pubs.acs.org/Langmuir © 2010 American Chemical Society
Micromechanics Model of Liquid Crystal Anisotropic Triple Lines with Applications to Self-Assembly Alejandro D. Rey* and E. E. Herrera Valencia Department of Chemical Engineering, McGill University, 3610 University Street, Montreal, Quebec, Canada H3A 2B2 Received May 17, 2010. Revised Manuscript Received June 25, 2010 Directed self-assembly of mesophases at three phase contact lines has been reported for a variety of solutions, including micelles, tobacco mosaic virus, DNA, silk, and others, through the action of capillary forces, wetting processes, and/or evaporation. This communication presents a new micromechanical line-excess model of the anisotropic contact line tension for nematic liquid crystal phases, which incorporates well characterized liquid crystal interfacial tensions. The anisotropic line tension is then used to formulate the contact line torque that promotes the azimuthal orientation, widely reported experimentally. The dependence of the line torque strength on the contact angle reveals the conditions that promote azimuthal orientational ordering close to the contact line. This work is limited to anisotropic line-excess tension, and wetting and evaporation processes are outside its scope.
1. Introduction Anisotropic triple lines arise when one of the three contacting bulk phases is anisotropic,1 such as a liquid crystal in contact with a solid substrate and a vapor.2 Anisotropic triple line phases are described by its geometry, including curvature and torsion, and its orientational order.2 This is analogous to anisotropic liquid crystal interfaces that are described by geometric bending and torsion in addition to the interfacial orientation.3 An essential feature of liquid crystal interfaces is the anisotropic component of the interfacial tension, known as the anchoring energy,4-9 that generates interfacial bending stresses and intrinsic interfacial torques.2 The interfacial torques vanish at a preferred state known as the easy axis when the director orientation description is adopted.2 The easy axis has been shown to generate a directed self-assembly mechanism in biological chiral liquid crystals.10 For example, the so-called plywood architecture found in collagen, cellulose, and chitin fiberbased natural materials self-assemble at the substrate, when the rodlike components exceed a critical concentration threshold; this occurs when the reduction in free energy due to excluded volume overcomes the orientational entropy effect.10 Analogously, significant evidence from various material systems indicates that the triple line is also involved in directed self-assembly, such that monodomain defect-free mesophases align their characteristic orientational order axis along the contact line.11 Examples of contact line self-assembly include hexagonal, cubic, and lamellar *To whom correspondence should be addressed. E-mail: alejandro.rey@ mcgill.ca. (1) Rusanov, A. I. Colloids Surf., A 1999, 156, 315. (2) Rey, A. D. Soft Matter 2007, 2, 1349. (3) Rey, A. D. Langmuir 2006, 22(1), 219. (4) Sonin, A. A. The Surface Physics of Liquid Crystals; Gordon and Breach Publishers: Amsterdam, 1995. (5) Jerome, B. Surface Alignment. In Handbook of Liquid Crystals; Demus, D., Goodby, J., Gray, G. W., Spiess, H.-W., Vill, V., Eds.; Wiley-VCH:Weinheim, 1998; Vol. 1. (6) Yokoyama, H. Handbook of Liquid Crystal Research; Oxford University Press: New York, 1997. (7) Sluckin, T. J.; Poniewierski, A. In Fluid Interfacial Phenomena; Croxton, C. A., Ed.; Wiley: Chichester, 1986. (8) Brake, J. M.; Mezera, A. D.; Abbott, N. L. Langmuir 2003, 19(16), 6436. (9) Amundson, K. R.; Srinivasarao, M. Phys. Rev. E 1998, 58, 1211. (10) Rey, A. D. Soft Matter 2010, published online April 9, http://dx.doi.org/ 10.1039/b921576j. (11) Lee, Y. S. Self-Assembly and Nanotechnology: A Force Balance Approach; John Wiley & Sons, Ltd.: Hoboken, NJ, 2008; pp 42-44.
Langmuir 2010, 26(16), 13033–13037
surfactant liquid crystals,11 nematic order in DNA solutions,10,12-14 tobacco mosaic virus,15 and solutions of gold nanorods;16 Figure 1 shows a schematic side view of self-assembled rods at the contact line with (left) nematic order and (middle) hexagonal order, and a (right) top view of a drop indicating that the orientational order is azimuthal and parallel to the contact line (Ctl); the dots in the left and middle panels are the cross sections of the rodlike molecules. In most reported cases,10-15 the self-assembly is driven by evaporation, the contact line is pinned, the contact angle decreases with time, and the mesophase arises when the concentration exceeds a critical value; the Onsager threshold10 for nematic order is jL/D ≈ 4, where j is the volume fraction, L the length, and D the diameter of the rodlike components. For nematic self-assembly of drying droplets of DNA solutions over solid substrates, it has been shown13 that the azimuthal orientation minimizes the bulk elasticity as well as the planar anchoring at the DNA/substrate and at the DNA/vapor interfaces. Related multiple coffee ring patterns that form from evaporation are the result of flow processes with depinned contact lines.16,17 In this communication, (i) we quantify the contact line driven by directed nematic liquid crystal self-assembly by first developing a micromechanical model18,19 for the contact line excess energy, whose anisotropic contribution is given in terms of well-established interfacial anchoring energy;2,4-6 (ii) the anisotropic contact line free energy is then used to demonstrate that elastic torques reorient the rods along the azimuthal coordinate; orientation effects acting on a nanocylinder at a fluid interface have been characterized.20 We (12) Morii, N.; Kido, G.; Suzuki, H.; Morii, H. Biopolymers 2005, 77, 163. (13) Smalyukh, I. I.; Zribi, O. V.; Butler, J. C.; Lavrentovich, O. D.; Wong, G. C. L. Phys. Rev. Lett. 2006, 96, 177801. (14) Smalyukh, I. I.; Butler, J. C.; Shrout, J. D.; Parsek, M. R.; Wong, G. C. L. Phys. Rev. E 2006, 78, 030701. (15) Wargacki, S. P.; Pate, B.; Vaia, R. A. Langmuir 2008, 24, 5439. (16) Sharma, V.; Park, K.; Srinivasarao, M. Colloidal dispersion of gold nanorods: Historical background, optical properties, synthesis, separation and self-assembly; Material Science and Engineering Reports, 2009, 65, 1 (17) Bhardwaj, R.; Fang, X.; Attinger, D. New J. Phys. 2009, 11, 075020. (18) Edwards, D. A.; Brenner, H.; Wasan, D. T. Interfacial Transport Processes and Rheology; Butterworth-Heinemann: Boston, MA, 1991. (19) Slattery, J. C.; Sagis, L.; Oh, E.-S. Interfacial Transport Phenomena, 2nd ed.; Springer-Verlag: New York, 2007. (20) Lewandowski, E. P.; Searson, P. V.; Stebe I, K. J. J. Phys. Chem. B 2006, 110, 4283.
Published on Web 07/14/2010
DOI: 10.1021/la1019668
13033
Letter
Rey and Herrera Valencia
Figure 1. Schematic of contact line (Ctl) directed self-assembly promoting azimuthal orientation in nematic (left) and hexagonal (middle) phases. Azimuthally oriented nematic phases have been reported for DNA solutions,12-14 and hexagonal phases11 are expected to be found in surfactant liquid crystals. (Right) Top view highlighting the azimuthal alignment effect of the contact line; short rods indicate mesophase orientation.
Figure 2. (Left) Schematic of the geometry and nematic orientation in the contact line region. See text. (Right) Cross-section of the contact line region used to calculate line excess properties; S(i) denotes the area and L(ij) the interfacial length within the contact line region. V, vapor; N, nematic; S, subtrate.
assume that orientational order is well-defined and that the three phase contact line is a large circle lying on a flat rigid substrate. The main objective of this communication is to develop an expression for the contact line tension that contains both isotropic and anisotropic contributions and is capable to describe contact line azimuthal self-assembly (see Figure 1, right). The presented work builds on a generic line tension expression presented in ref 10. The formulated line tension is then used to derive the contact line torque and to identify the anchoring parameters that drive the azimuthal self-assembly. Lastly, the dependence of the line torque on the contact angle is established, demonstrating that the larger the contact angle the stronger the torque promoting azimuthal self-assembly. The analysis of the orientation-dependent line excess properties assumes a given contact angle; couplings between line shape and line structure, between line/interfaces/bulk, and flow effects are outside the scope of this communication. This Letter is organized as follows. Section 2 presents the micromechanical liquid crystal theory, the derivation of the anisotropic contact line energy, and its application of contact line azimuthal selfassembly. The conclusions in section 3 emphasize the novelty of the results, including the connection between wetting and orientation, and the role of interfacial anchoring on contact line self-assembly.
2. Theory and Self-Assembly Model Figure 2 (left) shows the geometry in the triple line region (R(i); i = vapor, nematic, substrate) and the intersection of the nematic/ vapor (NV), the vapor/substrate (VS), and the nematic/substrate (NS) interfaces, ν(ij)(ξ(ij)) are interfacial unit tangent (normal) 13034 DOI: 10.1021/la1019668
vectors, and 2R is the contact angle. Since in this model the orientation n at the contact line is well-defined, we adopt the assumption2 that in the contact line region the orientation is homogeneous and its projection on the plane normal to the contact line (b-p plane) is R. The geometry of the flat circular contact (or triple) line is described by the Frenet-Serret formulas. The principal frame is (t, p, b), were t = p b is the unit tangent, p = b t is the unit principal normal, and b = t b is the binormal unit vector.2 Nematic ordering is defined by the symmetric traceless tensor order parameter:2 Q ¼ μn nn þ μm mm þ μl l l ;
Q ¼ QT ;
trðQÞ ¼ 0
ð1Þ
where the isotropic state corresponds to zero eigenvalues (μn = μm = μl = 0), the nematic uniaxial (U) state to two equal eigenvalues, and the nematic biaxial (B) state to three different eigenvalues. The orientation is defined completely by the orthogonal director triad (n, m, l). The uniaxial and biaxial scalar order parameters that measure molecular alignment are, respectively S ¼ 3ðnn : QÞ=2 ¼ 32 μn ; P ¼ 3ðmm - l l Þ : Q=2 ¼ 3ðμm - μl Þ=2 2.1. Torque and Force Balance. The triple line force and torque balance equations that define the shape (contact angle) and structure (orientation and molecular order) are2 0 B shape equations : @rl :Tl þ
X jun
I ν 3 Ts þ
¼ 0, v ¼ p, b
1 C ðξ 3 Tb Þ dl A 3 v
jun
ð2aÞ
Langmuir 2010, 26(16), 13033–13037
Rey and Herrera Valencia
Letter
structure equation: - ε : Tl þ rl 3 Cl þ
X jun
I ν 3 Cs þ
ðξ 3 Cb Þ dl ¼ 0 jun
ð2bÞ where Ti, i = b, s, l are the 3 3 bulk, 2 3 interface, and 1 3 line stress tensors; Ci; i = b, s, l are the bulk, interface, and line couple stress tensors, ε is the alternator tensor, Γl = -ε:Tl þ rl 3 Cl is the line torque, jun stands for the junction of the three interfaces at the contact line, ξ(ν) is an interface unit normal (tangent), and rl = tt 3 r is the line gradient. The total free energy F of the nematic liquid crystal due to bulk fb, surface fs, and contact line fl is assumed to have these classical dependencies:2 Z
Z
F ¼
f ðijÞ Q, ξðijÞ dA s
f b ðQ, rQÞ dV þ bulk
Z
f l ðQ, tÞ dl
S ðVNSÞ
Z
- tt 3
TðVSÞ s¥ ds - tt 3
ðVSÞ
L
ð3Þ
contact line
L
ð4aÞ
2 f bh ðQÞ ¼ a1 tr Q2 - a2 tr Q3 þ a3 tr Q2
ð4bÞ
ðijÞ
ðijÞ
ð4cÞ
ðijÞ
f ðijÞ san ¼ z11 ξ 3 Q 3 ξ þ z20 Q
ðijÞ
ðijÞ
: Q þ z21 Q 3 ξ 3 Q 3 ξ þ z22 ðξ 3 Q:ξÞ2
ð4dÞ
where the subscript h (g) denotes homogeneous (gradient), {ai}, i = 1, 2, 3 are the Landau coefficients (energy/volume), Li are gradient elasticity coefficients, f(ij) siso is the isotropic interfacial is the interfacial anchoring energy, and z(ij) tension, f(ij) san 11 are the anchoring coefficients (energy/area). With the adopted interfacial energies, the line and interface couples Cl = Cs = 0 and the structure eq 2b simplifies to -ε:Tl = 0; eq 3 describes a contact line in which rl Q, rsQ have negligible energetic contributions. 2.2. Micromechanical Theory. Next we use micromechanics to find the expression for the excess line tension fl (Q,t).10 According to micromechanics of contact lines,17 under negligible curvature, a line excess quantity Φl is given by Z
ðΦb - Φ¥b Þ dA -
Φl ¼ SðABCÞ
Z LðACÞ
ΦðACÞ s¥ ds
Z
Z
Df l Df ¼ t l Dn Dt
ð9Þ
which implies that locking the orientation may result in contact line reshaping and that locking the shape may result in reorientation. Comparing eqs6 and 7, we find the line stress tensor coefficients (fl, Bp, Bb) in terms of the bulk and interface stresses: f l ¼ tt 8 > Z Z : SðVNSÞ ðVNÞ ðVSÞ L
Z L
TðNSÞ s¥ ds
ðNSÞ
L
9 > =
ð10aÞ
> ;
Df l ¼ - pt Bp ¼ p 3 Dt 9 8 > > Z Z = > ; : S ðVNSÞ ðVNÞ ðNSÞ L
L
ð10bÞ
LðABÞ
ð5Þ
LðBCÞ
where S(ABC) is the cross-sectional area of the three-phase region and L(ij) {ij = AB, AC, BC} are the lines defined by the intersection of the three interfaces Σij{ij = AB, AC, BC} with S(ABC). Adapting the scalar eq 5 to the excess stress tensor Langmuir 2010, 26(16), 13033–13037
ð8Þ
The principle of rotational invariance yields the equivalence between the torque on the contact line (n ∂fl /∂n) and the torque on the director (t ∂fl/∂t):
ΦðABÞ s¥ ds
ΦðBCÞ s¥ ds
ð6Þ
ðNSÞ
where the first term is the usual line tensor of isotropic materials and (Bp, Bb)are the coefficients of the bending stresses due to anisotropic contributions (∂fl /∂t 6¼ 0) that give rise to the line torque:2
-n
ðijÞ f ðijÞ s ðQ, ξÞ ¼ f siso þ f san ,
TðNSÞ s¥ ds
Γl ¼ - ε : Tl ¼ Bp b - Bb p
f b ¼ f bh ðQÞ þ f bg ðrQÞ
L1 L2 tr rQ2 þ ðr 3 QÞ 3 ðr 3 QÞT þ 2 2 L3 þ Q : rQ : ðrQÞT 2
LðVNÞ
The excess line stress tensor Tl is decomposed into tension (tt) and bending components (tp, tb):2 Df Df Tl ¼ f l tt þ Bp tp þ Bb tb, Bp ¼ p 3 l , Bb ¼ b 3 l Dt Dt
2 The bulk fb and interfacial f (ij) s energy densities are
f bg ðrQÞ ¼
Z
ð7Þ
interfaces
þ
(A, vapor; B, nematic; C, substrate) gives the anisotropic excess line stress tensor Tl:10 Z Z Tl ¼ tt 3 ðTb - Tb¥ Þ dA - tt 3 TðVNÞ ds s¥
Df b 3 l ¼ - bt Dt 8 9 > > Z Z > : S ðVNSÞ ; ðVNÞ ðNSÞ
Bb ¼
L
L
ð10cÞ DOI: 10.1021/la1019668
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Rey and Herrera Valencia
where we assumed that the VS interface has no anchoring contribution. Recalling that the nematic bulk Tb¥ and inter2 facial stress tensors T(ij) s¥ are Df b¥ : ðrQÞT Tb¥ ¼ f b¥ I ð11aÞ DrQ ðijÞ TðijÞ s¥ ðQ, ξ Þ
¼
ðijÞ f ðijÞ s Is
- IðijÞ s
3
Df ðijÞ s DξðijÞ
we find the tensions (eq 12b,c) as excess interfacial terms Z Z ðVNÞ ðVSÞ χl iso ¼ ðf siso - f ðVNÞ ðf siso - f ðVSÞ s¥ Þ ds þ s¥ Þ ds LðVNÞ
ðijÞ
ξ ; ðijÞ
L
Z
ð11bÞ
f l ¼ χl iso þ χl an
L
ð12aÞ
ZZ 0 B -@
Z
L
ðVNÞ f siso¥
S
Z
ðVSÞ f siso¥
ds þ
ðVNÞ
L
ds þ
ðVSÞ
ðVNÞ
L
ðijÞ
C dsA
ðijÞ
ð12bÞ
L
χl an ¼
S
VNS
B ðf ban - f ban¥ Þ dA - @
Z
f ðVNÞ san¥
C f ðVSÞ san¥ dsA
ds þ
LðVNÞ
ðijÞ
where χliso (χlan) is the isotropic (anisotropic) contribution, where the distortion stress component along t is zero, and where the bulk energy is decomposed into isotropic and anisotropic parts: fb = fbiso þ fban. In eq 12, we assumed that that the bulk and surface stress tensor close to the contact line have the same functionality with respect to the free energies as indicated in eq 11. According to micromechanics, the anisotropic line anchoring χlan (eq 12c) is then given in terms of excess interfacial anchoring energy. 2.3. Azimuthal Self-Assembly. Next we specify this generic expressions (eq 12) to the case of homogeneous (Figure 2a) orientation (rn = 0), to uniaxial states (P = 0), and to the director confinement in the contact line region.2 We take the projection of n on the (b-p) plane to be2 cos R ¼ p 3 ðI - ttÞ 3 n
S ðVNSÞ
ðf biso - f b¥ ÞdA ¼
ðVNÞ
f siso ds þ LðVNÞ
ZZ
Z S
ðVNSÞ
Z
f ban dA ¼
ðVSÞ
LðVSÞ
LðVNÞ
13036 DOI: 10.1021/la1019668
Z
f siso ds þ
f ðVNÞ san ds þ
Z LðVSÞ
ðijÞ
μ2
ðNSÞ
f siso ds LðNSÞ
f ðVSÞ san ds
ð1þtan - 2 RÞ2
,
ðijÞ
ðijÞ
¼ -
1 ðijÞ ðijÞ ðijÞ ðijÞ ðijÞ ¼ z11 S e þ ðz21 - 2z22 ÞS e 2 , μ4 ¼ z22 S e 2 ð17a-fÞ 3
Using eq 4d, the surface free energy far from the contact line f (ij) b¥ becomes f ðijÞ s¥
¼
1 ðijÞ ðijÞ ðijÞ ðijÞ 2 2 - z11¥ S e þ 6z20¥ S e þ ðz21¥ þ z22¥ ÞS e 3 3
1 ðijÞ f siso¥ þ
ð18Þ where z(ij) nm¥ denote values far from the contact line. Replacing the interfacial energies eqs 17 and 18 in eqs 8 and 16, we find the line energies χliso, χlan, and torque Γl:
ð13Þ
Thus, we consider that the director n glides in a plane whose inclination from p is half the contact angle, that is, R (see Figure 2, left). We assume that at the contact line Q = S e(nn - I/3), where S e is the fixed equilibrium value of S. Since far from the contact line we consider planar orientation at both interfaces, fban¥ = 0. Expressing the bulk contributions (fbiso - fb¥) and fban in terms of interfacial f(ij) sm terms: Z
ðijÞ
μ0
ðijÞ
μ2 þ μ4
μ2 μ4 ðijÞ , , β ¼ ð1 þ tan - 2 RÞ 4 ð1þtan - 2 RÞ2 1 1 ðijÞ ðijÞ 2 ðijÞ ðijÞ ðijÞ 2 ¼ f siso þ - z11 S e þ 6z20 S e þ ðz21 þ z22 ÞS e , 3 3
ðijÞ
β2
LðVSÞ
ð12cÞ
ZZ
βðijÞ o ¼ μ0 þ
1
Z
ð16Þ
where the coefficients β(ij) 2k are given in terms of the anchoring coefficients z(ij) mn and the half contact angle R: ðijÞ
ZZ
ð15bÞ
ðVSÞ
ðijÞ
ðNSÞ
0
ðf ðVSÞ san Þ ds
ds þ
f ðijÞ ¼ β0 þ β2 ðt 3 nÞ2 þ β4 ðt 3 nÞ4 s
1 ðNSÞ f siso¥
Z
Using eq 4d, the interfacial free energy densities f(ij) s [(ij) = VS, SN], appearing in eq 15, can be written in terms of (t 3 n):
ðf biso - f biso¥ Þ dA Z
ð15aÞ
ðNSÞ
ðf ðVNÞ san Þ
χl an ¼
the excess line tension free energy fl (eq 10a) becomes
ðVNSÞ
ðNSÞ
ðf siso - f ðNSÞ s¥ Þ ds
þ
¼ ðNVÞ, ðSNÞ
χl iso ¼
LðVSÞ
Z
χl iso ¼ βo , χl an ¼ β2 ðn 3 tÞ2 þ β4 ðn 3 tÞ4 Z
ðVSÞ
ðf siso - f ðVSÞ s¥ Þ ds þ
β0 ¼ LðVSÞ
ðSNÞ
þ
ðβ0 ðSNÞ
ðSNÞ
β2 ðSNÞ
L
- f ðNSÞ s¥ Þ ds, β2 ¼
L
þ
ðVNÞ
ðβ0 LðVNÞ
Z
Z
Z
Z ds, β4 ¼ ðVNÞ
L
Z
ðVNÞ
ðVNÞ
β2 L
β4
- f ðVNÞ s¥ Þ ds
ðVNÞ
Z
ds þ
ðSNÞ
β4 L
ds
ds,
ðSNÞ
ð14aÞ
Γl ¼ 2β2 ðt:nÞ þ 4β4 ðt 3 nÞ3 ðp 3 nÞb - ðb:nÞp ð19a-dÞ
ð14bÞ
Introducing n 3 t = cos ψ in eq 19, we find the three equilibria states that minimize fl: (a) orientation along t:ψ = 0, sin ψ = 0, β2/2β4 > 0, β2 < 0; (b) orientation normal to t:ψ = π/2, cos Langmuir 2010, 26(16), 13033–13037
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Letter
ψ = 0, β2/2β4 > 0, β2 > 0 ; (c) oblique orientation, 0 < n 3 t < 1: cos2 ψ = -β2/2β4, -1 < β2/2β4 < 0, β2 < 0. At the three orientation equilibria, we find that the line torque (eq 19d) Γl = 4β4(β2/2β4 þ cos2 ψ) sin ψ cos ψ(cos Rb - sin Rp) vanishes. Since planar orientation at the VN and SN interfaces are (ij) obtained when (β(ij) 2 ,β4 ) are positive, the present model predicts that azimuthal self-assembly at the contact line (ψ = 0, β2/2β4 > 0, β2 < 0) is promoted by this interfacial anchoring effect. In addition, the interfacial effect that promotes azimuthal contact line director orientation increases with increasing contact angle and achieves its maximum at Γl max ðR ¼ π=2Þ ¼ - 4β4 ðβ2 =2β4 þ cos2 ψÞ sin ψ cos ψp ð20Þ at R = π/2; the minimum torque is Γlmin(R = 0) = 0. Hence, for droplet evaporation under pinned contact lines where R decreases with time, the early stage regime is predicted to have the strongest effect on azimuthal assembly.
3. Conclusions In summary, a new micromechanical tensorial model (eqs 6, 14, and 15) for liquid crystalline phases was formulated to obtain an expression for the anisotropic contact line tension (eq 19). Although eq 19a can be stated as a phenomenological expansion, this work reveals its interfacial origin (eq 17) through the micromechanical approach (eq 15). According to eqs 17 and 19, the anisotropic line
Langmuir 2010, 26(16), 13033–13037
tension coefficients β2 and β4 that control azimuthal self-assembly are functions of the interfacial anchoring coefficients μ2 and μ4, and the Landau anchoring coefficients (eq 4d), and therefore, the extensive existing knowledge of liquid crystal interfaces offers a rational approach to control 1D self-assembly. The model was used to derive an expression for line torques (eq 19d) that is essential to describe reorientation at a contact line. The analysis of line torque (eq 19d) reveals that planar interfacial anchoring promotes the azimuthal orientation, a fact not previously assigned to a line tension effect. The analysis also reveals that the torque that drives the reorientation is a maximum at the nonwetting limit (eq 20) and a minimum at the total wetting limit. This relation between wettability and azimuthal self-assembly can be used to manipulate orientation through the substrate, chemistry, and geometry. For contact line directed azimuthal selfassembly in droplet evaporation under pinned contact lines, it is predicted that the early stage of the process has the strongest effect on reorientation. Lastly, it is noted that the present model and predictions of azimuthal self-assembly by taking into account anisotropic contact line tension completes the model and predictions of ref 13 which are based on bulk elasticity and interfacial anchoring. Acknowledgment. This work is supported by the Natural Science and Engineering Research Council of Canada. E.E.H. V. gratefully acknowledges financial fellowship support from CONACYT-MEXICO (Postdoctoral Grant 000000000124600).
DOI: 10.1021/la1019668
13037
