Molecular Orientation in Langmuir Monolayers under Shear

Langmuir 2001, 17, 3017-3029

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Molecular Orientation in Langmuir Monolayers under Shear Jordi Igne´s-Mullol† and Daniel K. Schwartz*,‡ Department of Chemistry, Tulane University, New Orleans, Louisiana 70118 Received December 26, 2000. In Final Form: March 14, 2001 The coupling between the structure of the tilted condensed phases of docosanoic acid monolayers (L2, L2′, and Ov) and an external shear flow has been studied using Brewster angle microscopy. Generally, the coupling results in a reorientation of the alkyl tails, although different kinematics are observed depending on the thermodynamic phase, shear rate, and surface pressure. The coupling can result in continuous orientational changes or in abrupt reorientation of the molecular tilt angle. These molecular-level effects are connected to the macroscopic texture as well; shear can result in either domain fragmentation or annealing. A detailed quantitative analysis of phenomena reminiscent of the “flow alignment” and “tumbling” mechanisms found in nematohydrodynamics is presented. The analysis reveals that the kinematics of the reorientation is different from that in nematic liquid crystals. The underlying lattice plays a crucial role, and the observed phenomena can be explained with geometrical arguments.

1. Introduction Langmuir monolayers are important model systems for the study of dynamics and molecular interactions in biomembranes and have fundamental interest because of the ubiquitous nature of hexatic liquid crystalline (LC) phases in two dimensions. The past decade has seen great advances in the understanding of the structure and thermodynamics of fatty acid monolayers.1 The application of optical techniques such as fluorescence microscopy (FM)2 and Brewster angle microscopy (BAM)3,4 reveals a mosaic structure with macroscopic domains giving evidence of long-range orientational order, particularly in the tilted condensed phases. The molecular structure of these phases involves a strong coupling between the tilt azimuth of the alkyl tails and the lattice bond orientation,1,5,6 which has been studied in detail through the application of synchrotron X-ray scattering (see ref 1 and references therein). On the other hand, the understanding of molecular dynamics, transport, and rheology in monolayers is still incomplete. The presence of long-range orientational order has raised questions of whether the molecules in a hexatic monolayer might behave like a 2D nematic under flow conditions; i.e., either become aligned at a certain angle with the flow (flow alignment) or describe periodic orbits about an axis perpendicular to the interface (tumbling). Recent experiments show evidence of a flow-induced orientation of the molecular azimuth in L2′-phase monolayers.7,8 A series of shear domain cycles resulted in * To whom correspondence should be addressed. E-mail: [email protected]. Phone: 303-735-0240. Fax: 303492-4341. † Current address: Departament de Quı´mica Fı´sica, Universitat de Barcelona, Martı´ i Franque`s 1, E-08028 Barcelona, Spain. ‡ Current address: Department of Chemical Engineering, University of Colorado, Boulder, CO 80309. (1) Kaganer, V. M.; Mo¨hwald, H.; Dutta, P. Rev. Mod. Phys. 1999, 71, 779-819. (2) Riviere, S.; Henon, S.; Meunier, J.; Schwartz, D. K.; Tsao, M.-W.; Knobler, C. M. J. Chem. Phys. 1994, 101, 10045. (3) He´non, S.; Meunier, J. Rev. Sci. Instrum. 1991, 62, 936. (4) Ho¨nig, D.; Mobius, D. J. Phys. Chem. 1991, 95, 4590. (5) Nelson, D. R.; Halperin, B. I. Phys. Rev. B 1980, 21, 5312-5328. (6) Selinger, J. V.; Wang, Z.-G.; Bruinsma, R. F.; Knobler, C. M. Phys. Rev. Lett. 1993, 70, 1139-1142.

annealing of the domain structure, so that an initially rich mosaic of domains with different reflectivities evolved toward larger domains with only two, highly contrasted, reflectivities. This was interpreted as molecular alignment with the flow. Propagation of fronts, identified as shear bands, was also reported. It was not clear, however, whether the flow effects originated from distortions in the underlying lattice or from a direct coupling with the tilted alkyl tails. The latter scenario is in analogy with nematic flow, where the average orientation of the molecules is determined by the flow.9 In two recent publications we presented evidence of two different types of coupling for the L2′ 10 and Ov11 phases, which arise despite the strong similarities in the structures of the two phases. Although a continuous precession of the molecules was found in the L2′ phase,11 only propagation of fronts was observed in the Ov phase.10 Detailed analyses of both phenomena show that the similarities with nematohydrodynamics are only qualitative; the coupling of the hexatic bond-orientation order to the flow dominates the response. In this paper, we present a detailed description of the coupling between simple shear flow and the structure of the tilted condensed phases of docosanoic acid (L2, L2′, and Ov) as a function of shear rate, surface pressure, and temperature. We divide the observed flow-induced changes into two possible types: (1) continuous temporal changes in domain reflectivity and (2) nucleation and growth of regions with a new reflectivity through the propagation of fronts. One feature, present in all of the experimental observations, is worth noting here: there is no relaxation of the reflectivity upon cessation of flow, regardless of the state of evolution of the monolayer when the flow was interrupted. Because of this, it can be assumed that the instantaneous structure of the monolayer under flow conditions closely resembles that of a monolayer in (7) Maruyama, T.; Fuller, G.; Frank, C.; Robertson, C. Science 1996, 274, 233-235. (8) Maruyama, T.; Lauger, J.; Fuller, G. G.; Frank, C. W.; Robertson, C. R. Langmuir 1998, 14, 1836. (9) Chandrasekhar, S. Liquid Crystals, 2nd ed.; Cambridge University Press: New York, 1992. (10) Igne´s-Mullol, J.; Schwartz, D. K. Phys. Rev. Lett. 2000, 85, 14761479. (11) Igne´s-Mullol, J.; Schwartz, D. K. Nature 2001, 410, 348-351.

10.1021/la0017983 CCC: $20.00 © 2001 American Chemical Society Published on Web 04/19/2001

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Figure 1. Schematic of experimental apparatus. A p-polarized (P) laser beam is directed to the air/water interface at the Brewster’s angle (θB) by a mirror (M). After reflection, the resulting beam is focused through an analyzer (A) onto a CCD camera. The four rotating rollers drive two parallel bands that are in contact with the interface, creating a two-dimensional shear flow in the monolayer.

equilibrium. In particular, this implies that changes in the monolayer structure caused by the flow will be compatible with the strong coupling between the tilt azimuth and the lattice bond orientation in each phase. For example, molecules must remain tilted in the nearestneighbor (NN) direction in the L2 phase and in the nextnearest-neighbor direction (NNN) in the L2′ and Ov phases. This is a remarkable fact, and it will be crucial in understanding the nature of the flow-induced changes in the monolayer structure. Although we observe no apparent relaxation in the distribution of domain reflectivities in the shear experiments, recent studies have revealed viscoelastic relaxation of the shape of domains under shear; effects such as slippage between adjacent domains and elastic recovery have been reported.12 The observations presented in this article will be discussed from a mainly qualitative standpoint. Two particular cases will be analyzed in detail, namely, the continuous precession observed in the L2′ phase and the flow-induced alignment observed in the Ov phase, resulting in models that reproduce the observations. The methods developed for the analysis of these two cases can be used as a starting point for a detailed investigation of other types of coupling. 2. Experimental Procedure Monolayers of docosanoic acid, CH3(CH2)20 COOH, are prepared by depositing drops of a chloroform (Fisher Spectranalyzed) solution (∼1.0 mg/mL) on the surface of pure water (Millipore Milli-Q UV+) contained in a custombuilt Teflon Langmuir trough. The temperature in the subphase is controlled to within (0.5 K using a combination of a recirculating water bath (Neslab) and thermoelectric Peltier elements. A Teflon-encapsulated K-type thermocouple probe (Omega) serves to monitor the temperature of the water subphase. The surface area of the trough is adjusted by means of a motor-driven barrier, allowing compression or expansion of the monolayer. The surface pressure is monitored using a filter paper Wilhelmy plate and an R&K electrobalance. Shearing of the monolayer is achieved by means of two parallel bands that drag the monolayer in opposite directions (see Figure 1). The bands are Buna-N O-rings, each with a circular cross section of 0.16 cm, stretched between a pair of Delrin-covered stainless steel rods. The bands remain submerged in the water subphase while the chloroform solution is deposited on the interface, and they are subsequently lifted until they pierce through the (12) Ivanova, A.; Ignes-Mullol, J.; Schwartz, D. K. Langmuir, manuscript in press.

Igne´ s-Mullol and Schwartz

monolayer. Rotating the four rods in the same direction generates a simple shear flow in the region between the two moving bands. This region is 6.5 cm long and 1.5 cm wide. The observations are made near the stagnation line, which is roughly equidistant between the bands. The flow cell is driven by a computer-controlled DC motor. Shear rates in the range of 0.05-0.5 s-1 were used. Imaging of the monolayer is performed by means of a custom-built Brewster angle microscope (BAM).3,4 Light from a 30-mW 670-nm diode laser is p-polarized by a GlanThomson prism before being directed to the air-water interface with an incident angle equal to the Brewster angle of water (θB ≈ 53°). The plane of incidence is perpendicular to the velocity field applied to the monolayer. The incident beam is reflected from a mirror and directed toward the monolayer (Figure 1). The light reflected from the monolayer is focused onto a CCD camera by a 4× microscope objective after passing through an analyzer (a second Glan-Thomson prism). Because the monolayer is not parallel to the focal plane of the microscope, only a region of about one-third of the field of view is in focus. Moreover, images obtained with this configuration are distorted, and the dimension parallel to the incidence plane (X axis, in our choice of coordinates) is scaled by a factor of cos(θB) ≈ 0.6. Unless otherwise specified, images presented in this paper have been corrected for this geometrical distortion. The analyzer is oriented at 60° with respect to the plane of incidence throughout the experiments, which is found to optimize the contrast in the images. The laser-plus-microscope ensemble rests on a joystick-controlled motorized XY translation stage, which allows for tracking of domains that are not precisely on the stagnation line. Images are recorded on VHS tape and later digitized frame-by-frame using a Scion LG-3 frame grabber for further analysis with the public domain software NIH-Image and ImageJ. Quantitative analysis is performed with the software packages Mathematica and IgorPro. The gray level of the 8-bit digitized BAM images is linearly related to the monolayer reflectivity. Given the correspondence between the monolayer reflectivity and the orientation of the alkyl tails, our study of the coupling between the flow and the structure of the monolayers is based on the observed changes in reflectivity. The relationship between molecular orientation and reflectivity can be established using the formalism for propagation of light through birefringent media and Fresnel’s relations (which are elegantly implemented using the 4 × 4 matrix formalism of Berreman13-15). At a given surface pressure (Π), the polar tilt angle (θ) of the molecules in the monolayer is uniform (see Figure 2). Distinct boundaries exist between domains, within each of which the molecules share a common tilt azimuth, φ. Differences in the reflectivity of domains are directly related to differences in φ, with a maximum reflectivity contrast between domains that are roughly aligned with the flow at φ ≈ 90° (to the left in the BAM images) and φ ≈ 270° (to the right in the BAM images). Figure 2 shows the result of calculating the reflectivity of a Langmuir monolayer as a function of φ for different values of θ, assuming that the monolayer is a uniaxial anisotropic medium. The length of the C22 molecules and the dielectric tensor of the monolayer were measured previously using multiple-angle ellipsometry.16 (13) Berreman, D. W. J. Opt. Soc. Am. 1972, 62, 502. (14) Bethune, D. S. J. Opt. Soc. Am. B 1991, 8, 367. (15) Tsao, M.-W.; Fischer, T. M.; Knobler, C. M. Langmuir 1995, 11, 3184. (16) Paudler, M.; Ruths, J.; Riegler, H. Langmuir 1992, 8, 184.

Molecular Orientation in Langmuir Monolayers under Shear

Figure 2. Reflectivity curves, r(φ), corresponding to the fraction of light intensity reflected from a C22 acid monolayer as a function of the tilt azimuth (φ) of the fatty-acid molecules for different values of the polar tilt (θ). The numerical computations assume the experimental conditions described in the text (BAM conditions and the analyzer set at 60°). The length of the molecules is taken to be L ) 27.5 Å, and the elements of the dielectric tensor are taken to be ⊥ ) 2.16 and | ) 2.37 (ref 16). The plane of incidence is XZ, and the incident beam travels in the -X direction. The maximum contrast exists between domains that are close to being aligned with the flow, tilted toward either side of the incidence plane.

Compression of the monolayer results in a reduction of the value of θ and reduces the contrast of the BAM images (Figure 2). This puts an upper bound on the values of Π for which quantitative information can be extracted from the images. Moreover, optical artifacts, such as interference fringes and small illumination gradients are unavoidable in practice. To relate the domain reflectivity directly to shear strain, the local shear rate (γ˘ ) is experimentally determined from the measured rate of strain of the domain under study. This is more accurate than using an average value for γ˘ across the flow field, as domain slippage causes the local and average shear rates to differ.12 Moreover, measuring the local γ˘ value is often necessary because of the lack of a stationary reference frame (when the microscope is translated to follow a domain under study). 3. Results 3.1. Observations in the L2′ Phase. Coupling between the domain structure and the flow depends dramatically on γ˘ and Π. We have identified four types of phenomena, which appear at increasingly high Π, namely, (i) coarsening and annealing of the domain structure (C in Figure 3) with (ii) continuous reflectivity changes and occasional jump discontinuities, (iii) propagation of fronts that span single domains (F in Figure 3), and (iv) avalanche-like fronts that propagate across multiple domains (A in Figure 3). We have mapped the occurrence of the different behaviors on a schematic Π-γ˘ “phase diagram” (Figure 3). The boundaries are difficult to determine accurately, because different phenomena can occur simultaneously. Because only a limited region of the monolayer is viewed at any given time, the onset of a new phenomenon, such as the propagation of fronts, might not occur in the field of view unless Π is high enough for this phenomenon to occur throughout the monolayer. This means that, for a given γ˘ , the value of Π at which the boundary line is drawn is likely to be somewhat overestimated, particularly when higher shear rates are considered. The propagation of fronts is not observed at high temperatures. Whereas the two types of fronts mentioned above can be observed at T ) 17 °C, provided a high enough Π or γ˘ is used, only

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Figure 3. Different types of coupling between the orientation of the fatty-acid molecules, in the L2′ phase, and shear flow can be observed depending on the surface pressure (Π) and the shear rate (γ˘ ). Observations are performed at T ) 17 °C, where the L2′ phase spans from Π ≈ 17 mN/m to Π ≈ 29 mN/m. The region marked “C” corresponds to the coarsening and annealing of the domain structure described in the text. At higher γ˘ or higher Π (C+F) values, the propagation of fronts confined inside a single domain is also observed. At the highest values of Π (C+F+A), the behavior is dominated by avalanches of fast propagating fronts that span multiple domains.

continuous changes in the reflectivity are seen at T ) 22 °C, even at Π near the L2′-S transition and γ˘ ≈ 0.5 s-1. The observations reported in this section on L2′-phase monolayers correspond to measurements performed at T ) 17 °C. Temperature effects will be discussed again at the end of the section. 3.1.1. Coarsening and Annealing of the Domain Structure. The domain structure of the monolayers becomes coarser (domains increase in size) and the contrast of the images is enhanced upon application of shear-reversal cycles (Figure 4). After only a few cycles, the rich mosaic of reflectivities in a fresh monolayer (Figure 4a) changes so that only two highly contrasted values remain (Figure 4c). The shape of the reflectivity curve (see Figure 2) suggests that the alkane tails become oriented toward the direction of the flow, with dark domains toward the left and bright domains toward the right of the BAM images. Although this effect has been reported in the literature,7,8 the actual mechanism has not been explained, including whether the alignment process is driven by reorientation of the alkane tails or the underlying lattice. We propose that this annealing process takes place through the continuous changes in the domain orientation described in the next paragraph. 3.1.2. Continuous Precession and Jump Discontinuities of the Tilt Azimuth. Although the end result of shear cycles in the low-Π, low-γ˘ regime is the coarsening and annealing of the domain structure, close inspection of the local evolution reveals continuous changes in the reflectivity of individual domains.11 Figure 5 provides a clear example of this process. As shear is applied, the shape of the initially dark central domain is skewed, and its reflectivity is continuously increased. Simultaneously, the reflectivity of its surrounding domain decreases. At some point between parts c and d of Figure 5, the contrast of the images is lost, as the reflectivities of the two domains become equal. Qualitatively, this corresponds to a counterclockwise precession of the alkane tails from left to right in the central domain and from right to left in the surrounding domain. At higher strains, the contrast is recovered, but it has been inverted. Application of even higher strains has no observable effect on the reflectivities. Our observations, therefore, imply that the orientation of

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Figure 5. Continuous change in the reflectivity of an L2′-phase monolayer (T ) 17 °C, Π ) 23 mN/m, and γ˘ ) -0.2 s-1). The line segment in frame a is 100 µm long. The dark, fish-shaped domain in the center of frame a exhibits a continuous change of its reflectivity, while the surrounding domain darkens as strain increases, corresponding to continuous counterclockwise precession of the molecular orientation in both domains. Elapsed times after frame a are as follows: (b) t ) 1.7 s, (c) t ) 2.9 s, (d) t ) 4.3 s, (e) t ) 5.4 s, and (f) t ) 6.8 s. The contrast drops to zero between frames c and d when the reflectivities of both domains become equal.

Figure 4. Shear cycles are applied to an L2′-phase monolayer (at T ) 17 °C, Π ) 20 mN/m, and γ˘ ) (0.4 s-1), resulting in an increase in the domain size and annealing of the domain reflectivity. (a) Monolayer before shear is applied. (b) After 7 shear cycles. (c) After 16 shear cycles. The total (net) strain is γ ≈ 0 in all images. The line segment in frame a is 100 µm long.

the tails first rotates in the direction of the shear and approaches an asymptotic value at high strains. The asymptotic values of the reflectivities are either the maximum or the minimum in the monolayer, consistent with the alkyl tails turning toward the direction of the flow. In most of the experimental observations, this continuous process is interrupted by discontinuous changes in the reflectivity of particular domains. The discontinuities appear as diffuse fronts that propagate in arbitrary directions changing the reflectivity of the affected domains. The kinetics of this process is analyzed in detail in section 4. We show that the observations can be explained with a geometrical model in which the orientation of the alkane tails follows the distortions of the lattice.

Moreover, we propose that this process is the mechanism that leads to the observed coarsening and annealing of the domain structure. 3.1.3. Propagation of Fronts Spanning Single Domains (Type-I Fronts). Although the previous two mechanisms can be observed at any Π and γ˘ , an increase in Π or γ˘ (see Figure 3) results in the nucleation and growth of regions of a different reflectivity inside a given domain. These regions propagate as straight fronts (see Figure 6) that are oriented either parallel or perpendicular to the flow, i.e., at 45° relative to the principal strain axes of the shear flow (for simple horizontal shear flow, the principal strain axes are in the x ) y and x ) -y directions). The changes in reflectivity are confined to a single domain, although several domains can change simultaneously within the field of view. If a large enough strain is applied, the fronts propagate until the reflectivity of the affected domain is uniform again. This process is reversible: upon inversion of the flow direction, the front propagates backward, restoring the reflectivity of the domain to its initial configuration. Although the speed of propagation of the fronts appears to depend on the shear rate, it is typically on the order of several tens of micrometers per second. This type of flow-induced phenomenon agrees with similar observations reported by Fuller’s group,7,8 who described these advancing fronts as shear bands resulting from a plastic flow that accompanies the molecular tilt reorientation. The final reflectivity after the propagation of the front is consistent with one of the asymptotic values


Figure 6. Changes in reflectivity appear as sharp fronts that propagate within a single domain in an L2′-phase monolayer (T ) 17 °C, Π ) 22.5 mN/m, and γ ) -0.26 s-1). The fronts are either parallel or perpendicular to the flow. The arrows on frames a-c indicate the directions of advance of the fronts. Times elapsed after frame a are as follows: (b) t ) 0.27 s and (c) t ) 0.6 s. The line segment in frame a is 100 µm long.

described in the previous section, i.e., the gray level is either the darkest or the brightest observed in the monolayer. 3.1.4. Propagation of Avalanche-like Fronts (Type-II Fronts). The rearrangements in the monolayer structure described in the previous two sections roughly preserved the domain boundaries. In the case of propagating fronts, the changes in reflectivity were confined to a single domain. For high enough Π (Figure 3), however, a new type of front nucleates and propagates across domain boundaries (Figure 7). The reflectivity of the region affected by the fronts changes but, in general, does not become uniform. When two highly contrasted neighboring domains are affected by the same front, the contrast between them decreases. On the other hand, when one of these events occurs in a region with intermediate reflectivity (low contrast), highly contrasted domains might result. The kinematics is also quite different from that for

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Figure 7. Changes in reflectivity appear as sudden events that take the form of sharp fronts that propagate across domain boundaries in an L2′-phase monolayer (T ) 17 °C, Π ) 23 mN/ m, and γ˘ ) -0.55 s-1). A front grows in (b) across two domains of different reflectivities. Elapsed times after frame a are as follows: (b) t ) 0.03 s and (c) t ) 0.17 s. The arrows indicate the advance of the fronts. The line segment in frame a is 100 µm long.

the fronts described in the previous section. These avalanche-like fronts propagate much faster, with typical velocities on the order of hundreds of micrometers per second, that do not depend strongly on γ˘ . Their life is very short, typically on the order of a few tenths of a second. During the time when the steady-state shear flow is applied, several instances of this type of flow nucleate, grow, and cease to propagate, apparently with random occurrence. The size of the affected region grows with Π. The orientation of the fronts is not related to the flow direction (unlike that of the fronts described in the previous section). In the example shown in Figure 7, the front that propagates is the combination of two straight fronts oriented in directions about 30° apart. In general, the shape of the region affected by these fronts has smooth

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Figure 8. Propagation of fronts in an Ov-phase monolayer (T ) 46 °C, Π ) 21 mN/m, and γ˘ ) -0.2 s-1). A freshly prepared monolayer undergoes a single shear cycle. The orientation of the flow is as illustrated in frame a for the sequence a-d, and as illustrated in frame h for the sequence e-h. The line segment on frame (a) is 100 µm long. Images are not corrected for the geometrical distortion (section 2). The elapsed times after frame a are as follows: (b) 0.1 s, (c) 0.2 s, and (d) 5 s. The flow is subsequently inverted e-h. The elapsed times after frame e are as follows: (f) 0.13 s, (g) 0.23 s, and (h) 0.25 s. Arrows on frames e-g indicate the propagation of fronts. In frame g, the four observed reflectivities are labeled.

curved contours. The fact that the reflectivity changes affect domains with different molecular orientations suggests that these fronts are the result of a rearrangement of the underlying lattice. The apparent randomness in their occurrence and duration, along with the fast dynamics, suggests, qualitatively, that the flow triggers an avalanche of rearrangements in the lattice, which results in the reorientation of the tails in the affected sites in order to satisfy the NNN orientation favored in the L2′ phase. 3.1.5. Effect of Temperature. An increase in the temperature causes the boundary between different types of phenomena (see Figure 3) to shift to higher Π. For example, at T ) 22 °C coarsening and annealing and continuous changes in the reflectivity are dominant for all Π and γ˘ applied. Only at Π near the L2′-S transition are type-I fronts observable at this temperature. It is reasonable to expect that the viscoelastic properties of the monolayer depend on temperature. Our observations suggest that, whereas the liquidlike flow behavior is dominant at high temperatures, a viscoelastic response can be more easily triggered at low temperatures, resulting in local rearrangements (front propagation) in high-stress conditions. 3.2. Observations in the Ov Phase. The observed shear-induced changes in the structure of Ov-phase monolayers arise from the propagation of fronts. We do not observe continuous changes in the reflectivity as in the L2′-phase monolayers. This is true at all surface pressures, from about Π ) 13 mN/m up to about Π ) 30 mN/m (which is the range of the Ov phase at the working temperature of T ) 46 °C) and for all shear rates (γ˘ g 0.05 s-1). Small changes in temperature do not have a significant effect on the observed phenomena. The application of shear-reversal cycles leads, in general, to fragmentation of the domain structure in this phase. 3.2.1. Alignment of the Hexatic Lattice. Application of strains as small as γ ≈ 0.1 in Ov-phase monolayers results in the nucleation and propagation of fronts across all domains. If high enough strains are applied (γ ≈ 0.5), the propagation continues until only two highly contrasted values of the reflectivity are observed (Figure 8). These fronts can be either horizontal or vertical, i.e., at 45° relative to the principal axis of the strain tensor. Figure 8b shows the intersection between two such bands resulting in an angle close to 90°. This feature is shared with the type-I fronts observed in the L2′ phase. The bands resulting from the propagation of the fronts in the Ov phase span across domains of different reflectivities. This

is in contrast with the type-I fronts in the L2′ phase, which were constrained within the domain boundaries. The final reflectivity of each domain following the passage of a front depends on its initial value. The high contrast between the two resulting reflectivities suggests that the alkane tails are aligned in directions close to the flow on either side of the incidence plane (Figure 2). Inversion of the flow direction triggers the nucleation and propagation of the same type of fronts, resulting in a change in the reflectivity of all domains, both bright and dark. When the rotational component of the shear flow is clockwise, all domains are slightly darkened; domains become lighter in color when the flow is counterclockwise. These changes, however, are not as dramatic as those observed on a fresh (unsheared) monolayer. Domains that were darker prior to the inversion of the flow still remain darker afterward (Figure 8e). As shear-reversal cycles continue, so do these slight alternating changes in reflectivity. The domain boundaries remain roughly unaltered by these processes as long as the inversion of the flow direction occurs when the fronts have already disappeared. If the inversion of the flow direction takes place while front propagation is under way, fragmentation of the domains can occur, with the part of a domain affected by the front propagation adopting one reflectivity and the part yet unaffected by the front propagation changing to the opposite reflectivity. The fact that the reflectivity of all domains changes upon flow inversion suggests that the alkane tails are not aligned with the flow direction, but rather maintain a nonzero angle with it. This is qualitatively similar to the shear alignment in some bulk nematic LCs, raising the intriguing possibility that a nematic-like flow-induced orientation might be possible in these monolayers. In section 4, we describe the hypotheses and methods used to extract meaningful values for the alignment angle and show that the alignment process is different from the one in bulk nematic LCs. 3.2.2. Secondary Fronts at High Strains and Domain Fragmentation. If large strains (γ > 5) are applied to an Ov-phase monolayer, a new type of front is observed after the fronts described in the previous section have disappeared and only two reflectivities remain. These new fronts propagate at roughly (45° to the flow direction; that is, they are parallel to the principal axes of strain (Figure 9). The orientation and direction of advance of the fronts do not depend on the orientation of the shear flow. For instance, fronts propagating in the vx ) vy and vx ) -vy directions can be observed during both clockwise and counterclockwise shear. These fronts affect regions of the


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Figure 9. Propagating fronts appear in an Ov-phase monolayer at high strains (T ) 46.5 °C, Π ) 27 mN/m, and γ˘ ) 0.2 s-1). (a) After a total strain γ ) 6, a front parallel to the principal strain axis (∼45° from the horizontal) begins to propagate across the central light domain. (b) Upon propagation of the front (indicated by the arrow) the domain is divided in two regions with different reflectivities. Between frames b and c, the flow direction is inverted. (c) The two regions resulting from the propagation of the front reorient in different ways through the mechanism described in section 3.2.1 and illustrated in Figure 8. (d) After 0.3 s, the two domains resulting from the fragmentation of the initial domain are clearly observable. The position of the front upon inversion of the flow becomes the boundary between the two new domains. The line segment in frame a is 100 µm long.

monolayer spanning domain boundaries. The combination of flow inversion and secondary fronts leads to domain fragmentation. In Figure 9b, propagation of a secondary front has resulted in the division of the initially uniform domain in Figure 9a into two subdomains with different reflectivities. Flow is stopped before the front has propagated across the whole domain. Because of this, the two subdomains behave differently upon inversion of the flow direction (Figure 9c). Whereas subdomain 2 remains bright, subdomain 1 becomes dark, following the mechanism described in section 3.2.1. The result is more clearly observable in Figure 9d, after the flow alignment is complete. This process, along with the one described in the previous section, leads to the fragmentation of the domain structure. The fragmentation is more dramatic at low Π, whereas large domains persist after a series of shear-reversal cycles at high Π. 3.2.3. Formation of Labyrinthine Structure. The fragmentation of the domain structure in Ov-phase monolayers, which takes place as described in the previous sections, leads to a reduction of the average domain size. The application of a series of shear cycles results in a reorganization of the smaller domains, leading to the formation of labyrinthine structures, including domains that elongate and fold forming spirals and concentric sequences of dark and bright domains (Figure 10). 3.3. Observations in the L2 Phase. Earlier studies of flow in L2-phase monolayers7,8 reported the observation of fluidlike behavior, with reversible affine distortions of the domain shape but no change in the orientation of the molecules. In this section, we report the results of our observations at temperatures in the range 17 °C < T < 36 °C. Below T ≈ 28 °C, compression of the L2 monolayers

is limited by the transition to the L2′ phase, whereas at higher temperatures, compression of the L2 monolayers leads to a transition into the untilted LS phase. We have observed a strong coupling between the structure of the monolayers and the flow at all temperatures, provided that a high enough Π is applied. The qualitative results presented here correspond to measurements performed at T ) 32 °C and γ˘ ≈ 0.3 s-1. The effect of temperature is discussed in section 3.3.4. 3.3.1. Continuous Changes in Reflectivity without Annealing. Continuous changes in the domain reflectivity are occasionally observed for Π as low as 10 mN/m and they are found in all domains at Π ≈ 17 mN/m. At Π ≈ 17 mN/m, the phenomena observed are qualitatively similar to those reported in section 3.1.2, namely, continuous changes in the reflectivity with occasional jump discontinuities. Despite the similarity of the domain-level kinematics, coarsening and annealing are not observed in the L2 phase, as they were in the L2′ phase. 3.3.2. Propagation of Fronts. At Π g 18 mN/m, fronts that span single domains are observed. The structure and dynamics of these fronts are quite different from those of the type-I fronts described in the L2′ phase (section 3.1.3). Here, fronts are not straight or aligned in any particular orientation, and they propagate in all directions, ultimately filling the entire domain. Under these conditions of Π and γ˘ , reflectivities of all values are still observed, and fronts can appear in domains of any reflectivity. 3.3.3. Annealing and Fragmentation. For Π > 27 mN/ m, flow induces a sudden annealing of the domain structure, similar to the process described for Ov-phase monolayers. The rich mosaic of reflectivities transforms so that only two reflectivities remain. The details of the

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not reveal front propagation, despite the fact that Π > 20 mN/m can be applied to L2 monolayers. A similar Π value results in significant front propagation at T ) 32 °C (see above). Temperatures above T ) 32 °C do not alter the types of phenomena observed. The appearance of propagating fronts for T > 30 °C might be related to the observed change in viscoelastic behavior of monolayers in the L2 phase.18,19 Under two-dimensional Poiseuille flow, the parabolic velocity profile on the monolayer becomes triangular at T > 30 °C under conditions of γ˘ and Π that are similar the ones reported here. 4. Analysis and Discussion

Figure 10. Fragmentation and reorganization into a labyrinthine structure of an Ov-phase monolayer upon a sequence of shear cycles (T ) 46 °C, Π ) 21 mN/m, and γ˘ ) 0.2 s-1). Each snapshot portrays the same region of the monolayer (a) before any shear is applied, (b) after 6 shear cycles, and (c) after 18 shear cycles. The line segment in frame a is 100 µm long.

alignment mechanism cannot be clearly observed, because the contrast of the images is too low at such high Π. Inversion of the shear direction results in domain fragmentation and reorganization leading to the formation of patterns similar to the labyrinthine structures described for the Ov phase. 3.3.4. Effect of Temperature. The choice of temperature (T ) 15 °C) in earlier experiments7,8 limited the values of Π available within the L2 phase (Π < 15 mN/m). We observe that the flow-induced changes in the L2-phase monolayers become more significant for T > 30 °C, with the appearance of propagating fronts. Such a change in behavior might be related to the loss of herringbone order in the L2-phase monolayers predicted to occur at roughly T ) 30 °C for C22.17 Indeed, observations at T ) 22 °C do (17) Kaganer, V. M.; Loginov, E. B. Phys. Rev. B 1995, 51, 2237.

4.1. Alignment in the Ov Phase. As reported in section 3.2.1, the flow-induced reorientation observed in Ov-phase monolayers at small strains suggests the possibility of a flow-induced alignment of the alkane tails at a finite angle with the flow, in apparent analogy with flow alignment in bulk nematic LCs. Our observations of a reorganization leading to a mosaic of only two highly contrasted reflectivities (see Figure 8) suggests that two particular values of the tilt azimuth are induced by shear in a given direction and that two different azimuthal orientations are associated with shear in the reverse direction. A comparison of the relative changes in reflectivity observed (upon shear reversal) in these monolayers with the shape of the expected reflectivity curve (Figure 2) suggests that the two azimuth orientations stable for clockwise shear are φ ) 90° + R and φ ) 270° + R (φ ) 90° - R and φ ) 270° - R for counterclockwise shear), with R being the alignment angle. In Figure 8g, the azimuth φ ) 90° + R (φ ) 90° - R) corresponds to the reflectivity R1 (R2), and the azimuth φ ) 270° + R (φ ) 270° - R) corresponds to the reflectivity R3 (R4), where R1 is the reflectivity of the region labeled 1 in the figure, etc. To use this model to perform a meaningful calculation of R, one must transform the reflectivity curve (Figure 2) so that it relates the observed gray level in the 8-bit digitized images to φ. The reflectivity curve depends significantly on θ, which can be approximately related to the area per molecule with the simple expression cos(θ) ) A0/A,20,21 where A0 is the area per molecule at the transition to the untilted phase. Using a combination of Π-A isotherms and optical inspection, we find that, at 46 °C, the L2-Ov phase transition occurs at Π1 ) 13.8 ( 0.5 mN/m, and the Ov-LS (untilted) phase transition occurs at Π0 ) 31.5 ( 0.5 mN/m (Figure 11). Using X-ray diffraction, Durbin et al.20,21 measured the polar tilt at the L2-Ov phase transition (θ1) for a situation consistent with ours (we estimate θ1 ≈ 20.8° from their data). Because the Π-A isotherm is approximately linear in the Ov phase, one can express θ as a function of Π, cos(θ) ) (a - Π0)/(a - Π1), where a ) [Π0 - Π1 cos(θ1)]/[1 - cos(θ1)]. Because we lack an absolute calibration for the reflectivity in our images, we assume only that the 8-bit gray scale of the images [R(φ)] is linearly related to the calculated reflectivity curve, r(φ), i.e., R(φ) ) R0 + ∆R r(φ), where R0 and ∆R are unknown constants corresponding to the offset and gain of the digitization process, respectively. We extract four gray levels from our images (Figure 8g) and relate them to the reflectivity curve assuming that R1 ) R(90°-R), etc., where R is also unknown. The three unknown parameters (R0, ∆R, and R) are obtained (18) Kurnaz, M. L.; Schwartz, D. K. Phys. Rev. E 1997, 56, 3378. (19) Ivanova, A.; Kurnaz, M. L.; Schwartz, D. K. Langmuir 1999, 15, 4622-4624. (20) Durbin, M. K.; Malik, A.; Ghaskadvi, R.; Shih, M. C.; Zschack, P.; Dutta, P. J. Phys. Chem. 1994, 98, 1753. (21) Tippmann-Krayer, P.; Mohwald, H. Langmuir 1991, 7, 2303.


Figure 11. Π-A isotherm measured for a C22 monolayer (T ) 46 °C). The L2-Ov and Ov-LS transitions are marked. The dotted curve relates θ to Π and is obtained as described in the text.

Figure 12. Alignment angle of an Ov-phase monolayer under shear, estimated by fitting the reflectivity curve (solid line) to the four gray levels that can be extracted from a monolayer (Figure 8) according to the model discussed in the text. The three different symbols correspond to three independent experimental realizations.

by finding the best fit of the reflectivity curve to the four measured gray levels. As mentioned above, optical artifacts introduce a significant amount of noise in the images. To improve the accuracy of the fitting process, we combine reflectivity measurements obtained in independent experiments under the same conditions (therefore, corresponding to the same R). Different sets of data correspond to different, but linearly related, R(φ) values. We combine different sets of data by choosing a reference set and linearly transforming the others to best overlap the reference. The value of R is then extracted from the combined sets as shown in Figure 12. An analysis of the images obtained for different Π values reveals that the flow-induced reflectivity changes become less significant, when compared with the overall contrast in the images, with increasing Π. The reduction in contrast sets an upper limit to the values of Π for which we can measure the four reflectivities described above with an acceptable signal-to-noise ratio (Π < 28 mN/m for all of the data reported here). Because a lower value of R (Figure 12) will result in a decrease in the contrast, this trend appears to suggests that R decreases with increasing Π. However, a more careful analysis reveals that the decrease in θ (resulting from an increase in Π) actually accounts for the lower contrast at high Π (see Figure 2). Like those in the other tilted condensed phases, monolayers in the Ov phase have hexatic order. Despite this,

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Figure 13. Alignment angle R as a function of the surface pressure Π measured for Ov-phase monolayers. The solid line corresponds to tan(R) ) cos[θ(Π)] tan(30°). The dashed line corresponds to tan(R) ) [cos θ + (L/d) sin θ]-1.

one can define an in-plane director analogous to the director field of a two-dimensional nematic LC along the azimuthal tilt direction. Ignoring hexatic order, one can, therefore, attempt to use conventional nematohydrodynamics to explain the phenomena described in this section. The coupling between the nematic order parameter and an external flow is well described by the Ericksen-Leslie-Parodi theory.22 The flow behavior of a three-dimensional nematic is characterized by six viscosity coefficients, R1-R6. Application of this theory to simple shear flow predicts alignment of the nematic director field at an angle R, with the flow velocity given by tan2(R) ) R3/R2. Using geometrical arguments, Helfrich23 proposed a relationship between the aspect ratio of ellipsoidal nematic molecules and the ratio of two viscosity coefficients, namely, R3/R2 ) (a/b)2 (where a and b are the small and large axes of the ellipsoid, respectively). This model predicts that R approaches 45° in the limit of a circular molecule (a/b ) 1) and decreases with decreasing aspect ratio. In the analogy between the in-plane projection of the fatty acid molecules and a 2D nematic phase, the aspect ratio is a/b ) (cos θ + L/d sin θ)-1, where L/d ≈ 5.5 is the length-to-diameter ratio of the molecule. Therefore, this model suggests tan(R) ) (cos θ + L/d sin θ)-1, i.e., R is predicted to approach 45° at the transition to the untilted LS phase (see above and the dashed line in Figure 13). Our measured values of R as a function of Π (and, therefore, θ) are shown in Figure 13. Our data show only a marginal departure from a constant value of R ≈ 30°, i.e., only a weak dependence on Π. The measured values for R are, therefore, inconsistent with those predicted by the 2D nematic analogy (Figure 13). The model based on the analogy with 2D nematic LCs ignores the existence of a strong coupling between the tilt azimuth and the orientation of the local pseudohexagonal unit cell, the hexatic bond-orientational order parameter.1,5,6 The Ov phase is characterized by a lack of herringbone order in the backbone plane, by an azimuthal orientation toward next-nearest-neighbor (NNN) bonds, and by a lattice structure that has hexagonal packing in the plane perpendicular to the tails.1,20 As discussed in section 1, although shear flow distorts the shape of the underlying lattice, it is reasonable to expect that the tilt azimuth of a given molecule will prefer to remain in a position that approximates the NNN environment. The absence of observable relaxation of domain reflectivity upon cessation of flow implies that the monolayer is in a (22) de Gennes, P. G.; Prost, J. The Physics of Liquid Crystals, 2nd ed.; Oxford University Press: New York, 1993. (23) Helfrich, W. J. Chem. Phys. 1969, 50, 100.

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Figure 14. Diagram representing the projection of the alkane tails in an Ov-phase monolayer onto the air-water interface. We assume that the lattice orients so that the flow will be as shown by the arrows, resulting in an alignment angle R.

configuration that would be in equilibrium in the absence of shear. For Ov-phase monolayers, this means that the alkane tails are tilted toward a NNN “bond” at all times, or at least that the departure from this configuration is small. Clearly, the only way that this observation can be consistent with the shear-induced changes in tilt azimuth is if the shear is assumed to result in a reorganization of the hexatic lattice. The projection of the hexagonal lattice structure on the plane of the monolayer is a distorted hexagonal packing, expanded along the tilt direction (NNN) by a factor of 1/cos(θ). Motivated by our observations that R ≈ 30°, we asume that the monolayer is arranged by the external shear so that one of the degenerate symmetry lines of the lattice aligns with the flow direction. This leads to an analytical expression for R (see Figure 14), namely, R ) arctan{cos[θ(Π)] tan(30°)}. This results in a value for R that decreases slightly from 30° as Π is reduced. In Figure 13 our data are compared with the results of this model (solid line) and with predictions based on Helfrich’s geometrical argument for 2D nematic LCs (dashed line). The data are consistent with our model based on the flowinduced lattice alignment. It is important to note that a small uncertainty in θ(Π) has a very small impact on the value of R predicted from the above equation. Any change in θ, however, will result in a roughly equal change in the value of R determined experimentally. This leads to significant experimental uncertainty in the value of R. 4.2. Continuous Molecular Precession in the L2′ Phase. In section 3.1.2, we described the observation of continuous reflectivity changes in L2′-phase monolayers under shear flow, with occasional jump discontinuities. Qualitatively, we interpreted the observations as a precession of the alkane tails that was continuous most of the time. To perform a quantitative study of the kinematics of this process, we measured the evolution of the gray level in selected domains of the BAM images during 35 precession events for different values of γ˘ (both clockwise and counterclockwise) and different values of Π. In all cases, T ) 17 °C . After what can be long periods in an initial state where φ is approximately parallel to the flow (φ ≈ (90°), the gray level of a domain changes relatively quickly from dark to light or vice versa (Figures 5 and 15). This corresponds to a nearly 180° rotation of the molecules, after which the gray level remains essentially constant again. Higher values of γ˘ result in faster molecular precession, as can be seen in Figure 15a. In fact, the rate of change of the reflectivity is proportional to γ˘ , a clear indication that 1/γ˘ is a characteristic time scale of the precession. This is supported by the fact that data obtained with different γ˘ values can be collapsed when plotted as a function of strain, γ ) γ˘ t (Figure 15b).

Figure 15. Evolution of the gray level for three different domains under three different rates of shear for monolayers in the L2′ phase (Π ) 20 mN/m, T ) 17 °C). (a) The mean and standard deviation of the instantaneous distribution of gray levels of the domain under observation are represented as combinations of data points and error bars. (b) The data sets are superimposed when time is transformed into strain using the experimentally determined values of γ˘ . Arbitrary offsets are added to the strain axis to optimize the overlap of the data sets.

The conclusion is that, indeed, our observations correspond to a flow-induced reorientation of the alkane tails. The analysis presented in this section will compare the predictions of models for the kinematics of molecular precession with the observed changes in domain reflectivity. The model for the continuous evolution of the reflectivity, r(t), is obtained through the combination of a model for the continuous precession of the azimuth, φ(t), with the calculated reflectivity curve, r(φ), i.e., r(t) ) r[φ(t)] (see Figure 16). As described in section 4.1, a model for r(t) can be expressed in terms of the measured gray levels by linear transformation, R(t) ) R0 + ∆R r(t), where R0 and ∆R are the offset and gain of the digitization process, respectively, and must be determined empirically. The functional dependence θ(Π), needed for the evaluation of r(φ), can be determined in a way similar to that described for the Ov phase (section 4.1), using the measured values of Π at the L2-L2′ phase transition (Π1 ) 17 ( 0.5 mN/m) and at the L2′-S phase transition (Π0 ) 29.0 ( 0.5 mN/m) at T ) 17 °C and the corresponding values of θ, θ1 ) 17.5° (estimated from data in ref 24) and θ0 ) 0, for the respective transitions. The fast change in reflectivity preceded and followed by saturation (Figure 15a) is qualitatively similar to a Jeffery orbit,25 which describes the rotation of an elongated object under shear and the tumbling, or periodic changes in orientation, of nematic LC molecules. Using the analogy with 2D nematic LCs (discussed in the previous section), one might expect to observe a similar type of phenomenon in Langmuir monolayers under shear flow. The equation for a Jeffery orbit is re tan[φ(t)] ) -tan[(γ˘ ret)/(re + 1)], (24) Kenn, R. M.; Bohm, C.; Bibo, A. M.; Peterson, I. R.; Mohwald, H. J. Phys. Chem. 1991, 95, 2092. (25) Jeffery, G. B. Proc. R. Soc. London A 1922, 102, 161-179.


Figure 16. Model for the continuous evolution of the reflectivity (r, solid lines) as a function of the strain based on the geometrical model for the evolution of the azimuth, φ(t) ) arctan[-γ] (dashed lines). The reflectivity curve, r(φ), is obtained with the same parameters as in Figure 2 and with θ ) 16°.

where re is the aspect ratio for elongated particles and re2 ) -R2/R3 for nematic LCs9 (R2 and R3 are two of the Leslie coefficients; see section 4.1). Using Helfrich’s geometrical argument23 for ellipsoidal molecules (see section 4.1), this parameter becomes re ≈ b/a, where b and a are the major and minor axes of the ellipsoid, respectively. In the analogy between the in-plane projection of the alkane tails and a 2D nematic, b/a ≈ cos θ + L/d sin θ, which is close to unity for all Π in the L2′ phase. A Jeffery orbit is periodic, and predicts that the domain will eventually return back to its original intensity. With re ≈ 1, the model based on Jeffery orbit predicts that the intensity, in cases such as the central domain in Figure 5 should pass a maximum and start to decrease well within the long period of time for which the observed intensity remains constant. The existence of asymptotic regimes before and after a rapid change, instead of extrema followed by periodic behavior, indicates a clear incompatibility with the nematic tumbling model. In section 4.1, we noted the observed lack of relaxation in the domain reflectivities upon cessation of the flow, regardless of the stage in the evolution. This is common to all of the phases explored in this work. For the L2′ case, this means that the continuous change of φ(t), with occasional jump discontinuities, must remain consistent with the NNN orientation of the alkane tails. As a firstorder model for the continuous precession, we assume that a given molecule changes its orientation in order to follow its NNN during the distortion of the lattice due to shear flow (see Figure 17a,b). At time t ) 0, the alkane tail on a reference site is pointing to one of its NNNs, whose coordinates relative to the reference site are (x0, y0). At time t, the relative coordinates of the NNN will have changed to (x0 + γ˘ ty0, y0) when a strain of γ˘ t has been

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Figure 17. Geometrical model for the onset and magnitude of the jump discontinuities in the evolution of φ(t) in an L2′phase monolayer under shear. The diagram represents the projection on the plane of the monolayer of the lattice structure. The stretching of the rectangular unit cell results in a distorted hexagonal lattice. The NNN tilt is oriented perpendicular to the stretching direction.1 Our model assumes that the degenerate lattice lines align with the flow (horizontal clockwise shear, as illustrated by the arrows in a). The angular reference for φ(t) is illustrated in a. (a) A region that initially satisfies the NNN lattice-bond orientation is considered. The azimuth of the reference site 1 is toward the NNN 2. (b) After a strain of γ ) 0.6, the continuous reorientation results in a violation of the NNN condition, as 1 and 2 now become NNs. (c) A jump in the evolution of φ(t) results in a configuration closer to the NNN condition, with 1 tilted toward 3. In this example, the jump takes place between φ ) 221° and φ ) 194° (∆φ ) -26°), consistent with the data in Figure 18a.

applied by the horizontal shear flow. If the orientation of the molecule at the reference site follows the motion of the NNN, then φ(t) will satisfy the relationship tan[φ(t)] ) -γ˘ (t-t0), with t0 ) -x0/(γ˘ y0). This expression is qualitatively consistent with our observation of a steep change in the reflectivity of the domains preceded and followed by asymptotic regimes (Figure 16). Note the difference in the predicted evolution of the reflectivity for domains with -90° < φ < 90° and domains with 90° < φ < 270°, because of the lack of symmetry of the reflectivity curve (Figure 2). This continuous model, however, is topologically inconsistent with the NNN lattice-bond symmetry condition. Indeed, as the lattice is distorted by shear, the relative positions of different sites will change and sites that originally were NNN actually become NNs (see Figure 17) and are no longer stable tilt directions in the L2′ phase. The observed jump discontinuities in the evolution of φ(t) resolve this incompatibility. Our model proposes that, after being deformed by a certain strain, the structure of the monolayer relaxes into a more favorable configuration through the nucleation and propagation of jump discontinuities during the precession of a domain. Using these ideas, we have measured the magnitude of the experimentally observed jump discontinuities. For a given experimental realization, we measure the evolution

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Figure 18. Example of analysis of the evolution of domain reflectivity and measurement of jump discontinuities in L2′phase monolayers. The data (symbols) correspond to two neighbor domains whose reflectivities change in opposite ways during the same experiment (Π ) 22 mN/m, T ) 17 °C, and γ˘ ) 0.2 s-1). The two sets of data are fitted simultaneously, according to the model described in the text (solid lines). From the fit, the evolution of φ(t) can be extracted (dashed lines), and the jumps are characterized as follows: (a) φ ) 221° ( 7° to φ ) 188° ( 7° (∆φ ) -33° ( 10°); (b) φ ) 22° ( 8° to φ ) -26° ( 8° (∆φ ) -48° ( 11°).

of the gray level for two neigboring domains whose reflectivities evolve in opposite directions. In the example in Figure 18, the domain measured in a darkens whereas its adjacent domain, measured in b, brightens under shear. The evolution of both domains is fitted simultaneously to a model for the evolution of the gray level, R(t), that is a piecewise continuous function, with each continuous interval being constructed by composing R(φ) with the model for the continuous precession, φ(t), proposed above. Using information from the two domains simultaneously, we include information from R(φ) for all values of φ, as the domain in Figure 18a has 90° < φ < 270° and that in Figure 18b has -90° < φ < 90°. The offset (R0) and gain (∆R) of the digitization, which are fitted parameters, will be the same for both series. Each of the two series of data is characterized by one jump discontinuity, if necessary. We have not observed more than one jump discontinuity for a given domain in any experiment. The time at which the jump occurs is determined experimentally, and the fitted value is restricted to a narrow interval around the experimental time. The magnitude of the jump is extracted from a fit to the data. If the jump is determined to occur at t ) t1, then one of the data series is described by R(t) ) R0 + ∆R r(t-t0) for t < t1 and R(t) ) R0 + ∆R r(t-t0+∆t) for t g t1. A similar set of parameters is required to describe the second data series. The jump discontinuity does not always happen simultaneously in both domains. Once the gray level data are fitted, the jump discontinuity in φ(t) is characterized by a jump taking place at t ) t1 and occurring between the orientations φ(t1-t0) and φ(t1t0+∆t). In the two cases presented in Figure 18, both jumps are consistent with ∆φ ≈ (30° (with the jump in Figure 18a more clearly observable both in the images and in the data). The 35 precession events that were analyzed clearly reveal 23 instances of jump discontinuities. We were able to obtain an estimate for the magnitude of the jump in 17


of those cases, which resulted in 11 jumps of magnitude close to 30°. The observed jumps always take place in the region where the domain reflectivity is changing more rapidly with time. In all cases, the observed jump in φ(t) is in the same orientation as the flow (and the precession). A geometrical interpretation for the observed magnitude of these jumps is provided in Figure 17, based on a consideration of the lattice structure in the L2′ monolayer being distorted by shear and each alkane tail following the motion of its NNN. A situation clearly incompatible with the NNN symmetry occurs when φ ≈ 220° (Figure 17b), and a jump of ∆φ ≈ -26° results in a more favorable configuration (Figure 17c). These parameters are in good agreement with the jump measured in Figure 18a. When larger strains are applied, more configurations incompatible with the NNN symmetry will occur. Small jumps (-15° < ∆φ < 15°) occurring in the orientation opposed to the flow can reconcile the NNN lattice-bond orientation and the lattice distortion. These jumps would take place in the asymptotic region, where ∆R(φ)/∆φ is small in magnitude. Moreover, they would be smaller in magnitude than the jumps observed, making them more difficult to detect. No example of such jumps was observed experimentally. The combination of continuous precession and jump discontinuities in φ(t) provides a mechanism for the observed annealing of the domain reflectivity in the L2′ phase, so that, after a number of shear cycles, only two reflectivities are observed (section 3.1.1). Suppose a monolayer with domains of all reflectivities is sheared. If a large enough strain is applied, all of the domains will be oriented in the asymptotic region in φ(t). If no jumps took place, the process would, presumably, be reversible, and the “phase difference” between φ(t) in all domains should be recovered. Therefore, the rich mosaic of reflectivities should be restored by an equivalent strain in the opposite direction. However, if jump discontinuities occur in the asymptotic region, aimed at preserving the NNN symmetry, after a large enough strain, the “phase information” of each domain will be lost. A sequence of shear-reversal cycles will enhance this effect and result in domains of only two reflectivities being observed. Although it is plausible that the kinematics of φ(t) changes for different Π, only a small range of Π is available for exploration at T ) 17 °C before the onset of type-II fronts (18 < Π < 23, see section 3.1.3). No variations are observed for Π in that range. 5. Conclusions The application of a shear flow has a significant effect on the structure of Langmuir monolayers of fatty acids in the different tilted condensed phases. Observation of the monolayers using BAM has allowed us to visualize the changes in molecular orientation due to the flow, which we have explored for different experimental conditions. The observed changes can be divided into two types: continuous changes in domain reflectivity and discontinuous changes, the latter of which take the form of propagating fronts. Whereas both types of phenomena are observed in the L2 and L2′ phases, only propagating fronts are observed in Ov-phase monolayers. The application of a sequence of shear-reversal cycles can result in a variety of quite different effects. It can result in fragmentation of the domain structure (Ov phase) or in an increase in domain size (L2′). Although no a priori model exists to predict the type of coupling that should be expected in a certain configuration, we have analyzed two cases in quantitative detail, namely, the propagation of


fronts and flow alignment in Ov-phase monolayers and continuous changes in reflectivity in L2′-phase monolayers. In both cases, a plausible explanation of the mechanism leading to the observed transformations is presented. Both models are based on a reorganization of the structure of the underlying lattice, which is followed by a realignment of the alkane tails. This is different from the mechanism expected for a 2D nematic LC, where flow alignment and tumbling are two possible phenomena. The essential distinction is that flow acts directly on the molecular orientation (director) in a nematic phase, whereas the local unit cell orientation dominates the behavior in a hexatic monolayer. The structural details of the monolayers, including the ordering of the underlying lattice and the ordering of the alkane tails (polar tilt, azimuth, lattice-bond coupling, and herringbone order) must be taken into account to understand the effects of flow. As

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an example, the only symmetry difference between the Ov and L2′ phases is the presence of herringbone order in the latter. Yet the couplings between flow and orientation are remarkably different in the two phases. Our analysis of the coupling between flow and structure of Langmuir monolayers has focused on a microscopic interpretation of the observed phenomena. The different types of coupling are likely to be correlated with the viscoelastic properties of the monolayers; we suggest that the results presented here might ultimately lead to a molecular-level picture of hexatic rheology. Acknowledgment. This work was supported by the National Science Foundation (Award 9733281), and the Camille Dreyfus Teacher-Scholar Award Program. LA0017983

Molecular Orientation in Langmuir Monolayers under Shear

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