mohamed laradji, an-chang shi, jaan noolandi and rashmi c. desai- stability of ordered phases in...

8/3/2019 Mohamed Laradji, An-Chang Shi, Jaan Noolandi and Rashmi C. Desai- Stability of Ordered Phases in Diblock Copolymer Melts

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Olvera de la Cruz15 to include the leading wave vectordependence of the vertex functions but keeping thesaddle-point approximat ion a nd r emaining in th e weaksegregation limit. More recently, Muth uku mar 16 stud-ied composition fluctuation effects on the phase diagramof diblock copolymer melts through the incorporationof composition fluctuations into a density functionaltheory. However, the chain conforma tions ar e treat edapproximat ely in this th eory due to the t run cated formof the free energy functional. It is importa nt t o remarkthat the fluctuation theories, cited above, are character-ized by the isotropic correlation functions which arederived from Leiblers theory.4 Recently, we havedeveloped a theory for anisotropic fluctuations in or-dered phases of diblock copolymer melts by means of aself-consistent expansion around the exact mean-fieldsolution. The lowest correction to the mean -field ap-proximation is in the form of Gaussian fluctuat ionsaround t he mean -field solution. These fluctuations arecharacterized by the anisotropic density-density cor-relation functions. The theory has been applied to thesimple lam ellar p ha se.17 A description of the theoreticalframework and the nature of the anisotropic f luctua-tions is given in a n ea rlier publication.18 In the presentpaper, the theory for an isotropic fluctuations is devel-

oped into a convenient scaled form and then applied toall ordered phases in diblock copolymer melts in thew eak s egr egat i on l im i t . Som e of our r esul t s w er erecently r eported by us in a brief communication.19

The ma in thr ust of the t heory is tha t, in order to studythe effect of fluctuations in an ordered broken-symm etryphase, a self-consistent expansion must be performedar ound t he exact mean -field solution. The chain con-form at ions ar e th erefore d escribed by th ose of a flexiblechain in a periodic poten tial, i.e., by a diffusion equationin a periodic potential . Since in quantu m mechanicsth e Schr odinger equa tion is a diffusion equ at ion wit himaginary t ime, th e chain conformation problem isequivalent to the problem of an electron in a periodic

potential. Therefore, the theory bears an importa nt an dus eful anal ogy w i t h t he t heor y of ener gy bands i ncrystalline solids. This an alogy ena bles u s to exploitthe symmetry of the ordered phases and to use manytechniques developed in solid state physics.20 In thispaper, we present results on the scattering functions,the spinodal lines, and th e least stable or most u nstablemodes of the ordered phases in diblock copolymer melts.In part icular, we show that the HP L phase is unstablealong the lamellar -cylindrical phase boundaries, inaccordance with the recent experiments of Hajduk etal .11 The least-stable or most-unstable fluctuat ionm odes ar e us ed t o i nfer i nfor m at i on on t he ki net i cpathway of the order -order phase t ransit ions an d th eepitaxial relations between t he different ordered phases,

leading to results which are in good agreement withexper i m ent s and t he r ecent ki net i c s t udy of Q i andWang21 us ing a t i m e-dependent G inzbur g-L andautheory.

T he s t r uct ur es of t he or der ed phas es of diblockcopolymers are mostly identified from the scatteringintensities, which are obtained from small-angle X-rayor neutron scattering. The scattering function of thehomogeneous phase was calculated for the first time byLeibler.4 However, the theoretical calculation of thescattering functions for ordered diblock copolymer phaseshas been lacking. While the calculation of the scatt eringfunctions for the lamellar phase has been given in anearlier paper,17 the scattering functions of all ordered

phases are given in th e present paper.

The pa per is organized a s follows. A brief accoun t ofth e th eory for d iblock copolymer melts is given in sectionI I . T he t heor y of ani sot r opic fl uct uat i ons and i t sreciprocal space formulation is presented in section III.The r esults on the three classical phases ( lamellar,cylindrical, and spherical) are given in section IV, andthose on t he t wo recently ident ified complex pha ses, i.e.,the H PL an d th e DG phases, are presented in sectionsV and VI, respectively. Finally, the last section isdevoted t o concluding remar ks. Some of the deta ils ofthe theory and the numerical techniques ar e given inAppendices A a nd B.

I I. T h e o r y o f D i b l o c k C o p o l y m e r M e l t s

For t he st udy of the diblock copolymer phase behavior, weadopt the stan dard model in which the har d-core monomer -monomer interaction is modeled by a n incompressibilityconstraint and the polymer chains are assumed to be fullyflexible. We consider N polymer chains in a volume V. Asingle polymer chain consists of an A-block and a B-block,each consisting offRZ statistical segments, where fR ) f or 1 -f for R ) A or B, respectively. We will use t he convention inw h ich al l l en gt h s are s cal ed b y t h e G au s si an rad i u s o f gyration, R g ) b(Z/6)1/2, where b is the monomer statisticalK u h n l en g t h (w e con s id er t h e cas e w h ere t h e A- an d B-mon omers h a v e t h e s ame s i ze) an d t h e ch ai n arc l en gt h i s

scaled by the chain degree of polymerization Z. T h e localvolume fraction is used as a collective variable,

where F0 i s t h e n u m be r d en s it y of t h e m e lt a n d {riR()}

describes the cha in conforma tions. The par tition function ofthe st andard model is given by

where the -function accounts for the incompressibility con-stra int. The connectivity of the m onomers with in each chainis ensured through the functional P, which, for fully flexiblediblock copolymers, is given by

The functional integration over the chain conformat ions isreplaced by a functional integration over the local volumefractions and two au xiliary fields, A an d B22

where the free energy functional F is given by

Qc({R}) is the partition function of an independent single

R(r) )1

F0i)1

N

0

fRd (r - r i

R()) (1)

Z )i)1

N

D r i()P ({r i()})r

[1 -R

R(r)]

exp

[-

F0R g3

Z dr ZA(r) B(r)](2)

P ({r()}) )i)1

N

[r iA(fA) - r i

B(fB)]R

exp [- 32b

2

0

fRd

(dr i

R()

d )2

](3)

Z )R

{DRDR}r

[R

R(r) - 1] exp[-F ({R},{R})]

(4)

F ({R},{R}) )F0R g

3

Z{dr [ZA(r) B(r) -

R

R(r) R(r)] - V ln Qc({R})} (5)

Macromolecules, V ol. 30, N o. 11, 1997 Ordered Phases in Diblock Copolymer Melts 3243


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chain subject to the field R. It can be written as a pr oduct oftwo propagators

where r 1, r3, an d r 2 are the positions of the free ends of theA-block, th e B-block, an d the junction, respectively. Thepropagator QR(r,t|r) sat isfies th e following modified diffusionequation23

with the boundary condition QR(r,0|r ) ) (r - r).In principle, a ll therm odynamic informat ion can be ex-

tracted from the partition function Z. However, methods ofeval u at in g t h e fu n ct i on al i n t eg ral exact ly h av e n ot b eendeveloped. Therefore, a va riety of appr oximat ions ha ve to beadopted. In par ticular, evaluat ing the partition function (eq4) by the sa ddle-point appr oximation leads to the widely usedself-consistent mean-field theory. The mean-field theorydescribes the int eracting chains by an en semble of independen tchains in the self-consistent mean fields created by all otherchains. Mathem atically, the saddle-point m ethod correspondsto th e extremization of the free energy functional (eq 5),

resulting in t he following mean -field equations

where t he field (r) is a local Lagrange multiplier which isi n t rod u ced t o en force t h e i n comp res si bi li t y con d i t ion ,

RR(0)(r) ) 1. The mean-field propagator, QR

(0)(r,t|r ), satisfiesthe modified diffusion equation

We notice that the parameters entering the mean-field theoryare combinat ions ofZ a n d f. Therefore, the use of these twoparameters for describing the phase diagram is justified withinthe m ean-field approxima tion. Because of the complexity ofthe theory even within mean-field approximation, a varietyof further approximations have been adopted to solve themean -field equations. To obtain a n exact solution, the mean-field theory is reformulated in reciprocal space in order toexploit t he full symmetries of the ordered str uctures.24 In thisreciprocal space method, the mean-field quantities are ex-p an d ed i n t erms of p l an e w av es . T h e mean -fi el d p h as ediagram is then obtained by comparing t he free energies ofall stru ctures. Whereas the self-consistent equations ensur e

that the free energy is extremized, they do not necessarilyensur e its minimization. It is therefore possible that a givenmean -field solution corresponds to a saddle point.

In order to investigate the stability of the ordered phases,we ha ve to include t he effect of fluctuat ions in t he t heory. Inthis paper we study the effect of composition fluctuations atthe Gaussian level. Let R(r) and R(r) be the local fluctua-tions ofR(r) a n d R(r) away from their mean-field values,

R(0)

(r) and R(0)

(r), respectively. After carrying out the expan -s ion i n t erms of t h e fl u ct u at i n g f i el ds , t h e free en ergyfunctional (eq 5) is written as 18

where F (0) is the mean-field free energy, F (1) ) 0 because t hemean -field solution extrem izes the free energy functional given

by eq 5, an d t he second-order Gau ssian correction is given by

Higher order corrections are explicitly functionals ofR only.

The two-point cumulants in eq 12, CR(r ,r ), are completelydetermined by the m ean- field propagators and are given by

an d

with CBB(r ,r ) an d CBA(r ,r) obtained by exchanging A and Bin eqs 13 and 14, respectively. If we truncate t he fluctua tionsat the Gaussian level, the partition function of the melt isapproximated by

Because of the incompressibility constraint, the fluctuationsin the compositions ar e not independent . We thus introducea new collective variable (r) ) A(r) - B(r) to describethe composition fluctuat ions. It is desirable t o obtain an

effective free energy functional in t erms of only. This canbe achieved by performin g the functional in tegra l over A andB in eq 15, leading to t he effective free energy functional

where t he t wo-point den sity-density correlation function canbe written in the following matrix form

where I is the identity matrix and C ) C - 1-12. Th eoperators C, , 1, an d 2 ar e combinations of the two-point

cumulants, CR :

It should be noted that, in general, the integration of Rcannot be carried out exactly, and a random-phase approxima-tion (RPA) is usua lly adopted. This corresponds to a m ean-field approximation for the A a n d B, while keeping

fixed. At the Gaussian level, such an a pproximat ion is exact.

Qc({R}) )1

Vdr1 dr 2 dr3 QA(r 1,fA|r 2) QB(r2,fB|r3) (6 )

tQR(r,t|r) )

2QR(r ,t|r ) - R(r) QR(r,t|r) (7)

R(0)(r) ) Z

(0)(r) + (r) (8)

R(0)(r) )

1

Qc(0)0

fRdtdr 1 dr2 dr3 QR(0)(r1,t|r)

QR(0)(r ,fR-t|r 2) Q

(0)(r 2,f|r3) (9)

tQR(0)(r,t|r) ) 2QR(0)(r ,t|r ) - R(0)(r) QR(0)(r ,t|r ) (1 0)

F ) F (0) + F (1) + F (2) ... (11)

F(2)({R},{R}) )

F0R g3

Z {dr [Z A(r) B(r) -

R

R(r) R(r)] -1

2Rdr dr CR(r,r) R(r) (r)}

(12)

CAA(r ,r) )1

Qc(0)0

fAdt

0

tdtdr1 dr 2 dr3 [QA(0)(r1,fA-t|r)

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

(0)(r,t|r 2) QB(0)(r 2,fB|r 3) + QA

(0)(r1,fA-t|r )

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

(0)(r,t|r 2) QB(0)(r 2,fB|r 3)] (13)

CAB(r ,r) )1

Qc(0)0

fAdt

0

fBdtdr1 dr 2 dr3 QA(0)(r1,fA-t|r)

QA

(0)(r,t|r2) Q

B

(0)(r2,t|r ) Q

B

(0)(r,fB

-t|r3) (14)

Z exp(-F (0))D R D Rr

(R

R(r))

exp(-F (2)({R},{R})) (15)

F(2)({}) )

1

2dr dr [CRP A]-1(r ,r) (r) (r) (1 6)

CRP A )

F0R g3

Z [I -Z

2C]

-1C (17)

C ) CAA - CAB - CBA + CBB

1 ) CAA + CAB - CBA - CBB

2 ) CAA - CAB + CBA - CBB

) CAA + CAB + CBA + CBB (18)

3244 Laradji et al. Macrom olecules, V ol. 30, N o. 11, 1997


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Integrating out the composition fluctuat ions leads to thepart ition function

It is important to emphasize that the mean-field solution andthe two-point cumulants are functions of the two parameters

Z an d f only. The inclusion of fluctuat ions int roduces a third

p aramet er, Zh ) F02R g

6/Z2 ) (F02

b6/63)Z, which quantifies fluc-tua tion effects. This is in accordance with previous st udies.13

The density-density correlation function, CRP A(r ,r), can beused to obtain informat ion about several important thermo-dynamic properties: (1) The mean -field solution for a givenphase is linearly stable when all the eigenvalues of CRP A a repositive. The bounda ry in the (Z,f) plane where the smallesteigenvalue becomes zero defines the spinodal l ine for thatphase. (2) The experimenta lly observed scatt ering intensitiescorrespond to the F ourier tra nsform ofCRP A(r,r). Our theoryaccounts for an isotropic fluctuations a nd t herefore en ables usto calculate the scattering functions of the periodic orderedphases. (3) The elastic moduli of the ordered phases can beextracted from the scattering functions, a s demonstrat edearlier by Yeung et al.17

I II . Re c i p r o c a l S p a c e T h e o r y o f A n i s o t r o p i cF l u c t u a t i o n s

The theory developed above is formulated in realspace. However, calculating the density-density cor-relation within this formalism can be carried out onlyfor very simple geometries, such a s th e lamellar ph ase.17

In order t o apply the th eory to th e more complex phases,novel methods other tha n t he direct real space calcula-t i on have t o be devel oped. I n par t icul ar , t he ful lsymmetries of the ordered phases can be exploited tocast t he th eory into a tra ctable form. The key observa-tion in this problem is that the mean-field solution isperiodic, and the modified diffusion equation for thechain pr opagator is equivalent to the Sh rodinger equ a-t i on of a n elect r on i n a cr yst al li ne s oli d w it h t he

Hamiltonian HR ) -2 + R(0)(r). It is well-known from

solid-stat e physics th at th e electronic energy in a crysta lforms bands according to Blochs theorem;20 i.e., t h eeigenmodes ofHR can be labeled by the band index, n ,and a r educed wave vector, k , within the first Brillouinzone. The n ormalized eigenfunctions ofHR are Blochfunctions of the form

w her e t he s et {G} constitu tes the reciprocal lattice

vectors. The coefficient s u n kR (G) are solutions of the

following set of linear equations

where R(G) are th e Fourier components of the au xiliary

mean fields; i.e., R(0)(r) ) GR(G) exp(iGr). I n pr in-

ciple, this eigenvalue problem has an infinite dimension,but since th e coefficient s R(G) are very small for largereciprocal wa ve vectors, th e coefficients can be t ru ncat edat a finite nu mber N, leading to an N N eigenvalueproblem. The solution of this pr oblem can be carr iedout using standa rd l inear algebra software packages.A gener ic feat ur e of t he eigenvalues of HR is t h epresence of band gaps at the zone boundar ies. Figure1 shows the low-index eigenvalue ba nds for the la mellar

and homogeneous phases, at Z ) 11 and f ) 0.5. It is

obvious from this figure that the band gaps appear onlybecause of a finite periodic R.

Because the eigenfunctions of HR form a completeorthogonal basis, they can be used for t he expan sion ofthe fluctuat ing fields. The explicit expressions of thet w o-poi nt cum ul ant s ar e found i n Appendix A. I n

particular, Cn ,n R (k ,k ) becomes diagonal in k -space du e

to the constr aint of the wave vectors k within the firstBrillouin zone. These two-point cumulant s become N

N matrices at each wave vector k . Since CRP A is acombination of these two-point cumulant s, it becomesdiagonal in k . Thus f inding the eigenvalues of CRP A

becomes a n N N eigenvalue pr oblem at each redu cedwave vector k .

Our study of the stabili ty of an ordered phase pro-ceeds as follows:

(1) The mean-field solution at a certain point of thephase diagram, (Z, f), is obtained by solving the self-consistent equations, eqs 8-10, in reciprocal space. Inpart icular, th e mean -field pha se diagram which we havecalculated is in complete a greement with t he ea lier workof M at s en and Schi ck.24 Details of the mean-fieldcalculations ar e found in Appendix B. The mea n-field

calculations are used to obtain the phase diagram atweak segregations and to provide the Fourier compo-nent s R(G) needed for the fluctuations calculation.

(2) The mean-field Fourier components of R(G) arethen used to constru ct th e Ham iltonian HR. The eigen-values n k and the components of their corresponding

eigenvectors u n kR (G) are then calculated by solving the

set of linear equations, eq 21, at the wave vectors, k ,which are constrained within the first Brillouin zone.We exploit the symmetries of the first Brillouin zoneby considering only a portion of the Bril louin zone,which is then discretized into a fine mesh.

(3) The eigenvalues, n kR , and eigenvectors, u n k

R , a r ethen used to construct t he matrix elements of Cn ,n (k )

using t heir expressions, eqs A1-A4, in Appendix A.(4) The eigen values ofI - (Z/2)C are calculat ed usingeqs 17 and 18.

(5) The stability of the structure is investigated byexamining the eigenvalues of I - (Z/2)C (the eigen-values of are a lways positive). If all the eigenvaluesare positive, the ph ase is linearly stable or meta stable.However, if at least one eigenvalue is negative, thephase is linearly unstable. The boundary between themetastable and the unstable regions defines the spin-odal line. For illustra tion, Figure 2 displays the eigen-values, n k of I - (Z/2)C(k ) for the lamellar pha se atZ ) 10.8 a nd f ) 0.46 along the path M on the qz-axis (the lamellae are perpendicular to the x-axis) an dal ong t h e pat h M XY, i.e. along the edge of the f irst

Brillouin zone. The importan t observation wh ich can

F i g u r e 1 . Eigenvalues n ,kzA of the chain Hamiltonian HA forthe lamellar phase (thick lines) and the homogeneous phaseA(r) ) 0 (thin lines) at Z ) 11 and f ) 0.5. Notice that thebroken symmetr y of the lamellar phase leads t o the forma tionof band gaps which are n ot present for t he homogeneous case.

Z exp(-F (0)){de t[(F0R g3

2Z)

3

(I - Z2 C)]}-1/2

(19)

n kR (r) )

1

V

G

u n kR (G)e i(k +G)r (20)

[(k + G)2 - n kR ]u n k

R (G) +G

R(G - G)u n kR (G) ) 0 (21)



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be m ade f r om t hi s f i gur e i s t hat t he m os t uns t abl eeigenm ode occurs at th e first Brillouin zone bound ar ies,in a ccordance with t he earlier calculations of Shi et al.using perturbation theory.18

W e now t u r n t o t h e cal cul at ion of t he s cat t er ingfunctions within our theoretical framework. The ex-

perimentally observed scattering intensity is propor-tional to the Fourier transform of the density -densitycorrelation function,

where (r) ) A(r) - B(r). For an ordered phase,

Inserting this form into eq 22, the scattering intensitybecomes

where S (q) is the scatter ing function due to compositionfluctuations and is given by

Therefore, th e scatt ering int ensity is composed of twocontributions: The f irst one corresponds to a set of Bragg peaks located at the reciprocal wave vectors,whose amplitudes are completely determined by themean -field solution. The second contr ibution is due t ocomposition fluctuations and has to be calculated froma theory of fluctuations. The pa ir-correlation function

of t he densi t y fl uct uat i ons i s t he R PA cor r elat i onfunction which can be expanded in terms of the Blocheigenfunctions

Using th is form, t he scatt ering function is given by

I V. S t a b i l i t y , S c a t t e r i n g F u n c t i o n s , a n dF l u c t u a t i o n M o d e s

In this section we present results on the stability, the

scattering functions, and the fluctuation modes of the

ordered pha ses in diblock copolymer m elts. We con-centrate on the weak segregation region, where fluctua-tions are most significant; most of our analyses areconfined to the region defined by Z < 11.2. Withinmean-field theory, the lamellar, cylindrical, and spheri-cal phases are stable in this region. On the other han d,the HPL phase is found to be metastable within thisregion.24 Our results on the more complex case corre-sponding to the double-gyroid phase, which appearsabove a tr iple point located at Z ) 11.14,24 will bepresented in section VI.

The results of the stability analysis of the lamellar,cylindrical, spherical, and HPL structures are sum-marized in the generalized phase diagram of Figure 3.In this figure, the mean -field one-phase r egions of thelamellar, cylindrical, and spherical phases are indicatedby the shaded light-gray, dark-gray, and medium-gray

area s, respectively. The spinodal lines of these ph asesar e shown by lines with t heir corresponding gray levels.The HPL structure is metastable within i ts spinodalline, shown by the broken line. The most significantas pect of t hi s phas e di agr am i s t hat t he one- phas eregions of the so-called classical phases are encapsulatedwithin their spinodals, indicating the robustness of these t hree ph ases which are easily observed in experi-ments. On the other ha nd, the HPL phase is metastablein the center of the lamellar one-phase region and isunstable in the rest of the phase diagram. Therefore,the HPL phase is thermodynamically unstable along thelamellar -cylindrical pha se boundar y, where it ha s beenident ified in several experimen ts. We will ar gue below

that the experimentally observed modulated layeredstates are simply an intermediate stru cture due to theinfinitely degenera te fluctuat ion modes of the lam ellarphase. Another interesting featur e of this phase dia-gram is that the cylindrical and spherical phases havetwo symmetrically un related spinodal lines, whereas th elamellar a nd the HPL phases ha ve two symmetricallyr elat ed s pi nodal l ines. T his i s due t o t he fact t hemajority and the minority domains of the lamellar andHPL phases are topologically equivalent (the HPLstru cture being bicontinuous). In th e rest of this section,we will present details of the scattering functions andthe fluctuation modes leading to the instability of thephases, which a re used t o infer the kinet ic path ways ofthe order -order transitions and the epitaxial relations

between the various structures.

F i g u r e 2 . Plot of the eigenvalues of the operator CRP A, n k,

for the lamellar phase in the first Brillouin zone at Z ) 10.8an d f ) 0.46. The paths M a n d M X Y are shown by the insetof the figure. The arrow indicates where t he smallest eigen-value occurs. Notice that the instability occurs at the edge ofthe Brillouin zone (kz ) /D and a finite kxy (arrow)). At thisp oi n t o f t h e p h as e d i agram t h e l amel lar p h as e i s al mos tunstable. As is discussed in the text, this corresponds to ani n st ab il it y of t h e l amell ar p h as e ag ain s t an i n t ermed iat emodulated-layered state.

I(q) )1

Vdr1 dr 2 e-q(r1-r2)(r 1)(r2) (22)

(r) )G

(0)(G)eiGr + (r) (23)

I(q) )

G|

(0)(G)|2q ,G + S (q) (24)

S (q) )1

Vdr 1 dr2 e-q(r1-r 2)(r1) (r2) (25)

(r) (r) ) CRP A(r ,r) )k

n ,n

Cnn RP A(k ) n k

A*(r) n kA (r )

(26)

S (q) )G

nn

Cnn RP A(q - G)u n ,q-G

A* (G) u n ,q-GA (G) (27)

F i g u r e 3 . Pha se diagram of diblock copolymer m elts an d th e

spinodal lines. The sh aded r egions correspond to the one-phaseregions of the lam ellar (light gray), cylindr ical (dar k gra y), andspherical (medium gray) phases. The spinodal l ines of theordered pha ses ar e shown by their corr esponding gray levels.The spinodal line of the hexagonally perforated lamellar phaseis indicat ed by the broken line. The white r egion correspondsto th e homogeneous phase. The outer spinodal l ine of th espherical phase is not obtained due to numerical difficultiesin finding unstable mean-field solutions of the sphericalstructure within the homogeneous one-phase region. Thevarious symbols correspond to the points of the phase diagramat which the scattering functions and the real space profilesof the various phases are calculated.



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Before the detailed presentat ion of t he fluctuat ionmodes, it is appr opriate to briefly discuss their n atu re.For h omogeneous phases the fluctuat ion modes a resimply plane waves. On the other ha nd, for an orderedphase, the fluctuat ion modes ar e complex and can bewritten as l inear combinations of Bloch waves. Wedefine the eigenvectors ofCRP A by

with 1 e i e N. The eigenmodes are ther efore indexedby the reduced wave vector, k , and the band label, n .The effect of these fluctuation modes can be revealedin Fourier space th rough th e scattering functions, or inr eal s pace t hr ough t he s t r uct ur e of t he i nt er facesbetween the A-rich and the B-rich domains. In r ealspace, the effect of the least s table or most unstablemode can be obtained by perturbing the mean-fieldsolution with the corresponding eigenfunction. Thepert ur bed den sity profile of the A-blocks can be writ tenas

where a is the amplitude of the eigenmode with thesmallest eigenvalue (indicated with t he su perscript 1).Depending on whether the ordered phase is s table orunstable, this mode is termed the least stable or mostun stable eigenmode. The contour plot defined by A(r)) 0.5 defines th e inter faces in th e melt, a nd the effectof th e leading fluctuat ing m ode can be observed bytuning a.

A . S c a t t e r i n g F u n c t i o n s a n d F lu c t u a t i o n M o de so f t h e L a m e l l a r P h a s e . The scattering functions ofthe lamellar phase prior to i ts spinodal l ine (at Z )

10.8, f ) 0.462) is shown in Figure 4 for the planes qx )

0 a n d qz ) 0 (the lamellar planes are parallel to thex-y plane). At t hi s poi nt of t he phas e diagr am , t helamellar pha se is in fact metast able, an d the equilibriumsta ble pha se is cylindr ical. In agreemen t with previouswork,17 the scattering function has two strong Braggpeaks at (qz ) (k0, qx ) 0), where k0 ) 2/D a n d D isthe lamellar periodicity, and two higher order Braggpeaks located at (qz ) (2k0, qx ) 0). These strong peakssignal th e relatively strong lamellar ordering. We alsoobserved four additional peaks at (qz ) (k0/2, qx (3k0 /2). As has been shown before17 and w i l l bedemonstrated below, these four peaks correspond to in-plane fluctua tions with hexagonal ordering. The cal-culated scatt ering function in th e qx-qz plane is in good

agreement with the experimentally observed one.5

Inpart icular, th e ma in experimental featu res correspond-ing to the appeara nce of the four weak peaks du e to thein-plane fluctuations, their hexagonal ordering, and the10% difference in qz and qx are r eproduced by the th eory.A real space represent ation of the least-stable fluctua-tion mode is given in Figure 5, where the interfacesbetween the A-rich and the B-rich microdomains areshown for the density profiles corresponding to differenta values (see eq 29). These fluctua tion m odes lead toundulations of the lamellae in a such a way that theminority-rich domains pinch and bulge for small valuesofa. For lar ger values ofa, the minority lamellae breakup to form elongated cylinders with their principal axesarra nged in a hexagonal latt ice with th e same periodic-

ity as that of the equilibrium cylindrical phase at this

F i g u r e 4 . Scattering function of the lamellar phase at Z )10.8, f ) 0.462 prior to its spinodal line (shown by the symbolO in Figure 3). The t wo cuts, corresponding to the qx-qz planean d t h e qx-qy plane, are shown in the top part a nd the bottompart , respectively. The two left gr aphs show the surface plotsof the structure factor, and the two right graphs correspondto their contour plots. Notice th at the qy direction in the upperpart of this figure could be any in-plane direction.

i(k ) ) (1i(k ), 2

i(k ), ..., N

i(k )) (28)

A(r) ) A(0)

(r) + an

n1

(k ) n kA

(r) (29)

F i g u r e 5 . Three-dimensional contour plots of the lamellarphase showing the effects of the least stable mode on thelamellae at Z ) 10.8, f ) 0.462 (shown by the symbol O inFigure 3). The contour plots are defined by A(r) ) A

(0)(r) +an(1)(k ) n k

A(r) ) 0.5, where a is the amplitude of the least-

stable fluctuat ion mode: (a) a ) 0; (b) a ) 0.05; (c) a ) 0.15.Notice that, for a relatively large amplitude, the fluctuationmode leads to hexagonally-packed cylinders whose axes areparallel to the lamellae.



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point of th e phase diagra m. Therefore, this observationindicates that these fluctuation modes are responsiblefor driving the lamellar phase to the cylindrical phasewith the cylinders oriented parallel to the lamellae. Theepitaxial relation between the lamellar and the cylindri-cal s tructures which emerges from the theory is inagreement with the recent time-dependent Ginzburg-Landau simulation of Qi and Wang21 and experiments.8

We now discuss the nature of the in-plane fluctuationmodes in the lamellar phase. Becau se the mean -fieldprofile is isotropic in the x-y plane, the scatteringfunction is a lso isotropic in t he qx-qy plane as shown

b y t h e r in g s t ru ct u r e in F igu r e 4 . T h er e for e t h efluctuation modes of the lamellar phase are infinitelydegenerate in the x-y plane. If only one direction int he qx-qy plane is selected, as in Figure 5, this fluctua-tion m ode leads to th e formation of cylinders. In a realsituat ion, however, it is most likely that man y directionswill be simulta neously selected. Figure 6 shows tha twhen m ore tha n one direction is selected, the fluctuatingmodes lead to a modulated layered structur e. Since theequilibrium st ate corresponds t o a cylindrical pha se, thismodulated layered structure will eventually evolve intocylinders. However, due to the convoluted str ucture ofthe modulat ed layers, the t ran sformation from this stat eto the cylindrical structure will involve the transportof a l ar ge am ount of m at er i al. C ons equent ly, t he

kinetics of this process is expected to be very slow. It

is therefore plausible that when the lamellar phasebecomes unst able, the polymer m elt will be trapped in toan intermediate modulated layered state which mightbe identified as an equilibrium phase in an experiment.This scenario is in a greement with the recent experi-ment of Hajduk et al.11 which shows that the m odulated

layered structure is a kinetically trapped stat e. Like-wise, in the recent work of Qi and Wang, 21 the modu-lated layered state is also found to be a kineticallytra pped state. It should be noted tha t the origin of theintermediat e stat e is the convoluted perforat ed layereds t r uct ur e. For t he cas e of t he l am ell ar phas e, t hi sconvoluted stru cture is a consequence of the infinitelydegenerate fluctuation modes. However, it is possibleto have a convoluted structure without infinite degen-eracy, as will be shown later for the spherical phase.

B . S c a t t e r i n g F u n c t i o n s a n d Fl u c t u a t i o n Mo d e sof the Cylindrical Phase . Due t o the two-dimensionalnat ure of the cylindrical phase, th e scattering functionis more complex than that for t he lamellar case. In

order to be specific, we define the z-axis a long thecylinders, the y-axis along the [01] direction of t hehexagonal latt ice, an d the x-axis perpendicular to the

y-axis. We choose to plot th e scat terin g fun ctions in th eform of three perpendicular cuts in q-space correspond-ing to the planes defined by qz ) 0, qy ) 0, and qx ) 0.B ecaus e t he cyli ndr i cal phas e has i nner and out erspinodal lines, we present t he results on t he scatteringfunctions and fluctuation modes of this phase close tot hes e t w o s pi nodal l ines. Fi gur es 7 and 8 s how t hescattering functions of the cylindrical phase prior to itsouter and inner spinodal l ines at (Z ) 10.9, f ) 0.43)a n d (Z ) 10.9, f ) 0.476), respectively. The genericfeatures of the cylindrical scattering functions aredescribed by th e th ree following observations: (1) In th e

qz ) 0 plane, there a re six strong hexagonally arra nged

F i g u r e 6 . Three-dimensional contour plots of the lamellarphase showing th e effect of the degenera cy of the least-stablemodes, with a ) 0.05, a t Z ) 10.8, f ) 0.462 (shown by t hesymbol O in F igure 3): (a) selection of a single direction fort h e fl u ct u at i on mod e (as i n Fi gu re 4 ); (b ) s i mu l t an eou sselection of two directions for the fluctuation mode; (c)simultaneous selection of three directions of the fluctuationmode. We emphasize that this figure shows a very special casein which the three selected directions are threefold symmetric.Notice that the selection of more than one mode leads to amodulated-layered structure, reminiscent of the experimen-tally observed hexagonally perforated lamellar phase.

F i g u r e 7 . Scattering function of the cylindrical phase at Z) 10.9, f ) 0.43 prior to the outer spinodal line of this phase(shown by the symbol 4 in Figure 3). The three cuts corre-sponding to the qx-qy, qy-qz, an d qx-qz planes are shown inthe top, middle, and bottom graphs.



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Bragg peaks, corresponding to the hexagonal orderingof th e cylinders. The six weaker seconda ry Bragg peaksare a lso visible. (2) In th e qx ) 0 plane, there are twostrong Bragg peaks located at qz ) 0, qy ) (2k0/3,where k0 ) 2/D a n d D i s t he per iodici t y of t hehexagonal lattice. These two peaks indicate the layered

ordering of the cylindrical structure along the y-axis.(3) In the qy ) 0 plane, there are two weak secondaryBragg peaks located at qz ) 0, qx ) (2k0. T he l eas t -stable fluctuation modes correspond to four weak scat-tering peaks which are observed in the qx ) 0 and qy )0 planes. The exact locations of fluctuat ion peaksdepend on the values of Z a n d f, i m plying t h at t heleast-stable fluctuation mode near the inner spinodalline mu st be different from tha t n ear t he outer spinodalline.

Near t he outer spinodal line (Z ) 10.9, f ) 0.43), th efour fluctua tion pea ks a re locat ed at qx (k0/3, qz (22k0/3. A real space representat ion of these fluc-tua tion modes is given in F igure 9 for different values

ofa

. For small values ofa

, the fluctuation modes leadto the bulging and pinching of the cylinders. Largervalues of a lead to a breakup of the cylinders to formspheres which are centered on a body-centered-cubiclattice with a periodicity which is identical to that ofthe equilibrium spherical phase at this point of thephas e di agr am . An epit axial r elat i on bet w een t hecylindrical and spherical phases therefore emergesna tur ally from the fluctua tion modes. The cylinders ar eoriented along th e [111] direction of the bcc latt ice. Thefluctua tion peaks of th e cylindrical ph ase, together withthe six primary Bragg peaks in the qx-qy plane, cor-respond to the 12 primary Bragg peaks of the body-centered-cubic lattice.

Close to the inner spinodal l ine of th e cylindrical

phase (at Z ) 10.9, f ) 0.476), the scattering function

in the qy-qz plane exhibits extended wings close to t hetwo Bragg peaks. A real space representat ion of these

fluctuation modes is given in Figure 10 for differentvalues of a. I n par t icul ar , w e not ice t hat , f or s m allvalues of a, the fluctuation modes lead to a flatteningof the cylinders along th e [01] direction. For largervalues of a, these ellipsoidal cylinders merge to formmodulated layers, with a periodicity which is ident icalto that of the equilibrium lamellar phase at this pointof the pha se diagram. The epitaxial relation betweenthe lamellar an d t he cylindrical pha ses, emerging fromthe least-stable eigenmode of the cylindrical phase isidentical to the one we obtained from the least-stablemodes of the lamellar phase.

Because the least-stable f luctuation modes of t hecylindrical phase are only threefold degenerate, we

expect that the kinetics of the cylindrical-lamellar andcylindrical-spherical transitions proceed directly with-out a n intermediate state. The absence of an interme-diate state in the kinetics of the cylindrical-to-sphericaltransit ion has also been found in a recent numericalstudy by Qi and Wang.21 The a bsence of an inter medi-ate state during the cylindrical-to-lamellar transitionhas been reported in the experiments of Sakurai et al.9,10

In th e previous subsection, we found th at the lam ellar-to-cylindrical transition proceeds through a long-livedi nt er m ediat e m odulat ed-l ayer ed s t at e. O ur r esul t stherefore imply that the lamellar-to-cylindrical andcylindrical-to-lamellar transitions are not reversible intheir kinetic pathways.

C . S c a t t e r i n g F u n c t i o n s a n d F l u c t u a t io n M o d e s

o f t h e S p h e r i c a l P h a s e . For the spherical phase, we

F i g u r e 8 . Scattering function of the cylindrical phase at Z) 10.9, f ) 0.476 prior to the inner spinodal line of this phase(shown by the symbol 3 in Figure 3). The thr ee cuts shownare the same as those in Figure 8.

F i g u r e 9 . Three-dimensional contour plots of the h exagonalphase, at Z ) 10.9, f ) 0.43 (shown by t he sym bol4 in Figure3), showing the effects of the least-stable mode on the orderedcylinders. The contour plots have the same definition as inFigure 5: (a) a ) 0, i.e. the mean-field solution; (b) a ) 0.02;(c) a ) 0.075. Notice that the effect of a relatively largeamplitude leads to the formation of spheres which are ar-ranged in a body-centered-cubic lattice. The [111] direction ofthe cubic unit cell of this lat tice is along the cylindrical a xis.



9/14

use th e sta nda rd cubic unit cell to describe its geometr y.We choose to plot the scattering function along threecut s i n q-s pace: (i ) t he plane defi ned by qz ) 0corresponding to t he scatt ering from the (001) plan es;(ii) the plane defined by qx + qy + qz ) 0 correspondingto scattering from the (111) planes; ( ii i) th e planedefined by qx ) qy corresponding to t he scattering fromthe (110) planes. The scattering function of the spheri-cal phase close to its inner spinodal line at (Z ) 10.8,

f ) 0.448) i s s hown i n Fi gur e 11. T he s cat t er ingfunction in the qz ) 0 plane shows four strong Braggpeaks and several higher-order Bragg peaks signalingthe fourfold symmetry of this phase along the [100] (orequivalent) direction. On the other h and, the scatteringfunction exhibits six strong Bragg peaks in the planedefined by qx + qy + qz ) 0. These strong peaks signalthe t hreefold symmetry of the spherical stru cture in t he

[111] direction. In t he plan e defined by qx ) qy, thereare two strong Bragg pea ks corresponding to th e twofoldsymmetry a long th e [110] directions. The least-stablefluctuation modes of the spherical phase close to i tsspinodal line are manifested by the four wings in theplane qx ) qy. A r eal s pace r epr es ent at i on of t hes efl uct uat i on m odes i s s how n i n Fi gur e 12 for t hr eedifferent values of a. F or s m a ll v a lu e s of a, t h efluctuation modes lead to th e flatt ening of the spheresalong the [110] direction. Increasing a l eads t o t hem er gi ng of t hes e dis t or t ed s pher es , r es ult i ng i n aperforat ed layered stru cture perpen dicular to the [110]direction. Since the equilibrium pha se at th is point ofthe phase diagram corresponds to the cylindrical phase,this convoluted structure will appear as an intermediate

s t at e dur i ng t he ki net i cs of t he t r ans i t i on f r om t he

spherical to th e cylindrical phase. Therefore, the iden-tification of the least-stable modes of the spherical phase

close to its inner spinodal lines leads to the conclusiont hat t he cyli ndr ical-s p h er ica l a n d t h e s p h er i ca l-cylindr ical tra nsitions ar e not reversible. We found th atwhereas the cylindrical-spherical tra nsition proceedsdirectly, the spherical-cylindrical transition proceedsthrough an intermediate modulated layered state. Toour best knowledge, experimental st udies of the kinet icsof the spherical-cylindrical transition are not yet avail-able.

V. S c a t t e ri n g F u n c t i o n s o f t h e H P L P h a s e

From the stabili ty analysis, the H PL pha se is meta-stable in the middle of the lamellar phase. We haveargued in section IV that the experimentally observed

modulated layered phases are long-lived intermediatestru ctures a s a consequence of the infinite degeneracyof the least-stable f luctuation modes of the lamellarphase. Despite the instability of the HP L phase in mostof the phase diagram, we present in this section thecalculated scattering functions of this phase for com-pleteness. We choose the coordinat e system in such aw ay t hat t he z-axis is perpendicular t o the layers. I tshould be noted tha t, due t o the h exagonal symmetryof this phase, the two in-plane directions along the

x-axis and th e y-axis are not equivalent. The scatt eringfunction of the HPL pha se, prior t o its spinodal line (atZ ) 10.8, f ) 0.476), is plotted in Figure 13 for thethree cuts defined by qz ) 0, qx ) 0, and qy ) 0 . T h ethreefold symmetry of the perforated structure is evi-

dent in the scattering function in the qz ) 0 plane. We

Figure 10. Three-dimensional contour plots of the hexagonalp h as e, a t Z ) 10.9, f ) 0.476 (shown by the symbol 3 inFigure 3), showing the effects of the least-stable m ode on theordered cylinders: (a) a ) 0, i.e. the mean-field solution; (b) a) 0.06; (c) a ) 0.08. Notice that the effect of a relatively largeamp li t u de l ead s t o t h e format i on of mod u lat ed l amel laeparallel to the cylinders axes.

F i g u r e 1 1 . Scattering function of the spherical phase at Z) 10.8, f ) 0.448 prior to the inn er spinodal line of this ph ase(shown by the symbol 0 in Figure 3). The top graphs of thisfigure correspond t o the cut along the qx-qy plane. The m iddlegraphs correspond to a cut along the plane defined by qx + qy+ qz ) 0; qx a n d qy are defined by qx ) (qy - qx)/2 an d qy )(qx + qy - 2qz)/6. The bottom graph corresponds to a cutalong the plane defined by qx ) qy; qx is defined by qx )2qx.



10/14

should notice tha t th ese are not Bragg peaks. Six weaksecondary peaks are clear in this plane. The stacking

of the h oles is revealed by th e four strong Bragg pea ksin the scattering function in the qx ) 0 p la n e . T h elamellar ordering of the system is best reflected in th etwo strong Bragg peaks in the scattering function in theqy ) 0 plane. The least-sta ble modes correspond to th efour weak scattering peaks in the qy ) 0 plan e. It is ofinterest to make the comparison between the calculatedscattering function of the HPL phase and the corre-sponding experiments. Experimenta lly, for shea r ori-ent ed s am ples l ocat ed i n t he phas e di agr am at t helamellar-cylindrical ph ase boun dary,5,25 the measuredscattering functions, presumably from the HML/HPLstru ctures, show sixfold symm etric scatt ering peak s int he qz ) 0 pl ane. For t he dir ect i ons par al lel t o t he

layers, two strong Bragg peaks a nd four satellite peak sare observed. These two features are indeed presentin our calculat ed scattering functions for th e HPL ph asein the qz ) 0 and qy ) 0 planes. Therefore i t is ratherdifficult to distinguish, on the basis of the scatteringfunctions alone, between t he experimentally observedHML/HPL pha ses and t he model HPL phase with thespace group R 3hm , which we have used in our calcula-tion. Nevertheless, we notice that t he measur ement ofthe scattering functions on the t hird plane 25 is differentf r om t he cal cul at ed one on t he pl ane qx ) 0 . T h issuggests tha t th e observed HML/HPL st ructur e does notcorrespond to the space group R 3hm . A r ea l s pa cerepresentat ion of the most un stable mode in t he HP L

phase is given in Figure 14.

V I. D o u b l e - G y ro i d P h a s e

In this work we focused on the weak segregationregion of the phase diagram, where self-consistentmean-field theory predicts that the double-gyroid phaseis not an equilibrium phase for Z < 11.14. I n t hi ss ect i on w e pr esent r es ult s on t he s t abil it y of t hi ss t r uct ur e up t o Z ) 12. For this complex structure,the nu mber of plan e waves needed for t he calculationsincreases ra pidly with the degree of segregation. Thefollowing results were obtained for a limited, but large,number of plane waves (up to N ) 783). The stabilityan alysis along th e cylindrical-lamellar phase boundaryshows that surprisingly the double-gyroid structure isunst able for Z e 12, although the mean -field calcula-tion shows that this phase is stable in a na rrow chan nelalong the cylindrical-lamellar phase boundary for Zg 11.14.24 The convergence of the smallest eigenvalue1 of CRP A with the number of plane waves is demon-strated in Table 1 for two points along the lamellar -

cylindrical phase boundar y corresponding to (Z )10.75, f ) 0.47) and (Z ) 12 , f ) 0.43). It is evidentfrom this table that 1 ha s converged for Z ) 10.75 forabout 400 plane waves. On the other ha nd, for Z )12 there is still a slight variation in the value of1, evenup to 783 plane waves. This clearly demonstrat es thatthe double-gyroid structure is thermodynamically un-stable at Z ) 10.75. O n t he ot her han d, t he r es ult salso indicate th at the double-gyroid stru cture is un stablea t (Z ) 12 , f ) 0.43), where mean-field theory predictsa stable double-gyroid phase.24 Although we cannotexclude the possibility of not having used enough planewaves for the larger values ofZ, it is most likely that1 will remain negative for larger values of N. T hisobservation leads us to believe that the double-gyroid

phase is indeed an unstable phase at this point of the

F i g u r e 1 2 . Three-dimensional contour plots of the spher icalp h as e, a t Z ) 10.8, f ) 0.448 (shown by the symbol 0 inFigure 3), showing the effects of the most unstable mode onthe ordered phase. The view is along th e [110] direction: (a)a ) 0, i.e. the mean-field solution; (b) a ) 0.025; (c) a ) 0.05.Notice tha t th e effect of th e least-sta ble fluctua tion mode leadsto a layered structure perpendicular to the [110] direction.

Figure 13. Scattering function of the hexagonally-perforatedlamellar phase at Z ) 10.8, f ) 0.476, prior to the spinodalline of this phase (shown by the symbol + in Figure 3). Thetop graphs of this figure correspond to the qx-qy plane, themiddle graphs correspond to the qy-qz plane, and the bottomgraphs correspond to the qx-qz plane.



11/14

phase diagram, where m ean-field t heory predicts asta ble double-gyroid phase. Another interest ing obser-vat ion i s t hat 1 becomes more negative as Z isincreased, suggesting that the double-gyroid phasebecomes even more unstable for stronger segregation(larger Z).

We have not calculat ed th e scat tering functions of th edouble-gyr oi d phas e, s ince t hi s phas e i s uns t able.However, we have identified the most unstable fluctua-

tion modes of this phase. In par ticular, we found theresult t hat , for Z ) 10.75, f ) 0.469, the most u nsta blemodes drive the double-gyroid stru cture int o modulat edcylinders which are arranged in a hexagonal lattice asshown in Figur e 15. These cylinder s are orient ed alongthe [111] direction of the cubic un it cell of the double-gyroid structure. The most unsta ble fluctuation modeleads to the epitaxial relation between the cylindricaland the double-gyroid phases, which is identical to thatrecently observed by Schultz et al.26 This geometricrelation between th e two pha ses was a lso noticed frommean-field calculations.27

In order t o put t he a bove results in per spective, it ishelpful to sum mar ize th e salient featur es of the experi-mental phase diagram as given by Bates et al.2 On the

basis of the recent experiment of Hajduk et al.,11 t h e

time-dependent Ginzburg-L andau t heor y of Q i andWang,21 and the present theoretical s tudy, the HML/HPL stru ctures in the experimental ph ase diagrams canbe attributed to intermediate sta tes. The three classicalpha ses ar e presen t in all diblock copolymer system s. Onthe other h an d, the double-gyroid pha se is only observed

in diblock copolymers at small Zh ) (F02

b6/63)Z a n d a tweak segregations.2 This revised phase behavior stronglysuggests that the double-gyroid phase is intimatelyassociated with fluctuations. This observation cor-relates well with our theoretical result showing that themean-field solution of the double-gyroid phase, corre-sponding to Zh f , is not stable. The appearance of th e double-gyroid pha se is th erefore du e to higher-orderfluctuations.

V II. C onclusion

In this paper, we have presented a theory for aniso-tropic fluctuations in ordered phases of diblock copoly-mer melts. Becau se of the conn ectivity of the polymerchains and the periodicity of the ordered phases, thechain conformations are described by a modified diffu-sion equat ion which is a nalogous to the Schrodingerequat ion of an electron in a periodic poten tial. Exploit-ing th e well-kn own Blochs th eorem, we formula ted t hetheory in reciprocal space and determined the eigen-values of the chain diffusion operator, from which a

useful a nalogy to th e th eory of energy bands in crystal-

F i g u re 1 4. Three-dimensional contour plots of t he HPLp h as e, a t Z ) 10.8, f ) 0.476 (shown by the symbol + inFigure 3), showing t he effect of the most unst able fluctua tionmode: (a) a ) 0, i.e. the mean -field solution; (b) a ) 0.05; (c)a ) 0.10.

T a b le 1 . S m a l l e s t Ei g e n v a l u e , 1, o f CRP A a s a F u n c t i o n o f t h e N u m b e r o f P l a n e W a v e s , N, fo r T w o C a s e s : ( 1 ) (Z )

10.75, f ) 0.469); (2) (Z ) 12 , f ) 0.43)

N ca se 1 ca se 2

13 9 -0.5521 10 -2 -1.7778 10 -225 9 -0.5216 10 -2 -0.9891 10 -2

41 5 -0.5099 10 -2 -1.3447 10 -2

63 9 -0.5099 10 -2 -1.2999 10 -2

78 3 -1.2969 10 -2

F i g u r e 1 5 . Three-dimensional contour plots of the double-gyroid phase, at Z ) 10.75, f ) 0.469 (shown by the symbol i n Fi gu re 3 ), s h ow in g t h e effect of t h e mos t u n s t ab lefluctuat ion mode. The view is a long t he [111] direction of thecubic unit cell of this stru cture: (a) a ) 0, i.e. the mean-fieldsolution (Notice the similarity between this stru cture an d theseveral r eported pa ttern s obtained from t ran smission electronmicroscopy); (b) a ) 0.025; (c) a ) 0.15. Notice that the mostunst able m ode leads to h exagonally-packed modulated cylin-ders wh ich are oriented a long t he [111] direction of the cubicunit cell.



12/14

l ine solids emerged. In particular, th e origin of theanisotropic fluctuat ions is intima tely conn ected to th epresence of eigenvalue bands similar to electronicenergy bands. As the energy bands are important forth e electr onic propert ies of solids, the eigenvalue ba ndsare crucial for calculating the elastic properties of ordered diblock copolymer melts. The t heory is usedto calculate the stability (spinodal) lines of the orderedphases in the weak segregation regime. We found tha tthe lamellar, cylindrical, and spherical one-phase re-gions a re en capsulated by their spinodal lines, revealingthe robustness of these thr ee classical phases wh ichar e observed in diblock copolymer melts. An import an tresult of the theory is tha t t he HP L phase is unsta blealong the lamellar -cylindrical pha se bounda ry, whereit ha s been identified experimenta lly; this layered phaseis only metastable in the middle of the lamellar one-phase region.

The investigation of the least-stable eigenmodes of thevarious pha ses shows that when th e lamellar phase isdriven into the hexagonal phase, the kinetics of thetransition proceeds through a long-lived intermediatemodulated-layered state which we believe correspondsto the experimentally observed hexagonally modulated/perforat ed lamellar pha se. This finding is in agreemen t

with the recent experiment of Hajduk et al.11 a n d t h etime-dependent Ginzburg-Landau calculations of Qiand Wang.21 The least-stable mode of the hexagonalphase close to its inner spinodal line corresponds, ont he ot her han d, t o a l am el lar phas e. T her efor e, t helamellar-cylindrical tran sition is not reversible. Thean alysis of the m ost un sta ble eigenmode of the cylindri-cal phase when i t is driven into the spherical phaseshows that the cylinders transform directly into a body-cent er ed-cubic ar r angem ent of s pher es w it hout ani nt er m ediat e s t at e. I n cont r as t , w hen t he s pher i calphase is driven into the cylindrical phase, the tra nsfor-mation proceeds through an intermediate stat e. Fromthis we concluded that the cylindrical-spherical tra nsi-

tion is also irreversible. The a vailability of the eigen-modes from t he theory is exploited to calculate thescattering functions of the ordered phases.

We also an alyzed the stability of the recently-identi-fied double-gyroid pha se. Surpr isingly, we found tha tthis phase is unsta ble around th e tr iple point at Z )11.14. At stronger segregations, we found that thisphas e i s uns t able, al t hough w e do not excl ude t hepossibility of not having used enough plane waves inour expans i on ( w e have us ed up t o N ) 783). Theresults seem to indicate that the mean-field solutioncor r esponds t o a s addl e poi nt . T he r es ult s for t hedouble-gyroid phase are in fact in agreement with therecent experiments of Bates et al.2 which find that thedouble-gyroid phase is only stable for polymer chainswith low molecular weight and at weak segregation,l eadi ng t o t he s pecul at i on t hat t hi s phas e m i ght beindu ced by fluctua tions. The inclusion of higher -ordercomposition fluctuations is possible within the theoryand could be used to see if the double-gyroid phase willbe stabilized. However, th is is beyond the scope of thepresent work. The most unst able mode of the double-gyroid phase corresponds to hexagonally-packed cylin-ders a long t he [111] direction of th e double-gyroid pha se.The epitaxial relation between the double-gyroid phaseand t he cyli ndr i cal phas e i s i n agr eem ent w it h t herecent experiment s of Schultz et al.26

A c k n o w l e d g m e n t . We tha nk Prof. C. Yeung, Dr.

D. A. Ha jduk, an d Dr. M. W. Matsen for useful discus-

sions. M.L. and R.C.D. acknowledge the support fromthe NSERC of Canada.

A p p e n d i x A : R e c i p r o c a l S p a c e T w o - P o i n tC u m u l a n t s

I n t hi s appendix, w e give t he expr ess ions of t h ereciprocal space forms of the two-point cumulants in ourscaled units. The derivation of these cumulan ts can befound in ref 18.

where th e various fun ctions in the equations above aredefined by

A p p e n d i x B : C o m p u t a t i o n a l M e t h o d s f o rR e c i p ro c a l S p a c e

The study of anisotropic fluctuations starts with theidentification of the space groups for the ordered phases.

The ordered diblock copolymer phases considered in this

Cn ,n AA

(k ) ) fA

2

Qcn 1

n 2

n 3

n 4

g(n 2kA

fA - n 10A

fA) - g(n 30A

fA - n 10A

fA)

n 30A

fA - n 2kA

fA

e-An 10fA-

Bn 40

fB

[n 1An ,n 1,n 2

A(k ) n ,n 3,n 2

A*(k ) n 3,n 4

AB(0)n 4

B* +

n 1A*n ,n 3,n 2

A*(k ) n ,n 1,n 2

A(k ) n 3,n 4

AB*(0) n 4

B] (A1)

Cn ,n AB

(k ) )fAfB

Qcn 1

n 2

n 3

n 4

g(n 2kA

fA - n 10A

fA)g(n 3kB

fB -

n 40B

fB)e-A

n 10f

A-B

n 40f

Bn 1A

n ,n 1,n 2

A

(k ) n 2,n 3AB

(k )

n ,n 4,n 3

B*(k ) n 4

B*(A2)

Cn ,n BA

(k ) )fAfB

Qcn 1

n 2

n 3

n 4

g(n 2kB

fB - n 10B

fB)g(n 3kA

fA -

n 40A

fA)e-An 40fA-

Bn 10

fBn 1Bn ,n 1,n 2

B(k ) n 2,n 3

BA(k )

n ,n 4,n 3A* (k ) n 4

A* (A3)

Cn ,n BB

(k ) )fB

2

Qcn 1

n 2

n 3

n 4

g(n 2kB

fB - n 10B

fB) - g(n 30B

fB - n 10B

fB)

n 30B

fB - n 2kB

fB

e-Bn 10fB-

An 40

fA

[n 1Bn ,n 1,n 2

B (k ) n ,n 3,n 2B* (k ) n 3,n 4

BA (0) n 4A* +

n 1B*n ,n 3,n 2

B*(k ) n ,n 1,n 2

B(k ) n 3,n 4

BA*(0) n 4

A] (A4)

g(x) )1 - e-x

x(A5)

nR ) V u n 0R (0) (A6)

R

n ,n (k ) ) G

u n kR*

(G) u n k

(G) (A7)

n 1,n 2,n 3R

(k ) )1

VG1

G2

u n 1kA*

(G1) u n 20R*

(G2)u n 3kR

(G1+G2)

(A8)



13/14

w or k ar e t he one-dim ens ional l am el lar phas e, t hehexagonal ph ase with space group p6m , the BCC phasewith space group Im 3m , th e double-gyroid ph ase withspace group Ia 3hd, and the HML/HPL phase with spacegroup R 3hm . For a given ordered phase, the reciprocallattice vectors Gn are ident ified. These reciprocal latt icevectors ar e ordered according to t heir m agnitudes |Gn|2

) n, starting with |G1|2 ) 0, 0 ) 1 < 2 e 3 e .... Theplane waves corresponding to these reciprocal latticevectors, exp(iGnr), are used as the basis functions forthe eigenproblem defined by eq 21. For the comput a-tions, th e n umber of the reciprocal lat tice vectors N islimited to large N. Th e N N eigenproblem for thepolymer chains (eq 21) is then solved using standardsoftware packages.

For the calculation of the anisotropic correlationfunctions, it is necessary to obtain an exact mean-fieldsolution for the ordered phase. For th e m ean-fieldsolutions, further simplification of the basis functionscan be made on the basis of the observation that the

fields R(0)(r) and R

(0)(r) and the end-integrated propa-

gators, qR()(r), are periodic functions. The symmetr y of

the ordered structure ensures that the Fourier coef-ficients of these periodic functions for the reciprocallatt ice vectors within one star are related.28 A set of

new basis functions can be defined using t his observa-tion. Each function is a l inear combination of plane

waves with wave vectors Gin satisfying the condition

|Gin|2 ) n:

where S in as s um es t he values (1 accor ding t o t he

space group specifications and Nn is the total numberof reciprocal la tt ice vectors belonging t o the sta r n . T h e

values of Nn a nd S in can be found from the Interna-

tional Table of Crystallography.28 This set of basis

functions is used in the mean-field theory of diblockcopolymers.24

Within mea n-field th eory it is convenient to define t heend-integrated propagators

It can be sh own th at the t wo propagators a bove satisfythe diffusion equations

with th e boundar y conditions qR(r,0) ) 1 and qR(r,0) )

q(r,f). If one expan ds the propagators in terms of thebasis functions fn(r), t h e com p on e n t s of t h e e n d-integrat ed propagators sat isfy the different ial equations

where the matrix elements of AR are given by

and the coefficients nm l are defined by

Because AR is Her mitian, its n ormalized eigenvectors,

{R(n )}, form a complete orthonormal basis, and their

corresponding eigenvalues {R(n )} ar e r eal. T he coef-

ficients qR,i(t) a n d qR,i (t) can be expanded as a l inear

combination of the eigenvectors ofAR,

The problem therefore reduces to finding the coefficients

qR(m )(t) a n d qR

(m )(t). E quat ions B 6 and B 7 com binedwith the boundary conditions lead to

where qR(n )(0) is deter mined from t he inital condition

From the self-consistent mean-field equation (eq 9), anexpression for the components of the volume fractionsis obtained

and the single-chain partition function is given by Qc )qA,1

(f) ) qB,1 (1-f).

Since the free energy of the system is invariant undera consta nt shift in th e au xiliary fields, we will furt heruse the convention dr R(r) ) 0, leading to R,1 ) 1) 0. Thus the self-consistent equat ions eq 8) become

with R,1 )fR. The mean -field free energy per chain lessthat of the homogeneous phase is given by

fn(r) )1

Nnin

eiGi

n rS i

n(B1)

qR(r ,t) ) dr1 QR(0)

(r ,t|r1) (B2)

qR

(r,t) ) dr1 dr2 QR(0)

(r,t|r1) Q(0)

(r1,f|r2) (B3)

tqR(r,t) )

2qR(r,t) - R(r) qR(r,t) (B4)

tqR

(r,t) ) 2qR

( r,t) - R(r)qR

(r,t) (B5)

dqR,n(t)

dt) -

m

A nmR

qR,m (t) (B6)

dqR,n

(t)

dt

) -m

A nmR

qR,m

(t) (B7)

A nmR ) -

n

D2nm -

l

nm lR,l (B8)

nm l )1

NnNmNl

in

jm

kl

Gin +Gj

m ,Gkl S i

nS j

mS k

l(B9)

qR,i(t) ) n

qR(n )

(t)R,i(n )

(B10)

qR,i

(t) ) n

qR(n )

(t)R,i(n )

(B11)

qR,i(t) ) n

R,1(n )R,i

(n )e

-R(n )t

(B12)

qR,i

(t) ) n

qR(n )

(0)R,i(n )

e-R

(n )t(B13)

qR(n )

(0) ) m

,1(m )

e-

(m )f

i

,i(m )R,i

(n )(B14)

R,l )1

qR,1

(fR)

n

m

nm lp

q

R,1(p)R,n

(p)R,m

(q)qR

(q)(0)

1 - e-fR(R(p)+R

(q))

R(p) + R

(q)(B15)

A,i ) Z(B,i - fBi,1) + iB, i ) Z(A,i - fAi,1) + i

(B16)

F(0) ) Z

n

A,n2 - ln qA,1

(fA) (B17)



14/14

The reciprocal space self-consisten t mea n-field meth odfor a broken symmetry phase in a diblock copolymersystem is implemented numerically as follows:

(1) At a certain point of the phase diagram parameterspace (Z,f) and for a given ph ase with a specified space

group, determine the reciprocal vectors {Gin} a n d t h e

coefficient {S n}28 and calculate the coefficients nm lusing eq B9.

(2) Choose a lattice periodicity D.(3 ) Make an init ial guess for R,i and calculate the

initial values ofR,i using eq B16 and assuming i ) 0.(4) Compute the matrix elements A nm

R and solve the

eigenvalue problem ARR(n) ) R

(n )R

(n ). Then calculatethe volume fractions using eq B15.

(5) Using eqs B16, calculate t he Lagran ge mu ltiplieri as i ) (A,i + B,i)/2, and then calculate

If the values ofi, A,i, and B,i are smaller than a

certain convergen ce criter ion (e.g. 10-6

), proceed to step6. Otherwise, calculate n ew values ofR,i using

where an d are small positive num bers (we used ) ) 0.1), and return to step 2.

(6) Compute the free energy density using eq B14.The procedure is then repeated for a different lattice

periodicity, D, un til the lowest free energy density isobtained.

R e f e re n c e s a n d N o t e s

(1) Bat es, F. S.; Fredr ickson, G. H. Annu. Rev. Phys. Chem . 1990 ,41 , 525.

(2) Bates, F. S.; Schulz, M. F.; Khan dpur, A. K.; Forster , S. F .;Rosedale, J. H. Faraday Discuss. 1994, 98 , 7.

(3) Seul, M.; Andelman, D. S cience 1995, 26 7, 476.(4) Leibler, L. Macromolecules 1980, 13 , 1602.(5) Hamley, I. W.; Koppi, K. A.; Rosedale, J. H.; Bates, F. S.;

Almdal, K.; Mortensen, K. Macromolecules 1993, 26, 5959.(6) Hajduk, D. A.; Har per, P. E.; Gruner , S. M.; Honeker, C. C.;

Kim, G.; Thomas, E. L.; Fetters, L. J. Macromolecules 1994,27, 4063.

(7) Cohen, R. E.; Bates, F. S. J. Polym . S ci.: Polym . Ph ys. Ed .1980, 18 , 2143.

(8) Hajduk, D. A.; Gruner, S. M.; Rangarajan, P.; Register, R.A.; Fett ers, L. J .; Honeker, C. C.; Albalak, R. J .; Thomas, E.L. Macromolecules 1994, 27, 490.

(9) Sakurai, S.; Momii, T.; Taie, K.; Shibayama, M.; Nomura,

S.; Hashim oto, T. Macromolecules 1993, 26, 485.(10) Saku rai, S.; Umeda, H.; Taie, K.; Nomura, S. J. Ch em . Ph ys.1996, 105 , 8902.

(11) Hajduk, D. A. Private communication.(12) Matsen , M. W.; Bat es, F. S. Macromolecules 1996, 29 , 1091.(13) Fredrickson, G. H.; Helfand, E. J. Chem. Phys. 1987 , 87, 697.(14) Brasovskii, S. A. Zh. E ksp. Teor. Fiz. 1975 , 68 , 175 (Sov. Phys.

J E T P 1975, 41 , 85).(15) Mayes, A. M.; Olvera de la Cruz, M. J . C h em . P h y s. 1991,

95 , 1670.(16) Muthukumar, M. Macromolecules 1993, 26, 5259.(17) Yeung, C.; Sh i, A.-C.; Noolandi, J .; Desai, R. C. Macromol.

Th eory S im u l. 1996, 5, 291.(18) Shi, A.-C.; Noolandi, J .; Desai, R. C. Macromolecules 1996,

29 , 6487.(19) Laradji, M.; Shi, A.-C.; Desai, R. C.; Noolandi, J. Ph ys. Rev.

Lett. 1997, 78 , 2577.(20) Ashcroft, N. W.; Mermin, N. D. Solid S tate Physics; Saunders

College: Philadelphia, PA, 1976; Chapters 8 a nd 9.(21) Qi, S.; Wang, Z.-G. Phys. Rev. Lett. 1996, 76, 1679; Phys. Rev.E 1997, 55 , 1682.

(22) Hong, K. M.; Noolandi, J . Macromolecules 1981, 14 , 727.(23) Doi, M.; Edwards, S. F. The Theory of Polymer Dynamics ;

Oxford University Pr ess: London, 1986.(24) Matsen, M. W.; Schick, M. Phys. Rev. Lett. 1994, 72 , 2041.(25) Ham ley, I. W.; Gehlsen, M. D.; Khan dpur, A. K.; Koppi, K.

A.; Rosedale, J. H.; Schultz, M. F .; Bates, F. S.; Almdal, K.;Mortensen, K. J . Phys. II 1994, 4, 2161.

(26) Schultz, M. F.; Bates, F . S.; Almdal, L.; Morten sen, K. Phys.Rev. Lett. 1994, 73 , 86.

(27) Matsen , M. W.; Schick, M. Macromolecules 1994, 27, 6761.(28) International T ables of X-ray Crystallography; Henry, N. F.

M., Lonsdale, K., Eds.; Kynoch P ress: Birmingham , U.K.,1965; Vol. I. The space group for the hexagonal phase, p6m ,is found on 372; that of the BCC phase, Im 3m , on 524; thatof the H PL phase, R 3hm , on 524; and t ha t of the double-gyroid

phase, Ia 3hd, on 524.MA9618437

A,i ) ZB,i + i - A,i (B18)

B,i ) ZA,i + i - B, i (B19)

i ) A,i + B, i (B20)

R,i ) R,i + R,i - i (B21)


mohamed laradji, an-chang shi, jaan noolandi and rashmi c. desai- stability of ordered phases in...

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