Kinetics of phase transitions in weakly segregated block ...€¦ · the fundamental interest in...

PHYSICAL REVIEW E FEBRUARY 1997VOLUME 55, NUMBER 2

Kinetics of phase transitions in weakly segregated block copolymers:Pseudostable and transient states

Shuyan QiDivision of Physics, Mathematics and Astronomy, California Institute of Technology, Pasadena, California 91125

Zhen-Gang Wang*

Division of Chemistry and Chemical Engineering, California Institute of Technology, Pasadena, California 91125~Received 2 August 1996!

We study the kinetics of order-disorder and order-order transitions in weakly segregated diblock copolymersusing a time-dependent Ginzburg-Landau~TDGL! approach. In particular, we investigate the microstructuralchange as well as the order-parameter evolution after a sudden temperature jump from one phase to another.Direct numerical simulation of the TDGL equations shows that depending on the extent of the temperaturejump, these transitions often occur in several stages and can involve nontrivial intermediate states. For ex-ample, we find that transition from the lamellar phase to the hexagonal cylinder phase goes through a perfo-rated lamellar state within a certain temperature range. The numerical results are elucidated by a multimodeanalysis under the single-wave-number approximation. The analysis reveals that the geometric characteristicsof the free energy surface, particularly saddle points and ridgelike features, are responsible for the nontrivialintermediate states on the kinetic pathways. On the basis of this analysis, a generalized kinetic ‘‘phase dia-gram’’ is constructed, which is able to account for all the different scenarios observed in the numericalsimulation. Our results are discussed in connection with available experimental observations. In particular, wesuggest the possibility that the perforated-modulated lamellar structures obtained by Bates and [email protected]. Hamley, K. A. Koppi, J. H. Rosedale, F. S. Bates, K. Almdal, and K. Mortensen, Macromolecules26, 5959~1993!; S. Forster, A. K. Khandpur, J. Zhao, F. S. Bates, I. W. Hamley, A. J. Ryan, and W. Bras, Macromol-ecules 27, 6922 ~1994!# may be kinetic, intermediate states rather than new equilibrium phases.@S1063-651X~97!10301-4#

PACS number~s!: 64.70.2p, 81.30.Hd, 83.70.Hq

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I. INTRODUCTION

One of the most fascinating properties of block copomers is their ability to self-assemble into a variety of ordemicrostructures. The simplest block copolymer system isundiluted diblock copolymer with two incompatible blockwhere upon decreasing the temperature and/or increasinmolecular weight, a variety of ordered microstructures, sas body-center-cubic~BCC! spheres, hexagonally orderecylinders~HEX!, lamellae~LAM !, and more complex bicontinuous structures have been obtained@1#. The equilibriummorphological behaviors of such simple diblock copolymare now well understood. Relevant to the current work is tfact that in the weak segregation limit~i.e., near the order-disorder transition!, phase transitions can be effectedchanging the temperature. Thus an interesting and naquestion is what are the kinetic pathways during the variorder-order and order-disorder transitions after a suddenperature change? More specifically, how does the systransform from one microstructure to another~of differentsymmetry!? Are there interesting intermediate states durthe transition? If so, what is the nature of these intermedstates?

In this paper, we address these questions by a ti

*Author to whom correspondence should be addressed.

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dependent Ginzburg-Landau~TDGL! approach. Apart fromthe fundamental interest in understanding the kineticsphase transitions involving spatially modulated phases,swers to these questions are also helpful in designing suitprocessing routes for obtaining well-ordered structuresself-assembling systems. For example, our preliminarysults@2# indicate that hexagonal cylinders obtained by a teperature jump from a defect-free lamellar structure docontain many structural defects, whereas the same strucobtained by a temperature quench from the disorderedeven the BCC phase will usually be laden with defects. Tin turn raises an interesting question about the role ofinitial symmetry in directing the structural evolution of another phase.

To be concrete, we shall useA-B diblock copolymers asthe context, although the same analysis can be eaextended to other systems. For diblock copolymers,appropriate order parameter isc(rW)52( f2fA(rW))5fB(rW)2fA(rW)2(122 f ), wherefA(rW) andfB(rW) are, re-spectively, the local volume fractions ofA andB monomers,and f is the fraction ofA monomers in the diblock. The timeevolution of the order parameterc is governed, in the spiritof linear irreversible thermodynamics, by the following timdependent Ginzburg-Landau equation@3#:

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dc D1h~rW,t !, ~1!

1682 © 1997 The American Physical Society

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55 1683KINETICS OF PHASE TRANSITIONS IN WEAKLY . . .

where we have used the fact thatc is a conserved ordeparameter as is appropriate for diblock copolymers. In~1!, M is a mobility coefficient, which we assume to beconstant;h(rW,t) is a random force, which for a systemequilibrium at temperatureT, satisfies the fluctuationdissipation relation

^h~rW,t !h~r 8W ,t8!&522MkBT¹2d~rW2r 8W !d~ t2t8!; ~2!

andF@c(rW)# is the free energy functional which we takebe of the form

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2E dr1W E dr2WG~r 1W2r 2W !c~r 1W !c~r 2W !. ~3!

This form of the free energy can be considered as an exsion in the order parameterc and 122 f . The long-rangeinteraction in the last intergral arises from the connectivitythe two blocks and is typical of amphiphilic systems@4–6#,whereG(r 1W2r 2W ) satisfies¹2G(r 1W2r 2W )52d(r 1W2r 2W ). t isrelated to the order-disorder transition temperature andcoefficientsa,b,c,u, andv can be computed by evaluatinthe appropriate vertex functions given by Leibler@7# togetherwith the local approximation for the higher~third and fourth!vertex functions@5,8#. We point out that a large class ophysical systems with competing short-range and long-rainteractions can be described by a free energy functionathe type in Eq.~3! @9#. Thus we believe that the phenomenoogy we are studying is quite general and, hence, we allowfreedom of choosing the parameters in Eq.~3! as phenom-enological constants not specifically limited to diblock cpolymers.

Derivation of the TDGL equations from the microscopchain dynamics generally yield a nonlocal mobility coefcient @10–12#. Here we ignore this nonlocality, by assumina constant mobilityM . This should not be a serious approxmation in the weak-segregation regime since the orderrameter in this regime is dominated by a single wave nuber. By the same token, the distinction between conserand nonconserved order-parameter dynamics is not esseand, therefore, most of our results for the case of conseorder parameters should be applicable also for nonconseorder parameters.

Several dynamical aspects of systems that can bescribed by the free energy funtional Eq.~3! have been ad-dressed in the literature. Desai and co-workers, using a cbination of numerical and analytical approaches, haextensively investigated the kinetics of microstructure fmation from the disordered phase@13#. Oono and co-workers@14,15# developed a cell-dynamics approach for simulatthe formation of microstructures in diblock copolymerChakrabarti, Toral, and Gunton using the cell-dynamicsproach combined with simulated annealing, showed tlarge, defect-free lamellar domains can be obtained witsufficiently slow cooling rate@16#. A number of researcher

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have addressed the rheological properties of block copmers near the order-disorder transition@17,18# and in theordered phases@19,20#. In recent years, flow-induced ordeing and alignment of microstructures by flow in block cpolymers, have attracted much attention both theoretic@21–24# and experimentally@25#. While these dynamical is-sues are all important and interesting, we feel that the spler but no less important question of the kinetics of trantions between different ordered phases and betweenordered and the disordered phases in the quiescent condneed to be addressed. The kinetics turn out to be far frtrivial, even in transitions where there is no free energy brier: as we will demonstrate in this paper, pseudostabletransient states on the kinetic pathway, which are due,spectively, to saddle and ridgelike features on the free enesurface, give rise to interesting intermediate structures duthe transition and to nonsimple temporal behaviors. Soresults for the hexagonal to BCC and hexagonal to disdered phase transitions were reported by us recentlybrief communication@26#. The present paper is a more etended study that includes other transitions and also encpasses a broader range of issues.

The TDGL equation will be analyzed by two methoddirect numerical simulation and a simplified mode analyusing the single-wave-number approximation. The tempeture change is mimicked by a change in the parametert froman initial value favoring one phaset i to a value favoring adifferent phaset f . Broadly speaking, an increase int cor-responds to a decrease in temperature and vice versa. Inwork we focus on temperature-jump (T jump! kinetics be-cause in temperature-quench cases, the kinetics usuallvolve the formation of complicated topological defects acompanying the lowering of symmetry in a system. Inclusiof such defects requires a much larger system in the dinumerical simulation and more sophisticated analyticalproaches than employed here. Also, we avoid the issuenucleation and consider cases where theT jumps are beyondthe spinodal of the initial phase in question. Thus the trantions we study are barrierless and the microstructuchanges are global. Finally, we limit our study to transitioinvolving the lamellar~LAM !, hexagonal cylinder~HEX!,body-center-cubic~BCC!, and disordered~DIS! phases. Weomit the gyroid phase@27# since this phase is not stablethe immediate vicinity of the order-disorder transitio@28,29#.

This paper is organized as follows. In Sec. II, results frodirect numerical simulation of the TDGL equation are prsented for several order-order and order-disorder transitiThe temporal evolution of the system is monitored by defiing a global order parameter to indicate the different staof the kinetic pathways. Three-dimensional gray-level plare used to show more directly the microstructural evolutiThese numerical results are elucidated in Sec. III by a mtimode analysis under a single-wave-number approximatWe highlight two types of intermediate states: the psdostable state@30# and the transient state which corresponrespectively, to saddle and ridgelike features in the freeergy surface. A generalized phase diagram which includevarious metastability and pseudostability limits as wellregions in which other interesting intermediate structuresto be observed, is constructed. Our results are discusse

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1684 55SHUYAN QI AND ZHEN-GANG WANG

Sec. IV in connection with available experiments and alsothe context of some broader issues in the behavior of blcopolymer mesophases.

II. DIRECT NUMERICAL SIMULATION

In this section, we present results from direct numerisimulation of Eq.~1!. The parameters are choosen tou50.5, v51.5, b50.5, c50.02, anda51.5. To facilitatethe numerical procedure, we bring Eq.~1! into a cell-dynamics form suggested by Oono and co-workers@14,15#.The numerical simulations are performed on a 48348350simple cubic lattice. An initial ordered structure~LAM,HEX, BCC! is prepared at the initial temperaturet i ; then wechanget to t f and follow the subsequent evolution of thsystem after this sudden change in temperature. To besistent with the mean-field level mode analysis in Sec.we will ignore the thermal noise term in Eq.~1!, except toinclude it as providing an initial randomness~for possiblesymmetry breaking!. We have conducted studies that incluthe fluctuation during the simulation and do not find itsfects to be essential. The effects of fluctuations will be dcussed further in Sec. IV.

Two different procedures are used to generate the indefect-free ordered phases. One is to set a sinusoidal wwith the appropriate symmetry of the state~with the wave-length determined from the free energy! and then let thesystem equilibriate for a suffciently long time. Another prcedure is to let the ordered phase evolve from the disordstate by adding a random but anisotropic initial perturbatdetermined by the symmetry of the state. The two schelead to the same ordered structures. In both cases, we msure that the initial ordered phase is the equilibrium structat the temperature (t) from the mean-field diagram.

To quantify the global temporal behavior of the syste

we introduce an order parameterQ5A(sitesc(rW)2

5A(kckc2k. At the mean-field level,Q is zero in the dis-ordered phase and is positive in the ordered phases. Theevolution ofQ is used to demonstrate different stages inkinetics. Where there involve interesting intermediate stawe also show three-dimensional gray-level plots of the orparameters or two-dimensional cross sections. The diffetransitions are listed below. Cross references are madetween this section and Sec. III in order to corroboratetween the numerical findings and results from the simplifimode analyses.

A. LAM to HEX transition

A transition from LAM to the hexagonal cylinder~HEX!phase is possible only for asymmetric diblocks. Herechoosef50.45. The initial value of the temperaturelike prametert i50.35. The mean-field equilibrium structure coresponding to this set of parameters is the lamellar structThe structure is shown in Fig. 1~a!, with the layer normalalong thez direction. Three temperature jumps are studwith the finalt f equal 0.25,0.23, and 0.22, respectively,within the HEX phase in the mean-field phase diagram.physical grounds, we expect the cylinders to pinch off frothe lamellar layers containing the minority blocks, with theaxes lying within the initial lamellar layers.

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Figure 2 shows the temporal evolution of the global ordparameterQ. For all three temperature jumps, the order prameter reaches a final steady-state value consistent withequilibrium HEX phase. However, in all three cases, the elution is nonmonotonic. In the cases of the two lower teperatures~or largert f values!, there is a clear plateau following the initial rapid decrease in the order parametsuggesting that the system reaches some kind of stagnpoint. The duration of the plateau is clearly longer for tlower temperature (t f50.25) than for the higher temperatur(t f50.23). Fort f50.22 the plateau is less pronounced.

Direct visual inspection of the gray-level plot of the mcrostructures during the first plateau region does not revany specific structure within the lamellar layers. Thus tsystem remains in the lamellar phase, with a decreased oparameter. A multimode analysis using the single wanumber approximation in Sec. III shows that the first platein Fig. 2 corresponds to a saddle-point in the free enesurface.

Of special interest is the appearance of a second plateathe two lower temperature cases. A gray-level ordparameter plot@Fig. 1~b!# shows that at the end of the firsplateau, modulation within the lamellar layers begins to dvelop. The structure becomes most pronounced in thepart of the plateau; see Fig. 1~c!. There seems to be cleacorrelations between different layers. At the end of the pteau, cylinders evolve by connecting the dark regions in oof the diagonal directions of the xy plane. The final HEstructure is shown in Fig. 1~d!. It is remarkable that thisstructure is nearly perfect, i.e., free of any structural defealthough a symmetry breaking is clearly required to selthe orientation of the cylinders within the lamellar layers.

The structural characteristics of the intermediate structresemble in many respects the so-called perforated or mlated lamellar phase reported in literature@31–33#. One maytentatively identify the structure in Fig. 1~b! as the modu-lated lamellae~ML ! and that in Fig. 1~c! as the perforatedlamellae ~PL! although the evolution from ML to PL is acontinuous process. It will be shown in Sec. III that the sond plateau is due to another saddle point in the free enesurface. The PL structure corresponds to this saddle powhereas the ML is a transient state between the LAM andstates. Analyses there show that the perforated lamellarteau exists only when the final temperature is within a cert‘‘phase boundary’’ in the HEX part of the phase diagraThis saddle point disappears when the temperature jumoutside of this ‘‘phase boundary’’ as is the case ft f50.22. In that case, the cylinder phase emerges direwithout going through a distinct PL stage, although a shtransient ML state is still involved when the plateau LAdecays.

B. LAM to BCC transition

The initial state is the same as in the LAM to HEX trasition. The temperature parameter is changed fromt i50.35to t f50.216. The temporal evolution of the global order prameter is shown in Fig. 3. Notice the large initial decreain the order parameter. This initial fast decrease is followby a long valley where the order parameter increases oslightly but overall remains very weak. Gray-level plo

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FIG. 1. Microstructural evolution after aT jump from LAM to HEX at f50.45 (t i50.35,t f50.25); data taken at various stages of tsimulation.

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@with the same contrast level as in Fig. 1~a!# of the spatialdistribution of the order parameter shows no specific strture within the lamellar layers at the bottom of the curvcorresponding tot55000 @34#. However, upon enhancinthe contrast in the images, some weak traces of the Bwave superposed on the lamellar structure can be deteThe BCC waves appear at an accelerated pace at at515 000 and the final BCC structure is completed withinshort period of time thereafter. It will again be shown ththe long valley is due to the existence of saddle regionsthe free energy surface. Att520 000, the system reachesstable BCC microstructure, see Fig. 4. Note again thesence of defects in the structure.

C. LAM to DIS transition

Using the same initial condition as in Sec. II A and SeIIB @Fig. 1~a!#, we studied two temperature jumps, fro

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during the LAM to HEX transition atf50.45 for T jumps fromt i50.35 tot f50.25, 0.23, and 0.22, respectively.

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t i50.35 to t f50.21 andt f50.202, respectively. Botht fscorrespond to the disordered~DIS! phase. In both cases, theglobal order parameter decreases monotonically. Howevvisualization of the gray-level plots of the order parametduring the melting process reveal that melting proceethrough two different routes for the two cases: while in thcase of large temperature jump the lamellae melt direc~i.e., the amplitude of the lamellar wave simply decrease!,in the case of the smaller temperature jump, the lamellmelt by first developing modulation within the lamellar layers and melt as this three dimensionally ordered structuFigure 5 shows the gray-level plots in thexy cross section~i.e., in the lamellar layers!. Although the exact symmetry ofthis intermediate structure is difficult to ascertain owing tthe small system size used in the simulation and the weaness of the structure, it is reasonable to believe that thstructure is the same as the intermediate, modulated lamestructure observed during the transition from lamellae

FIG. 3. Temporal evolution of the global order parameterQduring the LAM to BCC transition atf50.45 for aT jump fromt i50.35 tot f50.216.

FIG. 4. Final microstructure after aT jump from LAM @sameinitial state as in Fig. 1~a!# to BCC at f50.45 (t i50.35,t f50.216).

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hexagonal cylinders. Using this assumption, we can shthat this transient structure results from a ridge-like lanscape in the free energy surface, similar to the transsphere structure seen during the melting of the hexagocylinders@26#; see Secs. IIIC and III E. On the other hanthe melting for the large temperature jump follows a simpdownhill path in a simple parabolic free energy surface.

D. HEX to BCC transition

The phenomenology for the HEX to BCC and HEXDIS transitions has been reported in our previous publica@26#. Here we discuss these transitions both for completenand to include other details that were omitted in our previopaper. For these two transitions, we choosef50.4 instead off50.45 used in the previous transitions@35#.An initial equilibrium hexagonal cylinder phase is pr

pared att i50.30 @see the initial structure in Fig. 7~a!#. Twotemperature jumps tot f50.25 and 0.246, respectively, arstudied. The evolution of the global order parameterQ isshown for the two cases in Fig. 6.

As with temperature jumps from the LAM phase, a temperature jump from the HEX phase to the BCC phase leto a rapid initial decrease in the order parameter. This radecrease is followed by a plateau which is again due tsaddle point in the free energy. At this saddle point,system remains a HEX phase. Thus there is a time lag bethe BCC structure appears. The sphere structure beginemerge from the cylinders at the end of the plateau; see7~b!. The final state is shown in Fig. 7~c!. Both temperaturejumps exhibit the same qualitative behavior, althoughduration of the plateau shortens for the higher tempera~small t). This implies that the saddle point becomes mounstable. Indeed, we have determined that there exisboundary within the BCC phase, such that the saddle pno longer exists for a temperature jump beyond this bouary. It is interesting to note that, as in the case of LAMBCC transition, existence of such a saddle point does

FIG. 5. Cross-section view of the in-layer (xy plane! structure att52000 during the LAM to DIS transition atf50.45 fort f50.21.

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E. HEX to DIS transition

The same initial state as used in Sec. II D is used as tinitial state for the HEX to DIS transition. Two temperaturejumps tot f50.24 and tot f50.22, respectively, are studied.The evolution of the global order parameter is depictedFig. 8. For the larger temperature jumpt f50.22, the orderparameter decays in a simple manner, suggesting direct ming of the cylinders. Gray-level plots of the order-parameteat various times show no structural change as the cylindemelt.

In contrast, the behavior ofQ in the case of the smallertemperature jump~to t f50.24) suggests a nontrivial meltingprocess. The slower decay at intermediate times is accomnied by the formation of structures along the cylinders: thcylinders seem to first break into spheres and the meltiproceeds via this modulated hexagonal cylinder~MHC!phase. This transient structure is clearest between the tinflection points on the curve, because at long times thstructure has all but melted and at short times the structuredominated by the hexagonal cylindrical structure of the intial state. Figure 9 shows the transient MHC state.

To study the effects ofin situ thermal fluctuations, wehave performed simulations both with and without thesfluctuations and found that the same qualitative behaviorobserved.~The results presented in this subsection for thsmaller temperature jump have been obtained with thermfluctuations: the final nonvanishing value ofQ in Fig. 8 isdue to these fluctuations.! However, the inclusion of thermalfluctuations in the case of the smaller temperature jumpfound to enhance the transient modulated cylinder structuThis point will be discussed further in Sec. IVA.

F. BCC to DIS transition

For this transition, we chosef50.45 and use the finalBCC structure obtained in the LAM to BCC transition andthe correspondingt i50.216; see Fig. 4. Two temperaturejumps are studied witht f50.21 andt f50.202, respectively.The behavior of the global order parameter shows som

FIG. 6. Temporal evolution of the global order parameterQduring the HEX to BCC transition atf50.4 (t i50.30).

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slight difference for the two temperature jumps. For tsmaller temperature jump (t f50.21), the curve has two inflection points, while for the larger temperature jum(t f50.202), the decay is rather fast and is nearly expontial. This difference for the larger and smaller temperatujumps in this case is rather similar to the difference in tbehavior ofQ for the larger and smaller temperature jumin the LAM to DIS and HEX to DIS transitions. However, ithis case, there is no free energy ridge. In Sec. III F we wshow that there exist two inflection points along the kinepath on the free energy surface. Although this does not lto any new intermediate structures~the system melts asperfect BCC structure!, it changes the temporal behavior othe melting process.

III. MODE ANALYSES USING ASINGLE-WAVE-NUMBER APPROXIMATION

To understand the observations of the numerical resand to gain insight into the transition mechanisms, in tsection we perform a simplified single-wave-number, amptude equation analysis@36#. This analysis is motivated by ouobservation that to a good approximation the kinetics ofvarious transitions can be described by the developmentdisappearance of various density waves. In fact nearly alltransient structures studied in this paper can be represeby an appropriate superposition of the LAM, HEX, and BCwaves, and the time evolution of the system correspondstime evolution of these waves with generally changing pportions. This conclusion is reached both by direct visinspection of the structures and by Fourier analyses of seral transient microstructures into the dominant modes.single-wave-number assumption is justified because denmodulations in the weak segregation limit is nearly sinsoidal.

We seek a minimal set of wave vectors which can repsent the LAM, HEX, BCC, as well as intermediate statobserved in the preceding section. To this end, we chooseset of wave vectors corresponding to the first smallest wavectors of a bcc lattice whose magnitude we take to beof the optimal wave vectork*5(c/b)1/4 @37# determinedfrom the quadratic term of the free energy. The order paraeter is then represented byc5(n51

6 Ancos(knW •rW), where

$knW % where theAns ~generally not equal to each other! are theamplitudes associated with these respective wave vecWe set our coordinate system such that

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FIG. 7. Microstructural evolution during the HEX to BCC transition atf50.4 with t i50.30, andt f50.246.

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Thus, in this single-wave-number approximation, the LAstructure is represented byA1Þ0 andAn50 (n52–6!. TheHEX structure can be obtained by choosing any threeplaner waves that includek1W with equal amplitude, while theBCC structure requires all sixAns to be equal in magnitud@38# and nonvanishing. In this construction, the ‘‘epitaxiarelationship between the initial and final structures@39#, isautomatically satisfied. For example, the HEX phase grofrom the LAM phase will have the cylinder axes lying withthe initial lamellar layers, and the spheres of the BCC phemerging from the HEX phase will reside on the initial cyinder axes.

In terms of the amplitudesAns, the free energy can bwritten as

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Correspondingly, the dynamic equation Eq.~1! becomes aset of equations for the various amplitudes@36#

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Note that in the single-wave-number approximation, thLaplacian operator in Eq.~1! is transformed to2k* 2 @37# , aconstant factor. Thus Eq.~6! simply describes the steepesdescent of the order parameters on the free energy surfgiven by Eq.~5!. This allows us to investigate the kinetics othe transitions by studying the landscape of the different frenergy surfaces corresponding to the different transitions

Further symmetry considerations require thatA25A6 andA35A5. Thus we group the six waves into four subsets:~i!k1W , with amplitudeA15A; ~ii ! k2W and k6W , with amplitudesA25A65B; ~iii ! k3W and k5W , with amplitudesA35A55C;and ~iv! k4W , with amplitudeA45D. The various transientstructures can be obtained by the appropriate superposiof these four subsets of waves.

The perforated-modulated lamellar~PL-ML! structureshown in Sec. II A deserves some further remarks. Withthe bcc-based, single-wave-number approximation, periodensity modulations in the lamellar layers can be represenby a structure with2A.B5C.2D.0. Such a structuredoes not have perfect in-layer sixfold symmetry whichbelieved to be associated the PL-ML structure@31#. We havealso used a hcp~hexagonal-close-packing! based wave vectorset to describe the LAM to HEX transition and found thsame qualitative behavior as we did using the bcc-basedtice in all cases where a transient perforated lamellar strture is involved. Thus, we will adhere to the bcc-based reresentation in most part of this paper as it enables to treatthe structures considered in a unified framework. In fathere is reason to believe that experimentally observPL-ML structures may be either hcp or bcc based@40#.

A. The LAM to HEX transition: pseudostable LAM and PL

In this phase transition, all the four groups of wavesA,B, C, andD are involved, where as mentioned above,A isthe amplitude of the initial lamellar wave. EitherA52B orA52C ~where the minus sign ensures the proper phaselationship! can form a HEX structure given that the otheparameters are zero, and whenA, B, C, andD are all non-

FIG. 8. Evolution of the global order parameterQ for the Tjump from the HEX to DIS phase atf50.4 with t i50.30.

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zero, a three-dimensional periodic structure is obtained.particular,2A.B5C.2D.0 represent a structure withdensity modulations within the lamellar layer; we will iden-tify this structure as the perforated-modulated lamellar~PL-ML ! structure.

The kinetic pathway from the LAM phase to the HEXphase after changingt from 0.35 to 0.25 is shown by thetime evolution of the four waves, see Fig. 10. InitiallyA520.3 and all the other three amplitudes are set to ze~some small perturbations are included as initial drivinforces for these waves to emerge.!. The final state withA52B andC5D50 clearly corresponds to the HEX struc-ture. Note, however, the existence of two plateau regions.the first plateau region,A is nonzero, although it has de-creased significantly from its initial value. But the other threwave amplitudes are still nearly zero. Therefore the structucorresponding to this plateau is still lamellar. The length othis plateau depends on the initial perturbation in the oth

FIG. 9. Intermediate state during the HEX to DIS transition af50.4 (t i50.30,t f50.24), taken at t51000.

FIG. 10. Evolution of the wave amplitudes during the LAM toHEX transition atf50.45 (t i50.35, t f50.25). Note the appear-ance of two plateaus beside the final HEX state.

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three waves and on the temperature. At the end of theplateau,A has another quick drop, and at the same timeother three waves take off. Then the system reaches theond plateau. On this plateau,D remains quite small, butBand C have increased considerably while maintainiB5C. The structure corresponding to this second plateashown in Fig. 11. It is rather reminiscent of the perforatlamellar structure Fig. 1~c! observed in the numerical simulation. The results in Sec. II A, including the behavior of tglobal order parameter and the microstructural featuresing the transition, are all consistent with the foregoing anasis.

The plateaus in Fig. 10 suggest that the system panear saddle points on the free energy surface. Using theenergy Eq.~5!, we can locate these saddle points and

FIG. 11. Microstructure generated using the six wave amtudes with values corresponding to the second plateau in Fig.

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region of the phase diagram in which they exist. We find tthe first saddle point evolves from the local free energy mmum corresponding to the LAM phase, i.e., the metastablimit of the LAM phase is reached by turning the local minmum into a saddle point. This scenario is shown in theduced parameter space ofA ~the lamellar wave! and B~which, together withA, consititutes the HEX wave! in Figs.12~a! and 12~b! with C5D50. The value oft where thishappens marks the spinodal of the lamellar phase. We sterm states corresponding to saddle points in the free ensurface pseudostable states@30#. The range oft in which thissaddle point exits is shown schematically in Fig. 15. Becaof the positive curvature along theA axis in Fig. 12~b!, thedecay of the initial LAM structure~with A more negativethan its saddle point value! in this temperature range, wilfollow a path that drives the system to the saddle point. Tsaddle point is responsible for the near stagnation~plateau!in both Fig. 2 and Fig. 10. As the temperature increasestdecreases!, the negative curvature in theB direction in-creases and the positive curvature in theA direction de-creases, thus the time spent by the system near the sapoint gets shorter. This is indeed the case; see Fig. 2.

A similar saddle point is responsible for the appearancethe second plateau in Fig. 10. At this saddle point, howevall the three amplitudesB, C, and D are nonzero, butB5C. Therefore the structure has three dimensional ratthan one-dimensional order. The saddle point is unstawith respect to bifurcation fromB5C. This is shown in Fig.12~c! where we setA520.117 808 andD50.007 061 attheir respective saddle point values. The system eventuleaves this saddle point following the unstable directiwhich leads to eitherA52B, C5D50, or A52C,B5D50, both respresenting a perfect HEX structure wthe cylinder axes lying within the original lamellar layerThe range oft where this saddle point exists is shown schmatically in Fig. 15, which lies within the HEX part of thequilibrium phase diagram.

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FIG. 12. Free energy contour plot in reduced order parameter space illustrating~a! the metastable LAM (t f50.28!, ~b! the pseudostableLAM ( t f50.27), and~c! the pseudostable PL (t f50.25! states. In~a! and~b! C andD are both set to zero whereas in~c! A andD are setto the saddle point values (A520.117 81,D50.007 06).f50.45 for all three figures.

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Although the pseudostable LAM structure itself hasinteresting structural features, the decay from this structurnontrivial. It turns out the most unstable direction is therection of increasingB andC with B5C. This implies thatwhen the system leaves the pseudostable LAM, it does sgoing through a transient ML state. If a pseudostablestate exists, the ML will evolve to the PL state which evetually decays to the HEX. If the temperature is outside ofpseudostable PL region, the ML will evolve continuouslythe HEX structure without a distinct intermediate stagThese conclusions are indeed consistent with the resulSec. II A.

B. LAM to BCC transition: pseudostable LAM states

As in the LAM to HEX transition, all the four wavesA,B,C, andD are required to describe this transition. Thenetic pathway displays two different behaviors dependingthe extent of the temperature jump. For the smaller tempture jump fromt i50.35 to t f50.216, the kinetic pathwayshows a plateau similar to the first plateau in the LAMHEX transition. At this plateau the structure is again lamewith a reduced wave amplitude. The other three waemerge almost equally at the end of this plateau whileAcontinues to decrease, reaching the final va2A5B5C52D50.028 24 corresponding to the equilibrium BCC structure at this temperature.

For the larger temperature jump fromt i50.35 tot f50.212, no plateau is seen in the pathway: there isteady decrease ofA from its initial value and a steady increase ofB, C, andD, reaching the steady-state value0.024 88. Incidently, we find that relaxation to the final eqlibrium BCC structure is slower for the case of the largerTjump; this is to be expected because the final temperatucloser to the~mean-field! spinodal~of the DIS phase!.

The LAM saddle point for the smallerT jump in the LAMto BCC transition has the same origin as the LAM sadpoint in the LAM to HEX transition. Beyond the spinodal othe LAM phase, the LAM saddle point~stable with respect tovariation in the lamellar wave amplitudeA but unstable withrespect to perturbations in at least one of the other amtudes! persists over a rather wide region in the phase dgram; this region has overlap with the lower temperature pof the equilibrium BCC phase. Thus if theT jump is into thisregion, transition from the LAM to the BCC phase will proceed by first going through the LAM saddle point with thconsequence of a time lag in the growth of the BCC strture. Beyond this region, transition will be a direct one witout a time lag. The scenario described in this section is fuconsistent with the simulation results of Sec. II B.

C. LAM to DIS transition: transient ML state

We again start with the equilibrium LAM structure at i50.35 with A520.3. Two T jumps to t f50.21 andt f50.202 are studied by numerically solving the focoupled amplitude equations with small perturbations inB,C, andD of the order of 0.3% of the initial value ofA. Theevolution ofA, B, C, andD at t f50.21 is shown in Fig. 13.All these four amplitudes eventually decay to zero asequilibrium phase at thist is the DIS phase. However, notthe transient increase inB andC. At the maximum ofB and

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C, the structural feature is similar to that obtained during thetransition from the LAM to HEX, although the densitymodulation within the layer~as measured, for example, byuB/Au or uC/Au) is much weaker here. For this reason weterm this transient structure also ML. Although the structurafeatures between this ML and the ML observed in the LAMto HEX transition are the same, the mechanisms which givrise to this structure in the two cases are different. Recall thain the case of the LAM to HEX transition, the ML is due toa saddle point on the kinetic path: it is the most unstablmode for the pseudostable LAM to decay, and if a pseudostable PL state exists, it will further evolve into such astate. Here, however, no such saddle points exist. Rather, ttransient growth of the amplitudeB or C is due to deflectionfrom the path along the axisB5C5D50 caused by a ridge-like feature in the free energy landscape. This is shown bycontour plot of the free energy in the reduced parametespaceA, B, see Fig. 14. We have found analytically thatthere exists a small finite windowt l,t,tu in the phase

FIG. 13. Temporal evolution of the different modes during thetransition from LAM to DIS atf50.45 (t i50.35,t f50.21).

FIG. 14. Free energy contour plot in the reduced order parameter space showing the ridgelike feature during the LAM to DIStransition atf50.45 (t f50.21).C andD are both set to zero.

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diagram for this transient state to exist. The upper boundtu coincides with the BCC-DIS phase boundary. Ff50.45, t l50.2075. Therefore, whent f50.202,t l , allfour amplitudes decrease monotonically, and melting ofLAM will proceed directly without involving a transient MLstate. This compares well with the simulation results.

D. HEX to BCC transition: pseudostable HEX state

Assuming that this transition involves only the initiHEX waves and another set of waves which are part ofBCC structure, the four groups of waves can be furtherduced to two, denoted byA(5B) andC(5D), respectively,where A consititutes the HEX density wave, witA5A15A25A6 andC5A35A45A5.

The temporal evolution ofA andC for a T jump fromt i50.30 to t f50.25 at f50.4 was reported in our earliepublication, and will not be shown here. It suffices to nothat there is a long plateau after the rapid decrease inA fromits initial value. Within the duration of this plateau,C re-mains close to zero. Thus there is a time lag for the Bstructure to appear, and when it does, the structure becothat of a modulated hexagonal cylinder~MHC! with a con-tinuous growth of the BCC component, reaching the fiequilibrium state withA52C. The plateau arises fromsaddle point which lies on theC50 axis. Thus we have apseudostable HEX structure in the HEX to BCC transitiosimilar in its origin to the pseudostable LAM state. Onagain, there is a region in the phase diagram for the psestability. The lower temperature~largert) side of the bound-ary is the spinodal of the HEX phase, i.e., when the Hreaches its metastability limit, it does so by turning the lominimum of free energy into a saddle point. The situationsimilar to how the LAM phase loses its metastability, sSec. III A. The high-T ~lower t) boundary lies within theBCC part of the phase diagram: forf50.4, tc50.248.Therefore aT jump to at f value beyond this boundary, i.et f,t l , but still in the BCC part of the phase diagram leato a direct transition from the HEX to the BCC. This eplains the observations in Sec. IID.

E. HEX to DIS transition: transient MHC state

In this case, the kinetics is again described by the torder parametersA andC, as defined in the last subsectioFor a largeT jump (t i50.30 to t f50.22), A decreasesmonotonically to zero; also any initial perturbation froC50 is rapidly damped. This corresponds to a direct meltof the HEX phase.

For a smallerT jump (t i50.30 tot f50.24), whileA stilldecreases monotonically, an initial perturbation fromC50first grows before it decays again to zero. Thus theretransient growth of the BCC component of the density waThis wave, superposed on the dominant HEX wave, resin a modulated hexagonal cylinder~MHC! state. These results are in full agreement with the simulation results.

By studying the free energy landscape in the order paretersA and C, we find that there exists a window in thphase diagram in which the landscape exhibits a ridgefeature. The transient appearance of the BCC feature duthe melting of the HEX is thus caused by the deflectionthe kinetic path from theC50 ridge.

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The degree of the deflection from the ridge, and hencestrength of the modulation and the length of its durationclosedly related to the magnitude of the fluctuations. Inabsence of any fluctuations~including initial noise!, a paththat starts withC50 will keep C50 even though theC50 axis is a ridge. Deviation fromC50 is caused byperturbations~either given initially or includedin situ!. Thesame considerations apply to other transient phenomcaused by ridgelike free energy landscapes. This issuebe discussed more fully in Sec. IV.

F. BCC to DIS transition

The transition from BCC to DIS is relatively featurelesHere, we do not find saddle or ridgelike features in the fenergy landscape. The numerical simulation above doesshow any distinctive intermediate structure. However, upcloser investigation of the free energy, we find two inflectipoints on the otherwise downhill path in a temperature wdow beyond the spinodal limit of the BCC phase. The higT ~low t) boundary of this window corresponds tt l50.2097 atf50.45. Fort f,t l the order parameter wildecay as a simple exponential, whereas fort f.tc the melt-ing will be somewhat slowed down between the two infletion points. We believe this is the reason for the nonsimdecay of the order parameter seen at the largert f in Sec. II F.

G. Generalized phase diagram

The different scenarios described in the previous sectican be summarized in a generalized phase diagram.phase diagram is obtained by a detailed study of theenergy landscape in the four order parametersA,B,C, andD, because under the assumption that the kinetics ofvarious order-order and order-disorder transitions can bescribed by a few dominant modes with a single wave nuber, the kinetics reduces to a steepest descent path on theenergy surface. We have calculated analytically all the mefield ‘‘phase boundaries’’ for the transitions studied in thpaper in terms oft and f . However, there are too mancurves in this phase diagram to leave any space for labethe different scenarios. Therefore, we choose to show othe different behaviors at a singlef50.45. The qualitativebehavior is the same for other values off . The diagram is notdrawn to scale; only the correct sequence of these phboundaries int is maintained.

The phase diagram is shown in Fig. 15. The three thsolid lines represent the equilibrium phase boundariestween the LAM and HEX, HEX and BCC, and BCC anDIS, respectively. These equilibrium phase boundaries arcourse independent of the initial state of the system. Hoever, all other boundaries describe the different scenaduring the transition from one phase to another, and heone must specify the initial state of the system. From leftright, the three columns describe theT-jump transition kinet-ics starting from the LAM, HEX, and BCC phases, respetively.

We now illustrate the richness of the phase diagramfollowing a T-jump sequence from the equilibrium LAMphase. Seven different kinetic scenarios can be identifidepending on the final temperature (t f). If t f lies in regionI L ~the subscriptL refers to the initial LAM phase!, the

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lamellar phase is metastable. Transition to the HEX phcan only occur via nucleation of the HEX within the LAMphase. The dotted line indicates the spinodal of the LAphase. In regionII L , the LAM phase will first become pseudostable, then passes through another pseudostable~which we identify as the PL!, and finally reaches the HEXstate. In regionIII L , the LAM state will again first becomepseudostable. However, after some time lag spent atpseudostable state, the HEX phase will emerge gradu~via a short transient ML! but directly from the LAM struc-ture without another clear intermediate stage. In RegIVL , the LAM will spend some time at the pseudostabLAM state, and then transform continually to the BCC struture. This transition will again necessarily involve a shtransient ML structure, but no distinct stages can be idefied. It is interesting, and not coincidental, that the lowboundary of the pseudostable LAM is identical to the spodal limit of the HEX phase. In RegionVL , the transitionfrom LAM to BCC will be a direct one. Again, the transitiowill involve the ML state, but the process is a continuous owith no distinguishable stages. If the finalt f lies in RegionIVL , the LAM phase will melt by going through a transieML state, caused by a ridgelike feature in the free enelandscape which deflects melting of the LAM from following a direct path. Finally, in RegionVII L , the LAM phasewill melt directly without involving any other structures.

The different kinetic scenarios for transitions startifrom the HEX or BCC phases can be similarly identififrom reading the generalized phase diagram.

Whereas equilibrium states correspond only to the glofree energy minimum, the kinetics of the various transitioinvolve states that are caused by other features of the

FIG. 15. Generalized phase diagram: the different regions icate different transition scenarios and the intermediate statevolved, starting~from left to right! from LAM, HEX, and BCCphases, respectively.

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energy surface. Besides the thermodynamically stable sttures, there are metastable states corresponding to localenergy minima, pseudostable states corresponding to sapoints, and transient states either arising from deflectionmore direct path or serving to connect to distinct states.nally unstable states in our study refer to those which posa nonzero thermodynamic driving force and decays uformly and monotonically. Clearly, inclusion of these statgreatly broadens the phase diagram.

IV. DISCUSSION

In this paper, we have used a TDGL approach to invegate the kinetics of several order-order and order-disortransitions after aT jump in weakly segregated block copolymers. Both direct numerical simulation of the TDGL equtions and a simplified mode-analysis reveal that the kineis far from trivial even in the absence of any free enerbarriers. In particular, we propose that the kinetic pathwoften involve saddle and ridgelike features in the free enesurface, which can lead to interesting intermediate statesnonsimple temporal evolution of the system. A multitudekinetic scenarios have been identified. In this section,discuss the experimental relevance of our findings. Sincepresentation of our results in this paper has been inframework of a mean-field theory, we start with a brief dcussion of the effects of fluctuations.

A. Effects of fluctuations

It is well known that thermal fluctuations have rather prfound effects on the microphase transitions near the ordisorder transition temperature for systems described byenergies of the type Eq.~3!. The phenomenology of the fluctuation effects was first worked out by Brazovskii@41#. Fre-drickson and Helfand@42# applied the Brazovskii theory todiblock copolymers, where it was shown explicitly that thmolecular weight of the diblock plays the role of the Gizburg parameter: the effects of fluctuations decrease wincreasing molecular weight. The main effects of thermfluctuations are~1! the mean-field second order transition fa symmetric diblock is renormalized to a weakly first ordone; and ~2! near the symmetric composition, the LAMphase and the HEX phase can be directly accessible fromDIS phase. One would expect these results to have cebearing on the kinetics. However, as we mentioned earwe have conducted studies which include thermal flucttions in the form of Eq.~2!, and did not find the kinetics tobe qualitatively different from simulations which only include an initial randomness. It is possible, that the twof ’sused in our studyf50.4 andf50.45 are already asymmetrienough for the fluctuation effects to become unimport@43#. More systematic studies are needed to further addthis issue. A simulation of the full equilibrium behavior ofree energy Eq.~3! by the Monte Carlo method or the TDGapproach used here would seem a welcome effort.

Provided that the qualititative features identified in tmean-field framework are correct, fluctuations~either ther-mal fluctuations or other externally imposed perturbatio!have some systematic quantitative effects on the psdostable and transient states. Since the pseudostable scorrespond to saddle points on the free energy surface,

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tuations will lead to a faster decay from the saddle point ahence the lifetime of these pseudostable structures becshorter with the inclusion of fluctuations. On the other hatransient structures that are caused by ridgelike landscapthe free energy will be enhanced since fluctuations helpsideway pushes which are responsible for the appearancthese transient structures in the first place. The oppositefects of fluctuations on the pseudostable states and transtates caused by ridgelike free energy landscapes couldceivably be exploited to experimentally distinguish betwethese two types of intermediate states.

B. Comparison with experiments: The nature and stabilityof the PL-ML structure

The number of experiments that directly probe the kinics of microphase transitions described in this paper is lited at present. Hashimoto and co-workers studied the tsition from the LAM phase to the DIS phase after aT jumpusing time-resolved x-ray scattering@44#. However theirstudy was focused on the diffusion of the chains acrossorder-disorder boundary and not on the microstructuchanges. More recent experiments by Singhet al. @45# stud-ied the ordering kinetics of block copolymers after a teperature quench, with a focus on nucleation and growth.

Of more direct relevance to our study is a recent workSakuraiet al. @46#. These authors studied the HEX to LAMtransition in poly~styrene-b-butadiene-b-styrene! triblock co-polymers. They used solvent casting to produce a nonelibrium HEX cylinder phase which was transformed into tLAM phase upon annealing. Based on small-angle x-scattering ~SAXS! and transmission electron microscop~TEM! results, these authors suggested that the HEXLAM transition occurs via lateral coalescence of cylindeimplying a perforated or connected lamellar intermedistructure. Unfortunately, the transition they studied isreverse of that studied here and in general the two mayshare the same intermediate state. Also solvent castingin their work could add additional complications into thtransition mechanism. So a direct comparison between texperiment and our theoretical results is difficult to draThermally induced HEX to BCC transition was also studiby Sakuraiet al. @47#, but the focus there was more to demonstrate the thermal reversibility of this transition thanelucidate the mechanism of the transition.

One of our main results is the prediction that the PL-Mstructure serves as an intermediate during the transition fLAM to HEX in certain temperature ranges. We now dcuss, in some more detail, the nature and stability ofPL-ML structure. Thomas and co-workers@32# were the firstto propose a perforated lamellar~which they term thecatenoid-lamellar! structure as a model for a microstructuthat appears during the rearrangement of a lamellar morpogy in thin films of poly~styrene-b-butadiene! diblocks. Themodel they proposed is a bicontinuous structure where bthe PS and PB components are three dimensionally cnected. Bates and co-workers, using SANS and Ttechinques, have demonstrated the existence of a bulk prated lamellar structure in the weak-~to-intermediate! segre-gation regime in both poly~ethylene-propylene-bethylethylene! and poly~styrene-b-isoprene! diblock

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copolymers@31# @Based on rheological measurements, theauthors further distinguished between a hexagonally molated lamellar~HML ! at the lower temperature and a hexagnally perforated lamellar~HPL! at a higher temperatureHowever, the two structures do not seem to have differsymmetry; their difference seem to be simply a matter ofdegree of the in-plane density modulation.# The structurethey proposed is monocontinuous, where the perforatiwithin the lamellar layers have sixfold symmetry, and whethe perforations between adjacent layers are staggered.eral calculations@48,49# have addressed the thermodynamstability of this structure, however, they fail to capture somof the essential structural features which are now believebe associated with the ML-PL structure@31,33,29,50#. Morerecent state-of-the-art self-consistent field calculatio@28,29# using a hexagonal-close-packed~hcp! lattice @51#representation of the PL-ML phases have conclusivshown that within mean-field theory, these structures arethe lowest free energy states. Hamley and Bates@50# hadreached the same conclusion earlier based on the hcp pretation by keeping modes corresponding to the first two loest wave numbers.

On the other hand, the structural nature of the PL-Mphase has not been fully resolved. Hamleyet al. @50, 31#suggested that PL-ML structure is analagous to the hexphase in liquid crystals@52# which consists of weaklycoupled layers, with in-plane hexagonal bond-orientatioorder. This suggestion is based in part on the unusually wsixfold scattering pattern~and lack of higher harmonics! intheqz50 plane of theq space~wherez is the layer normalof the lamellae!. Here we offer an alternative proposal. Wbelieve that the PL-ML structurecan be represented by aunderlying hcp lattice. The first three lowest groups of wavectors are shown in Fig. 16. These are the same set of wvectors conceived by Hamley and Bates@50#. However,these authors took thefirst group of wave vectors to correspond to the optimal wave numberk* and only includedharmonic corrections due to the second group. We propthat the density waves are dominated rather by the secand third groupswhosemagnitude, rather than that of thfirst, are closest to the optimal wave number. Assuming fther thatk(2)5k(3)5k* , we find a stationary structure withA(2)Þ0, A(3)Þ0 and uA(2)u@uA(3)u@uA(1)u. A three-dimensional gray-level plot is shown in Fig. 17. Except f

FIG. 16. The first three groups of wave vectors of a hcp strture.

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some minor details, our proposed structure is consistentthe known structural characteristics of the PL-ML phase.particular, the weak sixfold scattering spots in theqz50plane is seen to arise from the very small amplitudeA(1). In addition, from Fig. 16, assumingk(2)5k(3)5k* , itfollows that k(1)5(A3/2)k* , leading to an in-layer periodwhich is 1.15 times the interlayer spacing; this compawell with the factor 1.1 deduced by Hamleyet al. @31# fromthe small-angle neutron scattering~SANS! data. Further-more, the ratiouA(3)/A(2)u is found to increase with increasing temperature, again in agreement with the observationthe same group. Calculation of the stability matrix shothat in the weak-segregation limit~characterized by structures that are dominated by a single, optimal wave numb!,the PL structure is always a saddle point where a stationpoint exists and hence is only pseudostable. Inintermediate-segregation regime, the PL structure cancome metastable with positive local curvatures in the fenergy but does not have the lowest free energy. Thestructure is either a PL with weak in-layer density modutions, or is the transient state via which the pseudostaLAM state decays.

Recently, Yeunget al. @53# studied the anisotropic fluctuations about the lamellar phase. By performing a linstability analysis, these authors find that upon increastemperature, the lamellar phase becomes unstable withspect to fluctuations with wave vectors that form a ring wkz51/2k* , anduku5k* in k space. Note that bothk(3) in Fig.16 and the wave vectors associated with the amplitudeBandC in Sec. III lie on this ring. Therefore, we conclude ththe ML-PL structure are formed from the anisotropic flutuations of the lamellar phase which first become unstawith respect to these wave vectors.

Based on this analysis, we suggest two possibilities ccerning the PL structure observed in experiments. One psibility is that these structures are indeed kinetic, intermeate states en route from the LAM to HEX and other hightemperature phases. Results from more recent experimstudies seem to be leaning toward such a possibility@54#.

FIG. 17. The pseudostable PL structure constructed usingthree groups of wave vectors in Fig. 16 (f50.45,t50.25).

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Another possibility is that thermal fluctuation or othsymmetry-breaking mechanisms~such as shear! can lead tostabilization of a pseudostable or metastable state. Wecurrently exploring these possibilities.

Thermally induced LAM to HEX transition has been stuied by Hajduket al. @55# for the poly~styrene-b-ethene-cobutene! diblock copolymer. However, these authors did nobserve the appearance of the intermediate PL-ML structWe suspect that the thermal annealing used in this studythe long intervals between observations is responsiblemissing this intermediate structure: since the PL structura saddle point in the free energy, its lifetime is shortenedthermal fluctuations associated with the annealing, makthe pseudostable structure to disappear at an early stagealso possible that theT jump is too large for the PL state tappear~cf. Fig. 15!.

As regards other transitions, Bates and co-workers hstudied the HEX to BCC@39# and HEX to DIS transitions@56# for a series of poly~ethylenepropylene-b-ethylethylene!.These authors showed that BCC spheres pinch off fromHEX cylinders such that the cylinder axis is the~111! direc-tion of the resulting BCC structure which they term the ‘‘eitaxial’’ relationship. They also suggested that the BCphase form by way of an undulating cylinder structuThese are in agreement with our observed mechanism;Fig. 7~b!. More interestingly, these authors reported obsertion of a transient sphere structure during the melting ofHEX in shear cessation experiments. In one of these expments, an initially disordered phase of asymmetric PEP-Pclose to the order-disorder boundary is subjected to a cstant shear which induces a transition to the HEX ph~with the cylinders aligned along the shear direction!. Theshear is suddenly stopped, and the system is now in adition favoring the DIS phase. These authors observedthe cylinders first break into spheres before melting toDIS phase, much akin to what we find after a temperatjump. Indeed insofar as a HEX phase is created and thencondition is changed to favoring the DIS phase, the shecessasion experiment can be likened to a temperature jexperiment. We believe the mechanism described in thisper is responsible for their observations.

In conclusion, on the basis of numerical simulation amode analysis in the framework of the TDGL approach,have provided a comprehensive picture of the differmechanisms in the order-order and order-disorder transitof block copolymer and other weakly segregated microstrtured systems. The inclusion of different kinetic scenarconsiderably broadens the phase diagram for these systWe hope that our results will provide important insightsexperiments that are directly or indirectly related to mcrophase transition kinetics in block copolymers and otrelated systems.

ACKNOWLEDGMENTS

This research is supported in part by the National ScieFoundation~Grant Nos. ASC-9217368 and DMR-953191!and the Camille and Henry Dreyfus Foundation~Award No.TC-96-063!.

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@1# See F.S. Bates, Science251, 898 ~1991!; F.S. Bates and G.HFredrickson, Annu. Rev. Phys. Chem.41, 525 ~1990!; G.H.Fredrickson and F.S. Bates, Annu. Rev. Mater. Sci.26, 501~1996!.

@2# S.Y. Qi and Z.-G. Wang~unpublished!.@3# J.D. Gunton, M.S. Miguel, and P.S. Sahni,Phase Transition

and Critical Phenomena, edited by C. Domb and J.L. Lebowitz ~Academic, New York, 1983!, Vol. 8.

@4# F.H. Stillinger, J. Chem. Phys.78, 4654~1982!.@5# T. Ohta and K. Kawasaki, Macromolecules19, 2621~1986!.@6# D. Wu, D. Chandler, and B. Smit, J. Chem. Phys.96, 4077

~1992!; M.W. Deem and D. Chandler, Phys. Rev. E49,~1994!; 49, 4268~1994!.

@7# L. Leibler, Macromolecules13, 1602~1980!.@8# A.M. Mayes, M.O. de la Cruz, and W.E. McMullen,Macro-

molecules26, 4050~1993!.@9# M. Seul and D. Andelman, Science267, 476 ~1995!, and ref-

erences therein.@10# K. Kawasaki and K. Sekimoto, Physica A143, 349 ~1988!;

Physica A148, 361 ~1988!.@11# V. Kumaran and G.H. Fredrickson, Physica A204, 378

~1994!.@12# See, also, M. Doi and S.F. Edwards,The Theory of Polymer

Dynamics~Clarendons, Oxford, 1986!.@13# C. Roland and R.C. Desai, Phys. Rev. B42, 6658 ~1990!; C.

Sagui and R.C. Desai, Phys. Rev. Lett.71, 3995~1993!; Phys.Rev. E 49, 2225 ~1994!; 52, 2807 ~1995!; 52, 2822 ~1995!;Phys. Rev. Lett.74, 1119~1995!.

@14# S. Puri and Y. Oono, Phys. Rev. A38, 1542~1988!.@15# Y. Oono and Y. Shiwa, Mod. Phys. Lett. B1, 49 ~1987!; M.

Bahiana and Y. Oono, Phys. Rev. A41, 6763~1990!.@16# A. Chakrabarti, R. Toral, and J.D. Gunton, Phys. Rev. E47,

3025 ~1993!.@17# G.H. Fredrickson and R.G. Larson, J. Chem. Phys.86, 1553

~1987!; R.G. Larson and G.H. Fredrickson, Macromolecu20, 1897~1987!.

@18# G.H. Fredrickson and E. Helfand, J. Chem. Phys.89, 5890~1988!.

@19# K. Kawasaki and A. Onuki, Phys. Rev. A42, 3664~1990!.@20# M. Rubinstein and S.P. Obukhov, Macromolecules26, 1740

~1993!.@21# A. Onuki, J. Chem. Phys.87, 3692~1987!.@22# M.E. Cates and S.T. Milner, Phys. Rev. Lett.62, 1856~1989!.@23# G.H. Fredrickson, J. Rheol.38, 1045~1994!.@24# T. Ohta, Y. Enomoto, J.L. Harden, and M. Doi, Macromo

ecules26, 4928~1993!; 26, 4935~1993!; H. Kodama and M.Doi, ibid. 29, 2652~1996!.

@25# For an excellent review of theoretical and experimental won the dynamics of block copolymers, see G.H. Fredricksand F.S. Bates@1# ~the third paper!.

@26# S.Y. Qi and Z.-G. Wang, Phys. Rev. Lett.76, 1679~1996!.@27# D.A. Hadjuk, P.E. Harper, S.M. Gruner, C.C. Honecker,

Kim, E.L. Thomas, and L.J. Fetters, Macromolecules27, 4063~1994!.

@28# M.W. Matsen and M. Schick, Phys. Rev. Lett.72, 2660~1994!; Macromolecules27, 4014~1994!.

@29# M.W. Matsen and F.S. Bates, Macromolecules29, 1091~1996!.

@30# R. Kikuchi, T. Mohri, and B. Fultz, MRS Symposium Pro

kn

.

ceedings205, 387 ~1992!; B. Fultz, Philos. Mag. B67, 253~1993!.

@31# I.W. Hamley, K.A. Koppi, J.H. Rosedale, F.S. Bates, K. Almdal, and K. Mortensen, Macromolecules26, 5959 ~1993!; S.Forster, A.K. Khandpur, J. Zhao, F.S. Bates, I.W. HamleA.J. Ryan, and W. Bras,ibid. 27, 6922~1994!.

@32# E.L. Thomas, D.M. Anderson, C.S. Henkee, and D. HoffmaNature334, 598 ~1988!.

@33# M.M. Disco, K.S. Liang, S.K. Behal, R.J. Roe, and K.J. JeoMacromolecules26, 2983~1993!.

@34# The time unit we use in this paper isd2/(200MkBT), whered is the period of the microstructure corresponding to thetimal wave number,M is the mobility coefficient, and(MKBT) can be thought of as the collective, mutual diffusiocoefficent.

@35# In general it is desirable to be in the more asymmetric parthe phase diagram since the windows for the different phaget wider. In the temperature jumps involving the lamelphase, however, increasing the asymmetry also increasetemperature difference between the different phases andlarge a temperature difference can sometimes lead to numcal instabilities in the simulation. This is the reason wf50.45 was used in the cases involving the lamellar phas

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

@37# The dominant wave vector determined from the dynamic eqtion is slightly different from that determined from the freenergy. However, consistent with the single wave lengthproximation, we ignore this difference.

@38# The sign ofAns are determined by the miniminization of thfree energy and there are several equivalent choices. Inpaper, we shall chooseA1 andA4 to be negative, while all theotherAns are positive.

@39# M.F. Schulz, F.S. Bates, K. Almdal, and K. Mortensen, PhRev. Lett.73, 86 ~1994!; K.A. Koppi, M. Tirrell, F.S. Bates,K. Almdal, and K. Mortensen, J. Rheol.38, 999 ~1994!.

@40# S. Qi and Z.-G. Wang~unpublished!.@41# S.A. Brazovskii, Zh. E´ksp. Teor. Fiz.68, 175 ~1975! @Sov.

Phys. JETP41, 85 ~1975!#.@42# G.H. Fredrickson and E. Helfand, J. Chem. Phys.87, 697

~1987!.@43# M. Muthukumar, Macromolecules26, 5259~1993!.@44# T. Hashimoto, K. Kowsaka, M. Shibayama, and S. Sueh

Macromolecules19, 750 ~1986!; T. Hashimoto, K. Kowsaka,M. Shibayama, and H. Kawai,ibid. 19, 754 ~1986!.

@45# M.A. Singh, C.R. Harkless, S.E. Nagler, R.F. Shannon, aS.S. Ghosh, Phys. Rev. B47, 8425~1993!; C.R. Harkless, M.A. Singh, S.E. Nagler, G.B. Stepheson, and J.L. Jordan-SwPhys. Rev. Lett.64,2285~1990!.

@46# S. Sakurai, T. Momii, K. Taie, M. Shibayama, S. Nomura, aT. Hashimoto, Macromolecules26, 485 ~1993!.

@47# S. Sakurai, H. Kawada, T. Hashimoto, and L.J. Fetters, Mromolecules26, 5796~1993!.

@48# G.H. Fredrickson, Macromolecules24, 3456~1991!.@49# M. Olvera de la Cruz, A.M. Mayes, and B.W. Swift, Macro

molecules25, 944 ~1992!.@50# I.W. Hamley and F.S. Bates, J. Chem. Phys100, 6813~1994!.@51# The term hcp is used even though the actualc/a ratio differs

from that corresponding to the closest packingc/a5A3; seeC. Kittel, Solid State Physics~Wiley, New York, 1976!.

ol

,

.J

ro-

4


@52# P.S. Pershan,Structure of Liquid Crystals~World Scientific,Singapore, 1988!.

@53# C. Yeung, A.-C Shi, J. Noolandi, and R.C. Dasai, MacromTheory Simul.5, 291 ~1996!.

@54# D.A. Hajduk, H. T. Takenouchi, M. A. Hillmyer, F. S. BatesM. E. Vigild, and K. Almdal ~unpublished!.

@55# D.A. Hajduk, S.M. Gruner, P. Rangarajan, R.A. Register, L

.

.

Fetters, C. Honeker, R.J. Albalak, and E.L. Thomas, Macmolecules27, 490 ~1994!.

@56# K.A. Koppi, Ph.D. thesis, The University of Minnesota, 199~unpublished!; F.S. Bates, K.A. Koppi and M. Tirrell, Macro-molecules27, 5934 ~1994!; K. Almdal, K. Mortensen, K.A.Koppi, M. Tirrell, and F.S. Bates, J. Phys.~France! II 6, 617~1996!.

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