an-chang shi, jaan noolandi and rashmi c. desai- theory of anisotropic fluctuations in ordered block...

8/3/2019 An-Chang Shi, Jaan Noolandi and Rashmi C. Desai- Theory of Anisotropic Fluctuations in Ordered Block Copolymer

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Theory of Anisotropic Flu ctu at ions in Ordered Block Copolymer

Phas es

A n - C h a n g S h i * a n d J a a n N o o l a n d i

Xerox Research Centre of Canada, 2660 Speakman Drive,Mississauga, Ontario, Canada L5K 2L1

R a s h m i C . D e s a i

Department of Physics, University of Toronto, Toronto, Ontario Canada M5S 1A7

Received March 18, 1996; Revised Man uscript R eceived J un e 12, 1996X

ABSTRACT: A gener al th eoretical fram ework for th e stud y of anisotropic composition fluctu at ions aboutan ordered block copolymer phase is developed. The approach is based on t he idea that , in order tostudy the effects of fluctuations around an ordered broken symmetry phase, the theory must be formulatedas a self-consistent expansion around t he mean-field solution of this ordered sta te. A random phaseapproximation treatm ent of the theory leads to a nisotropic correlation functions for th e system. It isshown that the calculation of the polymer correlation functions in an ordered phase is equivalent to thecalculation of the energy bands a nd eigenfunctions for an electron in a periodic potent ial. This generalmeth od is applied to the lam ellar pha se of block copolymers. The calculat ed anisotropic scatter ing intensitycaptures the main features observed experimentally, including the secondary peaks due to fluctuationswith hexagonal symmetr y. The origin of the an isotropic fluctuations can be tr aced to the forma tion ofenergy bands, similar to electronic sta tes in solids.

I . I n t ro d u c t i o n

Block copolymers are fascinating materials withunique stru ctural an d mechanical properties.1,2 Due totheir amph iphilic na tur e, AB diblock copolymers self-assemble into a variety of ordered microphases.1-3 Athigh tem perat ures, A an d B blocks mix homogeneouslyt o for m a dis or der ed phas e. As t he t em per at ur e i sdecreased (or the Flory-Huggins par am et er i s i n-creased), the blocks undergo a spatial segregation.However, a macroscopic phase separation cannot occurbecau se th e AB blocks are chemically connected at the jun ctions. The pha se separa tion is, therefore, neces-

sarily on a mesoscopic scale, forming A- a nd B-richdomains separat ed by intern al int erfaces. The competi-tion between the spontan eous curvatu re of the inter nalinterfaces and the entr opic stretching (or p acking) of theA and B blocks dictates t he symm etry of th e equilibriumphases. The symmetry of the ordered pha ses is con-trolled by the degree of segregation and the chemicalcomposition of th e copolymers. The segrega tion of th eblocks is quantified by the product Z, where Z ) ZA +

ZB is the degree of polymerization of the copolymers,an d ZA a n d ZB are the degrees of polymerization of theA and B blocks, respectively. The chem ical compositionof the copolymers is qu ant ified by th e ra tio f ) ZA/Z. Adecrease in Z drives the melt from an ordered phaseto the disordered phase. Chan ges in f affect the shape

and packing symmetry of the ordered structure. Be-sides the well-known lamellar, cylindrical, and sphericalphases,1,3,4 more complicated structures such as thebicontinuous cubic phase and the hexagonally modu-lated an d perforat ed layered phases have been recentlyidentified.3,5,6

Theoretically, the study of diblock copolymer meltshas been based on a simplified, s tandard model, inwhich the diblock copolymer chains a re a ssumed to beflexible chains obeying Gaussian statistics.7 The poly-mer conforma tions are specified by t he position of themonomer, R(t). Each monomer is furth er assumed to

have a statist ical length bR. The ha rd-core repulsiveinteractions between the molecules are accounted forby an incompressibility constraint where the averagemonomer concentra tion is required to be uniform. Therema ining interactions a re modeled by a local entha lpyterm A(r)B(r), where A(r) and B(r) are the volumefractions of the A and B monomers a t position r. T hissimple model captures the three most important fea-tures of the system: chain conformation ent ropy, in-compressibility, and immiscibility between unlike mono-mers. The th ermodynam ics of a diblock copolymer meltcan, therefore, be described by the partition function of

the model. The part ition function ma y be cast in severalequivalent forms. On particularly simple and conve-nient formulation is to write the partition function ofthe melt as a functional integral over the monomerconcentrations R(r) and two auxiliary fields R(r) (R )A, B).8,9 The functional integra l is carr ied out with th eincompressibility constraint and with an integrand ofth e form exp{-F({}, {})}. The free energy fun ctiona lis the sum of three terms: an enthalpic contribution,an entropic contribution, and a coupling term,

where V is the volume of the system, and Qc i s t hepartition function of a single diblock copolymer chainin th e externa l fields R(r).9 It is important to note tha tQc({}) is a fun ctiona l ofR(r). Specifically, the single-chain pa rtition function Qc is determined by

where th e propagators QR(r, t|r) areX Abstract published in Advance ACS Abstracts, August 1, 1996.

F ) dr [A(r)B(r) -R

R(r)R(r)] -

V

Zln Qc({R}) (1)

Qc )1

V dr1dr2dr3QA(r1, ZA|r2)QB(r 2, ZB|r3) (2)

6487Macromolecules 1996, 29 , 6487-6504

S0024-9297(96)00411-1 CCC: $12.00 1996 Amer ican Ch emical Society


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Physically QR(r, t|r) is the probability distribution ofmonomer t a t r, given tha t monomer 0 is at r, i n t hepresence of an extern al field R(r). From the expressionfor the free energy functional, it is obvious that there

are two kinds of inhomogeneities in th e system. Thefirst is the inhomogeneity in the polymer concentrations,R(r), which leads to an enthalpic contribution to th efree energy. The second is the inhomogeneity in thefield R(r), which leads to non-Gaussian polymer con-formations. These two k inds of inhomogeneities arecoupled. In pr inciple, the pa rtition function of a d iblockcopolymer melt could be obtained by evaluating thefunctional integral, and the thermodynamics of thes ys t em w ould be det er m ined. I n pr act i ce, t hi s i sintractable at present, and a variety of approximationm e th od s h a ve b ee n d ev elop ed t o s t u dy t h e p h a sebehavior of the system.

The simplest method for estimating the functional

integra l is to use th e saddle-point a pproximation, whichamounts to replacing the functional integral by themaximum of the integran d. Physically, this approxima-tion is equivalent to the mean-field approximation,which ignores t he composition fluctuat ions of the sys-tem. The mean -field th eory of block copolymers is, inits m ost general form, a self-consistent field theory(SCFT). Becau se of the complexity of the theory evenwithin the mean-field a pproximation, numerous ad-ditional approximations have been developed to solvethe mean-field equations for diblock copolymer melts.The approximation methods can be roughly classifiedinto the strong segregation theory10-12 a n d t h e w ea k segregation theory.13,14 The strong segregation theory

introduced by Semenov10

makes th e furth er approxima-tion of ignoring chain conforma tion fluctua tions, thu srestricting the theory to the infinite Z limit. The weaksegregation th eory int roduced by Leibler,13 on the otherhand, is developed as a Landau expansion of the fullSCFT based on the random phase approximation (RPA)around the high-temperatu re, homogeneous phase; thu sthe theory is restricted to the weak segregation regime(Z 10). A density functiona l theory in t he form of atru ncated density expansion ba sed on th e RPA has alsobeen developed by several a uth ors.15-19 These densityfun ctiona l th eories give qua lita tively correct results, bu tthey are quantitatively inaccurat e. Other approaches,which involve neith er a priori assumptions about theshape of the equilibrium density profiles nor a tru ncatedfree energy, correspond to solving the mean-field equa-tions numerically.20-24 Ear lier nu merical meth ods20-22

involved further approximations for solving the equa-tions for th e propagators. However, there is no furt herapproximation involved when the mean-field equationsare cast in terms of the basis functions with the fullsymmetry of the ordered pha ses.23 With the develop-ment of this plane wave expansion method by Matsenand Schick,23 the mean-field theory of diblock copoly-mers can be solved exactly for weak and intermediatesegregation regimes. The mean -field pha se diagram ofthe standard model for diblock copolymer melts h asbeen calculated and i t captures most of the featuresobserved experimentally.24

I I. N a t u r e o f An i s o t r o p i c F l u c t u a t i o n s i nO rd e r e d P h a s e s

Compa ring with t he m ean-field results, very little ha sbeen done on the effects of fluctuations on the phasebeha vior of block copolymers. The composition fluctua -tions a re ignored in th e mea n-field a pproximation, anda proper treatment of the f luctuations leads to non-mea n-field results. The first work in th e block copoly-mer literature to treat composition fluctuations is byFredr ickson and H elfan d,25 who extended the Brasovskii

theory26 to diblock copolymers. In t his work, th e Lan-dau free energy functional of Leibler 13 was reduced toa form considered by Brasovskii,26 and a saddle-pointapproximat ion (the self-consistent Har tree approxima-tion) was made for the fluctuat ions. However, thistheory is confined to the weak segregation region dueto the u se of the Lan dau free energy functional. Thiswork was extended by Mayes and Olvera de la Cruz 27

to include the leading wave vector dependence of thevertex functions but keeping th e sa ddle-point approxi-mation and remaining in the weak-segregation l imit .More recently, Muthukum ar 28 studied composition fluc-tuation effects on the diblock copolymer phase diagramby incorporat ing composition fluctua tions into a densityfunctiona l theory. However, the chain conforma tionsare treated approximately due to the truncated form ofthe free en ergy functional.

In this paper, we develop a theory for the anisotropicfluctuations in the ordered phases of diblock copolymersby a self-consistent expansion around the exact m ean-field solution of the system. The lowest correction tothe mean -field approximat ion is in t he form of Gaussia nfluctuat ions around the mean-field solution. The a p-plicat ion of a genera lized RPA for th e system yields th edensity-density correlation functions. The RPA hasproven to be a valuable tool in understanding polymersystems. Essentially the RP A is a self-consistent ex-pan sion aroun d the mean -field sta te. The RPA can givethe stabili ty, and the magnitude of Gaussian fluctua-

tions in an equilibrium state. For example, with theRPA one can predict the scattering function and phasebounda ry of homopolymer 29 and copolymer melts.13

However, the usual formulation of the RPA assumes aspatially uniform mean-field state,29,13 w her eby t hefluctuations are assu med to be isotropic. If the t ran s-lational symmetry is broken, t his assumption is nolonger valid. Instead, the RPA must be reformulatedas an expansion around the broken-symmetry mean-field state. I t is obvious that the correct mean-fieldsolut ion is crucial for t he va lidity of the RPA, oth erwiseone cannot distinguish between approximations to thezeroth-order solution an d contr ibutions arising fromcomposition fluctuations. The essential r esult of this

approach is the RPA density-density correlation func-tion CRP A(r,r), characterizing the anisotropic composi-tion fluctua tions. The a vailability of the RPA correla-tion function enables several importan t applications:30

(1) The experimentally observed scattering intensity,w hich i s t he Four i er t r ans for m of CRP A(r ,r). Ou rm et hod account s for t he ani sot r opic nat ur e of t hefluctuat ions, and the calculated scattering functioncaptures the main features found experimentally.6 (2)The mean-field solution is l inearly unsta ble if anyeigenvalue of [CRP A]-1 is negative. The bounda ry wherean eigenvalue first becomes zero is the limit of meta-stability of a particular phase, i.e., the spinodal line forthat phase. The least s ta ble mode at this point domi-nates the fluctuat ions. (3) The elastic moduli of the

QR(r, t|r) )

R(0))r

R (t))rDR(t) exp[-0ZR dt( 32bR2(

dR(t)

dt )2

+ R[R(t)])](3)

6488 Shi et al. Macrom olecules, V ol. 29, N o. 20, 1996


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ordered phase can be extracted from the behavior of thescatt ering fun ction near th e Bragg peaks. Significantcorrections to the l imiting strong stretching theoryresult are found for the experimentally relevant param-et er s . (4) T he effect of fl uct uat i ons on t he phas ediagram can be computed by calculating the Gaussianfluctuation contribution to the free energy, as demon-s t r at ed by M ut hukum ar 28 using the approximate freeenergy functional.

The calculation of the polymer correlation functions

i n an or der ed phas e i nvol ves t he eigenvalues andeigenfunctions of the chain Ham iltonian

where R(0)(r) are the self-consistent mean fields of the

ordered phase. Because the chain Hamiltonian ha s thesame form as the Hamiltonian of electrons in periodicpotentials, the eigenvalue problem for the diblockcopolymers corresponds to the quantu m mechan icalproblem of electrons in periodic potent ials. In part icu-lar, the eigenvalues correspond to the energy bands insolid-state physics.

A s a f i r s t s t ep i n t hi s s t udy, w e have appl i ed t hegeneralized RPA to the diblock copolymer lamellarphase and communicated the main results in a shortpublication.30 T h e r e su lt s a r e t h a t t h e a n is ot r op icnature of the fluctuations is revealed in the scatteringfunctions, which can be obtained via small-angle neu-tron scat tering. For the lamellar phase, Hamley et al.6

have obs er ved t he fol low ing fr om s hear -or i ent edsamples: (1) At low temperatur e, the scattering patternshows meridional Bragg peaks a t a wave vector k0 withhigher order reflections at 2k0 and 3k0, indicat ive of verysmall layer un dulations. (2) At higher temperatures,four weak peaks corresponding to the layer undulationsare observed, which shows an approximate hexagonal

order. The periodicity of the in-plane undulations isabout 10% l ar ger t han t he i nt er l ayer s paci ng. O urcalculated scattering functions for the lamellae captureall these features.30 In this paper, we give a detailedexposition of the theoretical framework and illustratet he physi cs by s t udyi ng a s im ple w eak s egr egat edlamellar pha se. We will leave the st udy of fluctuat ioneffects on diblock copolymer pha se behavior for thefuture.

In order to understand the origin and nature of theanisotropic fluctua tions in a n ordered pha se, we applythe theory to the weakly segregated lamellar phase.Close t o the order-disorder tra nsition, th e m ean-fieldsymmetric diblock copolymer densities can be approxi-mat ed by sinusoidal profiles. The resulting eigenvalueproblem ha s the form of the Mathieu equation. Per-tur bation calculations reveal tha t t here a re two sourcesof an isotropy: (1) The change in the eigenvalues leadsto a renorma lization of the polymer r adius of gyrationalong the direction perpendicular to the lamellas, and(2) the appea ran ce of the gap in t he eigenvalue spectru mat the Bra gg points leads to the str ong resonance peaks.

The organ ization of th is paper is as follows: SectionIII describes th e model system, the derivation of the freeenergy functional, the generalized RPA for diblockcopolymers, and methods for calculating the polymercorrelation fun ctions in an ordered ph ase. Section IVapplies the theory to the simplest broken symmetryphase, the weakly segregated lamellar pha se. Section

V includes discussions a nd conclusions. Some mat h-ematical details are included in the Appendix.

I II . T h e o r e t i c a l F r a m e w o r k

A. T h e M o de l a n d F u n c t i o n a l In t e g r a l F o rm u -l a t i o n . We as s um e t hat t he pol ym er chai ns ar e de-scribed by the many-chain Edwards Hamiltonian.31 Th edegrees of polymer ization of th e chains ar e ZA, ZB, with

Z ) ZA + ZB being the total degree of polymerization ofthe diblock copolymers. We assum e tha t ther e are Nc

diblock copolymer chains contained in a finite volumeV. We associate each block R () A, B) with a Kuhnstatistical length bR and a bulk monomer density F0R.Let R R

i (t) denote th e position of monomer t at block R ofchain i. Le t r denote a spatial posit ion. The densityprofile of the polymers is chara cterized by the volumefraction of block R at position r, which can be expressedin t erms of the chain conforma tion,

where the chevron denotes that R(r) is a functional of

the chain conformations R Ri (t).

The partition function of the diblock copolymer meltcan be written in term s of a functiona l integral over a llthe chain conformations

where zc is the partition function of a copolymer chaindue t o the kinetic energy, and W({}) t V({})/kBT isthe intermolecular potential . The integral is over a ll

space curves R Ri (t). The probability density functiona l

P0({R(t)}) is assumed to be of the form

where the function ensures that the copolymer chainsare connected at one end, and the single-chain prob-

ability distribution p0({R Ri (t)}) has the standard Wien-

er form

We have also introduced, in eq 6, a function, [1 -RR(r)], to ensure the incompressibility constraint forthe system. The interaction potential W({}) i s a s -sumed to have the two-body Flory-Huggins form

For simplicity, we will assume that all the referencedensities are the same and set to unity, F0R ) F0 ) 1.We also assume tha t both blocks ha ve the same Kuhnlength bR ) b.

We now introduce the monomer concentra tions R(r)and two a uxiliary fields R(r) (R ) A, B) followingHelfand 8 and Hong and Noolandi.9 The pa rtition func-

HR ) -bR

2

6

2 + R(0)

(r) (4)

R(r) ) R(r ,{RRi

(t)}) )1

F0Ri)1

Nc

0ZR

dt(r - RRi

(t)) (5)

Z )zc

Nc

Nc!DR()P0({R()})

{r}

[1 -R

R(r)]

exp[-W({})] (6)

P0({R(t)}) )i)1

Nc

{[R Ai

(ZA) -

R Bi

(ZB)]p0({R Ai

(t)})p 0({R Bi

(t)})} (7)

p 0({RRi

(t)}) exp[- 32bR20ZR

dt(dRRi

(t)

dt)

2

] (8)

W({}) ) drA(r)B(r) (9)

Macromolecules, V ol. 29, N o. 20, 1996 Anisotropic Fluctuations in Block Copolymers 6489


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tion of a diblock copolymer melt in a volume V can bewritten as a functional integral over the volume frac-tions R(r) an d fields R(r),

where the functions ar e intr oduced to ensu re incom-pressibility RR(r) ) 1. The free energy functionalF

({}

,{}

) has the form

where Qc is the partition function of a single diblockcopolymer chain in the external fields, R(r).8,9 Th echain conformation contribution to the partition functionis contained in the single-chain partition function Qc,w hich i s r elat ed t o t he pr opagat or s of t he diblockcopolymer chains, defined by

which is the probability distribution of monomer t a t r,given that monomer 0 is at r, in the presence of anexterna l field R(r). I t is e as y t o s h ow t h a t t h es epropagators satisfy the modified diffusion equations 8

with the initial conditions QR(r, 0|r) ) (r - r). I nterms of these propagators, we have

Note that s ince the propagators depend on the f ieldsR(r), the single-chain partition function Qc is a func-tional ofR(r). In order to treat the imcompressibilityand the monomer conservation constraints, we intro-duce two sets of Lagran gian multipliers 9 and write thepartition function in the form

where the gran d potential is given by

where hR ) ZR/Z is th e aver age volume fra ction of blockR.

The ther modyna mics of a diblock copolymer meltwould be completely determined if the partition functionof the system could be evaluated. The exact evalua tionof th e functiona l integra l is, however, a formida ble task .There are essentially two sets of difficulties involved

here. The first is the calculation of the single-chainpartition function Qc to express it explicitly in terms ofthe fields R(r), so tha t the chain conformation contr i-butions to the free energy functional are obtained interms of R(r). T he s econd i s t he eval uat ion of t hefunctiona l integrals over the polymer concent rat ions R-(r) and the field R(r). The simplest met hod by whichto evaluate the functional integral is the saddle-pointappr oxim at i on, w hich r eplaces t he i nt egr al by t hemaximum of integrand a nd yields the m ean-field equa-tions. In what follows, we will ta ke the mean-fieldsolution as the zeroth order solution and express thefr ee ener gy funct i onal as an expans ion ar ound t hemean-field solution. We then trun cate the expansionat t he lowest nontrivial order. The functional integra lis Gaussian at this order an d can be evaluated.

As m entioned above, because the single-chain pa rti-tion fun ction Qc depends on the fields R(r) i n a nimplicit form, it is useful to write Qc explicitly in termofR(r). This can be a chieved by writing R(r) in theform

where R(0)(r) are known functions which will be deter-mined later as the mean-field solutions of the system.The propagator can then be cast in the form

where th e zeroth order propagators are the solutions of

with t he initial conditions QR

(0)(r, 0|r) ) (r - r). T hefluctuat ion part QR(r, t|r ) of t he pr opagat or s i s asolut ion of

with t he initial conditions QR(r , 0|r) ) 0. In order toobtain a form al solution for QR(r , t|r), it is conven ientto define Green functions in the zeroth-order fields

R(0)(r)

It is easy to see that the Green functions defined this

way are related to the propagators in t he f ield R(0)(r)

where (t) is the step function. In terms of the Greenfunctions, th e forma l solution of QR(r, t|r) c an b eobtained:

Z ) R

{DRDR}{r}

[1 -R

R(r)]

exp[-F({}, {})] (10)

F ) dr[A(r)B(r) -R

R(r)R(r)] -V

Zln Qc (11)

QR(r, t|r) )

R(0))rR (t))r

DR(t) exp

[-0

ZR

dt

(3

2bR2

(dR(t)

dt )2

+ R[R(t)])](12)

tQR(r, t|r) )

bR2

6

2QR(r, t|r) - R(r)QR(r, t|r) (1 3)

Qc ) 1V dr1dr2dr3QA(r1, ZA|r2)QB(r2, ZB|r3) (14)

Z ) R

{DRDRdR}D exp[-({}, {},

{}, R)] (15)

) dr[A(r)B(r) -R

R(r)R(r)] -V

Zln Qc -

dr[(r)(1 -R

R(r)) +R

R(hR - R(r))] (16)

R(r) ) R(0)

(r) + R(r) (17)

QR(r, t|r ) ) Q(0)

(r , t|r) + QR(r, t|r) (18)

tQR

(0)(r, t|r) )

bR2

6

2QR

(0)(r, t|r) - R

(0)(r)QR

(0)(r, t|r)

(19)

tQR(r, t|r) )

bR2

6

2QR(r , t|r ) -

R(0)

(r)QR(r, t|r) - R(r)QR(0)

(r, t|r) -R(r)QR(r, t|r) (20)

{ t

-bR

2

6

2 + R(0)

(r)}GR(r , t|r , t) ) (t - t)(r - r)(21)

GR(r, t|r, t) ) (t - t)QR(0)

(r, t - t|r) (22)



5/18

The formal solution allows us to obtain the solution forQR by itera tion

where the n th-order solution is

Using the per tur bation solutions ofQR(r, t|r), the single-chain partition Qc can be obtained as a series in theRs. Appendix A gives details of th e perturbationcalculations. The single-chain pa rtition function termin the free energy can then be cast in the form of anexpansion

where the coefficients CR1,...,Rn(r1, ..., rn) are th e n th-ordercumulant correlation function for a noninteracting

diblock chain in the field R(0)(r). The lowest order

cumulant correlation functions are given in AppendixA.

I n or der t o pr oceed, w e expand al l t he var iablesaround the zeroth-order solution of the system,

In t erms of the zeroth-order solution a nd t he variables,the grand free energy functional can be expanded inthe form

where the zeroth-order term (0) is given by

The first-order term has the form

and the second-order term has the form

The higher-order terms have the generic form

It should be noted that there is no explicit dependencein (n ) for n > 2. H ow ever , t her e i s an i m pl ici t dependence because the correlation functions are ob-

tained in terms of the mean fields R(0)(r).

B . M e a n -F i e l d A p p ro x i m a t i on . The mean-fieldapproximation reduces the interacting chain system toa system of independent chains in the self-consistentfields of the other chains. Technically, this amounts toevaluating the functional integral over a nd by thesaddle-function m ethod, i.e., replacing the integral bythe m aximum value of the integran d.9 This extremiza-tion yields a set of mean -field equations, with the

solutions R(0)(r) a n d R

(0)(r). Specifically, the mean-

field solution is that for which the first-order term(1)

vanishes. Since the variables are a rbitrary functions,(1) ) 0 results in the mean-field equations

In order to obtain exact solutions, th ese equat ions mustbe solved self-consistently using numerical methods. Forthe more complex pha ses, an expansion in t erms of acomplete set of basis functions with the desired sym-metry is required.

The mea n-field free energy is then obtained by insert-ing the mean-field solution into the free energy expres-sion. When we expand ar ound th e mean -field solution,the first-order contr ibution to th e free energy van ishes,and the zeroth-order contr ibution to the free ener gy isjust the mean -field free en ergy F(0)

QR(r, t|r) )

- dr1 dt1GR(r, t|r1, t1)GR(r 1, t1|r, 0)R(r1) -

dr1 dt1GR(r, t|r1, t1)QR(r1, t1|r)R(r1) (23)

QR(r, t|r) ) QR(1)

(r, t|r) + QR(2)

(r, t|r ) + ... (2 4)

QR(n )

(r, t|r) ) (-)n dr1...drn dt1...dtnGR(r, t|rn, tn)GR(r n, tn|rn-1, tn-1)...

GR(r2, t2|r1, t1)GR(r1, t1|r , 0)R(r1)...R(r n) (25)

V

Zln Qc )

V

Zln Qc

(0) + n)1

(-1)n

n !

R1,...,Rn

dr1...drnCR1,...,Rn(r1,...,rn)R1(r1)...Rn(rn)

(26)

R(r) ) R(0)

(r) + R(r)

R(r) ) R(0)

(r) + R(r)

(r) ) (0)(r) + (r)

R ) R(0) + R (27)

) (0) + (1) + (2) + ... (28)

(0) ) dr[A

(0)(r)B

(0)(r) -

R

R(0)

(r)R(0)

(r)] -

V

Zln Qc

(0)

- dr[(0)(r)(1 -R

R(0)

(r)) +R

R(0)

(hR - R(0)

(r))]

(29)

(1) ) dr{[B

(0)(r) - A

(0)(r) + (0)(r) +

A(0)

]A(r) + [A(0)

(r) - B(0)

(r) + (0)(r) +

B(0)

]B(r) -R

[R(0)

(r) - CR(r)]R(r) -

[1 -R

R(0)

(r)](r) -R

[hR - R(0)

(r)]R} (30)

(2) ) dr{A(r)B(r) -

R

R(r)[R(r) -

(r) - R]} -1

2R, drdrCR(r ,r)R(r)(r)

(31)

(n ) ) -

(-1)n

n !

R1,...,Rn

dr 1...dr nCR1,...,Rn(r1,...,rn)R1(r1)...Rn(rn) (32)

A(0)

(r) ) B(0)

(r) + (0)(r) + A(0)

B(0)

(r) ) A(0)

(r) + (0)(r) + B(0)

R(0)

(r) ) CR(0)

(r)

1 )R

R(0)

(r)

0 ) dr[hR - R(0)

(r)] (33)



6/18

The partition function of the melt can now be writtenin the form

C. Ga u s si an F lu c tu a ti on s a n d t h e R an d o mP h a s e A pp r o x im a t i o n . S o f a r t h e r e h a s b e e n n oappr oxim at i on m ade i n t he t h eor y. W hat w e ha veaccomplished is th e r ewriting of the pa rtition functionin th e explicit form given above. In what follows wewill choose the zeroth-order solution as the mean-fieldsolution and keep only the expansion to the lowest(second-order) order; th us we are only dealing withGaussian fluctuations around the mean-field solution.We will also use th e fact t hat the addition of a consta ntto the f ield R(r) i s equi val ent t o t he addit i on of aconstant to the free energy, thus the constants R ca nbe neglected in the free energy (or absorbed into (r)).These simplifications lead to th e a pproximat e pa rtitionfunction of the system

where we have replaced the functional integral over(r) by the explicit incompressibility condition, and theGaussian fluctua tion free energy functiona l (2) is given

by

In order to treat the incompressibil ity conditionexplicitly, we u se t he condition A(r) + B(r) ) 0, andintroduce three new variables

In terms of these new variables, the partition functionof the diblock copolymers has the form

where the Gaussian fluctuation free energy functionalis given by

In the above expression for the free en ergy functional,w e have i nt r oduced t hr ee new quant i t i es w hi ch ar ecombinations of the diblock copolymer correlation func-tions

To proceed, we invoke the RPA, which is technicallyequivalent to extremizing the free energy functionalwith respect to (r) and (r). To eliminat e the (r)variable, we use the relation (2)/(r) ) 0, whichleads to

T his r elat i on can be us ed t o el im i nat e (r) i n t h et heor y. I n or der t o achieve t hi s, w e i nt r oduce t heinverse functional operation -1(r ,r) through the r ela-tion dr-1(r ,r)(r,r ) ) (r - r). In terms of thisinverse relation, we have

Inserting this relation back into (2), we have

It is clear from this expression that the incompressibilitycondition contributes to the chain correlation function.We can introduce an incompressibility corrected cor-relation function by

and write the Gaussian fluctuat ion free energy in th esimpler form

We now use the same approximation to eliminate thefield var iables (r). Minimizing th e free energy func-tion with respect to (r) leads to the relation (2)/(r) ) 0. Explicitly, we have

Introducing t he inverse operator of C (r,r) t hr oughdr C -1(r,r )C (r,r) ) (r - r ), we can write

F(0) ) dr[A

(0)(r)B

(0)(r) -

R

R(0)

(r)R(0)

(r)] -

V

Zln Qc

(0)(34)

Z ) exp[-F(0)] R

{DRDR dR}D

exp[- n)2

(n )

] (35)

Z exp[-F(0)] R

{DRDR}{r}

[R

R(r)]

exp[-(2)] (36)

(2) ) dr{A(r)B(r) -

R

R(r)R(r)} -

1

2R, drdrCR(r, r )R(r)(r) (37)

(r) ) A(r) - B(r)

(r) ) 12

[A(r) - B(r)]

(r) )1

2[A(r) + B(r)] (38)

Z exp[-F(0)]DDD exp[-(2)] (39 )

(2) )

1

2 drdr {-

2(r - r)(r)(r) -

2(r - r )(r)(r) - C(r,r )(r)(r) -

2(r,r)(r)(r) - (r ,r)(r)(r)} (40)

C(r,r) CAA(r,r) - CAB(r ,r) - CBA(r,r) + CBB (r,r)

(r,r) CAA(r,r) + CAB(r ,r) - CBA(r,r) - CBB (r,r)

(r,r) CAA(r,r) + CAB(r,r ) + CBA(r,r) +CBB(r,r) (41)

dr{(r,r )(r ) + (r,r)(r)} ) 0 (42)

(r) ) - drdr -1(r,r)(r,r)(r) (43 )

(2) ) 1

2 drdr {-

2(r - r)(r)(r) -

2(r - r )(r)(r) - (r)(r)[C(r,r) - dr1dr2(r,r1)

-1(r1,r2)(r2,r)]} (44)

C (r ,r) )

C(r,r) - dr1dr2(r,r1)-1(r1,r2)(r 2,r ) (45)

(2) ) 1

2 drdr {-[2(r)(r) +

2(r)(r)](r - r) - C (r,r )(r)(r)} (46)

drC (r,r)(r ) + (r) ) 0 (47)



7/18

Inserting this expression into (2), we get

where the RPA correlation function [CRP A]-1(r,r) is

defined by

which can be written in a imple form

where I ) (r - r) is the unit operator.T he u ps hot of t he above calcul at ion i s t hat t he

part ition fun ction of the diblock m elt can be written asa Gau ssian integral within t he RPA. The RPA correla-tion function can be calculated once the exactmean-field

solution is known. Because of the complex na ture of th e problem, the mea n-field equations ha ve to be solvednumerically.

Technically, the RPA is simply the mean-field theoryapplied to an ensemble in which (r) is held fixed. Tothe lowest order, t he RPA free energy becomes

T he R PA for m al is m can al so be der ived us ing t heconcept of response fun ctions. Formally the equat ionsabove ar e identical to th ose for the high-temperature

RPA. The essential difference is tha t th e independentchain correlation functions CR,(r,r) arise from the fullmean-field solution and hence include the anisotropicnat ur e of t he s yst em . I f t he CR, are replaced by thecorrelation function for Gaussian coils we recover theresult of Leibler.13

D . M e th o d o f S o l u t io n : R e c i p ro c a l S p a c e F o r -m a l i s m . The above t heoretical fram ework was devel-oped in real space. In order to exploit the fact th at t heordered pha ses are broken symmetry phases, we nowformulat e the th eory in reciprocal space. To begin with,w e n o te t h a t t h e s ym m e tr y of a n or d er e d p h a s e iscompletely specified by a space group with a set of realspace latt ice points {R n} or th e corresponding reciprocalspace latt ice points {G}. T he equil ibr ium densi t y

profiles and the mean fields are periodic functions withthe periodicity of the latt ice. Because the modifieddiffusion equation can be regarded as a Schrodingerequat i on w it h i m agi nar y t i m e, w e can m ake us e of analogy with quantum mechanics by defining a Hamil-tonian

where R(r) are t he m ean-field potentials. The sym-metry of the ordered phase dictates tha t t he mean fieldsare periodic functions, R(r) ) R(r + R n). Accordin gto Blochs theorem,32 the eigenvalues n

R(k ) o f t h e

Hamiltonian HR have a band structure, which can belabeled by n an d a wave vector k , where k is restr ictedto the first Brillouin zone. The corresponding eigen-functions have the form of a modulated plane wave

w h er e t h e u n kR (r) are periodic functions, which are

solut ions of the Schrodinger equa tion

Because the Hamiltonian is Hermitian, the eigenfunc-tions are orthogonal and form a complete basis set

Because th e eigenfun ctions form a complete set, we canuse them as basis functions to formulate the theory.This will al low u s t o take advantage of the symmetryof the ordered phases. In particular, the mean-fieldpropagators QR(r, t|r) can be written in th e simple form

The RPA correlation function can, therefore, be ex-pressed in t erms of the eigenfunctions a nd eigenvaluesof the H amiltonian. Moreover, the a bove result showsthat the single-chain propagators QR(r, t|r) are diagona l

if we use the eigenfunctions n kR (r) as our basis func-

tions, i.e.,

With these basis functions, the cumulant correlationfunctions can be cast in the form

Using the eigenfunctions of the chain Ham iltonians HRas the basis functions, it can be shown (see AppendixB) tha t t he cumulan t correlation functions a re diagonalin t he lattice wave vector k space

The details of calculating the n k space correlation

functions CnR a n d Cn ,n

R (k ) are given in Appendix B. Theone- and two-point correlation functions in real spaceare t hen given by

(r) ) - dr C -1(r,r)(r) (48)

(2) )

1

2 drdr{C -1(r,r) -

2(r - r)}(r)(r)

)1

2 drdr[CRP A]-1(r,r )(r)(r) (49)

[CRP A

]-1(r,r) ) {C -1(r,r) -

2(r - r )}

-1(50)

CRP A

(r,r ) ) {[I - 2C ]-1

C }(r ,r) (51)

F ) F(0) +1

2 drdr [CRP A(r,r)]-1(r)(r) +

O(3) (52)

HR ) -bR

2

6

2 + R(r) (53)

n kR

(r) )1

Ve

ik ru n k

R(r) (54)

[- bR2

6(ik + )2 + R(r)]u n kR (r) ) nR(k )u n kR (r) (55)

1

Vn k

eik (r-r )

u n kR (r)u n k

R* (r) ) (r - r)

1

V dr ei(k -k )ru n k

R(r)u n k

R*(r) ) n ,n (k - k ) (56)

QR(r , t|r ) )n k

e-n

R(k )tn k

R(r)n k

R*(r) (57)

Qn k ,n k R

(t) dr dr n kR*

(r)QR(r, t|r)n k R

(r )

) e-nR

(k )tn ,n (k - k ) (58)

CR(r) ) Vn k

Cn kR n k

R (r)

CR(r,r) )n k

n k

Cn k ,n k R n k

R(r)n k

* (r) (59)

Cn kR ) Cn

Rk ,0

Cn k ,n k R ) Cn ,n

R(k )k ,k (60)



8/18

E . S p e c i a l Ca s e : E x p a n s i o n a r o u n d t h e H i g h -T e m p e r a tu r e S o l u t io n . T he s im plest cas e i s t hehom ogeneous phas e at hi gh t em per at ur es ; t he R PAcorrelation function in this case has been obtained by

Leibler.13 For th e homogeneous pha se, the mea n fieldsare constants, which can be taken a s zero. The chainHam iltonian becomes the free-part icle Ha miltonian.The eigenfunctions and eigenvalues of the Hamiltonian

in this case are u n kR (r) ) 1 and R(k ) ) bR

2k

2 /6. There isonly one energy band for the homogeneous phase, sothat t he band index n becomes redundan t. The Fourier

components of the Bloch fun ctions become u n kR (G) )

G,0. The reduced integrals needed in the correlationfunctions reduce to (see Appendix B)

We also have Qc ) 1 for the homogeneous phase. With

these expressions, t he single-chain correlat ion fun ctionscan be easily obtained, Cn

R ) ZR/Z ) hR. In real space,w e have CR(r) ) hR, as expected for a homogeneousphase. The two-point correlation functions are

Here we have introduced the radii of gyration for thetwo blocks R R ) b2ZR /6 and used the explicit form ofthe function g(x) ) [1 - exp(-x)]/x. These correlationfunctions for the homogeneous phase have been ob-tained earlier .13 The RPA correlation function in kspace is

where the incompressibility-corrected nonintera ctingcorrelation function C (k) is

and the quantities C(k), (k), an d (k) are combinationsof CR(k)

The RPA correlation function obtained above is identical

to earlier work.13 For a block copolymer, the incom-pressibility correction gives C (k) f 0 a s k f 0.

I V. E f fe c t s o f A n i s o t r o p i c F lu c t u a t i o n s f o rL a m e l l a r P h a s e

A. L a m e ll a r P h a s e . We now apply the theory tot he l am el lar phas e. For fi xed , Z, f, and l am el l arperiodicity L , we first solve the mean-field equations.The free energy density is then minimized as a functionof L to obtain t he equilibrium per iodicity for the set of

, f, a n d Z. T he det ail ed num er ical s t udi es of t helamellar phases have been published elsewhere30 a n dwill not be repeated h ere. In what follows we willformulate the theory in the planar geometry appropriatefor t he lamellar phase. We then apply the theory to aweakly segregated lamellar phase of a symmetric f )1/2 diblock copolymer melt.

For the lamellar phase, the system is described by aplana r geometr y. In part icular , th e mean-field solut ionis homogeneous in the x - y direction and periodic int he z direction. The mean-field potent ials R(r) becomeperiodic functions along the direction perpendicular tothe layers, i.e.,

where m is an integer and L is the periodicity of thelayers. The polymer chain Hamiltonian m ay be sepa-rated into the parallel and perpendicular pa rts

where we have written the spatial vector in the form r) (x , z). The parallel an d perpendicular Ha miltoniansa r e

where HR(x ) corresponds to a two-dimensiona l H amil-tonian for a free pa rticle, and HR(z) is th e Ham iltonianin the z direction, which h as the same periodic proper-ties as the mean-field potential R(z). The eigenfunc-tions of the system can be separated into a n x -depen-dent par t and a z-dependent part

where we ha ve written th e wave vector in th e form k )(q , kz). The corresponding eigenvalues ha ve the form

The equations determining the eigenvalues and eigen-functions are

Th e x -dependent eigenfunctions and eigenvalues areeasily found

CR(r) )n

CnR

u n 0R

(r)

CR(r,r) )n ,n

1

V

k

Cn ,n R (k )e

ik(r-r )u n k

R (r)u n k* (r) (6 1)

qnR ) n ,n

R (k ) ) n 1,n 2,n 3R

(k ) ) 1 (62)

CAA

(k) )ZA

2

Z

2[(R Ak)2 - 1 + e-(R Ak)

2

]

(R Ak)4

CAB(k) ) CBA(k) )

ZAZB

Z

1 - e-(R Ak)2

(R Ak)2

1 - e-(R Bk)2

(R Bk)2

CBB

(k) ) ZB2

Z2[(R Bk)

2

- 1 + e-(R Bk)2

](R Bk)

4(63)

CRP A

(k) )C (k)

1 - C (k)/2(64)

C (k) ) C(k) - (k)((k))-1(k) (65)

C(k) ) CAA(k) + CBB(k) - 2CAB(k)

(k) ) CAA(k) - CBB(k)

(k) ) CAA(k) + CBB(k) + 2CAB(k) (66)

R(r) ) R(z) ) R(z + m L ) (67)

HR ) HR(x ) + HR(z) (68)

HR(x ) ) -bR

2

6

2

x2

HR(z) ) -bR

2

6

2

z2

+ R(z) (69)

n kR

(r) ) qR

(x )nkzR

(z) (70)

nR

(k ) ) R(q) + nR

(kz) (71)

HR(x )qR

(x ) ) R(q)qR

(x )

HR(z)nkzR

(z) ) nkzR n kz

R(z) (72)



9/18

where A 0 is the area of the system in th e x -y direction.The eigenvalue problem for the z-dependent part isnontrivial in that it corresponds to a particle moving ina p eriodic potent ial. Accordin g to Blochs th eorem, th eeigenvalues of a particle moving in a periodic potentialform energy bands, which can be labeled by a bandindex n an d a wave vector kz which is restricted to thefirst Brillouin zone (-/L < kz e /L ). The correspond-ing eigenfun ctions a re of the form

where we have defined the length of the system in thez direction as L 0. Therefore, the total volume of thesystem is V ) A 0L 0, and t he nu mber of unit cells in th e

z direction is L 0/L . u nkz(z) is a periodic function withperiod L and obeys

The eigenfunctions ofHR(z) form a complete basis setand are orthogonal with the normalization

where the notation nkz indicates a summation over n

and an integral over kz.T he f act t hat t he x -dependent wave functions are

plane waves leads to simple expressions for the coef-ficients required in the correlation functions (see Ap-pendix B)

The expressions for the cumulant correlation functionsand, therefore, t he RPA correlation functions can beobtained in ter ms of th ese coefficient s. In pr actice, th emean-field equations of the system have to be solvednumerically. The resulting mean fields can then beused to calculate the eigenvalues (energy bands) andthe eigenfunctions for th e A and B blocks. These energybands and Bloch functions can be used to compute thevariety of correla tion fun ctions, an d eventu ally the RPAcorrelation function. The scatt ering function, th e fluc-tuating modes, etc., of the ordered phases can then beobtained. This program ha s been implemented for thelamellar pha se and the results have been published.30

In what follows we will apply the theory to th e weakly

segregated lamellas by assuming that the mean-fieldsolution contains only one mode.

B . We a k ly S e g re g a t e d L a m e ll a s. I n or d er t oillustra te the origin of the an isotropic fluctuations, weconsider a weakly segregated symmetric diblock melts o t hat ZA ) ZB ) Z/2. Close to t he order-disorderpoint, Z 10.5, the polymer concentrations can beapproximated by

where , 1 is the amplitude and L is the period of thelamellar structure. Both a n d L are determined fromthe mean-field equations. This simple form of thepolymer concentr at ions produces a set of self-consistentmean fields of the form

where the amplitudes of the mean fields are obtainedfrom t he mean -field equations

I t s houl d be not ed t h at t he L agr angian fact or (z)ensuring the incompressibili ty condition has beenom it t e d. I t c a n b e s h ow n t h a t (z) h a s t h e for m cos(4z/L ), and the amplitude is much smaller than, so that it can be neglected in the mean fields.

With the simple form for the potentials R(z) , theSchrodinger equ at ion det erm ining th e eigenva lues an deigenfunctions has the form

This equation corresponds to the Mathieu differentialequation34

where the new variables are defined by

a nd R R ) bR2ZR /6 are the radii of gyration for the R

blocks. According to Floquet, th e Math ieu equat ion ha ssolutions of the form 34

where u (z) ) u (z + l) and l an int eger. According tothe values of parameters a an d q, there a re two typesof solutions: undamped waves modulated by u (z) (Bloch

qR(x) )

1

A 0e

iqx

R

(q) )bR

2q

2

6(73)

nkzR

(z) )1

L 0e

ikzzu n kz(z) (74)

[- bR2

6 (ikz +

z)2

+ R(z)]u nkz(z) ) nkzR u nkz(z) (75)

nkz

nkzR (z)nkz

R* (z) ) (z - z)

dznkzR

(z)n kzR*

(z) ) (kz - kz)n ,n (76)

qnR )

1

L0

Ldz u n 0

R(z)

n ,n R

(kz) )1

L0

Ldz u nkz

R*(z)u n kz

(z)

n 1,n 2,n 3

R(kz) )

1

L0

Ldz u n 1kz

R*(z)u n 20

R*(z)u n 3kz

R(z) (7 7)

A(z) )1

2+ cos

2z

L

B(z) )1

2- cos

2z

L(78)

R(z) ) R cos2z

L(79)

A ) -

B ) (80)

-b

R

2

6 d

2

R

(z)dz

2 + R cos 2zL R(z) ) RR(z) (8 1)

d2

dz2

+ (a - 2q cos 2z) ) 0 (82)

z ) z/L

a ) 4

RZR

(2R R/L )2

q ) 2RZR

(2R R/L )2

(83)

(z) ) u (z)ez + u (-z)e -z (84)



10/18

waves) for ) ik an d da mped wa ves for complex . Onlythe case of the u nda mped modulat ed waves is physicallyinteresting. The regions of existence of the m odulatedwaves form bands in t he a-q plane (see Figure 1). Forq ) 0 (homogeneous phase), undamped waves exist forall energy values a > 0. This case corresponds to theenergy spectrum of free particles R ) bR

2kz

2 /6, and theresulting correlation functions a re isotropic. For n on-zero q (periodic potent ials), only finite regions ofa a r epossible for u nda mped waves (Figure 1). This corre-sponds to the existence of energy bands, and the free

particle spectrum R ) bR2

kz2 /6 has to be modified to

include the band structure. The appearance of energybands is a n extremely importa nt phenomenon in solid-stat e physics in th at it forms t he basis for distinguishing

meta ls, semiconductors, and insu lators. In th e case ofdiblock copolymers, the formation of energy bands isassociated with t he broken symmetry in th e system andcorresponds to the deviation of the spectrum from thefree-par ticle isotropic beha vior. This nat ur ally leads toanisotropy in the correlation functions, reflecting thefact th at the fluctuations ar e inheren tly anisotropic fora broken symmetry phase.

1 . M e a n -F i e l d T h e o r y . The m ean -field solut ion ofthe weakly segregated symmetric lamellar phase canbe f ound by us i ng t he fact t hat al l t he m ean-fi el dequations involve only the kz ) 0 solutions of the system.The Bloch functions nkz

R (z) a t kz ) 0 reduce to periodicfunctions u n 0

R (z). B eca u s e o f t h e s ym m et r y of t h e

lamellar phase, the solutions u n 0(z) are even functionswith period L . Since the eigenfunctions are relat ed tothe Mathieu functions, the mean-field solution of theproblem requires Mathieu functions which are evenfunctions of period . Accor d in g t o t h e t h e or y of Mathieu fun ctions,34 these eigenfunctions are denotedby ce2n(z,q) with the corresponding eigenvalues a2n(q).The parameter q is proportional to th e am plitude , sot hat q , 1 for weak segregated lamellae. For smallvalues of q, the eigenvalues a2n(q) can be obtained a sexpansions ofq.34 The first few eigenvalues a re

The corresponding eigenfunctions ha ve th e form

where the coefficients A 2m2n (q) can be obt ai ned as ex-

pansions in terms of q.34 The normalization of thesefunctions is chosen so tha t

The eigenfunctions and eigenvalues for the A and Bblocks are then given in terms of the Mathieu functions

where the parameter X is defined by

The parameters qR a r e

Here we have u sed th e relation for symm etric diblocksR A )R B ) b2Z /12. With the a vailability of th e Math ieufunctions, the one-, two-, an d thr ee-point integrals qn

R,

n ,n R (0), and n ,n ,n

R (0) can be evaluated. The resultscan th en be u sed to calculate the single-chain pa rtition

function Q an d the one-body correlation functions CnR.

After some lengthy computat ions, we obtain for thesingle-chain partition function

where the function g2(X) is defined by

It should be noted that the independent-chain homo-geneous-phase RPA correlation function is given by C(k)) 2Z g2[(k R )2 /2]. The one-point correlat ion functionsCn

R can then be used to calculate the polymer concen-trations, and straightforward calculations yield

F i g u r e 1 . Stability diagram for solutions of t he Mathieuequation.34 The solid and dotted lines represent the charac-teristic curves a n(q) and bn(q), which divide th e a-q plane intoregions of stability and inst ability (shaded a rea). At q ) 0,all values of a(q) > 0 are allowed. For the cases of small q(dashed line), a gap occurs and the a(q) spectrum splits intobands.

a 0(q) ) -1

2q

2 +7

12 8q

4 + ...

a2(q) ) 4 +5

12q

2 -76 3

13824q

4 + ...

a4(q) ) 16 +1

30q

2 +43 3

864000q

4 + ... (85)

ce2n(z,q) ) m )0

A 2m2n

(q) cos(2m z) (86)

1

0

2ce2n(z,q)ce2n (z,q) ) n ,n (87)

u n 0R

(z) ) 2ce2n(z

L,q

R)

nR(0)ZR )

X

4a 2n(q

R) (88)

X ) (2R RL )2

)b

2

6 (2

L )2Z

2(89)

qA ) -

Z

X

qB )

Z

X(90)

Q ) 1 +1

2(Z)2g2(X) + ... (91)

g2(X) )2X - 3 + 4e -X - e-2X

4X2 (92)

A(z) )1

2+ [g2(X)(Z) - g4(X)(Z)

3] cos

2z

L+ ...

B(z) )1

2- [g2(X)(Z) - g4(X)(Z)

3] cos

2z

L+ ...

(93)



11/18

where the function g4(X) is

In order t o determine th e equilibrium period L of thesystem, we need to calculate th e free energy density andthen to minimize the free energy density with respect

t o L or, equivalently, to X. Inserting the m ean-fieldsolution ofR(z), R(z), an d Q into th e m ean-field freeenergy, we obtain to lowest order

T he m i nim i zat i on of t he fr ee ener gy dens it y at t helowest order is th erefore equivalent to the maximum ofthe function g2(X). The maximum ofg2(X) occurs at X*

) 1.892 485 with g2(X*) ) 0.0952 847. The equilibriumperiod is then given by L * ) 3.2296R g ) 1.3185Z1/2b,where R g ) (b2Z/6)1/2 is the radius of gyration of thepolymer chains. The polymer concentra tions are nowgiven by

where the coefficients are g2(X*) ) 0.095 284 66, g4(X*)) 0.000 555 31. Self-consistency of the mean -fieldequations requires t hat

w her e t he cr i t ical value of Z i s given by (Z)c )1/g2(X*) ) 10.495. The solution of th is m ean-fieldequation for can be easily found by plotting th e left-hand s ide (f1() ) ) and the r ight-hand side (f2() )

Z/(Z)c - g4(X*)(Z)3) of the mean -field equat ion asfunctions of (Figure 2). The existence of a n onzerosolution for depends on the init ial s lope Z/(Z)c ofthe fun ction f2(). The mean -field equation for theamplitude has solution ) 0 if Z < (Z)c ) 10.495,and for Z > (Z)c ) 10.495, the stable solution for is

N ot e t hat t he exponent 1/2 is typical for mean -fieldth eories. This lowest-order m ean -field solution derivedabove could be obtained more conveniently using theLandau free energy functional of Leibler.13 It should

be emphasized that the above mean-field solution isobtained to the lowest order of , a n d t h e r e i s n o dependence in t he equ ilibrium period L at this order ofapp roximation. However, th e inclusion of higher -orderterms will introduce a dependence into L , as demon-strated by Olvera de la Cruz14,35 using the Landau freeenergy functional of Leibler, and by ma ny numericalsolutions of the mean-field theory.24

2 . T h e N a t u r e o f t h e A n i s o t r o p i c F l u c t u a t i o n s .Because the Mathieu functions are described in theliteratur e, we can in principle use th em a s t he eigen-functions and proceed to calculate the correlation func-tions. However, explicit forms for the eigenvalues a ndthe Mathieu functions for kz * 0 are not available.Fur t her m or e, becaus e w e ar e deali ng w it h a w eakpotential (R , 1), it is not necessary to use the generalsolution for arbitrary values ofR. I ns t ead, we us e aperturbation scheme developed by Brillouin 33 to calcu-late th e energy bands for th e weakly segregated poten-tial . We first write the potential R(z) i n t er m s of aFourier series

where VR ) R/2, A ) -, and B ). As mentionedearlier, the eigenfunctions for a periodic potential havethe Bloch form

The periodic modulation factor u kzR (z) can be expanded

in a Fourier series

Hence the eigenfunctions for the z direction have theform

T he fi r st t er m of t h i s equat i on cor r es ponds t o t he

g4(X) )1

4608X4(504X - 1647 + 2112Xe-X +

880e-X - 1056Xe-2X + 1776e-2X - 1152e-3X +

128e-4X + 16 e-5X - e-8X) (94)

f )1

L0

Ldz{(A(z) -

1

2)(B(z) -1

2) - A(z)A(z) -B(z)B(z)} -

1

Zln Q

1

2

2{1 - Z g2(X)} + ... (95)

A(z) )1

2+ cos

2z

L

2 + [g2(X*)(Z) - g4(X*)(Z)3] cos

2z

L+ ...

B(z) )1

2- cos

2z

L

2 - [g2(X*)(Z) - g4(X*)(Z)3

] cos

2z

L + ...(96)

) g2(X*)(Z) - g4(X*)(Z)3 + ...

(Z)c - g4(X*)(Z)3

(97)

)1

g4(X*)(Z)3( Z(Z)c - 1)

1/2

(98)

F i g u r e 2 . Solution of the m ean-field equation for th e am pli-t u t e . The dotted line is f1() ) , and the solid lines aref2() ) g2(X*)Z - g4(X*)(Z)3.

R(z) ) VR[ei(2z/L ) + e-i(2z/L )] (99)

kzR (z) )

1

L 0e

ikzzu kzR (z) (100)

u kzR (z) )

n )-

+

cnR(kz)e

i(2nz/L)

kz + n)-

n*0

+

cnR(kz)e

i(2nz/L )(101)

kzR

(z) )1

L 0c0

R(kz)e

ikzz +1

L 0

n)-n*0

+

cnR

(kz)ei(kz+(2n/L))z

(102)



12/18

solution for the homogeneous phase, the second termaccounts for the position-dependent amplitude of themean fields and can be taken as a perturbation term.The energy spectrum (eigenvalues) R(kz) is obtain ed bysubstitu ting t he solution int o the Schr odinger equ at ion.Manipulation of the equation leads to the equationsdetermining the coefficients

where th e new variables are defined by k0 ) 2/L , kz )kz/k0, R(kz) ) (b

2k0

2/6)R(kz), VR ) (b2k0

2/6)R. F or t h e

coefficients cnR(kz) of the Bloch function, we have

Because the periodic potent ial is weak, we can m ake aperturbation substitution in the denominator, R(kz)

0R(kz) ) kz

2, and discuss the behavior of cnR(kz). Accord -

ing to the value of kz, we must distinguish two cases.

For the first case where kz2 * (kz + n )2, the denomina-

tor is nonzero. In th e weak-potential approximat ion,

the coefficient R is small as well as the coefficientscn

R(kz) of the Bloch function, with the exception of c0R

(kz). In this case we have

The eigenfunctions have the form

where the coefficients c0R(kz) a r e d e t e r m i n e d b y t h e

normalization condition

The corresponding eigenvalues a re

For small values of kz 0, the eigenvalues become

R(kz) (1 - 8R2 )kz2 - 2R2 . From this expression i t isclear tha t the effect of a nonzero potent ial R for smallkz is t o r e nor m a liz e t h e K u hn le n gt h a lon g t h e zdirection into an effective bz < b,

The total eigenvalues are now of the form b2q2/6 +bz

2kz

2 /6, which is clearly a nisotropic. Because bz < b,the anisotropy introduced through R(q ,kz) ) b2q2/6 +bz

2kz

2 /6 breaks the circular symmetry of the scatteringfunction in th e q - kz plane so that the scattering r ingbecomes elongated in the kz direction. It should alsobe noted t hat the effective Kuh n length introduced

above is closely related to the effective electron massintroduced in solid-state physics.32

It is clear from these expressions that the perturba-tion results given above break down when kz f kc )(1/2, which leads us to consider t he second case wherekz approaches th e critical values kc ) (/L . Here someof the denominators in the perturbation expressions

become zero and the corresponding cnR(kz) assumes h igh

values. Fr om the beha vior of th e coefficient s c0R(kz) and

cnR(kz) in the neighborhood of the critical values kc, we

can dra w conclusions a bout the energy spectru m R

(kz)curve near kc. From t he equa tions for t he coefficients,we have near kz ) (1/2

where n ) (1. I ndependent of t he value of kz, t h ecoefficient c0

R(kz) is in principle very large because the

denominator always tends to zero. Also, because nR )

0 for |n | * 1, cnR(kz) are vanishingly small for n * (1.We therefore conclude that all the other coefficientscan be neglected for kz ) (/L , except th e coefficientsc0

R(kz) and c-1R (kz). Because th e eigenvalues and eigen-

f unct i ons have t he s ym m et r y t hat R(-kz) ) R(kz),cn

R(-kz) ) c-nR* (kz),

32 we only need to consider the caseswhere kz g 0. Considering the two modes n ) 0 and n) -1 only, the coefficients obey the equations

w her e w e have w r i t t en t he ei genval ues i n t he f or m

R(kz) ) kz2 + R(kz), and the wave vector in the form kz

) 1/2 + . These homogeneous an d linear equat ions forc0

R(kz) a n d c-1R (kz) h ave a nont r i vi al s ol ut ion i f t he

determina nt of the coefficients van ishes, leading t o theeigenvalues

According to this r esult, th e spectrum of the eigenvaluesR(kz) near kz ) k0/2 ) /L has two different values

That is, the eigenvalue spectrum becomes discontinuousa t kz ) /L , an d an energy gap, R(/L ) ) 2|VR| )|R|, appear s. T her e ar e only t w o gaps at kz ) (/Lobserved in our system since th e only nonzero Fouriercomponents in t he potential R(z) ar e for n ) (1. Thisabove ar gument applies to every |n | * 0. Energy gaps,however, need not appear for every n . If th e n t h

coefficient nR ) 0 , t h e R(kz) curve will become con-

tinuous a t t he points kz ) (n/L . For the special caseof a weak ly segregated lam ellar pha se, we have n

R ) 0for all |n | > 1. Therefore the eigenvalue spectrum of t he s ys t em has gaps onl y at kz ) (/L , as shown inFigure 3. Generically, the R(kz) function is single-valued a nd pa ra bolic, as in th e case of the homogeneous

[R(kz) - (kz + n )

2]cn

R(kz) - R[cn-1R (kz) + cn+1

R (kz)] ) 0(103)

cnR

(kz) )R[cn-1

R(kz) + cn+1

R(kz)]

R

(kz) - (kz + n )2

(104)

c1R

(kz) -Rc0

R(kz)

2kz + 1

c-1R

(kz) Rc0

R(kz)

2kz - 1(105)

u kzR

(z) c0R

(kz)

[1 -

R

2kz + 1eik0z

+

R

2kz - 1e-ik0z

](106)

c0R

(kz) 1 -4kz

2 + 1

(4kz2 - 1)2

R2

(107)

R

(kz) kz2 + 2R

2/(4kz

2 - 1) (108)

bz2 ) b2(1 - 8R

2) (109)

cnR

(kz) )Rc0

R(kz)

R(kz) - (kz + n )

2

c0R

(kz) )Rcn

R(kz)

R

(kz) - kz2

(110)

c-1R

(kz)[R

(kz) + 2] ) c0R

(kz)R

c0R

(kz)kzR ) c-1

R(kz)R (111)

R(kz) ) - ( 2 + R2 (112)

(R(k02 ) )

b2

6 (k0

2 )2

( |VR| (113)



13/18

phase, except for the critical points kz ) (/L wherethe R(kz) cur ve has discontin uities and t he ener gy splitsinto two values (Figure 3, th ick solid line). It sh ouldbe noted that we have used a n extended zone schemein our description of the eigenvalues.32 In this repre-sentation t he n th energy band is attr ibuted to the n t h

Brillouin zone. Because of th e periodicity of th e wave-number, the R(kz) cur ve ha s a periodicity of 2/L . Whenwe restrict ourselves to the reduced wavenumber kz, wecan assign all the energy bands to the f irst Bril louinzone. This makes the R(kz) relation multivalued a ndthe energy bands m ust be labeled by a corresponding

band num ber n ; therefore, R(kz) ) nR(kz). I n wh a t

follows we will use the extended zone scheme so thatthe band index n is not necessary. The eigenvalues arediscontinuous at kz ) k0/2 ) /L (see Figure 3),

The coefficients cnR(kz) ar e obt ai ned by i ns er t i ng t he

eigenvalues into th eir corresponding equations an d bythe normalization requirement

Because the eigenvalues R(kz) ) - - (2 + R2 )1/2 a r ederived un der the assumption that only the n ) 0 a n dn ) -1 modes are important, they are valid only whenkz /L , i .e., 0 . F or s m a ll v a lu e s o f , t h eeigenvalues have the form

The coefficients c0R a nd c-1

R a r e

The eigenfunctions of the system are then given by

The results obtained from our perturbation calcula-tions show th at t here exist critical kz points at (/L (theBragg points). The eigenvalues and eigenfunctions ofthe chain Hamiltonian have a qualitatively differentbehavior when kz i s clos e t o t he B r agg point s . I nparticular, when kz is away from (/L , the correctionsto the eigenfunctions and eigenvalues due to the weak

periodic potential R(z) are of the order O(R2 ). O n t he

ot her hand, w hen kz (/L , the corrections to theeigenfunctions a nd eigenvalues du e to th e weak p eriodicpotential R(z) are of the order O(R). We can ther eforeconclude that, to the lowest order of the perturbation,

the eigenfunctions of the system are well approximatedby th e homogeneous eigenfun ctions except when kz isclose to the Bragg point (/L . The eigenvalue spectru mof the diblocks in a weakly segregated lamellar phaseis well approximated by the homogeneous parabolicrelation except near the Bragg points k ) (/L , wherethe spectrum splits to form gaps of magnitude |R} ), i.e., the width of the gap is determined by the mean-field amplitude (Figure 3).

I n s um m ar y, t he behavi or of t he ei genval ues andeigenfunctions is, to the lowest order of the potential,controlled by the variable ) kz/k0 - 1/2. For l ar gevalues of ||, e.g., || g |R|, the eigenfunctions andeigenvalues a re well described by the relations for t hehomogeneous phase

For s m all values of ||, || < |R|, t he eigenvaluespectrum becomes discontinuous,

Because the variable is proportional t o kz - /L , t heeigenvalue s pect r um near t he B r agg poi nt i s a gainpara bolic. The corresponding eigenfunctions a re

F i g u r e 3 . Eigenvalue spectru m for t he weak potentia l eq 82.The dotted curve is the parabolic spectrum for the homoge-neous phase. The solid curves represent th e band stru cturewhen R ) * 0.

R

(kz) )b

2k0

2

6 [1

4+ 2 - 2 + R2 ] if kz e

L

R(kz) ) b2

k02

6 [14 + 2 +

2 + R2] if kz > L (114)

c0R(kz) )

R

R2 + [R(kz)]2

c-1R

(kz) )R(kz)

R2 + [R(kz)]2(115)

R(kz) - - |R| -2

2|R|+ ... (116)

c0R

1

2

R

|R|(1 -

2|R|-

2

8R2) + ...

c-1R -

1

2(1 (

2|R|-

2

8R2) + ... (117)

u kzR

(z) 1

2[R|R|(1 -

2|R|-

2

8R2) -

(1 (

2|R|

-2

8R2

)e-ik0z

](118)

R(kz)

b2

kz2

6+ O(R

2)

u kzR

(z) 1 + O(R) (119)

R

(kz) )b

2k0

2

6 [1

4- |R| - ( 12|R| - 1))

2] if kz e L

R(kz) )

b2

k02

6 [14 + |R| + ( 12|R| + 1))2

] if kz > L(120)

u kzR (z) )

1

2[R

|R|(1 -

2|R|-

2

8R2) -

(1 + 2|R| -2

8R2)e-ik0z] e 0



14/18

These eigenfunctions correspond to sta nding wa ves inthe system. Therefore, the a ppearance of the energygap corresponds to t he formation of standing waves inthe system. Because the energy gap appears at nonzero

Fourier components of the potential R(r), which cor-responds to the appearance of an ordered structure,there is a n int rinsic relationship between th e symmetrybreaking in the system an d the gaps in t he eigenvalues.

With the availability of the eigenvalues and eigen-functions, it is straightforward to compute the blockcopolymer correlation functions and RPA density-density corr elation functions a ccording to th e proceduregiven above. For the weak ly segregated lamellae withinthe one-mode approximat ion, we see that ther e are t wosources contribut ing to t he a nisotropic correla tion func-tions. The f irst is t he renormalization of the Kuhnlength arising from the correction to the eigenvalues;the second is th e strong resonan t scattering a t kz ) (k0/2, leading to the appearance of gaps in the eigenvaluespectrum. The scattering function changes from iso-tropic ringlike beha vior to an isotropic behavior with t wostrong peaks at the Bragg points k ) (/L . F u r t h er -more, the coupling between the x - y modes and the zmode through the non-Gaussian correlation functionsleads to additional scattering pea ks corresponding to th efluctuat ing layers. The lamellar pha se RPA scatteringfunctions ar e given in r ef 30.

V. C o n c lu s i o n s a n d D i s c u s s i on s

A general theoretical framework for the study of an isotropic fluctua tions in ordered block copolymersystems has been developed. The theory is based on asystematic expansion around the broken-symmetry

ordered m ean-field solution. The truncation of thetheory at the Gaussian f luctuation level and the ap-plication of the random phase approximation result in

a density-density correlation function Cn ,n RP A(k ), which

conta ins informa tion a bout the anisotropic compositionf l uct uat i ons i n t he s ys t em and can be m eas ur ed byscattering experiments. Methods of calculating th epolymer correlation functions in an inhomogeneousp h a se a r e d eve lop ed . I t h a s b ee n s h ow n t h a t t h eimportant quantities controlling the fluctuations are thespectra of the eigenvalues of the chain Ham iltonian HRan d the corresponding eigenfunctions. The eigenvaluesand eigenfunctions correspond to th e energy bands andwave functions for an electron in a periodic potential

R(r

). The calculation of th e RPA correlat ion functionsfor block copolymers is therefore equivalent to bandstru cture calculations in solid-stat e physics, and man ymethods developed for the band stru cture calculat ionscan be used for t he polymer pr oblem. The calculatedcorrelation function can be used to study the stabilityand the elastic properties of the ordered phases, as h asbeen demonstrat ed for the lamellar phase.30 The cal-culated scattering functions correspond to all the es-sential features of experimental measurements.

In order to illustrate the mean features of the aniso-tropic fluctuations, a weak ly ordered lam ellar pha se wasstudied in detail using pertur bation theory. The mosti m por t ant feat ur e em er ging fr om t he t heor y i s t heappearance of a band structure in the eigenvalues of

t he pr oblem . T he appear ance of gaps i n t he ener gyband is related to the broken symmetry of the system.For the lamellar phases, the f irst gap appears at kz )k0 /2, resulting in the most unstable fluctuation modesa t kz ) k0/2.30 Although the perturbation theory waspresent ed for th e lamellar ph ase, th e generic conclusionf r om t he t heor y appl i es t o any or der ed phas e.32 Inpart icular , when th e wave vector k lies on a Br agg planedefined by k 2 ) (k - G)2, the eigenvalue of the chainHamiltonian has the form 0(k ) ( |VG|, resulting in agap of or der 2|VG| if the Fourier component of th epotential at G is not zero (VG * 0).

It is appropriate to compare the theory developed inthis paper and the Brasovskii-Fredrickson -Helfand(BFH) theory.25,26 We start with the exact expansionof the free energy functional in terms of fluctuations (eqs28-32). The Gau ssian fluctua tion t heory developed inour paper is obtained by neglecting the higher-orderterms (n ) (n > 2) but treating the Gaussian fluctuationsexactly. The BFH theory is developed using severalapproximat ions: (1) the R(r) field variables are elimi-nat ed us i ng t he R PA ,13 r es ult i ng i n a L andau fr eeenergy functiona l; (2) the La nda u free ener gy fun ctiona lis reduced to the Brasovskii form;25 and (3) the fluctua-tions are treated using a self-consistent Hartree ap-

proximation,26 resulting in a ren ormalized Landa u freeenergy functional. The renormalization Landau freeenergy is then t reated using the mean-field a pproxima-tion. For th e symmetr ic diblock copolymer (f ) 1/2),mea n-field theory predicts a cont inuous order -disordertransition.13 The inclusion of the fluctuations withinthe BFH theory results in a weakly f irst-order phasetran sition in this case. On the other hand, the Gaussianfluctuation approximation cannot be used at the criticalpoint. The use of the Landau free energy functionalrestricts the validity of the BFH theory to the weaksegregation regime, while our Gaussian fluctuat iont heor y appli es t o t he w hol e phas e s pace except t hecritical point. Therefore, th e two th eories complementeach other.

The genera l th eoretical fram ework developed in ourpaper provides a promising technique for the study ofth e ph ysical pr operties of order ed m icropha ses of blockcopolymer melts, as demonstrated for the lamellarphase.30 The calculated RPA correlation functions forthe ordered phases characterize the anisotropic fluctua-tions. The origin a nd n atu re of the anisotropic fluctua -t i ons can be at t r i but ed t o t he for m at i on of a bandstructure for the eigenvalues of the correlation func-tions. The gaps in the spectrum lead to unst able modes,which are observed as scattering peaks in scatteringexperiments. Futu re applications of the theory willinclude calculations of the RPA correlation functions forthe complex block copolymer phases, and the effects of

fluctuations on the first-order phase boundaries of thesystem.

A c k n o w l e d g m e n t . We than k Dr. Chu ck Yeung forhis contributions in the early stages of this project, andDr. M. Lara dji for many useful discussions. R.C.D.acknowledges the support from the Natur al Sciencesand Engineering Research Council of Canada.

A p p e n d i x A : E x p a n s i o n o f Qc

I n t hi s appendi x w e out l i ne t he der i vat i on of t hesingle-chain partition function Qc as an expansion interms of the variables R(r). The forma l solution ofthe propagators allows us to obtain the solution for QR

u kzR (z) )

1

2[R

|R|(1 +

2|R|-

2

8R2) +

( 1 - 2|R| -2

8R2)e-ik0z] > 0 (121)



15/18

by itera tion

where the n th-order solution is

Therefore t he per tur bation solution of the propagatorshas the form

Using the pertu rbat ion solution ofQR(r , t|r), th e singlechain partition Qc can be obtained as a series in the R.The expan sion ofQc can be written in a more symmetricform by intr oducing th e symmetr ized n -point corr elat ion

functions CR1,...,Rn(n) (r 1,...,rn) in t he fields R

(0)(r),

where the CR1,...,Rn(n ) (r1,...,r n) are the appropriate combi-

na tions of the single-chain correla tion functions GR, andt he s um m at ion over Rn is for Rn ) A, B . T he secorrelation functions can be writ ten in terms of thezeroth order propagators QR(r, t|r ) using th e relation

N ot e t h a t w e h a ve n e gle ct e d t h e s u pe r scr ip t inQ(0)(r, t|r) to simplify the notation, and in what follows

Q(r, t|r) represen t the zeroth order propagator. We willalso drop th e superscript 0 in t he cumu lant correlationfunctions, with the understanding that these correlationfunctions ar e determined in the mean fields. The chainconformation contribution to the free energy functionalcan be obtained by using the relation

The single-chain partition function term has the freeenergy in the form of an expansion

where the coefficients CR1,...,Rn(r 1, ..., rn) are given in

terms of the single-chain correlation functions CR1,...,Rn(n)

(r1,...,rn). Physically the CR1,...,Rn(r , ..., rn ) i s t h e n t horder cumulan t correlation function for a n oninteracting

diblock chain in the field R(0)(r). The lowest-order

cumulant correlation functions are

where we have used the fact that , in the th ermodynamiclimit, the second term in CR (which is proportional to1/V) can be ignored. The single-chain mea n-field par ti-tion function Qc is given by

It should be n oted that these two-point correlation

QR(r, t|r) ) QR(1)

(r, t|r) + QR(2)

(r , t|r ) + . .. (A1 )

QR(n )

(r, t|r) ) (-)n dr1...drn dt1...dtnGR(r, t|rn, t4n )GR(r n, tn|rn-1, tn -1)...

GR(r2, t2|r1, t1)GR(r 1, t1|r, 0)R(r1)...R(rn) (A2)

QR(r, t|r) ) QR(0)

(r, t|r) + n)1

(-)n dr1...dr n

dt1...d tn GR(r, t|rn, tn)GR(rn,tn|rn -1, tn-1)...GR(r2, t2|r1, t1)GR(r 1, t1|r, 0)R(r1)...R(rn) (A3)

Qc ) Qc(0) + Qc

(1) + Qc(2) + ...

0 +1

Vn)1

R1,...Rn

(-)n

n ! dr1...drn CR1,...,Rn

(n )

(r ,...,rn)R1

(r1)...Rn

(rn) (A4)

GR(r, t|r, t) ) (t - t)QR(r, t - t|r ) (A5)

lnQc

Qc(0)

Qc(1)

Qc(0)

+Qc

(2)

Qc(0)

-1

2(Qc(1)

Qc(0))

2

+ ... (A6)

V

Zln Qc )

V

Zln Qc

(0) -1

Qc(0)

Z

R drCR

(1)(r)R(r) +

1

2Qc(0)

ZR, drdr{CR(2)(r , r ) -

1

Qc(0)

V

CR(1)

(r)C(1)

(r)}R(r)(r) + ... )V

Zln Qc(0) + n)1

(-1)n

n ! R1,...,Rn dr1...dr n

CR1,...,Rn(r1, ..., rn)R1(r1)...Rn(rn ) (A7)

CA(r) )1

QcZ0

ZAdt dr1dr 2dr3QA(r1, t|r)

QA(r, ZA - t|r 2)QB(r2, ZB|r3)

CB(r) )1

QcZ0

ZBdt dr1dr 2dr3QB(r1, t|r)

QB(r, ZB - t|r2)QA(r2, ZA|r3) (A8)

CAA(r , r) )1

QcZ0

ZAdt0

tdt dr1dr 2dr3

[QA(r1, ZA - t|r)QA(r, t - t|r)QA(r, t|r 2) QB(r2, ZB|r3) + QB(r1, ZB|r2)QA(r2, ZA - t|r)

QA(r, t - t|r)QA(r, t|r3)]

CAB(r ,r) )1QcZ

ZA

0dt0

ZBdt dr1dr2dr3QA(r 1, ZA -

t|r)QA(r, t|r2)QB(r2, t|r)QB(r, ZB - t|r3)

CBA(r , r) )1

QcZ0

ZAdt0

ZBdt dr1dr 2dr3QB(r1, ZB -

t|r)QB(r, t|r2)QA(r2, t|r)QA(r, ZA - t|r3)

CBB(r , r) )1

QcZ0

ZBdt0

tdt dr 1dr2dr3[QB(r1, ZB -

t|r)QB(r, t - t|r)QB(r, t|r2)QA(r 2, ZA|r3) +QA(r1, ZA|r2)QB(r2, ZB - t|r)

QB(r, t - t|r)QB(r, t|r 3)] (A9)

Qc )1

V dr1dr2dr3QA(r1, ZA|r2)QB(r2, ZB|r3) (A10)



16/18

functions have the symmetries CR(r, r ) ) CR(r, r),i.e., the matrix CR(r , r ) is real and symmetric.

A p p e n d i x B : R e c i p r o c a l S p a c e F o r m u l a t i o n

Here we give a detailed exposition of the correlationfunctions in terms of the eigenfunction and eigenvaluesof the chain Ham iltonian HR. Using the eigenfunctionsof HR as the basis functions, the one- and two-pointcumulant correlation functions can be cast in th e form

It will be useful to define one-, two-, and three-pointintegrals

The factors involving V1/2 in th e above expressions a reintroduced for convenience. As has been st ated in t hemain text, the mean-field propagators can be writtenin terms of the eigenfunctions. Inserting the expres-sions for QR(r, t|r) into th e correlat ion functions, we canperform the integrals over the t variables and obtainsimple expressions for th e correlation functions in t hereciprocal space. In terms of the eigenfunctions, t heone-body correlation functions can then be written as

where the function g(x) is defined by

The expressions for the two-body correlation functionsar e

The mean -field single-chain part ition function Qc ca nalso be cast in terms of these integrals

We now consider the most general case, in which theonly restr iction is that th e mean fields R(r) are periodicfun ctions. As we ha ve mentioned before, th e eigenfunc-tions for the chain H amiltonian a re Bloch functions of

the form

where u nkR (r) a re periodic functions, and the factor

1/(V)1/2 i s i nt r oduced for conveni ence. I n or der t o

proceed, we expand the periodic function u n kR (r) using

the reciprocal lattice vectors of the system

where {G} are the reciprocal lat tice points of the ordered

phase u nder consideration. Because t he m ean fieldsR(r) can be expanded in the form

th e S chrodinger equa tion becomes

This result for different reciprocal lattice vectors Ggenerates a n infinite nu mber of equations. The condi-tion of ha ving a n onzero solution is that th e determina nt

Cn kR

)1

V drn kR*

(r)CR(r) (B1)

Cn k ,n k R ) dr drn k

R*(r)CR(r , r )n k

(r)(B2)

qn kR

1

V drn k

R(r)

n k ,n k R

drn kR* (r)n k

(r)

n k ,n k ,n k R t V drn kR* (r)n k R* (r)n k R (r) (B3)

Cn kA )

ZA

QcZ

n1

k1

n

2k

2

n

3k

3

g[n 1A

(k 1)ZA - n 2A

(k 2)ZA]

e-n 2

A (k 2)ZA-n 3B (k 3)ZBqn 1k 1

An k ,n 1k 1,n 2k 2

An 2k 2,n 3k 3

ABqn 3,k 3

B*

Cn kB )

ZB

QcZ

n 1k 1

n 2k 2

n 3k 3

g[n 1B (k 1)ZB - n 2

B (k 2)ZB]

e-n 2

B (k 2)ZB-n 3A (k 3)ZAqn 1k 1

Bn k ,n 1k 1,n 2k 2

Bn 2k 2,n 3k 3

BAqn 3k 3

A*(B4)

g(x) 1 - e-x

x(B5)

Cn k ,n k AA )

ZA2

QcZ

n 1k 1

n 2k 2

n 3k 3

n 4k 4

g[n 2A

(k 2)ZA - n 1A

(k 1)ZA] - g[n 3A

(k 3)ZA - n 1A

(k 1)ZA]

n 3A

(k 3)ZA - n 2A

(k 2)ZA

e-n 1

A (k 1)ZA-n 4B (k 4)ZB

[qn 1k 1An k ,n 1k 1,n 2k 2

An k ,n 3k 3,n 2k 2

A*n 3k 3,n 4k 4

ABqn 4k 4

B* +

qn 1k 1A*n k ,n 3k 3,n 2k 2

An k ,n 1k 1,n 2k 2

A*n 3k 3,n 4k 4

AB*qn 4k 4

B]

Cn k ,n k AB )

ZAZB

QcZ

n 1k 1

n 2k 2

n 3k 3

n 4k 4

g[n 2A

(k 2)ZA -

n 1A

(k 1)ZA]g[n 3B

(k 3)ZB - n 4B

(k 4)ZB]e-n 1

A (k 1)ZA-n 4B (k 4)ZB

qn 1k 1An k ,n 1k 1,n 2k 2

An 2k 2,n 3k 3

ABn k ,n 4k 4,n 3k 3

B*qn 4k 4

B*

Cn k ,n k BA )

ZAZB

QcZ

n 1k 1

n 2k 2

n 3k 3

n 4k 4

g[n 2B

(k 2)ZB -

n 1B (k 1)ZB]g[n 3

A (k 3)ZA - n 4A (k 4)ZA) e -n 4

A

(k 4)ZA-n 1B

(k 1)ZB

qn 1k 1Bn k ,n 1k 1,n 2k 2

Bn 2k 2,n 3k 3

BAn k ,n 4k 4,n 3k 3

A*qn 4k 4

A*

Cn k ,n k BB )

ZB2

QcZ

n 1k 1

n 2k 2

n 3k 3

n 4k 4

g[n 2B

(k 2)ZB - n 1B

(k 1)ZB] - g[n 3B

(k 3)ZB - n 1B

(k 1)ZB]

n 3B

(k 3)ZB - n 2B

(k 2)ZB

e-n 1

B (k 1)ZB-n 4A (k 4)ZA

[qn 1k 1Bn k ,n 1k 1,n 2k 2

Bn k ,n 3k 3,n 2k 2

B*n 3k 3,n 4k 4

BAqn 4k 4

A* +

qn 1k 1B* n k ,n 3k 3,n 2k 2

B n k ,n 1k 1,n 2k 2B* n 3k 3,n 4k 4

BA* qn 4k 4A ] (B6)

Qc )1

V dr 1dr2dr3QA(r1, ZA|r2)QB(r2, ZB|r3)

n 1k 1n 2k2

e-n 1A (k 1)ZA-n 2

B (k2)ZBqn 1k 1A n 1k 1,n 2k 2

AB qn 2k2B* (B7)

n kR

(r) )1

Ve

ik ru n k

R(r) (B8)

u n kR (r) )

G

u n kR (G)e

iGr (B9)

R(r) )G

R(G)eiGr (B10)

[b2

6(k + G)2 - n k

R ]u n kR (G) +G

R(G - G)u n kR

(G) ) 0

(B11)



17/18

of the system vanishes

The order of this determinan t is infinite. In pra ctice,we can only deal with a finite determinant; e.g., if wetake 200 plane waves, we will have a determinant of order 200, and t he solution of the eigenvalue equa tion

gives 200 eigenvalues n kR (n ) 1, 2, . .., 200). T he

energy bands are obtained by changing the values of t he k s in th e S chrodinger equa tion.

T he r equi r ed i nt egr al s can now be cas t us i ng t heBloch functions. In par ticular, we have

Since k is restricted in the first Bril louin zone, th econdition k ) G leads to k ) G ) 0. Therefore we havefor the one-point integrals

where the reduced one-point integrals qnR are given by

Let us n ow consider th e two-point integrals. Usingthe Bloch functions and integrating out the r variable,

we have

T he delt a funct i on i n t he above expr ess ion k -k ,Krequires that k - k equals a reciprocal lattice vectorK. Since bot h k a n d k ar e r es t r ict ed i n t he fi r stBrillouin zone, t his condition puts a severe restrictionon t he val ues of K. We n ot e t h a t K ) 0 is alwaysallowed, which leads to k-k ,K ) k ,k K,0. It is possiblet o p r ov e t h a t K ) 0 i s t he onl y s olut i on for t he

requirementk

-k

)K

: (a) It is obvious tha t t he onlypossible Ks are the nearest and (possibly) the next-nearest neighbors to the latt ice point K ) 0; (b) thepossible k a n d k values ar e confi ned by t he t w operpendicular planes bisecting K a nd -K; (c) for anyvector k within the first Brillouin zone, the vector k +K is always outside the first Brillouin zone, except thatk is on the plane bisecting -K; (d) when k is on theplane bisecting -K, t h e s u m k + K is at the corre-sponding point on the plane bisecting K. However,these two points are degenerate and only one of themcan be i ncl uded i n t he s pect r um for a given band.Therefore it is not possible to have nonzero K so that k- k ) K, and the two-point integrals are diagonal int he k space

w her e t he r educed t w o-poi nt i nt egr als n ,n R (k ) a r e

given by

For the three-point integrals, the integral over thespatial variables leads to

Again, the delta function k 1+k 2-k 3+G1+G2-G3,0 requirest hat k + k 2 - k 3 ) K is a reciprocal lattice vector, whichputs severe restrictions on the possible values ofK. A

detailed considerat ion is needed t o obtain the possiblevalues ofK. Ther e is, however, a simplificat ion in th at ,for all the t hree-body integrals needed in the first- andsecond-order cumu lant corr elation functions, one of th ek s is always zero. Therefore, the t hr ee-body restrictionreduces to the two-body restriction, k 1 - k 2 ) K. T h efact that there is always one k ) 0 in can be seeneasily from t he expr essions for th e correla tion functionsby noting tha t t he one-body an d two-body integrals h avethe gener ic form given a bove. These expressions ensu rethat the needed three-point integrals have at least onek ) 0. Using the expressions for the s, we have

where the reduced three-point integrals n 1,n 2,n 3R (k ) are

given by

We can now use these reduced three-body integralst o e lim in a t e t h e s u m s ove r k s in the correlationfunctions. After some simple ma nipu lations, we obtainthe expressions for the cumulant correlation functionsand the single-chain parti t ion function. For the one-point correlation functions we h ave

de t|[b2

6(k + G)2 - n k

R ]G,G + R(G - G)| ) 0 (B12 )

qn kR

1

V drn k

R (r)

)G

u n kR

(G)1

V drei(k +G)r

)G

u n kR

(G)k +G,0 (B13)

qn kR ) u n k

R(0)k ,0

) qnRk ,0 (B14)

qnR ) u n 0

R(0) (B15)

n k ,n k R

drn kR*

(r)n k

(r)

)G

K

u n kR* (G)u n k

(G + K)k-k ,K (B16)

n k ,n k R )

G

u n kR*

(G)u n k

(G)k ,k

) n ,n R (k )k ,k (B17)

n ,n R

(k ) )G

u n kR*

(G)u n k

(G) (B18)

n 1k 1,n 2k 2,n 3k 3

R V drn 1k 1

R* (r)n 2k 2R* (r)n 3k 3

R (r)

)G1

G2

G3

u n 1k 1R*

(

an-chang shi, jaan noolandi and rashmi c. desai- theory of anisotropic fluctuations in ordered block...

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