Langmuir 2004, 20, 5141-5144
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Gravitational, Vertical Compression of Colloidal Crystals as Studied by the Kossel Diffraction Method Tadatomi Shinohara,*,† Hisashi Yamada,‡ Ikuo S. Sogami,† Norio Ise,§ and Tsuyoshi Yoshiyama† Department of Physics, Kyoto Sangyo University, Kita-ku, Kyoto 603-8555, Japan, Graduate School of Science, Osaka University, Suita, Osaka 565-0871, Japan, and Kyoto University, Yoshida-Honmachi, Sakyo-ku, Kyoto 606-8501, Japan Received August 1, 2003. In Final Form: March 15, 2004
In 1934, Kossel observed a diffraction cone from a copper single crystal using fluorescent X-rays.1 Kikuchi had observed a similar diffraction cone using electron beams.2 Although Laue pointed out the versatility of the Kossel method,3 it has not been widely used for ordinary crystals, probably because of the rather tedious analysis method required. Since visible light interacts more strongly with matter than X-rays and since colloidal dimensions are of about the same order of magnitude as the wavelength of visible light, Kossel analysis has proved to be a powerful technique for the structural determination of colloidal crystals by several groups.4-6 We carried out an analysis of the (Kikuchi-)Kossel lines for polymer latex crystals.7 Subsequently, we studied silica particles (density (F) ) 2.2 g/cm3) in light water.8 The lattice constant of the colloidal crystal could be determined with a precision of 0.2%, which is much higher than that of X-ray scattering or light scattering, and hence, a rather small gravitational effect could be detected. In this note, we discuss the characteristics of noncubic crystals under normal gravity in silica-light water systems. In the previous work,8 gravitational sedimentation of the particles was observed, supporting the findings of Crandall and Williams;9 the particle concentration decreased with increasing height, and the lattice constants correspondingly increased, while the lattice symmetry was not influenced. During this study, we occasionally observed peculiar Kossel images showing a noncubic structure. In the meantime, we have found experimental conditions under which long and stable pillar-shaped colloidal crystals of colloidal silica particles are formed, as shown in Figure 1. It seems that the formation of the pillar-type crystal is reproducible in light of the fact that it has also been observed independently by Yamanaka et al.10 The Kossel diffraction images obtained at various positions within the pillar-shaped crystal indicated that the local * To whom correspondence should be addressed. E-mail:
[email protected]. † Kyoto Sangyo University. ‡ Osaka University. § Kyoto University, Professor Emeritus. (1) Kossel, W.; Voges, H. Ann. Phys. (Leipzig) 1935, 23, 677. (2) Kikuchi, S. Jpn. J. Phys. 1928, 5, 83. (3) von Laue, M. Ann. Phys. (Leipzig) 1935, 23, 705; 1937, 28, 528. (4) Clark, N. A.; Hurd, A. J.; Ackerson, B. J. Nature 1979, 281, 57. (5) Pieranski, P.; Dubois-Violette, E.; Rothen, F.; Strzelecki, L. J. Phys. (Paris) 1981, 42, 53. (6) Monovoukas, Y.; Gast, A. P. J. Colloid Interface Sci. 1989, 128, 533. (7) Yoshiyama, T.; Sogami, I.; Ise, N. Phys. Rev. Lett. 1984, 53, 2153. (8) Shinohara, T.; Yoshiyama, T.; Sogami, I. S.; Konishi, T.; Ise, N. Langmuir 2001, 17, 8010. (9) Crandall, R. S.; Williams, R. Science 1977, 198, 293. (10) Yamanaka, J.; et al. Presented at the 55th Discussion Meeting on Colloid and Surface of the Chemical Society of Japan, Sendai, Japan, September 2002.
Figure 1. Pillar-shaped crystal for KE-E10 silica particles at a volume fraction of 0.0073. The photograph was taken on the 29th day after the dispersion was introduced into the cuvette and shows the portion of the cuvette that is 13-24 mm from the bottom. The sketch is given to facilitate recognizing the locations of the Kossel analysis in the pillar-shaped crystal.
Figure 2. Unit vector Kmm′ of the intersection of two Kossel cones with indices m and m′ and the respective reciprocal lattice vectors Gm and Gm′.
orientation of the crystal was almost the same everywhere, so that the crystal was believed to be a single crystal. We used this pillar-shaped crystal to study the gravitational effect on colloidal crystals. The laser diffraction experiments were conducted in the same manner as described earlier.7,8,11,12 On the supposition that the cuvette’s quartz plate is infinitely thin, we considered refraction at the boundary between the colloidal crystal surface and air. To correct for the refractive effect by using Snell’s law, we adopted the formula for the refractive index of colloidal dispersions (n(φ)) proposed by Hiltner and Krieger13
n(φ) ) nwater(1 - φ) + nparticleφ
(1)
where nwater and nparticle are the refractive indices of water and silica particles (nwater ) 1.33 and nparticle ) 1.4714,15), respectively, and φ is the volume fraction of the colloidal particles in the dispersion. Divergent beams emanating from a pointlike defect within the crystal are reflected by the lattice planes only at angles satisfying Bragg’s law. The reflected beams from a set of planes with the Miller index m ) (hkl) will then generate the surface of a cone (Kossel cone) whose central axis is parallel to the reciprocal lattice vector Gm (see Figure 2). Through the assignment of Miller indices to (11) Yoshiyama, T.; Sogami, I. S. Langmuir 1987, 3, 851. (12) Yoshiyama, T.; Sogami, I. S. In Ordering and Phase Transitions in Charged Colloids; Arora, A. K., Tata, B. V. R., Eds.; VCH Publishers: New York, 1996; pp 41-68. (13) Hiltner, P. A.; Krieger, I. M. J. Phys. Chem. 1969, 73, 2386. (14) Iler, R. K. The Chemistry of Silica; John Wiley & Sons: New York, 1979; p 19. (15) Riese, D. O.; Vos, W. L.; Wegdam, G. H.; Poelwijk, F. J.; Abernathy, D. L.; Gru¨bel, G. Phys. Rev. E 2000, 61, 1676.
10.1021/la030318f CCC: $27.50 © 2004 American Chemical Society Published on Web 05/07/2004
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Notes
Table 1. Height Dependence of the Lattice Constants experiment
height from the cuvette bottom (mm)
c (nm)
lattice constants c/a
days after introduction into cuvette
c/b
remark
1 2 3
23 22 19
714 ( 2 708 ( 2 703 ( 2
KE-E10 at φ ) 0.0073 0.9880 ( 0.0003 0.9903 ( 0.0004 0.9811 ( 0.0006 0.9813 ( 0.0004 0.9778 ( 0.0008 0.9806 ( 0.0007
4 5 6 7
24 22 20 18
730 ( 2 707 ( 2 703 ( 2 690 ( 2
KE-E10 at φ ) 0.0073 0.9932 ( 0.0007 0.9919 ( 0.0014 0.9839 ( 0.0010 0.9850 ( 0.0006 0.9821 ( 0.0010 0.9854 ( 0.0004 0.9849 ( 0.0003 0.9840 ( 0.0007
29 29 29 29
Figure 1a Figure 1b Figure 1c Figure 1d
8 9 10 11
24 22 20 18
728 ( 2 709 ( 2 702 ( 2 690 ( 2
KE-E10 at φ ) 0.0073 0.9905 ( 0.0007 0.9880 ( 0.0010 0.9849 ( 0.0006 0.9871 ( 0.0003 0.9838 ( 0.0019 0.9834 ( 0.0004 0.9821 ( 0.0013 0.9802 ( 0.0013
29 29 29 29
Figure 1e Figure 1f Figure 1g Figure 1h
12 13 14 15
10 10 8 5
KE-E20 at φ ) 0.0079 without Ion-Exchange Resins 850 ( 2 ∼1 ∼1 850 ( 2 0.9945 ( 0.0010 0.9954 ( 0.0003 823 ( 2 0.9844 ( 0.0007 0.9809 ( 0.0012 790 ( 2 0.9836 ( 0.0008 0.9810 ( 0.0011
4 4 4 4
Kossel lines photographed on films, we can determine the orientation and the structure of the crystal. Bragg’s condition inside the crystal is represented as
Gm )
2n(φ) cos Rm λ
(2)
using the magnitude Gm of the reciprocal lattice vector Gm and the semiapex angle Rm of a Kossel cone. Here, λ ()532.0 nm in the present experiment) is the wavelength of the incident laser beam in air. The Kossel images were photographed with a cylindrical camera with a diameter of 2Rcam ) 57.3 mm. The colloidal silica particles used in the present work were Seahoster KE-E10 and KE-E20 produced by Nippon Shokubai Co., Ltd. (Osaka, Japan). The average diameters of the spherical colloidal particles were 130 and 170 nm, respectively. The particle size distributions were about σ/D ≈ 5%, as shown in ref 16. Here, D is the particle diameter and σ denotes the standard deviation of the particle diameter distribution. After purification with Milli-Q water and deionization by ion-exchange resin particles, the effective surface charge densities of the silica particles were determined to be 0.14 and 0.19 µC/cm2, respectively, by the method described earlier.16-18 The purified dispersions were introduced into rectangular quartz glass cuvettes (height ) 45 mm; width ) 10 mm; inner thickness ) 1 mm). The lower part of the container was filled with ion-exchange resin particles. To avoid the influence of any ionic strength distribution (caused by relatively slow deionization processes by the ion-exchange resins) and prevent permeation of impurity ions from outside, we also used hermetically sealed quartz glass cuvettes (height ) 45 mm; width ) 10 mm; inner thickness ) 1 mm) with airtight lids, which were used without ionexchange resins (Table 1: experiments 12-15). The dispersions began to show iridescence due to microcrystals formed throughout the container within a few minutes of their introduction into the cuvettes. The cuvettes were kept standing vertically, and the temperature was held at room temperature. In dilute colloidal dispersions (φ below ∼0.01), microcrystals in the upper part soon (16) Yamanaka, J.; Ise, N.; Miyoshi, H.; Yamaguchi, T. Phys. Rev. E 1995, 51, 1276. (17) Yamanaka, J.; Yoshida, H.; Koga, T.; Ise, N.; Hashimoto, T. Phys. Rev. Lett. 1998, 80, 5806. (18) Yoshida, H.; Yamanaka, J.; Koga, T.; Koga, T.; Ise, N.; Hashimoto, T. Langmuir 1999, 15, 2684.
3 3 3
Figure 3a
Figure 4a Figure 4b
disappeared but pillar-shaped crystals grew instead, as shown in Figure 1. The pillars, which are located far away from the resin layer, were found to coexist with the colloidal fluid on the top of the pillars. Figure 3 is a diffraction image obtained from a pillarshaped crystal, which was formed in a KE-E10 silica particle dispersion at φ ) 0.0073 (experiment 2), together with the Kossel lines computer-calculated with assumed Miller indices. The Miller indices were determined by comparing the observed and calculated lines. It can be seen that the three Kossel lines do not meet at a point at points A and B. This indicates the noncubic nature of the crystal, as the following argument shows.12,19 Let us consider the three Kossel lines with indices m ) (hkl), m′ ) (h′k′l′), and m′′ ) (h′′k′′l′′). The corresponding axes to the first two indices are in the horizontal plane. The conditions necessary for these Kossel lines to meet at a point are that the reciprocal lattice vectors Gm, Gm′, and Gm′′ are coplanar
(Gm × Gm′)‚Gm′′ ) 0
(3)
the values of the radius rmm′ of the reflection circle determined by
rmm′ )
|Gm||Gm′||Gm - Gm′| 2|Gm × Gm′|
(4)
are the same for all sets of indices, and n(φ)/λ > rmm′ is satisfied. Generally, the unit vector Kmm′ of the intersection of two Kossel cones with indices m and m′ (the reciprocal lattice vectors are Gm and Gm′, respectively) is given by
Kmm′ )
{
λ 1 1 [|Gm|2Gm′ |G × G | 2|G n(φ) m m′ m × Gm′| 2
|Gm′| Gm] × (Gm × Gm′) (
x( ) n(φ) λ
2
}
2 - rmm′ Gm × Gm′
(5)
inside the colloidal crystal (see Figure 2). Let us now turn to Figure 3. For simplicity, we consider lattice constants a, b, and c, with a and b being in the (19) Chang, S.-L. Multiple Diffraction of X-Rays in Crystals; SpringerVerlag: Berlin, 1984.
Notes
Langmuir, Vol. 20, No. 12, 2004 5143
Figure 3. Kossel image of a colloidal crystal with a noncubic structure. The photograph was taken from a pillar-shaped crystal at an elevation of 22 mm from the bottom of the cuvette on the third day after the dispersion was introduced into the cuvette. Sample: KE-E10 silica particles at φ ) 0.0073.
horizontal plane and c being in the vertical direction. In the case of point A in Figure 3b, where the Kossel lines have indices m ) (200), m′ ) (101), and m′′ ) (101 h ) in the orthorhombic system with lattice constants a, b, and c, the reciprocal lattice vectors are given by
1 1 Gm ) (2rca 0 0 ), Gm′ ) (rca 0 1 ), and c c 1r c rca ) > 0 (6) Gm′′ ) ( ca 0 -1 ) c a As a result, we obtain the inner products between the unit vectors of the intersection as
Kmm′‚Kmm′′ ) 1 -
(1 - r2ca)2 2r2ca
Kmm′‚Km′m′′ ) Kmm′′‚Km′m′′ )
x[
1-
( ) ][
2 λ2 1 + rca n2(φ) 2rcac
2
1-
λ2 c2n2(φ)
(7)
(
)]
n2(φ)
2c2
2 2 λ2 1 + rca + n2(φ) 2c 2 λ2 1 + rca
(8)
When rca ) 1, all the inner products of eqs 7 and 8 are the same and the three Kossel lines meet at a point. In the case of point B in Figure 3b, where the Kossel lines have indices m ) (020), m′ ) (011), and m′′ ) (011 h ), lattice constant a in eqs 7 and 8 is interchangeable with b. A similar result is also obtained for other sets of Kossel lines that do not meet at a point (point C in Figure 3b, for example) in the case of a * c or b * c, but the analyses of points A and B could be done more precisely than those of other points. In the case of Figure 3a, we therefore concluded that body-centered-orthorhombic (bco) crystallites grow with their (110) plane parallel to the cuvette surface, thus the coincidental condition for the cubic system is broken, and we get c/a ) 0.9811 ( 0.0006, c/b ) 0.9813 ( 0.0004, and c ) 708 ( 2 nm.
Other examples of the Kossel lines of a noncubic crystal are given in Figure 4 for a pillar-shaped crystal of a different sample. A vertical contraction was also observed for another neighboring pillar-shaped crystal in the same cuvette, as can be seen from Table 1 by comparing experiments 4-7 with experiments 8-11. The lattice constants observed at various heights in the pillar-shaped crystals are tabulated in Table 1. Three tendencies are noteworthy. First, lattice constant c, and hence a and b, increases with ascending elevation. This is due to a concentration gradient caused by gravitational sedimentation, as observed previously by Crandall and Williams9 and ourselves.8 Second, c is always smaller than a and b. This indicates that the pillar-shaped crystal is compressed in the vertical direction due to gravity. Third, both c/a and c/b tend to increase with height. This is understandable because particles at lower heights are subject to the weights of the particles in the upper parts. The changes in c were ∼6% with reference to the lowest height in experiments 4-11, in which the measurements were conducted on the 29th day after the dispersion was introduced into the cuvette together with the ion-exchange resin particles. On the other hand, those for experiments 1-3 were only 2% on the third day. This difference in the change in c seems to reflect the fact that the ionic impurity level near the bottom of the cuvette was lower than that at elevated positions on the third day as a result of gradual deionization. According to the Sogami theory,20 the potential minimum position shifts toward larger distances as the ionic concentration is lowered. Thus, a larger value (703 nm) was obtained in experiment 3 than that expected from a purely gravitational effect. As time passed, purification proceeded, so that the ionic impurity level was homogeneously lowered by the 29th day and the c value became smaller (690 nm, as shown in experiments 7 and 11). Experiments 12-15, which were carried out on the fourth day for KE-E20 (without ion-exchange resins), showed a comparatively large change (8%) in c, in a small height range between 5 and 10 mm. This appears to be due to the larger diameter of the sample, which is expected
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Notes
Figure 4. Kossel images of a colloidal crystal with a noncubic structure. The specimen is a colloidal dispersion of KE-E20 silica particles with a volume fraction of 0.0079 at (a) 10 mm and (b) 5 mm from the bottom of the dispersion. The photograph was taken on the fourth day after the dispersion was introduced into the container. The scale is given in millimeters.
to demonstrate a larger gravitational influence and hence faster sedimentation. The large c values (790-850 nm) appear to be due to the higher charge density of the sample. Table 1 shows that the lattice constant within a plane also increases with height but at a different rate than it does along the c axis. It is tempting to suggest that the particle distribution is governed by the gravitational sedimentation and the opposing electrostatic interaction, though a quantitative description is not easy. The formation of orthorhombic crystals under normal gravity conditions is clearly due to their elastic constants being small, as has been reported.8,21 It is noteworthy that the cubic-orthorhombic deformation has been found for polystyrene latex particles in an electric field using Kossel line analysis.22 In conclusion, we observed crystals with a bodycentered-orthorhombic structure for KE-E10 and KE-E20 silica particles using the Kossel diffraction method. Pillar(20) Sogami, I. Phys. Lett. A 1983, 96, 199. Sogami, I.; Ise, N. J. Chem. Phys. 1984, 81, 6320. (21) Lindsay, H. M.; Chaikin, P. M. J. Chem. Phys. 1982, 76, 3774. (22) Cladis, P. E.; Garel, T.; Pieranski, P. Phys. Rev. Lett. 1986, 57, 2841. Pieranski, P.; Cladis, P. E. Phys. Rev. A 1987, 35, 355.
shaped crystals contracted vertically, due to the gravitational effect. The distortion of the crystals in the lower part of the dispersion was larger than that in the upper part. The gravitational effect is thus 2-fold. First, the lattice constants increase with height, while the crystal symmetry remains unaffected, which is due to the gravitational sedimentation of the particles. Second, the cubic-orthorhombic deformation takes place due to gravity, as discussed in the present note. The deformation is considered to be dependent on the surface charge density and size of the colloidal particles, their volume fraction, and the bulk modulus of the colloidal crystal. A systematic study is currently in progress. Acknowledgment. We would like to thank the reviewer for helpful suggestions and comments. We thank M. V. Smalley for his careful reading of the manuscript and suggestions. This study is partially funded by “Ground Research for Space Utilization” promoted by Japan Aerospace Exploration Agency (JAXA) and Japan Space Forum (JSF). LA030318F
