Interfaces of Modulated Phases - TAU

VOLUME 79, NUMBER 6 P H Y S I C A L R E V I E W L E T T E R S 11 AUGUST 1997

0any

ses,etricnal,

dary-1]

1058

Interfaces of Modulated Phases

Roland R. Netz,1,2 David Andelman,3 and M. Schick11Department of Physics, University of Washington, Box 351560, Seattle, Washington 98195-156

2Max-Planck-Institut für Kolloid- und Grenzflächenforschung, Kantstrasse 55, 14513 Teltow, Germ3School of Physics and Astronomy, Tel-Aviv University, Ramat Aviv 69978, Tel Aviv, Israel

(Received 22 November 1996)

Numerically minimizing a continuous free-energy functional which yields several modulated phawe obtain the order-parameter profiles and interfacial free energies of symmetric and nonsymmtilt boundaries within the lamellar phase, and of interfaces between coexisting lamellar, hexagoand disordered phases. Our findings agree well with chevron, omega, and T-junction tilt-bounmorphologies observed in diblock copolymers and magnetic garnet films. [S0031-9007(97)03772

PACS numbers: 61.25.Hq, 61.41.+e, 83.70.Hq

u

ne

r

l

b

o

atres).ibes

al

,g

oratl

mr

e-d

r-

e

Modulated phases are found in a surprisingly diverseof physical and chemical systems, including supercondtors, thin-film magnetic garnets and ferrofluids, Langmumonolayers, and diblock copolymers [1]. Such phasescharacterized by periodic spatial variations of the pertineorder parameter in the form of lamellae, cylinders, or cubarrangements of spheres or interwoven sheets. This sorganization results from competing interactions: a shoranged molecular one favoring a homogeneous state, along-ranged contribution, which can have a magnetic, eltric, or elastic origin, favoring domains. Because of thmodulation, interfaces between different phases or grboundaries within a single phase are most interesting,they have received much less attention than those occurin solids. Recently, experimental studies of grain boundries within lamellar phases of diblock copolymer havbeen carried out [2,3] which illustrate the rich interfaciabehavior exhibited by such systems. In particular, thrmorphologies of tilt boundaries [2(b)], denoted chevroomega, and T-junction, were observed. Such interfacare more difficult to describe theoretically than those btween uniform phases, which have been the subject of csic work [4]. In this article we study not only tilt grainboundaries within lamellar phases but also interfacestween coexisting modulated phases of different symmet

The dimensionless free-energy functional we use,

F ffg Z Ω

2x

2f2 1

1 2 f

2ln

1 2 f

2

11 1 f

2ln

1 1 f

2

212

s=fd2 112

s=2fd2 2 mf

ædV , (1)

includes an enthalpic term (proportional to the interactiparameterx) that favors an ordered state in whichjfj

is nonzero, an entropy of mixing preferring a disorderestate, f 0, and confiningjfj to be less than unity,and derivatives of the order parameter. The orderstate occurs with a modulation of a dominant wavvector qp 1y

p2 because of the competition betwee

0031-9007y97y79(6)y1058(4)$10.00

setc-ir

arenticelf-rt-d ac-e

ainyetinga-el

een,ese-as-

e-ry.

n

d

eden

the negative gradient square term (favoring domainslarge length scales) and the positive Laplacian squa(preferring a homogeneous state at small length scaleSuch a free-energy functional has been used to descrthe bulk phases of magnetic layers [5], Langmuir film[6], amphiphilic systems [7], diblock copolymers [8], andthe effects of surfaces on isotropic [9] and hexagonphases [10] of the latter.

In our numerical studies, we determine the minimumof (1) by directly using a conjugate-gradient methodwhich is more convenient than solving the correspondinEuler-Lagrange equation. We discretizef on a squarelattice, approximate the derivatives by nearest-neighbdifferences, and employ a mesh size sufficiently small thdiscretization effects are negligible within the numericaprecision. A typical grid contains 40 000 points. Figure 1shows the resulting two-dimensional bulk phase diagraas a function ofx and (a) the average order parameteF ; kflV and (b) the chemical potentialm. It is in goodagreement with previous calculations based on singlmode approximations [5,6]. In addition to the disorderesDd phase, there is a lamellarsLd and two hexagonal

FIG. 1. Two-dimensional bulk phase diagram, showing disoderedsDd, lamellar sLd, and hexagonalsHd phases, as a func-tion of the interaction strengthx, and (a) the average orderparameterF and (b) the chemical potentialm. Dashed linesin (a) denote triple lines and dashed lines in (b) denote th(metastable)L-D transitions which exhibit tricritical points (de-noted by solid circles).

© 1997 The American Physical Society


earshyve

segyle

.

sHd phases, which all join at the critical point atxC 3y4, m F 0. (In three dimensions, one expectadditional cubic phases, which are not studied here.) Flarger values ofx, the H andL phases each terminate aa triple point, whereas experimentally one sees that thmodulated phases exist even for very large values ofx.This unphysical part of the phase diagram is due tobreakdown of the gradient expansion in (1).

We first present results for grain boundaries in lamelar phases, and begin with the asymmetric tilt graboundary (GB) between two perpendicularL phases.This is theT-junction of Ref. [2(b)], for which the layercontinuity between the two adjoining phases is disrupteFigures 2(a)–2(c) show order-parameter profilesf fordifferent x and m 0 (symmetric stripes), whileFigs. 2(d)–2(f) show profiles for varying chemical potentialsm and x 1 (asymmetric stripes). Figure 2(bclearly shows the enlarged end caps noted in experim[1,2], and the series 2(a)–2(c) predicts that they becoless pronounced with increasingx. The GB interfacialenergygGB scales asgGB , sx 2 xCdmp with mp 3y2,

FIG. 2. Grain boundary (GB) between two perpendicullamellar phases. (a)–(c) Contour plots of the order parameprofiles form 0 andx 0.78, 1, 1.5; throughout the articlethe order parameter rangef21, 1g is represented by 20 grayscales. (d)–(f) Profiles forx 1 andm 20.02, 0.02, 0.06.(g) GB interfacial energygGB for m 0 as a function ofx,showing asymptotic scaling with an exponentmp 3y2 (inset).The dashed line gives the surface free energy density of olamellar layer for comparison. (h)gGB for x 1 as a functionof m for two GB configurations which are degenerate atm 0,but distinct otherwise.

sortese

a

l-in

d.

-)entme

arter

ne

Fig. 2(g), in accord with mean-field predictions [11]. Todemonstrate that the interfacial energygGB is indeedquite small, we show, with a dashed line, the differencin bulk free energies between the disordered and lamellphases multiplied by one wavelength of the latter. This iroughly the cost per unit area of disordering such a widtof lamellar phase. Clearly, the actual grain boundarbridges the two grains in a manner much less expensithan the insertion of a region of disorder. In Fig. 2(h)we plot the interfacial energies of two distinct GB struc-tures, which are degenerate atm 0, being related byorder-parameter reflection symmetryf ! 2f. Form fi 0 this symmetry is broken, and the two structurecorrespond to distinct local free energy minima, onmetastable and the other globally stable. The free-enerbarriers between such interfacial structures are responsibfor slow interface motion and, thus, long healing times inmultigrain lamellar samples [12].

In Figs. 3(a)–3(c) we show symmetric tilt-boundary(TB) configurations for a fixed angleu 90± between thelayer normals as a function ofx . Here the layer conti-nuity is maintained across the boundary, and thechevronmorphology is quite evident. In Figs. 3(d)–3(f), the tilt

FIG. 3. Tilt boundary (TB) between two lamellar phases(a)–(c) Profiles form 0, a fixed angleu 90± betweenthe layer normals, andx 0.78, 1, 1.5. (d)–(f) Profiles form 0, x 1, andu 28.08±, 53.14±, and126.86±. (g) TBinterfacial energygTB for m 0 as a function ofx, showingasymptotic scaling with an exponentmp 3y2 (inset). (h)gTBfor x 1 as a function ofu, showing agTB , u3 behavior forsmall u (dashed line) and linear behavior foru ! 180±.

1059


ras

in

ethenge

eytheisxic

ly;

hateftreeoxat

ial

his

ofonal

nt

ard:eeular

tedgy

tedoni-aresds

angle is progressively increased at fixedx, and one clearlysees the change from the chevron to theomegastructure.The omega shape of the layers at the boundary results frfrustration due to an imposed local lamellar wavelengmuch larger than the equilibrium value. Figures 3(d)3(f) resemble micrographs of undulating lamellar patternin garnet films [13], and the similarity of Fig. 3(f) to themicrographs [2(b)] of diblock copolymer TBs is striking.The scaling of the TB energygTB is again described byan exponentmp 3y2 [Fig. 3(g)]. Close to criticality,one finds pronounced reconstruction in terms of a squalike modulation, Fig. 3(a). For smallu, gTBsud , u3,Fig. 3(h), in accord with the bending behavior of elastic sheets; theu ! 180± limit is expected to be linear,gTBsud , 180± 2 u, in accord with a description in termsof decoupled dislocations of finite creation energy.

We now turn to interfaces between thermodynamicaldistinct phases. Interfacial structures between coexistilamellar and disordered phases for three different valuof the angleq between the lamellae and the interfacare shown in Figs. 4(a)–4(c) forx 1 (where thephases and, therefore, also the interfacial structure ametastable), and in 4(d)–4(f) forx 1.5 (where they

FIG. 4. Lamellar-disordered interface. (a)–(c) Profiles fox 1 and different anglesq 0±, 45±, and 90± betweenthe lamellae and the interface. (d)–(f) Profiles forx 1.5.(g) Interfacial energygLD for q 90± (dashed line) andq 0± (solid line), the latter scaling withmp 2 on approachto the metastable tricritical point (inset). (h)gLD as a functionof the angleq for x 1 (dashed line) and forx 1.5 (solidline).

1060

omth–s

re-

-

lyngese

re

r

are stable). The metastableL-D boundary is shown asa dashed line in Fig. 1(b). In Figs. 4(a) and 4(d), foq 0±, there is a relaxation of the outermost layersthe interface is approached, leading to a small increasethe wavelength, as in solids.

Before presenting the interfacial free energies, wdescribe our method of obtaining them becausecalculation of such free energies between coexistiphases which can be modulated is nontrivial [14]. Wcalculate the total free energy,FI, of phase I in a boxemploying periodic boundary conditions parallel to thleft and right faces, and reflecting (Dirichlet) boundarconditions on those faces themselves. We adjustlength of the box so that the free-energy densityminimized. This occurs when the length of the bois some integer number of wavelengths of the periodstructure of phase I. The volume of the box isV 0. By thismeans, there isno surface contribution to the total freeenergy, so that the bulk free energy is obtained directfI

b FIsV 0dyV 0. In a similar way we obtainfIIb

FIIsV 00dyV 00. For the system in I, II coexistence, wecalculate the total free energy in a box large enough tthe order parameters attain their bulk values on the land right faces at which reflecting boundary conditions aemployed, and we vary the length of the box to minimizthe interfacial excess free energy. The volume of the bis V . Again, there are no surface contributions so thwe obtain FI,IIsV , Ad V s fI

b 1 fIIb dy2 1 AgI,II. As

the bulk free energies are known, the desired interfacenergy follows directly.

The interfacial energygLD is shown in Fig. 4(g). Asx is decreased, the coexistence betweenL and D ispreempted by the hexagonal phase [see Fig. 1(b)]. Tis manifest in the behavior ofgLD for q 90± [dashedline in Fig. 4(g)] which becomes negative. This valuegLD is obtained by assuming an interfacial reconstructilocally resembling the energetically preferred hexagonphase [see Fig. 4(c)]. The value ofgLD for q 0± [solidline in Fig. 4(g)] remains positive down tox . 0.82and vanishes there with the classical tricritical exponemp 2 [15]. Strong anisotropy ofgLD as function ofqis observed [Fig. 4(h)], from which the shape of lamelldroplets in an isotropic phase can be qualitatively inferreFor x . 1.34 the drops are elongated parallel to thlamellae; forx , 1.34 theL andD phases are metastablat coexistence and the drops are elongated perpendicto the lamellae.

Hexagonal-disordered interfacial structures are depicin Figs. 5(a)–5(c); the corresponding interfacial energHD , shown in Fig. 5(g), scales again withmp 2.Structures for the hexagonal-lamellar interface are plotin Figs. 5(d)–5(f), where we chose the mutual orientatiin accord with the epitaxial relationship found for amphphilic systems [16]. The corrugation of the lamellae nethe interface, particularly evident in Fig. 5(d), resemblthat seen in experiments on diblock copolymer blen


ndddof

s

n

ne

,

.

.

s.

FIG. 5. (a)–(c) Hexagonal-disordered interfacial profiles fox 0.78, 0.9, and 1.2. (d)–(f) Hexagonal-lamellar profilesfor x 0.78, 0.9, and 1.2. (g) Interfacial energygHD alongHD coexistence as a function ofx, scaling withmp 2 (inset).(h) gHL as a function ofx, scaling withmp 3y2 (inset).

[17]. The interfacial free energy of this interface,gHL,is plotted in Fig. 5(h) and scales with the classical criticaexponentmp 3y2. This reflects the fact that, in contrastto the L-D and H-D interfaces, theH and L phasesare locked into a fixed relative position with respect totranslations perpendicular to theH-L interface.

At the triple point between disordered, lamellar, anhexagonal phases,x . 1.34, we find gLD . gHL 1

gHD . It follows that theL-D interface that we have cal-culated is not the thermodynamically stable one. Withiour model, theL-D interface is therefore wetted by thehexagonal phase at the triple point. As the occurrencesuch points between these three phases is not uncommexperimentally [18], it would be interesting to determinewhether this wetting does indeed occur.

Density fluctuations will decrease the interfacial criticaexponent to a valuemp , 3y2, but leave the classicaltricritical exponentmp 2 intact [11], while fluctuationsof the direction of the modulation normals will eliminatethe critical point [19]. By introducing uniaxiality, theselatter fluctuations can be suppressed.

In summary, we have employed a simple GinzburgLandau free-energy functional to calculate profiles anfree energies of several interfaces of modulated phasQualitative agreement with experiment is very good. Thobserved chevron and omega morphologies at lamel

r

l

d

n

ofon

l

-des.elar

grain boundaries emerge naturally, as do the expanded ecaps, characteristic of the T-junction. We also calculateprofiles and free energies of interfaces between disordereand modulated phases, and between modulated phasesdifferent symmetry. In all but the simplest cases, there isignificant reconstruction which leads to low interfacialenergies. An extreme example of such reconstructiois the lamellar-disordered interface, which we find to bewetted by the hexagonal phase at anL-H-D triple point.

This work was supported in part by grants fromthe United States–Israel Binational Science Foundatiounder Grant No. 94-00291, and the National SciencFoundation, Grant No. DMR9531161. We thank NedThomas for bringing the work of Gidoet al. to ourattention.

[1] M. Seul and D. Andelman, Science267, 476 (1995).[2] (a) S. P. Gido, J. Gunther, E. L. Thomas, and D. Hoffman

Macromolecules26, 4506 (1993); (b) S. P. Gido and E. L.Thomas, Macromolecules27, 6137 (1994).

[3] Y. Nishikawa et al., Acta Polym. 44, 192 (1993);T. Hashimoto, S. Koizumi, and H. Hasegawa, Macro-molecules27, 1562 (1994).

[4] E. Helfand and Y. Tagami, J. Chem. Phys.56, 3592(1972); L. Leibler, Macromolecules15, 1283 (1982);A. N. Semenov, Sov. Phys. JETP61, 733 (1985).

[5] T. Garel and S. Doniach, Phys. Rev. B26, 325 (1982).[6] D. Andelman, F. Brochard, and J.-F. Joanny, J. Chem

Phys.86, 3673 (1987).[7] G. Gompper and M. Schick, Phys. Rev. Lett.65, 1116

(1990).[8] L. Leibler, Macromolecules 13, 1602 (1980); G. H.

Fredrickson and E. Helfand, J. Chem. Phys.87, 697(1987).

[9] G. H. Fredrickson, Macromolecules20, 2535 (1987).[10] M. S. Turner, M. Rubinstein, and C. M. Marques, Macro-

molecules27, 4986 (1994).[11] J. S. Rowlinson and B. Widom,Molecular Theory of

Capillarity (Oxford University Press, New York, 1982).[12] Similar local free-energy minima for a moving interface

had been postulated in the context of crystal growth [J. WCahn, Acta Metall.8, 554 (1960)].

[13] M. Seul and R. Wolfe, Phys. Rev. Lett.68, 2460 (1992);Phys. Rev. A46, 7519 (1992).

[14] This problem has been formulated within the con-text of coexisting lamellar phases [A. Chatterjee andB. Widom, Mol. Phys. 80, 741 (1993)]; B. Widom,KNAW Symposium, Amsterdam, 1994 (to be published).

[15] B. Widom, Phys. Rev. Lett.34, 999 (1975); see alsoChap. 9.5 of Ref. [11].

[16] Y. Rancon and J. Charvolin, J. Phys. Chem.92, 6339(1988); K. M. McGrath, P. Kékicheff, and M. Kl´eman,J. Phys. II (France)3, 903 (1993).

[17] R. J. Spontaket al., Macromolecules29, 4494 (1996).[18] For an example, see R. Strey, Ber. Bunsen-Ges. Phy

Chem.97, 742 (1993).[19] S. A. Brazovskii, Sov. Phys. JETP41, 85 (1975).

1061

Documents

Interfaces of Modulated Phases - TAU