Langmuir 2004, 20, 11473-11479
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Thermodynamic Model of Surfactant Adsorption on Soft Liquid Crystal Interfaces Alejandro D. Rey† Department of Chemical Engineering, McGill University, 3610 University Street, Montreal, Quebec, Canada H3A 2B2 Received June 2, 2004. In Final Form: October 2, 2004 The Gibbs adsorption isotherm for planar liquid crystal/fluid interfaces is derived using the anisotropic Gibbs-Duhem equation. The Gibbs adsorption isotherm for planar interfaces is used to analyze the adsorption-driven orientation transition in aqueous solutions of anionic surfactants in contact with rodlike uniaxial nematic liquid crystal films. In qualitative agreement with experiments, the model predicts that, as the surfactant concentration increases, the tangential (planar) average molecular orientation of the liquid crystal with respect to the interface undergoes a transition to a normal (homeotropic) orientation. The anchoring coefficient or strength of anisotropic component of the interfacial tension is shown to depend on the surfactant’s concentration. Analyzing the response to addition of a co-cation, the model reveals that, as the fractional coverage of the surfactant’s chains increases, the interpenetration of liquid crystal molecules between the adsorbed surfactant tails promotes the orientation transition; at even higher surfactant chain concentrations, interpenetration is hindered because of lack of available space and a random surface orientation emerges. Thus, for aqueous surfactant solutions in contact with nematic liquid crystals, increasing the surfactant concentration leads to the following interfacial liquid crystal orientation transition cascade, planar orientation f homeotropic orientation f random orientation, which can lead to new sensor capabilities and surface structuring processes.
Introduction Two pillars of surface thermodynamics are the GibbsDuhem equation and the Gibbs adsorption isotherm:1,2
dγ + S h dΘ +
Fi dµi, ∑ i)1
-dγ )
Fi dµi ∑ i)1
(1)
where γ is the interfacial tension, S h is the interfacial entropy per unit area, Θ is the temperature, Fi is the interfacial excess concentration of component “i”, and µi is the corresponding interfacial chemical potential. The Gibbs adsorption isotherm is usually applied to calculate solute adsorption on liquid-fluid interfaces, because the interfacial tension can be measured. On the other hand, for rigid solid-fluid interfaces the Gibbs adsorption equation is used to calculate the decrease in interfacial tension because adsorption can be measured. The GibbsDuhem adsorption equation is based on the following dependency, γ ) γ(Θ, µi), and no average molecular orientation or geometric variables are taken into account. Nematic liquid crystals are anisotropic viscoleastic materials, characterized by orientational order and positional disorder. Nematic liquid crystal-fluid interfaces are also anisotropic, and the interfacial tension is a function of average molecular orientation and geometry:3
γ ) γ(Θ, µi, n, k)
(2)
where n is the director unit vector (n‚n ) 1) or average molecular orientation, k is the unit normal, and additional † Corresponding author. Tel.: (514) 398-4196. Fax: (514) 3986678. E-mail:
[email protected].
(1) Hunter, R. J. Foundations of Colloid Science; Clarendon Press: Oxford, 1986. (2) Lyklema, J. Fundamental of Interface and Colloid Science; Academic Press: London, 1995; Vol. 1. (3) Yokoyama, H. Handbook of Liquid Crystal Research; Collins, P. J., Patel, J. S., Eds.; Oxford University Press: New York, 1997; Chapter 6, p 179.
gradient elasticity contributions to be discussed below have been neglected for clarity. The dependency in eq 2 reveals that liquid crystal interfaces display adsorption-orientation-shape couplings absent in isotropic systems. For isothermal adsorption of one component at equilibrium γ ) γ(µ, ne) on a flat liquid crystal interface, we find4
|
|
∂γ(n, µ) ∂γ(n, µ) (I - nn)‚ ) 0 T F(ne, µ) ) n)ne ∂n n)ne ∂µ (3) demonstrating that adsorption is coupled to the average molecular orientation; I is the three-dimensional unit dyad. This chemo-mechanical coupling offers new routes to extract information from surfaces and to manipulate surface structures. The interfacial thermodynamics involving liquid crystals is an active area of research because electrooptical and mechanical properties are strongly influenced by surface conditions.3-8 Because optical response is a function of orientation, which in turn is a function of surface conditions, nematic liquid crystals have been shown to have unique capabilities to detect surface adsorption events as well as surface topography. Biological, medical, and material applications of liquid crystal surface sensor capabilities have been widely investigated. For example, Abbott and co-workers9 have shown that thin nematic liquid crystal films can be efficient biosensors, (4) Rey, A. D. J. Chem. Phys. 2004, 120, 2010. (5) Jerome, B. 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) Sluckin, T. J.; Poniewierski, A. In Fluid Interfacial Phenomena; Croxton, C. A., Ed.; Wiley: Chichester, 1986; Chapter 5. (7) Faetti, S. In Physics of Liquid Crystalline Materials; Khoo, I.-C., Simoni, F., Eds.; Gordon and Breach: Philadelphia, 1991; Chapter XII, p 301. (8) Barbero, G.; Durand, D. In Liquid Crystals in Complex Geometries; Crawford, G. P., Zumer, S., Eds.; Taylor and Francis: London, 1996; p 21.
10.1021/la048642d CCC: $27.50 © 2004 American Chemical Society Published on Web 11/18/2004
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able to detect ligand-receptor events on treated solid surfaces. Brake and Abbot10 have shown that liquid crystals are also able to optically detect ionic surfactant adsorption on liquid crystal-aqueous solution interfaces through changes in orientation. Brake and Abbot10 studied nematic liquid crystal films of 4-cyano-4′′-pentylbiphenyl (5CB) in contact with aqueous solutions of sodium dodecyl sulfate (SDS). Brake and Abbot10 found that for a SDSfree interface the director orientation is tangential, while for a SDS-laden interface the orientation is normal. The liquid crystal films were supported on a grid structure placed over a surface-treated solid substrate. The presence of grids and substrate added additional contributions to the adsorption-driven orientational continuous transition, but the main issue is that SDS adsorption, thus, results in the planar-homeotropic transition. In addition they proved that the main interaction of SDS and the liquid crystal is through the interaction between the tails of the surfactant and the liquid crystal. Similar effects are found by grafting surfactants, such as lecitin and stearic acid, on solid surfaces.5,11 In this case the long surfactant’s tails form a layer that the mesogenic molecules can interpenetrate. It is found that for solid substrates the effect of grafted chains dominates over the effects of the substrate, and the liquid crystal adopts the orientation of the chains, which results in homeotropic or sometimes conical anchoring. For sufficiently low surfactant adsorption, when the distance between the tails δ is larger than the liquid crystal molecular size r, penetration of the mesogens between the tails occurs. When the δ ∼ r penetraton is frustrated, a disordered surface layer emerges.11 The experiments of Brake and Abbot10 show that, at sufficiently high concentrations, the homeotropic anchoring is lost, and the alignment is highly nonuniform. This last effect was attributed10 to solubilization of 5CB into SDS micelles, and, hence, the δ ∼ r regime was not explored. Theoretical descriptions of interfacial thermodynamics include statistical mechanics and macroscopic models.3-8,12-20 Adsorption-driven transitions at solidliquid crystal interfaces have been described, usually neglecting any solid elasticity. Pieranski et al.20 studied adsorption of water on nematic liquid crystal-crystal (gypsum, muscovite, plogopite) interfaces, using a macroscopic model that neglects elasticity in the crystalline solid, and presented a theoretical model for the adsorptioninduced orientation transition. The interfacial thermodynamics of solid-liquid crystals and fluid-liquid crystals has been reviewed.3,5,11 The anchoring mechanisms of liquid crystals at solid, fluid, and free surfaces have been characterized experimentally and theoretically.3,5,11 A general Gibbs-Duhem equation for liquid crystal-fluid interfaces has been derived and used to describe the (9) Gupta, V. K.; Skaife, J. J.; Dubrovsky, T. B.; Abbott, N. L. Science 1998, 279, 2077. (10) Brake, J. M.; Abbott, N. L. Langmuir 2002, 18, 6101. (11) Sonin, A. A. The Surface Physics of Liquid Crystals; Gordon and Breach Publishers: Amsterdan, 1995. (12) Osipov, M. A.; Hess, S. J. Chem. Phys. 1993, 99, 4181. (13) Sen, A. K.; Sullivan, D. E. Phys. Rev. A 1987, 35, 1391. (14) Papenfuss, C.; Muschik, W. Mol. Cryst. Liq. Cryst. 1995, 262, 561. (15) Ericksen, J. L. In Advances in Liquid Crystals; Brown, G. H., Ed.; Academic Press: New York, 1979; Vol. 4, p 1. (16) Virga, E. G. Variational Theories for Liquid Crystals; Chapman Hall: London, 1994. (17) Jenkins, J. T.; Barrat, P. J. Appl. Math. 1974, 27, 111. (18) Rey, A. D. J. Chem. Phys. 1999, 110, 9769. (19) Barbero, G.; Durand, G. In Liquid Crystals in Complex Geometries; Crawford, G. P., Zumer, S., Eds.; Taylor and Francis: London, 1996; p 21. (20) Pieranski, P.; Jerome, B.; Gabay, M. Mol. Cryst. Liq. Cryst. 1990, 179, 285.
Rey
morphactant effect or adsorption-driven shape changes.4 The present paper expands the available theoretical descriptions by analyzing adsorption of ionic surfactants on soft liquid crystal-fluid interfaces, using the liquid crystal Gibbs adsorption isotherm. The objectives of this paper are (1) to present a general model equation for surfactant adsorption on soft liquid crystal-fluid interfaces based on the liquid crystal Gibbs adsorption isotherm; (2) to characterize the couplings between anchoring and surfactant adsorption; and (3) to qualitatively reproduce and characterize the adsorptiondriven orientation transitions observed by Brake and Abbott10 for aqueous surfactant solutions in contact with uniaxial nematic liquid crystals. Bulk and Interfacial Liquid Crystal Elasticity In this section we present and discuss the elastic energies associated with nematic liquid crystal ordering, including bulk and interface. For a volume V bounded by a smooth interfacial area A between a nematic liquid crystal and a fluid phase, the total free energy of the nematic liquid crystal is16,21
F ) Fb + Fs
(4)
where Fb is the total bulk free energy and Fs is the total surface free energy. The total bulk free energy Fb is given by16,21
Fb )
∫f dV,
f ) fiso + fg
(5a,b)
where f is the bulk energy density, fiso is the isotropic contribution, and fg is the gradient energy density known as Frank elastic energy. The energy density fiso is independent of the director orientation and plays no direct role in this paper. The Frank elastic energy density is given by16,21
1 1 1 fg ) K11(∇‚n)2 + K22(n‚∇ × n)2 + K33|n × ∇ × n|2 2 2 2 (6) where {Kii}; ii ) 11, 22, 33 are the splay, twist, and bend (Frank) elastic constants. The total interfacial energy is given by3,16,21-23
Fs )
∫γ dA,
γ ) γiso + γan + γg
(7a,b)
where γ is the interfacial free energy density, γiso is the isotropic interfacial free energy density, γan is the anchoring interfacial free energy density, and γg is the gradient interfacial free energy density. The isotropic interfacial tension γiso is independent of the director orientation n and of the unit normal k. In isotropic interfaces γiso is called the interfacial tension. The interfacial free energy density γan is known as the anchoring energy, and it represents the anisotropic contribution to the interfacial free energy density associated with deviations of the director from its preferred orientation due to the action of torques driven by bulk distortions or external fields. The preferred orientation or easy axis can be (i) parallel to the interface unit normal k, also known as homeotropic, (ii) tilted with respect to k, or (iii) tangential to the interface, also known as planar orientation. For the tilted and planar orientations, unique, multiple, or degenerate stability can arise depending on (21) de Gennes, P. G.; Prost, J. The Physics of Liquid Crystals, 2nd ed.; Oxford University Press: London, 1993.
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the nature of the material in contact with the nematic liquid crystal.3 In the present paper we consider unique stability, with the preferred orientation being either homeotropic or degenerate planar, meaning that when the preferred orientation is tangential to the interface all tangential directions are energetically equivalent. Under these conditions, the appropriate form of the anchoring γan energy is the generalized Rapini-Papoular energy:
A derivation of these equations for nonspecific solutes has been presented.4 For the sake of brevity, well-known steps are omitted (see eqs 6.1-6.13 of ref 24 for a derivation of the isotropic Gibbs-Duhem equation). The surface Helmholtz free energy per unit area A h is assumed to depend on
A h )A h (Θ, F1, ..., FN, n, ∇sn, k)
3,7,16,21-23
γan ) γ2(n‚k)2 + γ4(n‚k)4
(8)
where γ2 and γ4 are the anchoring energy coefficients or anchoring strengths that represent the polar anchoring strength for changes in the angle between n‚k. As discussed below, the director orientation depends on the signs of γ2 and γ2/2γ4.7 All the interfacial anisotropic effects are contained in γan. For fluid/fluid interfaces, no anisotropy is present, and γan ) 0. The gradient interfacial free energy density γg is given by16
(11)
where Θ is the temperature and Fi, i ) 1, ..., N, are the moles per unit area of component i. The surface Helmholtz free energy per unit mole A ˆ is assumed to depend on
A ˆ )
A h )A ˆ (aˆ , Θ, w1, ..., wN-1, n, ∇sn, k) F
(12)
where F ) 1/aˆ is the surface molar density, aˆ is the area per unit mole, and wi, i ) 1, ..., N-1 are the surface mole fractions of component i. With the use of the surface Helmholtz free energies the following quantities are defined:
|
|
|
∂A ˆ 1 ∂A ∂A h h S h ≡), µi ≡ , F ∂Θ aˆ ,wi,υ F ∂Θ aˆ ,Fi,υ ∂Fi Θ,Fj,υ ∂A ˆ (13a,b,c) γ≡ ∂aˆ Θ,wi,υ
1 γg ) (K22 + K24)k‚g, 2 g ) (n‚∇)n - n(∇‚n) ) (n‚∇s)n - n(∇s‚n) (9a,b)
S ˆ )
where K24 is the saddle-splay (Frank) elastic constant and g is the surface gradient energy density vector. Here and below we use the following tensor operation conventions: (∇n)ijT ) ∇jni, (A‚n)i ) Aijnj, (A‚B)ij ) Ai0B0j, where A, B are arbitrary tensors. A variational calculation including bulk and surface energies leads to the static interfacial director balance equation:4,16
where S ˆ is the interfacial entropy per unit mole, µi is the interfacial chemical potential, γ is the thermodynamic interfacial tension, and the vector υ ) (n, ∇sn, k) denotes the orientation-related variables and the unit normal. Computing dA h and dA ˆ and combining the obtained using the equality A h ) FA ˆ , we find the interfacial Euler and Gibbs-Duhem equations, respectively:
|
N
A ˆ )U ˆ - ΘS ˆ ) γaˆ +
∂fg ) λsn, h ) han + hg, han ) -h + k‚ ∂∇n ∂γg ∂γan ∂γg , hg ) + ∇s‚ (10) ∂n ∂n ∂∇sn
( )
where λs is a Lagrange multiplier included to enforce the unit vector constraint: n‚n ) 1. Equation 10 shows that bulk and interfacial director fields are coupled through gradient elasticity. The so-called easy axis3,16,21 or prefereed interfacial orientation is the director that minimizes the interfacial energy and, hence, satisfies -h ) λsn. In adsorption models, eq 10 is used to determine equilibrium orientation and solute adsorption, as discussed in the introduction. Gibbs-Duhem and Gibbs Adsoption Isotherm Equations for Soft Liquid Crystal-Fluid Interfaces In this section we present the surface Euler equation and the surface Gibbs-Duhem equation,24 for nematicfluid interfaces in a form compatible with surfactant adsorption. It should be noted that Teixeira and Slukin have also discussed adsorption-induced transitions on the basis of a microscopic model of liquid crystal mixtures.25,26 (22) Barbero, G.; Evangelista, L. R. An Elementary Course on the Continuum Theory for Nematic Liquid Crystals; World Scientific: Singapore, 2001. (23) Kleman, M.; Lavrentovich, O. D. Soft Matter Physics; Springer: New York, 2003. (24) Slattery, J. C. Interfacial Transport Phenomena; SpringerVerlag: New York, 1990. (25) Teixeira, P. I. C.; Slukin, T. J. J. Chem. Phys. 1992, 97, 1492. (26) Teixeira, P. I. C.; Slukin, T. J. J. Chem. Phys. 1992, 97, 1510.
µiwi ∑ i)1 N
A h )U h - ΘS h )γ+ N
dγ ) -S h dΘ ∂γg
|
∂γ
|
µiFi ∑ i)1
Fi dµi + In‚ ‚dn + ∑ ∂n Θ,µ ,∇ n,k i)1
|
i
s
∂γ :d(∇sn)T + Is‚ ‚dk (14a,b,c) ∂∇sn Θ,µi,n,k ∂k Θ,µi,∇sn,n where U h is the interfacial internal energies per unit area, U ˆ is the interfacial internal energies per unit mass, In ) I - nn, and ∇s ) Is. ∇ is the surface gradient vector, and Is ) I - kk is the 2 × 2 unit surface dyadic. Each of the adjacent bulk phases obeys a Gibbs-Duhem equation, which imposes a constraint on eq 14c. The difference between the two bulk Gibbs-Duhem equations gives N
(∆S h ) dΘ -
(∆Fsi ) dµsi ) 0 ∑ i)1
(15)
where the symbol ∆ denotes the difference between the two bulk values, and which shows that variations in temperature and chemical potential have one restriction. Assuming that the component i ) 1 is the major component in the solvent phase, using the zero adsorption plane24 of component 1 as reference and using eq 15 in conjunction with eq 14c yields the following form of the interfacial Gibbs-Duhem equation:
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|
N
dγ ) -S ˜ dΘ ∂γg
|
∂∇sn
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In interfacial models of fluid-liquid crystals, the surface Helmholtz free energy per unit area A ˜ (Θ, F˜ , φ) is assumed to first order to depend on3,20
∂γ
F˜ i dµi + In‚ ‚dn + ∑ ∂n Θ,µ ,∇ n,k i)2 i
s
∂γ
Θ,µi,n,k
|
:d(∇sn)T + Ik‚ ‚dk (16) ∂k Θ,µi,∇sn,n
where the interfacial entropy and interfacial concentrations are given by
S ˜ )S h-
F1 F1 ∆S h , F˜ j ) Fj ∆F , j ) 2, 3, ... ∆F1 ∆F1 j (17a,b)
where ∆(*) denotes the difference of (*) between the two bulk values. Model for Anionic Surfactant-Adsorption Couplings In this section we adapt eq 16 to develop a thermodynamic model of surfactant adsorption on a planar interface created by the contact of an aqueous surfactant solution and a nematic liquid crystal. We consider the adsorption of a fully ionized anionic surfactant solute of the common type:
NaAS f Na+ + AS-
(18)
where AS ) RSO3O is an alkylsulfate and where water is the solvent. Placing the aqueous solution in contact with a nematic liquid crystal phase, under certain specific conditions described by Brake and Abbot,10 forms a stable flat interface. Assuming that the water-LC and surfactant-LC are immiscible, that the surface is flat (i.e., k ) constant), and that no director gradients arise (i.e., ∇n ) 0), the Gibbs-Duhen eq 16 becomes -
-
dγ ) -S ˜ dΘ - F˜ Na+ dµNa+ - F˜ AS- dµAS- +
|
∂γ dφ ∂φ Θ,µ (19)
where F˜ Na+ is the mole per unit area of sodium ions relative to the zero adsorption plane5 of the water, F˜ AS- is the mole per unit area of alkylsulfate ions relative to the zero adsorption plane5 of the water, µNa+ is the surface chemical potential of sodium ions and µAS- is of the alkylsulfate ions, and φ is the angle of the director with respect to the fixed unit normal: φ ) cos-1(n‚k). Assuming that as a whole the interface is electrically neutral requires F˜ Na+ ) F˜ AS- ) F˜ , and eq 19 becomes
dγ ) -S ˜ dΘ - F˜ (dµNa+ + dµAS-) +
|
∂γ dφ (20) ∂φ Θ,µ
Using the classical approach to chemical potentials of electrolyte solutions1,2 we obtain
|
∂γ dφ, µ ) µNa+ + µASdγ ) -S ˜ dΘ - F˜ dµ + ∂φ Θ,µ (21)
A ˜ (Θ, F˜ , φ) ) A ˜ iso(Θ, F˜ ) + A ˜ an(Θ, F˜ , φ); τ(Θ, F˜ ) cos2 φ (23a,b) A ˜ an(Θ, F˜ , φ) ) 2 The anisotropic contribution (eq 23b) takes into account the deviation of the director from the unit normal and the equivalence between n and -n. Using the assumed surface Helmholtz free energy per unit area A ˜ (Θ, F˜ , φ), the surfactant surface chemical potential is
µ(Θ, F˜ , φ) )
|
∂A ˜ (Θ, F˜ , φ) ) µiso(Θ, F˜ ) + µan(Θ, F˜ , φ) ∂F˜ Θ,φ (24)
where the isotropic µiso(Θ, F˜ ) and anisotropic µan(Θ, F˜ , φ) surface chemical potentials of the surfactant are
µiso(Θ, F˜ ) )
|
∂A ˜ iso(Θ, F˜ ) , ∂F˜ Θ,φ µan(Θ, F˜ , φ) )
|
1 ∂τ(Θ, F˜ ) cos2 φ (25a,b) 2 ∂F˜ Θ,φ
By inverting the surfactant’s interfacial chemical potential, the interfacial molar density is obtained
µ ) µ(Θ, F˜ , φ) )
|
∂A ˜ (Θ, F˜ , φ) f F˜ ) F˜ (Θ, µ, φ) ∂F˜ Θ,φ (26a,b)
where the arrow denotes inversion. With the use of Euler’s eq 22 and the expression for F˜ ) F˜ (Θ, µ, φ) from eq 26b, the interfacial tension γ(Θ, µ, φ) is found to be
γ(Θ, µ, φ) ) A ˜ [Θ, F˜ (Θ, µ, φ)] -
[
| ]
∂A ˜ (Θ, F˜ , φ) ∂F˜
Θ,φ
F˜ (Θ, µ, φ) (27)
F˜ (Θ,µ,φ)
Regrettably in the general case it is not possible to use eq 27 to separate isotropic and anisotropic interfacial tension contributions because the orientation dependency appears ˜ an[Θ, F˜ (Θ, µ, φ), φ]. Because in both A ˜ iso[Θ, F˜ (Θ, µ, φ)] and A the objective of this paper is to describe adsorptionorientation transitions we can linearize the model around the transition state (F˜ *, µ*):
A ˜ (Θ, F˜ , φ) ) A ˜ *(Θ) +
|
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∂A ˜ iso (F˜ - F˜ *) + ∂F˜ F˜ *
|
2 ∂A ˜ an ˜ iso 1∂ A (F˜ - F˜ *)2 + (F˜ - F˜ *) + 2 ∂F˜ 2 F˜ * ∂F˜ F˜ *
|
2 ˜ an 1∂ A (F˜ - F˜ *)2 + ... ) A ˜ * + µ*(F˜ - F˜ *) + 2 ∂F˜ 2 F˜ *
1 R(Θ)(F˜ - F˜ *)2 + β(Θ)(F˜ - F˜ *) cos2 φ + ... (28) 2
where µ is the chemical potential of both the undissociated and the dissociated surfactant defined by µ ≡ ∂A h /∂F|Θ,υ. The Euler eq 14b becomes
where β(Θ) ) ∂τ(Θ, F˜ )/∂F˜ |Θ,φ. In this model the transition state (F˜ *, µ*) will remain undetermined. The surfactant interfacial chemical potential then becomes
A ˜ (Θ, F˜ , φ) ) U ˜ (S ˜ , F˜ , φ) - ΘS ˜ (Θ, F˜ , φ) ) γ(Θ, µ, φ) + µ(Θ, F˜ , φ)F˜ (22)
µ(Θ, F˜ , φ) ) µ* + R(Θ)(F˜ - F˜ *) + β(Θ) cos2 φ + ... (29)
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Using Euler’s equation the interfacial tension becomes
1 γ(Θ, µ, φ) ) A ˜ - µF˜ ) A ˜ *(Θ) - µF˜ * + R(Θ)F˜ *2 2 1 R(Θ)F˜ 2 - β(Θ)F˜ * cos2 φ + ... ) A ˜ *(Θ) - µF˜ * 2 (µ - µ*)2 1 1 + (µ - µ*)β cos2 φ - β2 cos4 φ (30) 2R R 2R It follows from eq 30 that the isotropic and anisotropic contributions to the interfacial tension are
˜ *(Θ) - µF˜ * γiso(Θ, µ) ) A
(µ - µ*)2 2R
(31)
γan(Θ, µ, φ) ) γ2 cos2 φ + γ4 cos4 φ, β β2 γ2(Θ, µ) ) (µ - µ*), γ4(Θ) ) (32a,b,c) R 2R Using eq 10 it is found that the equilibrium condition in the absence of bulk elastic torques and tangential interfacial torques is given by
∂γan ∂γan ) λsn f (I - nn)‚ )0 ∂n ∂n
(33a,b)
|
∂γ(Θ, µ, φ) )0 ∂φ Θ,µ
(34)
and the equilibrium director angle that satisfies this condition is a function of the surfactant chemical potential and the temperature: φe ) φ(Θ, µ). For the chosen form of γ (eqs 30-32), we find that the stable and unstable equilibrium angle satisfies7
[
]
γ2 + cos2(φe) ) 0 2γ4
x
γ2 2γ4
The stable equilibrium angles that satisfy eq 34, according to the values of γ2 and γ4, are7
(i) planar anchoring: φe ) π/2, 0 < γ2/2γ4, γ2 < 0 (ii) homeotropic anchoring: φe ) 0, 0 < γ2/2γ4, 0 < γ2
x
-
F˜ ) -
|
|
|
∂γiso ∂γ2 ∂γ )cos2 φe ∂µ Θ,φe ∂µ Θ ∂µ Θ ∂γ4 (µ - µ*) β 2 e - cos φ (37) cos4 φe ) F* + ∂µ Θ R R
|
With the introduction of the fractional coverages θAS- and θNa+ οf AS- and Na+ ions, 0 0 θNa+ ) F˜ NAσNa θ ) F˜ NAσ0, θAS- ) F˜ NAσAS -, +,
0 0 where NA is Avogadro’s number and σAS - and σNa+ are the + area per molecule οf the AS and Na ions, respectively, we find that the fractional coverages obey
θAS- ) θ*AS- +
0 0 βNAσAS (µ - µ*)NAσAS cos2 φe R R (39)
θNa+ ) θ*Na+ +
0 0 βNAσNa (µ - µ*)NAσNa + + cos2 φe R R (40)
θ ) θ* +
(36a,b,c)
(iii) oblique anchoring: φe ) cos-1
The anchoring transition occurs at µ ) µ*, at which γ2 ) 0. Thus, in the absence of surfactant the anchoring is planar, while in the presence of sufficient adsorbed surfactant such that µ > µ*, the orientation is homeotropic. In this paper we are concerned with discontinuous transitions. However, continuous transitions should in principle also be possible. At equilibrium (φ ) φe) in the absence of temperature changes, the Gibbs-Duhem eq 21 simplifies to
(35)
The three solutions to this equation are
(i) φe ) 0, (ii) φe ) π/2, (iii) cos φe )
(iii) homeotropic anchoring: µ > µ*, φe ) 0, µ > µ*, γ2/2γ4 > 0, γ2 < 0
0 0 θ ) θAS- + θNa+, σ0 ) σAS - + σNa+ (38a,b,c,d,e)
Because the only relevant angle is that between the unit normal k and the director n, equilibrium is given by
sin(φe) cos(φe)
(i) planar anchoring: µ < µ*, φe ) π/2, µ < µ*, -1 < γ2/2γ4 < 0, γ2 > 0
γ2 , 2γ4 γ2/2γ4 < 0
A parametric set that satisfies the experimental results is β < 0 and R > 0. Using eq 32 it follows that the following regimes close to the transition appear:
(µ - µ*)NAσ0 βNAσ0 cos2 φe R R
(41)
showing that the increase of adsorbed chains eventually leads to the orientation transition, by changing the sign of µ - µ*. It is customary to express the above results in terms of the following activities:1,2
µNaAS ) µ0NaAS + 2RT ln a(, a( ) a+a- (42a,b) 0 where µNaAS is the standard chemical potential, a( is the mean activity of NaAS, a+ and a- are the activities of Na+ ions and AS- ions, respectively. The ion activities are defined by
0 0 µAS- ) µAS µNa+ ) µNa + + RT ln a+, - + RT ln a(43a,b) 0 0 where the standard chemical potentials µNa + and µASmay be identified with the energy of solvation. The ionic activity coefficients are
a+ ) yNa+xNa+, a- ) yAS-xAS-, y(2 ) yNa+yASwhere yNa+, xNa+ and yAS-, xAS- are the activity coefficients
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and mole fractions for Na+ and AS-, respectively. In terms of activity coefficients, the surfactant chemical potential is
µ ) µ0 + 2RT ln y(x
(44)
where x ) xNa+ ) xAS- is the surfactant’s mole fraction in the bulk solution. Replacing these results in the Gibbs adsorption equation yields
|
|
| ( )
∂γiso ∂γ2 ∂γ )cos2 φe ∂µ Θ,φe ∂µ Θ ∂µ Θ ∂γ4 y(x 2RT β ln cos4 φe ) F* + - cos2 φe (45) ∂µ Θ R y*(x* R
F˜ ) -
|
Furthermore, assuming ideality (y( ) yNa+ ) yAS- ) 1) gives
F˜ ideal ) F* +
Hence, measuring adsorption jumps and interfacial tension jumps allows for the determination of the model parameters R and β, respectively:
x 2RT β ln - cos2 φe R x* R
( )
(46)
According to eqs 35, 36, and 46, the model predicts two adsorption-orientation regimes. (i) Planar Orientation-Low Adsorption Regime (µ < µ*, φe ) π/2). For sufficiently low surfactant concentration, the director is parallel to the interface and the adsorption F˜ | and fractional coverage θAS-| are independent of the anisotropic contributions:
( )
|
β)
4(∆F˜ ) 2 2∆F˜ , R)∆γ 2(∆γ)3
(53a,b)
In the presence of a large excess of NaCl, the aqueous surfactant solution can be held at constant ionic strength.1,2 The chloride ion has insignificant interfacial activity, and as the surfactant concentration changes, the chemical potential of the Cl- and Na+ remains constant. The Gibbs adsorption isotherm (eq 19) becomes
-dγ ) F˜ AS- dµAS- +
|
∂γ dφ ∂φ Θ,µ
(54)
Following exactly the same procedure presented above, using eqs 37-54, we find that the adsorption of the surfactant’s alkyl tails is
F˜ AS- ) F*AS- +
θAS- ) θ*AS- +
∂γiso y(x (µ - µ*) 2RT ) F* + ln ) F* + F˜ | ) ∂µ Θ R R y*(x*
(
)
yAS-xASRT β ln - cos2 φe (55) R y*AS- x*ASR
(
)
0 RTNAσAS yAS- xASln R y*AS- x*AS0 βNAσAS -
R
cos2 φe (56)
(47)
|
∂γiso (µ - µ*)NAσ0 ) θ*| + ) θ*| + θ| ) ∂µ Θ R 2RTNAσ0 y(x ln (48) R y*(x*
( )
(ii) Homeotropic Orientation-High Adsorption Regime (µ > µ*, φe ) 0). For sufficiently strong adsorption, the director is normal to the interface and the adsorption F˜ ⊥ and fractional coverage θ⊥ are dependent on the anisotropic contributions:
F˜ ⊥ ) F* +
(µ - µ*) β R R 0
θ⊥ ) θ*⊥ +
(49) 0
βNAσ (µ - µ*)NAσ R R
(50)
(iii) Transition State. At the transition µ ) µ*, the director jumps by π/2 rad and the surfactant adsoprtion and fractional coverage increase by 0
βNAσ β ∆F˜ ) F˜ ⊥ - F˜ | ) - , ∆θ ) θ⊥ - θ| ) R R (51a,b) The decrease in interfacial tension or change in surface pressure πm of the adsorbed surfactant monolayer at the transition is
πm ) ∆γ ) γ| - γ⊥ ) -
β2 2R
(52)
Increasing xAS- leads to an orientational transition at µAS) µ*AS-, the director jumps by π/2 rad, and the fractional coverage due to the adsorbed alkyl tails increases by
∆θAS- ) θAS-⊥ - θAS- | ) -
0 βNAσAS R
(57)
where as before the subscripts ⊥ and | denote normal and parallel interfacial director orientation, respectively; in eq 57 the coverage increases, ∆θAS- > 0, because β < 0. Comparing the jump in coverage in the absence of NaCl with the respective jump in its presence, we find
Υ)
0 0 0 σNa ∆θNaA σNa + + σAS+ ) ) 1 + 0 0 ∆θASσ σ AS
(58)
AS
This linearized model predicts that the interfacial coverage jump ratio Υ is greater than 1. The result is a consequence of the difference between eq 19 and eq 54. With increasing surfactant concentration as the tails continue to pack above the anchoring transition, the interpenetration becomes more and more hindered.11 Experiments show that when the tail-tail distance δ is j 0LC the of the order of the liquid crystal molecular sizeω director orientation is no longer homeotropic, and surface C disorder emerges.11 The critical bulk concentration xAS for this adsoprtion-driven anchoring transition from homeotropic to a random state is found by using eq 56 in conjunction with conservation of the total interfacial area balance evaluated at δ ) $oLC. This procedure gives
Surfactant Adsorption on Crystal Interfaces Model C xAS - )
y*AS- x*ASyAS-
{ [
Langmuir, Vol. 20, No. 26, 2004 11479
×
0 2 R (At(1 - θ*AS-) - (ωLC) ) β exp + RT R A N σ0 t
A AS
]}
(59)
where At is the total area. To find eq 59 we use eq 56, set cos2 φe ) 1, and define the transition when At(1 - θAS-) ) (ω0LC)2. By increasing the coverage beyond a critical value close to the characteristic size of the liquid crystal, the ability of the liquid crystal to interact with the adsorbed chains decreases.11 Equation 59 predicts that increasing anchoring strength (βv) delays the randomization because the chain-mesogen interaction increases. In the absence of NaCl, the critical surfactant concentration for the homeotropic-random orientation transition is obtained by replacing in eq 59 AS- by NaAS and RT by 2RT. In partial summary, by increasing the surfactant concentration the model predicts the following interfacial anchoring transitions: planar orientation (φe ) π/2) w homeotropic orientation (φe ) 0) w random orientation. Altough the last regime was not explored by Brake and Abbott,10 they did find that above the critical micelle concentration the orientation was randomized, and the homeotropic anchoring was completely lost. Conclusions The classical Gibbs adsorption isotherm is based on an isotropic model and fails to describe soft anisotropic interfaces, such as liquid crystal-fluid interfaces. The interfacial tension of soft anisotropic liquid crystal-fluid interfaces has an orientation-dependent contribution,
whose magnitude depends on solute adsorption. Thus, changes in solute concentration lead to interfacial orientation changes. The liquid crystal Gibbs adsorption isotherm applied to aqueous solutions of anionic surfactants in contact with nematic liquid crystals predicts that as the bulk surfactant concentration increases an orientation transition from planar to normal occurs, in qualitative agreement with Brake and Abbot’s results.10 The model shows that increasing the surfactant interfacial coverage and, hence, the mesophase-alkyl chain interactions leads to a common alignment normal to the interface. The fundamental role of alkyl chains on the orientation transition is elucidated by analyzing an aqueous surfactant solution with constant ionic strength by addition of NaCl. It is also found that, at sufficiently low bulk surfactant concentration and stable planar orientation, adsorption and fractional coverage are independent of the anchoring coefficient; at sufficiently high bulk surfactant concentration and stable normal orientation, adsorption and fractional coverage are proportional to the anchoring coefficient. Above a critical packing fraction, no penetration of liquid crystal molecules between surfactant molecules is possible, and a homeotropic-random orientation transition occurs. Reversible adsorption-driven orientation transitions in soft anisotropic liquid crystal-fluid interfaces are a chemo-mechanical effect that can be used to control adsorption through external orienting fields or to control bulk orientation through monolayer formation. Acknowledgment. This work is supported by a grant from the donors of the Petroleum Research Fund (PRF) administered by the American Chemical Society. LA048642D
