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Langmuir 2003, 19, 3542-3544
Comparison of Steady-State Shear Viscosity and Complex Shear Modulus in Langmuir Monolayers Michael Twardos and Michael Dennin* Department of Physics and Astronomy, and Institute for Interfacial and Surface Science, University of California at Irvine, Irvine, California 92697-4575 Received December 4, 2002. In Final Form: January 17, 2003
The flow behavior of liquid crystal phases of monolayers of simple long-chain fatty acids confined to the air-water interface (Langmuir monolayers) is highly nonlinear.1,2 This raises the important question of how to characterize the rheological response of these materials, as standard linear viscoelastic relationships may not apply. Recent studies of the flow behavior of these phases have focused on measuring flow profiles in channels,2-5 domain orientation,6-9 and the complex shear modulus,1,10-12 G*(ω) ) G′(ω) + iG′′(ω). The complex shear modulus is defined by the relationship σ(ω) ) G*(ω)γ(ω), where σ(ω) is the stress in response to a strain γ(ω). These studies have found strong amplitude and frequency dependence of G*(ω),1,10 unusual velocity profiles,4,5 including dependence of the velocity profile on strain,2 and anomalous peaks in G*(ω) as a function of surface pressure.1,13 The surface pressure, Π, of a monolayer is the surface tension of the pure water-air interface minus the surface tension of the air-water-surfactant interface. Measurements of the anomalous peak using alternate techniques13 found that the peak in the steady-state viscosity, η ) σ/γ˘ , where γ˘ is the rate of strain, corresponds with the peak in G*(ω). The correspondence between G*(ω) and η as a function of pressure raises the question of whether there is any correspondence between G*(ω) and η as a function of frequency and shear rate. An example of such a relationship is the Cox-Merz rule for polymer systems.14 This is an empirical relationship that is used to convert a measurement of the steadystate viscosity, η(γ˘ ), to the modulus of the dynamic (or complex) viscosity, η*(ω): η(γ˘ ) ) |η*(ω)|ω)γ˘ , where the dynamic viscosity is defined in terms of the real and imaginary parts of G*(ω):
η*(ω) )
G′(ω) G′′(ω) +i ω ω
The modulus of the dynamic viscosity is then * To whom correspondence should be addressed. E-mail:
[email protected]. (1) Ghaskadvi, R. S.; Ketterson, J. B.; Dutta, P. Langmuir 1997, 13, 5137. (2) Ivanova, A. T.; Schwartz, D. K. Langmuir 2000, 16, 9433. (3) Schwartz, D. K.; Knobler, C. M.; Bruinsma, R. Phys. Rev. Lett. 1994, 73, 2841. (4) Kurnaz, M. L.; Schwartz, D. K. Phys. Rev. E 1997, 56, 3378. (5) Ivanova, A.; Kurnaz, M. L.; Schwartz, D. K. Langmuir 1999, 15, 4622. (6) Maruyama, T.; Fuller, G.; Frank, C.; Robertson, C. Science 1996, 274, 233. (7) Ignes-Mullol, J.; Schwartz, D. K. Phys. Rev. Lett. 2000, 85, 1476. (8) Ignes-Mullol, J.; Schwartz, D. K. Langmuir 2001, 17, 3017. (9) Ignes-Mullol, J.; Schwartz, D. K. Nature 2001, 410, 348.
|η*(ω)|ω)γ˘ t
[ ( )] |
G′′(ω) G′ 1+ ω G′′
2 0.5
ω)γ˘
Due to the strongly strain dependent behavior of the system,1 one does not expect the Cox-Merz rule to hold in detail. However, one might expect that η(γ˘ ) ∝ |η*(ω)|ω)γ˘ , with the two exhibiting the same shear-rate (frequency) dependence. Another option is that G′′(ω)/ω and η(γ˘ ) have the same dependence on ω and γ˘ , respectively. We report tests of both of these possibilities. The complex dynamic viscosity, the steady-state viscosity, and G′′(ω)/ω were measured for two Langmuir monolayers: heneicosanoic acid (C21) and 2-hydroxytetracosanoic acid (2OH TCA). For C21, we studied the flow behavior in both the L2 and the L′2 phase.15,16 The main difference between these two phases is the orientation of the tilt of the molecules. In the L2 phase, the molecules are tilted toward their nearest neighbor, and in the L′2 phase, they are tilted toward their next-nearest neighbor. Past studies have revealed differences in these two phases with regard to flow alignment and velocity profiles.2,6,8,9 For 2OH TCA, we consider two different surface pressures. Both values of surface pressure correspond to states of the L′2 phase, but on either side of an anomalous peak in the viscosity.17 We measure the mechanical properties of the Langmuir monolayers using a two-dimensional Couette viscometer that is described in detail in ref 18. The apparatus consists of two concentric cylinders oriented vertically. The outer cylinder consists of 12 individual pieces that can be expanded and compressed between 6 and 12 cm to adjust the surface pressure. This barrier can be rotated in either direction at a constant angular speed in the range from 0.0005 to 0.1 rad/s. The inner cylinder is a Teflon rotor that contacts the film with a knife-edge. The rotor has a diameter of 3.84 cm and is suspended by a torsion wire with a torsion constant κ ) 370 dyn cm/rad. Angular displacements of the coil were measured magnetically and used to determine the torque, τ, on the inner cylinder. When measuring G*(ω), angular rotations of the inner cylinder were less than 5 × 10-4 rad. Because we used a range of phases, our apparatus is not always in the narrow-gap limit.19 Therefore, care is needed in computing the steady-state viscosity. Past measurements18 of steady-state shear in these systems have shown that the viscosity of Langmuir monolayers is well described by a power-law model, η ) Rγ˘ n-1. This behavior was confirmed for all of the monolayers reported on here by measuring the torque on the inner cylinder as a function of rotation rate, Ω. For the Couette geometry and a fluid described by a power-law viscosity (η ) Rγ˘ n-1), the torque on the inner cylinder is proportional to Ωn. (10) Ghaskadvi, R. S.; Bohanon, T. M.; Dutta, P.; Ketterson, J. B. Phys. Rev. E 1996, 54, 1770. (11) Brooks, C. F.; Fuller, G. G.; Frank, C. W.; Robertson, C. R. Langmuir 1999, 15, 2450. (12) Ghaskadvi, R. S.; Carr, S.; Dennin, M. J. Chem. Phys. 1999, 111, 3675. (13) Ghaskadvi, R. S.; Dennin, M. Langmuir 2000, 16, 10553. (14) Cox, W. P.; Merz, E. H. J. Polym. Sci. 1958, 28, 619. (15) Knobler, C. M.; Desai, R. C. Annu. Rev. Phys. Chem. 1992, 43, 207. (16) Kaganer, V. M.; Mohwald, H.; Dutta, P. Rev. Mod. Phys. 1999, 71, 779. (17) Twardos, M.; Dennin, M.; Brezesinski, G. In preparation. (18) Ghaskadvi, R. S.; Dennin, M. Rev. Sci. Instrum. 1998, 69, 3568. (19) Bird, R. B.; Armstrong, R. C.; Hassuage, O. Dynamics of polymer liquids; Wiley: New York, 1977.
10.1021/la026952v CCC: $25.00 © 2003 American Chemical Society Published on Web 03/08/2003
Notes
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Figure 1. The dependence of the steady-state shear viscosity (η(γ˘ ), solid squares) and the complex shear viscosity (|η*(ω)|, open squares) on shear rate (γ˘ ) and angular frequency (ω), respectively. Also shown is G′′(ω)/ω (solid circles). The results are given for two different states of 2OH TCA. The symbols are the data, and the lines are guides to the eye. (a) The results for Π ) 32 dyn/cm. The solid line has a slope of -0.6, and the dashed line has a slope of -0.2. (b) Results for Π ) 48 dyn/cm. The solid line has a slope of -0.6, and the dashed line has a slope of -0.4.
Once the exponent n is determined, the rate of strain is computed from
γ˘ (Ω) )
2/n 2 Ωro n r 2/n - r 2/n o i
and the viscosity from
η)
τ(ro2/n - ri2/n)n 4πΩ ri2 ro2/n
Here ri is the radius of the inner barrier, and ro is the radius of the outer barrier. The measurement of the complex shear modulus was accomplished using the procedures outlined in ref 20. The inner coil is driven with a known sinusoidal torque, and the measured in-phase and out-of-phase response allows for determination of G*(ω). This is then used to compute η*(ω). The C21 monolayers were formed using 10 mM solutions of heneicosanoic acid (purchased from Acros and used without further purification) dissolved in chloroform. The monolayer was relaxed for 30 min before measurements were made. The 2OH TCA material was acquired from the Knobler lab at UCLA. The trough was filled with Millipore filtered water with a resistivity of 18.2 Ω cm that was adjusted to a pH of 2.5 with the addition of HCl for all experiments. In the 2OH TCA system, we made measurements at Π ) 32 and 48 dyn/cm. The peak in viscosity occurs roughly at 40 dyn/cm. For the C21 system, we made the following series of measurements for the same monolayer: (a) T ) (20) Ghaskadvi, R. S.; Ketterson, J. B.; MacDonald, R. C.; Dutta, P. Rev. Sci. Instrum. 1997, 68, 1792.
Figure 2. The dependence of the steady-state shear viscosity (η(γ˘ ), solid squares) and the complex shear viscosity (|η*(ω)|, open squares) on shear rate (γ˘ ) and angular frequency (ω), respectively. Also shown is G′′(ω)/ω (solid circles) and G′(ω)/ω (open triangles). The results are given for three different phases of C21. The lines are guides to the eye. (a) Results for the L2 phase (T ) 18 °C, Π ) 10 dyn/cm). The solid line has a slope of -0.7, and the dashed line has a slope of -0.2. (b) Results near the transition from L2 to L′2 (T ) 18 °C, Π ) 20 dyn/cm). The solid line has a slope of -0.5, and the dashed line has a slope of -1.0. (c) Results for the L′2 phase (T ) 14 °C, Π ) 20 dyn/cm). The solid line has a slope of -0.6, and the dashed line has a slope of -0.8.
18 °C, Π ) 10 dyn/cm (L2 phase); (b) T ) 18 °C, Π ) 20 dyn/cm (approximately at the transition between L2 and L′2); (c) T ) 14 °C, Π ) 20 dyn/cm (L′2 phase). At each point, the complex shear modulus was measured first and the steady-state viscosity was measured second. The results for each situation are summarized in Figure 1 for the 2OH TCA monolayers and Figure 2 for C21. For each phase, both the steady-state viscosity, η(γ˘ ), and the modulus of the dynamic viscosity, η*(ω), are shown. In Figure 1, we also plot G′′(ω)/ω, which is the purely dissipative part of |η*(ω)|. In Figure 2, we plot both G′′(ω)/ω and G′(ω)/ω. In this case, we see that there is actually a significant contribution from the elastic part of |η*(ω)|, in the L′2 phase. A number of features are independent of the phase that is studied and consistent with past reports on the materials. First, the steady-state viscosity is always significantly lower than either |η*(ω)| or G′′(ω)/ω. This is not surprising given the reported dependence on strain of G*(ω).1 The behavior of both |η*(ω)| and η(γ˘ ) is consistent with the monolayers being shear-thinning, that is, each decreases with increasing ω and γ˘ , respectively. For C21, the values of |η*(ω)| and η(γ˘ ) are larger in the L′2 than in the L2 phase, as expected.1 Finally, the changes in magnitude of |η*(ω)| and η(γ˘ ) roughly correspond in each
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phase. This is evident in Figure 2, where each quantity increases as one goes from state a to state c for the C21 monolayer. It is interesting to note that the three different quantities exhibit different sensitivities to the change in phase, with the changes in |η*(ω)| being the largest. The fact that the magnitudes of the quantities track each other as a function of Π is not surprising. This is expected based on the measurements of the peak in viscosity as a function of Π, where the magnitudes of viscosity and G′′(ω) also tracked each other.13 The interesting features are the frequency and shearrate dependence of the two quantities. The shear-rate dependence of η(γ˘ ) is relatively consistent from phase to phase. One finds η(γ˘ ) ∝ γ˘ n-1 with n in the range 0.3-0.5. The most striking result is that the frequency dependence of |η*(ω)| is unrelated to the shear-rate dependence of η(γ˘ ). Therefore, there is no equivalent of the Cox-Merz relationship. In addition, the frequency dependence of |η*(ω)| is significantly more sensitive to the thermodynamic phase. Taking |η*(ω)| ∝ ωn, we find -0.2 > n > -1.0 for C21, depending on the phase. For the L2 phase, one observes a value of n consistent with -0.2, and in the L′2 phase, the value is between -0.8 and -1.0. The large difference between the two phases is presumably due to the contribution from G′(ω). This can be seen by comparing |η*(ω)|, G′′(ω)/ω, and G′′(ω)/ω (see Figure 2). In the L2 phase, |η*(ω)| and G′′(ω)/ω have essentially the same order of magnitude and frequency scaling. Here, G′(ω)/ω also has a similar frequency scaling but is smaller than G′′(ω)/ω. As one approaches the L′2 phase, the importance of G′(ω)/ω increases (Figure 2b), until in the L′2 phase it is the dominant contribution to |η*(ω)| (Figure 2c). Also interesting is the frequency dependence of |η*(ω)| for 2OH TCA. Here one observes different behavior for the frequency dependence of |η*(ω)| on either side of the peak in viscosity (n ) -0.2 below the peak and n ) -0.4 above the peak). This is in obvious contrast to the behavior of the viscosity, for which the scaling is essentially the same for both pressures. At the higher value of pressure, the frequency and shear-rate dependence of |η*(ω)| and
Notes
η(γ˘ ), respectively, are closer to each other. Therefore, there may be a value of pressure for which they agree. However, this would not represent the generic behavior. Also, the differences revealed by measuring both quantities may prove useful for understanding the source of the peak. Notice, in this case, G′′(ω)/ω basically tracks |η*(ω)| for both situations. If one considers simply G′′(ω)/ω, there also appears to be no generic correlation between the frequency and shearrate dependence for G′′(ω)/ω and η(γ˘ ). There are indications for C21 that there might exist special values of the surface pressure for which they agree, even though η(γ˘ ) and |η*(ω)| do not. For example, in Figure 2c, one sees that the scaling for G′′(ω)/ω is close to that for η(γ˘ ). In summary, we have shown that for simple fatty acid monolayers there is no generic relationship between shearrate and frequency dependence of the steady-state viscosity and the complex shear modulus. This manifests itself in two ways. For a given value of surface pressure, the dependence of viscosity on shear rate is different from that of |η*(ω)| on frequency. Also, the measured exponent for the viscosity as a function of shear rate is less sensitive to changes in surface pressure (and thermodynamic phase) than the measured exponent for |η*(ω)| as a function of frequency. However, changes in the magnitudes of the two quantities do track each other. Careful modeling and studies of these similarities and differences may be able to shed light on some of the interesting flow behavior in monolayers. Clearly, in trying to understand the flow behavior of monolayers, it is essential that both the complex shear modulus and the steady-state viscosity be measured separately. Acknowledgment. The authors thank the National Science Foundation, Grant CTS-0085751, for funding. Additional funding was provided by the Research Corporation and the Sloan Foundation. LA026952V
