j.m. carlson, s.a. langer and j.p. sethna- frustration in modulated phases: ripples and boojums

8/3/2019 J.M. Carlson, S.A. Langer and J.P. Sethna- Frustration in Modulated Phases: Ripples and Boojums

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EUROPHYSICS LETTERSEurophys. Let t . , 5 (4), pp. 327-331 (1988)

15 February 1988

Frustration in Modulated Phases: Ripples and Boojums.J. M. CARLSON,S. A. LANGER nd J. P. SETHNALaboratory of Atomic and Solid State PhysicsCornel1 University - Ithaca, N . Y . 14853, U S A(received 29 September 1987; accepted 30 November 1987)PACS. 61.30 - Liquid crystals.PACS. 64.70M - Transitions in liquid crystals.Abstract. - Many exotic modulated phases can be described in term s of regular ar ra y of defects.We have studied frustrated continuum elastic theories for several such liquid crystallinesyste ms. While in each case th e natu re of th e modulation and th e defects is quite different, w enote striking similarities in the structure of the free energies. In each case, frustration arisesfrom th e local packing prop erties of th e molecules, and is linked to tota l divergence te rm s in th efree energy. These term s can be integrated exactly to produce surface terms , which producenegative contributions to th e free energy a t th e defects. In each case, th e modulated phase ispreferred when the net defect energy is sufficiently negative.

We pre sen t several exotic mod ulated phases which sh ar e a common feature-ach has astable phase (or phases) which possess reg ular a rra ys of defects (fig. 1).Certain lipid bilayersystems exhibit a ripple phase [l],which can be described in terms of a one-dimensionalperiodic density modulation composed of high-density pinched regions separated by gaps.Chiral sm ectic liquid-crystal films form two-dimensional pa tte rn s composed of boojum s (socalled because of their similarity to the boojums of superfluid 3He-A)[2]. The metallicglasses [3 ,4] and t h e blue ph ases of cholesteric fluids [4] have been described as networks ofdisclination lines.In this le tte r we discuss th e similarities between th e continuum elastic theories for eachof these phases. These theories are frustrated because the locally energetically idealconfiguration of molecules cannot be extended to fill all space (fig. 2)(l). Locally, the lipidbilayers display spontaneous splay in one dimension; the smectic film displays spontaneousbend in tw o dimensions; and th e blue phases display spontaneous twist in th re e dimensions.In each case, strain builds up when th e ideal stru ctu re is extended. As we will show, d efects(places wh ere th e molecular orientation shifts abruptly) can act to relieve this str ain (fig. 3).Typically, in a physical system the defects will spread out over a few intermolecularspacings; howev er, th e size of th e defect is small compared to t h e length scale on which th emolecular orientation varies. For this reason, most of th e essential features ar e captured inour models which describe the limit in which the size of the defect is equal to theintermolecular spacing.

('1 For diagrams of the metallic glasses and blue phases, see ref. [41.


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328 EUROPHYSICS LETTERS

Fig. 1. - a) Freeze-fracture micrograph of the r ipple phase of DPPC showing the top view of' th ecorrugated bilayer. A cross-section perpendicular to the stripes produces a sawtoothlike pattern.(Courtesy of J. Zasadzinski and M. Schneider.) b ) Micrograph through crossed polarizers of t h eboojums in HOBACPC. The film looks dark whenever the molecular axis is either vertical o rhorizontal. (Courtesy of N. A. Clark, D. H. Van Winkle and C. Muzny.)Our first example is the smectic r ipple phase P3, which is observed in hydratedphospholipid bilayers [5]. Each molecule consists of a hydrophilic polar head group, and apair of hydrophobic hydrocarbon chains. For certain ranges of temperature and

concentration th e amphiphilic na tur e of thes e molecules causes them to form bilayers. In th eripple phase the layers are corrugated, so there is a one-dimensional periodic densitymodulation of the chains. Figure la) is a typical freeze-fracture electron micrograph of theP s f phase of dipalmitoylphosphatid lcholine (DPPC) [6]. The ripples have a sawtoothlikesys tem is frus trated because the cross-sectional are a of the head groups i s g rea te r than tha tof the associated chain pair, making it impossible to fill space without introducing somestrain into the bilayer configuration, or the molecule itself.

shape and th e wavelength is - 130K corresponding to approximately 15mo lecules [71. The

a)vp tz

%)---Fig. 2 . - Local ideal configurations. a) Schematic DPP C molecules show ing spontaneous splay of thetails. b ) Projections e of th e optical axis of th e H OBA CPC molecules into th e plane of th e film, showingan inclination to bend.


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J. M . C A R L S O N et al.: F R U S T R A T I O N I N M O D U L A T E D P H A S E S : R I P P L E S A N D B O O J U M S 329

Fig. 3. - Defect configurations. a ) A ripple of length A. Bulk regions of closely packed chains areseparated by gaps . b ) A boojum. 6 is nearly parallel to the bound ary almost everyw here, and th ere ar eno point defects anywhere. e) Simulated view through crossed polarizers of the boojum in b) .In our one-dimensional model [l]of the r ipples th e scalar ord er param eter $ r ep re sen t sth e t i l t of th e chains with respect to th e normal to the head group layer (see fig. Za)),which,for simplicity, we will assum e is rigid an d flat ('). As shown in table I, th e f ree energy of thismodel sys tem is a f rus t ra ted + [4] heory with a t e rm which repres en ts the core energy of t h edefects. Because the heads are larger than the chains, adjacent molecules will haveminimum e nergy when th ey t i l t tow ards one another , g iving rise t o a positive gradi ent in th eord er param eter . In th e modulated phase th is resul ts in regions of spontaneous inward splayof the chains, as illustrated in fig. 3a). The bulk regions are of finite length A , and a resepara ted by gaps ( i . e . defects) which relieve th e stra in which builds up as + becomes large.

I n t h e free en erg y, t h e local low-energy configuration satisfies a$ = q , where q is a strictlyTABLE .

P , Phases Chiral films Blue phases Metallic glassesFrustrated as-g a,+-sct a,n?-9%knk d,xu.A7:- J,,derivativecovariant (2 = 1 , 2 ) ( z , J , ~ =1-3) (t=1-3; u,v,;=1-4)Free energy (D$)*+WZ$*+ 4 K,(D. e)'+ K,(D X 2)' (Dn)' (DX)*Total divergence g d t 98x2 ga.(n.an-nd.n) 93,( [23J 'Ik,-- [ X d J 'Iik)Defects Gaps Boojums s=-112 Six o1dDefect lines coordinatededges

( 2 ) In this model, the rippling is associated only with modulation of the o rder parameter 5 . W ebelieve th at out-of-plane r ippling is driven by th e modulation of the chain angle when th e rigidity of th elayers is somewhat relaxed.


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330 EUROPHYSICS LETT ERSpositive constant which increases as the size mismatch of the head groups and chains isincreased. The fru stration is associated w ith th e total divergence ter m - d$, and it may bethoug ht of in two ways. F irs t, as we have ju st discussed, it produces a spontaneous splay ofth e chains. Second, after integrating by parts, we see tha t it makes a negative contributionto t he energy at t he defect. Viewing the frustration the second way, when th e net defectenergy becomes sufficiently negative, the modulated phase is preferred. Otherwise auniform (+= constan t) phase is th e global minimum of th e free ene rgy. The phase diagramfor this model has been solved analytically. For these and other details see ref. [l].Our second example is the chiral liquid crystal, HOBACPC (R(-) hexyloxybenzylidenep'-amino-2-chloropropyl cinnamate, pronounced ho-baps), which can form freely suspendedfilms only a few m onolayers thick [8-111. The HOBACPC molecule is long and thin, and inthe smectic C and I phases th e molecular axes ar e tilted with respect t o the normal to th efilm. When a sam ple is cooled through t h e C to I transition, th e smectic I phase nucleates indroplets. Viewing the sample through crossed polarizers (fig. lb ) ) indicates that themolecular axes ar e almost everywhere parallel to th e borders of th e droplets (3). The droplettex tur e is not stabl-the droplets grow, merge, and form a regular patte rn of stripesseparated by defect lines.We describe th e sy stem in ter m s of a two-dimensional unit vector field e ( x ,y), which isth e projection of th e local averag e molecular orientation onto th e plane of t h e film (th e (x, )-plane). Because the HOBACPC molecules are chiral, the system does not have mirrorsymm etry, and 6 likes t o bend (4 ) (fig. 2b)). If + is the angle between I? and th e x-axis, thenth e local ideal configuration is given by d+ = qe, or, equivalently, by a x 13= q, where q is ameasure of the chirality of the system. The free energy is given in table I.Like the rippled phase of the lipid bilayer, the HOBACPC film is frustrated, and thefrustration may be viewed in two ways. First, a unit vector field in two dimensions cannothave constant curl, so th e ideal local configuration of molecules cannot be ex tend ed t o coverthe whole film. Second, when the total divergence term - 2qd x e in the free energy isintegrated by parts, it becomes the surface integral [email protected]. This te rm makes itsmaximum negative contribution to the en ergy when th e orde r param eter lies parallel to th eboundary of the droplet and points counterclockwise (assuming q > 0). The condition for alocal minimum in the free e ner gy dens ity, therefore, becomes a boundary condition for th edroplets. If the boundary condition were to be satisfied everywhere, then the re would haveto be a point defect within or on the edge of the droplet. Since the core energy of such adefect is large, the defect is expelled from the droplet, violating the boundary conditionslightly in the process, and producing a boojum. The boundaries of the droplets are nowdefect l ines, because 8 changes orientation by - 180"in m oving from one droplet to anoth er,and it is these defect lines which relieve th e stra in of frustration. The d efect lines ar e alsoth e dominant feature of the stable stripe te xtu re, which, ap art from th e high-order term s inthe free energy, is driven by phenomena similar to the ripples in DPPC.Two other systems in which frustration leads to defects via total divergences in th e freeene rgy a re th e cholesteric blue phases and th e metallic glasses. Details of these system s a regiven in ref. [41. In th e blue phases th e order param eter is a headless unit vector n ( x ,y, ) ,repr ese ntin g t h e ave rage local molecular orientation. In th e locally ideal configuration n hasa nonzero twist, because molecules placed end to end prefer to be parallel, but moleculesplaced side to side prefer t o have a small angle between them. The resulting defects are

( 3 ) They could also be perpend icular, but this is probably not the case. See ref. [2]. It also does not(4) We do not know the microscopic mechanism involved here. All we can say is th at t he effect ismatter .not forbidden by symmetry.


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J . M. CARLSON et al.: FRUSTRATION IN MODULATED PHASES: RIPPLES AN D BOOJUMS 331s = - disclination lines. In th e locally ideal configuration of metallic glasses all atom s lie onth e vertices of perfect tetrah ed ra. Te trah ed ra cannot fill space in 3 dimensions, bu t they canpack on the surface of a sp here in 4 dimensions. The o rder pa ram eter for th e metallic glassesis th e 4 x 4 otation matrix 2 that maps the local configuration of atoms into the ideal 4-dimensional template. The defects in the metallic glasses are - 2" disclination lines wheresix, rather than five, tetrahedra meet along an edge.Our goal is a general theory of frustrate d systems in the continuum limit. Table I outlinesthe key features we have identified so far. In each of these systems the microscopicproperties of constituent molecules prescribe a locally ideal configuration which cannot fillspace. This local minimum energy configuration is exp ressed as a preferred deriv ative of theorde r parameter . The covariant derivative DY of the order parameter !" is th e differencebetween the actual derivative and the preferred derivative (see the table for specificforms)(5>, and is a measure of how close the system comes to achieving its idealconfiguration (6) . In each case, the free en ergy density contains a te rm which is simply th esqua re of th e covariant derivative. This is sensible, because these te rm s a re minimized whenthe system is in its ideal configuration. In the first two systems, when the covariantderivative is squared, the cross ter m is a total divergence. When the free energy density isintegrated over the whole sample, the total divergence can be integrated exactly andbecomes a surface term. This surface term is evaluated at the boundaries of the sample,where it is insignificant in t he thermodynamic limit. How ever, this te rm also contributesaround all defects in th e sample, and th ereb y effectively renormalizes th e defect energy . Ifth e net defect energy is sufficiently negative, th e sam ple will have a finite den sity of defects.These defects ar e the gaps in the ripple phase, th e boojums in H OBAC PC, and th e networksof disclination lines in the blue phases and metallic glasses.

* * *This work was supported by National Science Foundation Grant No. DMR-8451921.( 5 ) The derivative has one index for the dimensions of space and one for the components of theorder parameter.( 6 ) Given the form of the order parameter and the symmetries of the system, it is straightforwardto write down the most general allowed covariant derivative. For the chiral films, the most generalcovariant derivative is ai+- qRijcj, where Rij is a rotation matrix. Using something other thanRij = 8ij simply shifts the frustration from the bend to the splay term in the free energy, withoutchanging the basic physics. For the blue phase the covariant derivative in table I is the most general,given spherical symmetry.

R E F E R E N C E S[l] CARLSONJ . M. and SETHNA. P ., to be published in Phys . Rev. A.[21 LANGERS. A. and SETHNA. P . , Phys . Rev. A , 34 (1986) 5035.[31 NELSOND. R. , Ph ys . Rev. B, 28 (1983) 5515.[41 SETHNA. P., Ph ys . Rev. B, 31 (1985) 6278.[5] See, for example, SACKMAN. , RUPPLE D. and GEBHARDTC . , in Liquid Crystals of One and

TwoDimensional Order, edited by W. HELFRICHand G. HEPPKE (Springer-Verlag, New York,N . Y . ) 1980, p. 309.[6] ZASADZINSKIJ. and SCHNEIDER. , to be published.[7] JANIAK M. J., SMALL. M. and SHIPLEY. G., Biochem., 15 (1976) 4575.[8] YOUNG C. Y. , PINDAK ., CLARK N . A. and MEYER R. B., Ph ys . Rev. Lett., 40 (1978) 737.[9] R O S E N B L A ~ . , MEYER R . B., PINDAK . and CLARKN . A ., Ph ys . Rev. A , 21 (1980) 140.[lo] PINDAK ., YOUNG C. Y. , MEYER R . B. and CLARKN. A. , Ph ys . Rev. Lett., 45 (1980) 1193.[ll] VAN WINKLE D. H . and CLARK N . A ., Ph ys . Rev. Lett., 53 (1984) 1157.

j.m. carlson, s.a. langer and j.p. sethna- frustration in modulated phases: ripples and boojums

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