Langmuir 2006, 22, 219-228
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Mechanical Model for Anisotropic Curved Interfaces with Applications to Surfactant-Laden Liquid-Liquid Crystal Interfaces Alejandro D. Rey* Department of Chemical Engineering and McGill Institute of AdVanced Materials, McGill UniVersity, Montreal, Quebec, Canada H3A 2B2 ReceiVed July 21, 2005. In Final Form: NoVember 4, 2005 A mechanical model for anisotropic curved interfaces, applicable to thermodynamically closed surfactant-laden liquid-liquid crystal interfaces is developed. The model takes into account the mechanical effects due to surface bending and surface tilting (anchoring) and incorporates liquid crystal anisotropy into classical fluid membrane mechanics. In the absence of the aligned liquid crystal, the model converges to the fluid membrane mechanical model, and in the absence of surfactant, it converges to the nematic interface mechanical model. Use of the well-known HelfrichRapini-Papoular surface energies leads to the Laplace equation for anisotropic curved interfaces, whose material limits are the vesicle shape equation and the liquid crystal Herring equation. Applications of the model to shape selection in liquid drops embedded in aligned nematic liquid crystals illustrates the competition between anchoring and bending and shows how anisotropic surface tension distorts the droplet and how bending tends to restore the spherical shape. This theory presented in this article shows that the interaction of interfacial anchoring and bending creates new regimes in classical fluid membrane mechanics.
1. Introduction Surfactant-laden interfaces between isotropic liquids and nematic liquid crystals1-4 are examples of anisotropic curved interfaces, where both the bending energy arising from the surfactant layer and the anchoring energy emerging from the relative orientation between the liquid crystal and the unit surface normal play important roles. Curved anisotropic interfaces are of increasing interest from fundamental and applied points of view. Practical applications of curved anisotropic surfaces include the well-known surfactant ability to control liquid crystal surface orientation.1-4 It is well known that surfactants have the ability to promote specific orientations (anchoring) at liquid crystalsubstrate interfaces; in certain surface concentration ranges, surfactants promote a homeotropic (along the unit surface normal) average orientation of the rigid-rod molecules that form the nematic liquid crystal phase.1-5 More recently, detailed studies in liquid-liquid crystal interfaces have demonstrated that surfactant adsorption leads to surface anchoring transitions from planar (parallel to the interface) to homeotropic.5 In addition to surface orientation control, the role of surfactants in reducing the interfacial tension offers new opportunities in the formation of polymer-dispersed liquid crystals,6 offering a pathway to droplet size control. The formation and stability of liquid crystalline emulsions is a current topic of interest in multiphase liquid crystalline materials.7 Fundamental issues of curved anisotropic interfaces are of current interest because of the need to predict droplet shapes in nematic emulsions.4 Current work has been based on free-energy minimization of thermodynamically closed liquid drops embedded * E-mail:
[email protected]. (1) Gupta, V. K.; Skaife, J. J.; Dubrovsky, T. B.; Abbott, N. L. Science 1998, 279, 2077. (2) Brake, J. M.; Abbott, N. L. Langmuir 2002, 18, 6101. (3) Lockwood, N. A.; de Pablo, J. J.; Abbott, N. L. Langmuir 2005, 21, 6805. (4) Lishchuk, S. V.; Care, C. M. Phys. ReV. E 2004, 70, 011702. (5) Rey, A. D. Langmuir 2004, 20, 11473. (6) Drzaic, P. S. Liquid Crystal Dispersions, World Scientific Publishing Company: Singapore, 1995. (7) Loudet, J. C.; Richard, H.; Sigaud, G.; Poulin, P. Langmuir 2000, 16, 6724.
in aligned nematic matrixes.4 The challenge in this area is to integrate the classical mechanical and thermodynamical models of membranes8-11 and liquid crystals.12-25 For membranes, the important ingredient in the formulation of these models8-11,26 is the curvature energy, and for liquid crystals, it is the anchoring energy.12 Hence, for surfactant-laden liquid-liquid crystal interfaces both surface bending and tilting must be included. For curved interfaces, it has been shown that both the thermodynamic and mechanical liquid and solid membrane models give equivalent descriptions.9 For liquid crystals, it has also been demonstrated that thermodynamic and mechanical models give compatible descriptions.13 In this article, we use a static mechanical model (8) Ljunggren, S.; Eriksson, J. C.; Kralchevsky, P. J. Colloid Interface Sci. 1997, 191, 424. (9) Kralchevsky, P. A.; Eriksson, J. C.; Ljunggren, S. AdV. Colloids Interface Sci. 1994, 48, 19. (10) Kralchevsky, P. A.; Nagayama, K. Particles at Fluid Interfaces and Membranes; Elsevier: Amsterdam, 2001. (11) Pozrikidis, C., Ed. Modeling and Simulation of Capsules and Biological Cells; Chapman and Hall/CRC Press: Boca Raton, FL, 2003. (12) Yokoyama, H. Handbook of Liquid Crystal Research; Collins, P. J., Patel, J. S. Eds.; Oxford University Press: New York, 1997; Chapter 6, p 179. (13) Rey, A. D. J. Chem. Phys. 2004, 120, 210. (14) Jerome, B. Handbook of Liquid Crystals; Demus, D., Goodby, J., Gray, G. W., Spiess, H.-W., Vill, V., Eds.; Wiley-VCH: Weinheim, Germany, 1998; Vol. 1,. (15) Sluckin, T. J. and Poniewierski, A. In Fluid Interfacial Phenomena; Croxton, C.A., Ed.; Wiley: Chichester, U.K., 1986; Chapter 5. (16) Faetti, S. In Physics of Liquid Crystalline Materials; Khoo, I.-C., Simoni, F., Eds.; Gordon and Breach: Philadelphia, 1991; Chapter XII, p 301. (17) Barbero, G.; Durand, D. In Liquid Crystals in Complex Geometries; Crawford, G. P., Zumer, S., Eds; Taylor and Francis: London, 1996; p 21. (18) Sonin, A. A. The Surface Physics of Liquid Crystals; Gordon and Breach Publishers: Amsterdan, 1995. (19) Osipov, M. A.; Hess, S. J. Chem. Phys. 1993, 99, 4181. (20) Sen, A. K.; Sullivan, D. E. Phys. ReV A 1987, 35, 1391. (21) Papenfuss, C.; Muschik, W. Mol. Cryst. Liq. Cryst. 1995, 262, 561. (22) Ericksen, J. L. In AdVances in Liquid Crystals; Brown, G. H., Ed.; Academic Press: New York, 1979; Vol. 4, p 1. (23) Virga, E. G. Variational Theories for Liquid Crystals; Chapman Hall: London, 1994. (24) Jenkins, J. T.; Barrat, P. J. Appl. Math. 1974, 27, 111. (25) de Gennes, P. G.; Prost, J. The Physics of Liquid Crystals, 2nd ed.; Oxford University Press: London, 1993. (26) Eriksson, J. C.; Ljunggren, S. In Surface and Interfacial Tension; Hartland, S., Ed.; Surfactant Science Series 119; Marcel Dekker: New York, 2004; pp 547-614.
10.1021/la051974d CCC: $33.50 © 2006 American Chemical Society Published on Web 12/03/2005
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based on force and torque balance equations in conjunction with constitutive equations derived using thermodynamic functions. Membrane mechanical models are based on interfacial force and torque balance equations governing the unit surface normal k, augmented by constitutive equations for curvature-dependent interfacial tension. These and equivalent thermodynamic models have been widely studied, and the models have been used to predict droplet and vesicle shapes, interfacial waves, and phase stability, among other things.9-11,26 Liquid crystal interfacial mechanical models are based on force and director torque balance equations augmented by constitutive equations for the anchoring energy that takes into account preferred liquid crystal surface orientation;23 here director denotes the average molecular orientation and is a unit vector: n‚n ) 1.23,29 Mechanical and thermodynamical models have been used to predict surface orientation5,12 and droplet shapes,3,23 among other things. Hence, it is clear that to describe membranes in contact with liquid crystals both the torques and couples acting on the director n and the surface unit normal k have to be considered. In this article, we contribute to the eventual formulation of the general theory for membranes in contact with liquid crystals by first developing a simpler model in which the director orientation is fixed in space and hence only the surface unit normal needs to be considered as a variable. When considering liquid crystal emulsions, the fixed director condition may be attained for sufficiently small droplets, typically in the submicrometer range,3 when bulk distortions are energetically more costly that surface tilting or when a sufficiently strong magnetic field aligns the liquid crystalline phase because of its anisotropic magnetic susceptibility.25 For curved interfaces, bending gives rise to the moment tensor M, and gradients of the moment tensor (∇s‚M) contribute to bending stresses.9,10 In addition, gradients of the bending stresses (∇2s M) give rise to capillary pressure.9,10 For liquid crystal interfaces, the interfacial tension is anisotropic (γ(k)), changes in surface tilting (∂γ/∂k) create bending stresses,29 and surface gradients of bending stresses (∇s(∂γ/∂k)) give rise to capillary pressure; the term (∂γ/∂k) is known as the Cahn-Hoffman capillary vector.30,31 Thus, for curved anisotropic interfaces the term (∇s‚M) must be integrated with (∂γ/∂k) into a generalized capillary vector. In addition, for curved anisotropic interfaces the capillary pressure (∇2s M) must be integrated with (∇s(∂γ/ ∂k)). This article develops a mechanical model that integrates curvature and anisotropy. The objectives of this article are (i) to develop a mechanical model for curved anisotropic interfaces that combines membrane mechanics and liquid crystal anchoring; (ii) to generalize the Cahn-Hoffman vector to curved anisotropic interfaces; and (iii) to demonstrate the role of curvature and anchoring in shape selection in liquid droplets embedded in an aligned nematic liquid crystal matrix. The article builds on previous extensive work on membrane mechanics9-11,27,28,33,34 and liquid crystal interface mechanics.12-24,35-39 The organization of the article is as follows. Appendix I presents the differential geometry used. Section 2 presents the interfacial force balance and torque balance equations and introduces the interfacial stress tensor and interface molecular field appearing (27) Povstenko, Y. Z. J. Colloid Interface Sci. 1991, 144, 497. (28) Waxman, A. M. Stud. Appl.Math. 1984, 70, 63. (29) Rey, A. D. Phys. ReV. E 2003, 67, 011706. (30) Cheong, A. G. Rey, A. D. J. Chem. Phys. 2002, 117, 506. (31) Cheong, A. G.; Rey, A. D. Phys. ReV. E 2002, 66, 021704. (32) Eliassen, J. D. Ph.D. Thesis, University of Minnesota, 1963. Eliassen, J. D. University Microfilms, Ann Arbor, MI, 1983. (33) Sagis, L. M. C. Physica A 1997, 246, 591. (34) Jenkins, J. T. SIAM J. Appl. Math. 1977, 32, 755.
Rey
in the balance equations. The interface considered in this article is embedded in a 3D space. The principle of rotational invariance is used in conjunction with the torque balance equation to develop expressions for the interfacial stress tensor, the moment tensor, the couple tensor, and the liquid crystal torque; the details of the derivations are given in Appendix II. Section 3 presents a reformulation of the interfacial stress tensor in terms of normal, shear, and bending stresses. Section 4 presents a generalization of the Cahn-Hoffman capillary vector for anisotropic curved interfaces and gives a compact form of the interfacial stress tensor. Section 5 present the final forms of the normal (Laplace equation) and tangential interfacial force balance equations. The Laplace equation is shown to have a compact form when using the generalized Cahn-Hoffman capillary vector. Under constant density, the tangential force balance equation is shown to imply the thermodynamic surface Euler equation. Section 6 presents generic constitutive equations needed to use the force balance equation presented in section 5. Section 7 presents particular constitutive equations for the Helfrich-Rapini-Papoular interface and derives the Laplace equation for this interface. The derived Laplace equation is shown to include the well-known vesicle shape equation and the liquid crystal Herring equation. Section 8 presents an application of the theory developed in sections 2-7 and analyzes, using the Laplace equation, the shape of liquid drops immersed in an aligned nematic liquid crystal matrix. Section 9 presents the conclusions.
2. Interfacial Balance Equations In this article, we consider the isothermal mechanics of a thermodynamically closed surfactant-laden interface S between an isotropic fluid and an undistorted nematic liquid crystal phase. As mentioned above, the nematic liquid crystal order is described by the director n.25 The director n points everywhere along, say, the z axis of a reference rectangular coordinate system, n ) δˆ z, where δˆ z denotes a unit vector. The outward unit normal to S is k, directed from the isotropic phase into the nematic phase. The isotropic phase is denoted by the index “1”, and the nematic phase, by “2”. Appendix I presents the differential geometry used in this article. (i) Interfacial Stresses and Interfacial Force Balance Equation. In this section, the basic force balance is introduced,27,28 and the nature of the stress tensors is defined. The 3 × 3 stress tensors in the bulk phases are denoted by Tb(1) and Tb(2), and the 2 × 3 interfacial stress tensor Ts is given by the sum of a tangential contribution and a bending contribution:
Ts ) Ts| + Ts⊥
(1a)
Ts| ) TRβ s aRaβ
(1b)
Ts⊥ ) TR(n) s aR k
(1c)
R, β ) 1, 2
(1d)
where | denotes the tangential plane and ⊥ denotes the interface normal direction. The 2 × 2 tangential stress tensor TRβ s describes normal (stretching) and shear stresses, wheras TR(n) is s the 2 × 1 bending stress tensor.40 The interfacial static force balance equation is the balance between interfacial forces and the bulk stress jump40,41 (35) Rey, A. D. Langmuir 2003, 19, 3677. (36) Rey, A. D. Liq. Cryst. 2001, 28, 549. (37) Forest, M. G.; Wang, Q. Phys. Lett. A 1998, 245, 518. (38) Yue, P. T.; Feng, J. J.; Liu, C.; Shen, C. J. Fluid Mech. 2004, 515, 293. (39) Cermelli, P.; Fried, E.; Gurtin, M. E. Arch. Ration. Mech. Anal. 2004, 174, 151.
Mechanical Model for Anisotropic CurVed Interfaces
∇s‚Ts + k‚[Tb(2) - Tb(1)] ) 0
Langmuir, Vol. 22, No. 1, 2006 221
(2)
where ∇s(•) ) Is‚∇(•) is the surface gradient operator and Is ) I - kk is the surface projection tensor. Expressing ∇s‚Ts in component form, eq 2 becomes9,10,27,28 R(n) β R(n) bR)aβ + (TRβ (TRβ s | R - Ts s bRβ + Ts |R)k + k‚[Tb(2) - Tb(1)] ) 0 (3)
where |R denotes the covariant derivative, bRβ represents the components of the surface curvature tensor b ) -∇sk, and aβ represents the covariant basis vectors. Projecting eq 3 into tangential and normal components yields the interfacial force balances:9,10,27,28 Rβ nn nn (TRn s |R + Ts bRβ) + [Tb(2) - Tb(1)] ) 0
(4a)
nβ nβ Rn β (TRβ s |R - Ts bR) + [Tb(2) - Tb(1)] ) 0
(4b)
R, β ) 1, 2
(4c)
Equation 4a is used to derive the Laplace equation. Equation 4b is the tangential force balance equation. (ii) Interfacial Molecular Field and Torque Balance Equation. In this section, (1) we show that the interfacial torque balance equation imposes certain restrictions on the interfacial stress tensor Ts because its components are involved in generating couples and torques, and (2) we derive the relation between the bending stress and molecular field. The classical concept of molecular field used in liquid crystal theories42 is introduced and is shown to be a direct link between stresses and torques. The general interfacial torque balance equation is given by the sum of the torques acting on the surface unit normal k
Γs + ΓLC ) 0
ΓLC )
ΓRLCaR
R ) 1, 2
where aλR represents the components of the surface matrix tensor and βR represents the covariant components of the surface alternator Es. The normal component of the dual Tsx is nonzero if its tangential component Ts| is asymmetric, whereas the tangential component of the dual Tsx is nonzero if the bending stress Ts⊥ is not zero. By substituting eqs 7 and 8 into eq 5, the torque balance equation finally reduces to
Es‚(Ts⊥‚k) - (Es:Ts|)k + ∇s‚Cs + ΓLC ) 0
Aˆ ) Aˆ (F, k, ∇sk)
(6c)
dAˆ ) -
ξ| γ M dF - :d∇sk + dk 2 F F F
Tsx ) -:Ts
(7b)
where Tsx is the vector dual of Ts, is the 3D alternator tensor, and Cs is the 2 × 3 surface couple arising from surface interactions. As shown in Appendix II (eq II.20), the dual Tsx has tangential and normal components (40) Edwards, D. A.; Brenner, H.; Wasan, D. T. Interfacial Transport Processes and Rheology; Butterworth-Heinemann: Boston, 1991. (41) Slattery, J. C. Interfacial Transport Phenomena; Springer-Verlag: New York, 1990. (42) Rey, A. D. Phys. ReV. E 2000, 61, 1540.
∂Aˆ ∂F ∇sk,k
(
)
γ ) -F2
(12a)
( ) ( )
∂Aˆ M)- F ∂∇sk
∂Aˆ ∂b
) F
F,k
(12b)
F,k
∂Aˆ ∂k F,∇sk
(
ξ| ) FIs‚
)
(12c)
The bending moment tensor M is a symmetric tangential tensor: M‚k ) k‚M ) 0 and M ) MT. Using the surface Euler equation41 for a thermodynamically closed interface, we find
Aˆ (F, k, b) )
γ(F, k, b) + constant F
(13)
where γ(F, k, b) is the interfacial tension. The bending moment tensor M and the tangential capillary vector ξ| are also defined by
(∂b∂γ) ∂γ ξ ) (I ‚ ) ∂k M)
|
(7a)
(11)
where γ is the interfacial tension,41 M is the bending moment tensor,9,10 and ξ| is the tangential component of the capillary vector,36 given by
where ΓnLC ) 0 because any director orientation around k is equivalent. The interfacial torque vector Γs is given by surface stress asymmetry and by gradients of surface couple stresses Cs,40
Γs ) Tsx + ∇‚Cs
(10)
where F is the surface mass density, its differential gives
(6a) (6b)
(9)
Next we separate the relations between the stress components (Ts⊥, Ts|) and the torques (∇s‚Cs + ΓLC) using the concept of the molecular field or vector direction along which torques vanish.42 Positing a Helmholtz free energy per unit mass Aˆ of the form34
(5)
where Γs is the interfacial torque generated from surface interactions and ΓLC is the torque acting on the unit normal k by the contacting nematic liquid crystal phase. The general forms of these two interfacial vectors are
Γs ) ΓRs aR + Γns k
Rβ βλ Tsx ) (TR(n) s aλR)aβ - (Ts βR)k ) Es‚(Ts⊥‚k) - (Es:Ts|)k (8)
(14a)
F,k
s
F,∇sk
(14b)
Following previous work on anisotropic interfaces,42 the tangential surface molecular field h| is defined as the variational derivative of the Helmholtz free energy A:
(
)
δA F∂Aˆ F∂Aˆ ) -(ξ| + Is‚(∇s‚M)) ) Is‚ + ∇s‚ δk ∂k ∂∇sk (15)
h| ) -Is‚
Using the principle of rotational invariance23 of the free energy to arbitrary rotations of the entire system, it is shown in Appendix II (eq II.29) that the following expression relating h|, M, and ξ| holds:
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s‚(h|k)‚k + ks:(M‚b) - ∇s‚(M‚s) + k × ξ| ) 0 (16) Comparing term by term the torque balance equation (eq 9) and expression 16 from rotational invariance, we find (Appendix II) the following expressions for stresses (Ts⊥, Ts|), couples (Cs), and torques (ΓLC)
Ts⊥ ) h|k
(17a)
Ts| ) -M‚b + γIs
(17b)
Cs ) -M‚Es
(17c)
ΓLC ) k × ξ|
(17d)
where the symmetric tangential contribution γIs is the usual40,41 normal stress tensor. The derivation of eq 17a uses the facts that k‚Ts⊥ ) 0 and Ts⊥ ) Ts⊥‚kk. The derivation of eq 17b uses the fact that Ts| ) (Ts|)T. Equation 17c is the definition of the interfacial couple stress. Equation 17d is the classical torque expression given in terms of the capillary vector ξ|. The molecular field h| is shown to generate bending stresses Ts⊥, and according to eq 17a, it is just the bending stress vector
h| ) -(ξ| + Is‚(∇s‚M)) ) Ts⊥‚k
(18)
In Boussinesq fluid interfaces,41 bending stresses are absent, and no molecular field exists. In the absence of surfactant, eq 18 is consistent with the expression for the pure liquid crystal interface. Next, we comment on the restrictions imposed by the torque balances on the form of the stresses. Inserting eqs 17a-d into the torque balance equation (eq 9) yields
Es‚(Ts⊥‚k) - (Es:Ts|)k - ∇s‚(M‚Es) + Es‚ξ| ) 0 (19) Re-expressing the third term of eq 19 gives
∇s‚(M‚Es) ) Is‚∇s‚(M‚Es) + (M‚Es:b)k
(20)
interface tensor, we find
1 1 γtension ) Is:Ts ) γ - M:b 2 2
The nature of γtension can be elaborated further by using thermodynamic arguments that eventually lead to a more transparent form of eq 23. Using the Euler surface equation (eq 13), the differential of the interfacial tension reads
dγ ) M:db + ξ|‚dk
Equations 21 show that the torque balance equation imposes the following restrictions on the interfacial stress tensor components:
Ts⊥‚k + ξ| + Is‚∇s‚(M) ) 0
(22a)
Ts| + M‚b ) (Ts| + M‚b)T
(22b)
In partial summary, in this section we have used the interfacial torque balance equation and rotational invariance to develop expressions for the stress, moment tensor, couple tensor, and liquid crystal torque and have found the relation between the molecular field and the bending stresses.
3. Interfacial Stress Tensor This section reformulates the expressions (eqs 17a and 17b) of the interfacial stress tensor into a format that is needed when discussing interfacial tension. According to eqs 1a, 17a, and17b, the final expression of the interfacial stress tensor is
Ts ) γIs - M‚b + h|k
(23)
Because the interfacial tension is defined by half the trace of the
(25)
This differential indicates that the interfacial tension can be expanded as follows
1 γ ) γ0 + γ1(H) + M:b + γan(n‚k) 2
(26)
where subscripts (0, 1) indicate the order in H and where γan(n‚k) is the anchoring energy present in liquid crystal interfaces;12 the exact functionality {γ1; M; γan} is given by the adopted constitutive equations. In the absence of, respectively, liquid crystals and surfactants, the expressions are
1 γ ) γ0 + γ1(H) + M:b 2
γ ) γ0 + γan(n‚k)
(27)
where H is the average curvature defined in eq I.14 in Appendix I. According to eq 23, the interfacial stress tensor Ts| can be decomposed into a trace, a traceless part, and the bending stress
1 Ts ) γ - (M:b) Is - M‚b + h|k 2
(
)
(28)
where the overbar on the second term (M‚b) indicates the traceless tensor. Under limiting conditions in the absence of both liquid crystals and surfactants, the interfacial stress becomes, respectively
Replacing this result in eq 19 gives the following normal and tangential equations:
Es‚(Ts⊥‚k + ξ| + Is‚∇s‚(M)) ) 0 Es:[(Ts|) + (M‚b)]k ) 0 (21)
(24)
Ts ) (γ0 + γ1)Is - M‚b - Is‚(∇s‚M)k
(29a)
Ts ) (γ0 + γan)Is - ξ|k
(29b)
and
Equation 29a is consistent with membrane stress,9,10 and eq 29b, with liquid crystal interface stress.42 The most significant difference between surfactant-laden and anisotropic interfaces is the presence of shear stress in the former. The origin of bending stresses is due to curvature in the former and orientation in the latter.
4. Generalized Cahn-Hoffman Capillary Vector for Curved Interfaces The Cahn-Hoffman capillary vector ξ was introduced to describe anisotropic interfacial tension effects without considering curvature effects, primarily for crystal-crystal interfaces.43 The capillary vector approach to anisotropic interfaces was then adapted to describe soft nematic liquid crystal interfaces30,31 and was subsequently used to analyze Rayleigh capillary instabilities in liquid crystal filaments.30 In this article, we generalize the Cahn-Hoffman capillary vector by including curvature effects because they are significant in surfactant-laden liquid-liquid crystal interfaces. The generalized capillary vector is denoted by Ξ. (43) Cahn, J.W.; Hoffman, D.W. Acta Metall. 1974, 22, 1205.
Mechanical Model for Anisotropic CurVed Interfaces
Langmuir, Vol. 22, No. 1, 2006 223
Γ ) -k × h|
In the present formulation, the capillary vector Ξ is first decomposed into normal Ξ⊥ and tangential Ξ| components:
Ξ ) Ξ⊥ + Ξ|
(30)
The relation between interfacial tension γ and the capillary vector Ξ is then given by
∫Sγ dS ) ∫SΞ‚k dS ) ∫SΞ⊥ dS
(
∂γ k + r∇sγ ∂r
)
Ξ⊥ ) (γ0 + γ1)k
(32)
1 ∂γ ∂γ 1 ∂γ ∂H ∂γ ∂K ∂γ 1 )r ) r +r H - K ) - M:b ∂r 2 ∂H ∂r ∂K ∂r 2 ∂H ∂K 2 (33)
)
where the last equality follows from eqs I.14 and I.15. Expressing ∇sγ in eq 32 as a function of b, we get
∇sγ ) -(ξ| + (∇s‚M))‚b + {Is‚(∇s‚(M‚b))}
(34)
where the term in brackets is an edge term. For a sphere of radius r, the curvature tensor is b ) HIs. Introducing the equality (rb) ) -Is into eq 34 gives
r∇sγ ) ξ|‚Is + (∇s‚M)‚Is
(35)
Introducing eqs 33 and 35 into eq 32 finally yields the capillary vector Ξ:
1 Ξ ) γ - (M:b) k + (ξ| + (∇s‚M))‚Is 2
(
)
(36)
Equation 36 was efficiently derived using a locally spherical surface, but its final tensor expression is valid for any surface; more general derivations that start with nonspherical surfaces must lead to the same result but with much lengthier manipulations. The normal component of the capillary vector Ξ⊥ is
1 Ξ⊥ ) γ - (M:b) k 2
(
)
(37)
According to this expression, the free-energy density associated with the creation of a unit interface is
1 Ξ⊥‚k ) γ - M:b 2
(38)
which agrees with eq 24. Equations 15 and 36 show that the tangential component of the capillary vector Ξ| is the negative of the molecular field:
Ξ| ) -h| ) (ξ| + (∇s‚M))‚Is
Ξ| ) (∇s‚M)‚Is
(41a,b)
Ξ| ) ξ|
(42a,b)
and
For a sphere, H ) -1/r and K ) 1/r2, and hence
(
which is consistent with the meaning of the interfacial molecular field.36 The limiting expressions of the capillary vector components in the absence of both liquid crystals and surfactants, are, respectively,
(31)
The normal component Ξ⊥ describes the free-energy cost on increasing the surface area.30 However, the torque acting on the unit surface normal k is given by the tangential component Ξ|: Γ ) k × Ξ| and indicates that the free energy is anisotropic and a function of interface orientation k.30,31 Next, consider a locally spherical surface of radius r, unit normal k, and position vector r ) rk. The capillary vector Ξ for curved anisotropic interfaces is defined by
Ξ ) ∇(rγ) ) γ + r
(40)
(39)
The torque Γ ) k × Ξ| acting on the unit normal k is then
Ξ⊥ ) (γ0 + γan)k
Equations 42a and 42b are consistent with the liquid crystal Cahn-Hoffman ξ vector.30,31 The main distinction between the capillary vector of these two limiting interfaces is best illustrated in flat interfaces (H ) 0, b ) 0), where eq 41b predicts Ξ| ) 0, but eq 42b predicts Ξ| * 0. Next, we express the interfacial stress tensor Ts in terms of the capillary vector Ξ. In practical applications, the most useful aspect of the capillary vector Ξ is that it allows us to use a compact form of the interfacial stress tensor leading to a transparent expression for the capillary pressure. Introducing the capillary vector Ξ into the interfacial stress tensor Ts (eq 28) gives
Ts ) Ξ⊥Is - Ξ|k - M‚b ) Ξ‚P - M‚b
(43)
where the geometric tensor
Ρ ) kIs - Isk
(44)
maps the capillary vector Ξ into the tangential stress Ξ⊥Is and bending stress Ξ|k. As shown below, the capillary pressure is simply ∇‚Ξ. In partial summary, in this section we have generalized the Cahn-Hoffman ξ capillary vector to include curvature effects. It was found that the equation relating the capillary vector of anisotropic-curved interfaces and the Cahn-Hoffman ξ vector is30,31
{
Ξ ) ξ + (∇s‚M)‚Is -
(21(M:b))k}
(45)
In addition, the interfacial stress tensor expression (eq 43) given in terms of the capillary vector results in a compact form that separates energy (Ξ) from geometry (P).
5. Final Form of Force Balance Equations This section presents the most compact form of (1) the Laplace equation in terms of the capillary vector and (2) the tangential force balance. (i) Laplace Equation. Assuming that the bulk stresses are pure pressures
T1 ) -P1I
(46a)
T2 ) -P2I
(46b)
the normal force balance equation (eq 4a) becomes
∆P ≡ P2 - P1 ) TRβbRβ + TR(n)|R ) Ts|:b - ∇s‚Ξ|
(47)
Substituting the tangential stress Ts| expression (see text line after eq 44) into eq 47 gives a compact form of the Laplace equation
224 Langmuir, Vol. 22, No. 1, 2006
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∆P ) -DM‚b:q - ∇s‚Ξ
(48)
where D is the deviatoric curvature and q is the deviatoric curvature tensor, defined in Appendix I. For spherical drops (D ) 0) embedded in a liquid crystal, the equation reduces to the simple divergence form
∆P ) -∇s‚Ξ
Laplace equation (eq 48). In most cases, it is customary to develop constitutive equations using the average curvature H and the Gaussian curvature K as variables.34,45 (i) Moment Tensor M. We first express M in terms of derivatives of the free energy with respect to average curvature H and Gaussian curvature K:
F∂Aˆ ∂H ∂K ) C1 + C 2 ∂b ∂b ∂b
(49)
In the absence of liquid crystallinity, the Laplace equation becomes
ˆ (F∂A ∂H ) F∂Aˆ C )( ∂K ) C1 )
∆P ) 2H(γ0 + γ1) - DM‚b:q - ∇s‚(Is‚(∇s‚M)) (50)
2
and in the absence of surfactant, the equation becomes
∂γ ‚I ∂k s
( )
∆P ) 2H(γ0 + γan) - ∇s‚
(52)
(53)
(54)
and in the isotropic curved-interface limit (i.e, no liquid crystal), the equation becomes
∇s(γ) - (M:(∇sb))‚Is ) 0
∇s(FAˆ ) ) (M:∇sb)‚Is + ξ|‚b
(56)
Replacing this result in eq 53 shows that under constant density the tangential force balance equation implies the surface Euler equations
∇s(γ - FAˆ ) ) 0 f Aˆ )
γ + constant F
(57a) (57b)
as already pointed out.34 In partial summary, a compact and revealing form of Laplace’s equation was derived (eq 48) in terms of the generalized CahnHoffman capillary vector. At constant density, the tangential force equation (eq 4b) implies the surface Euler equation (eq 13) for a closed interface.
6. Constitutive Equations In this section, we present constitutive equations for (i) the moment tensor M, (ii) the capillary vector Ξ, and (iii) the interfacial stress tensor Ts and then use them to derive (iv) the
(59a)
∂K ∂ ) (det(b)) ) det(b)b-T ) Kb-T ∂b ∂b
(59b)
M)
F∂Aˆ 1 ) C1Is + C2Kb-T ∂b 2
(60)
Inserting the inverse of b (b-T ) (2HIs - b)/K) into eq 60 gives
M)
(
)
C1 F∂Aˆ ) + 2C2H Is - C2b ∂b 2
(61)
(ii) Capillary Vector Ξ. Using eqs 36 and 61, we find the following components of the capillary vector:
C1 H - C2 K 2
(62a)
C1 - (b - 2HIs)‚∇sC2 + ξ| 2
(62b)
Ξ⊥ ) γ -
(55)
showing the correspondence between the moments (-M, ξ|) and curvatures (∇sb, b) in generating tangential forces in the isotropic curvature and anisotropic interface limits, respectively. Computing the surface gradient of FAˆ assuming constant density gives
∂ 1 1 ∂H ) I :b ) Is ∂b ∂b 2 s 2
With these results, the moment tensor M (eq 2b) becomes
In the anisotropic interface limit (i.e, no surfactant), the equation becomes
∇s(γ) + ξ|‚b ) 0
(58c)
H,k
( )
Performing the divergence operation in ∇s‚(M‚b) yields the tangential balance
∇sγ - (M:(∇sb))‚Is + ξ|‚b ) 0
(58b)
K,k
Using the fact that H ) Is:b/2 and K ) det(b) gives
(51)
(ii) Tangential Force Balance Equation. Introducing expression 23 for the stresses into the tangential component of the force balance equation (eq 4b) gives
∇sγ - Is‚(∇s‚(M‚b)) + (ξ| + (∇s‚M))‚b ) 0
(58a)
()
Ξ | ) ∇s
(iii) Interfacial Stress Tensor Ts. Using eqs 43, 61, and 62, the total interfacial stress tensor is
(
Ts ) γ -
) {[ ( )]
C1 C1 H - C2K Is - Dq 2 2 C1 ∇s - [(b - 2HIs)‚∇sC2] + ξ| k (63) 2
}
(iv) Laplace Equation. Introducing eqs 61 and 62 into eq 48 finally yields
∆P ) 2H(γ0 + γ1 + γan) - ∇s‚ξ| + C1(K - H2) C1 + (b - 2HIs):∇s∇sC2 (64) ∇2s 2
()
Assuming that the anchoring energy is a function of (n‚k), we find that the divergence term (-∇s‚ξ|) in eq 64 is the sum of geometric curvature and director curvature terms
-∇s‚(ξ|) )
∂x| :b ) γ′′(b:nn - 2H(n‚k)2) ∂k
(65)
(44) Boruvka, L.; Neumann, A. W. J. Chem. Phys. 1977, 66, 5464. (45) Ljunggren, S.; Eriksson, J. C.; Kralchevsky, P. J. Colloid Interface Sci. 1993, 161, 133.
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Langmuir, Vol. 22, No. 1, 2006 225
where we denoted γ′′) d2γ/d(n‚k)2. Insering eq 65 into eq 64, we find the Laplace equation
()
C1 + 2 2 (b - 2HIs):∇s∇sC2 + (b:nn - 2H(n‚k) )γ′′ (66)
∆P ) 2H(γ0 + γ1 + γan) + C1(K - H2) - ∇2s
The material limits of eq 71 are the Ou-Yang-Helfrich vesicle shape equation27,46 for curved interfaces
∆P ) 2Hγ0 + 4kc(H - Ho)(K - H2 - HoH) - 2kc∇2s H (72) and the liquid crystal Laplace equation30
The limiting expressions of eq 66 in the absence of liquid crystals and surfactants, respectively, are
()
C1 + 2 (b - 2HIs):∇s∇sC2 (67a)
∆P ) 2H(γ0 + γ1) + C1(K - H2) - ∇2s
∆P ) 2H(γ0 + γan) + (b:nn - 2H(n‚k)2)γ′′ (67b) Equation 67a is consistent with the Boruvka and Neumann equation,44,45 and eq 67b is consistent with the nematic Laplace equation.30 Comparing the Laplace equations (67a and 67b), we see the correspondence between energy gradients in physical and orientational space {∇2s Ci, γ′′} and between geometric factors {(b - 2HIs), (b:nn - 2H(n‚k)2)}. In partial summary, in this section a generalized Laplace equation (eq 66) for anisotropic curved interfaces was formulated, and its material limits (i.e., no surfactant and no liquid crystallinity) are shown to be consistent with previous equations for curved interfaces and for anisotropic interfaces.
7. Laplace Equation for the Helfrich-Rapini-Papoular Interface In this section, we derive a specific Laplace equation (eq 66) using two well-known models for liquid crystal surfaces and surfactant-laden interfaces. The Rapini-Papoular anchoring energy density that describes the anisotropic contribution to interfacial tension is given by12
FAˆ anchoring(k) )
W (n‚k)2 2
(68)
where W is the anchoring coefficient. When W < 0 (W > 0), the energy is minimized when n ) k, known as homeotropic (planar) anchoring. The most studied cases in liquid crystals is when the substrate is a rigid solid and hence k is fixed. In the present case, n is constant, but k is variable. For concentrated surfactant-laden liquid-liquid crystal interfaces, the preferred orientation is homeotropic and hence W < 0.5 The Helfrich free energy26 widely used to describe the elasticity of membranes and surfactant-laden interfaces reads
FAˆ curvature(H, K) ) 2kc(H - Ho)2 + khcK
(69)
where kc is the bending elastic modulus, Ho is the spontaneous curvature, and khc is the torsion elastic moduli. Using the surface Euler equation, the surface tension γ is
γ ) γ0 + 2kc(H - Ho)2 + khcK +
W (n‚k)2 2
(70)
The coefficients Ci in eqs 58b and 58c are C1 ) 4kc(H - Ho) and C2 ) khc. Comparing eqs 26 and 70, it is found that γ1 ) -2kcHo(H - Ho). The Laplace equation (eq 66) according to this model is
∆P ) 2H(γ0 - 2kcHo(H - Ho)) + 4kc(H - Ho)(K - H2) 2kc∇2s H + W(b:nn - H(n‚k)2) (71)
∆P ) 2Hγ0 + W(b:nn - H(n‚k)2)
(73)
Equation 73 can be put in the form of the liquid crystal Herring equation30
{
∆P ) γ0 +
}
W (n‚k)2 + W [(n‚e1)2 - (n‚k)2] c1 + 2 W γ0 + (n‚k)2 + W [(n‚e2)2 - (n‚k)2] c2 (74) 2
{
}
where c1 and c2 are the principal curvatures. In partial summary, this section presents the Laplace equation (eq 71) for a Helfrich-Rapini-Papoular interface whose material limits are consistent with previous theoretical results.27,30,46
8. Application Particular numerical solutions to the Laplace equation (eq 71) are beyond the scope of this article. In this section, we present an application of this equation as a guide to shape selection when the matrix phase is a perfectly aligned nematic liquid crystal, the droplet phase is a low-molar-mass solvent, and the interface is laden with surfactant or has no surfactant. In previous work,4 it has been shown that the shape of small enough droplets, typically in the submicrometer range, embedded in aligned nematic mesophases adopt an ellipsoidal shape. For the converse problemswhere the droplet is nematicsit has also been shown to adopt nonspherical shapes.47 In the absence of intrinsic curvature (Ho ) 0), the emergence of the ellipsoidal shape is the effect of anisotropic interfacial tension. The specific question we wish to address in this section is the shape change of an initially stable and spherical droplet of an isotropic fluid of radius Ro, originally embedded in the isotropic phase of a mesogenic material (i.e., the matrix is in the isotropic phase), when the temperature is lowered below the clearing temperature and the matrix undergoes a phase transition and becomes a uniaxial rodlike nematic liquid crystal (i) in the absence and (ii) in the presence of surfactant; we neglect the small temperature effect on interfacial tension and consider the Ho ) 0 case. We restrict the discussion to homeotropic anchoring (W < 0) because it is preferred in the presence of sufficiently high surfactant surface concentrations. Because homeotropic anchoring is preferred, any deviation from a spherical shape will result in an ellipsoid whose major axis is normal to the aligned director field; for planar anchoring (not treated here), the ellipsoid will be aligned with its major axis along the director. The initial stable state of the droplet of radius Ro in the isotropic matrix is defined by
∆P ) -
2γ0 Ro
(75)
(i) Temperature Quench in the Absence of Surfactant. This is an application of the liquid crystal Herring equation (eq 74). Here we consider a droplet of radius Ro in the absence of surfactant and hence under no curvature effects. After quenching the system into the nematic phase, the spherical droplet will deform into an (46) Ou-Yang, Z.; Helfrich, W. Phys. ReV. A 1989, 39, 5280. (47) Prinsen, P.; van de Schoot, P. Phys. ReV. E. 2003, 68, 021701.
226 Langmuir, Vol. 22, No. 1, 2006
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ellipsoidal drop with major axis R (normal to the director) and minor axis r because of the anchoring energy.4,47 We consider two limiting cases: -W/γ0 , 1 and -W/γ0 . 1. (a) Case -W/γ0 , 1. This regime gives rise to weak droplet deformation, R/r ≈ 1. Using a spherical coordinate system where 1 represents the polar trajectories, we approximate the term (n‚ k) appearing in eq 74 by
(n‚k) ≈ 1 2
(n‚e1) ≈ 0
(n‚e2) ≈ 0
2
2
(
0 ≈ 2H γ0 -
W - 2kcD2 2
)
(82)
where we assumed that ∆P is close to zero. The deviatoric curvature D (eq I.18a), which is a measure of deviation from the spherical shape, is
4kcD2 ) 2γ0 - W
(83)
(76) Inserting the definition for D, we find
Using the liquid crystal Herring equation (eq 74) in the absence of surfactant gives
{
}(
2γor W r ) - γo 1+ Ro 2 R
)
(77a)
2
2
a (r/R) (n‚e2)2 ≈ + 2 2 (78)
(n‚e1)2 ≈ 0
where the constant a is a < 2/3. Using the liquid crystal Herring equation (eq 74) in the absence of surfactant gives
-
{
( ( ) )}(- R1) + {γ + W(Rr ) }(- 1r)
2γ0 r 2 W ) γ0 - a Ro 2 R
2
0
(79) In the limit of R/r . 1, the effective tension (γo + W(r/R)2) corresponding to the diverging curvature (-1/r) must vanish, and the droplet aspect ratio becomes
(Rr) ) x-W γ
(80)
0
in agreement with previous results.47 Increasing the -W/γ0 ratio increases the droplet anisotropy. (ii) Temperature Quench in the Presence of Surfactant. This is an application of the full Laplace equation (eq 71). Here we consider a droplet of radius Ro in the presence of surfactant and hence under curvature effects. After quenching the system into the nematic phase, the spherical droplet will deform into an ellipsoidal drop with major axis R and minor axis r because of the anchoring energy effect. We restrict the analysis to cases when kc/(γ0Ro2) ≈ 1 and -W/γ0 , 1, thus expecting a weak droplet deformation: R/r ≈ 1. Using a spherical coordinate system, where 1 represents the polar trajectories, we find
(n‚k)2 ≈ 1
(n‚e1)2 ≈ 0
(n‚e2)2 ≈ 0
1-
W ) 2γ0
(77b,c)
where in eq 77b we used r ≈ Ro. The aspect ratio R/r is a linear function of W/γ0, in agreement with previous results.4 Because -W/γ0 < 1, eq 77b does predict an ellipsoid (r/R < 1). (b) Case -W/γ0 . 1. We consider the case of -W/γ0 . 1 giving rise to a strong droplet deformation, R/r . 1, whose major axis (R) is oriented normal to the fixed director. Using an elliptical coordinate system, where 1 represents the polar trajectories, we roughly estimate the director field appearing in eq 74 by
(Rr )
x x Ro2γ0 2kc
1+
2γ0r W r W R ) -1≈1+ f ≈1R Ro(γ0 - W/2) γ0 r γ0
(n‚k)2 ≈ a -
R ≈1+ r
x
Ro2γ0 W 2kc 4γ0
x
Ro2γ0 (84) 2kc
Decreasing the ratio of interface tension to curvature Ro2γ0/2kc leads to a spherical shape, in agreement with previous results.4 Increasing -W/4γ0 increases deviations from sphericity. In partial summary, in this section we show applications for the Laplace equation (eq 71) for a Helfrich-Rapini-Papoular interface to predict shape changes when an isotropic liquid droplet is immersed in a matrix that undergoes an isotropic-to-nematic phase transition, in the absence and presence of surfactant. Anchoring energy drives droplet shape anisotropy, whereas bending energy tends to quench it. The results are consistent with the literature.4,47
9. Conclusions This article presents a mechanical model for curved anisotropic interfaces, applicable to thermodynamically closed surfactantladen liquid-liquid crystal interfaces. The model incorporates the mechanical effects due to surface bending and surface tilting (anchoring) and systematically incorporates liquid crystal anisotropy into classical membrane mechanics. A generalized molecular field is formulated and used to express bending stresses. A generalized Cahn-Hoffman capillary vector for curved anisotropic interfaces is derived and used to express normal and bending stresses and is also used to derive a compact form of the Laplace equation (eq 66). The material limits of this equation are consistent with the Boruvka and Neumann equation44 and the nematic Laplace equation.30 Use of the Helfrich-RapiniPapoular interface leads to a Laplace equation whose material limits are the Ou-Yang-Helfrich vesicle shape equaiton (eq 72) and the liquid crystal Herring equation (eq 74).30,46 The application of the model to shape determination in isotropic drops embedded in aligned nematic matrixes indicates that whereas anchoring increases droplet shape anisotropy, bending tends to retain spherical shapes. Interaction and competition between bending and anchoring leads to anisotropic membrane mechanics. Acknowledgment. This research was supported in part by the National Science Foundation under grant no. PHY99-07949.
Appendix I This Appendix summarizes the differential geometry used in the article.41 Consider an interface whose points are located in 3D space by a position vector R, given parametrically by
(81)
Using the Laplace equation (eq 71) in the presence of surfactant gives
R ) R(uR)
R ) 1, 2
(I.1)
The surface coordinates induce two tangential base vectors aR
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Langmuir, Vol. 22, No. 1, 2006 227
defined by
b)∂R aR ) R ∂u
R ) 1, 2
(I.2)
∂k ) -∇sk ∂R
∇s(*) ) Is‚∇(*) )
(I.12a)
∂(*) ∂(*) ) aR R ∂R ∂u
(I.12b)
The surface metric tensor aRβ is defined by
aRβ ) aR‚aβ
R, β ) 1, 2
(I.3)
where ∇s(*) is the surface gradient. The components of b obey
whose determinant is
a ) det(aRβ) > 0
(I.4)
The corresponding reciprocal base vectors aR and metric tensor are
aR ) aRβ ) aR‚aβ
∂uR ∂R R, β ) 1, 2
)
(I.5b)
1 1 K ) - s:(b‚s‚b) ) RβγδbRγbβδ ) (c1c2) (I.15) 2 2 where c1 and c2 are the radii of curvature. The relation between K and H is
(I.6a)
{ } { }
γ a + bRβk R β R
(I.7a)
1 K ) 2H2 - b:b 2
The curvature tensor b can be decomposed into a trace and a deviatoric curvature8-10
b ) HIs + Dq
(I.7b)
where the surface Christoffel symbols are
)
∂aRδ ∂aδβ ∂aβR γ 1 ) aγδ + R- γ R β 2 ∂uβ ∂u ∂u
Is‚q ) 0
(I.17)
1 D ) (c1 - c2) 2
(I.18a)
1 D2 ) H2 - K ) b:b - H2 2
(I.18b)
Appendix II
s ) -k × Is ) -Is × k ) -k × I ) -I × k ) aRaβeRβ )
This Appendix presents the derivation of eqs 8, 16, 17a, 17b, 17c, and 17d. (i) Derivation of the Interface Stress Dual Equation (Eq 8). The dual of a surface tensor Tsx is defined by
aRaβeRβ (I.9) where γδ )
Is‚Is ) q‚q ) 2
(I.8a,b)
The counterclockwise rotation of a vector around the unit normal k is given by the dyadic surface unit alternator Es
aRγaβδRβ.
(I.16)
where D, the deviatoric curvature, is8-10
R ∂aR )aγ + bRβ k β γ β ∂u
{ } (
(I.13b)
∂k 2H ) Is:b ) -∇s‚k ) -aR‚ R ) bRR ) (c1 + c2) (I.14) ∂u
where aRβaβγ ) δRγ . The surface derivatives of the base vector are
∂uβ
bγβ ) aγRbRβ
The average curvature H and the Gaussian or total curvature K are
Is ) aβaRδβR ) aRaβδβR ) aRβaRaβ ) aRβaRaβ (I.6b)
∂aR
(I.13a)
(I.5a)
The base and reciprocal base vectors define the surface unit tensor δβR, and the dyadic surface idem factor Is
aR‚aβ ) δβR
bγδ ) aδβbγβ
The surface unit normal k is given by
1 1 1 1 k ) Es:E ) ERβaR × aβ ) (a × a2 - a2 × a1) 2 2 2 xa 1 (I.10) where E is the triadic spatial unit alternator. Other useful relations involving the surface unit normal k are
aR × aβ ) kRβ
(I.11a)
aR × k ) -k × aR ) aββR
(I.11b)
Is × Is ) aRkaβRβ ) -E + Esk + kEs
(I.11c)
The symmetric curvature dyadic b is a measure of the change in k with changes in R
R(n) Tsx ) -E:Ts ) -E:(TRβ s aRaβ + Ts aRk)
(II.19)
Using eqs I.9, I.10, and I.11c, the dual Tsx becomes βR R(n) Tsx ) -E:Ts ) (-kEs:TRβ s aRaβ + aβkaR:Ts aRk) ) Es‚(Ts⊥‚k) - (Es:Ts|)k (II.20)
which is eq 8. (ii) Derivation of Rotational Invariance Equation (Eq 16). Using the principle of rotational invariance23 gives
Rβχ
(
)
∂Aˆ ∂Aˆ ∂Aˆ ∂Aˆ n + k + ∇ k + ∇ k )0 ∂nχ β ∂kχ β ∂∇sδkχ sδ β ∂∇sχkδ sβ δ (II.21)
Using the symmetry of ∇sk ) (∇sk)T, the last term is rewritten as
228 Langmuir, Vol. 22, No. 1, 2006
(
Rey
)
∂Aˆ ∂Aˆ ∂Aˆ ∇ k ) ∇ k ) ∇sδ k ∂∇sχkδ sβ δ ∂∇sδkχ sδ β ∂∇sδkχ β ∂Aˆ ∇sδ k (II.22) ∂∇sδkχ β
-E:∇s‚(Mk) ) -∇s‚(M‚E‚k) ) -∇s‚(M‚Es)
Because Aˆ is an even function of (n‚k), the first term in eq II.21 is
Inserting these results into eq II.25 gives eq 16:
(
)
∂Aˆ ∂Aˆ n ) kχ ∂nχ β ∂kβ
(II.23)
Substituting eqs II.22 and II.23 into eq II.21 yields the following vector equation:
((
- : -
)
∂A ˆ ∂A ˆ ∂A ˆ k- ∇s‚ ‚∇ k ∂k ∂∇sk ∂∇sk s ∂A ˆ ∂A ˆ ∇s‚ ) 0 (II.24) k -k ∂∇sk ∂k
( )
)
Using eqs 12 and 15, eq II.24 reads
E:(M‚b) ) kEs:(M‚b)
-E:h|k ) (βRaβkaR:hR| aRk) ) (βRaβaR‚(hR| aRk)‚k) ) Es‚h|k‚k (II.26)
(II.27b)
E:kξ| ) Es‚ξ| ) k × ξ|
(II.28)
s‚(h|k)‚k + ks:(M‚b) - ∇s‚(M‚s) + k × ξ| ) 0 (II.29) (iii) Derivation of Equations 17a-17d. Comparing eqs II.29 and 9 term by term, we find the following expressions for the interfacial stresses (Ts⊥,Ts|), moment tensor (M), and torque (ΓLC)
Es‚(Ts⊥‚k) ) Es‚(h|k)‚k f Ts⊥ ) h|k -(Es:Ts|)k ) (Es:(M‚b))k f Ts| ) -M‚b + Z ∇s‚Cs ) -∇s‚(M‚s) f Cs ) -M‚s
(II.30) (II.31)
M ) Cs‚s (II.32)
ΓLC ) Is‚ΓLC ) Es‚ξ| ) k × ξ|
-E:(h|k - M‚b) - E:∇s‚(Mk) + E:kξ| ) 0 (II.25) Working term by term in eq II.25 gives
(II.27a)
(II.33)
where according to eq 22b Z is a symmetric tangential tensor that must be equal to the normal stress γIs. Equations II.30II.33 are eqs 17a-17d. LA051974D
