Article pubs.acs.org/Langmuir
Dislocation-Mediated Deformation in Solid Langmuir Monolayers: Plastic Bending and Tilt Boundary E. Hatta* Nanoelectronics Laboratory, Graduate School of Information Science and Technology, Hokkaido University, Sapporo, 060-0814, Japan ABSTRACT: The shear response of three types of textures (mosaic, striation, and stripe) in 10,12-pentacosadiynoic acid solid Langmuir monolayers has been investigated with Brewster angle microscopy. Low temperature mosaic textures respond to an applied stress elastically. Upon the application of shear the change of contrast appears in the form of propagation of fronts roughly perpendicularly to the shear direction within a single domain reversibly, while the domain shape keeps constant since it is presumably frozen kinetically. The striation and stripe textures at high temperatures show a viscoplastic behavior (plastic bending) in its rheological response, being consistent with the formation of a dislocation wall (tilt boundary) through dislocation dynamics (dislocation glide and climb). The stress-induced formation of a tilt boundary provides a manifestation of the collective motion of a number of dislocations.
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melting in Langmuir monolayers17 and LB films (trilayers)18 were discussed with the dislocation-mediated melting scenario.19 Shear thinning has been observed in experiments involving Langmuir monolayers.20 This phenomenon can be explained by the increase in unbound dislocations in the presence of shear stress.21 Shear-induced dislocation proliferation can thus become a major factor affecting their viscous responses. Dislocation-mediated phenomena that may occur in a Langmuir monolayer to an applied stress, however, have been still less explored. There may be significant differences in dislocation motion between two and three dimensions. The pinning mechanism of dislocations is in fact quite different between them since in three dimensions a dislocation is a line object while in two dimensions it is a point-like one. The barriers to dislocation motion in two dimensions thus generally small compared to those in three ones due to the difference of their pinning mechanisms. It is, however, still large compared to a few times kBT far from the melting temperature Tm and thus the barriers must be overcome by an applied external force (see Discussion). The stress relaxation can be elastic and recoverable (i.e., the low-frequency shear modulus is finite) or plastic and nonrecoverable (i.e., the low-frequency shear modulus is zero). Our aim of this study is to explore the manifestation of dislocation motion in solid Langmuir monolayers caused by a shear stress. In this paper we report the stress relaxation revealed by the texture changes to an applied shear stress in solid Langmuir monolayers and consider the significance of the collective motion of dislocations on the stress relaxation of viscoplastic solid monolayers.
INTRODUCTION Insoluble monomolecular films at the air−water interface (Langmuir monolayers) exhibit a variety of optical textures observed with polarizing fluorescence microscopy and Brewster angle microscopy, showing the underlying organization of the molecular tilt azimuth over mesoscopic (and even macroscopic) length scales. They include mosaic,1 stripe,2 spiral,2 star,3 and boojum4,5 ones. Some studies have been carried out on the disclination dynamics in Langmuir monolayers.6−8 Changes of textures were reported between tilted liquid condensed phases by two different thermodynamic processes (heating−cooling and compression−decompression).1,9 It was revealed by BAM observations that a reorientation of the alkyl tails in the tilted condensed phases of monolayers occurs also by the application of an external shear flow.10,11 The observed flow-induced textural changes in monolayers could be classified into several types; (i) shear annealing of the domain structure, (ii) continuous precession of the tilt azimuth, and (iii) propagation of fronts within a single domain and across domain boundaries. It was shown that the underlying lattice plays a crucial role for the observed phenomena and that the coupling between the structure of the tilted condensed phases of monolayers and an external flow causes a reorganization of the structure of the underlying lattice followed by a realignment of the alkyl tails.11 It is thus an interesting question whether the presence and the motion of defects such as free dislocations in the underlying lattice of a monolayer play a significant role in the kinetics of textural transitions. The presence of defects such as isolated and paired dislocations in single- and multilayered Langmuir−Blodgett (LB) films has been revealed with an atomic force microscope.12,13 The existence of defects may be correlated with the complex rheological behaviors in Langmuir monolayers.14−16 The experimental results of two-dimensional © 2015 American Chemical Society
Received: June 21, 2015 Revised: August 17, 2015 Published: August 21, 2015 9597
DOI: 10.1021/acs.langmuir.5b02249 Langmuir 2015, 31, 9597−9601
Article
Langmuir
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EXPERIMENTAL PROCEDURE
We purchased 10,12-pentacosadiynoic acid (PCA) from Wako Pure Chemical Industries, Ltd., and PCA monomers were spread onto pure water (Millipore Milli-Q at 18 MΩ) contained in a custom-built Teflon trough. The surface pressure was monitored by a filter paper Wilhelmy plate and an R&K electrobalance. The subphase temperature was adjusted in the range of 0−25 °C by a combination of a circulating water and thermoelectric Peltier elements. Benzene was chosen as a solvent because larger domains can be obtained with it. A 10 mMol solution of PCA was spread onto the confined water surface (68 cm2) until an average molecular area of 23 Å2 was reached. The use of a concentrated solution in a confined small spreading area was crucial for the formation of large mosaic, striation, and stripe textures where the appearance of those textures shows the long-range orientational order of the molecules in the monolayers. Lower concentrations (for example, ∼0.5−1 mMol) resulted in small-sized mosaic textures only. In this study the rheological response of the monolayer upon the application of shear force was observed by the changes in reflectivity of BAM images. The modulation in intensity contrast of domains can be a result of the rotation of the tilt azimuth of the molecules and/or the underlying lattice relative to the plane of incidence of the BAM. It was found that the molecules in the PCA monolayer are orientationally ordered with a significant tilt angle in the solid (S) phase.22 For this reason the BAM images of this material show much contrast between adjacent domains with uniform tilt azimuths even in the S phase, as is clear also from the relationship between molecular orientation and reflected intensity, established using the Berreman formalism.23,24 This makes the monolayer under study easier to observe the rheological responses of the monolayer to shearing deformations in the S phase This situation is quite different from the case of the BAM images of usual fatty acid and ester monolayers in which the tilt angle of the molecules is nearly zero in the S phase. A home-built Brewster Angle Microscopy (BAM) visualizes the Langmuir monolayer. A laser light (30 mW, 660 nm, He−Ne) is p-polarized by a Glan-Thomson prism and is directed to the water surface at the Brewster’s angle (θB ≈ 53°). The light reflected from the monolayer is focused onto a CCD camera by a 5× microscope objective after passing through an analyzer (a second Glan-Thomson prism). The orientation angles of analyzer were adjusted to optimize the contrast in the images with respect to the polarizer. There is some ambiguity in molecular tilt azimuth as determined by BAM. We lifted it with the help of the results of ref 22. Shear deformations of the monolayers were achieved using a simple, two-belt channel cell, which consists of two parallel stretched belts between two Delrin rollers that drag the monolayer in the opposite directions.10,11 In a typical shearing experiment we applied a constant shear rate of 0.2 s−1 for a finite period of time. For simple shear flow the shear rate refers to that estimated as the velocity of the bands divided by the gap width from the flow cell geometry. The local shear rate in the monolayer in the S phase may be, however, different from an average one, because of the coexistence of elastic and plastic regions10 and/or slippage between domains.25 The local shear rate γ̇ was thus determined from the measured rate of strain of the domain under investigation. The striation edge point tracking analysis was performed with Video Spot Tracker v.05.07 (CISMM at UNC−CH). The trajectories were taken simultaneously and during the evolution of the boundary. Each of the trajectories followed positions of one edge point at different times.
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Figure 1. Three typical textures of monomeric pentacosadiynoic acid (PCA) monolayers: (a) mosaic (T = 4.8 °C); (b) striation (T = 12 °C); and (c) stripe (T = 18 °C). π = 20 mN/m. The scale bar is 50 μm.
high temperature range (15−25 °C). We show the shear response of mosaic texture upon an applied shear in Figure 2.
Figure 2. Shear response of a mosaic texture in a monomeric PCA monolayer. The elapsed times after frame (a) are as follows: (b) 0.33 s, (c) 3.84 s, (d) 4.35 s. (T = 4.6 °C, π = 20 mN/m, γ̇ = 0.2 s−1, α = 110°). The flow directions are shown as white arrows. The black arrows in (a) and (c) indicate the direction of an advanced front. The scale bar is 50 μm.
The direction of applied shear is perpendicular, and the velocity gradient is horizontal. A shear band propagation clearly occurs in a sea horse-shaped domain during the applied shear. The curved front moves roughly perpendicularly to the flow (i.e., at π/4 to the principal strain axis of the shear strain) (Figure 2b and c). The change in reflectivity by a propagation front is confined within a single domain and the reversal of shear direction restores the original reflectivity in the texture (Figure 2d). The domain shape does not change substantially during shear. The contrast of neighboring domains does not change simultaneously. The orientation of propagation fronts observed in our mosaic textures are generally either parallel or perpendicular to the shear, that is, ± 45° relative to the principal strain axis of the shear. The evolution of nucleation and growth of a discontinuous boundary in a curved, striation texture is shown in Figure 3. The texture includes many fine striations inside, indicating local fluctuations of growth directions in monolayer. The curved texture was created naturally in the growth process of solid monolayer by spreading the solution. The applied shear initiates the nucleation of a tilt boundary (, shown in the region surrounded by a dashed red elliptic line, Figure 3b). It moves in the form of front propagation in the opposite direction of applied shear (Figure 3c and d). As the boundary moves, the boundary shape changes from a diffuse (Figure 3c) to a sharp one (Figure 3d). We carried out the point tracking26,27 of several points of the texture to observe the flow dynamics of the monolayer near the boundary (Figure 3e). It can be seen from this tracking that tracking points flow nearly parallel to each other through the boundary. All of trajectories seem to have approximately the same discontinuity at the boundary within the resolution of their tracking interval. The variations of molecular tilt azimuths
RESULTS
In Figure 1 we show three typical textures observed in monomeric PCA solid monolayers. The textures depend strongly on the subphase temperature in our preparation procedure. A mosaic texture (Figure 1a) was observed mainly in the low temperature range of 0−5 °C. One dimensional (1D), anisotropic striation texture (Figure 1b) was observed in the intermediate temperature range (10−15 °C), while another 1D anisotropic stripe texture (Figure 1c) was observed in the 9598
DOI: 10.1021/acs.langmuir.5b02249 Langmuir 2015, 31, 9597−9601
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Figure 3. Shear response of a bent striation texture in a monomeric PCA monolayer. The elapsed times after frame (a) are as follows: (b) 0.39 s, (c) 0.99 s, (d) 1.83 s. (T = 12.0 °C, π = 20 mN/m, γ̇ = 0.2 s−1, α = 155°). The white arrow indicates the direction of the applied shear. The region surrounded in a dashed red elliptic line in (b) indicates the nucleation of a tilt boundary. Note that the boundary shape changes from a diffuse ((c)) to a sharp ((d)) one as the growth of the tilt boundary. (e) Trajectories of several tracking points near the boundary upon shear. The red arrows indicate the moving direction of tracking points. Each point was tracked every 1/30 s. The boundary is shown in the blue dotted line as a guide for the eye. (f) Schematic of the tilt azimuths of the molecules in the texture and the corresponding contrast change calculated by the BAM formula (the lower panel). The symbols ⊥ indicate the dislocations. The scale bar is 100 μm.
phase transitions in this material were investigated on pure water by in situ synchrotron grazing incidence X-ray diffraction (GIXD).22 It was demonstrated in that study that in the monomer state the monolayer constitutes of untilted carboxylterminated hydrophilic chains in hexagonal arrangement and highly tilted (≈40° from the normal to water toward the nearest neighbor (NN)) methyl-terminated hydrophobic chains that break the hexagonal symmetry of the hydrophilic part. Therefore, the appearance of remarkable intensity contrast of the textures in the PCA solid monolayer is consistent with the above GIXD result. In the mosaic texture of the S phase (Figure 2) the contrast within a single domain changes largely during the application of shear although domain shape is not deformed substantially. This indicates that the domain shape is presumably frozen kinetically during the rapid growth of solid monolayer by spreading the solution and that the shear force couples directly with the molecular orientation and not via the coupling of the shear with the monolayer lattice in this case. It was shown that in both the tilted (L2’ and Ov) and untilted (S) phases a reorganization of the underlying lattice plays a significant role for the shear-induced reorientation of the monolayers.10,11 In the tilted L2’ and Ov phases of docosanoic acid monolayer the appearance of propagation fronts of shear bands is mainly due to a reorientation of the molecular tilt azimuth through a reorganization of a distorted hexagonal lattice and in the S phase the appearance of shear bands results only from the reorientation of the underlying lattice on which the molecules are located vertically to the water surface.28 The manner in which the molecular orientations are coupled with shear in our PCA mosaic texture thus seems to be quite contrast to the previous results. Front propagation does not occur simultaneously in neighboring domains around the sea horse-shaped domain (Figure 2). This indicates that the neighboring domains respond to the applied shear independently. Intensity of light reflected from the bending region in the striation (Figure 3) and the stripe (Figure 4) textures changes discontinuously, meaning that molecular tilt azimuth changes discontinuously in this region. The jump in the variations of tilt azimuths in the bending region suggests the formation of a tilt boundary mediated by a group of dislocations alignment. The
calculated by the BAM formula from the reflected light intensity of the observed texture are shown in Figure 3f. The formation and movement of a discontinuous boundary revealed by the discontinuous change of contrast along the stripes can be seen also in stripe texture upon bending (Figure 4). After
Figure 4. Bending induced tilt boundary formation and movement in the stripe texture (a-c) and schematic of discontinuous change of molecular tilt azimuths and the corresponding contrast change along and across the stripes calculated by the BAM formula (d). The symbols ⊥ indicate the dislocations located at the tilt boundary. In this schematic the continuous small change of curvature along the stripes on the both sides of the tilt boundary has been left out of consideration. Discontinuous change of contrast due to the sudden change of molecular tilt azimuths across the boundary is shown only. The elapsed times (applied strains); (a) 0.0 s (0.0), (b) 1.8 s (0.33), (c) 4.1 s (0.79). (T = 18.0 °C, π = 20 mN/m, γ = 0.2 s−1, α = 30°). The scale bar is 100 μm.
formation the boundary moves toward the curved region as shear increases. In this figure we can see discontinuous contrast change along the stripes in the bending region of the texture (corresponding to discontinuous change of the molecular tilt azimuths in the each side of the boundary) as well as continuous contrast change within individual stripes that corresponds to the continuous change of the molecular tilt azimuths of the molecules (Figure 4b and c).
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DISCUSSION In this study the applied shear affects the tilt azimuth and not the magnitude of the tilt angle itself because the intensity contrast of domains present and the domain textures created during shear does not change upon cession of shear. In our monomeric PCA monolayers the pronounced textural anisotropy suggests that molecules have a significant tilt angle even in the S phase. Recently monomers to blue to red chromatic 9599
DOI: 10.1021/acs.langmuir.5b02249 Langmuir 2015, 31, 9597−9601
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dislocation movement occurs spontaneously for the tilt boundary formation in the absence of an applied external force. To observe the bent solid texture with a radius of curvature dislocations must be distributed randomly or form a single tilt wall or multiple dislocation walls with an average spacing between them. Let us consider the stability of such dislocation structures. For randomly distributed dislocations, their total energy including the self-energies of the dislocations and the interaction energy is given by Er = Aρln(dr/r0), A = μb2/2π(1−ν), where dr is an average spacing between them (= ρ−1/2, ρ is the dislocation density (=1/bR)), r0 is the effective radius of the dislocation core, μ is the shear modulus, ν is the Poisson ratio.29 If the dislocations form multiple dislocation walls or a single tilt wall, their total energies are represented by Em = Aρln(dm/r0) and Es = Aρln(ds/r0), with dm, ds being the dislocation spacing between dislocations in multiple walls and in the single tilt wall, respectively. Using the relations ρ = (dmΛ)−1 = (dsL)−1 (, Λ is an average spacing between walls and L is the crystal width) and L > Λ > dm > ds, we get Er > Em > Es. The dislocations thus favor to array in the form of a single tilt wall (Figure 5d) rather than multiple walls (Figure 5e). The above result means that dislocation glide and climb (Figure 5c) plays a significant role for the decrease of the interaction energy due to the decreasing spacing between dislocations in the walls, dm > ds. In Figure 4 there is a small mismatch between the locations of the tilt boundary and the curved region, that is, on the right of the boundary the curved region is located a little away from the boundary (≈ some tens of μm, the schematic of Figure 4 has been drawn with the exception of curved region). We guess that this is because not enough shear was applied for the boundary to reach the central part of the curved region.
formation of a tilt boundary can be described as a dislocationpatterning process (dislocation wall formation) occurring in deformed crystals generally.29 It was found out that the monomer phase of PCA has an expanded low packing structure and that molecules located in hexagonal positions can slide along the NN directions of their highly tilted alkyl chains during polymerization.22 Therefore, the creation of dislocations by monolayer bending and the feasibility of chain sliding along the monolayer lattice may facilitate the formation of a tilt boundary (dislocation wall). The bending experiment such as carried out in this study may provide a most direct way of studying the collective motion of a number of dislocations. Let us consider the curvature of a bent solid monolayer below (Figure 5). The deformed crystal generally contains a great
Figure 5. Schematic illustration of stages in the dislocation alignment for the formation of a tilt wall in a bent solid monolayer. (a) Dislocations are piled up on the glide line immediately after bending and dislocations of one sign are collected in the neutral region and those of the opposite sign are moved in opposite directions. As the result an excess of dislocations of the required sign remain. (b) Bent crystal with randomly distributed dislocations of one sign on the glide lines. (c) Rearrangement of dislocations through both glide and climb motions. Further bending leads to the enhancement of dislocation alignment, resulting in the formation of a single wall ((d)) or multiple walls ((e)).
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CONCLUSIONS
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AUTHOR INFORMATION
Upon the applied shear we have observed three types of textural changes in the monomeric PCA monolayers with different mechanical (elastic and plastic) properties. In the elastic (reversible) mosaic texture the applied shear couples with the tilt azimuth of the molecules directly. The domain shape keeps its original shape during the application of shear. Striation and stripe textures, on the other hand, exhibit an irreversible, plastic bending followed by the formation of a tilt boundary (dislocation wall). The formation and the stability of a dislocation wall have been discussed from the viewpoint of collective dynamics of dislocations. The characteristic textural changes in the plastic regime seem to be based mainly on the extended lattice structure and the resultant chain sliding in the monomer S phase of this material.
number of dislocations distributed randomly in the crystal. In the stress field, the Peach-Koehler force f PK(y) = σb causes a dislocation with the Burgers vector b to move. Here σ = Ey/R is the stress at a distance y from the neutral line in a bent crystal with Young’s modulus E and radius of curvature R. Dislocations move either to the neutral line or to the boundary depending on their Burgers vector and on the applied stress (Figure 5a). As the result there will be an excess of edge dislocations of one sign on the glide lines (Figure 5b). On a simple dimensional argument we can estimate the mechanical energy for the dislocations to move to the neutral line in a bent crystal under the application of stress in two dimensions. We put the distance of the dislocations in the slip lines to the neutral line equal to l, the required energy for the dislocations to reach the neutral line, F(l), is F(l) = ∫ l0 f PK(x)dx = σbl = El2b/2R. Putting E ≈ (10−2−10−3) N/m, l ≈ 10−5 m, b ≈ 10−10 m, and R ≈ 10−4 m, we get F(l)≈ (10−18−10−19) J. This value is much larger compared to the thermal energy at room temperature, kBTR ≈ 10−21 J (TR = 295 K). We can thus not expect that the
Corresponding Author
*E-mail:
[email protected]. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS
This work was supported by a Grant-in-Aid for Exploratory Research 16656001 from the Ministry of Education, Culture, Sports, Science and Technology, of Japan. 9600
DOI: 10.1021/acs.langmuir.5b02249 Langmuir 2015, 31, 9597−9601
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(24) Tabe, Y.; Yokoyama, H. Fresnel formula for optically anisotropic Langmuir monolayers: an application to Brewster angle microscopy. Langmuir 1995, 11, 699−704. (25) Ivanova, A. T.; Ignes-Mullol, J.; Schwartz, D. K. Microrheology of a sheared Langmuir monolayer: Elastic recovery and interdomain slippage. Langmuir 2001, 17, 3406−3411. (26) Hatta, E. Division and fusion, extension, adhesion, and retraction of vesicles created through Langmuir monolayer collapse: Cell-like behavior. J. Phys. Chem. B 2007, 111, 10155−10159. (27) Hatta, E. Anomalous diffusion of fatty acid vesicles driven by adhesion gradients. J. Phys. Chem. B 2008, 112, 8571−8577. (28) Maruyama, T.; Fuller, G. G.; Frank, C. W.; Robertson, C. R. Flow-induced molecular orientation of a Langmuir film. Science 1996, 274, 233−235. (29) Friedel, J. Dislocations; Pergamon Press, New York, 1964.
REFERENCES
(1) Schwartz, D. K.; Knobler, C. M. Direct observations of transitions between condensed Langmuir monolayer phases by polarized fluorescence microscopy. J. Phys. Chem. 1993, 97, 8849−8851. (2) Ruiz-Garcia, J.; Qiu, X.; Tsao, M. W.; Marshall, G.; Knobler, C. M. Splay stripe textures in Langmuir monolayers. J. Phys. Chem. 1993, 97, 6955−6957. (3) Qiu, X.; Ruiz-Garcia, J.; Stine, K. J.; Knobler, C. M.; Selinger, J. V. Direct observation of domain structure in condensed monolayer phases. Phys. Rev. Lett. 1991, 67, 703−706. (4) Overbeck, G. A.; Hönig, D.; Möbius, D. Observation of bond orientational order in floating and transferred monolayers with Brewster angle microscopy. Thin Solid Films 1994, 242, 26−32. (5) Rivière, S.; Meunier, J. Anisotropy of the line tension and bulk elasticity in two-dimensional drops of a mesophase. Phys. Rev. Lett. 1995, 74, 2495−2498. (6) Hatta, E.; Fischer, Th. M. Splitting of an s = 1 point disclination into half-integer disclinations upon laser heating of a Langmuir monolayer. J. Phys. Chem. B 2003, 107, 6406−6410. (7) Hatta, E.; Fischer, Th. M. String defects connecting pairs of halfinteger disclinations and tilting transition of a Langmuir monolayer. J. Phys. Chem. B 2005, 109, 2801−2804. (8) Brugués, J.; Ignés-Mullol, J.; Casademunt, J.; Sagués, F. Probing elastic anisotropy from defect dynamics in Langmuir monolayers. Phys. Rev. Lett. 2008, 100, 037801−037804. (9) Fischer, B.; Tsao, M. − W.; Ruiz-Garcia, J.; Fischer, Th. M.; Schwartz, D. K.; Knobler, C. M. Observation of a change from splay to bend orientation at a phase transition in a Langmuir monolayer. J. Phys. Chem. 1994, 98, 7430−7435. (10) Maruyama, T.; Lauger, J.; Fuller, G. G.; Frank, C. W.; Robertson, C. R. Orientation in a fatty acid monolayer: Effect of flow type. Langmuir 1998, 14, 1836−1845. (11) Ignés-Mullol, J.; Schwartz, D. K. Molecular orientation in Langmuir monolayers under shear. Langmuir 2001, 17, 3017−3029. (12) Bourdieu, L.; Silberzan, P. Chatenay. Langmuir-Blodgett films: from micron to angstrom. Phys. Rev. Lett. 1991, 67, 2029−2032. (13) Kajiyama, T.; Oishi, Y.; Suehiro, K.; Hirose, F.; Kuri, T. Direct Observation of Edge Dislocation in Lignoceric Acid Monolayer Based on Atomic Force Microscopy. Chem. Lett. 1995, 2, 241−242. (14) Ivanova, A.; Kurnaz, M. L.; Schwartz, D. K. Temperature and flow rate dependence of the velocity profile during channel flow of a Langmuir monolayer. Langmuir 1999, 15, 4622−4624. (15) Ghaskadvi, R. S.; Ketterson, J. B.; Dutta, P. Nonlinear shear response and anomalous pressure dependence of viscosity in a Langmuir monolayer. Langmuir 1997, 13, 5137−5140. (16) Ghaskadvi, R. S.; Dennin, M. Alternate measurement of the viscosity peak in heneicosanoic acid monolayers. Langmuir 2000, 16, 10553−10555. (17) Pauchard, L.; Bonn, D.; Meunier, J. Dislocation-mediated melting of a two-dimensional crystal. Nature 1996, 384, 145−147. (18) Sikes, H. D.; Schwartz, D. K. Two-dimensional melting of an anisotropic crystal observed at the molecular level. Science 1997, 278, 1604−1607. (19) Ostlund, S.; Halperin, B. I. Dislocation-mediated melting of anisotropic layers. Phys. Rev. B: Condens. Matter Mater. Phys. 1981, 23, 335−358. (20) Kurnaz, M. L.; Schwartz, D. K. Dislocation-mediated melting of anisotropic layers. Phys. Rev. E: Stat. Phys., Plasmas, Fluids, Relat. Interdiscip. Top. 1997, 56, 3378. (21) Bruinsma, R.; Halperin, B. I.; Zippelius, A. Motion of defects and stress relaxation in two-dimensional crystals. Phys. Rev. B: Condens. Matter Mater. Phys. 1982, 25, 579−604. (22) Lifshitz, Y.; Golan, Y.; Konovalov, O.; Berman, A. Structural Transitions in Polydiacetylene Langmuir Films. Langmuir 2009, 25, 4469−4477. (23) Berreman, D. Optics in stratified and anisotropic media: 4 × 4matrix formulation. J. Opt. Soc. Am. 1972, 62, 502−510. 9601
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