Langmuir 1991, 7, 460-468
460
A Study of Colloidal Crystal Morphology and Orientation via Polarizing Microscopy Yiannis Monovoukas and Alice P. Gast* Department of Chemical Engineering, Stanford University, Stanford, California 94305-5025 Received April 30, 1990. In Final Form: August 13, 1990 We present microscopic observations of colloidal crystals grown in thin capillary cells. Observation between crossed polarizers reveals striations and polarization-dependent crystal colors. We employ dynamical diffraction theory to account for the depolarization of diffracted light in colloidal crystals and to correct Bragg diffraction wavelengths and intensities. Using Jones calculus, we predict crystal colors and intensities transmitted through crossed polarizers. We construct theoretical conoscopic images that provide a summary of the effect of crystal orientation on diffraction wavelength and intensity. Both striated and unstriated high volume fraction face-centered cubic (fcc) crystals often appear to orient with the (110) planes parallel to the cell walls. We show that striated crystals are twin fcc structures by determining the relative orientation between successive bands. We illustrate that as the volume fraction and cell thickness decrease, the tendency of the suspensions to crystallize with the (111)planes parallel to the cell walls increases while the number density of striated crystals decreases.
Introduction Suspensions of charged colloidal spheres are of fundamental interest because of their similarity to molecular When suspended in water, colloidal polystyrene spheres acquire a negative surface charge due to dissociation of surface sulfate groups and interact with a repulsive Coulomb potential screened by counterions in solution. This screening can be easily modified by altering the concentration of electrolyte ions leading to a range of repulsion forces and thus to a range of mutual sphere separations. E ~ p e r i m e n t a l and ~.~~ theoretical ~ investigations'O have shown that manipulation of the forces between particles leads to a disorder-to-order transition at high particle concentrations and high surface charge and a t low ionic strengths. A variety of crystal structures and morphologies have been observed microscopically and with light diffraction te~hniques.'l-'~Body-centered cubic (bcc) crystal structures are dominant at low volume fraction and ionic strengths, and face-centered cubic (fcc) structures are observed a t higher particle concentrations. Microscopic observations of colloidal crystals have revealed structural details as well as optical properties. When observed between crossed polarizers with a transmission optical microscope, the crystals appear colored. Upon rotation of the crystal in a plane perpendicular to the incident light, the transmitted wavelength and intensity change. The color and intensity vary further when the polarizer is removed. We have used dyamical X-ray
* To whom correspondence should be addressed.
(1)Hiltner, A. P.; Krieger, I. M. J . Chem. Phys. 1969, 73, 2386. (2) Hachisu, S.; Kobayashi, Y.; Kose, A. J. Colloidlnterface Sci. 1973, 42, 342. (3) Clark, N. A.; Hurd, A. J.; Ackerson, B. J. Nature 1979, 281, 57. (4) Pieranski, P. Contemp. Phys. 1983,24, 25.
( 5 )Aastuen, D. J. W.; Clark, N. A,; Cotter, L. A.; Ackerson, B. J. Phys. Reu. Lett. 1986, 57, 1733. (6) Ottewill, R. H. Langmuir 1989, 5, 4. (7) Lekkerkerker, H. N. W. Phase Transitions 1990, 21. (8) Sirota, E. B.; Ou-Yang, H. D.; Sinha, S. K.; Chaikin, P. M.; Axe, J. D.; Fujii, Y. Phys. Rev. Lett. 1989, 62, 1524. (9) Monovoukas, Y.; Gast, A. P. J. Colloid Interface Sci. 1989, 128, 533. (10) Kremer, K.; Robbins, M. 0.; Grest, G. S. Phys. Rev. Lett. 1989, 57, 2694. (11) Monovoukas, Y.; Gast, A. P. Phase Transitions 1990, 21, 183. (12) Carlson, R. J.; Asher, S. A. Appl. Spectrosc. 1984, 38, 297. (13) Yoshiyama, T. Polymer 1986, 27, 828.
0743-7463/91/2407-0460$02.50/0
diffraction theory,14 modified for visible light scattering from colloidal particles, to interpret diffraction phenomena in colloidal crystals15 and have predicted an anisotropic attenuation of the polarization components of the incident light due to coherent and incoherent scattering within the crystal. In this work we use Jones calculus16in conjunction with dynamical diffraction theory to predict both the corrected Bragg wavelength and intensity of light transmitt,ed through the top polarizer (analyzer) of the microscope in terms of the crystal volume fraction and orientation. We summarize our results in theoretical conoscopic images that we compare with experimental images. In the last part of this study, we examine the effect of cell thickness and suspension volume fraction on the structure and orientation of colloidal crystals. Pansu et al.17 and Van Winkle and Murray18 have reported a sequence of structural phases exhibited by colloidal suspensions in very thin glass cells that allow stacking of only a few (less than 20) crystal layers. In this work, we show that the glass surfaces can influence the crystal morphology and orientation well beyond the first few layers.
Experimental Section We prepared and characterized aqueous suspensions of negatively charged polystyrene spheres of diameter 133 nm as described in ref 9. We introduce suspensionsof moderate ionic strength (3 X 10" M KC1) and volume fraction ranging from 0.02 to 0.10 into thin glass cells of path length L = 50, 100, and 200 pm and dimensions 1 X 50,2 X 50, and 4 X 50 mm, respectively, where they form fcc crystals of various colors and sizes. As demonstrated," samples of volume fraction 0.016 < 9 < 0.067 lie in the coexistence region of the phase diagram and phase separate into a crystalline phase of & = 0.067 and a disordered phase of @d
= 0.016.
We studied the crystallites with a Zeiss Axioplan optical polarizing microscope in transmission. In Figure la, we show a set of parallel crystal planes between a polarizer, along the Y (14) Zachariasen, W. H. Theory of X-ray Diffraction in Crystals; J. Wiley: New York, 1945. (15) Monovoukas, Y.; Fuller, G. G.; Gast, A. P. To be submitted for publication. (16) Azzam, R. M. A.; Bashara, N. M. Ellipsometry and Polarized Light; North-Holland The Netherlands, 1987. (17) Pansu, B.; Pieranski, P.; Pieranski, P. J. Chem. Phys. 1984,45, 331. (18) Van Winkle, D. H.; Murray, C. A. Phys. Reu. A 1986, 34, 562.
0 1991 American Chemical Society
Colloidal Crystal Study via Polarizing Microscopy
Langmuir, Vol. 7, No. 3, 1991 461
a
ORTHOSCOPY
b
conoscoplc image
black
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jlass
/i
re'
1/11
analyzer
planes
d P CONOSCOPY \ A
a=O
a=36"
crystal planes
\
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Figure 1. (a) A family of crystal planes between a polarizer along the Y axis and an analyzer along the X axis (crossed polarizers).
Incident light along the 2 axis forms a plane of incidence with OA, the normal to the planes. The components of the polarization vector are either normal ( u ) or parallel ( T )to the plane of incidence. Crystal orientation is defined by angles a,formed by the incident light direction and the normal to the planes, and @,between the projection of the intersection of the plane of incidence with the crystal planes, BC, and the X axis. Conoscopicimages are formed at the back focal plane of the objective lens of the microscope and provide a summary of diffraction phenomena from crystals at a variety of angles a and @.(b) Orthoscopic and conoscopic illumination modes. . cone angle is defined by the numerical aperture Crystals with alll = 0 are at extinction and are colored when a # 0 and j3 # n ( ~ / 2 )The of the objective lens (0.75) and the refractive index of the crystal (-1.35). axis, and an analyzer at SOo to it, along the X axis. The normal to the crystal planes, OA, forms a plane of incidence with the direction of the incident light (2 axis). Crystal orientation is defined by angles a, the complementary Bragg angle formed between the positive 2 axis and OA, and j3, the angle between the positive Y axis and BC, the projection on the X Y plane of the intersection of the plane of incidence with the crystal planes. Figure l a also shows the u and r components of the polarization vector in the coordinate system of the crystal, defined as normal and parallel to the plane of incidence, respectively. Orthoscopic observation of the crystals between crossed polarizers reveals structural and orientational details11 allowing investigation of the effect of crystal rotation on the wavelength and intensity of transmitted light. Diffraction images can be observed at the back focal plane of the objective lens by inserting a Bertrand lens between the objective lens and analyzer and opening the condenser diaphragm below the sample to allow a cone of rays to fall on the crystal planes. The two illumination modes are illustrated in Figure Ib together with the inclination angle a and angle B defining the orientation of the crystallite relative to the crossed polarizers. The cone angle, defined by the refractive index of the crystal and the numerical aperture of the
objective lens,19 is approximately 3 6 O for the conoscopic images presented here using a 40X, numerical aperture = 0.75 objective lens. Conoscopic images thus provide a summary of the effect of crystal orientation on the diffraction colors for 0 I a I 36" and 0 5 j3 I3 6 0 O . We photograph orthoscopic and conoacopic images for further analysis. We quantify orthoscopicimagesrecorded on a Sony VO-5800H videocassette recorder with a VSP Labs video camera and a Sony Trinitron monitor, using Werner Frei Associates Imagelab and Imagetool interactive digital processing software on an IBM PC/ A T computer. We first define the crystal boundary from the pixel intensity difference between the crystal and the background and then paint the area within the boundary with a chosen intensity. The number of pixels with this intensity provides a crystal area fraction. Our predicted conoscopic images were photographed from a Silicon Graphics Personal Iris with a Hitachi high-resolution color monitor. (19)Bloss, F . D. A n Introduction to the Methods of Optical Crystallography; Holt, Rinehart and Winston: New York, 1961.
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Theory Light diffraction phenomena in colloidal crystals are best described by dynamical diffraction theory to account for the polarization dependent multiple scattering within the crystal. We have shown in ref 15 that the IJ and x components of the polarization vector of the incident light are attenuated anisotropically due to diffraction and to incoherent scattering losses from lattice imperfections and impurities. Bragg's law predicts the wavelength of the diffracted light, AB, for planes (hkl) of interplanar separation Dhkl, a t an angle a defined in Figure l, in a crystal of refractive index n as AB = 2ndhklsin (90 - a )
(1) but does not consider interference between incident and diffracted beams. Dynamical diffraction corrects the diffracted wavelength
where $of is the real part of the crystal polarizability given by
(m2- 1) (3) (m2 2) with m the ratio of the refractive index of polystyrene to that of water and 4J the particle volume fraction. In this system, the refractive index of the crystal is n = 1 . 6 0 + ~~ 1.33(1 - pC)and m = 1.60/1.33. We calculate the intensity of light diffracted from a set of planes H in the thick crystal limit, as developed by Zachariasen14 as
$0' = 34J-
4aa
u =x sin (90 - a )
(9)
provided that 4aa(m - 1) < A, a condition well satisfied by our 67-nm polystyrene particles. A fcc crystal oriented with the (111)planes parallel to the cell walls has CY = 0, K , = K , 1, and IH,, = IH,, resulting in equal attenuation of both polarization components of the transmitted light. The transmitted polarization vector retains its original direction along the Y axis; when observed between crossed polarizers, such crystals are at extinction and appear black. For a # 0, however, IH,, # ZH,,, the magnitude of the components of the transmitted polarization vector are attenuated unevenly and the direction of the polarization vector changes. This allows light to pass through the analyzer and results in colored crystals. The anisotropy in transmitted light can be expressed as a ratio of transmitted intensities
of the ir and u components. This ratio is defined in terms of the light attenuation anisotropy 8"
+
-
through a Beer's law relationship: I e6" where n, is the average index of refraction of the suspension. We use Jones calculus'6 to predict the state of the electric vector of light leaving the analyzer. The electric vector of magnitude EOleaving the polarizer can be represented as Ei = E 0 / 2 ~ / ~ [ 11. 0 , The electric vector leaving the analyzer, Et, is given by the optical train
Et = aP'b?X'
(11)
where A is the Jones matrix for the analyzer and b is the Jones matrix for the diffracting sample that considers attenuation anisotropy of the amplitude of the electric vector through 6". l' and p1are the rotation matrix and its inverse used to align BC with the X axis (see Figure la, i.e., to orient the crystal in the X Y Z coordinate system of the microscope through rotation by &@degrees. Hence eq 11 becomes
where with
and (7) Here X is the incident light wavelength and accounts for intensity losses due to incoherent scattering and determines the magnitude of the diffracted intensity. Note that y = 0 corresponds to the corrected Bragg peak maximum. The polarization dependence enters the diffraction intensity expression through K , = 1 and K , = lcos 2(90 - C Y ) ! for the polarizations normal and parallel to the plane of incidence. Polystyrene spheres are strong scatterers of light due to both their size and refractive index difference with water. This scattering is described by the Rayleigh-Debye scattering factor for a sphere of radius u20 3 f = ,(sin
u - u cos u )
(8)
U
where (20) van de Hulst, H. C . Light Scattering by Small Particles; Dover: New York, 1981.
[cos@ -sin@][!] sin@ cos@
EO
= -sin (2@)sinh
2112
The intensity of the transmitted light is obtained by squaring eq 12 to give
Thus, whereas the crystal color depends only on a as shown in eq 2, the color intensity depends on both a and (3 as shown in eq 13.
Result s Transmitted Wavelength and Intensity. As we have shown with dichroism measurements,15 the primary role of anisotropic attenuation of light propagating through
Colloidal Crystal Study via Polarizing Microscopy
650
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,
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Langmuir, Vol. 7, No. 3, 1991 463
,
,
,
7 02
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I
_ -_
-
/
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Figure 2. Diffraction wavelength, AB,^,,^^, as corrected by dynamical diffraction, and relative transmitted intensity, It/&,, against the orientation angle of the (111)crystal planes. These results consider diffraction from only the (111) planes.
fcc colloidal crystals is diffraction from the close-packed (111)crystal planes. In Figure 2, we consider diffraction from these planes and show the effect of crystal orientation, through the inclination angle ~ ~ 1 1 and 1 , the volume fraction 4 on the corrected Bragg wavelength and transmittedlight intensity for a crystal observed between crossed polarizers. We plot eq 2 for the corrected wavelength and eq 13 for the transmitted light intensity for samples of volume fraction 0.0945, well above the coexistence regime, and 0.067 corresponding to the crystal volume fraction of suspensions within the coexistence region. In Figure 2 we show that the corrected Bragg wavelength decreases with increasing ~ ~ 1 1 the 1 ; crystal color shifts toward the blue as the diffracting planes make a greater angle with the cell walls. Similar behavior is observed for both volume fractions, with 4 = 0.0945 always at shorter wavelengths than 4 = 0.067 because of their smaller interplanar spacing. As eq 2 suggests, the wavelength of the transmitted light depends only on the crystal orientation angle a and the volume fraction and that there is no effect of P on Xp,corr. The corrected wavelength curves describing diffraction from the (200) planes lie at shorter wavelengths than the (111) while those of higher index planes lie outside the visible region for the particle concentrations and ionic strength studied here. The relative transmitted intensity, also plotted in Figure 2, exhibits a rapid increase with CY and depends strongly on P. Maximum transmitted intensity is observed at = (2n + 1 ) ( ~ / 4 )where , n = 0, 1,2,3, ...,whereas at small a111 or /3 = n(x/2) no light is transmitted through the analyzer and the crystal appear black. Though the intensity plots in Figure 2 were obtained by using 4 = 0.067, nearly identical curves would represent the transmitted intensity for all volume fractions studied here. Orthoscopic Images. Magnification of the crystallites with a polarizing microscope allows us to study colloidal crystal morphology in thin cells and to relate colors to crystal structure and orientation. We present typical orthoscopic images of crystals photographed between crossed polarizers in Figure 3 for volume fractions 0.0945 and 0.0403 in cells 50 and 200 pm, respectively. The average lateral crystal size when 4 = 0.0945 is roughly 150 pm and is 300 pm in the 4 = 0.0403 sample. The concentrated sample in Figure 3a exhibits cellular space filling crystallites appearing mostly blue with small regions of disorder confined between them. Small disordered regions persist in polycrystalline samples above the saturation volume fraction, +c = 0.067, indicating disordered regions not observable with suspensions of the
same volume fraction in large cuvettes. These small disordered regions may be glassy domains or could contain impurities and particles of different sizes; they deserve further study. A few of the crystallites are striated with parallel lamellae of alternating color and varying width. Crystallites overlap as we increase the cell path length and the final size of both striated and unstriated crystals decreases suggesting that the number density of nucleation sites increases with cell path length. Only minor changes occur over a period of several months indicating that, although very slow growth may persist, the structures are essentially frozen in space by neighboring crystallites. Crystals of volume fraction between 0.067 and 0.0945, lying above the upper boundary of the coexistence region, are also space-filling and undergo little change with time due to contact with neighboring crystallites. The crystal color extends to longer wavelengths as the interplanar spacing increases with 4. We observe similar crystal overlapping and size decrease with crystals in wider cells as with the 4 = 0.0945 samples. Samples of a lower volume fraction, 4 = 0.0403 in Figure 3b, produce larger and rounder crystallites, especially in 200 and 400 pm cells, with colors extending to even longer wavelengths covering a range from blue to red, while 50pm cells favor elongating black crystallites. According to Figure 2, crystals of longer wavelengths are a t smaller inclination angles relative to the cell walls. Rotation of the microscope stage between crossed polarizers results in crystals color and intensity changes. The crystallites in Figure 3a upper mostly blue but a great many of them fade to white four times upon a full rotation of the microscope stage, Le., keeping a constant and varying P from 0 to 360', and four times they appear nearly black. The white appearance is caused by the intense transmission; spectral analysis indicates that the wavelength actually remains unchanged upon rotation. Referring to Figure 2, we infer that these crystals lie a t large inclination angles ( a 2 35O) and they are brightest when P = (2n 1)(7r/4). The angle between fcc (111)and (110) crystal planes is 35.26'; thus white crystals may be oriented with the (110) planes parallel to the cell walls. Some black crystallites, on the other hand, remain a t extinction at all P. As Figure 2 shows, no light is transmitted through crossed polarizers when the inclination angle CY is zero. We therefore conclude that crystallites a t extinction are oriented with their (111)planes parallel to the cell walls. In samples of lower volume fraction we observe similar intensity changes; however, these are often accompanied by changes in color. A bright red crystallite, for instance, in a sample within the coexistence region of the phase diagram with 4 = 0.0403, turns green as it goes through extinction and is bright red again at 90'. A dark red crystal, however, turns black a t a 45' rotation returning to its original color a t 90'. The dynamical diffraction results plotted in Figure 2 do not predict a change in wavelength as P varies when diffraction from only the (111)planes is considered. The color change observed with the low volume fraction samples is due to contribution to the (200) planes to the depolarization of the incident light due to diffraction in colloidal crystals. According to eq 13 the intensity transmitted through the analyzer increases 1; a bright red crystallite is at a sharply with ~ ~ 1 1hence, greater a than a dark red. Contribution from the (200) planes becomes noticeable a t moderate inclination angles and at positions of the rotation stage where pzm is close to 45' corresponding to maximum (200) contribution. At Pzm = 45*, Pill is small and according to Figure 2 the transmitted intensity due to the (111)planes is small. At
+
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464 Langmuir, Vol. 7, No. 3, 1991
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Figure 3. (a, left) Orthoscopic image of a sample of suspension volume fraction 0.0945 in a cell 50 pm thick, as observed between crossed polarizers. Striated crystallites exhibit similar behavior upon rotation of the microscope stage as the unstriated ones. The average lateral crystal dimension is approximately 150 pm. (b, right) Orthoscopic image of a suspension of volume fraction 4 = 0.0403 in a 200-pm cell observed between crossed polarizers. The average lateral crystal dimension is approximately 300 pm.
large inclination angles, the change in color is washed out by the large intensity of transmitted light. The effect of volume fraction and crystal orientation on the depolarization of light in colloidal crystals is summarized in conoscopic images of crystallites a t extinction, aligned with their (111) planes parallel to the cell walls, and are presented below. Experimental Conoscopic Images. Conoscopic illumination of crystallites a t extinction allows us to relate color and intensity to crystal orientation. As illustrated in Figure lb, the center of the conoscopic image observed at the back focal plane of the objective lens corresponds to zero inclination angle and the edges to a = 36'. Parts a and b of Figure 4 show experimental conoscopic images of crystals of volume fraction 0.0945 and 0.0403 corresponding to samples whose orthoscopic images are shown in parts a and b of Figure 3, respectively. Both images display a dark cross concentric with the center of the image with the dark branches indicating the directions of the crossed polarizers. The center of the image corresponds to a crystallite with a111 = 0 illuminated with light parallel to the 2 axis and shows that the polarization of the incident light is unchanged upon diffraction, and the crystal it at extinction a t all 0 as the analyzer blocks all transmitted light. The edge of the conoscopicimage provides the same information on the depolarization of incident light due to diffraction in colloidal crystals as an orthoscopic image of a crystal oriented with its primary diffraction planes at a = 36' to the cell walls, i.e., with the (110) planes parallel to the walls.
The high volume fraction 9 = 0.0945 image displays mainly blue and white colors in agreement with the orthoscopic colors of Figure 3a. The dark cross illustrates that at angles p = n(7r/2)the polarization components of the incident light normal and parallel to the crystal planes are equally diffracted by the crystal and the attenuation anisotropy is zero and thus all emerging light is blocked by the analyzer. White regions are observed a t a = 35' and p ==: (2n + 1)(7r/4) in agreement with Figure 2. Between a volume fraction of 0.07 and 0.09 conoscopic images from crystallites a t extinction exhibit a green region at small inclination angles. At even lower volume fractions, samples in the coexistence region with & = 0.067 produce images with a red region near the center of the image a t small a , as shown in Figure 4b. A white crystallite with a ==: 36' and p ==: 45' turns blue upon rotation to p = 75' and is a t extinction a t /3 = 90'. A bright red crystal, however, may turn green or blue before it goes through extinction depending on which direction the microscope stage rotates. At a position where it appears green, the (200) planes offer a maximum contribution to the trans==: 20'. mitted intensity because , & =~45' and Rotation of the microscope stage on a horizontal plane has no effect on the images of Figure 4 (parts a and b). Conoscopic images of colored crystallites, on the other hand, display a dark cross off the center of the image. The dark cross maintains its orientation upon varying 0,but its center describes a circle about the center of the image. The radius of this circle gives the angle of the primary diffracting planes, a , interpolated between 0 and 36'.
Langmuir, Vol. 7, No. 3, 1991 465
Colloidal Crystal S t u d y via Polarizing Microscopy
L
A
A
Figure 4. (a (top left) and b (top right)) Experimental conoscopic images of a crystal at extinction in samples of 4 = 0.0945 (Figure sa) and & = 0.067 (Figure 3b), using a Bertrand lens and a 40X objective lens of numerical aperture 0.75. The center of the image is at cy111 = 0 corresponding to the (111) planes parallel to the walls, and the edge at CY = 36". (c (bottom left) and d (bottom right)) Theoretical conoscopic images of crystals at extinction in samples of volume fraction 4 = 0.0945 (c) and 4 = 0.067 (d). The bright regions correspond to B = (2n + l ) ( r / 4 ) and the dark ones to 0 = n ( x / 2 ) .
Conoscopic images, therefore, represent a summary of diffraction phenomena in colloidal crystals oriented with 0 5 cy 5 36" and 0 i p F 360" and observed between crossed polarizers, providing information on transmitted wavelength and intensity and allowing us to relate crystal color to orientation. Below, we use dynamical diffraction equations (2 and 13) to predict the wavelength and intensity of light leaving the analyzer and to construct theoretical conoscopic images for comparison with the experimental images of Figure 4 (parts a and b). Theoretical Conoscopic Images. In parts c and d of Figure 4 we show graphical representations of the dynamical diffraction predictions of the transmitted wavelength and intensity for crystals @ = 0.0945 and @ = 0.067, respectively. These theoretical conoscopic images are
computed on a Silicon Graphics Personal Iris and photographed from a Hitachi high-resolution color monitor. Transmitted wavelength and intensity are plotted on a 128 X 128polar grid with an RGB (red, green, blue) raster graphics color model mapped to an HSV (hue, saturation, value) model.21 RGB color is linearly interpolated between grid points using Gouraud shading22that allows a smooth color transition from a polygonal to a curve surface. The center of the image corresponds to all1 = 0 and the edge to a111 = 36'; therefore these images represent conoscopic patterns observed microscopically when light of cone angle (Bl).Foley,J. D.;Van Dam, V. Fundamentals oflnteractive Computer Graphics; Addison-Wesley: New York, 1982. (22) Hearn, D.; Baker, M . P. Computer Graphics;Prentice-Hall: Englewood Cliffs, NJ, 1986.
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466 Langmuir, Vol. 7,No. 3, 1991 a
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b
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Figure 5. Effect of rotation on the color of successive lamellae
in striated crystallites. Neighboringlamellae exchangecolor upon rotation by 20" or 70" indicating that striated crystals are twin fcc structures with a (110) plane parallel to the wall. 36" is incident on a crystal oriented with its (111)planes parallel to the cell walls. In agreement with Figure 2, the dark bands correspond to p = n(7r/2) and the brightest regions are at the edge, all1 = 36", at p = (2n + 1)(7r/4), that we set to white corresponding to saturation:O and va1ue:l in the HSV model. The hue in the HSV model is defined by the corrected Bragg wavelength given by eq 2. Comparison of the predicted conoscopic images with the experimental images of Figure 3 suggest that high volume fraction images are in excellent agreement with predicted transmitted wavelength and intensity, indicating that indeed the close-packed (111)crystal planes are the major source of diffraction. Experimental images from samples of lower 4, however, show a rapid change from red to white, or red to blue, with increasing ~ ~ 1 1 1These . trends are not predicted theoretically. Figure 4d shows a gradual change to shorter wavelengths as the inclination angle increases because only diffraction from the (111) planes is considered. The white region next to the red region in the experimental image of Figure 4b is due to superposition of the contribution from the fcc (200) diffracting planes, as described before.15 The transmitted intensity at the outer regions of the experimental image is especially high due to superposition of intensities from all diffracting planes that washes out the gradual change from red to yellow and green. Improved theoretical conoscopic images of low-density crystals should include contributions from all diffracting planes. Striations. The relative orientation of the bands in striated crystallites is obtained by rotating the stage until neighboring bands alternate color. In Figures 5, we show a striated crystallite in the coexistence region of the phase
Figure 6. (a) Area fraction of crystallites at extinction aligned with the (111)planes parallel to the cell walls, relative to the total crystalline area, plotted against particle volume fraction. Crystals in 50-pm cells and 4 = 0.025 are entirely oriented with a111 = 0. (b) Area fraction of striated crystallites relative to the total area of colored crystallites, against volume fraction in cells 50,100,and 200 pm thick. Fewerstriated crystallitesare observed with low volume fraction samples in thin cells.
diagram with crystal volume fraction 4c= 0.067 in a 100pm cell surrounded by black crystallites. The lamellae are of random width but are parallel to each other throughout the crystallite. We observe similar parallel striations with all the samples investigated. Upon rotation of the sample by 20" clockwise or by 70" counterclockwise, neighboring bands alternate color. We also notive that the black crystallites in Figure 5 remain a t extinction as angle p varies from 0 to 360". The difference in p of 20' (or 70') between lamellae is indicative of crystal twinning in fcc crystals as can be seen by projecting the particles in the (111)twin planes onto the (110) planes at approximately 35". Moreover, the lamellae are parallel to each other suggesting that the crystallites are formed by parallel stacking of the (111)planes. Effect of Glass Walls and Density. Dilute colloidal suspensions in thin cells crystallize to larger structures with smoother edges than more concentrated suspensions and exhibit a variety of colors, as we have shown in Figure 3b. Decreasing the size of the glass cell from 400 to 50 p m results in mainly black crystallites and fewer striated colored ones. We quantify the effect of cell thickness and suspension volume fraction on the orientation and morphology of colloidal crystals by recording orthoscopic images on a Sony videocassette recorder and using interactive digital processing software to measure crystal areas as they appear on a Sony Trinitron monitor. In Figure 6a, we plot the area of crystals oriented with the (111)planes parallel to the cell walls, relative to the total crystal area, against the suspension volume fraction.
Colloidal Crystal Study via Polarizing Microscopy Crystals in low volume fraction suspensions orient almost entirely with the close-packed (111)planes along the cell walls. Similar, but less pronounced trends are observed in cells 100and 200pm thick. Black crystals are extinction in suspensions of high volume fraction constitute less than 8 % of the total crystalline area, and this fraction is almost the same in all cells. A t high volume fractions, the nucleation barrier is lower than those in more dilute suspensions due to the greater supersaturation, and the crystals nucleate and grow rapidly in the bulk of the cell volume. At lower volume fractions, the repulsive forces on the colloidal particles from the cell walls lower the nucleation barrier in an area next to the walls as the particle motion is restricted in the direction normal to the walls. Moreover, the particles near the walls are repelled by both particles in the bulk and charges on the wall surfaces. These interactions are minimized by registration of the highest density (111)planes parallel to the walls. This wall influence is especially strong in narrow 50-pm cells because of the close proximity of the two walls. Van Winkle and Murray's have observed a &fold two-dimensional structure near the walls in extremely narrow gaps that developed into a bulk bcc structure, of no 3-fold symmetry, as the gap increased, indicating that crystal structure in narrow cells is determined by minimization of the particlewall and particle-particle interactions. In our case the bulk structure, determined by Kossel line analysis,ll is fcc, and in 50-pm cells the interactions are minimized by the alignment of the high density planes with the walls as observed. The effect of the walls on the crystal morphology is also shown in Figure 6b, where we plot the area of striated crystallites relative to the total area of colored crystals, against the suspension volume fraction, for various cell thickness. The area fraction of striated crystals decreases with both volume fraction and cell thickness, from about 0.4 in all cells to zero in 50-pm cells. We have observed twinned crystals with their (111) planes parallel to the walls;" however, this twinning would not be apparent in orthoscopic images since such crystals will remain a t extinction. Diffraction studies indicate that crystals grown slowly from low volume fractions in thin cells may have fewer twins. Thus the sharp drop in the fraction of striated crystals in 50-pm cells may reflect the ordering effect from the cell walls. We note that many crystals are not circular in cross section but rather oblong with the long axis perpendicular to the striations. This may reflect transport limitations in the crystal growth process as growing planes await the arrival of particles. The preferred growth direction remains the (111);however, theselectionof one (111)plane to twin and grow fastest remains unexplained. It is important to mention that these oblong crystals and striaton orientations are random throughout the sample, indicating that no large scale gradients exist.
Discussion Depolarization of light propagating through colloidal crystals due to anisotropic attenuation of the IJ and a polarization components leads to colored crystallites when observed between crossed polarizers. This anisotropy is successfully described by dynamical diffraction theory that accounts for coherent and incoherent scattering within the crystal. Dynamical diffraction predicts a corrected Bragg diffraction wavelength that depends on the crystal volume fraction and a,the inclination angle of the crystal planes relative to the cell walls. Using Jones calculus, we treat diffraction as an anisotropic attenuation and calculate
Langmuir, Vol. 7, No. 3, 1991 461 a transmitted intensity that depends strongly on both CY and p and weakly on 4. Orthoscopic observation a t high volume fraction crystals combined with dichroism mea~urementsl~ suggests that many of them orient with the (110) crystal planes parallel to the cell walls. We also observe this orientation in striated crystals. The bands are all a t CY i= 36O, but differ from their neighbors by p i= 70°,corresponding to twin fcc crystal structures with the twin (111)planes a t 36" to the walls with their (110) planes parallel to the walls. We observe a change in both the intensity and the wavelength by varying p in crystals of lower volume fraction. Here diffraction from the (200) planes contributes to the overall depolarization of incident light. This leads to a red-to-black or a red-to-green change upon rotation of the stage. The contribution to the attenuation anisotropy from the (111)planes is always greater than the (200) contribution when a111 i= C Y ~ ~ O OTherefore .~~ the transmitted intensity from the (111) planes dominates when pllli= ( 2 n + l)(a/4) and the crystal appears red. As pill moves away from this value, moves closer to it and the (200) transmitted intensity dominates, and thus the crystal appears colored with wavelength given by XB,corr,SOO-
Diffraction wavelengths and intensities are summarized in conoscopic images of crystallites a t extinction as shown in Figure 4. The center of the image corresponds to a crystal oriented with its (111)planes normal to the incident light a t a111 = 0 that appears black a t all positions of the microscope stage because the transmitted intensity is zero at all as shown in Figure 2. As we move radially outward from the center, the inclination angle a111 increases and the wavelength of the anisotropy decreases, but the transmitted intensity increases sharply and bright white crystallites are observed. Experimental conoscopic images of crystallites a t extinction thus provide information on the depolarization due to diffraction from crystals oriented with the primary diffraction planes at 0 ICY I 36' and 0 I p I360' and indicate that contribution from the (200) family of planes is responsible for the color changes observed upon varying p. Theoretical conoscopic images considering only diffraction from the primary diffraction (111)crystal planes are in excellent agreement with the experimental images for high volume fraction crystals. As the volume fraction decreases, interparticle spacing increases and contribution to conoscopic images from diffraction from (200) diffraction is possible. This contribution affects both the wavelength and the intensity of the crystals. The rapid change from red to white with increasing CY observed in low volume fraction experimental images is due to the high transmitted intensity, a result of contribution from multiple planes, that washes out the yellow and green regions. The effects of the suspension volume fraction and cell thickness on the crystal orientation and morphology are shown in Figure 6. As the suspension volume fraction decreases, repulsive forces from the negatively charged walls lower the nucleation barrier near the walls. In the very thin, 50-pm cells, wall repulsions force the particles into a two-dimensional 3-fold close-packed array near the walls. Additional close-packed planes are deposited on top of the first one, and so on, leading to a threedimensional fcc structure oriented with the (111)planes parallel to the cell walls. These crystals appear black a t all rotations of the microscope stage when viewed between crossed polarizers. The interplay between the crystal spreading and growth
468 Langmuir, Vol. 7,No. 3, 1992 process and resulting shape and orientation deserves further study. At this point we can say that cell walls exert a subtle influence on growing crystallites; however, transport limitations and growth rates may dominate final morphology.
Conclusion Corrected Bragg diffraction wavelengths and transmitted intensities of light propagating through colloidal crystals observed between crossed polarizers are successfully predicted by dynamical diffraction theory. Orthoscopic observations reveal crystal morphology. We identify striated crystallites as twin fcc structures. Conoscopic images summarize diffraction phenomena for a variety of crystal orientations. Theoretical conoscopic predictions match experimental results for high volume fraction crystals as the (111) planes are the major contributors to the attenuation anisotropy. Contribution from all dif-
Monovoukas and Gust fracting planes should be considered a t low volume fractions to improve predicted conoscopic images and to interpret orthoscopic color and intensity changes upon rotation of the microscope stage. Strong wall repulsions promote alignment of the densest crystal planes parallel to the 50-bm cell walls.
Acknowledgment. We gratefully acknowledge support of this work through the NSF PYI Program Grant CTS 85-52495. We appreciate the help of Rosemarie Koch and Ken Carlson with the microscopic studies. We are grateful to Will Zimmerman for his help with the conoscopic computations and to Professor Bud Homsy for use of the Silicon Graphics Personal Iris. Additional support for this research has been provided by the Xerox Foundation and by the Ford Corporation. Registry No. Polystyrene, 9003-53-6.
