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Orientation in a Fatty Acid Monolayer: Effect of Flow Type T. Maruyama, J. Lauger, G. G. Fuller,* C. W. Frank, and C. R. Robertson Department of Chemical Engineering, Stanford University, Stanford, California 94305-5025 Received September 18, 1997. In Final Form: January 12, 1998 The two-dimensional fluid dynamics of the different phases of a fatty acid monolayer (docosanoic acid) were examined. Using Brewster angle microscopy, we studied the polydomain structure of two liquid condensed phases (the L2 and L′2 phases) and the “solid”, S, phase in situ during the application of extensional flow and simple shear flow. We found that only the L2 phase deformed nearly reversibly with a liquid-like response. Nonsymmetric domain deformations were, however, found for that phase at the lowest rate of strain studied with a four-roll mill. At higher strain rates, plots of the evolution of strain against time collapsed onto a single curve so that the rate of strain was independent of roller speed. Furthermore, a critical strain γc exists above which the rate of strain shows a stepwise increase despite the constant velocity of the rollers. The two other phases, the L′2 and S phases, experienced flow-induced reorientation of the lattice onto which the molecules are arranged. The reorientation process was accompanied by the appearance of shear bands in the monolayers at (45° to the extension axis of both types of flow. The shear bands observed in the L′2 phase were modeled as a plastic flow accompanying the molecular tilt reorientation developed within elastic regions of the monolayer. This model describes the time evolution of the bandwidth quite well and provides strong evidence of the existence of an additional phase within the conventional L′2 phase region. In simple shear flow, the velocity profile for the L2 phase across the gap of the shearing cell showed a nonlinear distribution of shear rates, which were highest at the center of the gap. Flowinduced breakup of domains was observed in the L′2 phase subject to simple shear.
1. Introduction Saturated fatty acids, in the form of Langmuir monolayers, have long been one of the most intensively studied systems with respect to their isotherms and surface viscosities.1 However, it has been within the past decade, with the application of X-ray diffraction2,3 and the invention of Brewster angle microscopy (BAM),4,5 that the detailed in situ lattice structure of the many phases of these materials and their extremely complex polymorphism have become available. Consequently, there is presently considerable knowledge of the long-range correlation of fluctuations of orientational order, such as bond or lattice orientation, tilt, tilt-bond coupling, and herringbone structures.6 These powerful techniques have elucidated the distinctive features of the phases and hence even led to the discovery of new phases that are not evident from analyzing monolayer isotherms.7,8 The subject of this investigation was docosanoic (C22) acid. According to X-ray diffraction studies,9 at the temperature used in this study (15 °C) the pressure-area isotherm involves three phases, the L2, L′2, and S phases, divided by two phase transition points: one at a surface pressure of π ) 14.5 mN/m and the other at π ) 27.5 mN/m. The L2 phase has the head groups of the molecule (1) Gaines, G. L. Insoluble Monolayers at Liquid Gas Interfaces; Interscience: New York, 1966. (2) Dutta, P.; Peng, J. B.; Lin, B.; Ketterson, J. B.; Prakash, M.; Georgopoulos, P.; Ehrlich, S. Phys. Rev. Lett. 1987, 58, 2228. (3) Kjær, K.; Als-Nielsen, J.; Helm, C. A.; Laxhuber, L. A.; Mo¨hwald, H. Phys. Rev. Lett. 1987, 58, 2224. (4) He´ron, S.; Meunier, J. Rev. Sci. Instrum. 1991, 62, 936. (5) Ho¨nig, D.; Mo¨bius, D. J. Phys. Chem. 1991, 95, 4590. (6) Bibo, A. M.; Knobler, C. M.; Peterson, I. R. J. Phys. Chem. 1991, 95, 5591. (7) Overbeck, G. A.; Mo¨bius, D. J. Phys. Chem. 1993, 97, 7999. (8) Durbin, M. K.; Malik, A.; Ghaskadvi, R.; Shih, M. C.; Zschack, P.; Dutta, P. J. Phys. Chem. 1994, 98, 1753. (9) Kenn, R. M.; Bo¨hm, C.; Bibo, A. M.; Peterson, I. R.; Mo¨hwald, H.; Als-Nielsen, J.; Kjaer, K. J. Phys. Chem. 1991, 95, 2092.
arranged on a distorted, hexagonal lattice with the alkyl chains’ tails tilted collectively away from the film surface normal at an angle toward their nearest lattice neighbors. As the monolayer is compressed, the collective tilt angle diminishes in magnitude, and across the transition to the L′2 phase at π mN/m, the tilt direction switches toward the next nearest neighbor. As the monolayer enters the S phase, the tilt angle tends to zero and the alkyl chains are perfectly upright, but the head groups remain arranged on a distorted, hexagonal lattice, leading to a herringbone order. The long-range order in tilt orientation and lattice distortion causes a domain structure in a docosanoic acid monolayer, where each domain is comprised of molecules ordered along the same orientation. The resultant macroscopic optical anisotropy of the monolayer makes BAM an ideal method by which to view the distributed domains of this system. This optical probe uses light polarized in the plane of incidence that is reflected from the interface at the Brewster angle θΒ for the substrate (water, in our case). The presence of a thin film at the interface will cause the Brewster condition to be violated, and light will be reflected. If the film consists of domains characterized by unique refractive indices, the reflected light will reveal the film morphology. The monolayer can be imaged as a mosaic of domains or “polydomain” structure in which each gray level of the image corresponds to a different direction of the tilt azimuth or lattice distortion. In two recent papers by this laboratory, we have reported on evidence of the flow-induced orientation of the tilt azimuthal angle of the L′2 phase of docosanoic acid.10,11 Using Brewster angle microscopy to image the polydomain structure of this amphiphile, it was revealed that exten(10) Friedenberg, M. C.; Fuller, G. G.; Frank, C. W.; Robertson, C. R. Langmuir 1996, 12, 1594. (11) Maruyama, T.; Fuller, G.; Frank, C.; Robertson, C. Science 1996, 274, 233.
S0743-7463(97)01047-0 CCC: $15.00 © 1998 American Chemical Society Published on Web 03/04/1998
Orientation in a Fatty Acid Monolayer
sional flows of sufficient magnitude were able to couple to the anisotropic morphology of the monolayer and orient it in a manner that had the tilt azimuth aligned perpendicular to the extensional axis.11 It was not clear, however, whether the coupling with the flow originated from the distorted, hexagonal lattice onto which the head groups are placed or from the tilted aliphatic tails. The distortion was also nonaffine and irreversible for this phase. The lower pressure L2 phase did not display this reorientation process but only underwent distortion of its domains. However, this distortion was basically reversible, and the initial domain pattern could be nearly restored upon reversing the flow direction. The “solid” S phase was also examined and was found to yield under application of extensional deformations. In other words, it resisted deformation until a strain of sufficient magnitude was applied, after which its domain pattern was observed to rapidly distort. Although the domain contrast in the S phase was considerably reduced compared to those of the L2 and L′2 phases, it was evident that the distortion of this phase was also accompanied by a reorientation of its lattice by 90°. It is important to note that the reorientation processes in both the L′2 and S phases were preceded by the propagation of “shear” bands at (45 to the principal axes of strain. As the deformation ensued, these shear bands thickened until they covered the entire area of view in the BAM images. From these observations, it was concluded that extensional deformations are capable of global reorientation of the tilt azimuth in the L′2 phase toward an orientation perpendicular to the axis of strain. In this paper the process of flow-induced orientation in docosanoic acid monolayers is further explored. First, a detailed analysis of the process of shear band propagation is presented along with studies of the nature of nonaffine deformation in extensional flow. Second, the influence of flow type is examined by measurements using simple shear flow, which is a superposition of purely extensional and purely rotational flow. Since the majority of interfacial rheological measurements utilize simple shear flow, it is of interest to understand the nature of deformation and orientation processes when a significant amount of rotation is added to the flow kinematics. 2. Experimental Methods 2.1. Materials. Docosanoic acid from Sigma was used without further purification. Water as a subphase was produced by a Milli-Q system (Millipore) and had a resistivity of 18.2 Ω‚cm. Chloroform solutions of docosanoic acid at 1.0 mg/mL were spread onto the purified water in the Langmuir trough (KSV System 3) at 20 °C; then the subphase temperature was cooled to 15 °C, at which the influence of flow was investigated. 2.2. Brewster Angle Microscopy. The setup of the Brewster angle microscopy used in this study was similar to that described by Ho¨nig and Mo¨bius. A 5-mW HeNe laser light at 632.8 nm was introduced onto the monolayer through a GlanThompson polarizer at the Brewster angle for water, and the reflected beam was focused onto a CCD camera (Sanyo VDC3825) by a 50-mm focal length lens through a second Glan-Thompson polarizer as an “analyzer”. The analyzer was set at 60° with respect to the p-plane of the incident laser light, which was found to optimize the contrast in the polydomain structure. 2.3. Flow Cells. Two types of flow cells were used in this study: a four-roll mill and a two-belt channel cell. Extensional deformations were applied to the monolayers using a four-roll mill, which consists of four cylinders set on the corners of a square (Figure 1a). The distance between adjacent rollers was 31.3 mm, and the diameter of the rollers was 19.6 mm. The geometry and size of the device were chosen after the suggestions of Higdon12 to maximize the spatial extent of homogeneous, linear flow. The (12) Higdon, J. J. L. Phys. Fluids A 1993, 5, 274.
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Figure 1. Two types of flow cells and the induced flow fields used in this study: a four-roll mill (a) and a two-belt channel cell (b). Note that the simple shear flow produced by a two-belt channel cell is a superposition of a pure extension, with the extension axis at 45° with respect to the velocity gradient, and a pure rotation. rollers were constructed of black Delrin, and the upper part of the rollers was notched so that leveling the water surface with the monolayer at the notch minimized the apparent meniscus effect among the rollers, which was found to considerably affect the monolayer flow in some cases.13 Rotation of the rollers as shown in the figure produced hyperbolic streamlines with the direction of the extension axes determined by the sense of rotation. Simple shear deformations of the monolayers were studied using a two-belt channel cell, which consists of two parallel stretched belts with an inner channel width of 12.3 mm, set within a bath of fluid of length of 81.0 mm (Figure 1b). The belts were made of poly(ethylene terephthalate) and were held between two rollers covered by Teflon. Rotation of the rollers moved the belts in a manner creating a simple shear deformation that was applied to a monolayer in the channel region. The flow is characterized by a stagnation line along the center axis. Tension could be applied to the belts to produce a smooth and slipless motion. The top edges of the belts were made almost level with the water surface to minimize the presence of a meniscus in the channel. The flow cells were suspended above a Langmuir trough with the mill cylinders or the belts protruding through the air-water interface. The laser beam of the BAM was reflected at the geometric center of the mill or on the center line of the channel, where one finds either a stagnation point or stagnation line of the flow where the velocity is zero but the velocity gradient is finite. The plane of incidence was coincident with the horizontal midplane of the mill or the plane perpendicular to the channel. The flow cells were driven by a stepping motor (Compumotor Model M57-51), by which the rate and the direction of roller/belt rotation were precisely controlled to apply the strain and strain rate to be applied to the monolayers. The isotherm and the domain structure of the monolayers were not affected by the presence of the flow cells.
3. Results 3.1. Orientation and Deformation Processes in Extensional Flow. 3.1.1. The L2 Phase: Nonaffine Deformation in Extensional Flow. Deformation of the L2 phase under extensional flow was observed with roller speeds between 0.126 and 1.26 rad/s and a surface pressure of 12 mN/m. The deformation process appeared to be reversible except for the lowest strain rate studied. Furthermore, the domains retained their respective light intensities, which means that the tilt directions of the molecules do not couple to the flow for this phase. There was little measurable tendency of deformed domains to relax following the cessation of flow, suggesting that both the coupling of the monolayer flow with subphase (13) Maruyama, T.; Friedenberg, M.; Fuller, G.; Frank, C.; Robertson, C.; Ferencz, A.; Wegner, G. Thin Solid Films 1996, 273, 76.
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Figure 2. Two line segments at time t, lx(t) and ly(t), on a BAM image defined to be parallel to the principal axes of extensional flow (x for stretching and y for compression), using the domain boundaries with distinctive shapes.
convection and the line tension between adjacent domains are considerably small. The influence of the line tension, however, can become significant when the straining process is fairly slow, which will be discussed later in this section. The deformation of the polydomain pattern was quantitatively analyzed through the sequentially recorded BAM images of deforming domains under flow. Two line segments at time t, lx(t) and ly(t), were defined on a BAM image to be parallel to each principal axis of extensional flow (x for stretching and y for compression). In Figure 2 examples of such lines are shown drawn between distinctive domain boundaries. Under ideal planar extensional flow, these lines would deform as (lx, ly) ) (lx0e˘ t, ly0e-˘ t) or log [lx(t)/lx0] ) -log[ly(t)/ly0] ) ˘ t, where lx0 and ly0 are the initial lengths of each line segment and ˘ is the rate of strain. By analyzing successive frames, we tracked the stretching and compression of the two line segments as γ′x(t) ) log[lx(t)/lx0] and γ′y(t) ) -log[ly(t)/ly0] through the recorded BAM images, frame by frame, for each experiment with different roller speeds. It was found that both γ′x(t) and γ′y(t) evolve in nearly the same manner for roller speeds ω between 0.252 and 1.26 rad/s. This type of response is shown in Figure 3a, where the data for ω ) 0.504 rad/s are plotted. In Figure 4, the average values of the strain, (γ′x + γ′y)/2, are plotted with respect to time for all of the flow rates. The strain evolved distinctively for the two axes only for the lowest roller speed studied (ω ) 0.126 rad/s), as shown in Figure 3b, and thus the plot for 0.126 rad/s in Figure 4 should be treated with care (plotted using the open symbols). For each roller speed the strain is initially linear in time, meaning that the domains deform with a constant rate of strain ˘ , as expected for ideal planar extensional flow. Surprisingly, it was found that each evolution curve of strain shows a clear kink at a finite time, which means that there exists a critical strain γc above which the rate of strain (defined as the slope of the strain with respect to time) shows a stepwise increase despite the constant velocity of the rollers. In Figure 5, the rate of strain both below and above the critical strain is plotted as a function of the roller speeds. It was found that the three plots of the strain evolution for the higher roller speeds (for ω of 0.504-1.26 rad/s) collapsed onto a single curve with rates of strain that are independent of roller speed both below and above the kink of a strain evolution curve. For ω ) 0.126 rad/s, the strain along the compression axis evolves with a higher rate of strain than that along the stretching axis (see Figure 3b), which implies that the flow field is not symmetric along the x and y directions. Thus it is assumed that the curve for ω ) 0.252 rad/s is of a transitional nature between a nonsymmetric behavior and a self-similar one. Nonsymmetric flow behavior are normally associated with the flow of compressible materials, since the trace of the rate of strain tensor should be zero for incompressible flow. Discrepancies from the assumption of incompress-
Figure 3. Time evolution of strain, γ′x(t) (0) and γ′x(t) (O), for the two line segments defined in BAM images of domains. The data in part a were taken at ω ) rad/s, and similar data were obtained for roller speeds ω between 0.252 and 1.26 rad/s. In part b the orthogonal strains are plotted for the lowest roller speed of ω ) 0.126 rad/s, where the two strains followed different functions of times.
Figure 4. Time evolution of the average strain, (γ′x + γ′y)/2, for the two line segments defined in BAM images of domains for the roller speeds ω between 0.126 and 1.26 rad/s (0, ω ) 0.126 rad/s; b, ω ) 0.252 rad/s; 2, ω ) 0.504 rad/s; [, ω ) 0.756 rad/s; 9, ω ) 1.26 rad/s).
ible flow, however, are normally significant at higher strain rates, instead of lower ones, as is the case here. Further examination of the BAM images at ω ) 0/.126 rad/s reveals the true origin of nonsymmetric domain deformations for lower rates of strain. In Figure 6, BAM images with two orthogonal line segments (the white lines), deformed from Figure 6a to Figure 6b due to an extensional flow field of ω ) 0.126 rad/s for 4 s, were compared against Figure 6c. Figure 6c was generated by a uniform rescaling of Figure 6a so that the length of the stretching line segment (the horizontal line) became equal to that of the same segment in Figure 6b. This required a strain of 0.363 to produce the best fit to the data. It should be noted that both images, Figure 6a and b, were taken in the regime of high rates of strain for ly(t). It is clear that, in Figure 6b, the vertical, compressing line segment is shifted to the left relative to the stretching line segment when compared to Figure 6c. This nonsymmetric flow phenomenon is thought to be
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Figure 5. Rate of strain for different roller speeds (9, below the critical strain; b, above the critical strain). The rates of strain for higher roller speeds are shown to be independent of roller speed both below and above the kink in a strain evolution curve. Those for ω ) 0.126 rad/s are plotted by an open square and circle to denote that the values are averaged over nonuniformly evolving strains.
Figure 7. BAM image of a docosanoic acid monolayer at zero surface pressure. The dark part of the image is the bare water surface, and the docosanoic acid appears as irregular, highly structured domains. The images are roughly 1 mm2.
Figure 6. Comparison of BAM images of the domain structure with two orthogonal line segments (two white lines). Part b is a BAM image of domains deformed from an initial structure (part a) due to an extensional flow field of ω ) 0.126 rad/s for 4 s. Part c is a uniformly rescaled image from part a using a strain of 0.363 so that the length of the stretching line segment (a horizontal white line) becomes the same as that in part b. The data for both parts a and b were taken in a time regime after the strain curve kink for a compressing line segment (a vertical white line). The scale of the images is roughly 1 mm2 for part a.
caused by nonuniform deformation or localized thinning and bending of domains. Nonuniform deformation of domains has also been observed to occur in the hydrodynamic distortion of circular domain shapes in some monolayers in regions of coexistence between separate phases. In such cases, the circular domains distort not only to ellipse-like shapes in weak flow but also to bola-shaped domains for stronger flow fields.14,15 It has been shown that the difference in dipole (14) Mann, E. K.; He´ron, S.; Langevin, D.; Meunier, J. J. Phys. II 1992, 2, 1683. (15) Benvegnu, D. J.; McConnell, H. M. J. Phys. Chem. 1992, 96, 6820. (16) Khan, S. A.; Armstrong, R. C. J. Non-Newtonian Fluid Mech. 1986, 22, 1.
density in neighboring domains, µ, and the line tension between the domains, λ, play an important role in determining the properties of domains. In the present case, where the domains exist due to the difference in molecular tilt azimuth, µ is zero, and λ is related to the crystallographic mismatch between domains or “lattice defects” rather than a chemical or property difference between phases. It is hypothesized that the nonsymmetric flow occurs when the characteristic time scale for the applied flow field is of the same order as, or larger than, the relaxation time τλ for distortion caused by λ. Since we observed the nonuniform deformation only at ω ) 0.126 rad/s, the transition between two regimes, with and without a significant effect of λ, lies somewhere between ω ) 0.126 rad/s and ω ) 0.252 rad/s. Thus from Figure 5, τλ is estimated to be between 1/0.049 ∼ 20 s and 1/0.027 ∼ 37 s. The reason for the kink to appear in the strain evolution curve is not clear, but we can speculate on its origins through experimental observations. In the case of docosanoic acid with a C22 alkyl chain, the BAM image taken during spreading the monolayer at zero surface pressure shows that the fatty acid molecules exist as molecularly assembled islands with a domain structure and corresponding tilt order. An example of such an island surrounded by bare water is shown in Figure 7. From this observation, we can envision that molecules with relatively long alkyl chains exert attractive forces among themselves that are strong enough to cause spontaneous assembly of the molecules at zero surface pressure on water, instead of an ideal “gas” phase1 where the molecules exist independently with their alkyl chains flexing freely. Such an idealization is depicted in the cartoon in Figure 8. Therefore we expect the monolayer to possess some elastic response caused by the attractive nature of the
1840 Langmuir, Vol. 14, No. 7, 1998
Figure 8. Model for amphiphilic molecules as an assembled phase and a “gas” phase during compression for short-chain (a) and long-chain (b) molecules. In the low-surface-pressure regime, short-chain molecules experience a gas phase where the molecules exist independently with their alkyl chains flexing freely, while long-chain molecules exert attraction forces among themselves strong enough to induce self-assembly into domains.
molecules before it starts to flow under the application of a stress. We hypothesize that the kink corresponds to the critical strain γc for a transition between elastic deformation and viscous flow of the monolayer. Under the movement of rollers, the monolayer responds by elastically deforming itself up to the strain of γc, and above this strain the monolayer yields and starts to flow with the rate of strain ˘ ∼ τc/η, where τc is a yield stress and η is the viscosity of the monolayer. In principle, this rate of strain should be determined by the physical properties of a monolayer and independent of how the flow is driven and applied, which would explain why ˘ becomes constant for higher roller speeds. An actual value of a critical strain for a given roller speed is dependent on the geometry of rollers, which prescribes how the applied stress field is transmitted or relaxed between the roller surfaces and the measurement location. The surface pressure of a Langmuir monolayer is defined as the difference in pressure between the monolayer and a water surface. It should be noted that when a monolayer material has a self-assembling property, as in the present case, the isotherm with respect to the molecular area normally obtained from the area surrounded by trough rims and barriers is valid only below the molecular area correspondent to that inside self-assembled islands coexisting with bare water on the surface. The existence of highly stable domains in BAM images at zero surface pressure suggests that molecular migration within this layer is very slow, and the isotherm implies that the monolayer is a compressible body. Thus it is plausible to assume a regime where a monolayer responds to an applied stress as an elastic body up to a certain critical strain,
Maruyama et al.
which was found to have a maximum value of 0.3 at ω ) 0.504 rad/s in Figure 4. This magnitude of strain seems too large to be related to the intramolecular attraction forces, even though the area of a domain remains constant under pure extensional deformation. The critical strain could, however, be described by the elastic properties of domain boundaries such as the transition in domain distortion from ellipse-like shapes for a small strain to bola-shaped domains for a larger strain. The phenomena could also be further analyzed in terms of theoretical work about a two-dimensional model for foams, where elastic deformation of boundaries is predicted until a yield strain is reached. 3.1.2. The L′2 Phase: Shear Band Propagation in Extensional Flow. The appearance and propagation of shear band structures were observed by BAM at surface pressures Π between 16 and 26 mN/m, where the docosanoic acid monolayer at 15 °C is in the L′2 phase. The speed of the rollers, ω, was varied between 0.126 and 1.26 rad/s, as in the L2 phase experiments, which corresponds to strain rates between 0.1 and 1.0 s-1, as estimated using a simple geometric argument. It was previously reported that the flow behavior of the L′2 phase was found to be qualitatively different than the dynamics of the L2 phase.11 For domain deformation during a flow reversal in which identical strains in each direction were imposed, the process was not reversible and the final pattern was not similar to the initial pattern. The tilt azimuth of monolayer molecules was strongly coupled with the applied flow field, and that resulted in a drastic change in the intensity contrast between domains, as revealed through BAM images. When the Brewster angle microscope was set so that the incident plane of the laser beam, which includes an incident beam path and a reflected laser beam path, was parallel to the principal axis of extension in the flow, the applied extensional flow field produced BAM images showing high contrast between domains. All the domains were characterized by one of two gray scales. For flows where the plane of incidence was parallel to the compression axis, the contrast between individual domains could not be resolved by BAM, and the entire field of view was characterized by a single, intermediate gray scale. Close inspection of the domains during a flow reversal experiment where the plane of incidence was switched from being initially parallel to the extension axis to parallel to the compression axis showed the development of shear bands that first appeared at (45° to the stretching direction. The bands cut across the original domains and thickened as time progressed, until the entire image was transformed. From the intensity analysis discussed in ref 11, it was argued that the shear bands contain molecules with tilt axes oriented perpendicular to the stretching direction. The term “shear band” is used here because the band appears along one of two principal shear lines in a planar pure extensional field. As the deformation proceeds, a band-shaped region of a hexatic lattice undergoes a homogeneous shear that produces the original hexatic lattice structure in a new tilt orientation. Thus the flowinduced reorientation of molecules observed in this study is reminiscent of the deformation twinning in crystalline materials, which suggests that the L′2 phase deforms more as a crystalline solid with plasticity than as a fluid. Plastic straining in many crystalline materials is accomplished by crystallographic slip, resulting by the gliding of many dislocations along slip planes which are preferred with respect to the crystalline structures of the materials. A slip plane and a slip direction in the plane
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of deformation constitute a slip system. In a single crystal with N slip systems, of which the R slip system is defined by its slip plane normal nR and slip direction sR, the traceless deformation rate generated by all available slip systems is given by17 N
D)
γ˘ RRR ∑ R)1
(3.1)
where γ˘ R denotes the shear rate on the R slip system and RR is the symmetric, traceless Schmid tensor associated with the slip system R and is given by
sRi nRj + sRj nRi 2
RRij )
(3.2)
For a polycrystalline aggregate subject to a macroscopic plastic deformation rate D h , the self-consistent conditions (global compatibility and equilibrium) of the polycrystal require
D h ) 〈D〉
(3.3)
where the angular brackets designate a volume average over the polycrystalline aggregate. From the linearity of the equations above, one can consider each possible slip plane independently without any loss of generality. Assuming that a shear band corresponds to a region of plastic flow accompanying the molecular tilt reorientation while the remainder of the monolayer remains elastic, the time evolution of a shear bandwidth can be calculated. Let us consider a shear band developed between y ) y1 and y ) y2 within a sufficiently large area of width l, y0 < y < y3 ) y0 + l, such that we can define a macroscopic shear rate γ˘ . This model is described in Figure 9. The positions of four points, (0, yn) for 0 e n e 3, at time t ) 0 when the flow field is applied, move to the new positions so that
x0 ) γ˘ y0t
(3.5)
x 1 - x0 x3 - x2 ) ) γc y1 - y0 y3 - y2
(3.6)
where γc is the critical strain below which there is a linear, elastic response. Then the strain γ(t) within the shear band is calculated as
γ˘ l(t - tc)
(3.7)
W(t)
where W(t) ) y2 - y1 is the shear bandwidth at time t and tc is the critical time at which an induced strain reaches γc. Invoking a stress balance between the elastic region characterized by a surface shear modulus G and a plastic shear band with a surface plastic viscosity η,
dγ dt
Gγc ) η )
ηlfd>W(t)
(3.8)
{
‚ γ˘ -
(γ˘ t - γc) W(t)
W(t) is obtained by solving eq (3.8) for t > tc ) γc/γ˘ and W(t) < ηl/Gtc as
(3.4)
x3 ) γ˘ (y0 + l)t
γ(t) ) γc +
Figure 9. Plastic band model for evolving shear bands under simple shear flow. The positions of four points, (xn, yn) for 0 e n e 3, on a deforming film are shown at t ) 0 (a), t ) tc (b), and t ) tc + t′ (c). Note that the plastic flow occurs for t g tc only between y ) y1 and y ) y2 and that the strains of the regions below and above the flowing zone remain the same as that at t ) tc.
‚
d W(t) dt
}
(17) Ahzi, S.; Lee, B. J.; Asaro, R. J. Mater. Sci. Eng. 1994, A189, 35.
W(t) )
( )(
)
t - tc γ˘ ηl ‚ Gγc t - tc + ∆t
(3.9)
∆t ) η∆γ/Gγc is the characteristic time scale related to an initial strain jump ∆γ at time tc when an induced strain reaches the critical strain and plastic flow is initiated so that γ(tc) ) γc + ∆γ. From eq 3.9, we can derive a universal expression for a reduced bandwidth at time t ) tc + t′ with respect to a final bandwidth at time tf ) tc + t′f which involves a single material constant ∆t,
W(t′) 1 + ∆t/t′f ) W(t′f) 1 + ∆t/t′
(3.10)
Note that this function takes the same form for measurements taken up to the same t′f, independent of the applied shear rate γ˘ , when the surface pressure and temperature are fixed so that ∆t is assumed to be constant. It can also be shown by a similar argument that eq 3.10 holds even when multiple shear bands are generated simultaneously, as is often observed in our experiments. In Figure 10, reduced bandwidths, W(t′)/W(t′f), for Π ) 16 mN/m (Figure 10a) and Π ) 21 mN/m (Figure 10b), are plotted with respect to the reduced time, t′/t′f, for each Π up to the same t′f for different roller speeds. The solid line represents a theoretical curve expressed by eq 3.10 fitted to the data by a constant ∆t. It is evident that the time evolution of the bandwidths is well-described by this
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Figure 10. Reduced bandwidth, W(t′)/W(t′f), for Π ) 16 mN/m (a) and Π ) 21 mN/m (b), plotted with respect to reduced time, t′/t′f, for each Π up to the same t′f for different roller speeds (9, ω ) 0.126 rad/s; b, ω ) 0.252 rad/s; 2, ω ) 0.504 rad/s; [, ω ) 0.756 rad/s; 0, ω ) 1.26 rad/s). The solid line was calculated using eq 3.10 fitted to the data by a constant value of the parameter ∆t.
simple model. It should be noticed that at each surface pressure it is occasionally observed that the reduced bandwidth grows linearly with time, such as for ω ) 0.756 rad/s at Π ) 16 mN/m. This situation corresponds to the case where ∆t is infinity or γc is zero. These responses do not occur often, and we hypothesize that such cases take place when an elastic deformation up to γc is not developed in an ideally uniform manner, as assumed in our simple model. Therefore, in Figure 11a, where the fitted values of ∆t are plotted against surface pressure, the fitted values of 1/∆t take on one of two values: a maximum value denoted by the solid squares and a value close to zero given by the solid circles. The solid squares represent average values of 1/∆t determined for several different strain rates at each surface pressure for experiments where the reduced bandwidth produced a nonlinear response as a function of reduced time. The solid circles represent average values of this parameter for experiments where this relationship was roughly linear. The upper envelop of this fitted parameter shows a distinct minimum in 1/∆t around Π ) 21 mN/m. It is important to point out that this surface pressure corresponds to a discontinuity in the unilateral compressional modulus K′ ) A(dΠ/dA) obtained from the isotherm for this monolayer, where A is a mean molecular area. In principle, this modulus should be in the form of a tensor, since the samples under consideration are intrinsically anisotropic. However, under normal conditions in an experiment to acquire a pressure-area isotherm, the monolayer will be a polydomain in structure with domains characterized by a tilt azimuth that is broadly distributed in orientation. For this reason, the following discussion will assume a simple, scalar compressional modulus. This modulus is shown plotted in Figure 11b. K′ can also be related to G as
K′ ) G + λL
(3.11)
Figure 11. Comparison between 1/∆t for a variety of applied strain rates as a function of surface pressure Π (a) and the unilateral compressional modulus K′ (assumed to be a scalar function) obtained from the isotherm (b) plotted against surface pressure Π. Note that the surface pressure at which 1/∆t becomes a minimum corresponds to the location of the discontinuity in the plot of K′. The meaning of the square and circular symbols in part a is explained in the text.
where λL is one of two Lame’s constants which express the free energy F of a deformed isotropic body under external forces as
1 1 F ) F0 + λL(uii)2 + Guikuki 2 2
(3.12)
where F0 is the free energy of the undeformed body and uik is the strain tensor.18 To determine both G and λL, it is necessary to undertake an additional elasticity measurement with a different deformation condition such as hydrostatic compression. However, the surface pressure dependence of 1/∆t, which is proportional to G, looks similar to that for K′, except for the pressures around Π ) 21 mN/m, where 1/∆t shows a minimum of zero while K′ shows a discontinuous change. The discontinuity of K′ in an isotherm arises from a slight kink in the pressure-area isotherm, at which a second-order phase transition is thought to take place. The existence of this relatively small kink has often been neglected in many references,19,20 including an X-ray diffraction study of docosanoic acid,9 which established the notation we have followed in this paper. However, Bibo et al. recently reported the claim of a phase transition within the conventional L′2 phase region as a result of a detailed phase miscibility study.6 Furthermore, recent X-ray diffraction data also suggest that, between the L2 and L′2 phases, a new, “I” phase exists in which the (18) Behroozi, F. Langmuir 1996, 12, 2289. (19) Peterson, I. R.; Brzezinski, V.; Kenn, R. M.; Steitz, R. Langmuir 1992, 8, 2995. (20) Schwartz, D. K.; Knobler, C. M. J. Phys. Chem. 1993, 97, 8849.
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Figure 12. BAM images of deformation of the domain structure of the L2 phase of docosanoic acid subject to a simple shear flow of 2 s-1.The flow is from left to right, and the velocity gradient is vertical. Each frame is separated in time by 1/3 s, so that the total strain applied to the sample in frame f is 4. Note that the intensity contrast of portions of the pattern is modulated as time proceeds, indicating that the tilt azimuthal angle rotates in time with this flow.
molecular tilt azimuth orients intermediately between nearest-neighbor and next-nearest neighbor directions.21 According to this X-ray study, the phase transition we observed at Π ) 21 mN/m can be interpreted as a secondorder transition between the “I” and L′2 phases. In the vicinity of a second-order phase transition, spatial fluctuations in the system increase in magnitude and the system becomes considerably vulnerable to externally applied fields, such that the susceptibility associated with the field diverges at the transition point. The evolution of (1/∆t) f 0 at π ) 21 mN/m, where the second-order transition occurs, can be linked to an increase in spatial inhomogeneities which would not allow an ideal elastic deformation to evolve in the system. It is of interest to evaluate the initial strain jump ∆γ using ∆t ) η∆γ/Gγc. Although we were not able to find any published data for docosanoic acid to provide G and η for steady flow, an earlier work treated the docosanoic acid monolayers as viscoelastic films and measured their dynamic response by resonance rheometry using a forcedoscillation ring placed at the air-water interface.22 We can use the dynamic values obtained under conditions most similar to our own to make a rough estimation of ∆γ. From ref 22, the dynamic modulus is estimated to be G′ ) 50 mN/m and the dynamic viscosity is roughly η′ ) 0.18 sP. Since γc is on the order of 0.1 from Figure 4 and ∆t is also on the order of 0.1 from Figure 11a, we obtain ∆γ ) Gγc∆t/η. This order of magnitude is reasonable to describe the initial strain jump as the strain caused by swiveling of molecules by π/2 as an elementary step of crystallographic slip in a film under applied stress. 3.2. Orientation and Deformation Processes in Simple Shear Flow. 3.2.1. The L2 Phase. In Figure 12, a sequence of frames is shown where the L2 phase is subjected to a simple shear flow of shear rate 2 s-1. In the case of simple shear flow, the shear rate and total strain refer to those estimated as the velocity of the bands divided by the gap width unless otherwise noted. In the series shown in Figure 12, the monolayer is subject to a total strain of 4 and each frame is separated by a time of 1/3 s. (21) Durbin, M. K.; Malik, A.; Richter, A. G.; Ghaskadvi, R.; Gog, T.; Dutta, P. J. Chem. Phys., in press. (22) Buhaenko, M. R.; Goodwin, J. W.; Richardson, R. M. Thin Solid Films 1988, 159, 171.
As in the case of extensional flow, this phase did not show any evidence of shear band propagation in simple shear. However, due to the rotational component of this flow, there is a time-dependent variation in the intensity contrast of the domain pattern that does not occur in irrotational, extensional flow. This modulation in intensity contrast is a result of the rotation of the tilt azimuth of the fatty acid chains relative to the plane of incidence of the BAM. Upon reversal of the flow (not shown here), the original domain pattern is recovered, indicating that the deformation and orientation processes are nearly reversible for this particular phase. Sequentially recorded frames of BAM images for the deforming domains produced almost straight and parallel streamlines, that trace the trajectories of points marking distinctive boundaries at the corner of domains. However, the streamlines revealed a spatial distribution in shear rate, instead of a constant, uniform value throughout the channel between the belts. In Figure 13a, the velocity along the channel for such traced points, dx/dt, is plotted over a range of channel width positions y when the average shear rate obtained from the flow cell geometry was 0.214 s-1. The slope for the average shear rate is also shown as a dashed line in the figure. It is apparent that the shear rate is a function of position and is highest toward the center of the channel. At the center it approaches a value up to four times the average shear rate. Since the BAM images analyzed here were obtained close to the channel center, a boundary region with fairly low shear rates exists near the moving belts, and the velocity profile in the channel has the shape shown schematically in Figure 13b. Such a velocity profile can be expected to occur if the monolayer is characterized by both elastic and viscous properties. Such a monolayer can sustain an applied shear stress at the moving boundaries while the middle zone between them undergoes plastic deformation and sustains most of the deformation. This response in simple shear flow coincides with the assumption of an elastic-plastic transition in deformation used to explain the results in extensional flow, although the rotational component of simple shear flow could result in more complicated processes of deformation than those in irrotational, extensional flow.
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perpendicular to its principal extension axis of strain. Then in a flow reversal experiment where the extension axis is switched between two orthogonal ones, the alternating sets of allowed tilt azimuth are (φ, φ + π) and (φ + π/2, φ + 3π/2); φ ) 0 for extensional flow, and φ ) π/4 for simple shear flow. Then if we define the contrast by
C)
Figure 13. The velocity, dx/dt, across the gap in simple shear flow plotted as a function of position (a). The velocities were measured by tracing the trajectories of points in the monolayer as functions of time. The average shear rate was 0.214 s-1. A schematic of the overall velocity profile in the channel is shown in part b. The slope corresponding to the average shear rate is shown by the dashed line in both figures. The center of the channel is in the vicinity of y ) 0.
3.2.2. The L′2 Phase. The result of subjecting the L′2 phase to a simple shear flow is shown in Figure 14. In this figure, a flow of 2 s-l is shown being applied for a strain of 4. Shear band propagation clearly occurs during simple shear flow, as shown by the arrows in the figure. It was observed that the bands propagate across domains with sharp boundaries oriented either parallel or perpendicular to the flow direction (the arrows in Figure 14 follow a shear band with boundaries parallel to the flow direction). It is important to note that since the principal axis of strain is at 45° relative to the flow direction, the propagation of the shear bands is at 45° with respect to the strain component of the simple shear flow. As discussed earlier in the section on extensional flow, the presence of the shear bands is evidence of flow-induced orientation of the tilt azimuth perpendicular to the principal axis of strain. However, unlike extensional flow, which was able to orient molecules in all of the domains in the field of view of the Brewster angle microscope, in simple shear flow only isolated domains showed the existence of shear bands. It was also observed that a particular domain might contain a shear band for a number of cycles of flow reversals before the band would disappear. As the flow cycling continued, shear bands would intermittently appear and disappear in various domains throughout the field of view. This suggests that the appearance of the bands is sensitive to the relative orientation of the tilt azimuth of a domain and the direction of flow. The intensity calculation of light imaged in the Brewster angle microscope, presented in ref 11, allows us to evaluate the image contrast expected for a given rotation of the tilt azimuth. Our data indicate that an applied flow reorients the tilt azimuth of molecules to φ or φ + π, where φ is chosen for the applied flow so that both φ and φ + π become
Imax - Imin Imax - Imin
where Imax is the intensity of the brightest domain and Imin is that of the darkest one23 and the contrast is determined to switch between 0 and 0.73 in an extensional flow study by reversing flow direction but between 0.51 and 0.59 in simple shear. This result is in accordance with our observation that shear flow does not induce as dramatic a change in contrast compared with extensional flow. 3.2.3. Flow-Induced Breakup of Domains in the L′2 Phase. The phenomena described in section 3.2.2 were observed for flow reversal cycles of simple shear flow where the maximum shear rate was never greater than a value of 2 s-l. When sufficiently large shear rates were applied to the L′2 phase, however, a strain hardening and breakup of the monolayer domains were observed. This phenomenon was manifested by a dramatic fracturing and refinement of the domain pattern. This is shown in Figure 15. In the first frame of the sequence, the monolayer is shown immediately following the imposition of a strain of magnitude 4 at a shear rate of 4 s-1. In the subsequent frames, the monolayer was subjected to strains in the opposite direction. As the deformation proceeds, the length scales associated with the domain pattern are observed to markedly decrease. The result is a highly refined defect structure compared with the initial pattern. It is notable that the fracturing process occurs almost immediately upon reversal of the flow. Upon cessation of the flow, the fractured domain pattern does not reheal over time scales up to 60 min. Indeed, the initial domain pattern, characterized by domains of length scales comparable to those shown in Figure 15a, was only reestablished by expansion and recompression of the monolayer. 4. Conclusions In this paper the two-dimensional fluid dynamics of the different phases of docosanoic acid monolayers have been examined. Using Brewster angle microscopy, we studied the polydomain structure of two liquid condensed phases (the L2 and L′2 phases) and the “solid” S phase in situ during the application of extensional flow or simple shear flow. We found that only the L2 phase could deform nearly reversibly and that the tilt azimuthal orientation was unaffected by the flow. The two other phases experienced flow-induced reorientation of the molecular tilt azimuth, globally for extensional flow and in isolated domains for simple shear flow. The reorientation process for both types of flow accompanied the appearance of shear bands in the monolayers at (45° to the extension axis of each flow. The deformation of the polydomain pattern was quantitatively measured in the L2 phase by tracking the stretching and compression of two orthogonal line segments. Nonsymmetric domain deformations were found at the lowest rate of strain, while the plots of the strain evolution for higher flow rates were self-similar with a (23) Hosoi, K.; Ishikawa, T.; Tomioka, A.; Miyano, K. Jpn. J. Appl. Phys. 1993, 32, 135.
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Figure 14. BAM images of the response of the L′2 rad/s; phase to a simple shear flow of 2 s-l. The sequence shows frames separated by 1/3 s for a total strain of 4. In this series, shear band propagation can be identified within the domain located at the center of the frames. The shear band is oriented vertically and propagates and thickens horizontally in time (see the arrow in frame b). The flow is from left to right, and the velocity gradient is vertical.
Figure 15. BAM images showing the breakup of the L′2 phase subject to a simple shear flow of 4 s-1. In this sequence, each frame is separated by a time of 1/6 s for a total strain of 4. Note the fracture of the domain pattern into a fine, grainy appearance.
rate of strain independent of roller speed. A critical strain was observed above which the rate of strain shows a stepwise increase although the velocity of the rollers remains constant. The kink observed in the plots of strain versus time was assumed to result following a critical strain γc, representing the transition between elastic deformation and viscous flow of the monolayers. Assuming that the shear bands observed in the L′2 phase correspond to a plastic flow of material that accompanies the molecular tilt reorientation within elastic regions of the monolayer, a model was proposed to predict the time evolution of the width of shear bands. The time evolution
of the bandwidth was well-described by this model under ideal conditions, and the comparison between the model parameter obtained by fitting the data and the unilateral compression modulus provides strong evidence of the existence of an additional phase within a conventional L′2 phase region. Acknowledgment. This work was supported by the NSF-MSERC Center for Polymers at Interfaces and Macromolecular Assemblies (CPIMA). T.M. is grateful for support in the form of a fellowship from Bridgestone Corp. LA971047+
