arXiv:cond-mat/9605129v2 21 May 1996
Chiral Lyotropic Liquid Crystals: TGB Phases and Helicoidal Structures
Randall D. Kamien and T.C. Lubensky Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, PA 19104
The molecules in lyotropic membranes are typically aligned with the surface normal. When these molecules are chiral, there is a tendency for the molecular direction to twist. These competing e?ects can reach a compromise by producing helicoidal defects in the membranes. Unlike thermotropic smectics, the centers of these defects are hollow and thus their energy cost comes from the line energy of an exposed lamellar surface. We describe both the twist-grain-boundary phase of chiral lamellar phases as well as the isolated helicoidal defects.
20 May 1996
1. Introduction Lyotropic liquid crystals share a number of features with their thermotropic cousins. In lyotropic lamellar systems, however, it is believed that because the molecules are amphiphillic the molecular axes will align with the layer normal in an untilted, Lα phase. Chiral molecules are well known to exert torques on each other  leading to chiral mesophases in thermotropic liquid crystals. In chiral lyotropics then, there will be a frustration between normal alignment and the tendency of the molecules to twist . In thermotropic smectic-A phases, this frustration can be relieved via the formation of a twist-grain-boundary phase (TGB)  analogous to the Abrikosov ?ux line lattice of type-II superconductors. In this case the defects are screw dislocations with cores made of the associated nematic liquid crystal phase. In this letter we will describe a similar screw dislocation structure for lyotropic lamellae and propose an Lα TGB phase. We will also discuss a possible defect mediated phase transition  between the Lα phase and a normal cholesteric phase .
2. Helicoidal Defects in Membranes We model the free energy of an isolated bilayer membrane as a sum of contributions. The free energy for membrane ?uctuations is  Fm κ = 2 dS 1 1 + R1 R2
dS (? · N)2 ,
mean curvature. The free energy for ?uctuations of the molecular axis n is given by the membrane version of the chiral Frank free energy, namely:
? Fn =
where R1 and R2 are the principal curvatures, N is the layer normal and ? · N is the
dS K1 (?⊥ · n)2 + K2 (n · ?⊥ × n ? q0 )2 + K3 [n × (?⊥ × n)]
the non-linear terms arising from properly constructed, covariant derivatives do not come in. We couple these two free energies together by adding a term which favors the alignment of the layer normal with the molecular axis: Fmn = γ dS 1 1 ? (n · N )
where ?⊥ refers to a gradient in the tangent plane of the membrane. At quadratic order,
? The total free energy is the sum F = Fm + Fn + Fmn . In the Monge gauge, we may
represent the surface as a height function h(x, y ) over the co¨ ordinates x and y . In this case the layer normal is N = [??x h, ??y h, 1] / 1 + (?⊥ h)2 . Expanding the nematic director √ about the ground state direction n = (? z + δ n) / 1 + δn2 , the free energy to quadratic order in the deviations from the ground state becomes F = 1 2 dxdy γ (?⊥ h + δ n) + κ ?2 ⊥h
2 2 ? + Fn .
?⊥ h (which follows when γ → ∞), or, in other words, if Note that if we require δ n = ?? It is only by relaxing the constraint that n be parallel to N that we see manifestations of local membrane geometry will never show the e?ects of chiral molecules. The Lα phase is In the Lβ ′ phases, there is an equilibrium projection c of the the director n onto the tangent plane, and there are manifestations of chirality in the free energy involving c [8,9,10], for example, through a term of the form for a number of interesting modulated structures in and of the membrane. c · ?⊥ × c. These terms are responsible we require n||N, the resulting e?ective free energy has no chiral term since ?⊥ × δ n = 0. chirality. Thus, free energies of the Lα phase of membranes (with n||N) based only on the the analog of the smectic-A phase in which all twist and bend have been expelled .
The free energy (2.4) is very similar to that for smectic-A liquid crystals, although the functions are restricted to depend only on the membrane co¨ ordinates. If we consider a screw-like con?guration of the surface h(x, y ) = b arctan(y/x) (where b is the “Burgers vector” of the defect), then we can calculate the energy of the associated con?guration, after relaxing δ n0 = ?⊥ h (i.e., n = N). This con?guration, however, is singular at the origin and is not allowed. To facilitate the screw-like defect the membrane must cease to exist in the core, or, in other words, there will be exposed membrane edges around a solvent-?lled cylindrical core. Thus, in the presence of a screw defect we must now add a line energy. This line energy comes from two pieces: the ?rst is the energy cost of having an exposed edge, while the second comes from any surface terms in the free energy, for instance the Gaussian curvature. We group these energies together into a term: Fe = ?
where C is the inner edge of the helicoidal surface. radius ξ and the twist penetration depth λ = 2 Far away from the cylindrical core, δ n = b [y, ?x, 0] /(x2 + y 2 ). Between the core K2 /γ , the nematic director relaxes to its
large radius value. This means that the core has a double-twist texture, as found in liquid crystalline blue phases. A perfect helicoidal surface has no mean curvature, thus our defect does not contribute to Fm . The free energies per turn are (with λ = Fmn = λ2 1 2 γb π ln 1 + 2 2 ξ (2.6) K2 /γ ):
Fn = ?K2 q0 b Fe = ? b2 + (2πξ )2,
can be negative: this chooses the sign of b. Minimizing the sum of the constituent free energies, we ?nd that there will always be a non-zero value of b for which the energy is minimized. In the two limits (b ? ξ or b ? ξ ) we ?nd: ? 2πξK2 q0 ? ? 2 2 2 ? ? 2π ξγ ln λ /ξ + 1 + ? ? ? ? ? K2 q0 ? ? πγ ln λ2 /ξ 2 + 1 b? ? ξ (2.7) b? ? ξ.
where, in the London (type II) limit, ξ ? λ. To minimize the sum, note that only Fn
At the same time, ξ is determined also via minimizing the free energy. We thus have γ 2 λ4 (b? )4 (b? )2 + (2πξ ? )2 = 16π 2 ?2 λ2 + (ξ ? )2
(ξ ? )4 .
The optimal value of ξ = ξ ? will determine the size of the hole in the membrane. Unlike a layered system where b is quantized in units of the spacing, b varies continuously with K2 q0 . When a defect appears, the membrane must continue wrapping around into a many-layered helicoidal surface. Not to do so would create a line of exposed molecules leading out from the center of the defect r = 0 to the boundary r = ∞. The line energy ? presumably contains both energetic and entropic factors. In princi-
ple, ? can be negative at the temperatures we are considering. If that were the case, even the non-chiral membrane, made of racemic mixture of molecules would unbind, leading to a randomly defective, layered structure. A lyotropic with a negative line tension would never really form a single layer structure and would have defects proliferating at the molecular level leading to a random unstructured collection of amphiphillic molecules. In the absence of any interactions that keep the membrane away from itself , there is no lowest energy state since the defect strength b need not be quantized for a single membrane. Entropic repulsion will balance against the energy gain of the defect, setting 3
the value of b. The entropic repulsion will scale as b?2 leading to some b? that will minimize the total free energy above the lower critical chirality. However, the entropic interaction will scale as the area of the bilayer whereas the line energy will scale like the length. The detailed balance between entropic repulsion and defect energy will thus be size dependent. It is also possible that a preferred, spontaneous torsion along the free edge would lead to a speci?c value for b . Thus it should be possible for chiral lyotropic lamellae to exhibit isolated helicoidal fragments.
3. Lamellar Melting and TGB Phases When membranes are stacked together, they can form multi-layered, lamellar structures, similar to smectic-A phases in thermotropic liquid crystals. In highly swollen systems, the layer spacing d is determined by entropic repulsion . As in thermotropic smectics, the Lα phase excludes twist . Likewise, a su?ciently strong chirality can favor the entry of twist into the Lα phase in the form of a TGB phase consisting of periodically repeated twist-grain boundaries composed of periodic arrays of screw dislocations . Though we know of no report of an experimental observation of this phase, we see no reason why it could not exist. Transitions in lyotropic systems are driven predominantly via changes in concentration rather than changes in temperature. We can determine the surfactant concentration, or equivalently the layer spacing d, at which the Lα phase ?rst becomes unstable with respect to a proliferation of dislocations by calculating the point at which the total energy per unit length of dislocation ?rst becomes negative. This calculation is analogous to the calculation of HC 1 in a superconductor – it establishes the mean-?eld instability of the Lα phase to the formation of a TGB phase. In the limit d ? ξ , we ?nd from (2.6) the total free energy per unit length of a dislocation is F = 1 d λ ξ
γd2 π ln
? K2 q0 d + ?|d| ,
where the pre-factor of 1/d converts from free energy per turn to free energy per unit length . Note that in (2.2), the elastic constants are membrane elastic constants, related to the bulk elastic constants by a factor of 1/d, the inverse layer spacing, and consequentially they do not change as the density changes (assuming that density changes are made most 4
easily by changes in the layer spacing). The density is inversely proportional to the layer spacing, and so, against simple intuition, the energy of the defect (with b = d) decreases with increasing density. This is a simple result: as the layer spacing increases, the elastic energy of each distorted lamella grows since it must be distorted more to connect the consecutive layers. We thus predict that with increasing density the lamellar structure will there will be a critical spacing d? (critical density ρ? ∝ 1/d? ) below which (above which) d? = K2 q0 ? ? . πγ ln(λ/ξ ) be pocked with defects above a lower critical chirality. For a given value of K2 q0 ? ? > 0
the defects will penetrate. From (3.1), we ?nd:
In general, a transition will occur when d is reduced to twice the preferred Burger’s vector of the free membrane (2.7). If ?uctuations are unimportant, then d? is the layer spacing at which there is a second-order transition from the Lα to the TGB phase. Of course, it is possible that ?uctuations are strong enough to destroy any TGB phase
? that might form and that the Lα phase transforms to a chiral NL phase with twisted
orientational order like that of a cholesteric. This phase is the liquid crystal analog of the melted vortex lattice of superconductors in a magnetic ?eld , which may intervene between the Meissner phase and the vortex phase of a type-II superconductor. It is a cholesteric phase formed by melting the TGB dislocation lattice while still retaining shortrange smectic order. Heat capacity measurements provide strong evidence of this phase in thermotropic systems . Possible phase diagrams based on the analogy between this system and high-TC superconductors are shown in Figure 2. We propose that the layers melt via dislocation loop unbinding, as in the smecticA-to-nematic transition . Here, however, the chiral bias will cause one handedness of screw dislocation to be preferred over the other. This bias should change a secondorder-like unbinding transition to a ?rst-order transition: microscopic defect loops can no longer unbind smoothly since a loop must contain an equal number of left and right handed screws. Either the defect loops will unbind and the dissident part of the loop will move to the boundary, or, equivalently, the appropriate dislocations will nucleate from the boundary. The chiral interaction also changes the simple unbinding picture in another way: from the above we can see that the free energy di?erence per unit length between a wronglyhanded screw and a correctly-handed screw will be: ?F/L = 2|K2 k0 b|. 5 (3.3)
We thus expect that, in the presence of thermal ?uctuations, at the unbinding transition of the defect loops, the largest size dislocation loop will have a characteristic length L? = kB T . 2 K 2 | k0 | d (3.4)
As the chirality of the molecules becomes small, i.e. as k0 → 0, the screw dislocations will grow to in?nite size and will proliferate, as in the usual non-chiral scenario. We can estimate the latent heat per molecule via a scaling argument. If we assume that without chirality there is a second-order smectic-A-to-nematic transition (or nearly second order), then the transition will be rounded out when the correlation length is on the order of L? . In this ?uctuation dominated “type II” limit the transition from smectic order to isotropic order would proceed via an intermediate cholesteric-like phase (either TGB or
? NL ) as shown in Figure 2(a). We believe that the most likely candidate for this phase is ? the NL phase  of thermotropic smectics . It is also possible that the isotropic phase
intervenes before a TGB phase ever forms, as is shown in the phase diagram of Figure 2(b). For ?xed d? we may calculate λ2 /ξ 2 . In the limit that d? ? ξ ? and λ ? ξ ? , we have: π 3 K2 γ 1 = 3 2 3. (ξ ? ) ln (λ/ξ ) (K2 q0 ? ?) λ2 Thus we expect extreme type II behavior, i.e.
large. The transition from the lamellar to TGB phases would presumably be second-order strong ?uctuations.
large when K2 q0 ≈ ? or when K2 γ is very
? while the transition from the lamellar to the NL phase would be ?rst-order due to the
In the latter case we can estimate the scaling behavior of the free energy per unit volume at the transition by cutting o? a the transition at the size L? . Scaling gives us for the entropy per unit volume: ? = TC ?f ? (L? )?1/ν ?T
? |K2 k0 |(1?α)/ν ,
where f is the free energy density, t is the reduced temperature, ν is the correlation length exponent and α is the usual speci?c heat exponent of the nearby second-order transition. Note that this result based on dislocation loop unbinding gives the same scaling result in terms of k0 as that obtained previously via Landau theory . Since the defects in the lamellar phase are necessarily accompanied by holes in the layers, X-ray scattering should provide a view of the dislocation unbinding transition leading 6
to a chiral TGB or cholesteric phase. As the dislocation loops grow to L? , the scattering √ will show liquid like peaks at q⊥ = 4π/( 3L? ) ∝ k0 d, where q⊥ is the wavevector perpen-
dicular to the layer normals. Though there will be a proliferation of holes in each layer, only the holes coming from the same dislocation loop will be correlated, leading to an
X-ray structure with peaks at the characteristic inverse length scale. This could be tested by changing the concentration of chiral molecules (or, equivalently, adjusting the ratio of left-handed to right-handed enantiomers), as well as by adjusting the layer spacing. Finally we mention that it is possible for the lamellar stacks to be in a “type I” limit. In this case the melting from the stacked phase to the isotropic phase will occur at one place: there will be no intervening nematic or cholesteric-like phase. We would expect this transition to be ?rst-order and would not occur via dislocation loop unbinding. Instead we would expect this transition to occur via dislocations nucleating at the boundary. We also note that in the zero chirality limit there are no known transitions from lamellar structures to nematic structures. In the chiral case, however, we have seen that chiral mesophases arise via the presence of chiral topological defects.
4. Acknowledgments It is a pleasure to acknowledge stimulating discussions with R. Bruinsma, S.T. Milner, D. Nelson, P. Nelson, T. Powers, D. Roux, C. Sa?nya and J. Toner. This work was supported in part by NSF Grants DMR94-23114.
References  A.B. Harris, R.D. Kamien and T.C. Lubensky, in preparation (1996).  T.C. Lubensky, R.D. Kamien and H. Stark, to appear in Mol. Cryst. Liq. Cryst. (1996) [cond-mat/9512163].  S.R. Renn and T.C. Lubensky, Phys. Rev. A 38 (1988) 2132; 41 (1990) 4392.  W. Helfrich, J. Phys. (Paris) 39 (1978) 1199; D.R. Nelson and J. Toner, Phys. Rev. B 24 (1981) 363.  D. Roux, private communication (1995).  W. Helfrich, Z. Naturforsch. 28C (1973) 693; P. Canham, J. Theor. Biol. 26 (1970) 61.  P.G. de Gennes, Solid State Commun. 14 (1973) 997.  S. Langer and J. Sethna, Phys. Rev. A 34 (1986) 5035; G.A. Hinshaw, R.G. Petschek and R.A. Pelcovits, Phys. Rev. Lett. 60 (1988) 1864.  J.V. Selinger and J.M. Schnur, Phys. Rev. Lett. 71 (1993) 4091; J.V. Selinger, Z.G. Wang, R.F. Bruinsma and C.M. Knobler, Phys. Rev. Lett. 70 (1993) 1139.  P. Nelson and T. Powers, Phys. Rev. Lett. 69 (1992) 3409; J. Phys. II (Paris) 3 (1993) 1535.  W. Helfrich, Z. Naturforsch. 33A (1978) 305.  W. Helfrich, J. Chem. Phys. 85 (1986) 1085.  We thank J. Toner for discussions on this and many other points.  D.R. Nelson, Phys. Rev. Lett. 60 (1988) 1973; D.R. Nelson and H.S. Seung, Phys. Rev. B 39 (1989) 9153.  T. Chan, C.W. Garland and H.T. Nguyen, Phys. Rev. E, 52 (1995) 5000; L. Navailles, unpublished (1995).  R.D. Kamien and T.C. Lubensky, J. Phys. I (Paris) 3 (1993) 2131.  T.C. Lubensky, J. Phys. (Paris) C1 (1975) 151.
Figure Captions Fig. 1. A screw dislocation in a lyotropic lamellar phase. Note that the core is devoid of all molecules and is ?lled with solvent. In thermotropic smectics, the core is nematic and contains the mesogens. Fig. 2. Possible phase diagrams for the lyotropic system as a function of chirality q0 and surfactant density ρ. Solid lines indicate ?rst-order transitions while dashed lines indicate second-order (or weakly ?rst-order) transitions. (a) Lyotropic with ? four distinct phases: lamellar, TGB, NL and isotropic. (b) It is possible that the isotropic phase intervenes before the TGB phase occurs. The phantom TGB region is shown with hatched lines: it will never appear.
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* NL /Cholesteric
* /Cholesteric NL
(a) Figure 2