Langmuir 2007, 23, 7907-7910
7907
Low-Frequency Dilational Elasticity of the Nematic 4′-Pentyl-4-biphenylcarbonitrile (5CB)/Water Interface Radwa Ibrahim El-Sadek,† M. Roushdy,† and Jules J. Magda*,‡ Department of Physics, Faculty of Science, Cairo UniVersity, Giza, Egypt 12613, and Department of Chemical Engineering, UniVersity of Utah, Salt Lake City, Utah 84112 ReceiVed March 24, 2007. In Final Form: May 29, 2007 Axisymmetric oscillating pendant drop shape analysis has been used to study the interfacial rheology of the liquid crystal 4′-pentyl-4-biphenylcarbonitrile (5CB) in water with homeotropic anchoring. Nearly spherical 5CB droplets were subjected to low frequency (1-5 mHz) volume oscillations, and the increase in tension with surface dilation was used to calculate the complex modulus. The droplet interface response is completely elastic, with no relaxations occurring on the experimental time scale. This surprising result is attributed to droplet storage of elastic energy in the form of distorted orientational distributions within the bulk (Frank elasticity) and on the surface (anchoring elasticity).
1. Introduction When a surface phase is shifted out of equilibrium by a sudden expansion in surface area, re-equilibration subsequently occurs by a variety of relaxation processes differing in time scale.1 For example, relaxation may occur by diffusion and adsorption of surfactants2 and/or by conformational changes of adsorbed molecules.3 One can probe the differing time scales of these relaxation processes by varying the rate at which the surface is expanded and then contracted in an oscillatory experiment. This is the essence of an increasingly popular surface characterization technique known as dilational interfacial rheology. Dilational interfacial rheology has been used to measure the complex dilational modulus E* as a function of frequency for a variety of both adsorbed2,4-8 and spread3,9-11 surface layers. Interface dilational properties are thought to be highly relevant to dynamic † ‡
Cairo University. University of Utah.
(1) Ravera, F.; Ferrari, M.; Liggieri, L. Modeling of dilational visco-elasticity of adsorbed layers with multiple kinetic processes. Colloids Surf., A 2006, 282283, 210-216. (2) Lucassen, J.; Van Den Tempel, M. Dynamic measurements of dilational properties of a liquid interface. Chem. Eng. Sci. 1972, 27, 1283-1291. (3) Monroy, F.; Rivillon, S.; Ortega, F.; Rubio, R. G. Dilational rheology of Langmuir polymer monolayers: Poor-solvent conditions. J. Chem. Phys. 2001, 115, 530-539. (4) Freer, E. M.; Yim, K. S.; Fuller, G. G.; Radke, C. J. Shear and Dilational Relaxation Mechanisms of Globular and Flexible Proteins at the Hexadecane/ Water Interface. Langmuir 2004, 20, 10159-10167. (5) Monteux, C.; Fuller, G. G.; Bergeron, V. Shear and Dilational Surface Rheology of Oppositely Charged Polyelectrolyte/Surfactant Microgels Adsorbed at the Air-Water Interface. Influence on Foam Stability. J. Phys. Chem. B 2004, 108, 16473-16482. (6) Caseli, L.; Masui, D. C.; Furriel, R. P. M.; Leone, F. A.; Zaniquelli, M. E. D. Adsorption Kinetics and Dilational Rheological Studies for the Soluble and Anchored Forms of Alkaline Phosphatase at the Air/Water Interface. J. Braz. Chem. Soc. 2005, 16, 969-977. (7) Aske, N.; Orr, R.; Sjoblom, J. Dilational elasticity moduli of water-crude oil interfaces using the oscillating pendant drop. J. Dispersion Sci. Technol. 2002, 23, 809-825. (8) Dicharry, C.; Arla, D.; Sinquin, A.; Graciaa, A.; Bouriat, P. Stability of water/crude oil emulsions based on interfacial dilational rheology. J. Colloid Interface Sci. 2006, 297, 785-791. (9) Kwok, D. Y.; Vollhardt, D.; Miller, R.; Li, D.; Neumann, A. W. Axisymmetric drop shape analysis as a film balance. Colloids Surf., A 1994, 88, 51-58. (10) Kwok, D. Y.; Tadros, B.; Deol, H.; Vollhardt, D.; Miller, R.; CabrerizoVilchez, M. A.; Neumann, A. W. Axisymmetric Drop Shape Analysis as a Film Balance: Rate Dependence of the Collapse Pressure and Molecular Area at Close Packing of 1-Octadecanol Monolayers. Langmuir 1996, 12, 1851-1859. (11) Miano, F.; Winlove, C. P.; Lambusta, D.; Marletta, G. Viscoelastic properties of insoluble amphiphiles at the air/water interface. J. Colloid Interface Sci. 2006, 296, 269-275.
processes such as film rupture and drop coalescence,5,7,8 perhaps even more relevant than equilibrium interfacial tension measurements.8 However, liquid crystal (LC) drops differ from ordinary isotropic drops in significant ways. LC drops have a preferred direction of surface orientation and an “anchoring elasticity” that acts to maintain it.12,13 The preferred surface alignment may be planar (i.e., parallel to the interface) or homeotropic (i.e., perpendicular to the interface), depending on the system studied, and the anchoring strength impacts LC device operation.14 A liquid crystal drop may also store elastic energy within its interior by virtue of the Frank elasticity.12 Frank elasticity characterizes the bulk driving force for obtaining a uniform director field, where the director is a unit vector in the direction of average molecular orientation.15 For a small LC drop, a uniform director field is inconsistent with the surface anchoring, and the resulting distortion in the director field is a source of stored elastic energy. Both the Frank elasticity and the anchoring elasticity affect the apparent surface tension value of a nematic LC as measured by the Wilhelmy method, as shown theoretically by Rey.12 Thus, if the surface area of an LC drop is suddenly expanded and if the average surface orientation thereby changes, one would also expect the Frank elasticity and anchoring elasticity to profoundly affect the interface rheological response. As far as we know, this expectation has not been tested experimentally, although generalized constitutive models have been derived for the viscoelasticity of nematic/fluid interfaces.16,17 Therefore, the goal of this short note is to investigate possible effects of orientation distortions on the dilational surface rheology of a low molar mass nematic LC, 4′-pentyl-4-biphenylcarbonitrile (5CB). The LC droplets are suspended in an aqueous solution containing a surfactant known to induce homeotropic surface anchoring, (12) Rey, A. D. Modeling the Wilhelmy Surface Tension Method for Nematic Liquid Crystals. Langmuir 2000, 16, 845-849. (13) Rey, A. D. Young-Laplace equation for liquid crystal interfaces. J. Chem. Phys. 2000, 113, 10820-10822. (14) Taguchi, D.; Hamatsu, M.; Kitazawa, K.; Manaka, T.; Iwamoto, M. Orientational ordering process of 4′-pentyl-4-cynanobiphenyl molecules deposited on polyimide Langmuir-Blodgett films. Colloids Surf., A 2006, 284-285, 263266. (15) Larson, R. G. The Structure and Rheology of Complex Fluids; Oxford University Press: New York, 1999; Chapters 10-11. (16) Rey, A. D. Viscoelastic theory for nematic interfaces. Phys. ReV. E 2000, 61, 1540-1549. (17) Rey, A. D. A rheological theory for liquid crystal thin films. Rheol. Acta 2001, 40, 507-515.
10.1021/la700864k CCC: $37.00 © 2007 American Chemical Society Published on Web 06/14/2007
7908 Langmuir, Vol. 23, No. 15, 2007
cetyltrimethylammonium bromide (CTAB),18 and the dilational rheology is measured via oscillating pendant drop shape analysis (OPDSA).19 This type of interface has also been recently studied using ellipsometry.20 Although OPDSA has not to our knowledge been previously applied to LC drops, results from other experimental techniques suggest that small LC droplets may have anomalous dynamic surface properties. Lee and Denn21 measured the bulk rheology of blends consisting of small droplets of a liquid crystal polymer (LCP) dispersed in a flexible polymer matrix. LCPs, unlike low molar mass LCs, are rife with orientational defects that give rise to a domainlike microstructure.15 Lee and Denn were only able to fit their bulk linear rheology results with the Palierne emulsion model22 if they assumed that the interfacial tension value was zero for LC droplets smaller than the domain size (∼3 µm).21 The droplet retraction method has been applied to both LCP droplets23 and nematic 5CB droplets24 suspended in flexible polymer matrixes. In the latter study, the 5CB droplet diameter was of the order of 1 mm, similar to that studied here. For both the LCP droplet and the 5CB droplet, the apparent interfacial tension value was observed to decrease with time as droplet retraction occurred, a surprising result attributed to the evolution of the droplet microstructure during retraction.23,24 Droplet retraction interfacial tension measurements are superficially similar to the OPDSA measurements employed here, but there are some significant differences as well. In the former method, the droplet as a whole deforms from a sphere into an ellipsoid under the action of external flow forces. Droplet retraction back to a spherical shape takes place on a time scale of 60 s, at least for 5CB droplets dispersed in poly(dimethylsiloxane).24 In the current OPDSA study, the 5CB drop is perturbed by injecting or withdrawing material from its interior, and the drop shape remains nearly spherical at all times. The deformation (i.e., dilation) is expected to be linear and to primarily occur at the interface, which is subject to low-amplitude oscillations in the surface area (3-10%). Most importantly, the chosen period of surface area oscillation exceeds 15 min, which allows us to probe the slow relaxation processes of the interface. 2. Experimental Methods CTAB (Sigma-Aldrich) and 5CB (Frinton Laboratories, Vineland, NJ) were used as received. 5CB melts to form a nematic liquid crystal at 21 °C, and it clears to an isotropic liquid at 35.2 °C.25 The values for its density were taken from Tintaru et al.26 Aqueous CTAB solutions we prepared using distilled deionized water produced by a commercial water purification system (Milli-Q from Millipore Inc.). Interfacial tensions and dilational interfacial moduli were measured using a home-built pendant drop tensiometer described in several earlier publications.27,28 Since our last publication, this (18) Brake, J. M.; Mezera, A. D.; Abbott, N. L. Effect of Surfactant Structure on the Orientation of Liquid Crystals at Aqueous-Liquid Crystal Interfaces. Langmuir 2003, 19, 6436-6442. (19) Loglio, G.; Pandolfini, P.; Miller, R.; Makievski, A.V.; Ravera, F.; Ferrari, M.; Liggieri, L. Drop and Bubble Shape Analysis as a Tool for Dilational Rheological Studies of Interfacial Layers. In NoVel Methods to Study Interfacial Layers; Mobius, D., Miller, R., Eds.; Elsevier: London, 2001; pp 439-483. (20) Bahr, Ch. Surfactant-induced nematic wetting layer at a thermotropic liquid crystal/water interface. Phys. ReV. E 2006, 73, 030702/1-030702/4. (21) Lee, H. S.; Denn, M. M. Rheology of a viscoelastic emulsion with a liquid crystalline polymer dispersed phase. J. Rheol. 1999, 43, 1583-1598. (22) Palierne, J. F. Linear rheology of viscoelastic emulsions with interfacial tension. Rheol. Acta 1990, 29, 204-214. (23) Wu, J.; Mather, P. T. Interfacial Tension of a Liquid Crystalline Polymer in an Isotropic Polymer Matrix. Macromolecules 2005, 38, 7343-7351. (24) Wu, Y.; Yu, W.; Zhou, C. A study on interfacial tension between flexible polymer and liquid crystal. J. Colloid Interface Sci. 2006, 298, 889-898. (25) Gannon, M. G. J.; Faber, T. E. The surface tension of nematic liquid crystals. Philos. Mag. A 1978, 37, 117-135. (26) Tintaru, M.; Moldovan, R.; Beica, T.; Frunza, S. Surface tension of some liquid crystals in the cyanobiphenyl series. Liq. Cryst. 2001, 28, 793-797.
Letters
Figure 1. Apparent interfacial tension (top) and interfacial area (bottom) versus time for a pendant drop of 5CB suspended in 2 mM CTAB solution and oscillating at 1.1 mHz at room temperature. The continuous curves give the best fits to sinusoidal functions. instrument has been modified to accommodate OPDSA measurements of the complex dilational interfacial modulus following the design of Neumann and co-workers.10,29 Pendant drops of 5CB were formed using a 22 gauge needle in an optical cell filled with aqueous CTAB solutions. The needle was part of an airtight syringe (Hamilton, Inc.), and the drop volume was oscillated by manipulating the position of the syringe plunger using a computer-controlled linear stepper motor (Oriel Instruments, Stratford, CT). Pendant drops were aged for 15 min before commencing oscillations. Images of the oscillating drop were captured using a charge-coupled device (CCD) camera and a frame grabber and then stored on a Macintosh computer. To calculate the surface free energy density Γ, we require an equation appropriate for anisotropic interfaces that relates the surface curvature to the pressure jump ∆P, such as the liquid crystal Herring equation.30 According to this equation, the pressure jump for a given curvature depends on the director field at the surface, the easy axis of the surface, and the anchoring elasticity. As a first order approximation, we assume that the surface director field is uniform and homeotropic, as expected for 5CB droplets in CTAB solutions, and use the Rapini(27) Tripp, B. C.; Magda, J. J.; Andrade, J. D. Adsorption of Globular Proteins at the Air/Water Interface as Measured via Dynamic Surface Tension: Concentration Dependence, Mass-Transfer Considerations, and Adsorption Kinetics. J. Colloid Interface Sci. 1995, 173, 16-27. (28) Kim, J.-W.; Kim, H.; Lee, M.; Magda, J. J. Interfacial Tension of a Nematic Liquid Crystal/Water Interface with Homeotropic Surface Alignment. Langmuir 2004, 20, 8110-8113. (29) Susnar, S. S. Ph.D. Thesis, University of Toronto, 1999. (30) Cheong, A.-G.; Rey, A. D. Cahn-Hoffman capillarity vector thermodynamics for curved liquid crystal interfaces with applications to fiber instabilities. J. Chem. Phys. 2002, 117, 5062-5071.
Letters
Langmuir, Vol. 23, No. 15, 2007 7909
Figure 2. Apparent interfacial tension (top) and interfacial area (bottom) versus time for a pendant drop of 5CB suspended in 2 mM CTAB solution and oscillating at 3 mHz at room temperature. The continuous curves give the best fits to sinusoidal functions.
Figure 3. Apparent interfacial tension (top) and interfacial area (bottom) versus time for a pendant drop of 5CB suspended in 2 mM CTAB solution and oscillating at 5 mHz at room temperature. The continuous curves give the best fits to sinusoidal functions.
Papoular form31 of the anchoring elasticity. In this case, the liquid crystal Herring equation reduces to30
oscillation. These quantities were obtained by a least-squares fit of the time-dependent Γ and A results to sinusoidal curves (LevenbergMarquardt algorithm), a process that also yields the phase angle δ between the oscillations in Γ and A. The complex modulus can be decomposed into an elastic component E′ ) |E*| cos δ and a viscous component E′′ ) |E*| sin δ.4
∆P ) 2H(Γiso + 0.5Γan) ) 2HΓ
(1)
where Γiso and Γan are the isotropic and anchoring contributions, respectively, to the surface free energy density and H is the mean surface curvature. Equation 1 has the form of the conventional Laplace-Young equation, but the effective interfacial tension Γ includes an explicit anchoring contribution Γan that can be either positive or negative. The time-dependent interfacial area A and the time-dependent (apparent) interfacial tension Γ were obtained by using the Simplex algorithm to fit the experimental drop shape to a theoretical drop shape,32 where the latter was calculated using eq 1 as applied to axisymmetric drops. In all cases, an accurate fit was possible using a single value of Γ. At a given drop oscillation frequency ω, the magnitude of the complex dilational surface modulus was calculated using the equation4 |E*| ) dΓ/d ln A ) A0∆Γ/∆A
(2)
where A0 is the average interfacial area, ∆A is the amplitude of the area oscillation, and ∆Γ is the amplitude of the interfacial tension (31) Yokoyama, H. Handbook of Liquid Crystal Research; Collings, P. J., Patel, J. S., Eds.; Oxford University Press: New York, 1997; Chapter 6, p 179. (32) Tripp, B. C. Ph.D. Thesis, University of Utah, 1993.
3. Results and Discussion Figure 1 shows the time-dependent value of the (apparent) interfacial tension Γ and the interfacial area A, as obtained for a pendant drop of 5CB suspended in an aqueous CTAB solution (2 mM) and oscillating at a frequency of 1 mHz at 25 °C. Figures 2 and 3 show Γ and A results obtained under the same conditions at the drop oscillation frequencies ω ) 3 and 5 mHz, respectively. Each data point in Figures 1-3 was obtained by fitting experimental drop edge coordinates to eq 1, with the additional assumption of axisymmetric drops.32 At each point in time, we found it possible to accurately fit the drop shape with a single value of the effective tension value Γ which presumably includes an anchoring contribution. As shown in a previous study,28 pendant drops of 5CB in pure water are too spherical to accurately apply the pendant drop method (Bond number Bo < 0.1). However, addition of 2 mM CTAB lowers the equilibrium tension Γ0 sufficiently (Γ0 ) 2.5 dyn/cm ( 0.1) so that Bo increases into
7910 Langmuir, Vol. 23, No. 15, 2007
Figure 4. Magnitude of the complex dilational surface modulus versus frequency for a pendant drop of 5CB in 2 mM CTAB solution at room temperature.
an acceptable range,33 0.11-0.14, and also switches 5CB surface anchoring to homeotropic.18 It is immediately apparent from Figures 1-3 that the interface rheological response is almost completely elastic; that is, the phase angle δ and the dilational viscosity η′ ) E′′/ω are both very nearly zero. This is quite surprising, because the frequency range investigated is very low, among the lowest that can be found in the literature for an oscillating drop study. If as expected the static interfacial tension value Γ0 is independent of drop volume, then at low ω the amplitude of the tension oscillation (∆Γ) must approach zero, but the results in Figures 1-3 show no indication of this. Oscillatory drop experiments at such low frequencies are difficult4 because they must be of long duration (>20 min) and a certain amount of instrumental drift is probably inevitable. This may explain the relatively large scatter in the calculated value of |E*| (eq 2), which is plotted against ω in Figure 4. Nonetheless, we can ascertain that |E*| has a value in the range (0-5 mN/m) expected for an interface containing a single small-molecule surfactant.5 Furthermore, the |E*| values are not noticeably smaller at the lower frequencies studied. Thus, either Γ0 depends on drop size or the LC interface has some very slow relaxation process (33) Rai, P. K.; Denn, M. M.; Maldarelli, C. Interfacial Tension of Liquid Crystalline Droplets. Langmuir 2003, 19, 7370-7373.
Letters
with a time scale that exceeds 15 min. CTAB diffusion and adsorption is an unlikely candidate for such a process, because surface relaxation by small-molecule diffusion and adsorption usually occurs on a time scale of 30 s, at least for isotropic fluids.2,4 Alternatively, the low-frequency response observed in Figure 1 may be a consequence of LC surface anchoring elasticity or bulk Frank elasticity, which are believed to be important in other experimental studies of LC droplets (see Introduction). In particular, the results presented here are reminiscent of LC droplet retraction experiments at much higher deformation rates in which Γ decreases with time as the droplet retracts from an ellipsoid into a sphere (see Introduction). Several research groups have applied the static pendant drop method to nematic liquid crystals,25,26,28,33-35 yet none have reported that the equilibrium interfacial tension value Γ0 depends on drop size in the range of drop diameters studied here (1.2-1.5 mm). Therefore, we believe that the Γ value of the 5CB interface becomes independent of drop size at sufficiently low oscillation frequencies. However, such a proposition is difficult to prove, because we do not know how low the frequency needs to be or, equivalently, the length of a static experiment required for equilibrium to be achieved, and instrumental drift is likely to become an issue.
4. Conclusions The shape of oscillating pendant nematic 5CB drops can be accurately described using an approximate form of the liquid crystal Herring equation30 that resembles the conventional Laplace-Young equation. However, even at very low oscillation frequencies (1-5 mHz), the interface dilational response is almost perfectly elastic (elastic modulus E′ ≈ |E*| ) 4.0 ( 1.0 mN/m), with no relaxations occurring on the experimental time scale. This unusual behavior is tentatively ascribed to LC surface anchoring elasticity and bulk Frank elasticity. Apparently, surface dilation changes the amount of elastic energy stored in a small LC drop, and the excess energy relaxes very slowly, if it relaxes at all. LA700864K (34) Krishnaswamy, S. Experimental determination of the surface tension of two liquid crystals. Liq. Cryst., Proc. Int. Conf. 1980, 487-489. (35) Song, B.; Springer, J. Surface phenomena of liquid crystalline substances: temperature-dependence of surface tension. Mol. Cryst. Liq. Cryst. 1997, 307, 69-88.
