Langmuir 1998, 14, 2455-2466
2455
Determination of the Thickness and Optical Properties of a Langmuir Film from the Domain Morphology by Brewster Angle Microscopy Marc N. G. de Mul*,† and J. Adin Mann, Jr. Department of Chemical Engineering, Case Western Reserve University, 10900 Euclid Avenue, Cleveland, Ohio 44106-7217 Received December 2, 1997. In Final Form: February 16, 1998 A method employing Brewster angle microscopy was developed to measure the relative thickness of monolayers and multilayers at the air-water interface. Results proved that multilayers in ultrathin films of 8CB (4′-n-alkyl[1,1′-biphenyl]-4-carbonitrile; nCB, n ) 8) are built up from one or two stacked bilayers on top of a monolayer at the air-water interface covered by a continuous bilayer. In addition, the fan texture of the mixed splay-bend disclination defects, which are formed in 10CB monolayers on reverse collapse of a continuous trilayer, was used to obtain estimates for the components of the anisotropic dielectric constant of the monomolecular film. This estimate was made by comparing experimental images with computationally generated model images based on an assumed director distribution around the defect. The results support the validity of the folding mechanism of multilayer generation and reverse collapse as opposed to the condensation mechanism. Finally, comparison of the experimental and theoretical values of the ratio of the multilayer thickness and the monolayer thickness indicates that at this level the water-film interface cannot be assumed to be sharp or well defined.
Introduction The morphologies of surfactant monolayers at the airwater interface are striking examples of the spontaneous self-organization of materials in nature. The patterns formed by these two-dimensional (2D) films can be remarkable, ranging from the intertwined spiral-shaped islands seen in phospholipid films to the fan textures formed by fatty acids and liquid crystalline materials. Observation of a simple droplet of insoluble surface-active material spread on a water surface can reveal intricate textures of a kind never seen before. Microscopic observation of the textures of Langmuir monolayers became possible one and a half decades ago, first by fluorescence microscopy1-6 and more recently by ellipsometric microscopy7-9 and Brewster angle microscopy (BAM).10,11 These methods have been used to clarify the surface phase diagram of long-chain fatty acid monolayers and to identify characteristic domain structures of its various phases.12 Direct observations of the collapse of the 2D films to form three-dimensional (3D) crystals13,14 and multilayers15,16 have provided new in* To whom correspondence should be addressed. † Current address: The Gillette Company, 1 Gillette Park 7G-1, Boston, Massachusetts 02127-1096. (1) von Tscharner, V.; McConnell, H. M. Biophys. J. 1981, 36, 409. (2) Lo¨sche, M.; Sackmann, E.; Mo¨hwald, H. Ber. Bunsen-Ges. Phys. Chem. 1983, 87, 848. (3) Knobler, C. M. Adv. Chem. Phys. 1990, 77, 397. (4) Mo¨hwald, H. Annu. Rev. Phys. Chem. 1990, 41, 441. (5) McConnell, H. M. Annu. Rev. Phys. Chem. 1991, 42, 171. (6) Schwartz, D. K.; Ruiz-Garcia, J.; Qiu, X.; Selinger, J. V.; Knobler, C. M. Physica A 1994, 204, 606. (7) Cohn, R. F.; Wagner, J. W.; Kruger, J. Appl. Opt. 1988, 27, 4664. (8) Beaglehole, D. Rev. Sci. Instrum. 1988, 59, 2557. (9) Reiter, R.; Motschmann, H.; Orendi, H.; Nemetz, A.; Knoll, W. Langmuir 1992, 8, 1784. (10) He´non, S.; Meunier, J. Rev. Sci. Instrum. 1991, 62, 936. (11) Ho¨nig, D.; Mo¨bius, D. J. Phys. Chem. 1991, 95, 4590. (12) For example: Ruiz-Garcia, J.; Qiu, X.; Tsao, M.-W.; Marshall, G.; Knobler, C. M.; Overbeck, G. A.; Mo¨bius, D. J. Phys. Chem. 1993, 97, 6955. (13) Mann, E. K.; He´non, S.; Langevin, D.; Meunier, J. J. Phys. II France 1992, 2, 1683.
sights into these processes. A variety of materials has been probed for morphological information with the objective of correlating microscopic information with macroscopic film properties.17 In most of these studies, conclusions were based on a qualitative analysis of the textures seen in the films. The main tool used for quantitative characterization of thin films is ellipsometry,18 which can be applied to monomolecular films on fluid surfaces to obtain values of the refractive indices and thickness. The analysis is complicated by the film texture, by the inherent roughness of liquids due to capillary waves, and by the optical anisotropy caused by the homogeneous orientation of surfactant molecules at the interface. The incident light beam diameter must be much larger than the monolayer domain size for the signal to be independent of position and time. Ellipsometric microscopy (EM) is the analogous technique wherein a microscope objective and ocular are substituted for the detector. A further variation is BAM, in which the incident angle is set to Brewster’s angle for the substrate. At this angle, the reflectance of the filmsubstrate system is minimal for p-polarized light and image contrast is at a maximum. In ellipsometric experiments, the measured variable is the ellipticity F, which is the ratio of the Fresnel reflection coefficient for p-polarized light to that of s-polarized light. This parameter can be written as a function of the two ellipsometric angles ∆ and Ψ
F ) tan Ψei∆
(1)
For films much thicker than a monolayer, values for both (14) Siegel, S.; Ho¨nig, D.; Vollhardt, D.; Mo¨bius, D. J. Phys. Chem. 1992, 96, 8157. (15) de Mul, M. N. G.; Mann, J. A., Jr. Langmuir 1994, 10, 2311. (16) Friedenberg, M. C.; Fuller, G. G.; Frank, C. W.; Robertson, C. R. Langmuir 1994, 10, 1251. (17) For example: Mann, E. K.; Langevin, D.; He´non, S.; Meunier, J.; Lee, L. T. Ber. Bunsen-Ges. Phys. Chem. 1994, 98, 519. (18) Azzam, R. M. A.; Bashara, N. M. Ellipsometry and Polarized Light, 1st ed.; North-Holland: Amsterdam, 1992; p 340.
S0743-7463(97)01315-2 CCC: $15.00 © 1998 American Chemical Society Published on Web 04/04/1998
2456 Langmuir, Vol. 14, No. 9, 1998
angles can be measured, and the film thickness d and refractive index n are uniquely determined. In the case of very thin films such as monolayers, only ∆ is measurable, so that a single measurement is not sufficient.19 Because surfactant monolayers are always uniaxial or biaxial, two or more components of the refractive index, or equivalently the dielectric tensor, must be measured in addition to d. A number of approaches have been used to circumvent this problem, for example making measurements at varying incident angles and wavelengths.20 The best solution is to seek independent measurements of one or more parameters. For monomolecular films, measurements of the thickness by X-ray reflection have been used successfully.21 BAM is an adaptation of ellipsometry, so the intensity at each point in the image depends on the local thickness and optical properties through eq 1. In theory, these parameters can be measured quantitatively by determining the light intensity at the camera and by analyzing the polarization state of the reflected light. The analysis uses a 4 × 4 matrix method based on the Fresnel reflection equations and will be reviewed in the next section. In principle, it is possible to compute the tilt angle of the molecules,22,23 the refractive indices, and the thickness of the thin films. As Overbeck and co-workers24-26 have demonstrated, assuming a model molecular orientation distribution for certain monolayer textures allows one to calculate theoretical microscope images. In their work, a thickness was assumed and the refractive indices of bulk films were used. By comparing the calculated image with experimental images, the model orientation distribution can be tested.27 In this paper, we will attempt to take this analysis one step further by using it to calculate values of the monolayer thickness and dielectric tensor directly based on a model for the observed morphology. We report measurements of the thickness and optical properties of thin films of 4′-n-alkyl[1,1′-biphenyl]-4-carbonitriles (nCB, n ) 8, 10). Our interest in these materials is twofold: first, the structure of the nCB films at the air-water interface provides information on the mechanism of bulk alignment in liquid crystal displays, induced by the boundary conditions at the surface; second, they serve as model compounds for surfactant films because of their rigidity and high dipole moment. The nCB materials are thermotropic liquid crystals that form well-ordered monolayers and multilayers at the air-water interface.15,16,28 Based on the surface pressure versus area per molecule isotherms and on the structure of the bulk smectic-A mesophases of 8CB and 10CB, it was deduced that monolayers of these materials collapse reversibly to trilayers on compression, which consist of an interdigitated bilayer on top of a tilted monolayer at the water interface. Further compression leads to coalescence of the circular bilayer domains, formation of a continuous trilayer, and (19) McCrackin, F. L.; Passaglia, E.; Stromberg, R. R.; Steinberg, H. L. J. Res. Natl. Bur. Stand. A 1963, 67A, 363. (20) Casson, B. D.; Bain, C. D. Langmuir 1997, 13, 5465. (21) Paudler, M.; Ruths, J.; Riegler, H. Langmuir 1992, 8, 184. (22) Hosoi, K.; Ishikawa, T.; Tomioka, A.; Miyano, K. Jpn. J. Appl. Phys. 1993, 32 2, L135. (23) Tsao, M.-W.; Fischer, T. M.; Knobler, C. M. Langmuir 1995, 11, 3184. (24) Overbeck, G. A.; Ho¨nig, D.; Wolthaus, L.; Gnade, M.; Mo¨bius, D. Thin Solid Films 1994, 242, 26. (25) Overbeck, G. A.; Ho¨nig, D.; Mo¨bius, D. Thin Solid Films 1994, 242, 213. (26) Weidemann, G.; Gehlert, U.; Vollhardt, D. Langmuir 1995, 11, 864. (27) de Mul, M. N. G.; Mann, J. A., Jr. Langmuir 1995, 11, 3292. (28) Xue, J.; Jung, C. S.; Kim, M. W. Phys. Rev. Lett. 1992, 69, 474.
de Mul and Mann
collapse of the trilayer to multilayer domains. Because BAM observations show that the multilayer domains have a homogeneous reflected intensity, we proposed that they are built up from a variable number of stacked interdigitated bilayers on top of the continuous trilayer. We have since obtained additional evidence from molecular mechanics calculations29 that the bilayers are strongly interdigitated. The collapse of the continuous trilayer is reversible, and when it is expanded, the trilayer undergoes reverse collapse during which it breaks up to a 2D foam structure. This 2D foam consists of trilayer Plateau borders with holes or foam cells containing only a monolayer. The mechanism of reverse collapse has not been determined. However, reverse collapse of trilayers of 10CB induces formation of one of two types of splay and bend defects (disclinations) in each foam cell with a molecular orientation (director) distribution similar to the schlieren textures observed in thick liquid crystalline films.30,31 We have previously determined that the kind of defect that is centered in a foam hole has a mixed splay-bend director structure, with a bend orientation in the defect center giving way to a splay orientation at the foam cell edge.27 Splay and bend defects exist also in 10CB monolayers outside of collapse regions in the pressure-area isotherm. In this paper we will present quantitative evidence that the bright homogeneous domains formed in nCB films on compression consist of an integer number of stacked bilayers. Moreover, we will confirm our model for the mixed splay-bend defect structure formed during reverse collapse of 10CB trilayers and in the process obtain estimates for the components of the dielectric tensor of the film.32 Optical Properties of Monolayers and Multilayers Reflectance of Thin Films. The reflectance R () Ir/ Ii, with Ir the reflected and Ii the incident intensity) of thin films on a substrate depends on the thickness, surface roughness, and optical properties of the films and substrate.33 In general, the surface roughness contribution is assumed to be small (i.e., the interfaces are sharp at the molecular level and the Fresnel equations apply). The relation between R and the film thickness d and refractive indices n2, n1, and n0 of the substrate, film, and ambient at Brewster’s angle, respectively, was derived more than a century ago by Drude34 for an isotropic film. If a film is anisotropic and if more than one film is present, a 4 × 4 matrix method can be applied to calculate R.18,35 However, the thickness dependence in the thin film limit is identical to that derived by Drude. Dependence of the Reflectance on the Thickness. Drude derived an equation describing the ellipticity F for a thin isotropic film (defined as F ) Rp/Rs, with Rp and Rs the Fresnel reflection coefficients of a film-substrate system for p-polarized and s-polarized light, respectively) at Brewster’s angle (incident angle φ0 determined by tan (29) Danko, C. A.; de Mul, M. N. G.; Mann, J. A., Jr., unpublished results. (30) de Gennes, P. G.; Prost, J. The Physics of Liquid Crystals, 2nd ed.; Clarendon: Oxford, 1993. (31) Lavrentovich, O. D.; Pergamenshchik, V. M. Int. J. Mod. Phys. 1995, 9, 2389. (32) de Mul, M. N. G. Ph.D. dissertation, Case Western Reserve University, 1995. (33) Meunier, J. Light Scattering by Liquid Surfaces and Complementary Techniques; Langevin, D., Ed.; Marcel Dekker: New York, 1992; Chapter 17. (34) Drude, P. Ann. Phys. Chem. N. F. 1891, 43, 126. (35) Berreman, D. W. J. Opt. Soc. Am. 1972, 62, 502.
Brewster Angle Microscopy of a Langmuir Film
Langmuir, Vol. 14, No. 9, 1998 2457
φ0 ) n2/n0). For a sharp interface, n1 is a constant, leading to
πd xn0 + n2 (n0 - n1)(n2 - n1) F) λ n2 - n2 n2 2
0
2
2
2
2
2
m
2
Because R ∝ F2, Ir is proportional to d2. The same result is obtained using the Fresnel equations for isotropic films and for anisotropic films with their optic axes perpendicular to the interface. The reflected intensity Ir is determined by the reflected electric field vector Er according to Ir ) E†r Er. In general, Er is computed by multiplying the incident field Ei by a complex-amplitude reflection matrix R (notation following Azzam and Bashara18)
]
R R Er ) REi ) Rpp Rps Ei sp ss
r01pp + r12ppe-j2β 1 + r01ppr12ppe-j2β
(4)
In this equation, β is the film phase thickness, which is proportional to d/λ, and r01pp and r12pp are the Fresnel reflection coefficients at the air-film and film-water interface, respectively. In the thin film limit, d , λ, so the exponents may be expanded. Moreover, at Brewster’s angle, r02pp ) 0, which leads to
r01pp ) r12pp
(5)
Therefore, to first order in d/λ, Rpp ∝ d/λ, and as before,
| |
R ) Rpp 2 ∝ d2
(6)
As will be shown later, the dependence of R on d2 is also valid as a first-order approximation when the optic axis of the films is not perpendicular to the surface. Therefore, eq 6 may be used to determine the relative thickness of film regions (e.g., of monolayers and multilayers) even if the optical properties of the film are unknown. Dependence of the Reflectance on the Optical Properties. The reflectance of stratified anisotropic planar structures is computed in the most general case with eq 3. The 4 × 4 matrix method may be used to determine the components of R, which depend on n0 and n2, on the incident angle φ0 and the refracted angle φ2, as well as on the 4 × 4 multilayer matrix L. In this context we will not discuss the calculation method at length because a comprehensive discussion can be found in Azzam and Bashara’s book.18 Rather, we will briefly review the important parts of the method for our analysis. Note that a simplified expression for the reflectance of a film consisting of a single layer can be derived analytically.36 (36) Tabe, Y.; Yokoyama, H. Langmuir 1995, 11, 699.
i-1
L(z + ∑hj,hi) ∏ j)1 i)1
(7)
In this equation L is the 4 × 4 layer matrix for a single i-1 hj to z layer of thickness hi which extends from z + ∑j)1 i + ∑j)1hj:
L ) e-jωhi∆ ≈ I - jωhi∆
(8)
Here ω is the angular frequency and ∆ is a 4 × 4 matrix that depends on the 6 × 6 matrix M (the optical matrix) of the layer ∆ ) ∆(M). All information on the optical properties of a layer is contained in M, which is determined by
(3)
For an isotropic film in general and for uniaxial and biaxial films in some special cases,18 Rsp and Rps vanish. Therefore, if the incident light beam is p-polarized, Ir depends only on Rpp
Rpp )
L)
(2)
1
[
For a film made up of m layers the multilayer matrix is:
M)
[
0E G G′ µ0µ
]
(9)
The components of this matrix are rank 2 tensors (3 × 3 matrixes). The magnetic permeability tensor µ is equal to the unit tensor for nonmagnetic media, whereas the optical rotation tensors G and G′ vanish in media that are not optically active. The dielectric tensor E contains the dielectric properties of the film. A coordinate system always exists in which E is diagonal37
[ ]
x 0 0 Exyz ) 0 y 0 0 0 z
(10)
Assuming that the principal axes of E are equivalent to the optic axes of the film, the components of E are equal to the dielectric constants of the film. Application to Brewster Angle Microscopy and Ellipsometric Microscopy (EM). The structures studied by BAM and EM are monolayers and multilayers. Therefore, E depends on the molecular orientation in each monolayer. The orientation can be taken into account by assuming that the principal dielectric axes are equivalent to the molecular axes.25 The dielectric constants of thick oriented films can be measured and set equal to the principal dielectric parameters x, y, and z. Equation 10 then gives E in molecular coordinates x′y′z′ (Figure 1), and this can be converted to beam coordinates xyz using a rotation matrix R(θ,φ), according to:
Exyz ) R(θ,φ)Ex′y′z′ R-1(θ,φ)
(11)
More generally, R(θ,φ) is given as a function of the Eulerian angles ψE, θE, and φE by38
[
]
cos ψE sin ψE 0 R(ψE, θE, φE) ) -sin ψE cos ψE 0 × 0 0 1 cos φE sin φE 0 1 0 0 0 cos θE sin θE -sin φE cos φE 0 0 -sin θE cos θE 0 0 1
[
][
]
(12)
In terms of the in-plane molecular orientation direction θ and the tilt angle with the surface normal φ (Figure 1), (37) Born, M.; Wolf, E. Principles of Optics, 6th ed.; Pergamon: Oxford, 1980. (38) Goldstein, H. Classical Mechanics; Addison-Wesley: Reading, MA, 1959.
2458 Langmuir, Vol. 14, No. 9, 1998
de Mul and Mann
Figure 1. Definition of beam coordinates xyz, molecular coordinates x′y′z′, tilt angle φ, and in-plane molecular orientation angle θ.
which are related to the Eulerian angles by θE ) π - φ, φE ) 1/2π + θ, and ψE ) 1/2, R(θ,φ) becomes
[
cos φ cos θ sin θ sin φ cos θ R(θ,φ) ) cos φ sin θ -cos θ sin φ sin θ sin φ 0 -cos φ
]
(13)
To measure the components of E or, if E is known, to determine the molecular orientation angle θ at different locations in the film, the BAM and EM can be operated in the PS or the PSA configuration. In the latter setup, a polarizer (P) is positioned in the path of the incident light beam to produce p-polarized light before the beam hits the optical system (S), in this case the water surface covered by the film, and a second polarizer, the analyzer (A), is used the modify the polarization state of the reflected light beam before it falls on the camera. The electric field at the camera is then equal to
EPSA ) KTAR′(A)Er ) K
[ ][
]
1 0 cos A sin A E (14) 0 0 -sin A cos A r
with Er given by eq 3, R′ a rotation matrix, TA the Jones matrix of an ideal polarizer, K an unknown gain constant, and A the angle of the transmission axis of the analyzer with the plane of incidence. In the PS configuration, the analyzer is omitted and EPS is simply equal to:
EPS ) K′Er
Figure 2. (a) Schematic diagram of the Brewster angle microscope used: A, analyzer; C, camera; FB, floating barrier; La, HeNe laser; Le, lens; M, mirror; MB, moving barrier; n, surface normal; P, polarizer; T, trough; W, water surface. (b) Schematic diagram of the setup used to calibrate the camera. Letter codes as before; FO, fiberoptic illuminator; ND, neutral density filter; S, screen.
the principal dielectric constants x′, y′, and z′ are equal to the coefficients of an ellipsoidal equation.37 The following expression for the average ⊥ in terms of x′ and y′ can then be derived:
⊥ )
(
π2 4 y′
dφ ∫0π/2 x1 - (1 - x′/y′) sin2 φ
)
-2
)
π2 4 y′
K2(x1 - x′/y′)
(16)
where y′ > x′ and K is the complete elliptic integral of the first kind. As a result, if ⊥ and | are known, as well as the film thickness and a model structure for the monolayer features which may be deduced from PS images, theoretical PSA images can be generated for a range of values of x′,25,27 which is related to ⊥ and y′ through eq 16. By comparing the computed images to measured images it is possible to estimate the actual values of x′ and y′. Accurate results may also be obtained by quantitative measurements of the reflected intensity as a function of the location in the film and fitting the results to the model parameters by a least-squares procedure. Experimental Methods
(15)
where K′ is an unknown gain constant. The reflectance R in the PS setup is less dependent on small changes in the components of E than in the PSA configuration.27 Therefore, it is reasonable to assume that the molecules in the film have cylinder symmetry so that x′ and y′ are both equal to the average ⊥. Experimental values of ⊥ and | ) z′ can often be obtained from the results of refractive index measurements on oriented thick films. However, in the PSA setup, this assumption is no longer valid and changes in the components of E perpendicular to the molecular axis can lead to significantly different results. In the system of principal dielectric axes,
Equipment and Materials. The film balance and Brewster angle microscope used have been described before.15 In brief, we used a Lauda Langmuir trough which was kept at 20.0 ( 0.1 °C with a circulating water bath. The trough is located in the Polymer Microdevice Laboratory at Case Western Reserve University, which is a class 100 clean room. The BAM was assembled in our laboratory from standard components following the general Mo¨bius design (Figure 2a). We used a 10 mW HeNe laser (Melles-Griot) and a Pulnix TM-7CN camera recording 768 × 494 pixels. During the experiments, images were not recorded on videotape, but captured directly from the camera output by a Perceptics Pixelbuffer frame grabber combined with an Apple Macintosh computer. The images were expanded to correct for the incident angle and filtered to reduce diffraction fringes caused by the coherent nature of the laser beam, unless otherwise noted.
Brewster Angle Microscopy of a Langmuir Film
Langmuir, Vol. 14, No. 9, 1998 2459 only a small part of the total noise. Therefore, we have not corrected i in eq 17 for the dark current. The calibration used in this paper assumed that each pixel responded following approximately the same constants A and B. Although this approach is sufficient for the work reported here, for higher accuracy, a pixel-by-pixel calibration is possible. Similarly, if a TEM00 laser is used as a light source, a correction for the Gaussian intensity distribution of the laser beam can be incorporated into eq 17. At a distance r from the center of the Gaussian intensity distribution that has a beam radius w, the intensity is given by:
I(r) ) I0e-r /w 2
2
(18)
with I0 the intensity at the center. Equation 17 then modifies to
Figure 3. Gray scale level response i of the CCD camera versus the relative incident intensity I. The liquid crystal 8CB was obtained from BDH Chemicals (Poole, U.K.), and 10CB was purchased from EM Industries (Hawthorne, New York). Both materials had a purity of at least 99.8%. The 8CB multilayers are formed at a surface pressure in excess of the equilibrium spreading pressure (ESP) of 5.2 ( 0.1 mN/m,27 and are therefore metastable with respect to 3D crystal formation. Although we have not observed crystallization in 8CB films, crystallization always occurs in 10CB because the ESP is equal to zero within experimental error. Therefore, the 2D foam can only be observed after minimizing all sources of nucleation by very carefully cleaning the trough with methanol and chloroform and extensive flushing with water from a Millipore Milli-Q Plus unit, pretreated by a Millipore Reverse Osmosis system. Camera Calibration. To determine the relationship between the reflected intensity incident on the charge-coupled device (CCD) chip and the gray scale value of the image, we calibrated our camera at different camera shutter speeds (i.e., integration times). The calibration setup is shown in Figure 2b. A collimated fiber optic illuminator was used as a light source. Light from the source was directed to two crossed polarizers, which made it possible to vary the light intensity by adjusting the angle between their optic axes, which had an accuracy of (1°. The camera was shielded from background light by a screen, and an OD 2.0 neutral density filter was inserted between the polarizers and the camera to further reduce the incident light intensity. Finally, the camera was found to be insensitive to the polarization state of the incident light beam. The CCD chip was homogeneously illuminated within experimental error and the mean gray scale value, which is in the range of 0 to 255, was found by averaging over the entire image. The response function of the CCD chip is shown in Figure 3 in terms of the gray scale value i recorded against the incident light intensity. The data were found to be fitted well by an equation of the form
i)
AI 1 + BI
(17)
with I the intensity in arbitrary units and A and B adjustable constants. Although it is possible in principle to obtain an absolute intensity value by repeating the procedure after careful calibration of the light source, we have used only relative intensity values, which is sufficient for our purpose. Note that the relationship between the gray scale value and the intensity is distinctly nonlinear at gray scale values >100, in contrast with the linear relation usually assumed in the literature.7,22,24,25 Moreover, we observe that the curve does not start at the origin, but levels off at a gray scale value of ∼9, which is the background noise level. The background noise is due to charge transfer inefficiencies in the charge-coupled device in the camera and to thermally generated charge carriers known as the dark current.39 Because the total background noise was not observed to change significantly for different integration times, the dark current is (39) Yang, E. S. Microelectronic Devices, 1st ed.; McGraw-Hill: New York, 1988.
AmnI0e-rmn/w 2
imn )
2
1 + BmnI0e-rmn/w 2
2
(19)
in which rmn is the distance of pixel (m, n) to the center of the Gaussian. Because w as well as the location of the center are generally not known accurately, they may be added as fitting parameters to the calibration problem. Thickness Measurements. To measure the relative thickness d of multilayer domains, >500 images were captured during the multilayer region in the π-A isotherm. The BAM was set up in the PS configuration for this purpose. Although we did not attempt to obtain a representative sample of all domain types and the number of domains of a certain size and thickness varies appreciably between experiments, we did obtain a statistically reliable number of domains of each thickness. The domains were analyzed without computer processing of the images, and the mean gray scale value of each of the domains, which were picked from the center of the field of view, was determined by averaging over a region drawn inside the domain to exclude the diffraction effects at the edges. The reflected intensity of the domain could then be calculated using eq 17, and d was determined by taking the square root in accordance with eq 6. We expressed each measured value of d as a multiple of the measured monolayer thickness. In this analysis we did not correct for the Gaussian intensity distribution of the laser beam, leading to a wide domain thickness distribution instead of a narrow range of d values. Evaluation of the Components of the Dielectric Tensor. The model of R as a function of d and E given in the Introduction can be used to estimate the components of E. We used literature values of ne ) 1.69 and no ) 1.53, which were measured on thick films of 7CB at 20.0 °C,40,41 and neglected the effect of the longer n-alkyl tail. The components of the dielectric tensor then follow from | ) n2e ) 2.86 and ⊥ ) n2o ) 2.34. We also set φ equal to42-45 71° and the monolayer thickness (to which the results are not very sensitive) d1 to 1.2 nm. Computed images of the splaybend structure27 of the defects in the 10CB monolayers were generated for a range of values of x′, which is related to y′ through eq 16. This procedure was done for a number of analyzer angles close to 90° using Mathematica running on a Macintosh computer. The estimated values of x′ and y′ are those for which the measured series of images corresponds most closely to the computed series. The estimated values of x′ and y′ can be refined if a correction is made for the Gaussian beam intensity distribution, or if the field is homogeneously illuminated. In that case, the gray scale values around the splay-bend structures can be measured (40) Davies, M.; Moutran, R.; Price, A. H.; Beevers, M. S.; Williams, G. J. Chem. Soc., Faraday Trans. 2 1976, 72, 1447. (41) de Jeu, W. H. Physical Properties of Liquid Crystalline Materials, 1st ed.; Gordon and Breach Science Publishers: New York, 1980. (42) Guyot-Sionnest, P.; Hsiung, H.; Shen, Y. R. Phys. Rev. Lett. 1986, 57, 2963. (43) Tang, Z. R.; McGilp, J. F. J. Phys. Condens. Matter 1992, 4, 7965. (44) Bartmentlo, M.; Vrehen, Q. H. F. Chem. Phys. Lett. 1993, 209, 347. (45) Enderle, Th.; Meixner, A. J.; Zschokke-Gra¨nacher, I. J. Chem. Phys. 1994, 101, 4365.
2460 Langmuir, Vol. 14, No. 9, 1998
Figure 4. Example of an unprocessed image used for relative film thickness measurements, showing multilayer domains with a range of thicknesses on top of a continuous trilayer. Only the domain in the center of the field of view was used for thickness measurements. The image was only scaled in intensity and reduced in size, and corresponds to an area in the monolayer of ∼700 µm in the horizontal dimension and 850 µm vertically as do all of the following experimental images. The image in this figure has not yet been corrected for the oblique incident angle of the laser beam, so that its vertical size appears compressed.
de Mul and Mann
Figure 5. Histogram showing the number of domains counted for each thickness interval. Note that the domain thickness distribution is not smooth, but has a number of peaks, proving that multilayers exist.
reliably, so that the dielectric tensor components may be fitted to the data using a least-squares procedure. For this purpose we plan to upgrade our BAM with a high-intensity diode laser, collimated to a large footprint, so that we can focus on a small area at the maximum of the Gaussian curve.
Results Thickness of 8CB Multilayers. In previous experiments we found that monomolecular films of the nCB materials at the air-water interface collapse on compression to form trilayers when n ) 8-10.15 Moreover, further compression leads to a second collapse, during which optically homogeneous domains with a variety of reflected intensities are formed on top of the homogeneous trilayer. An example of the type of morphology observed is shown in Figure 4. As discussed before, we proposed that the domains consist of an integer number of stacked bilayers, which may be topped by an additional monolayer. To obtain additional evidence for this model, we calculated the reflectance R of each of 562 domains in 8CB films from measurements of the average gray scale value, using a relationship determined by calibrating the CCD camera. R was expressed as the ratio of the reflected intensity of a domain and the average reflected intensity of a monolayer, which eliminates the gain constant in eq 15. Moreover, as the film thickness d scales with R following eq 6, the relative thickness may be found by taking the square root of R. The results are plotted as a histogram in Figure 5. As can be seen from the histogram, the domain thickness distribution is not smooth but has a number of peaks. Therefore, the domains have a number of discrete thicknesses, pointing to the existence of multilayers. Moreover, starting from the monolayer peak, the peak spacing is 0.8 ( 0.1 d1, where d1 is the monolayer thickness. Because the second peak is thought to represent a trilayer, the subsequent two peaks must be due to multilayers with thicknesses of five and seven layers, respectively. Domains with more layers exist but in much smaller numbers. This result confirms the stacked bilayer model.
Figure 6. Possible model structures for the centered defects (notation: c - clockwise director orientation around the center of a bend defect; a - anticlockwise orientation; i - inward orientation in a splay defect; o - outward orientation; kˆ is the light propagation direction).
In addition, at least up to seven layers the stacked bilayer domains are not covered by additional monolayers. Types of Orientation Defects and Dielectric Tensor Calculation. In our previous work we computed model images of the mixed splay-bend defects in the PS configuration. In this setup, the reflectance R is not very sensitive to changes in the components of the dielectric tensor E given by eq 10. Therefore, the molecules could be assumed to be cylindrically symmetrical, and the components of E could be calculated from the anisotropic refractive index. This procedure has the disadvantage that R is the same for the orientations θ and θ + π. Although a mixed splay-bend defect can have four varieties as shown in Figure 6 - the bend orientation can be clockwise or anticlockwise (c or a), molecules in the splay orientation can be pointed inward or outward (i or o) - in the PS setup we could not distinguish between the ci and ao defects and between the co and ai defects. Adding an analyzer makes R highly sensitive to the components of E. This relationship is demonstrated in Figure 7, in which calculated images are shown of a ci disclination as a function of the components of E and the analyzer rotation angle A. Small changes in x′ and y′
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2.0
2.1
2.2
2.3
Figure 7. Computed images of ci model defects as a function of the analyzer angle A and the dielectric tensor component in the x′ direction, x′. Comparison with Figure 8 shows that x′ is 2.20 ( 0.05. Note that images were computed at more values of x′ than are represented here.
lead to widely different images. This makes it possible to estimate x′ and y′ by comparing theoretically computed images with measured ones. Moreover, we may determine which of the model defects in Figure 6 occur in the monolayers. BAM images of the splay-bend defects are shown in Figure 8 for the same range of analyzer angles as in Figure 7. Two varieties of defects are observed experimentally, and on comparison with the model images in Figure 7, the components of the dielectric tensor E are found to have the following values:
x′ ) 2.20 ( 0.05 y′ ) 2.49 ( 0.05
(20)
z′ ) 2.86 ( 0.02 Once the components of E are known, theoretical images can be computed for each of the four defect types as a function of A. The computation results are shown in Figure 9. Again, by comparing Figure 9 with the experimental images in Figure 8, we find that only the ci and ai defects occur in practice. Apart from the ci and ai defects, other defect structures exist as well. First, we have observed co and ao defects in small amounts when the 2D foam is recompressed (i.e., when the monolayer supporting the 2D foam collapses). Moreover, during initial formation of the 2D foam during reverse collapse of the continuous trilayer, noncentered
defects are observed. Examples of the noncentered defects are given in Figure 10. Whether or not a defect is centered in the foam cell appears to be intimately connected to the collapse mechanism. During some experiments, exclusively noncentered defects were formed. However, in general, equal amounts of centered and noncentered defects are present. One way of finding the molecular orientation in foam cells containing a noncentered defect is by comparing the reflectance R at each location with the reflectance around a centered defect. This analysis is made easier by the fact that foam cells with centered and noncentered defects often occur side by side. For example, the director orientation in the noncentered defect shown in Figure 10b is as given in Figure 11a. The noncentered defect structure is analogous to that of the boojums observed in freely suspended smectic films46 and in monolayers of fatty acids.25,47 Note, however, that the foam cell contains a distorted boojum, because the boundary conditions at the border between the monolayer and the trilayer regions impose an inward orientation on the molecules, as in the case of the centered defects. Theoretical Thickness Calculation. Using the values of the components of the dielectric tensor already determined, the reflectance of multilayers as a function of the number of layers can be calculated. For the layer matrix L(0, d1) of the monolayer, a tilt angle φ of 71° and (46) Langer, S. A.; Sethna, J. P. Phys. Rev. A 1986, 34, 5035. (47) Schwartz, D. K.; Tsao, M.-W.; Knobler, C. M. J. Chem. Phys. 1994, 101, 8258.
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Figure 8. BAM images of centered disclination defects as a function of the analyzer angle A. Note that two varieties of defects occur with a different structure in the center of the defects.
E according to eq 20 were used. The layer matrix of the bilayer is L(d1, d2), with d2 the bilayer thickness. We assumed that the director in the bilayer is perpendicular
to the interface so that φ ) 0. Moreover, the reflectance of the monolayer was averaged over all θ, whereas molecules in the bilayer were taken as cylindrically
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Figure 9. Computed images of all four model defects against A for x′ ) 2.20. Comparison with Figure 8 proves that only ci and ai defects occur in practice.
symmetric; that is, ⊥ ) x′ ) y′ ) 2.34, which is equivalent to averaging over θ. The calculation results are shown in Figure 12 as (R3/R1)0.5, (R5/R1)0.5, and (R7/R1)0.5 versus d2/ d1 for d1 equal to 1.2 nm. Here, R1 is the reflectance of a monolayer, R3 is that of a trilayer, R5 is that of a monolayer covered by two bilayers, and R7 is that of a monolayer covered by three bilayers on the water surface. The calculation results depend only on the ratio d2/d1 and not on the value of d1 itself. The arrows in Figure 12 indicate the experimentally observed values from Figure 5 of R3/R1 ) 1.8 ( 0.1, R5/R1 ) 2.6 ( 0.1, and R7/R1 ) 3.5 ( 0.2. Moreover, we may determine values for the monolayer and bilayer thickness based on simple geometrical arguments. The bulk density of the smectic-A phase of 8CB is equal to48 1.0 × 103 kg/ m3. Therefore, the molecular volume of 8CB is 0.48 nm3. From the pressure versus area per molecule isotherm,15 the area per molecule in the monolayer is found to be 0.40 nm2, and therefore the monolayer thickness is 1.2 nm. (48) Leadbetter, A. J.; Durrant, J. L. A.; Rugman, M. Mol. Cryst. Liq. Cryst. 1977, 34, 231.
Finally, the thickness of an interdigitated 8CB bilayer is ∼2.6 nm.29 The bilayer thickness is then 2.2 times the monolayer thickness. Figure 12 shows that the reflectance of the trilayer is proportional to the square of the thickness to a reasonable approximation, which confirms the Drude result according to eq 2. However, d2/d1 is not equal to the theoretical value of 2.2 at the experimentally observed values of R3/ R1, R5/R1, and R7/R1. Rather, we find that the experimental value of d2/d1 is equal to 1.1 ( 0.3. The cause of this discrepancy between the experimental results and the theoretical computations will be discussed in the next section. Discussion Optical Properties of 10CB Films. The values measured for the dielectric parameters give direct information on the molecular arrangement in the first monolayer at the water surface. The dielectric tensor of a material depends on the electronic polarizability tensor
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Figure 10. BAM images of foam cells containing noncentered defects (A ) 80°). The defects show a fan structure originating in a single point on the monolayer-trilayer boundary line.
r by the Clausius-Mosotti equation:49
r E-I ) E + 2I 30ν
(21)
where ν is the molecular volume for spherical molecules, or alternatively a parameter that may be called the dielectric volume. For a phenyl ring system, the principal component of r perpendicular to the ring is larger than the components in the ring plane.50,51 Therefore, because E is inversely proportional to r according to eq 21, we may assume that the component of the dielectric tensor parallel to the phenyl plane is larger than the one perpendicular to it. Because for 10CB y′ > x′, we conclude that the biphenyl groups are predominantly lying flat on the water surface. Moreover, the difference between x′ and y′ is relatively large, indicating that the twist angle between the planes of the biphenyl rings is close to zero, in contrast to smectic bilayers where it is ∼30°.29 The molecular orientation described here maximizes the contact between the biphenyl group and the water surface. This situation is favorable because of hydrogen bonding between water molecules and the biphenyl ring system.52 More accurate values of E may be determined by fitting quantitative gray scale value measurements around a splay-bend defect directly to the model using a numerical (49) Smyth, C. P. Dielectric Behavior and Structure, Dielectric Constant and Loss, Dipole Moment and Molecular Structure, 1st ed.; McGraw-Hill: New York, 1955. (50) Le Fevre, C. G.; Le Fevre, R. J. W. Rev. Pure Appl. Chem. 1955, 5, 261. (51) Dunmur, D. A.; Tomes, A. E. Mol. Cryst. Liq. Cryst. 1983, 97, 241. (52) Suzuki, S.; Green, P. G.; Bumgarner, R. E.; Dasgupta, S.; Goddard, W. A., III.; Blake, G. A. Science 1992, 257, 942.
optimization procedure. However, in our current experimental setup, the noise in the images and the inaccuracies in some model parameters make an accurate fitting procedure impractical. We do not have reliable values of the tilt angle φ and the film thickness in 10CB films. Although the sensitivity of the results to the film thickness is small, a 10% change in the tilt angle leads to a 2% error in E. Image noise is caused by the rapidly fluctuating nature of the director around a defect, by the Gaussian beam intensity distribution of the laser beam, by not wellfocused regions in the image due to the oblique incident angle, and by diffraction rings around surface features. Monolayers of the nCB materials are special in the sense that the molecules are strongly tilted and the refractive index is very large. However, the method employed here to estimate the components of the dielectric tensor should also prove useful in monolayers of other compounds. Therefore, the method is a powerful tool to measure E of monolayers at the air-water interface, provided they exhibit a distinct molecular orientation morphology. Multilayer Structure and Stability. The relative thickness measurements reported here allow us to refine the stacked bilayer model for the multilayer domains. Based on the experimental results, we determined that the domains formed during collapse of the trilayer predominantly contain one or two bilayers, whereas domains with more bilayers are less numerous (Figure 13). The five- and seven-layer domains are therefore either more stable than domains with more layers, or the collapse mechanism favors formation of domains with a low number of bilayers. Because the multilayer domains do not undergo coalescence processes that change the number of layers (such as the spreading of a layer from one domain over another) until the multilayer domains are closepacked,15 we conclude that the predominance of domains
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Figure 13. Model structure of multilayer domains with five and seven layers. The domains contain one and two stacked bilayers, respectively, on top of a continuous trilayer.
Figure 11. Model structure for non-centered defects. The defects have a boojum structure which is distorted to satisfy the splay boundary condition at the boundary line between the monolayer and the trilayer.
Figure 12. Calculated relative reflectance Ri/R1 plotted as its square root against the ratio of the bilayer thickness and the monolayer thickness d2/d1. The arrows intersect the calculated curves at the measured values for the trilayer (i ) 3), 5-layer (i ) 5), and 7-layer domains (i ) 7).
with five or seven monolayers is a kinetic effect linked to the collapse mechanism. Because it leads to formation of
collapsed domains with few layers, the classical folding model is a strong candidate for the collapse mechanism.53 However, using the results of the dielectric tensor measurements, the observed ratios of the trilayer, 5-layer, and 7-layer reflectances to the monolayer reflectance could not be theoretically verified. For the observed reflectance ratios the calculated thickness of the bilayer is 1.1 ( 0.3 times that of the monolayer, which might indicate that it is not a bilayer at all but only a monolayer, and that domains with an even number of monolayers exist. This conclusion is not acceptable for several reasons. First, molecular area measurements based on the surface pressure versus area isotherm clearly indicate that the trilayer is built up from a monolayer at the surface covered by a bilayer. Second, a monolayer with the highly polar cyano groups exposed to the air would be unstable with respect to collapse to the interdigitated bilayers observed in bulk smectic A films, in which the molecules have a strong antiparallel association.54 It is more likely then that the theoretical thickness calculation is inaccurate, for example because the values of the tilt angle φ in the monolayer and the bilayer and the dielectric parameters contain errors. However, the unexpectedly large optical thickness of the monolayer relative to the bilayer is directly apparent from the experimental results, which indicated a d2/d1 ratio of 0.8 ( 0.1 (Figure 5), whereas a value of 2.2 is expected theoretically. Part of this discrepancy may be explained by assuming that the interface region between the monolayer and the water surface is not sharp because of capillary waves, because water molecules are retained in the bilayers after collapse, changing the optical properties of the bilayers, because the water substrate contains surface active species, or because of a disruption of the bulk water structure at the surface. None of these arguments is sufficient to fully explain the discrepancy. The most likely explanation we have at present is that the interface at the monolayer and submonolayer level is not sharp or well-defined, so that the Fresnel reflection equations do not give an accurate representation. We plan to investigate this effect further. Finally, it is possible to obtain absolute values of the film thickness by using a reference material with a film thickness and tilt angle that are known, for example from X-ray reflection measurements. The reference compound should not exhibit mesoscopic director gradients and it should form a homogeneous film. To eliminate all instrument gain constants, the thickness of the test (53) Ries, H. E., Jr. Nature 1979, 281, 287. (54) Leadbetter, A. J.; Frost, J. C.; Gaughan, J. P.; Gray, G. W.; Mosley, A. J. Phys. 1979, 20, 375.
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compound should be measured under the same conditions as the actual film. The 8CB film could serve this purpose because it forms homogeneous films that are thick enough to avoid problems with the roughness of the film-water interface. A second possibility are the alkoxycyanobiphenyl compounds terminated with siloxane groups, which have recently been studied by Ibn-Elhaj and coworkers.55-57 Molecular Alignment and Reverse Collapse Mechanism. The boundary line of trilayer regions with the monolayer aligns the molecules in the adjacent monolayer in 10CB films. This alignment was demonstrated before by observing the director patterns in 10CB films where the 2D foam is highly expanded so that the Plateau borders are very thin.27 Moreover, the director was seen to assume a splay orientation at the boundary of a foam cell with a centered or noncentered disclination defect. However, during the PSA observations reported here, almost exclusively ci and ai defects were recorded during expansion of the foam. In these defects, the director is oriented inward, that is, the cyano groups point away from the central disclination defect. It is likely that the predominance of ci and ai defects during expansion is caused by an orientation-specific process at the bilayer edge, which can be identified with the reverse collapse of the trilayer. Combining this observation with the evidence provided (55) Ibn-Elhaj, M.; Riegler, H.; Mo¨hwald, H. J. Phys. I France 1996, 6, 969. (56) Ibn-Elhaj, M.; Riegler, H.; Mo¨hwald, H.; Schwendler, M.; Helm, C. A. Phys. Rev. E 1997, 56 (2), 1844. (57) Ibn-Elhaj, M.; Mo¨hwald, H.; Cherkaoui, M. Z.; Zniber, R. Langmuir 1998, 14, 504.
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by the thickness distribution of the multilayer domains in 8CB, we conclude that the collapse and reverse collapse of nCB films at the air-water interface most likely follow a monolayer folding mechanism, as opposed to a condensation mechanism, in which molecules are returned from the bilayer to the monolayer one by one. Moreover, we expect that the folding mechanism applies in all collapse processes in which multilayers are formed, if the film material can form coherent bilayers as well. Conclusions For the first time, Brewster angle microscopy was used quantitatively to measure the relative film thickness of monolayers and multilayers at the air-water interface. Moreover, the anisotropic dielectric constant of 10CB monolayers could be measured by comparing experimental images of the fan texture around disclination defects with computationally generated images based on a known model director distribution. The results of the thickness measurements and of the dielectric constant measurements were shown to support the folding model of the collapse and reverse collapse mechanism. However, using the measured dielectric constant, the observed thickness ratio of the monolayer and the multilayers could not be reproduced theoretically, probably because the monolayer-water interface is not sharp or well-defined. Acknowledgment. Financial support for this work was provided by NASA Lewis Research Center (grant no. NCC3-266) and by the NSF Center for Advanced Liquid Crystalline Optical Materials. LA9713154
