1308
Langmuir 2001, 17, 1308-1309
Line Tension Anisotropy in Two-Dimensional Uniaxial Crystals F. N. Braun* Department of Chemistry, Heriot-Watt University, Edinburgh EH14 4AS, United Kingdom H. Yokoyama
The locking term underpins the anchoring potential predicted below. If we set g ) 0, reflecting an unlocked hexatic liquid crystal phase or a simple nematic phase, there is no associated anchoring effect (see also ref 5). To derive line tension from H, we consider the partition function for a system confined between two flat parallel surfaces at y ) 0 and y ) h in Cartesian coordinates r ) (x,y)
Zh )
Electrotechnical Laboratory, 1-1-4 Umezono, Tsukuba, Ibaraki 305, Japan Received May 22, 2000. In Final Form: October 31, 2000
Many two-dimensional molecular systems exhibit, in addition to uniform bulk phases, a range of microphaseseparated equilibrium textures with lamellar- or bubbletype morphology.1 A key parameter governing such textures is the line tension τ across the interface between coexisting constituent phases. In the case of a uniaxial crystalline phase, line tension at a domain edge is generally sensitive to the orientation φ of the uniaxial director c with respect to the cut. The surface may be said to “anchor” the director along an “easy axis” φe,2 such that for small deviations δφ ) |φ - φe| , 1, we can identify a harmonic component of the line tension ∼δφ2, i.e.
τ(φ) = τ(φe) +
W 2 δφ 2
(1)
Pettey and Lubensky3 have pointed out the relation to linear bulk elastic terms in the space derivatives ∂Rcβ of the director. If the director is everywhere uniform, such terms can be transformed by Gauss’s theorem into an effective anchoring potential. In this paper, we show that surface anchoring is also related to the so-called “Frank” elastic terms of bulk nematic symmetry. These are quadratic in ∂Rcβ and do not transform by Gauss’s theorem. Rather, we find that they generate an effective anchoring potential at the level of thermal renormalization of the line tension. We begin with a director fluctuation Hamiltonian of the form4
1 H) 2
∫ dr[Ks(∇‚δc)
2
2
where Ks and Kb are, respectively, the splay and bend Frank constants of nematic elasticity. H pertains to a uniaxial crystal in which the director has nematic inversion symmetry, i.e., c ) -c, and fluctuations are locally damped, or “locked”, to bond orientations of the crystal (by the term ∼δc2). It also describes locked hexatic liquid crystalline phases, in which bond-orientational order persists, despite the absence of translational order. (1) Seul, M.; Andelman, D. Science 1995, 267, 476. (2) This terminology is standard in analogous contexts at 3D nematic surfaces, see e.g.: Yokoyama, H. In Handbook of Liquid Crystal Research; Collings, P., Patel, J. S., Eds.; Oxford University Press: Oxford, 1996. (3) Pettey, D.; Lubensky, T. C. Phys. Rev. E 1999, 59, 1834. (4) Neglecting higher-order coupling to phonons and bond-orientational fluctuations, this is essentially the Hamiltonian adopted by Nelson and Halperin in their study of critical phenomena in tilted smectic monolayers: Nelson, D. R.; Halperin, B. I. Phys. Rev. B 1980, 21, 5312.
(3)
The fluctuation contribution τ˜ to line tension per unit length L follows as
τ˜ ) lim -kBT hf∞
∂ ln Zh ∂L
(4)
In the following we calculate Zh for arbitrary h, and then take h f ∞. The advantage of this approach, as opposed to calculating Z∞ directly, is that it allows us to exploit a mathematically useful analogy with the 1D quantum oscillator.6 We begin by expanding δc(r) in Fourier modes along x, parallel to the surface
δc(r) )
1 L
c˜ (qx,y) exp(iqxx) ∑ q x
Each mode contributes a partition function factor
Zh,qx )
∫D{c˜(qx,y)} exp(-Hh,q /kBT)
(5)
x
where, using eq 2, and defining φ with respect to the surface normal
Hh,qx )
1 2
∫0h dz{[(Ks cos2 φ + Kb sin2 φ)qx2 + g]c˜2 +
(∂y∂c˜ ) } (6)
(Ks sin2 φ + Kb cos2 φ)
2
We rewrite eq 7
Zh,qx )
2
+ Kb(∇ × δc) + gδc ] (2)
∫D{δc(r)} exp(-Hh/kBT)
∫ dc˜ 0 dc˜ 1Gh,q (c˜ 1,c˜ 0)
(7)
x
with
Gh,qx(c˜ 1,c˜ 0) )
˜ (h))c˜ ∫c˜c(0))c ˜
1
0
D{c˜ (z)} exp(-Hh,qx/kBT) (8)
The kernel Gh,qx satisfies an equation of the Schroedinger form for the 1D quantum oscillator
[
]( )
ωqx ∂2 ∂ Gh,qx(c˜ ,c˜ 0) ) βqx-1 2 - βqx c˜ 2 Gh,qx(c˜ ,c˜ 0) ∂h 2 ∂c˜
(9)
with initial condition
G0,qx(c˜ ,c˜ 0) ) δ(c˜ - c˜ 0)
(10)
(5) Braun, F. N.; Yokoyama, H. Eur. Phys. J. B 1999, 9, 377. (6) This analogy has been pointed out previously, in the context of director fluctuations in 3D confinement: Ajdari, A.; Duplantier, B.; Hone, D.; Peliti, L.; Prost, J. J. Phys. II 1992, 2, 487.
10.1021/la000696+ CCC: $20.00 © 2001 American Chemical Society Published on Web 01/20/2001
Notes
Langmuir, Vol. 17, No. 4, 2001 1309
and where we define
ωqx )
(
It is instructive to estimate the magnitude of W we might expect in an experimental situation. Let us assume on dimensional grounds
)
(Ks cos2 φ + Kb sin2 φ)qx2 + g Ks sin2 φ + Kb cos2 φ
1/2
g ∼ kBT/σ2
βqx ) (kBT)-1[(Ks sin2 φ + Kb cos2 φ)((Ks sin2 φ + 2
2
Kb cos φ)qx + g)]
1/2
(11)
with σ a characteristic molecular length. For the elastic constants, we adopt (see ref 8)
K ∼ η2F2
The solution is well-known (see, e.g., ref 7) p)∞
Gh,qx(c˜ ,c˜ 0) )
∑e
-ωqx(p+1/2)h
ψp(c˜ )ψp*(c˜ 0)
(12)
p)0
with eigenfunctions
ψp(c˜ ) )
( )
βqx 1 p 1/2 π (2 p!)
1/4
e-βqxc˜ /2Hp((βqx)1/2c˜ ) 2
(13)
Substitution of eq 12 into eq 7 yields after integration
Zh,qx )
[ (
)]
βqx 1 - e-2ωqxh 4π e-ωqxh
-1/2
(14)
Hence we obtain for the line tension, according to eq 4
τ˜ ) lim -kBT hf∞
)
kBT 2
∂/∂L
∂ ∂L
ln Zh,q ∑ q
gkBT 2L
∑q
ln(βq /4π) ∑ q x
(15)
2
(16)
which become in the thermodynamic limit L f ∞
kBTqc τ˜ *(φ) 4π
(17)
where qc is an ultraviolet elastic cutoff, and
τ˜ *(φ) )
tan-1(K h s cos2 φ + K h b sin2 φ)1/2 (K h s cos2 φ + K h b sin2 φ)1/2
(18)
is dimensionless, with K h R ) qc2KR/g. For Ks > Kb, τ˜ is a minimum at φmin ) 0. Conversely, for Kb > Ks, we have φmin ) π/2. Expansion about this minimum to lowest order in ∆K ) |Ks - Kb| yields
τ˜ (φmin + δφ) = τ˜ (φmin) +
W 2 δφ 2
(19)
with
W)
kBT g ∆K 2π K K
( ( ))
1/2
tan-1[qc(K/g)1/2]
ηF(/kBT)1/2
∼ kBT/σ ∼ O(10-7 dyn)
WPL/W ∼ (/kBT)1/2Fµ2/η
1 (Ks cos φ + Kb sin2 φ)q2 + g
kBT/σ
(21)
This is the same order of magnitude as one expects by a similar dimensional argument for the isotropic part of the line tension.5 We can also compare it with Pettey and Lubensky’s anchoring potential, WPL ∼ Kl, with Kl a linear bulk elastic constant. Writing WPL ∼ Kl ∼ µ2F2/σ, with µ a chiral or dipolar order parameter, we have
x
x
τ˜ (φ) )
W∼
x
To relate this to the surface anchoring form, eq 1, we focus on the φ-dependent part, τ˜ (φ). We calculate
τ˜ (φ) )
where 0 < η < 1 is the nematic order parameter, 0 < F < 1 is the molecular packing fraction, and is the anisotropic part of the molecule-molecule pair interaction energy. Typically for a system exhibiting nematic order, ∼ 100kBT.5,8 Hence, setting ∆K/K ∼ 1, kBT ∼ 10-14 erg, and σ ∼ 10-7 cm, we estimate
(22)
In a crystal exhibiting dipolar order µ, the nematic order parameter is also defined, η ∼ µ.8 Hence the two contributions are of comparable magnitude, WPL ∼ W. However, in the lower symmetry of a nematic crystal, only η is nonzero, such that W is the dominant anchoring effect. To summarize, we have used partition function statistical mechanics to calculate the contribution of thermal Frank elastic modes to line tension in 2D uniaxial crystals with nematic or dipolar order. We find that this contribution is anisotropic in the sense that it depends on the orientation of the director with respect to the cut. We have cast the anisotropy in the form of a surface anchoring potential, favoring orientation along a “thermal easy axis” which is either planar or perpendicular to the surface according to the relative magnitudes of the bend and splay elastic constants. The anisotropy is a consequence of coupling between the elastic modes and bond orientations of the crystal and contrasts with the situation in pure 2D nematic liquid crystals, where, in the absence of such coupling, there is no thermal surface anchoring effect.5 It presents a more general mechanism for surface anchoring than that proposed by Pettey and Lubensky, insofar as it persists in the absence of chiral or polar elastic terms. Hexatic liquid crystalline phases have been realized experimentally in Langmuir monolayers (reviewed in ref 9) and free-standing smectic films (reviewed in ref 10). However, we are not aware of any detailed experimental investigation of line tension in these systems as yet, with which to explicitly compare our predictions. LA000696+
(20)
(7) Feynman, R. P.; Hibbs, A. R. In Quantum Mechanics and Path Integrals; McGraw-Hill: New York, 1965.
(8) Braun, F. N.; Yokoyama, H. J. Chem. Phys. 1998, 109, 1196. (9) Kaganer, V. M.; Moehwald, H.; Dutta, P. Rev. Mod. Phys. 1999, 71, 779. (10) Stoebe, T.; Huang, C. Int. J. Mod. Phys. B 1995, 9, 2285.
