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Langmuir 2001, 17, 8010-8015
Measurements of Elastic Constants of Colloidal Silica Crystals by Kossel Line Analysis Tadatomi Shinohara,*,†,‡ Tsuyoshi Yoshiyama,† Ikuo S. Sogami,† Toshiki Konishi,‡ and Norio Ise‡ Department of Physics, Kyoto Sangyo University, Kyoto 603-8555, Japan, and Central Laboratory, Rengo Co., Ltd., Osaka 553-0007, Japan Received March 14, 2001. In Final Form: August 24, 2001 We made structure analyses of colloidal crystals in salt-free silica dispersions with water as dispersant by Kossel line analysis and estimated bulk moduli of static elasticity. Due to the density difference between silica particles and water, there were observed sedimentation effects and small but still detectable gravitational pressure effects on the colloidal silica crystals. Using laser diffraction by a parallel Ar laser beam, we observed intrinsic Kossel images of colloidal crystals to accurately probe symmetry structure. We photographed the Kossel images of colloidal crystals systematically at different heights and found that the crystals keeping cubic structure received uniform compression due to gravity. High precision measurement of a lattice constant as a function of height enabled us to estimate the elastic moduli of the colloidal crystals.
1. Introduction Investigation on ordering formation processes in colloidal dispersions is helpful for understanding the fundamental properties of colloidal crystals and elucidating the electrostatic interparticle interactions of macroions.1 Since the interparticle separation of colloidal crystal is of the same order as the wavelength of light in the visible region, laser diffraction is a convenient method to probe symmetry structures in the colloidal crystals. The purpose of this paper is to observe sedimentation effects and pressure effects on the colloidal silica crystals using the Kossel line analysis.2-4 We use parallel Ar laser beams and observe intrinsic Kossel images of colloidal crystals, which are the diffraction patterns from lattice planes of colloidal particles illuminated by divergent beams produced through random scattering of incident beams at a pointlike source existing inside the dispersion. We photograph the backward Kossel images at various heights since the colloidal dispersion is not transparent in most cases. From these Kossel images, we uniquely determine the crystal symmetry and precisely estimate the lattice constants and the elastic moduli of the colloidal crystals. In contrast to atomic crystals, it is possible to control characteristics of the colloidal crystals by adjusting parameters such as the size, surface charge and concentration of the colloidal particles, and concentration of added salts. Since the colloidal crystal has a fundamental length of about 103-104 times that of the atomic length scale, the physical properties of the colloidal crystal are different from those of the conventional solids. For example, the colloidal crystals show overdamping of the longitudinal sound modes and small elastic constant. The bulk modulus * To whom correspondence should be addressed. E-mail:
[email protected]. † Department of Physics, Kyoto Sangyo University. ‡ Central Laboratory, Rengo Co., Ltd. (1) Sood, A. K. Solid State Phys. 1991, 45, 1-73. (2) Yoshiyama, T.; Sogami, I.; Ise, N. Phys. Rev. Lett. 1984, 53, 21532156. (3) Yoshiyama, T.; Sogami, I. S. Langmuir 1987, 3, 851-853. (4) Yoshiyama, T.; Sogami, I. S. In Ordering and Phase Transitions in Charged Colloids; Arora, A. K., Tata, B. V. R., Eds.; VCH Publisher: New York, 1996; pp 41-68.
of the colloidal crystals lies in the range 10-103 Pa and the particle concentration is approximately 1012-1014 cm-3, while the bulk modulus of metals lies in the range 1010-1012 Pa and the atomic concentration is approximately 1023 cm-3. This means that the ratio of the elastic constant to the particle concentration of the colloidal crystal is of the same order as that of the metal. It is necessary to precisely measure the lattice constants in order to determine the small elastic constant of such a soft colloidal crystal. Due to the density difference of silica particles (2.2 g/cm3) and water (1.0 g/cm3), there is an effect of gravitational force on the particles, when the particle diameter is large. Because of the small size of the particles (∼1000 Å diameter), the gravitational effects are small but still measurable. The results showed that the lattice constants of the colloidal crystals varied as a function of height of the dispersion. Crandall and Williams5 derived first a simple expression relating the height dependence of the lattice parameter in a column of colloidal crystals to its elastic modulus by the light diffraction. Although their finding is of historical importance, the Kossel line analysis in the present paper provides much more precise information of the crystal symmetry and the lattice constants than the Bragg diffraction. 2. Experimental Section The colloidal silica dispersion used in this work was Cataloid SI-80P produced by Catalyst & Chemicals Co., Ltd., Tokyo, Japan. The average radius of the spherical colloidal particles was determined to be 500 Å by ultra-small-angle X-ray scattering.6 After purification and deionization by ion-exchange resin particles, the effective surface charge density of the silica particles was determined as 0.2 µC/cm2.7,8 By dilution with a Milli-Q water, we obtained several semidilute specimens of volume fractions 0.007-0.05. These specimens were introduced into quartz glass cuvettes (45 mm height,10 mm width, 1 mm inner thickness) (5) Crandall, R. S.; Williams, R. Science 1977, 198, 293-295. Williams, R.; Crandall, R. S. Phys. Lett. 1974, 48A, 225-226. (6) Konishi, T.; Ise, N. Phys. Rev. B 1998, 57, 2655-2658. (7) Yamanaka, J.; Yoshida, H.; Koga, T.; Ise, N.; Hashimoto, T. Phys. Rev. Lett. 1998, 80, 5806-5809. (8) Yoshida, H.; Yamanaka, J.; Koga, T.; Koga, T.; Ise, N.; Hashimoto, T. Langmuir 1999, 15, 2684-2702.
10.1021/la010393v CCC: $20.00 © 2001 American Chemical Society Published on Web 11/30/2001
Elastic Constants of Colloidal Silica Crystals
Langmuir, Vol. 17, No. 26, 2001 8011
Figure 1. Kossel image for the specimens with φ ) 0.0109 at (a) 2 mm from the bottom of the dispersion and (b) 13 mm from the bottom of the dispersion photographed 121 days later. together with ion-exchange resin particles. The upper and lower parts of the containers were filled with the resin particles. The dispersion began to show locally iridescence in several minutes, while the time for the dispersion to reach gravitational equilibrium is of the order of a few hours. The dispersion became iridescent throughout the container in a few hours. The cuvettes were allowed to stand vertically. The temperature was held at room temperature. The Kossel images were obtained as described before.2-4 The specimen cuvette was mounted upright on a goniometer head, and the incident Ar laser beam (wavelength 4880 Å) was incident normally to the wide surface of the cuvette. Backward Kossel images were photographed on Fuji FG films with a modified rotating crystal camera consisting of a cylindrical film holder with diameter 2Rcam ) 57.3 mm, a collimation slit of 1 mm diameter, and the goniometer head. Time of exposure was about 3 s. Figure 1 shows the Kossel images for a dispersion which were photographed at different heights on the same day. Miller indices and spacings of lattice planes were obtained for each Kossel line, and the crystal structure was determined uniquely from symmetry of the diffraction pattern. The Miller indices must be assigned to the Kossel lines so that all indices are even or odd integers for crystals of face-centered cubic (fcc) structure and the sum of all indices is even for crystals of body-centered cubic (bcc) structure. Thus the Kossel diffraction image provided accurate three-dimensional information on the ordering formation inside colloidal dispersions. Measurements of the coordinates of three points on a Kossel line determine the directions of beams on the Kossel cone relative to the incident laser beam and the semiapex angle R of the Kossel
cone.4 We assumed the pointlike source in the colloidal dispersion was generated at the boundary on the side of the laser source between the dispersion and container surface and the center of the cylindrical camera is set at the position of the pointlike source. 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 the refractive effect by using Snell’s law, we adopted the formula for the index of refraction of colloidal dispersion n(φ0) proposed by Hiltner and Krieger9
n(φ0) ) nwater(1 - φ0) + nparticleφ0
(1)
where nwater and nparticle are indices of refraction of water and silica particle (nwater ) 1.33 and nparticle ) 1.4710,11). φ0 is the volume fraction of the dispersion. Photographed Kossel lines were deformed owing to effects shown in the Appendix. We regarded these corrections as experimental errors.
3. Results and Discussion The colloidal dispersions were studied in quartz glass cuvettes with the shape of a rectangular prism. The spatial (9) Hiltner, P. A.; Krieger, I. M. J. Phys. Chem. 1969, 73, 23862389. (10) Iler, R. K. The Chemistry of Silica; John Wiley & Sons: New York, 1979; p 19. (11) Riese, D. O.; Vos, W. L.; Wegdam, G. H.; Poelwijk, F. J. Phys. Rev. E 2000, 61, 1676-1680.
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Table 1. Time Evolution of the Lattice Constant and the Bulk Modulus at O0 ) 0.0109 days
av lattice constant (Å)
bulk modulus (Pa)
4552 4550 4543
+1 42-9 +16 38-2 +3 36-5
45 75 121
coordinate system of this analysis consisted of an x axis being normal to the wide surface of the cuvette, a y axis being horizontal and parallel to the cuvette surface, and a z axis with a vertically upward direction. The direction of the gravity is along the z axis. In both images in Figure 1, three Kossel lines with indices (200), (101), and (101h ) ((020), (011), and (011h )) intersect at one point. This fact confirms that the crystals at different heights in the cuvette have the cubic bcc structure. Henceforth we make Kossel line analyses by assuming a bcc structure for all crystals in the present paper. It is of interest to compare our results with those reported by Williams and Crandall.5 These authors concluded bcc and fcc symmetries at lower and higher concentrations, respectively. However, it seems that Williams and Crandall came to the conclusion by comparing the observed interparticle spacing with that calculated for a bcc or fcc structure (See Table 1 in ref 5): the relative position of the diffraction peak positions characteristic for these crystal structures was not given. On the other hand, the Kossel line analysis allows us to infer the lattice symmetry in an almost unique manner. Furthermore, the error limits in the lattice constant were about 10% in the Williams and Crandall work. In our case it was possible to detect a 0.2% difference in the lattice constant by Kossel images. These considerations show that our Kossel line analysis represents a large improvement over the diffraction technique employed by Williams and Crandall. When the sedimentation effect can be ignored, the volume fraction φ of the colloidal particles is defined by φ ) (4πR3/3)n, where R is the radius of the colloidal particle and n is the particle number density. In the colloidal crystal the nearest-neighbor distance 2D0 of the particles is expressed by
2D0 )
{
a0/21/2 for fcc structure a031/2/2 for bcc structure
}
(2)
in terms of the lattice constant a0 of the crystal. The sedimentation effect leads to inhomogeneous distribution of particles and the volume fraction becomes a function of the vertical coordinate z. In the case where the whole dispersion is considered to be filled with a single crystal, at any given height z from the bottom of the dispersion part, the lattice constant a0(z) of the colloidal crystal and the volume fraction φ(z)are related by
a0(z) )
{
[16π/3φ(z)]1/3R for fcc structure [8π/3φ(z)]1/3R for bcc structure
}
(3)
We observed crystals with a cubic structure throughout the container, and no anisotropy of their lattice constants in both horizontal and vertical directions was detected in the present experiments. The lattice constants of the colloidal crystals were determined as a function of height of the dispersion. The volume of unit cell at the height z is represented as a3(z) using the lattice constant a(z). The gravitational force and the elastic restoring force are
related by12
Feffgφ(z) dz )
a3(z + dz) - a3(z) a3(z)
B
(4)
where Feff ) Fparticle - Fwater is the effective density of the particle and g is the acceleration due to gravity. B is the bulk modulus of static elasticity of the dispersion, and the height dependence of B has not been explicitly put in. Equation 4 can be rewritten as
d(a3(z)) )
Feffgφ(z) 3 a (z) dz B
(5)
The initial mean volume fraction φ0 of the dispersion is given as
hφ0 )
∫0h φ(z) dz
(6)
where h is the total height of the dispersion in the container. The crystal structure, the lattice constant, and the orientation of crystals in the salt-free dispersions were determined by Kossel line analysis. The results showed that the bcc crystallites grew with the (110) plane being parallel to the cuvette surface. The lattice constants had a tendency to decrease with time, as shown in Table 1. It was further shown that the lattice constants of the colloidal crystals increased from the lower to the upper parts of the cuvette and that the orientations of the crystals in the lower part tended to deviate from those in the upper part. These features which can be interpreted as effects of gravity appeared more prominent for the dilute dispersions. As shown from Figure 2a, the averaged lattice constants were found to be smaller than the values calculated by eq 3 under the assumption of the uniform space filling distribution, but the differences were small. The average was taken for observed distances at z ) 11 mm (φ0 ) 0.0109), z ) 14 mm (φ0 ) 0.0294), z ) 14 mm (φ0 ) 0.0360), z ) 16 mm (φ0 ) 0.0446), and z ) 14 mm (φ0 ) 0.0474). Thus the small difference mentioned cannot be taken as equivalent to the previously reported inequality relation (between the interparticle spacing and the average spacing) under the gravity-insensitive condition. Assuming C ) φ(z)a3(z) is constant, we get
a3(z) ) a3(0) + bz
(7)
from eq 5. Here, b ) FeffgC/B. It should be mentioned that the experimental data can also be expressed by a linear relation, instead of the cubic relation (eq 7). From existing experimental data, it is difficult to judge which of the relations are precisely valid. Further experiments are necessary and in progress. Figure 3 shows variation of the cube of the lattice constant (in Å) of the colloidal crystal as a function of the height z (in mm) from the bottom of the dispersion. The height z ranges from 0 to 22 mm. Error bars of Figure 3 were derived from the consideration in the Appendix. The specimen originally had a volume fraction φ0 of 0.0109. The observation was made 121 days after crystallization. From Figure 3, it is seen that the cube of the lattice constant a(z) follows eq 7. From eqs 6 and 7 we get (12) Kesavamoorthy, R.; Arora, A. K. J. Phys. 1985, A18, 33893398; 1986, C19, 2833-2846.
Elastic Constants of Colloidal Silica Crystals
Langmuir, Vol. 17, No. 26, 2001 8013 Table 2. Bulk Modulus for Various Volume Fractions
Figure 2. The averaged lattice constants (in Å) of the sample with the volume fractions 0.0109, 0.0294, 0.0360, 0.0446, and 0.0474 measured at 121, 85, 66, 77, and 85 days after crystallization, respectively. The curves show (a) eq 3 as a function of φ and (b) eq 12 for the charge number of the silica particle |Z| ) 790 (solid curve), 680 (dotted curve), and 900 (dot-dashed curve).
vol. fraction
particle concn (cm-3)
days
bulk modulus (Pa)
0.0073 0.0109 0.0360 0.0446 0.0474
1.4 × 2.1 × 1013 6.9 × 1013 8.5 × 1013 9.1 × 1013
113 121 66 77 85
20 36 191 172 224
1013
Pa for the specimen of φ0 ) 0.0109, 121 days after the preparation. The value of b from eq 7 was obtained from the slope of the best fit in Figure 3. Table 1 represents the time evolution of the bulk modulus for a given dispersion. At an early stage (say, several days after the preparation), the slope of the lattice constant a(z) versus the height z was so small, being within the experimental errors, that it could not be determined. As time elapsed, the slope took a definite value and the errors decreased. Therefore it was necessary to wait many days (about 100 days at a volume fraction of 0.0109) to obtain a reliable value for the elastic moduli of the colloidal crystals. It turned out that the elastic constant of colloidal crystals was very small compared with that of metals ranging 1010-1012 Pa. In colloidal crystals the particle concentration is approximately 1012-1014 cm-3, while the atomic concentration is approximately 1023 cm-3 in metals. Thus, it was ascertained that the ratio of the elastic constant to the particle concentration of the colloidal crystal is of the same order as that of the metal. Table 2 seems to indicate that the colloidal crystal becomes soft with the decrease of the concentration. It is worthwhile to emphasize that we determined the elastic constant of colloidal crystal by using a principle (diffraction images) quite different from the methods of usual measurements. Since the bulk modulus is related to the second derivative of the interaction potential energy, it is of some interest to discuss here about the interparticle interaction in colloidal dispersions. Various properties of colloidal dispersions have been studied using electrostatic repulsive potentials such as the Yukawa potential.13 The crystal structure is interpreted to be formed as the AlderWainwright effect14 with the colloidal particle dressed with an electrical double layer15,16 and the nearest-neighbor distance 2D(z) is described by
2D(z) ) 2R + 2Rκ(z)-1
(R e 1)
(10)
In the present case, the Debye screening parameter κ is defined by
κ2(z) )
Figure 3. The cube of the lattice constant a(z) (in Å) of the colloidal crystal as a function of the height z (in mm) from the bottom of the dispersion. The height z ranges from 0 to 22 mm.
φ(z) )
bhφ0
1 loge[1 + bh/a3(0)] a3(0) + bz
(8)
The concentration difference between the upper part and the bottom part of the dispersion lies in the range of the shift of at most (5% from φ0. B is obtained as
B)
Feffhgφ0 loge[1 + bh/a3(0)]
(9)
where Feffg ) 1.18 × 104 N/m3. As a result, we get B ) 36
4πe2 (n (z) + ns(z) + n0(z)) kT c
(11)
for monovalent small ions, where e, , and kT are the electronic charge, the dielectric constant, and the thermal energy, respectively. nc is the concentration of diffusible counterions, ns is the concentration of foreign salt, and n0 is the concentration of both H+ and OH- ions from the dissociation of water. From counterion dominance assumption, salt-free condition, and global charge neutrality (13) Chaikin, P. M.; Pincus, P.; Alexander, S.; Hone, D. J. Colloid Interface Sci. 1982, 89, 555-562. Shih, W. Y.; Aksay, I. A.; Kikuchi, R. J. Chem. Phys. 1987, 86, 5127-5132. Thirumalai, D. J. Phys. Chem. 1989, 93, 5637-5644. (14) Alder, B. J.; Wainwright, T. E. J. Chem. Phys. 1957, 27, 12081209. Alder, B. J.; Hoover, W. G.; Young, D. A. J. Chem. Phys. 1968, 49, 3688-3696. Hoover, W. G.; Gray, S. G.; Johnson, K. W. J. Chem. Phys. 1971, 55, 1128-1136. (15) Kose, A.; Hachisu, S. J. Colloid Interface Sci. 1974, 46, 460469. Hachisu, S.; Kobayashi, Y. J. Colloid Interface Sci. 1974, 46, 470476. (16) Wadachi, M.; Toda, M. J. Phys. Soc. Jpn. 1972, 32, 1147.
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condition, we get |Z| n(z) ) nc(z) and ns, n0 ∼ 0. Here n(z) is the particle number density at height z and Ze is surface charge of the colloidal silica particle. It seems that the observed tendency for 2D(z) to increase with z agrees well with eq 10. However, as pointed out earlier by Ito et al.,17 the basic concept of eq 10 was not adopted by Perrin,18 who obtained the correct Avogadro number with 2R. The use of 2D (instead of 2R) led to a physically unacceptable situation that the universal constant varied with ionic strength. Therefore we would not take the repulsion-only assumption.19 On the other hand, Sogami and Ise20-22 obtained an effective pair potential UG containing attractive and repulsive components and derived the interparticle distance Rmin of macroions by minimizing the potential UG. By equating Rmin to 2D, we get
2D(z) ) [C(z) + 1 + {(C(z) + 1)(C(z) + 3)}1/2]/κ(z) (12) where
C(z) ) κ(z)R coth κ(z)R
Each Kossel pattern was photographed on a film by the cylindrical camera with a camera radius Rcam. On the film we take the coordinate system with a horizontal X axis and a vertical Y axis. The origin is the point at which the incident laser beam passes through the film. Thus we fix the position of a point on a Kossel line on the film by its coordinates (X, Y). The position (X, Y) is plotted using a computer coordinating area curvimeter, X-PLAN 360i, produced by Ushikata Mfg. Co., Ltd., Tokyo, Japan. We analyzed the properties of electromagnetic waves inside the crystal under the approximation that incident waves diverging from a distant source fell on the crystal as plane waves.23 A beam on a Kossel cone in the colloidal crystal (air) is described by the angle i(r) between the x axis and the beam, where the angle i(r) is decomposed into the horizontal component ih(rh) and the vertical component iv(rv). Here cos i ) cos ih cos iv(cos r ) cos rh cos rv). Then this beam travels along the vector
(
(13)
Yoshida et al.8 found the following relationship between the effective charge number Ze and the analytical charge number Za for SI-80P silica dispersion:
log10 Ze ) 0.52 log10 Za + 1.16
Appendix: Error Estimation of Lattice Constant
(14)
This implies Za ) 950 for Ze ) 510. In Figure 2b we drew curves of eq 12 for |Z| ) 680, 790, and 900. The agreement of the theoretical 2D with observation using |Z| < 950 is gratifying and seems to justify the Sogami potential. 4. Conclusions Structure analysis was made of colloidal silica crystals under salt-free conditions using the Kossel line technique, which made high-precision determination of the lattice constant possible. Because of the density difference between silica particle and water (dispersant), small gravitational sedimentation effects were observed. By analysis of the intrinsic Kossel diffraction patterns, the cube of the lattice constant was found to increase with increasing height (0-22 mm) from the bottom of the dispersion cell. From there the elastic modulus of the colloidal crystals was determined. The ratio of the elastic constant to the particle number concentration was of the same order as that of the metals. The elastic moduli of the crystals of silica colloids thus measured are larger, by 10-100, than those of the crystals of polystyrene lattices obtained by Crandall and Williams.5 It should be noticed that the crystal structure was proved to be bcc and the crystal contraction was uniform in the present experiment. Acknowledgment. We wish to express our sincere thanks to Dr. Junpei Yamanaka, Faculty of Pharmaceutical Sciences, Nagoya City University, Nagoya, for advice. This study is funded in part by the fund for “Ground Research for Space Utilization” by NASDA and Japan Space Forum. (17) Ito, K.; Ieki, T.; Ise, N. Langmuir 1992, 8, 2952-2956. (18) Perrin J. Les Atomes; Libraire Felix Alcan: Paris, 1913; Chapter 4. (19) This does not mean that the Alder-Wainwright simulation14 is wrong. What is unacceptable is the use of the effective hard sphere concept (eq 10). (20) Sogami, I. Phys. Lett. 1983, 96A, 199-203. (21) Sogami, I.; Ise, N. J. Chem. Phys. 1984, 81, 6320-6332. (22) Ito, K.; Sumaru, K.; Ise, N. Phys. Rev. B 1992, 46, 3105-3107.
cos i sin ih cos iv sin iv
)
(A.1)
inside the crystal. From the positions of Kossel lines recorded on the film of the cylindrical camera, the values of (i;ih,iv) were obtained by means of ref 4. The three sets of estimated angle (i;ih,iv) determine the lattice constant of the crystal. Systematic errors of the estimated lattice constants mainly originated from the following three causes. (1) Refraction at the boundary between the colloidal crystal surface and an air occurs twice, that is, at the interface between the colloidal crystal surface and the cuvette’s inner surface and at the interface between the cuvette’s outer surface and the air. Although we assumed that the cuvette’s quartz plate is infinitely thin, we must consider the effect of the thickness of the container (wide surface about 1.2 mm thick, refractive index of quartz glass 1.5510). (2) We assumed that a pointlike source is produced at the boundary between the colloidal dispersion and the cuvette’s inner surface. However there is another possibility that the pointlike source exists at other points in the dispersion (at most 0.2 mm deviation). (3) The specimen cuvette is fixed on a goniometer head by clay. In section 2, we assumed the specimen cuvette was fixed so as to match the center of cylindrical camera with the position of the pointlike source which was generated at the boundary on the side of laser source between dispersion and container surface. The specimen cuvette could be settled with the deviation ((1.0 mm) in the direction of the incident laser beam from that appropriate position. As a result of these factors, we assumed a beam of a Kossel cone traveled with the direction (r;rh,rv) from the starting point (x0, y0, z0)on the air. Then this beam arrived at the position (X, Y) of the film as follows:
X ) Rcam(rh + θ)
(A.2)
Y ) z0 + (Rcam cos θ - x0 cos rh - y0 sin rh) tan rv (A.3) where (23) Yoshiyama, T.; Sogami, I. S. Phys. Rev. Lett. 1986, 56, 16091612.
Elastic Constants of Colloidal Silica Crystals
sin θ )
1 (-x0 sin rh + y0 cos rh) Rcam
Langmuir, Vol. 17, No. 26, 2001 8015
(A.4)
The angle i was corrected so that the beam from the pointlike source inside the dispersion reached to the observed position (X, Y) on film. Like the closed curve of Figure 4a, when the Kossel line was photographed on both (left and right) sides of film, the Kossel image was magnified (or reduced) on film. Like the closed curves of Figure 4b, when the Kossel lines were photographed on one (left or right) side of the film, a little parallel displacement of the beams merely happened. In parts a and b of Figure 4, errors lie within a tolerance of (1) 0.11% -0.22 % and and 0.05%, (2) -0.03% and -0.005%, and (3) +0.24 (0.06%, respectively. Errors were also derived from the uncertainty at fixing a coordinate system on the film. For estimation of rotation angle δ ()(10°) of the coordinate axis round origin and deviation X0 ()(2 mm), Y0 ()(2 mm) from origin for the right coordinate system, the corrected position (Xcor, Ycor) and the measured position (Xmea, Ymea) are related as
Xcor ) Xmea cos δ + Ymea sin δ + X0
(A.5)
Ycor ) -Xmea sin δ + Ymea cos δ + X0
(A.6)
In parts a and b of Figure 4, the errors from this effect -0.27 -0.28 % and +0.13 %, respectively. were in the range of +0.24 The thickness of the photographed Kossel line and the plotting by naked eye also cause errors. In this case, we repeated the plotting of the same line several times and regarded the variance of the estimated lattice constants as errors. In Figure 4a, the margin of error from this contribution is (0.018%. In Figure 3 we expressed these errors of the lattice constant as vertical error bars. We estimated the measured errors at (0.5 mm of the height from the bottom of the dispersion and used horizontal error bars. In addition, errors may originate from the factors that the incident laser beam was not normal to the container wide surface and that the film did not adhere closely to the cylindrical camera. Further, the uncertainties in the
Figure 4. Kossel image for the specimens with (a) φ ) 0.0073 at 10 mm from the bottom of the dispersion photographed 113 days later and (b) φ ) 0.0109 at 15 mm from the bottom of the dispersion photographed 75 days later.
particle concentration and the refractive index of the colloidal silica dispersion might be error sources, but these errors are negligibly small in comparison with the errors discussed above. LA010393V
