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Langmuir 1995,11, 3184-3188
Quantitative Analysis of Brewster-Angle Microscope Images of Tilt Order in Langmuir Monolayer Domains Mei-Wei Tsao, Thomas M. Fischer,t and Charles M. Knobler” Department of Chemistry and Biochemistry, University of California, Los Angeles, California 90095-1569 Received October 24, 1994. I n Final Form: March 13, 1995@ We describe an attempt at the quantitative evaluation by Brewster-angle microscopy of the molecular tilt azimuth in domains of two condensed phases of Langmuir monolayers of methyl eicosanoate. The domains,which are surrounded by the isotropicliquid-expandedphase, contain star defects in which there is an ordering of the tilt azimuth into six pie-shaped regions. The presence of this high symmetry allows the orientation to be determined. In one phase the molecules are tilted toward the domain boundary (splay texture) and in the other tilt is parallel to the boundary (bend texture). The relation between the change in texture and recent diffraction studies on esters is examined.
Introduction Brewster-angle microscopy1g2(BAM)is an increasingly important technique for the study of thin films a t interfaces. Like ellipsometry, the method is based on measurement of the change in the state of polarization when light is reflected from thin films between two isotropic media. Such changes depend on the thickness of the film and its optical anisotropy. If the incident beam is p-polarized and strikes the interface a t the Brewster angle of the bare surface, essentially all of the background related to the substrate is eliminated and the method is then sufficiently sensitive to allow images of monolayers a t the airlwater interface to be obtained. In principle, the optical anisotropy of the monolayer could be detected by observation of the reflectivity a t normal incidence. But the ratio of reflected p- and s-light is very small a t the Brewster angle, which, together with the polarizer used for the incident light, provides an extinction ratio very much higher than that which could be obtained with a single polarizer in ordinary polarized microscopy. There have now been several studies by BAM in which phase boundaries have been determined by observations of qualitative changes in monolayer texture^.^-^ Isotropic phases (such as the gas and liquid-expanded) can be distinguished by the difference in their thickness; transitions between a phase in which the molecules are tilted to one in which the molecules are normal to the surface or transitions between phases in which there are differences in the molecular tilt order can be detected by changes in anisotropy. The textures of monolayers, as revealed by polarized fluorescence microscopy (PFM)and BAM, are surprisingly complex. There is often large-scale organization of the molecular tilt azimuth into regular arrays of stripes, spiral^,^,^ and stars.g These textures are closely related Fakultat fiir Physik und Geowissenschaften, Universitat Leipzig, Linnestr 5, D-04103,Leipzig, Germany. Abstract published in Advance ACS Abstracts, July 15, 1995. (1) Honig, D.;Mobius, D. J.Phys. Chem. 1991,95,4590. (2) HBnon, S.;Meunier, J . Rev. Sci. Instrum. 1991,62,936. (3)Overbeck, G. A,; Mobius, D.J. Chem. Phys. 1993,97,7999. (4)HBnon, S.;Meunier, J. J . Chem. Phys. 1993,98,9148. ( 5 ) Riviere, S.;Hbnon, S.; Meunier, J . Phys. Rev. E 1994,49,1375. (6) Riviere, S.;Hbnon, S.;Meunier,J.; Schwartz, D. K.; Tsao, M.-W.; Knobler, C. M. J . Chem. Phys. 1994,101,10045. (7) Ruiz-Garcia,J.; Qiu, X.; Tsao, M.-W.;Marshall, G.; Knobler, C. M.;Overbeck, G. A,; Mobius, D. J . Phys. Chem. 1993,97,6955. (8) Overbeck, G. A.; Honig, D.; Wolthaus, L.;Gnade, M.; Mobius, D. Thin Solid Films 1994,242,26. (9)Qiu, X.; Ruiz-Garcia,J.;Stine, K. J.;Knobler, C. M.; Selinger, J. V.Phys. Rev. Lett. 1991, 67,703.
to those observed in liquid crystals, and theories analogous to those developed for liquid crystals are able to describe many of the patterns.lOJ1 Several qualitative comparisons have been made between experimental BAM images of such patterns and images calculated from assumed models of the tilt organizatiod2but there have been no quantitative tests. While it is not possible to determine the molecular organization in a monolayer film by a direct inversion of BAM images, it is possible, in principle, to make quantitative comparisons between experiment and theory. Hosoi et al.13 obtained BAM images of stearic acid monolayers along a n isotherm. They determined the molecular tilt angle as a function of molecular area by comparing the contrast (the normalized differencebetween the intensities of the brightest and darkest domains in the image) to a calculation14of the change in contrast as a function of angle. They made no attempt, however, to analyze the anisotropy and determine the tilt azimuth. The quantitative analysis of BAM images is difficult because absolute values of the reflectivity cannot be extracted from the video images. Given the variations in laser intensity and background, it is difficult as well to compare different images of the same monolayer. There is a greater likelihood of success if an analysis can be performed of a single image, for which many of the experimental variables are held constant. The chances can be improved if there is some relation imposed, for example, by symmetry, between the tilt azimuths in neighboring regions of the monolayer. A recently investigated phase transition in esters provides such an opportunity. Fischer et al.15 have described a “blooming”transition in domains of a liquidcondensed phase ofmethyl esters surrounded by the liquidexpanded phase. The blooming has been observed by optical measurements in the Cle-C22 methyl esters; it accompanies a transition between condensed phased6that
@
(10)Selinger, J. V.; Wang, Z.-G.;Bruinsma, R. F.;Knobler, C. M. Phys. Rev. Lett. 1993,70,1139. (11)Fischer, T.M.; Bruinsma, R. F.;Knobler, C . M. Phys. Reu. E 1994,50,413. (12)Overbeck, G. A,; H o n-k D.; Mobius, D. Thin Solid Films 1994, 242,213. (13)Hosoi, K.;Ishikawa, T.; Tomioka, A.; Miyano, K. Jpn. J . Appl. Phys. 1993,32,L135. (14)Berreman, D. J . Opt. SOC. Am. 1972,62,502. (15)Fischer, B.; Tsao, M.-W.; Ruiz-Garcia, J.; Fischer, T. M.; Schwartz, D. K.;Knobler, C. M. J.Phys. Chem. 1994,98, 7430. (16)For a recent review of monolayer phases, see Desai, R. C.; Knobler, C.M. Ann. Rev. Phys. Chem. 1992,43,207.
0743-746319512411-3184$09.00/00 1995 American Chemical Society
Langmuir, Vol. 11, No. 8, 1995 3185
Analysis of Brewster-Angle Microscope Images
satisfactory contrast is found. The best images are obtained when the polarizer is slightly displaced (*O.lo) from the p orientation. Images were digitized to an 8-bit gray scale with an image grabber (Scion,LG-3); a time-base corrector (HotronicsAP41)is used to preserve the quality of the image during the transfer from videotape. The only manipulation of the images was noise reduction with a low-pass filter (NIH Image 1.43). .
Figure 1. BAM image of a star defect in methyl eicosanoate. The reflectivities of the different regions change when the monolayer is cooled through the blooming transition, but the positions of the arms do not change.
was first detected from pressure-area isotherms. l7 At temperatures above the transition, the domains have the form of six-arm “star” defects; i.e., they are divided into six pie-shaped regions that correspond to different orientations of the molecular tilt azimuth, Figure 1. When the transition occurs, the new phase grows symmetrically from the center of the domain, much like the opening of the petals of a flower. (See Figure 6 in ref 15.) The star texture is preserved in the new phase, but the tilt azimuths are different, as evidenced by changes in the fluorescence intensity or the reflectivity. We describe the quantitative analysis of these textures above and below the transition and comment on the relaton between the changes in texture and the underlying structure.
Experimental Section The trough, the materials, and the experimental procedures have been described in ref 15. We will review here only the details of the imaging. The Brewster-angle microscope, which is of our own construction, is similar to that described by Honig and Mobius.’ An Ar+ laser (wavelength 514 nm) is used as the light source. It is operated at a power level of 500 mW, but after the beam has passed through the optics and the cover glass, it has been attenuated to 280 mW. The high intensity allows us to work with a small aperture, thereby increasing the depth of field. Noticeable convection is induced only when the nominal laser power exceeds 1W. The polarizer is an 8-mm Glan-Laser prism (Karl Lambrecht) with an extinction ratio of 5 x 10-5and the analyzer is an 8-mm Glan-Thompson prism (Karl Lambrecht) with an extinction ratio of The images are formed with a 16x ,strain-free microscope objective with a numerical aperture of 0.35(Leitz Wetzlar) on a CCD camera (Dage-DTI 72s) and are recorded on videotape. The polarizer is adjusted for minimum reflectivity before deposition of the monolayer. After deposition, the image is brought into the dynamic range of the camera by successively adjusting the positions of both the polarizer and analyzer until (17) Lundquist, M.Chem. Scr. 1971, I , 5.
Analysis Our goal is to determine the spatial variation of the orientation of the molecular tilt azimuth within the monolayer. This information cannot be obtained unequivocallyfrom techniques other than BAM. The precise relation between the structure of a monolayer and the BAM images follows from a classical analysis of the reflectivity of a layered structure.18 The problem has been discussed by H6nonlgand Overbeck;20our analysis, which is given in full detail elsewhere,21differs somewhat from their approaches. It is based on the 4 x 4 matrix method developed by Berreman14J8 and the transfer matrix technique similar to that of Bethune.22 The observed reflectivity .“depends on the molecular properties of the amphiphile, its orientation (tilt angle and tilt azimuth),the characteristicsof the incident beam, and the conditionsof detection. We assume the monolayer to be a stratified layer of axially symmetric molecules. The molecular dielectric tensor emolhas elements exlc= em f ezz. Euler angles are employed to transform the molecular system to the laboratory coordinate system:
Here 8is the molecular tilt angle and #is the tilt azimuth; a third rotation is unnecessary because of the assumed azimuthal symmetry. To describe the textures requires the specification of nine parameters: The dielectric constant of water E,,, (which we take as 1.771, ea, e=, e,#, the thickness of the monolayer d, the angle of incidence of the beam 0, the tilt of the polarization of the incident beam away from the p orientation 6i, and the analyzer angle 6,. (Note that 6i is not precisely zero because the maximum contrast in the image is obtained when a small amount of s polarization is mixed into the beam.) Because the reflectivities are not measured on an absolute scale, it is also necessary to include a single factor S that scales the calculated to the experimental values. It is not possible to perform a direct inversion of the measured reflectivity to obtain 8 and #, but Wcan be calculated for an assumed structure and compared with the experimental results. In principle, the validity of the structure can be tested by showing that it accounts for changes in the reflectivity produced by changing the analyzer angle or, equivalently, that it accounts for the textures of domains that have different orientations with respect to the incident beam. In practice, such comparisons may be difficult because the differences in reflectivity may be small and because the images do not have a uniform background and may be contaminated by the presence of experimental artifacts such as Newton’s rings (18)Azzam, R.M.A.; Bashara, N. M.“Ellipsometry and Polarized Light”;North Holland: Amsterdam, 1987, p 340. (19) Hbnon, S. Ph.D. Thesis, Physics, Universit.6 Paris VI, 1993. (20)Overbeck, G. A. Ph.D. Thesis, Mathematisch-NaturwissenschaftlichenFachbereich, Georg-AugustUniversitiit, Gattingen, 1993. (21)A detailed derivation of the equations and a descriptionof the Ph.D. Thesis, Department of analysis can be found in Tsao, M.-W. Chemistry and Biochemistry, University of California, Los Angeles, 1994. (22)Bethune, D.S.J . Opt. Soc. Am. 1991,8, 367.
Tsao et al.
3186 Langmuir, Vol. 11, No. 8, 1995 caused by a small tilt of the glass protective cover on the CCD sensor. There are two features of the domains that we have observed in esters that simplify the problem: (1) The symmetry of the six-arm star defects, which requires that the organization of the tilt azimuth be modulo 2 d 6 , limits the number of possible structures. (2) The blooming transition provides 12 different tilt orientations within the same domain, which reduces the problem ofvariations in the background between different parts of the microscope field. (3)The textures have also been investigated by polarized fluorescence microscopy, which allows the general nature of the tilt pattern (splay, bend) to be deduced. Our procedure is to assume values for the instrumental parameters 0. di, and 6, and the molecular parameters E=, czz, 8, and d and then determine S and the values of 4 in different regions of a domain by minimizing the square root of the sums of the squares of the differences between the observed and calculated reflectivities, and zeac l
o=
a
normalized by the average reflectivity &. In a typical instrument it is difficult to determine the instrumental parameters with high accuracy. For example, the orientation of the incident beam is generally known at best to 0.05-0.1"; the incident angle is set by determining the minimum in the reflectivity from a clean water surface and is not measured. Accurate measurements of the parameters could be carried out, but not trivially. We have not done so nor is this information easily obtained with the available commercial instruments. Instead, we have chosen to treat the instrumental parameters as variables, which allows us to examine as well the sensitivity of the tilt azimuth to them. We make our initial choice of parameters by assuming that the tilt azimuths in adjacent segments of a star differ by 60" so the specification of the tilt azimuth within one segment fixes the tilt azimuths in all other segments. This condition is relaxed in subsequent variations of the parameters; the parameters are varied systematically one a t a time to find their optimum values. The instrumental parameters 0 and 6iare used to calculate the components of the incident electric fieldazZ Then the molecular parameters en, E,, 8, and d , plus the variable tilt azimuth 4, are employed to calculate the transfer matrices for each tilt orientation. Finally, after the components of the reflected fields are calculated from the transfer matrices, the analyzer angle 6, is used to further select the combination of polarization states of the reflected fields, and the net reflectivity for each molecular orientation is obtained. This reflectivity corresponds to the signal detected by the camera. The values of cU:and cZz,2.19 and 2.37, respectively, are taken from the ellipsometric study of fatty acids by Paudler et al.,23and are consistent with the values determined by den E n g e l ~ e n . ~ ~ The results of a fit to reflectivity data from a single domain in the high-temperature phase are represented by the full circles in Figure 2. Each point is the reflectivity from one of the six pie-shaped regions in the domain. It is typically an average over 1000to 1500pixels, the number being limited by the presence of imperfections in the image, such as bright spots caused by stray reflections. There are Newton's rings within each of the regions and we have assumed that they represent a modulation of the signal (23) Paudler, M.; Ruths, J.; Riegler, H. Langmuir 1992,8, 184.
(24) den Engelsen, D. Surf. Sci. 1976, 56, 272.
I'
,
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50
100
150
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250
300
350
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Figure 2. Reflectivity as a function of tilt azimuth for several domains. The line represents the curve calculated by adjusting the parameters for the best fit to the points indicated by the full circles. Other points on the curve were taken from five other domains. They have been multiplied by scale factors chosen for a best fit to the curve. The error bars represent the standard deviationsof the reflectivitiesmeasured around each segment of a domain. A fit to the six data sets using the same scale factors produces a curve indistinguishable on this scale from the one shown here. and that the averaging minimizes their effect. The error bars represent standard deviations. We could increase the precision in the reflectivity by making background corrections; we have not done so because we do not believe that there is a completely objective way to treat the data. The Newton's rings could largely be removed by a Fourier transform procedure, which enhances the appearance of the image. The reflectivities obtained by this method do not differ significantly from those obtained from the unenhanced images, however, and we have therefore not removed the rings. We have also examined five other domains in both the high- and low-temperature phases and points representing them are also included in the figure. They have been calculated with the same values of the parameters and have been scaled to the fitted data by adjusting the value of S. These points do not fall at the same values of the tilt azimuth because the domains had different orientations of their arms with respect to the laser beam direction. In a nonlinear least-squares procedure there can be many minima in the parameter space, so we cannot state with certainty that the values that we have found represent a global minimum. We have, however, examined a broad range of parameters. For example, the tilt angle has been varied from 0 to 90" and the molecular length from 10 to 30 A. The instrumental parameters that have been examined generously bracket the values consistent with the uncertainties in the settings. The sensitivity of the least-squares minimum to variations in the parameters is shown in Figures 3-7 and the optimum values of the parameters are given in Table 1. The instrumental quantities, 0, di, and 6, all fall within the range of the nominal settings ofthe instrument. There have been no X-ray diffraction or reflectivity studies of ester monolayers with which we can compare the optimal chain tilt angle of 15.5", but there are measurements on fatty acids with which we can make contact. Lin et aLZ5 examined the diffraction from monolayers of docosanoic (25) Lin, B.; Shih, M. C.; Bohanon, T. M.; Ice, G. E.; Dutta, P. Phys. Rev. Lett. 1990,65, 191.
Analysis of Brewster-Angle Microscope Images
Langmuir, Vol. 11, No. 8, 1995 3187 I
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"
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-70
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.
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Figure 4. Variation of CJ with molecular tilt angle 8.
40
30 THICKNESS (A)
Figure 7. Variation of CJ with monolayer thickness d. Table 1. Parameters That Provide the Best Fit to the Measured Reflectivitiesa Oldeg
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= 2.19 and eYy = 2.37 and that
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Figure 5. Variation of CJ with angle of polarization 6i.
acid in the low-temperature, low-pressure region that corresponds to the Lfl phase of the esters (which they called the B phase) and found 6' = 17". The tilt angle 6' and the monolayer thickness d have been treated independently. A number of diffraction studies of condensed monolayers have shown that the tilt angle and thickness are related by the expression d = 1 cos 8, where 1 is the molecular length.26 If we calculate I from the tilt a n le and thickness determined from the Reflectivity and diffraction measurefit, we obtain 22 ments on monolayers of arachidic acid are consistent with the value 24 A for I, which is essentially the length of the hydrocarbon chain.26 Thus, the length that we obtain is about 10% short. As seen in Figure 7, however, there is a broad minimum in the residual plot ford, and a thickness
1.
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(26) Tippmann-Krayer, P.;Mohwald, H. Langmuir 1991, 7,2303.
consistent with a 24-A molecular length differs only negligibly from that at the minimum. If we assume the relation between d, I, and 8 and take 1 = 24 A (a fit with one less parameter), we find 6' = 15.5" as before and no significant change in any of the other parameters. The choice of one domain for the fitting is arbitrary. We would prefer to treat all of the domains equally, but we know of no way of incorporating the scale factors into the fit. As a n ad hoc procedure we have also fitted the 36 scaled points together, employing the values of S determined from the adjustment of the six fitted points. We are assuming that the tilt angle is the same in the two phases. This is unlikely to be true, but the differences are probably The values of the parameters determined this way are only negligibly different from those obtained previously. The key point is that we are not trying to determine the molecular parameters from the BAM studies; we argue only that the reasonableness ofthe values that provide the best fit lends some confidence to the analysis of the tilt organization. The tilt azimuths that provide the best fit correspond to those deduced from the PFM measurements. At high temperature the texture corresponds to a splay defect in which the tilt is uniform and outward within each wedge, with a 60" jump in the tilt azimuth a t each arm. The tilt azimuth changes by 90" during the blooming transition. In the low-temperature phase there are two choices
3188 Langmuir, Vol. 11, No. 8, 1995 corresponding to clockwise and counterclockwise tilt orientations and both are observed. If the domains were hexagonal, the tilt azimuth within each region of the hightemperature phase would be everywhere normal to the LE-LC interface, which appears to be the most favorable orientation." This condition cannot be met in more rounded domains and we cannot rule out the possibility that there is a small variation in the tilt azimuth within a region or near the boundary that cannot be resolved by the BAM. When Fischer et al.15analyzed the blooming transition, no structural measurements of ester monolayers had been performed. Their analysis of the phase transition was based on the miscibility studies of Bib0 et al.,27which allowed the phases in monolayers of ethyl esters to be related to those of acids, systems for which there have been extensive diffraction studies. According to the miscibility measurements, the higher temperature phase is identified as L2, a phase in which the tilt is toward nearest neighbors, and the lower-temperature phase as L2", in which the tilt is also toward nearest neighbors. The 90" change in tilt-azimuth accompanying the phase transition was attributed to the onset of herringbone order in the L F phase and its effect on the stability of the boundaries between the six pie-shaped segments within the domains. Recently, Foster et a1.28reported the first diffraction experiments on ester monolayers. They discovered that in methyl eicosanoate the higher-temperature phase is L2', in which the tilt is toward next-nearest neighbors, (27) Bibo, A.M.; Knobler, C. M.; Peterson, I. R. J.Phys. Chem. 1991, 95, 5591. (28) Foster, W. J.;Shih,M. C.; Pershan,P. S.ProceedingsofMateria1s Research Society, Symposium AA:Applications ofSynchrotron Radiation Techniques to Materials Science II (Boston, MA, 1994);Perry, D. L., Ice, G., Shinn, N., D'Amico, K., Terminello, L., Eds.; pp 187.
Tsao et al. rather than Lp. This assignment is in agreement with the phase diagram proposed by Lundquist17and has been corroborated by diffraction measurements carried out by Braslau et al.29 The conclusions of the miscibility study are apparently erroneous. It appears that the 90" change in tilt azimuth observed during the blooming is the result of the 30" change in molecular orientation from nextnearest neighbors toward nearest neighbors. (In a hexagonally ordered system, the change from next-nearest to nearest neighbors can be accomplished by a rotation of n16 or 7~12.1As discussed by Fischer et al.,15 the bend structure would in addition be stabilized by herringbone order, but it seems probable that the swivelingofthe chains is the principal cause of the macroscopic reorientation of the tilt azimuth. In conclusion, we have demonstrated that inversion of BAM images to obtain the tilt azimuth can in practice be carried out. Although we have been aided by the symmetry of the textures, we have been inhibited by the lack of a precise knowledge of the instrumental parameters and lack of information about the molecular tilt angle and the film thickness, quantitites that are increasingly available from diffraction studies. It therefore is apparent that the determination of orientations of the tilt azimuth from BAM analysis is feasible.
Acknowledgment. This work was supported by the National Science Foundation. T.M.F. acknowledges additional support under the Feodor Lynen program of the Alexander von Humboldt Foundation. LA9408331 (29)Braslau, A.;Daillant, J.;Fischer, B.; Knobler, C. M. In preparation.
