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Modeling the Wilhelmy Surface Tension Method for Nematic Liquid Crystals Alejandro D. Rey† Department of Chemical Engineering, McGill University, 3610 University Street, Montreal, Quebec, Canada H3A 2B2 Received March 4, 1999. In Final Form: September 9, 1999 A general model for the Wilhelmy (fiber) surface tension measurement method for uniaxial nematic liquid crystals has been derived using fundamental principles and classical liquid-crystal physics. A rigorous formulation of the contributions of surface and bulk nematic elasticity was implemented for the Wilhelmy method. The surface contribution is a function of the surface anchoring strength of the liquid-crystal free surface and the director surface orientation at the solid-nematic-vapor common line. The exact bulk elasticity contribution is a function of the director field in the meniscus, which is a function of the nematicfiber interactions, the size of the fiber, and the temperature, among other things. The specific form of the Wihelmy model equation for four typical nematic textures was developed and analyzed. It is found that the effect may depend on the size of the fiber, the presence of disclinations, and the representative Frank elastic constants. The expression representing the actual measurement or apparent surface tension was derived and used to identify the exact nature of the nematic contributions for typical nematic textures. Identification of material systems, experimental conditions, and geometric factors that enhance the impact of nematic ordering on the Wilhelmy surface tension measurements was given. A qualitative comparison of the model prediction with experimental data highlights the utility of the derived equations.
† E-mail:
[email protected]. Tel: (514) 398-4196. Fax: (514) 398-6678.
augmented by terms that account for the bulk and surface orientational elasticity of NLCs. The importance, magnitude, and mathematical description of the nematic ordering contributions have not been discussed in the literature but are certainly critical given the central importance of the Wilhelmy method in surface science and the need of interpreting experimental surface tension data of NLC.6 A unique model equation of the Wilhelmy method for NLC such as eq 1 does not exist because the measurements involve the selection of one out of several bulk disortion modes, depending on among other things the NLC-fiber surface properties. Thus, distortion mode selection is at the core of the problem. Because NLCs are anisotropic viscoelastic materials, the bulk distortions will involve different combinations of elastic modes, such as splay, twist, and bend.2,7 In addition, the bulk elasticity effect is temperature-dependent and, for thermotropic NLCs, only present at temperatures below the isotropic-nematic transition temperature. The orientation at the free surface of NLC can also be tangential, planar, or tilted, and thus the magnitude and even the sign of the orientationdependent part of the surface tension, known as the anchoring energy,3-5 will depend on the nature of the NLC in question. The approach taken in this work is to derive the governing balance equation for the Wilhelmy surface tension method and then to apply it to a number of realistic particular cases. No attempts at modeling the Wilhelmy method are currently available but they are certainly necessary to improve the current understanding of experimental data. The objectives of this paper are (1) to present a general model equation for the Wilhelmy surface tension method for uniaxial nematic liquid crystals at static, isothermal
(1) Hiemenz, P. C. Principles of Colloid and Surface Chemistry, 2nd ed., Marcel Dekker: New York, 1986. (2) de Gennes, P. G.; Prost, J. The Physics of Liquid Crystals; Clarendon Press: Oxford, U.K., 1993. (3) Je´roˆme, B. Rep. Prog. Phys. 1991, 54, 391. (4) Sonin, A. A. The Surface Physics of Liquid Crystals; Gordon and Breach: Amsterdam, The Netherlands, 1995.
(5) Barbero, G.; Durand, G. In Liquid Crystals in Complex Geometries; Crawford, G. P., Zˇ umer, S., Eds.; Taylor and Francis, London, 1996, p 21. (6) Gannon, M. G. J.; Faber, T. E. Philos. Mag. 1978, 37, 117. (7) Chandrasekhar, S. Liquid Crystals, 2nd ed.; Cambridge University Press: New York, 1992.
Introduction The Wilhelmy plate method is a standard and popular procedure to measure the surface tension of liquids.1 In this method a thin vertical plate or fiber is suspended at a liquid surface and the weight of the entrained meniscus around the plate or fiber is detected by a balance. For isotropic liquids the weight w of the static entrained meniscus is given by1
w ) Pγ cos θ
(1)
where P is the wetted perimeter of the plate or fiber, γ is the liquid surface tension, and θ is the contact angle. If the contact angle is zero, the weight gives for known perimeters the value of the surface tension. In practice, a clean and rough platinum plate or fiber can be used to ensure zero contact angle. Details of the method can be found in the literature.1 In the rest of this paper we only discuss fibers, because they seem to be the preferred geometry when using small liquid samples. For isotropic liquids the measurement involves only surface properties because the entrained meniscus does not store elastic energy. If a similar measurement is performed on orientationally ordered materials such as nematic liquid crystals (NLCs),2 the meniscus will store elastic energy because bounding surfaces introduce bulk orientation distortions that increase the elastic free energy of the system. The bounding surfaces in the Wilhelmy method are the surface of the fiber and the surface of the meniscus. In addition, the surface tension of the NLC is anisotropic.3-5 Thus, it is clear that eq 1 will have to be
10.1021/la9902542 CCC: $19.00 © 2000 American Chemical Society Published on Web 11/16/1999
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ti ) γiIi - Ii‚
[
tsv ) γsvIsv dγi i
]
(3a)
d(n‚k )
nki ; γi ) γii + γai (n‚ki)2; i ) slc, lcv (3b,c)
{ki}
Figure 1. Schematic of the Wilhelmy surface tension measurement method and sample geometry. The axisymmetric nematic liquid-crystal (NLC) meniscus has height h, and the static contact angle at the NLC-fiber-vapor common line is θ. The thin cylindrical fiber is suspended at the NLC surface from the arm of a tared balance.
conditions, (2) to characterize the nematic contributions to the measured apparent surface tension for the most likely nematic textures, and (3) to identify the material systems, experimental conditions, and geometric factors that enhance the impact of nematic ordering on the Wilhelmy surface tension measurements. Nonequilibrium phenomena are beyond the scope of this work. Model Equations In this section we present the derivation of the general balance equation for the Wilhelmy surface tension measurement method for a uniaxial nematic liquid crystal. Experimental Wilhelmy data for an nCB (n-cyanobiphenyl) thermotropic NLC are available.8 The geometry and schematic of the Wilhelmy method are given in Figure 1.1 The coordinate system is cylindrical {r, φ, z}, with z along the vertical direction. The meniscus of height h is considered to be axisymmetric. The bottom of the fiber is placed on the plane defined by the NLC-vapor interface far away from the fiber. The static contact angle between the fiber of diameter b and the NLC at the contact line is θ. The orientation of the NLC is given by the unit vector n, known as the director.2 It is assumed that before the meniscus formation the NLC is at equilibrium, with a homogeneous director field: ∇n ) 0. After the meniscus formation, the balance of forces along the vertical direction zˆ is
[
dAsv dAslc + (νslc‚tslc) + Φ ) mg - zˆ -(νsv‚tsv) dh dh dAlcv dFb (2) + (νlcv‚tlcv) dh dh
]
where Φ is the force on the balance, m is the mass of the cylindrical fiber, g is the acceleration of gravity, {νi} (i ) sv, slc, lcv) are the unit tangent vectors whose direction is into the three interfaces (sv, solid-vapor; slc, solidliquid crystal; lcv, liquid crystal-vapor), {ti} (i ) sv, slc, lcv) are the surface stress tensors,8 {Ai} (i ) sv, slc, lcv) are the three interfacial areas, h is the axial distance, and Fb is the total bulk elastic nematic energy. The three surface stress tensors {ti} (i ) sv, slc, lcv) are given by9-11 (8) Edwards, D. A.; Brenner, H.; Wasan, D. T. Interfacial Transport Processes and Rheology; Butterworth-Heinemann: Stoneheam, MA, 1991. (9) Virga, E. G. Variational Theories for Liquid Crystals; Chapman Hall: London, 1994. (10) Rey, A. D. Liquid Crystals 1999, 26, 913. (11) Rey, A. D. J. Chem. Phys. 1999, 110, 9769.
where (i ) sv, slc, lcv) are the three interfacial unit normals obeying {ki‚νi} ) 0 (i ) sv, slc, lcv). For interfaces involving NLC, the interfacial tension contains an isotropic contribution (γi) and an anisotropic contribution (γa), whose dependence on orientation is assumed to be given by the classical Rapini-Papoular expression.3-5,12 The magnitudes of the interfacial tension γi for NLCs are similar to those of isotropic liquids. The coefficient γa is known as anchoring energy, and its magnitude depends on the material system; for interfaces involving solids, it depends on surface treatments.4 When γa is equal to zero, the surface tension is isotropic, and when γa is not zero, the surface tension is anisotropic and its magnitude depends on the surface director orientation. In general, the anchoring energy is smaller than the isotropic surface tension by 1 or more orders of magnitude.4 In addition, when the preferred surface orientation or easy axis is tangential to the interface, the anchoring energy is positive (γa > 0), and when it is orthogonal (homeotropic), it is negative (γa < 0). For the nematic free surface, it is reported3 that for several common NLCs the anchoring observed can be homeotropic, planar, or tilted. For NLC the bulk free energy, known as Frank elasticity, is given by13
2Fb )
∫{K11(∇‚n)2 + K22(n‚∇ × n)2 +
K33|n × ∇ × n|2 + K24∇‚(n ∇‚n + n × ∇ × n)} dV (4) where Kii with ii ) 11, 22, 33, 24 are the temperaturedependent Frank elastic constants for splay, twist, bend, and saddle-splay, respectively. The magnitude of the first three constants is on the order of 10-11 J/m2. A frequently used2,4 simplifying assumption that allows one to estimate energies in the presence of complex or computationaly intractable director fields is to consider elastic isotropy (K ) K11 ) K22 ) K33) and K24 ) 0, in which case Fb simplifies to
∫
2Fb ) K {(∇‚n)2 + |∇ × n|2} dV
(5)
To simplify the contributions from the surface terms, we note that for a cylinder dAlcv/dh ) 0 and that dAslc/ dh ) -dAsv/dh. When these facts are used and the net force is defined as w ) Φ - mg, the force balance equation becomes
w ) -zˆ ‚[(µsv‚tsv) + (µslc‚tslc)]
dAslc dFb + dh dh
(6)
The balance of forces at a common line is14
(µsv‚tsv) + (µslc‚tslc) + (µlcv‚tlcv) ) 0
(7)
Using this force balance and the fact that dAslc/dh is the wetted fiber perimeter (dAslc/dh ) πb ) P), we find that w is given by (12) Rapini, A.; Papoular, M. J. Phys. Colloid 1969, 30, C454. (13) Frank, F. C. Faraday Soc. 1958, 25, 19. (14) Slattery, J. C. Interfacial Transport Phenomena; SpringerVerlag: New York, 1990.
Modeling the Wilhelmy Surface Tension Method
w ) Pzˆ ‚µlcv‚tlcv + dFb/dh
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(8)
Finally using the expression for the surface stress tensor for the lcv interface, we get
w ) P{γlcv(zˆ ‚νlcv) - [2γalcv(n‚klcv)](zˆ ‚klcv)(n‚νlcv)} + dFb/dh (9) Equation 9 properly reduces to the classical Wilhelmy equation (1) for isotropic fluids. For NLC the governing equation for the Wilhelmy method has three additional contributions, two arising from the interfacial energy and one from the bulk energy. In general, the magnitude of the additional nematic contributions scales with γalcv for the former and |K∇n| for the latter, and they should be assessed for each material system, experimental condition, and geometry. Equations 1 and 9 define the apparent surface tension γapp of a NLC:
γapp ) {γlcv(zˆ ‚νlcv) - [2γalcv(n‚klcv)](zˆ ‚klcv)(n‚νlcv)} + 1 dFb (10) P dh which shows that when using eq 1, the measured value γ ) γapp is not a material property. In the case of complete wetting, the model equation (9) and the apparent surface tension γapp (eq 10) simplifies to
w ) P(γilcv + γalcv(n‚klcv)2) +
dFb ; dh
γapp ) (γilcv + γalcv(n‚klcv)2) +
1 dFb (11a,b) P dh
For complete wetting slowly approaching the nematicisotropic transition temperature TNI from above, the change in surface tension is equal to
γapp - γilcv ) γalcv(n‚klcv)2 +
1 dFb P dh
(12)
where residual nematic ordering and/or fluctuations may contribute to the anchoring energy and to the Frank energy. Because the sign of the anchoring energy can be + or -, the change itself can be of either sign, in agreement with experiments.7 Effect of Bulk Elastic Modes on the Force Equation In this section we investigate the effect on the different possible bulk elastic modes on the Wilhelmy force equation (11) for the case of complete wetting. In the Wilhelmy setup, we must consider the orienting effects from the fiber-NLC interface and from the NLC-vapor meniscus surface, whose shape is unknown. To make the problem tractable, we can estimate Fb(h) by adopting the following simplifying assumptions: (1) the NLC is elastically isotropic2 (K ) K11 ) K22 ) K33) and K24 ) 0, (2) the equilibrium bulk director field is affected only by the fiberNLC interface, (3) at the fiber-NLC interface the director is strongly anchored2 and therefore its orientation is known, and (4) the NLC meniscus is axially symmetric. Using these simplifying assumptions, we can use the results of Brochard and de Gennes,15 who computed the characteristic director fields and elastic energies of cylindrical fibers submerged into NLCs. In what follows, (15) Brochard, F.; de Gennes, P. G. J. Phys. 1970, 31, 691.
Figure 2. Top view schematic of director field n around a cylindrical fiber displaying the bend (B) mode. The director field is planar (nz ) 0) everywhere, and far away from the fiber it is constant and along the x direction (φ ) 0). At the fiber surface, n is tangential: nφ ) 1. At (r ) b/2, φ ) 0, π) there are two surface disclination lines along the axial z direction. In Figures 2 and 4 the full lines around the fiber are tangential to the director field.
we compute the Wilhelmy force for the characteristic NLC deformation modes, its deviation from the isotropic viscous liquid, and the apparent surface tension. In all cases, we consider total wetting, with the unit normal klcv along the radial direction: klcv ) (1, 0, 0). That is, the angle θ in Figure 1 is zero. The names of the deformation modes used below reflect the main distortions involved, that is, splay, bend, and/or twist. In some cases, the nematic liquid crystal may exhibit complex textures because of the presence of defects, such as points, lines, and walls. In such cases the problem may be analyzed numerically, or by introducing averaging procedures. Textured nematics are beyond the scope of this paper. (A) Bend Mode. Figure 2 shows a schematic of the top view of the director field n around a fiber displaying the bend (B) mode. In this mode the director field is planar (nz ) 0) everywhere, and far away from the fiber it is constant and along the x direction (φ ) 0). At the fiber surface, n is tangential: nφ ) 1. At (r ) b/2, φ ) 0, π) there are two surface disclination lines along the axial z direction, of core radius a. The disclination core radius is of molecular size. The total bulk elastic free energy FBb for the bend mode is given by
FBb ) πKh ln(P/πa)
(13)
In this mode n‚klcv ) 0. The force expression for the bend mode wB, its difference ∆wB ) wB - wi from the isotropic fluid case, and the apparent surface tension γBapp are given by
wB ) Pγilcv + πK ln (P/πa); ∆wB ) wB - wi ) πK ln (P/πa) (14a,b) γBapp ) γilcv + (1/P)πK ln (P/πa)
(14c)
The difference is a function of NLC elasticity (K), the defect core radius (a), and the size of the fiber (P). (B) Twist Bend Mode. Figure 3 shows a schematic of the top view of the director field n around a fiber displaying the twist bend (TB) mode. In this mode the director field far away from the fiber is vertical (nz ) l) everywhere. At the fiber surface, n is tangential: nφ ) l. The total bulk elastic free energy FTB b for the twist bend mode is given by
FTB b ) π K h
(15)
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Figure 3. Top view schematic of director field n around the fiber displaying the twist bend (TB) mode. The director field far away from the fiber is vertical (nz ) l) everywhere, and at the fiber surface, it is tangential: nφ ) 1. In Figures 3 and 5 the dots mean that the director is vertical, and the short segments mean that the director is horizontal.
Figure 5. Top view schematic of director field n around a cylindrical fiber displaying the splay (S) mode. In this mode the director field is vertical (nz ) l) far away from the fiber. At the fiber surface, n is radial: nr ) 1.
isotropic fluid case, and the apparent surface tension γSB app are given by
wSB ) P(γilcv + γalcv) + πK ln(P/πa); ∆wSB ) wSB - wi ) Pγalcv + πK ln(P/πa) (17a,b) i a γSB app ) (γlcv + γlcv) + (1/P)πK ln(P/πa)
(17c)
The difference ∆ωSB is a function of NLC anisotropic surface tension (γalcv), bulk elasticity (K), the defect core radius (a), and the size of the fiber (P). (D) Splay Mode. Figure 5 shows a schematic of the top view of the director field n around a fiber displaying the splay (S) mode. In this mode the director field is vertical (nz ) 1) far away from the fiber. At the fiber surface, n is radial: nr ) 1. The total bulk elastic free energy FSb for the splay mode is Figure 4. Top view schematic of the director field n around a cylindrical fiber displaying the splay-bend (SB) mode. The director field is planar (nz ) 0) everywhere, and far away from the fiber it is constant and along the x direction (φ ) 0). At the fiber surface, n is radial: nr ) 1. At (r ≈ b, φ ) π/2, 3π/2) there are two disclination lines along the axial z direction.
In this mode n‚klcv ) 0. The force expression for the twist bend mode wTB, its difference ∆wTB ) wTB - wi from the isotropic fluid case, and the apparent surface tension γTB app are given by TB
w
)
Pγilcv
TB
+ πK; ∆w γTB app
)
γilcv
TB
)w
i
- w ) πK (16a,b)
+ πK/P
(16c)
The difference ∆wTB is only a function of NLC elasticity (K). (C) Splay-Bend Mode. Figure 4 shows a schematic of the top view of the director field n around a fiber displaying the splay-bend (SB) mode. In this mode the director field is planar (nz ) 0) everywhere, and far away from the fiber it is constant and along the x direction (φ ) 0). At the fiber surface, n is radial: nr ) l. At (r ≈ b, φ ) π/2, 3π/2) there are two disclination lines along the axial z direction, of core radius a. The total bulk elastic free energy FSB b for the splay-bend mode is the same as that in the B mode. In this mode n‚klcv ) 1. The force expression for the splaybend mode wSB, its difference ∆wSB ) wSB - wi from the
FSb ) πKh/4
(18)
In this mode n‚klcv ) 1. The force expression for the splay mode wS, its difference ∆wS ) wS - wi from the isotropic fluid case, and the apparent surface tension γSapp are given by
wS ) P(γilcv + γalcv) + πK/4; ∆wS ) wS - wi ) Pγalcv + πK/4 (19a,b) γSapp ) (γilcv + γalcv) + πK/4P
(19c)
The difference ∆ωS is a function of NLC anisotropic surface tension (γalcv), bulk elasticity (K), and the size of the fiber (P). Discussion The above analysis shows that the difference between the nematic and isotropic weight measurements ∆w depends on the bulk elasticity of the selected texture, and for radial surface anchoring (SB and S modes), it also depends on the surface anchoring energy (γalcv). The bulk elasticity contribution depends on the fiber size (P) when the director field displays disclination lines (B and SB modes). The surface elasticity contribution arises with the radial surface orientation (SB and S modes). Thus, specific quantitative estimates of the nematic ordering effect on the Wilhelmy plate measurements depend on the material system, the surface treatment of the fiber,
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and the size of the fiber, among other things. As mentioned above, if one applies eq 1 to data for a NLC, the surface tension reported will be an apparent surface tension. Next we discuss the magnitude of the nematic elasticity effects within the model assumptions enumerated above. For low molar mass rodlike NLC, the largest contribution would come from the surface anchoring strength which is generally at least 1 or more orders of magnitude smaller than the -isotropic surface tension. This is in qualitative agreement with the surface tension measurements6 of 5CB using the Wilhelmy method showing that across the nematic-isotropic phase transition the discontinuity in the apparent surface tension is approximately 0.5 mJ/ m2.6,7 The magnitude and sign of the discontinuity at the nematic-isotropic phase transition is, in general, material dependent; for order of magnitude estimates we refer the reader to Chandrasekhar’s book.7 It is likely that sensitivity to the rate of temperature changes when measuring surface tension may lead to hysteresis effects, but such effects do not appear to have been systematically studied.7 The fact that the surface tension in the nematic phase is lower than that in the isotropic phase is also in agreement with our model, because for n-CB nematic liquid crystals, the anchoring energy as defined in our model is negative; that is, the preferred anchoring is homeotropic. The bulk elasticity contribution may be important under certain experimental conditions and material systems. For example, as the temperature of a NLC decreases toward the nematic-smectic A transition temperature (TNA), the bend constant K33 increases rapidly, and at TNA it diverges: K33 f ∞.2,7 Thus, as T f TNA, the director field may undergo a structural transformation, to avoid diverging bend energies. This structural transition will coincide with a change in the behavior of the apparent surface tension as a function of temperature. This observation is not in disagreement with experimental data,4,6 which show a local maximum in the apparent surface tension as a function of increasing temperature, although it should be emphasized that the temperature dependence of the surface ordering effects also contributes to the effect.7 The bulk elasticity contribution may also be nonnegligible in other nematic material systems such as discotic nematic mesophases16 and liquid-crystalline polymers17 which are reported to exhibit higher values of some of the Frank elastic constants. For example, it is
reported16 that for discotic nematic mesophases the Frank elastic constants are on the order of 10-8 N. For a fiber of P ) 10-4 m, this will give a non-negligible contribution to the apparent surface tension for the bend and splaybend modes on the order of 1 mN. Thus, care must be taken before neglecting nematic bulk elasticity.
(16) Fleurot, O. Ph.D. Dissertation, Clemson University, Clemson, SC, 1998.
(17) Lee, S.-D.; Meyer, R. B. In Liquid Crystalline Polymers; Ciferri, A., Ed.; VCH Publishers: New York, 1991; p 343.
Conclusion A general model for the Wilhelmy surface tension measurement method has been derived using fundamental principles and classical liquid-crystal physics. It is shown that when compared to the equation valid for isotropic liquids, the equation for nematic liquid crystals contains additional contributions arising from nematic surface elasticity and from the nematic bulk elasticity. The surface contribution is a function of the surface anchoring strength of the liquid-crystal free surface and the director surface orientation at the solid-nematic-vapor common line. The exact bulk elasticity contribution is a function of the director field in the meniscus, which is a function of the nematic-fiber interactions, the size of the fiber, and the temperature, among other things. A catalog of four possible nematic textures was characterized with respect to their contribution to the Wilhelmy model equation. It is found that the effect may depend on the size of the fiber, the presence of possible disclinations, and the representative Frank elastic constants. The nematic contributions were characterized by defining a new apparent surface tension, which is the measured quantity when using the Wilhelmy method in conjunction with the classical equation (1). The model was used to analyze and explain the salient features of available experimental data such as the discontinuous change in surface tension at the nematic-isotropic transition temperature. Material systems where the bulk nematic elasticity should be included in the computations of the apparent surface tension were identified. This work has shown that a mathematical framework based on liquid-crystal surface physics and nematic elasticity is needed to use the Wilhelmy surface tension measurement technique for nematic liquid crystals. Acknowledgment. Financial support of the Natural Sciences and Engineering Research Council (NSERC) of Canada is gratefully acknowledged. LA9902542
