Langmuir 2006, 22, 3491-3493
3491
Anisotropic Fluctuation Model for 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 January 10, 2006. In Final Form: February 25, 2006 A model for thermal fluctuations on surfactant-laden liquid-liquid crystal interfaces is formulated and used to derive the expression of the mean square displacement as a function of four elastic moduli of the interface. The measurable liquid crystal contributions to thermal roughness include the average molecular orientation, the interfacial anchoring modulus, and the bulk elasticity modulus. Surfactant-driven interfacial orientation transitions provide an additional means to extract interfacial elastic moduli parameters in surfactant-laden liquid-liquid crystal interfaces in conjunction with thermal roughness measurements.
Surfactant-laden liquid-liquid crystal interfaces are novel soft anisotropic two-dimensional phases where both curvature and liquid crystal orientation play important roles.1 Currently, there is intense interest2-4 in using surfactant-laden liquid-liquid crystal interfaces as models for development of new biosensors based on the sensitivity of the optical response to polarized light due to interface orientation processes. Optical techniques such as reflectivity and ellipsometry have been used to characterize the interfacial tension and bending modulus of interfaces and membranes.5 These techniques are based on detection of interfacial roughness due to thermal fluctuations. For interfaces between isotropic liquids, the linear fluctuation model needed to extract elasticity moduli (interfacial tension and bending modulus) from the experimental signal is well-known. This letter presents the appropriate extension of a linear thermal fluctuation model when one of the phases is a uniaxial nematic liquid crystal far from any phase transition. The output of the model is an equation that allows determination of the tension and bending elastic moduli as well as two additional liquid crystal contributions: anchoring and gradient elasticity. A linear thermal fluctuation model involving liquid crystal interfaces has to take into account the anisotropic nature of this phase, the anchoring interactions, and the long-range elasticity in the subsurface region of the liquid crystal.6-9 In this letter, the anisotropy of the nematic phase is described by the average molecular orientation or director n. The anchoring interactions between the liquid crystal orientation and the interface unit normal decide whether the orientation is planar, tilted, or normal. For aqueous-liquid-crystal interfaces with sufficiently high (low) surfactant concentration, the interfacial liquid crystal orientation * To whom correspondence
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(1) Rey, A. D. Langmuir 2006, 22 (1), 219. (2) Gupta, V. K.; Skaife, J. J.; Dubrovsky, T. B.; Abbott, N. L. Science 1998, 279, 2077. (3) Brake, J. M.; Abbott, N. L. Langmuir 2002, 18, 6101. (4) Lockwood, N. A.; de Pablo, J. J.; Abbott, N. L. Langmuir 2005, 21, 6805. (5) Langevin, D.; Meunier, J. In Membranes, Microemulsions, and Monolayers; Gelbart, W. M., Ben-Shaul, A., Roux, D., Eds.; Springer-Verlag: New York, 1994; pp 485. (6) Yokoyama, H. In Handbook of Liquid Crystal Research; Collins, P. J. Patel, J. S., Eds.; Oxford University Press: New York, 1997; Chapter 6, p 179. (7) Sluckin, T. J.; Poniewierski, A. In Fluid Interfacial Phenomena; Croxton, C. A., Ed.; Wiley: Chichester, U.K., 1986; Chapter 5. (8) Virga, E. G. Variational Theories for Liquid Crystals; Chapman Hall: London, 1994. (9) de Gennes, P. G.; Prost, J. The Physics of Liquid Crystals, 2nd ed.; Oxford University Press: London, 1993.
tends to be along (normal to) the aliphatic chains.2,9 Finally, under strong anchoring,6-9 interfacial roughness couples to the subsurface orientation creating gradients and increasing the effective interfacial elastic energy.9 In this letter, we include anisotropy, and separately include anchoring, and gradient elasticity to the classical interfacial thermal fluctuation model; as shown below, the magnitude of the material properties allows the separation between anchoring and gradient elasticity, and hence, these asymptotic regimes can be formulated independently. In what follows, we start with the well-known model for isotropic liquids, then proceed to include anchoring, and finally consider gradient elasticity. The model is not intended to reproduce any existing measurements for particular material systems but to provide the simplest and most general expressions to analyze fluctuation data. Thermal fluctuations on a surfactant-laden liquid-liquid interface are described by the distance h(r,t) from an average plane using
h(r,t) )
∫ h˜ (q,t)eiq‚r dq
(1)
where the Fourier transforms, h˜ (q,t), are the amplitudes of the surface distortions and q ) (qx,qy) is the 2D wave-vector parallel to the interface average plane. For isotropic liquids, the mean square amplitudes of wavenumber q are10
〈|h˜ (q,t)|2〉 )
AkBT (∆F + γq2 + µq4)
(2)
where A is the interfacial area, ∆F is the density difference between the two liquids, γ is the interfacial tension, µ is the bending elastic modulus, and kBT is the thermal energy. In this isotropic model the crossover length Lγ ) (µ/γ)1/2 divides the spectrum such that when the fluctuation wavelength λ ) 2π/q > Lγ the resistance is interfacial tension and when λ < Lγ curvature dominates. Equation 2 follows from the linearized Laplace equation10
η(r,t) ) po + ∆Fgh - 2γH + 2µ∇s2H
(3)
where po is the reference pressure, H ) -∇s‚k/2 is the average curvature, ∇ s is the surface gradient, k is the unit normal, and (10) Kralchevsky, P. A.; Eriksson, J. C.; Ljunggren, S. AdV. Colloid Interface Sci. 1994, 48, 19.
10.1021/la060092r CCC: $33.50 © 2006 American Chemical Society Published on Web 03/10/2006
3492 Langmuir, Vol. 22, No. 8, 2006
Letters
η is the force density associated with thermal fluctuations. For small distortions, H ) ∇s2h/2. The Fourier transform of eq 3 gives
η˜ (q,t) ) (∆Fg + 2γq2 + µq4)h˜ (q,t)
(4)
and the work Ψ ˜ (q,t) associated with the thermal fluctuation is
Ψ ˜ (q,t) ) η˜ (q,t)h˜ (q,t) ) (∆Fg + 2γq2 + µq4)|h˜ (q,t)|2
(5)
Using equipartition,9 the average of the fluctuation work is 〈Ψ ˜(q,t)〉 ) AkBT, which in conjunction with eq 5 finally leads to well-known eq 2.5,10 A dissipative linear thermal fluctuation model that includes interfacial viscosity is also available.11 For surfactant-laden liquid-liquid crystal interfaces, eq 2, containing two elastic modulus (µ,γ), is incomplete because it fails to take into account the interfacial anchoring energy (Fanc) and the gradient Frank elasticity (Fg)6,8,9
Fanc )
W 2
∫(n‚k)2 dA;
Fg )
K 2
∫∇n‚(∇n)T dV
(6a,6b)
where we use the Rapini-Papoular expression6,8,9 for anchoring and the one constant approximation for Frank energy; here W is the anchoring modulus, and K is the Frank elastic constant. When W > 0, the preferred director orientation or easy axis that minimizes the anchoring energy is planar (n ⊥ k), whereas for W < 0, it is homeotropic (n//k). The anchoring energy Fanc is the anisotropic component of the interfacial tension.8,9 The internal length scale introduced by these two energies is the extrapolation length, lext ) K/W. When the characteristic geometric (external) length scale lg is smaller than the extrapolation length lext, the system stores energy by changes in k while n remains constant, a condition known as weak anchoring.8,9 Strong anchoring prevails when lg > lext, and anchoring energy is minimized while director gradients occur. In the present case, lg ) λ ) 2π/q, and we must know the magnitude of W, which depends on the surfactantliquid-crystal material system, to ascertain for a given fluctuation mode whether anchoring or gradient elasticity dominates. Given the lack of comprehensive experimental data for surfactantliquid-crystal systems, and the known12 5 orders of magnitude range in W, it is best to analyze both anchoring regimes. (a) Weak Anchoring. In this regime, the director is constant n ) no, whereas thermal fluctuations are affected by gravitation, interfacial tension, bending, and anchoring. The starting point is the generalized Laplace equation, recently derived for a constant director field1
po + ∆Fgh ) 2H(γo - 2kHo(H - Ho)) + 4k(H - Ho)(Υ 1 H2) - 2µ∇s2H + W nn - (n‚k)2Is :b (7) 2
(
)
where γo is the zero curvature interfacial tension, Ho is the spontaneous mean curvature, b ) -∇sk is the curvature tensor, Υ is the Gaussian curvature, and Is the unit tensor. The significance and implications of the last term were previously discussed.1 The last term is the capillary pressure Panc due to anchoring, which has two contributions
W Panc ) W(nn - (n‚k)2Is):b + (n‚k)2(Is:b) 2 area tilting area dilation
(8)
(11) Kralchevski, P. A.; Ivanov, I. B.; Dimitrov, A. S. Chem. Phys. Lett. 1991, 187, 129. (12) Sonin, A. A. The Surface Physics of Liquid Crystals; Gordon and Breach Publishers: Amsterdam, 1995.
from which it follows that homeotropic and planar conditions lead to
Panc,⊥ ) |W|H, Panc,| ) |W|nn:b
(9)
Hence for homeotropic regimes anchoring capillary pressure is director-independent, and it is due to area dilation, whereas planar anchoring is director-dependent and due to area dilation and tilting. Below we neglect Ho. Using the procedure involved in obtaining eqs 3-5, the director-dependent force due to thermal fluctuation is found to be in the linear approximation
η(r,t,n) ) po + ∆Fgh - 2Hγo + 2µ∇s2H 1 W nn - (n‚k)2Is :b (10) 2
(
)
where the last term is the liquid crystal anchoring contribution. For small distortions, b ) ∇s∇sh, and the fluctuation force in terms of h is
η(r,t,n) ) po + ∆Fgh - γo∇s2h + 2µ∇s2∇s2h W(nn:∇s∇sh - (n‚k)2∇s2h) (11) Examination of eq 11 shows that the anchoring contribution never vanishes if b * 0. Repeating the procedure outlined below eqns.(5), we find that for planar orientation (n‚k ) 0) the mean square displacement now is
〈|h˜ (q,t,n)|2〉| )
AkBT (∆F + γoq + kq4 + Wnn:qq) 2
(12)
where director anchoring introduces fluctuation anisotropy through Wnn:qq. For homeotropic orientation (n‚k)1) the linear model has no n dependence and gives
〈|h˜ (q,t,n ) k)|2〉⊥ )
AkBT (∆F + (γo - W)q2 + µq4)
(13)
As shown experimentally,2-4,12 since at low (high) surfactant loading the director is planar (homeotropic), the thermal roughness discontinuity ||‚|h˜ |2〉 - 1|| due to anchoring transition by increase of surfactant loading is
||〈|h˜ |2〉-1|| ) 〈|h˜ (q,t,n)|2〉|-1 - 〈|h˜ (q,t,n ) k)|2〉⊥-1 ) |W||nn:qq - |W⊥ |q2 AkBT
(14)
where the signs in W are taken into account and the director orientation is indicated by the subscripts on W. This function is again anisotropic and never vanishes. (b) Strong Anchoring. In this regime, the surface energy is always minimized and n ) k. Director gradients in the liquid crystal near the interface arise due to coupling to thermal roughness and because liquid crystals transmit torques through the gradient elasticity.9 For example for the homeotropic case with a director no originally aligned everywhere along the unit vector δz, an interface fluctuation of wave vector q changes the director to n ) δz - (∇sh)e-qz thereby increasing the subsurface
Letters
Langmuir, Vol. 22, No. 8, 2006 3493
elastic energy Fg.9 The gradient interfacial energy γg is then obtained by γg ) ∫Fg dz. For strong anchoring, the W contribution is not present, and the director gradient-dependent fluctuation force instead of eq 10 now is η(r,t,∇n) ) po + ∆Fgh - 2H(γo + γg) + 2µ∇s2H, where γg is the contributions due to Frank energy. According to previous work9,13 γg is given by γg,| ) Kq(cos ψ)2/2; cos ψ ) n‚q; γg,⊥ ) Kq/2, where the subscript again refers to n. The main distinction is the appearance of the factor (n‚q)2 in the planar anisotropic case, as already pointed out.13 Following the same procedure outlined above, the mean square displacements are
〈|h˜ (q,t)|2〉⊥ )
AkBT
; (∆F + γoq + Kq3/2 + µq4) AkBT 〈|h˜ (q,t)|2〉| ) (∆F + γoq2 + Kq3(cos ψ)2/2 + µq4) (15a,15b) 2
The jump ||〈|h˜ |2〉-1|| due to the anchoring transition from planar to homeotropic driven by an increases in surfactant loading is ||〈|h˜ |2〉-1|| ) Kq3((cos ψ)2 - 1)/2AkBT. Considering both anchoring regimes, we find the presence of three internal length scales: Lγ ) (µ/γo)1/2, LK ) µ/K, and lext; these three length scales are the boundaries of regimes were bending, interfacial tension, anchoring, and gradient elasticity dominate. Hence, the fluctuation spectrum will have, depending (13) Braun, F. N. Phys. ReV. E 1999, 59, R4749.
on the internal length scale ordering, at most three crossovers since long wavelength modes and short wavelength modes create different resistances. As an example, if we assume the following values µ ) 10-21 J, K ) 10-11 N, γ ) W ) 10-4 J/m2, then Lγ ) 3 nm > LK ) 10 nm > lextra ) 100 nm. Hence, fluctuations with short wavelengths λ ) 2π/q < Lγ create bending modes, and fluctuations with intermediate wavelengths Lγ < λ < LK give rise to interfacial tension and anchoring modes. In this scenario, fluctuations with relatively larger wavelength modes λ > lextra are quenched mainly by interfacial tension and weakly by gradient elasticity. Other crossover outcomes in the fluctuation spectrum, driven by different internal length scale orderings, are easily predicted by the combined weak-strong anchoring model. Last, the anchoring transitions are detectable if γo ∼ W and γo ∼ γg. In summary, a linear thermal fluctuation model that describes interfacial roughness was derived for surfactant-laden liquidliquid crystal interfaces and used to derive the expression of the mean square displacement as a function of the elastic moduli of the interface. The model provides a means to extract anchoring energies from experimental data, as well as an analytical way determine whether anchoring or gradient elasticity dominate. The present model is also applicable to fluctuations in nematic nanoemulsions materials of current interest.14 Acknowledgment. This research was supported by the Natural Science and Engineering Research Council of Canada. LA060092R (14) Lishchuk, S. V.; Care, C. M. Phys. ReV. E 2004, 70, 011702.
