simulation results demonstrate that the static prerties of the S-H equation closely follow the Brazovskii prdictions and, therefore, it seems appropriate to applyapproximation scheme to the dynamics. The correspondebetween statics and dynamics becomes particularly transent in a superfield formulation@17#, which shows that thedynamical perturbation theory in terms of the superfieldsexactly the same structure as the static perturbation thexcept for the appearance of a different ‘‘kinetic’’ termwhich leads to abarepropagator that is distinct from that othe static theory. The superfield correlation functions cancalculated by constructing diagrams as in the static thebut replacing the static bare propagator by the approprdynamical one@17#. The superfield correlation functions encode all the dynamical correlations of the fieldf and the

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latter can be extracted from well-defined relations@17#. Thedetails of this calculation and the results will be describeda separate publication@10#. In this paper we present the maresults, which can be compared to the numerical simulatio

The two correlation functions that appear in the dynamcal study are

G~ t,t8!5^f~ t !h~ t8!&

and

C~ t,t8!5^f~ t !f~ t8!&.

Both of these can be obtained from a single superfield crelation function@Q(1,2)# which is related toGdyn through

Q21~1,2!5d2Gdyn

dF1dF2, ~10!

where theF ’s are the superfields and the derivative is evaated with the source term set to zero. Treating Eq.~2! ac-cording to the formalism described above leads to an expsion forGdyn that involves two-point correlation functions othe fluctuations and is a natural extension of the staticG(f)to correlation functions involving time. The expression, hoever, does not lend itself to easy analysis except in two caearly times when a variant of linear theory can be appland late times when the system is stationary. The late tdynamics involves analysis of the nonlinear terms and wbe discussed in the concluding section. The early timenamics is where one is justified in retaining only quadraterms in the renormalizedGdyn . In this limit we obtain thefollowing equation for the equal-time correlation functioCq(t,t)5^fq(t)f2q(t)&, which is the structure factor that imonitored in the simulations:

Cq~ t,t !5E0

t

@G~ t,t8!#2dt8,

~11!

Gq~ t,t8!5expS 2Et8

t

@r ~ t9!1~q22q02!2#dt9D .

The mass parameterr (t), which is now time dependent, irenormalized in the dynamic theory using the same apprmation as in the static theory. The diagrams used are shin Fig. 5. These again are only to one loop. The mass tethen becomes

r ~ t !1~q22q02!25t1~q22q0

2!2

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~2p!d

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@t1~q22q02!2#

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~2p!d

exp$22@t1~q22q02!2#t%

@t1~q22q02!2#

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Again, the approximation is made self-consistent by repling thet with r in the integrand,

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Here D(q) is the renormalized propagatorD(q)5r eq1(q2

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rameter that is the solution to Eq.~7!.In Eq. ~13!, at long times the integrand in the second te

becomes small andr (t) approachesr eq , which is confirmedin Sec. III above. At time zero the subtraction of the secoterm is just a subtraction of the Hartree level correction ftor from the bare parameter introduced earlier@Eq. ~6!# and,therefore,r (t50) is just the bare parametert, while as t→` r (t) approaches the renormalized, equilibrium value

This scenario is confirmed by the simulations. The simlation results discussed in Sec. III B confirm that the equilrium valuer eq is consistent with the Hartree prediction. Anlyzing the early stage dynamics can verify the value ofr (t)at short times. Figure 6 shows the growth ofS(q0) as afunction of time for various values of the bare control prametert. To average over noise, five independent runs wtaken for each value of the control parameter. Ifr (t) is timeindependent, the early stage evolution~linear theory! is de-scribed by@18#

S~q,t !5Cq~ t,t !5S~q,0!exp$2D~q!t%11

D~q!

3@12exp$2D~q!t%#, ~14!

where D(q) is D(q)5r 01(q22q02)2 and r 0 is the time-

independent value ofr. At the peak of the structure factoq5q0, andD(q0) is just r 0, the value of which can be estmated by fitting the simulation results shown in Fig. 6 to tform given in Eq.~14!. The results of such fits are presentin Fig. 7. This figure shows thatr 0 andt are linearly relatedwith a slope that is nearly 1. This implies that for short timthe dynamically renormalized parameter is linearly relatedthe bare parameter and is not renormalized to positive va

FIG. 5. As with the static case, diagrams only up to orderl areused and then the bare parameter is replaced by the renormaparameter to give a self-consistent solution.

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for negative values of the bare parameter. The early stdynamics, therefore, is characteristic of a system exhibitunstable growth.

These fits must be considered with some care. In the cwherer is not time dependent, the linear theory describessystem only for short times, that is, times on the order ofnatural time scaler 21 ~see, for example,@19#!. For the fast-est growing mode, when the data are fitted to different tiscales, the results obtained are consistent for all time scup to r 21. For the S-H equation, however, the results otained are dependent on the time range. In particular, asrange of time increases the values obtained forr become lessnegative. This is consistent withr (t) growing as a functionof time and the results of the fits merely represent soaverage value ofr (t) for the time range involved.

As already stated, if the system has been quenchednegative value oft, Eq. ~13! indicates that the early timeevolution will exhibit unstable growth. As time evolves thsecond term in Eq.~13! decreases and the value ofr (t) ap-proachesr eq , which is always positive. Although the integral in Eq.~13! is hard to evaluate analytically, numerics caprovide some insight into its behavior. Figure 8 showsnumerical evaluation of the time evolution ofr (t) for vari-ous values ofr eq and l50.06. As confirmed in Sec. IIIabove,r (t) will eventually become positive no matter hodeep the quench is, although the time for this to happen mbecome very long. Whiler (t) is negative the system is expected to undergo unstable growth; although, since the mparameter is time dependent, the time evolution will be mcomplicated than the usual phase ordering scenario.

To know whether the system has time to develop a molated structure beforer (t) changes sign, we need to defintwo times, the crossover timetcross at which r (t) changessign, and an average growth timetave . The average growthtime is just the inverse ofr ave defined as

tave21 5r ave5

1

tcrossE

0

tcrossr ~ t8!dt8. ~15!

In Fig. 9, we show plots oftcross andtave as functions of thebare control parameter. For shallow quenchestcross is smallcompared totave and the system reaches a well-definmetastable equilibrium state, which is disordered. Iftcrossbecomes larger thantave , metastability becomes difficult todefine. The system takes longer to equilibrate in the me

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FIG. 6. The time evolution of the structurfactor peak~in arbitrary units! for the earliesttimes. Different sets are for different quencdepths as indicated on the graph. All quenchesbelow the mean-field transition temperature intially show unstable growth.

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FIG. 7. Growth time~in arbitrary units! ascalculated from fits to a linear theory plotteagainst the quench depth below the mean-fitransition. For short times the system shoulddescribed by linear theory, which predicts ustable growth for all quenches below the meafield transition.

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stable disordered phase than it takes to grow lamellar sttures. We can use the conditiontcross5tave to define a cross-over temperature tdyn* . Above this temperature thdisordered phase quickly becomes locally stable and a nuation event is needed for the formation of lamellar structurBelow tdyn* , the system is expected to evolve continuoutoward the lamellar phase without any evidence of metability of the disordered phase.

Figure 9 is in sharp contrast to standard Ginzburg-Lan~G-L! theory. For the time-dependent G-L equation, the prence of noise suppresses the critical point. In the regwhere the bare parameter is negative but the renormalparameter is positive,tcross is always smaller thantave andso there is no unstable growth until the true critical pointreached. For the system under consideration here, diffescenarios are possible depending on the relationship ofdynamical crossover temperature to the first-order transitemperaturet t* at which the free energy of the lamellaphase becomes lower than that of the disordered phase.t t*is higher thantdyn* , there will be a regime of temperatureover which the system will undergo nucleation and growOn the other hand, ift t* is lower thantdyn* , then there is nonucleating regime and one will observe only unstagrowth, albeit of an unusual nature since the free enesurface is evolving with time. Since the value oft t* 522.74 given by Hohenberg and Swift@9# is considerablybelow tdyn* the latter scenario is the one that best fits t

c-

le-s.ya-

u-ned

ntisn

.

ey

s

system. The results of our simulations, presented in the nsection, are in qualitative agreement with this scenario.would like to emphasize that the growth dynamics describabove is qualitatively different from rapid nucleationwhich the system quickly reaches the stable equilibristate. The value oftcross, on the other hand, indicates a typof ergodicity breaking as the system takes a very long timereach equilibrium.

All of our numerical results indicate thatt t* predictedfrom Hartree theory is lower thantdyn* and they are remarkably close to each other. We have been unable to comewith a simple relationship between these two temperatuand, therefore, can only interpret the similarity of the twoa remarkable coincidence. The dynamical crossover isduced from time scales that characterize the evolution offree energy surface while the transition temperature isduced from a comparison of the depths of the two wells. Inot clear why the two temperatures should be similar in mnitude. It can be argued that there is only a single paraml controlling the scale of fluctuations and, therefore, the ttemperatures should be related; however, there is no obvargument to suggest that they should be identical.

The picture emerging from the dynamical renormalizatiis a natural extension of the effect of fluctuations on tstatic results of the Brazovskii model. Following a quenfrom a relatively high temperature to a temperature whertis negative, the system is in a locally unstable region~top of

-a-

ldess

FIG. 8. The time evolution of the mass parameter as calculated by dynamic renormaliztion. r (t) ~dimensionless! eventually becomespositive for all quenches below the mean-fietransition but the time for that crossover increasdramatically for larger quenches. At large timer (t) approachesr eq .

oeled

heasst


FIG. 9. The calculated crossover time~in ar-bitrary units! plotted against quench depth. Alsplotted is a characteristic growth time for thlamellar structure. These results have been scausing Eq. ~8! so that they arel independent.When this time becomes much less than tcrossover time then the lamellar structure henough time to grow. Notice that this occurs juabove or at the transition temperaturet t .

arr-r

en

dmmo

pllonibreav

rot

-iteetthre

hots

entap-ofthe

rheft

lowv-

oes

ysisuatehderco-

tak-ingn-

thetheer-

a hill!. As the fluctuations grow with time, the nonlineterms characterized byl become important and they renomalize the curvature of the hill. This scenario is quite diffeent from the usual picture of evolution in an adiabatic pottial.

C. Simulation results for late times

Further evidence for the dynamical scenario presentethe previous section comes from examining the long tievolution of the peak of the structure factor obtained froour numerical simulations and correlating that to snapshof the system as it evolves. Figure 10 is a plot of the amtude of the structure factor peak as it evolves. For shalquenches,t>20.03 (t* >22.26), the peak grows to aequilibrium value and does not evolve further. The equilrium values are consistent with the equilibrium valuesported above in Sec. III. For deeper quenches the peakplitude grows quickly and then appears to saturate; howecareful examination shows that the value continues to gvery slowly. This is consistent with late stage domain growwhich was studied extensively by Elderet al. for two-dimensional~2D! systems@24#. The last feature in the system is a final rise to an equilibrium value. This rise is a finsize affect caused by the majority domain finally taking ovthe entire system. This being the case, the peak would noexpected to grow much since the difference betweenvalue before and after this final evolution should only repsent the surface area between the different domains.

--

ine

tsi-w

--m-erwh

rbee-

For a quench depth oft520.07 (t* 525.27) this timeevolution can be compared to a series of system snapsshown in Figs. 11–15. These figures representf50 isosur-faces which would be the boundaries between the differmicrophases of the system. For early times the systempears to be very disordered while at later times domainsordered lamellar structures begin to appear. Just beforefinal convulsive growth, at a timet54000, there still appeato be a few different ordered domains while just after tfinal growth, att55000, only one domain appears to be lein the system. Thus we still have a consistent picture of sbut continuous domain growth in the system which is goerned by the evolution of the second-order term as it gfrom negative values to positive values.

V. DISCUSSION AND FUTURE WORK

In the work presented above we have shown that analof the static free energy does not always provide an adeqdescription of the fluctuation-driven dynamics. AlthougFig. 10 shows features that are reminiscent of a first-orphase transition, dynamical analysis of the renormalizedefficients suggests that a more complicated evolution ising place. A similar situation occurs for the superconducttransition, which is also a fluctuation-driven first-order trasition @20#.

In the present work we have paid close attention totime evolution of the mass parameter and its effect onevolution of the system at early times. The dynamical p

r

e

ion

nrlyere

s

of

FIG. 10. The evolution of the structure factopeak~in arbitrary units! as a function of time fordifferent quench depths. For all of these runsl50.06. From the static work we know that thactual transition occurs for20.04,t,20.03.For shallow quenches above the actual transittemperatureS(q0) grows quickly to its equilib-rium value. For quenches to below the transitiotemperature the system grows quickly at eatimes and then reaches a late stage regime whit evolves by domain growth andS(q0) grows asa small power law in time. The highlighted line ia quench tot520.07. Snapshots of this systemas it evolves are presented in the next setfigures.

om-Asor

inthees

thene

hehes-s.

tioncalotr-n-


FIG. 11. Time51000.

FIG. 12. Time52000.

FIG. 13. Time53000.

turbation theory also becomes tractable at times large cpared totcross when the system has reached equilibrium.pointed out earlier, one enters this regime fairly quickly fshallow quenches. At this stage, the higher order termsGdyn become important. The most interesting aspect ofrenormalized theory is that the fourth-order term becomnonlocal in time@10#,

Gdyn~4!.E d1d2F2~1!K~122!F2~2!, ~16!

and the integral of the kernel over all times leads toBrazovskii result for the renormalized fourth-order term. Ocan derive an ‘‘effective’’ Langevin equation for the fieldsffrom the renormalizedGdyn , and Eq.~16! implies that thisLangevin equation has ‘‘memory’’ effects that appear in tnonlinear term. This would lead to unusual behavior of tequilibrium correlation functions and might provide a ditinctive signature of fluctuation-driven first-order transitionWe are in the process of investigating the effective equafor f numerically. One obvious consequence of the nonloterm is that the barrier to nucleation is dynamic and it is nvalid to think of nucleation as tunneling through a fixed barier. The situation is closer to the problem of quantum tu

FIG. 14. Time54000.

FIG. 15. Time55000.

a

asrywf tat

d

lear

son

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neling for many degrees of freedom where one also findseffective equation that is nonlocal in time@21#.

Another area of interest is the role of defects in the phtransition of the system. These will be important for veshallow quenches to just below the transition point andhave noted their presence when looking at snapshots osystem. These defects are reminiscent of the perforlamellar phase described in Ref.@22#. One way to study thisis to use a directional order parameter measure introduceChristensen and Bray@23# to study the 2D problem. Thisorder parameter is similar to the order parameter for compfluids and provides directional as well as density informtion. What it may show is that the local gradient terms aenhanced for negative values oft, suggesting that perhapthere is a disordered to nematic transition in this regiwhich others have suggested for 2D@24#.

The evidence we have presented here suggests thatime evolution of systems undergoing a fluctuation-drivefirst-order transition is different from standard nucleatioWe would like to close with a discussion of possible expemental systems where this dynamic scenario can be stuexperimentally. Since the effectivel is extremely small inRayleigh-Benard systems the regime where the disorde

e

a

ieonhecl

n

e

eheed

by

x-e

,

the-.-ed

d

state is metastable will probably be inaccessible@25#.Diblock copolymers are another experimental systemwhich the results described here may be observed. Onecial difference between this work and the polymer expemental system is that in a polymer melt the order paramis conserved. Inclusion of the conservation law in the wodescribed here would shift the magnitude of the wave vecpeak in the growth factor of the linear theory by an amouproportional to the quench deptht/k0

4; however, this shift issmall and it does not change the fact that growth occursan isotropic shell ofq vectors, which is crucial to the resultreported here. As mentioned in the Introduction, the dynaequations considered here apply only to polymers inweak segregation regime; that is where the conservation@13# and hydrodynamics are not important. Even so,diblock copolymers there should be a significant temperarange, below the mean-field transition, where the disordestate is locally stable, and a study of the equilibrium, twtime correlation function should reveal the existence ofnonlinear memory term. It might also be possible to obseunusual nucleation if the system parameters allow fort t* tobe higher thantdyn* .

.
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~1977!.@13# In block copolymers the conservation of monomers impl

that modelB dynamics is applicable. In the weak segregatilimit, however, a single wave number dominates and the sof unstable wave vectors is unchanged; therefore the con

m.

s

llu-

sions of this work will still apply@cf. S. Qi and Z. Wang, PhysRev. Lett.76, 1679~1996!.

@14# M.C. Cross and P.C. Hohenberg, Rev. Mod. Phys.65, 851~1993!.

@15# Y. Oono and S. Puri, Phys. Rev. Lett.58, 836 ~1987!.@16# C. Unger and W. Klein, Phys. Rev. B29, 2698~1984!.@17# J.P. Bouchaud, L. Cugliandolo, J. Kurchan, and M. Me´zard,

Physica A226, 223 ~1996!.@18# J.W. Cahn and J.E. Hilliard, J. Chem. Phys.31, 688 ~1959!;

H.E. Cook, Acta Metall.18, 297 ~1970!.@19# N.A. Gross, W. Klein, and K. Ludwig, Phys. Rev. E56, 5160

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3071 ~1991!.@21# J. Sethna, Phys. Rev. B24, 698 ~1981!; B. Chakraborty, P.

Hedega˚rd, and Mats Nylen, J. Phys. C21, 3437~1988!.@22# F. Bates and G. Fredrickson, Phys. Today52, 32 ~1999!.@23# J.J. Christensen and A.J. Bray, Phys. Rev. E58, 5364~1998!.@24# K.R. Elder, J. Vinals, and M. Grant, Phys. Rev. A46, 7618

~1992!.@25# G. Ahler ~private communication!.

kinetics of ordering in fluctuation-driven first-order transitions: simulation and theory

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