A technique for direct observation of particle motion under shear in a Langmuir monolayer M. Levent Kurnaz and Daniel K. Schwartz
Citation: Journal of Rheology 41, 1173 (1997); doi: 10.1122/1.550814 View online: http://dx.doi.org/10.1122/1.550814 View Table of Contents: http://sor.scitation.org/toc/jor/41/5 Published by the The Society of Rheology
A technique for direct observation of particle motion under shear in a Langmuir monolayer M. Levent Kurnaz and Daniel K. Schwartza)
Department of Chemistry, Tulane University, New Orleans, Louisiana (Received 4 April 1997; final version received 20 June 1997)
Synopsis We have used fluorescence microscopy to observe the effect of shear on rigid two-dimensional structures residing in a Langmuir monolayer. Monolayers of 12-NBD stearic acid undergo a two-dimensional liquid-to-solid transition on the water surface. The domains of the solid phase that form in the coexistence region are elongated ~needle-shaped! crystallites. We observe the effect of shear within the monolayer in two geometries: two-dimensional Poiseuille flow and simple shear flow created by moving bands. The technique allows us to determine the average orientational order parameter as well as the details of the rotational kinematics, which are, as expected, well-described by a Jeffery orbit. We propose that this technique of direct observation in Langmuir monolayers will be a useful method for the study of systems of rigid particles under flow. © 1997 The Society of Rheology. @S0148-6055~97!01605-2#
I. INTRODUCTION The behavior of rigid elongated particles in flow is an important component in the rheology of various complex fluid systems, such as liquid-crystalline polymers ~rigid-rod polymer solutions! @Chow et al. ~1992!; Wang et al. ~1993!#, suspensions of spheroidal particles @Hinch and Leal ~1973!; Okagawa et al. ~1973!; Frattini and Fuller ~1986!#, as well as in biological problems such as the transport of red blood cells @Bitbol ~1986!#. Theoretical models used to treat these systems are often based on a particle-level description of the suspension and involve rotation, alignment, and interactions between particles @Jeffery ~1922!; Anczurowski and Mason ~1967!; Hinch and Leal ~1973!; Okagawa et al. ~1973!; Rahnama et al. ~1995!#. Macroscopic predictions from these models are often tested using rheological or optical ~birefringence! measurements @Frattini and Fuller ~1986!; Chow et al. ~1992!; Wang et al. ~1993!#. Other experiments have used direct photographic methods to observe particle orientation and rotation @Anczurowski et al. ~1967!; Anczurowski and Mason ~1967!; Okagawa et al. ~1973!; Okagawa and Mason ~1973!; Stover et al. ~1992!#. One difficulty involved in the direct observation of individual particles is the three-dimensional nature of the systems. How are we to observe particles deep within the flow? Researchers have solved this problem either by limiting their observations to very dilute suspensions @Anczurowski et al. ~1967!; Anczurowski and Mason ~1967!; Okagawa et al.; Okagawa and Mason ~1973!# or by creating concentrated suspensions from particles that are carefully index-matched with the solvent and adding a few non-index-matched particles to observe @Stover et al. ~1992!#. In this paper, a!
Address correspondence to D.K.S. at Dept. of Chemistry, Tulane University, New Orleans, LA 70118-5698. Electronic mail:
[email protected]
© 1997 by The Society of Rheology, Inc. J. Rheol. 41~5!, September/October 1997
0148-6055/97/41~5!/1173/9/$10.00
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FIG. 1. Thermodynamic surface pressure vs molecular area isotherm for a monolayer of 12-NBD stearic acid at 21.5 °C.
we present an alternative, particles can be placed within a surfactant monolayer ~Langmuir monolayer! at the air/water interface. These particles can be easily and directly observed using various optical microscopy techniques while under the influence of the quasi-two-dimensional ~2D! monolayer flow. The behavior of the particles in these 2D model systems may provide useful insight into more complex rheological problems. To demonstrate the applicability of the technique, we examine a well-understood problem, tumbling of dilute, rigid, elongated particles in a shear flow, and quantitatively verify that the particles obey Jeffery’s equation in the 2D limit. A Langmuir monolayer is a single layer of water-insoluble amphiphilic molecules placed at the air/water interface, generally contained in a specially designed Langmuir trough, which allows the control of surface concentration ~by varying the area accessible to the molecules! and the measurement of surface pressure. Two-dimensional surface pressure versus molecular area isotherms ~the 2D analog of P – V isotherms! often suggest the presence of 2D phase transitions within the monolayer. For example, a horizontal region in an isotherm ~see Fig. 1! indicates the coexistence region of a first-order phase transition. More subtle features, such as kinks, may indicate higher-order transitions. In this paper, we utilize a fluorescently labeled amphiphile ~12-NBD stearic acid! that undergoes a surface pressure-driven 2D liquid-to-crystal transition @Muller and Gallet ~1991a!; Muller and Gallet ~1991b!#. The crystallites that form in coexistence with 2D liquid are elongated 2D fibers. The monolayer structure in the coexistence region is, therefore, a 2D suspension of rod-shaped particles where the suspending liquid is actually the melt of the substance that the particles are composed of ~analogous to small ice particles suspended in water at the freezing point!. Both the particles and suspending
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FIG. 2. A cartoon of the device used to create simple shear flow within the trough.
liquid are 1 monolayer thick, the difference between the two phases is in the molecular packing and order. Our suspension, therefore, is created in situ. II. EXPERIMENT 12-NBD stearic acid ~Molecular Probes! was deposited from chloroform solution ~approximately 1 mg/ml! onto a clean water ~Millipore Milli-Q UV1! surface contained in a custom-made Teflon Langmuir trough. After several minutes delay for the chloroform to evaporate, the monolayer was slowly compressed slightly into the coexistence region. The monolayer was monitored using fluorescence microscopy or Brewster angle microscopy ~BAM! during this procedure to assure that only a few crystalline nuclei were created. If the monolayer was compressed too quickly, many crystallites nucleated and grew into a rigid network; it was subsequently impossible to observe individual particles. By careful control of compression speed and waiting times, we were able to control the size and concentration of the particles. A typical particle aspect ratio was 762 and was fairly insensitive to the growth conditions at constant temperature. After the 2D suspension was created, surface shear was initiated in one of two geometries: surface pressure-driven flow through a channel ~2D Poiseuille flow! or simple shear created by counter-rotating parallel bands. To create the channel flow, the two barriers confining the monolayer were translated at the same velocity in the same direction. The monolayer was forced through a channel cut in a third, immobile, barrier placed between the two moving barriers. The channel was 2 mm wide and 25 mm in length. To create simple shear flow, Teflon bands, placed through the air/water interface, were driven by motorized Delrin rollers, as shown in Fig. 2. The gap was about 6 mm wide and
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FIG. 3. A Brewster angle microscope image showing the right half of a channel that is 2 mm wide. The edge of the channel is visible on the right. Elongated 2D crystallites flow from bottom to top. The long axis of the particles is aligned in the flow direction on average.
60 mm in length. A constant shear rate was observed in the gap between the two bands. The stagnation line at the center of the gap was particularly useful as it allowed us to observe individual particles for extended times. It was difficult to observe the monolayer near the moving bands because of surface curvature near the meniscus. A clever choice of band material might mitigate this problem. The monolayer was observed using an Olympus BH2-UMA fluorescence microscopy ~excitation light came from a 100 W Hg arc lamp!. The crystalline domains appear brighter in these images since the density of the fluorescent molecules is greater. Alternately, we used a custom built BAM to observe the monolayer. The observations using both techniques were consistent; however, fluorescence microscopy was preferred because of increased contrast and the absence of distortion due to tilting of the lens assembly at the Brewster’s angle. In both cases, the images of flow were videotaped and later digitized for analysis using a Scion frame grabber on a PowerMac 8500 controlled with public domain NIH Image software. III. RESULTS AND DISCUSSION Observation of channel flow made it clear that the crystallites were generally aligned in the shear direction ~see Fig. 3!. By tracking individual crystallites at various locations in the channel, we determined that the velocity profile agreed quantitatively with the semielliptical form expected for our geometry when the dissipation due to surface viscosity is negligible compared to the drag exerted by the subphase @Schwartz et al. ~1994!; Stone ~1995!#. Knowing the shear rate at any point in the channel, we were able to extract a rough dependence of the crystallite alignment upon the shear rate simply by measuring the angle of the major axis ~relative to the flow direction! for many particles positioned at various places within the channel. We define the usual 2D orientational order parameter, S, as S 5 2^cos2 u21/2& .
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FIG. 4. The orientational order parameter is plotted versus the shear rate of particles in the surface pressuredriven channel flow. The orientation angle of approximately 1000 particles was measured as a function of position across channels such as the one shown in Fig. 3. The velocity profile across the channel was measured simultaneously ~by tracking the motion of various particles! allowing us to directly calculate the shear rate at any point in the channel.
This is defined in such a way that S 5 0 for an isotropic distribution and S 5 1 for a perfectly aligned distribution. Figure 4 shows the dependence of S on the shear rate within the channel. Of course, there is a one-to-one correspondence between the shear rate and the position in the channel. As expected, the particles near the edges are better aligned due to the increased shear rate. The reason for the average alignment in the shear direction is that the particles are constantly tumbling; however, the rotational velocity is much slower when the particle is pointing in the direction of shear than when it is perpendicular to the shear @Jeffery ~1922!#. In order to verify this directly, we observed the behavior of particles on or near the stagnation line of the constant shear apparatus. One example is shown below ~Fig. 5!. The largest particle we observed to tumble was about 150 mm long ~shown in Fig. 5!. Figure 6~a! shows the time dependence of the major axis orientation angle relative to the shear direction. Unfortunately, the particle moved out of our field of view after about 15 s. The data points in Fig. 6~b! represent a numerical differentiation of the angle data in Fig. 6~a!. Figure 7 shows analogous time-dependent orientation data for a smaller particle ~about 40 mm long!, which stayed in our field of view for a longer period. The lines drawn through the data in Figs. 6 and 7 are derived from the equations proposed by Jeffery @Jeffery ~1922!#. In 2D, the differential equations describing the angular coordinates of a spheroidal particle in a shear flow ~Jeffery’s equations! reduce to the single equation
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FIG. 5. Fluorescence microscopy was used to observe the motion of a particle, 150 mm in length, near the stagnation line of the simple shear apparatus. A sequence of images captured from videotape is shown, displaying the typical tumbling behavior.
du dt
5
G
2 ~r r2e 11 e
sin2 u1 cos2 u!,
where G is the shear rate, and r e is the aspect ratio of the particle. Figure 6~b! is a direct comparison of d u /dt versus u, verifying Jeffery’s equation. The solid lines in Figs. 6~a! and 7 represent the corresponding integrated solution of Jeffery’s equation:
cot u 5 re cot
S D 2p T
1f ,
where the period of revolution, T 5 (2 p /G)(r e 11/r e ), and f is the arbitrary initial phase. The aspect ratio of the particle in Fig. 5 was 7.4 and the shear rate was 0.99 s21. The corresponding period of revolution is 47 s. Clearly, the match between the data and the expected form in Fig. 6~a! is excellent, especially considering that there are no adjustable parameters. The aspect ratio was 6.3 for the particle of Fig. 7. Combining this with the measured shear rate of 1.24 s21, yields a period of revolution of 33 s. Again, the agreement between the data and theory is excellent.
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FIG. 6. ~a! The orientation angle of the particle shown in Fig. 5 is plotted versus time ~dots!. The line drawn through the data represents the solution to the 2D Jeffery equation. Note that there are no variable parameters associated with the line. ~b! The data in part ~a! were numerically differentiated and plotted versus the angle as an explicit demonstration of Jeffery’s equation.
IV. CONCLUSIONS Two-dimensional elongated particles, embedded in a surfactant monolayer, were directly observed under shear flow using optical microscopy techniques. The average alignment in the shear direction was observed as a function of shear rate in the 2D channel
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FIG. 7. The orientation angle of a different particle ~40 mm in length! is plotted versus time along with a line representing the solution to the 2D Jeffery equation.
flow. In simple shear flow, individual particles were observed to tumble in quantitative agreement with the expected Jeffery orbit for the appropriate shear rate and particle aspect ratio. Direct observation of rigid particles under flow in a 2D surfactant monolayer has been shown to be a useful technique for the observation of particle-level phenomena, such as alignment and tumbling. ACKNOWLEDGMENTS The authors thank Gerry Fuller for many helpful conversations. This work was supported by the Center for Photoinduced Processes ~funded by the National Science Foundation and the Louisiana Board of Regents!, Louisiana Education Quality Support Fund Contract No. LEQSF ~1996-99!-RD-B-12, and the donors of the Petroleum Research Fund.
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Jeffery, G. B., ‘‘Motion of ellipsoidal particles immersed in a viscous fluid,’’ Proc. R. Soc. London, Ser. A 102, 161–179 ~1922!. Muller, P., and F. Gallet, ‘‘First measurement of the liquid–solid line energy in a Langmuir monolayer,’’ Phys. Rev. Lett. 67, 1106–1109 ~1991a!. Muller, P., and F. Gallet, ‘‘Shape anisotropy of ferroelectric domains in a Langmuir monolayer,’’ J. Phys. Chem. 95, 3257–3262 ~1991b!. Okagawa, A., R. G. Cox, and S. G. Mason, ‘‘The kinetics of flowing dispersions VI. Transient orientation and rheological phenomena of rods and discs in shear flow,’’ J. Colloid Interface Sci. 45, 303–329 ~1973!. Okagawa, A., and S. G. Mason, ‘‘The kinetics of flowing dispersions VII. Oscillatory behavior of rods and discs in shear flow,’’ J. Colloid Interface Sci. 45, 330–358 ~1973!. Rahnama, M., D. L. Koch, and E. S. G. Shaqfeh, ‘‘The effect of hydrodynamic interactions on the orientation distribution in a fiber suspension subject to simple shear flow,’’ Phys. Fluids 7, 487–506 ~1995!. Schwartz, D. K., C. M. Knobler, and R. Bruinsma, ‘‘Direct observation of Langmuir monolayer flow through a channel,’’ Phys. Rev. Lett. 73, 2841–2844 ~1994!. Stone, H. A., ‘‘Fluid motion of monomolecular films in a channel flow geometry,’’ Phys. Fluids 7, 2931–2937 ~1995!. Stover, C. A., D. L. Koch, and C. Cohen, ‘‘Observations of fibre orientation in simple shear flow of semi-dilute suspensions,’’ J. Fluid Mech. 238, 277–296 ~1992!. Wang, C., B. E. Vugmeister, and H. D. Ou-Yang, ‘‘Pretransitional orientational ordering of rigid-rod polymers in shear flow,’’ Phys. Rev. E 48, 4455–4459 ~1993!.
