Langmuir 2000, 16, 9433-9438
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Transient Behavior of the Velocity Profile in Channel Flow of a Langmuir Monolayer Ani T. Ivanova and Daniel K. Schwartz* Department of Chemistry, Tulane University, New Orleans, Louisiana 70118 Received May 31, 2000. In Final Form: August 25, 2000 We have observed the surface pressure-driven flow of Langmuir monolayers through a narrow channel by means of Brewster angle microscopy (BAM). The monolayers, composed of long chain carboxylic acids, were studied in a “hexatic” liquid crystalline mesophase (L2 phase). The velocity profile across the channel demonstrated transient behavior; i.e., it evolved from the expected parabolic shape to triangular as a function of flow rate, location along the channel, and time elapsed after start-up of flow. These transient effects appear to be related to the total strain in the system and suggest the importance of elasticity in the monolayer flow response. For situations in which the velocity profile was triangular, the distribution of domain widths across the channel indicated that the degree of domain stretching was systematically less than expected from the amount of strain the monolayer had experienced. This implied that elasticity of domain boundaries and/or slippage between domains resists domain stretching and may contribute to the observed non-Newtonian behavior of the monolayer.
Introduction Interfacial flow and the stability of multiphase systems can be significantly influenced by the presence of surfactants. Therefore, an understanding of how the presence of surfactants affects interfacial rheology is important for materials with high internal surface areas such as foams and emulsions.1 Applications of such materials are found in technologies including detergency, food technology, pharmaceuticals, and coating processes. Langmuir monolayers have often been used as model systems to study interfacial rheology because they allow for convenient control of thermodynamic conditions and molecular density as well as direct observation of the flow.2-5 However, Langmuir monolayers of even simple molecules such as n-alkylcarboxylic acids are known to display complex structures including a variety of liquid crystalline phases.6,7 These phases are characterized by a local lattice arrangement of the molecular headgroups as well as by longrange order of the molecular tilt direction that gives rise to a domain structure. Thus, a monolayer in a hexatic liquid crystalline state typically appears as a mosaic of domains with distinct boundaries. The complex structure of a fatty acid monolayer at both molecular and mesoscopic length scales is expected to affect its rheological behavior. Studies of flow in Langmuir monolayers of carboxylic acids have revealed non-Newtonian response within the liquid-crystalline phases. For example, flow-induced orientation of the tilt angle was observed by Fuller’s group in the L2′ phase in extensional flow.3 Further work by the same group suggested that within the L2 phase there is * To whom correspondence should be addressed. Tel 504/8623562; FAX 504/865-5596; email dks@ tulane.edu. (1) Edwards, D. A.; Brenner, H.; Wasan, D. T. Interfacial Transp. Processes Rheol. 1991. (2) Schwartz, D. K.; Knobler, C. M.; Bruinsma, R. Phys. Rev. Lett. 1994, 73, 2841. (3) Maruyama, T.; Fuller, G.; Frank, C.; Robertson, C. Science 1996, 274, 233. (4) Kurnaz, M. L.; Schwartz, D. K. Phys. Rev. E 1997, 56, 3378. (5) Maruyama, T.; Lauger, J.; Fuller, G. G.; Frank, C. W.; et al. Langmuir 1998, 14, 1836. (6) Knobler, C. M.; Desai, R. C. Annu. Rev. Phys. Chem. 1992, 43, 207. (7) Kaganer, V.; Mohwald, H.; Dutta, P. Rev. Mod. Phys. 1999, 71 (3), 779.
a critical strain for a transition between elastic deformation and viscous flow of the monolayer.5 Nonlinear shear response to an applied sinusoidal shear stress and an unexpected maximum in viscoelastic moduli in the L2 phase were also reported.8 Recent experiments in our group with surface pressure-driven monolayer flow through a narrow channel found unusual velocity profiles in the L2 phase.4 It was shown previously2 that in a situation where the subphase viscosity can be ignored (as in the L2 phase) one expects a parabolic velocity profile across the channel if the monolayer behaves as a 2-dimensional (2D) Newtonian fluid. However, under certain conditions, sharp (triangular) velocity profiles were observed during channel flow of arachidic (C20) acid, indicating a non-Newtonian response. In other work,9 we extended the channel flow observations of fatty acid monolayers to a systematic study throughout the different mesophases (L2, Ov, and L2′) as a function of temperature, flow rate, and alkyl chain length. These steady-state flow experiments showed that the unusual behavior of the velocity profile was typical for the other tilted phases (L2′ and Ov) as well as the L2 phase. The results4,9 also indicated that the flow behavior was related to the details of the molecular packing in the particular mesophases and not to the interaction of the monolayer with the subphase. In this study we further explore which aspects of the monolayer properties are playing a role in the unusual behavior by studying transient phenomena in channel flow. We have used Brewster angle microscopy (BAM)10,11 to visualize the flow and have observed transient effects in the measured velocity profile. In particular, we have studied the L2 mesophase, where the velocity profile evolves from the expected parabolic to a triangular as a function of flow rate, location along the channel, or time elapsed after the start-up of flow. Many of these phenomena are apparently related to the total strain and suggest that monolayer elasticity may play a role in the flow behavior. (8) Ghaskadvi, R. S.; Ketterson, J. B.; Dutta, P. Langmuir 1997, 13, 5137. (9) Ivanova, A.; Kurnaz, M. L.; Schwartz, D. K. Langmuir 1999, 15, 4622. (10) He´non, S.; Meunier, J. Rev. Sci. Instrum. 1991, 62, 963. (11) Ho¨nig, D.; Mo¨bius, D. J. Phys. Chem. 1991, 95, 4590.
10.1021/la000754r CCC: $19.00 © 2000 American Chemical Society Published on Web 10/21/2000
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Ivanova and Schwartz
Experiment A Langmuir monolayer was prepared by spreading stearic (C18) or arachidic (C20) acid (>99%, Sigma) from chloroform (Fisher Spectranalyzed) on a clean water surface (Millipore Milli-Q UV+) contained in a custom-built Langmuir trough. The pH of the pure water in equilibrium with atmospheric CO2 was 5.7 ( 0.1. The surface pressure was monitored by a filter paper Wilhelmy plate and a R&K electrobalance. The subphase temperature was adjusted by a combination of a recirculating water bath and thermoelectric Peltier elements and monitored by a Teflon-encapsulated thermocouple probe. The trough was equipped with two Teflon barriers, one of them motorized and the other one attached mechanically to it, such that both can move at the same controllable speed. This geometry allowed translation of the monolayer without compression. A stationary barrier was placed between the two movable barriers. It was constructed of glass and made hydrophobic by treatment with octadecyltrichlorosilane. The glass barrier contained two channels, 21 mm in length. One of them, 1 mm wide, was the microscope channel for BAM observations. The other channel had a variable width, 1-10 mm, and was used as a twodimensional flow divider to allow control of very low flow rates through the microscope channel. After bringing the subphase to the desired temperature, the monolayer was deposited at a surface pressure of π ) 10 mN/m and slowly compressed to the desired surface pressure of 21 mN/m. This surface pressure was chosen since it was the minimum value for which the unusual velocity profiles were previously observed.4 Then, the second Teflon barrier was coupled to the motorized one, and they were moved in concert to force the monolayer through the narrow channel. The barrier speed was controlled by Labview software. The monolayer flow in the channel was observed with a custombuilt BAM. Depending on the specifics of the experiment, the BAM was focused at different places along the 21 mm long channel (beginning, middle, exit). BAM allows observation of the monolayer due to differences in contrast caused by the tilt direction of molecules.10,12 Each shade of gray in a BAM image corresponds to a particular azimuthal tilt orientation; thus, a monolayer typically appears as composed of domains. Within a given domain, the molecules are all tilted in the same azimuthal direction. To determine velocity profiles of the monolayer during flow, we used as markers the distinctive features of domain boundaries. The videotaped BAM images were digitized and analyzed frame by frame with NIH-image software to extract the velocity profiles across the channel (see Figure 1). Typically, the image analysis resulted in velocity profiles that were symmetric with respect to the center of the channel within experimental errors. Occasionally, due to higher uncertainties as the flow rates increased, the velocity profiles appeared somewhat asymmetric. We have previously shown that the critical flow rate for the onset of triangular velocity profile depends on the temperature, surface pressure, and alkyl chain length, i.e., the location within the monolayer phase diagram.9 In this study, we observed the transient monolayer response in a particular phase, the L2 phase, for which the onset of unusual flow behavior (sharp velocity profiles as opposed to the expected parabolic) occurred at easily accessible flow rates. We monitored the monolayer flow at different flow rates, where the centerline velocity was in the range 30-800 µm/s, and at three different locations along the channel (2-3 mm from the beginning, in the middle, and 1-2 mm from the exit end of the channel). During channel flow the domains were deformed but retained their integrity (domain boundaries were clearly visible). Therefore, we were able to perform an analysis of the domain width at several locations across the channel under conditions where the velocity profiles were either parabolic or triangular. An object stripe of width w ) 10 pixels and height y ) 150 pixels was chosen from the desired location at distance x from the channel edge (see Figure 2). The characteristic domain width at position x was determined using the following difference correlation function: (12) He´non, S.; Meunier, J. J. Phys. Chem. 1993, 98, 9148.
Figure 1. A typical sequence of BAM images of C20 monolayer flow along the channel; (a) and (b) indicate different times during flow. The total elapsed time is about 2 s. The arrows indicate some distinctive features in the domains that are followed frame by frame and are used to map the velocity profile across the channel.
Figure 2. A BAM snapshot from a C20 monolayer during channel flow. The y-axis indicates a direction along the channel and the x-axis across the channel. The highlighted area shows an object stripe of width w that is used for domain width analysis. 5
Fx(n) )
150
∑ ∑[I(i,j) - I(i+n,j)]
2
i)-5 j)0
where I is the pixel intensity. Fx(n) was calculated for values of n varying from x - 20 to x + 20. For example, we chose from the image an object stripe with the specified dimensions and x ) 250 (pixels from one edge of the channel). We compared it consecutively (selecting each consecutive image stripe by shifting by 1 pixel in the x direction) to the intensities within stripes in the region between x ) 230 and x ) 270 pixels from the channel edge. In the situation when x ) 250, Fx(n)0) ) 0 because at that location the image stripe and the object stripe were identical. However, when the image stripe was shifted by even 1 pixel from the object stripe (in the (x direction), the value of Fx(n) increased, since the light intensity of the image stripe had changed due to the substituted 1 pixel column. In a BAM image, as mentioned before, the monolayer appears as a mosaic of domains with different reflectivity, and neighboring domains are distinguished by brightness. For small shifts of the object stripe away from its
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Figure 3. Fx(n) vs n for an object stripe starting x ) 250 pixels from the channel edge. The analysis was for monolayer flow with a parabolic velocity profile. The domain width was determined as described in the text. Dashed lines indicate the chosen minimum and maximum values of Fx(n). original position (small n) each piece of the object stripe remained in its original domain (where the light intensity was relatively constant), resulting in a small value of Fx(n). For larger values of n the object stripe began to overlap neighboring domains and Fx(n) increased. Eventually, the image stripe lay totally outside of the original domains (in neighboring ones). If n was further increased, Fx(n) remained almost constant for a number of n values until the image stripe “hit” another domain with similar reflectivity as the first one (then Fx(n) decreased). As a result, a typical curve for Fx(n) plotted as a function of n has the form of an inverted peak that flattens at both ends, as shown in Figure 3. Figure 3 demonstrates how a domain width was extracted from the plot of Fx(n) vs n. We usually ignored the minimum in the inverted peak since it corresponded simply to the situation in which the image stripe and the object stripe coincided and was not meaningful for our measurements. We considered the next lowest Fx(n) value as the curve minimum (Fmin) and as the maximum (Fmax) the value at which Fx(n) was approximately constant. Then, the domain width was determined by measuring the width (fwhm) of the inverted peak at a value corresponding to Fmin + (Fmax - Fmin)/2, as described in Figure 3. This correlation function-based method was judged more reliable than direct measurement of domain widths by manual image inspection since it incorporated statistical averaging over the image and avoided bias possibly introduced by arbitrary domain selection. By determining the domain width in a number of different locations across the channel, we obtained a domain width distribution. The image analysis was performed for four different images that were typical for monolayers displaying a parabolic velocity profile and four images from a typical triangular velocity profile. The analyzed image frames were captured ∼5 mm downstream from the channel entrance.
Results and Discussion A power law model was used previously4 to parametrize different types of velocity profiles in channel flow. In general, the velocity profile across the channel is given by u ) umax(1 - xj(1+R)/R) where xj ) 2x/d (d is the channel width and x is the distance across the channel with the origin at the center). The parameter R can describe a wide range of rheological behavior: R ) 1 describes a Newtonian fluid and is associated with a parabolic velocity profile, R > 1 indicates a sharpened profile (associated with shear thickening), and R > 3 indicates a nearly triangular velocity profile. The same model describes shear thinning fluids (blunt velocity profiles) when R < 1. Figure 4 shows the evolution of the velocity profile of a C18 monolayer as a function of flow rate in the L2 phase. The values of R and the best fit to a parabolic profile, as well as to a power law model, are presented for each of
Figure 4. Velocity profiles of a C18 acid monolayer at T ) 10 °C and π ) 21 mN/m. The velocity profiles were determined for a monolayer that was observed at mid-channel. The dashed lines show the best parabolic fit to the data, and the solid lines are fits to a power law model (as described in text). From (a) to (d) the velocity profiles gradually evolve from parabolic to sharpened and finally triangular as a function of flow rate.
the velocity profiles. At very low flow rates the velocity profile was parabolic, sharpened as the flow rate increased, and finally became triangular. This effect has been
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previously reported,4 and it was shown that sharpening of the profile did not correspond to an increase in viscosity of the system with increasing rate of shear (shear thickening). Results also indicated that the flow behavior was not related to the interaction of the monolayer with the subphase.4,9 Those experiments were performed for a steady-state flow in a single location along the channel (mid-channel). We monitored the monolayer flow response as a function of flow rate at different locations along the channels beginning, middle, and exit endsand determined the critical flow rate for which the onset of a triangular profile was first observed from the evolution of velocity profiles for each of the locations chosen (see Figure 5). At any given location, the velocity profile gradually sharpened with increasing flow rate. Figure 5a shows an example for a C20 monolayer observed at the exit end of the channel. However, the critical flow rate for the appearance of a triangular profile was location-dependent. Figure 5b summarizes the values of R for the beginning, middle, and exit end of the channel as a function of flow rate. It demonstrates that the critical flow rate decreases as the monolayer is monitored further downstream. When the monolayer flow was recorded near the channel entrance, the velocity profile became triangular at ∼600 µm/s, in the middle of the channel at ∼520 µm/s, and toward the exit end at ∼400 µm/s. To support the observation described above, we performed a channel flow experiment at a fixed flow rate of 500 µm/s and determined velocity profiles at different locations along the channel. The spatial evolution of the velocity profile at a flow rate of 500 µm/s is shown in Figure 6. As indicated in the figure, near the channel entrance it was nearly parabolic (R ) 0.94), sharpened toward the middle (R ) 2.8), and was triangular near the exit end of channel (R ) 7.9). These results indicate that the critical flow rate for the onset of a triangular profile changes as a function of the time the monolayer has spent in channel, suggesting a relationship between the velocity profile and the amount of total strain. When the monolayer is monitored further down the channel, it has spent longer time in the channel and experienced a higher strain. In addition to this spatially transient behavior, we also observed temporally transient behavior at the exit end of the channel following the start-up of flow (see Figure 7). For a flow rate of ∼400 µm/s, for which the monolayer was previously seen to display a steady-state triangular profile at the exit end of the channel, the velocity profile was initially parabolic (Figure 7a) and evolved to triangular (Figure 7c) after about 30 s. By using a pressure sensor on each end of the channel, we were able to measure the surface pressures π1 and π2 before and after the channel. The time for ∆π ) π1 - π2 to reach a constant value was about 3 s, much shorter than the time (∼30 s) required for the velocity profile to evolve to triangular for the same flow rate. This observation implied that the velocity profiles at the exit end of channel (Figure 7) were determined for a fully developed monolayer flow. This behavior is consistent with the interpretation that the sharpening of the velocity profile is related to increase in the total monolayer strain. In the “start-up” experiment, a section of monolayer imaged at a given time has traveled a certain distance (and experienced a related amount of shear strain) from its location when the flow was initiated. A section of monolayer imaged at a later time (in the same channel location) has traveled farther and experienced more shear strain. The connection between time after
Ivanova and Schwartz
Figure 5. (a) Velocity profiles of an C20 monolayer at T ) 18 °C and π ) 21 mN/m determined at the exit end of the channel, for increasing flow rates from I to III. The “critical” flow rate in this instance was determined to be ∼400 µm/s, for which a sharp velocity profile is clearly displayed (example III). (b) Values of R for an C20 monolayer (T ) 18 °C and π ) 21 mN/m) observed at the entrance (circles), in the middle (squares), and at the exit end (triangles) of the channel. The dashed lines are guides for the eye. The points labeled with Roman numerals refer to the velocity profiles in (a). The flow rate necessary for a triangular profile decreases systematically as one moves from the entrance to the middle to the exit of the channel.
start-up and strain in these experiments is analogous to the connection between location along the channel and strain in the steady-state experiments described above. For example, when a monolayer with a centerline velocity u ) 450 µm/s is monitored near the exit end of the channel at time t ) 33 s after the start-up of flow, it has an average
Channel Flow of a Langmuir Monolayer
Figure 6. Spacial evolution of velocity profiles of an C20 monolayer (T ) 18 °C and π ) 21 mN/m) at a flow rate of 500 µm/s. The velocity profiles are (a) nearly parabolic at the entrance of the channel, (b) sharpened in the middle of the channel, and (c) nearly triangular at the exit end of the channel.
strain of γ j ) (∂u/∂x)(t) ) (450 µm s-1/550 µm)(33 s) ) 27. This average strain is approximately equivalent to the amount by which the monolayer will be strained at steady state if it is observed at a distance l ) (450 µm s-1)(33 s) ≈ 1.5 cm, which is consistent with the experimental distance from the channel entrance in the previous experiment (γ ) (∂u/∂x)(l/u) ) 27). The observations of spatial and temporal transient effects in the velocity profile imply that the total amount of strain is an important factor, thus suggesting that elasticity of the monolayer plays a role in the flow behavior. Possible sources of elastic behavior can be found if one considers the monolayer structure in the relevant phases. The monolayer in the liquid-crystalline mesophases is characterized by a domain structure. Domain boundaries, with their line tension, are a possible source of elastic stress in the monolayer. Also, BAM images indicate that although the domains elongate during flow, they retain their integrity. One approach to study the possible effect
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Figure 7. Time evolution of velocity profiles of an C20 monolayer (T ) 18 °C and π ) 21 mN/m) that was observed at the exit end of the channel. (a) A parabolic velocity profile immediately after the flow started (2-3 s). (b) A sharpened velocity profile a few seconds after the flow started (5-10 s later). (c) A triangular velocity profile 30 s after the flow started.
of the elasticity of domain boundaries would be to estimate whether the amount of stretching of domains was related to different types of flow behavior. We have, therefore, performed an image analysis of domain widths from BAM images for different types of flow. A comparison of the domain width distributions for parabolic and triangular velocity profiles across the channel is shown in Figure 8. Each plot was obtained by averaging the results for domain widths for each particular location across the channel from four different measurements. The images for each measurement were captured for C20 monolayer flow at T ) 18 °C at a distance ∼5-6 mm from the channel entrance. The range of flow rates for the parabolic flow was 30-60 µm/s, and for the triangular flow it was 600-900 µm/s. We also calculated the expected distribution of domain widths for both parabolic and triangular flows (Figure 8) assuming that domains were deformed by shear and that
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Figure 8. Domain widths across the channel for a parabolic flow (circles) and triangular flow (triangles). The dotted and dashed lines in the inset represent a theoretical domain width distribution from the middle to the edge of the channel for a parabolic and triangular flow, respectively. Domain widths were calculated using the power law model for the velocity profile and assuming that the domain area of an initially square domain with dimensions 65 × 65 µm remained constant throughout the flow. The experimental parameters were as follows: the distance from the channel entrance was 5 mm, the channel width d was 1 mm, R ) 1 for the parabolic profile, and R ) 3.11 for the sharp profile.
domain area stayed constant during flow: The width
w≈
a2 a a2 ) ≈ length a(1 + |γ|) 1 + |γ|
where a is the side of an initially square domain and the strain, γ, was calculated using the power law model for the velocity profile:
|γ| )
lx1/R du l 1 + R ) dx v R 1 - x(1+R)/R
The expected domain width was calculated with parameters identical with the experimental conditions (i.e.,
Ivanova and Schwartz
distance traveled down the channel, R, channel width). Figure 8 indicates that domains are widest in the center and become narrower toward the channel edge. This narrowing is consistent with elongation due to shear strain under the assumption that domain areas stay nearly constant as directly observed for small strain. For conditions where the velocity profile was parabolic, the domain widths were qualitatively consistent with the model. The deviation near the center of the channel is to be expected since actual domains are finite in size and will always experience some strain. For conditions where a triangular profile was observed, the domain width decreased even more rapidly as one moves from the channel’s center toward the edge. However, the domain widths in triangular flow did not decrease as fast as expected from the model. This suggests that the domains either resist being stretched, slip along each other, or both. This provides additional evidence linking triangular velocity profiles to viscoelastic monolayer response. Conclusions We have observed transient behavior of the velocity profile in the channel flow of a Langmuir monolayer in one of the mesophases, the L2 phase, where the velocity profile evolves from a parabolic to triangular as a function of flow rate, location along the channel and time elapsed after the start-up of flow. These transient phenomena were related to the total strain and suggest the importance of elasticity in the monolayer flow response. Since the monolayer was within the L2 phase, elasticity of domain boundaries may play a role. Domain widths were measured as a function of distance across the channel. Although the domain widths decreased more rapidly from center toward the channel edge for triangular flow than parabolic flow, they did not fall off as rapidly as expected. This implied that the elasticity of domain boundaries as well as domain slippage may contribute to the observed non-Newtonian response of the monolayer. Acknowledgment. This work was supported by the National Science Foundation (Award 9733281), the donors of the Petroleum Research Fund, and the Camille Dreyfus Teacher-Scholar Awards Program. LA000754R
