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Kinetic Analyses of Colloidal Crystallization in a Wide Range of Sphere Concentrations as Studied by Reflection Spectroscopy
Journal of Colloid and Interface Science 228, 151–156 (2000) doi:10.1006/jcis.2000.6933, available online at http://www.idealibrary.com on
Kinetic Analyses of Colloidal Crystallization in a Wide Range of Sphere Concentrations as Studied by Reflection Spectroscopy Tsuneo Okubo1 and Hisanori Ishiki Department of Applied Chemistry and Graduate School of Materials Science, Gifu University, Yanagido 1-1, Gifu 501-1193, Japan Received January 24, 2000; accepted April 24, 2000
The nucleation and growth rates in the colloidal crystallization of silica spheres (103 nm in diameter) from 0.006 to 0.04 in volume fraction (φ) have been measured by reflection spectroscopy. Kinetics of the crystallization has been discussed in a wide sphere concentration range (from φ = 0.0005 to φ = 0.04) using the data of this work and the previous work (110 nm in diameter) in exhaustively deionized aqueous suspensions. The induction period for nucleation decreases sharply as the sphere concentration increases. The nucleation rate increases substantially from 1 × 10−3 to 1 × 107 mm−3 s−1 when φ increases from 0.0005 to 0.04. The crystal growth process consists of the fast growing step toward metastable crystals (rate v 1 ) and slow growth accompanied with the reorientation toward stable ones (rate v 2 ). The v 1 values increase first from 5 to 20 µm/s and then turn back to 5 µm/s after passing a maximum. v 1 above φ = 0.01 remains at 5 µm/s and is insensitive to sphere concentration. The slow step is observed in the high-sphere concentrations only, and v 2 decreases sharply from 3 µm/s to 0.7 nm/s when sphere concentration increases from 0.004 to 0.04 in volume fraction. Importance of the electrostatic intersphere repulsion by overlapping of the electrical double layers and the cooperative and synchronized fluctuation of colloidal spheres in the crystallization processes are supported strongly. °C 2000 Academic Press Key Words: colloidal crystal; nucleation rate; crystal growth rate; reflection spectroscopy; colloidal silica spheres.
1. INTRODUCTION
Recently, keen attention has been paid to the structural and dynamic properties of colloidal crystals (1–13). Suspensions showing crystal-like structures are ideal for model studies of crystals. Furthermore, phase transition phenomena of crystallization or melting takes place sharply. Several researchers have studied the mechanism of colloidal crystallization for two groups of colloidal crystals: (a) diluted and deionized aqueous suspensions, and (b) concentrated suspensions in the refractive index matched organic solvents. The former are very convenient models for both soft- and hard-sphere systems depending on the ionic concentrations of the suspension, i.e., soft crystals in the
1
To whom correspondence should be addressed.
completely deionized state and hard crystals in the presence of a rather large amount of sodium chloride, for example. Colloidal crystallization takes place for monodispersed colloidal particles in suspension. Many researchers have clarified that the colloidal crystals are formed by Brownian movement of colloidal particles and the interparticle electrostatic repulsion (1–13). In other words, colloidal particles form crystal-like distribution automatically with the help of Brownian movement of the particles for maximizing packing density and/or minimizing dead space. The lattice spacing of colloidal crystals changes by the particle concentration exclusively, which supports strongly the importance of the interparticle repulsion induced by Brownian movement of particles themselves and also the electrostatic interparticle repulsive forces. We have studied the nucleation and crystallization processes in the exhaustively deionized and highly diluted aqueous suspensions from the sharpening of the reflection peak (14–17). The crystallization process was unexpectedly fast. In the previous papers (16, 17), nucleation rates in aqueous media increased drastically as sphere concentration increased. Crystal growth rates also increased substantially as the sphere concentration increased, but only in a narrow concentrstion range. Furthermore, nucleation and crystallization rates were very sensitive to the degree of deionization of the aqueous suspensions. Recently, we further studied the kinetics of colloidal crystallization in a sinusoidal electric field (18), in the presence of salt (19), and in alcoholic organic solvents (20). Microgravity experiments using aircraft have been made for the kinetic analyses of colloidal crystallization (1–23). Recently, several researchers reported that crystallization rates for the hard-sphere system increase in diluted concentrations and then decrease after passing a maximum as the sphere concentration increases (24–26). In this paper, colloidal crystallization rates in the soft-sphere colloidal systems have been investigated in detail over a very wide range of sphere concentrations (from 0.0005 to 0.05 in volume fraction) to compare the kinetic data for the soft-sphere and hard-sphere systems. The crystallization rates of the former are, however, too fast to be followed by the conventional inverted mixing method. Thus, in the present work stopped-flowtype reflection spectroscopy has been applied for the kinetic analyses.
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2. MATERIALS AND METHODS
2.1. Materials Colloidal silica spheres of CS-82 were a gift from Catalyst & Chemicals Ind. Co. (Tokyo). The diameter (d0 ), standard deviation (δ) from the mean diameter, and polydispersity index (δ/d0 ) were 103 nm, 13.2 nm, and 0.13, respectively. The values of d0 and δ were determined by electron microscopy. Charge density of the spheres was determined to be 0.38 µC/cm2 for strongly acidic groups by conductometric titration with a Horiba model DS-14 conductivity meter. The sphere sample was carefully purified several times using an ultrafiltration cell (model 202, membrane: Diaflo-XM300, Amicon Co.). Then, the sample was treated on a mixed bed of cation- and anion-exchange resins [Bio-Rad, AG501-X8 (D), 20–50 mesh] more than 4 years before use since the newly produced silica spheres always released a considerable amount of alkali ions from the sphere surfaces for a long period of time. Water used for the purification and for suspension preparation was purified by a Milli-Q reagent grade system (Milli-RO5 plus and Milli-Q plus, Millipore Co., Bedford, MA). 2.2. Reflection Spectroscopy The cell system used, which was the same as that reported previously (21), consists of a quartz observation cell (40 × 10 × 2 mm), a column of cation- and anion-exchange resins (Bio-Rad), and a peristaltic pump (Masterflex 7524-10, Illinois) as shown in Fig. 1. The pump circulates the colloidal suspensions first to the resin column and then to the observation cell to deionize the suspension continuously. The flow rate was usually 3 ml/min. The rate was increased to 9 ml/min before the measurements were made. A light beam from a halogen lamp (Hayashi LA-150SX, Tokyo) hits the cell wall through a Y-type optical fiber cable and the reflection spectra at the angle of 180◦ are taken on a photonic multichannel analyzer (PMA-50, Hamamatsu Photonics, Hamamatsu). When the colloidal suspension is passed into the narrow observation cell, the crystals are melted away by the shear flow. After the flow was stopped nucleation and crystallization started. These processes were followed from the growth of the Bragg reflection peaks after a certain induc-
tion time. In our experimental conditions the suspension in the flow cell (2-mm path length) was transparent. Thus, the reflection spectroscopy in our experiments affords information on the crystal growth processes over the whole depth of the suspension in the cell. 2.3. Determination of the Nucleation and Crystal Growth Rates Most kinetic measurements on colloidal crystallization including this work have observed an induction period, after which the crystal growth starts especially in diluted suspensions. This observation supports the fact that the kinetics of colloidal crystallization is explainable by the classical diffusive crystallization theory including nucleation and crystal growth processes (17, 27, 28). The number of nuclei that germinate per unit time, the nucleation rate, vn , is given by vn = Nn /ti ,
[1]
where Nn is the total number of nuclei which are formed during the nucleation process and ti is the induction period. Here, we assume that the number of nuclei equals the number of single crystals formed. The number of sphere particles per mean size of single crystal, Nc , will be given for a cubic lattice by Nc =
√
2L 3 /l 3 ,
[2]
where L and l are the mean size of single crystals formed and the nearest-neighbor intersphere distance, respectively. The total number of colloidal spheres (NT ) in a unit volume is φ/[(4/3)π(d0 /2)3 ]. Then, vn is given by √ vn = NT /Nc ti = φl 3 /[(4 2/3)π (d0 /2)3 L 3 ti ].
[3]
The size of the colloidal single crystals from the homogeneous nucleation, L is estimated from the peak intensity of the reflection spectra (24), I ∝ Ncryst L 3 ∝ L 3 ,
[4]
where Ncryst is the number of single crystals in the reflecting volume, which is directly proportional to the number concentration of crystals in the final stages of the crystallization process, being equal to the total number of nuclei formed in the whole course of crystallization. The final size of the crystals formed was determined independently using a CCD microscope (type VH-5910, Keyence, Osaka). The first crystal growth rate of a metastable crystal, v1 is given by (16) v1 = v1∞ [1 − exp(−σ )]. FIG. 1.
Schematic representation of the apparatus.
[5]
Here, σ is the relative supersaturation given by (φ − φc )/φc . φc
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is the critical concentration of melting. v1∞ is the maximum crystal growth rate. Equation [5] is further simplified as v1 = v1∞ − v1∞ φc /φ.
[6]
Equation [6] means that the growth rate should decrease linearly as the reciprocal sphere concentration increases and was supported experimentally, but only in a very low-sphere concentration range (16). 3. RESULTS AND DISCUSSION
3.1. Reflection Spectroscopy Typical reflection spectra in the course of crystallization are shown in Fig. 2 at φ = 0.02. The reflection peak became sharp with time in the course of crystallization, which corresponds to an increase in crystal size. The background intensity decreased in the course of crystallization, which is ascribed to the diminution in the multiple scattering of the suspension in the course of crystallization. Figure 3 shows the peak shifts in the course of crystallization for the same sphere samples shown in Fig. 2. Assignment of the crystal lattice structure, whether the lattices are face-centeredcubic or body-centered-cubic, could not be made since the spectra gave the single peaks only. Clearly, the peak wavelength, λ, decreased in the course of crystallization. This peak shift reflects the fact that metastable and loose crystals are formed in the first crystallization step and then the crystals become stable and compact ones. Figure 4 shows the λp values observed at the final stage of crystallization in this work. The nearest-neighbor intersphere distance, l, is determined from λp using Eq. [7] in an aqueous suspension at 25◦ C (29).
FIG. 3. Change in the reflection peak wavelengh in the course of crystallization of CS-82 spheres at 25◦ C. s, φ = 0.004; X, 0.008; 1, 0.02.
tance, l0 , on the other hand, is obtained from Eq. [8] (29), l0 = 0.904d0 φ −1/3 ,
[8]
where d0 is the diameter of the spheres, 103 nm in this work. The solid curve in the figure shows the l0 values thus calculated. Excellent agreement between l and l0 demonstrates that the lattice spacing of the colloidal crystal changes by the sphere concentration exclusively and, furthermore, the crystals are formed by the intersphere repulsion induced by Brownian movement of spheres themselves and also by the electrostatic intersphere repulsive forces. 3.2. Nucleation Process
The calculated values of the nearest-neigher intersphere dis-
Figure 5 shows the induction periods (ti ) determined in this work from the intersection of the slope line and the time axis in the reflection peak intensity vs t plots (see Figs. 7 and 8, for example). ti increased sharply as the sphere concentration decreased. The nucleation rate, vn was estimated using Eqs. [1]–[3] and observed ti . In Fig. 6 the vn values obtained in the previous work (16, 17) are also cited by triangles and crosses with the values in this work denoted by open circles, respectively.
FIG. 2. Reflection spectra in the course of crystallization of CS-82 spheres at 25◦ C. φ = 0.02, Curve 1,0 s after stopped flow; 2, 4.0 s; 3, 8.0 s; 4, 12.0 s; 5, 16.0 s; 6, 19.4 s.
FIG. 4. Reflection peak wavelength of colloidal crystals formed finally of CS-82 spheres as a function of sphere concentration at 25◦ C. s, observed; -------, calculated.
l = 0.460λp
[7]
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FIG. 5. Induction period of colloidal crystallization of CS-82 spheres at 25◦ C. Large circles are the mean of the repeated experimental values shown by small circles.
The nucleation rate increased substantially as the sphere concentration increased. As is clear in Fig. 6, vn increased by 1010 fold when φ increased by 102 -fold! Thus, an increase in sphere concentration is followed by an increase in vn and number of nuclei (Nn ) and resulted in a decrease in the size of single crystals formed as discussed by Dhont et al. (24). It should be mentioned here that the size of single crystals, which is observed with the naked eyes, is polydisperse as has often been observed (30). Therefore, the clear-cut separation of the nucleation step from the crystallization process will be difficult, and the nucleation reaction may remain operative, even in the crystal growth period. 3.3. Crystal Growth Process Figure 7 shows typical examples of the time dependencies of the corrected peak intensities, which were obtained from the peak intensity observed subtracted by the background intensity. Intensity vs time curves show S-shaped characteristics, especially at low-sphere concentrations. The crystal growth rates, v1 and v2 , were evaluated from the cubed-root plot of the peak intensity using Eq. [4]. Typical plots are shown in Fig. 8.
FIG. 6. Nucleation rates of CS-82 (s: this work) and CS-91 (X, Ref. (25); 1, Ref. (24)) spheres plotted as a function of sphere concentration at 25◦ C.
FIG. 7. Reflection peak intensity in the course of crystallization of CS-82 spheres at 25◦ C. ———, φ = 0.004; ·········, 0.008; ------, 0.02.
Before calculation of the first and second growth rates, v1 and v2 , the apparent growth rates, v10 and v20 , were determined as the reciprocal periods where the initial linear lines intersect two horizontal lines, giving initial and final intensities, 1/(tf − ti ), where ti is the induction time. In the calculation of v1 and v2 , the final mean size of single crystals must be known. CCD camera and close-up photography were used to measure size, L, and the results are shown in Fig. 9. L was largest when the sphere concentration was slightly higher than the critical concentration of melting and decreased sharply when the sphere concentration increased. This dependency of L with φ has been observed often before [11, 15, 30, 31], though the sphere concentration where the size becomes largest depends strongly on the degree of deionization of the suspensions. Figure 10 shows the v1 values thus evaluated from v10 multiplied by L. Again, the experimental errors are rather large, as is usual for the kinetic parameters of soft colloidal crystals, which is quite sensitive to the degree of deionization. However, it is clear that v1 increases first from 5 to 20 µm/s and then decreases back to 5 µm/s, passing a maximum. The v1 values above φ = 0.01 remain at 5 µm/s and are quite insensitive to sphere concentration. It should be mentioned here that the v1
FIG. 8. Cube roots of the reflection peak intensity in the course of crystallization of CS-82 spheres at 25◦ C. s, φ = 0.02.
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FIG. 9. Size of crystals formed finally of CS-82 (s, this work) and CS-91 (X, Ref. (17); 1, Ref. (16) spheres plotted as a function of sphere concentration at 25◦ C.
values in the diluted suspensions decrease linearly as the reciprocal concentration of spheres increases (16). A clear-cut explanation for the crystallization rate first going up and then going down as the sphere concentration increases is not obtained yet. However, the main course will be the dynamic nature of phase transition; i.e., spheres in the colloidal lattices interact with each other strongly and further in a cooperative manner through the electrical double layers. This dynamic phase transition is different from the classical one, which is explained by the translational diffusion of monomers. It should be noted here that the importance of the dynamic phase transition has been reported for colloidal crystals (32) and other crystals such as metals (33) and polymers (34). Figure 11 shows the plots of log v2 versus φ, which demonstrates that the slow step of crystallization decreases very sharply as the sphere concentration increases. v2 decreases from 3 µm/s to 0.7 nm/s when φ increases from 0.004 to 0.04. The second slow crystallization step is ascribed to the reorientation of the metastable crystals formed in the first step toward the stable ones matched with the other crystals mediated by the grain bound-
FIG. 10. First-step growth rates, v1 of CS-82 (s, this work) and CS-91 (X, Ref. (17) 1, Ref. (16); u, Ref. (18)) spheres plotted as a function of sphere concentration at 25◦ C.
FIG. 11. Second-step growth rates, v2 , of CS-82 spheres plotted as a function of sphere concentration at 25◦ C.
aries and the cell wall. When the sphere concentration is high, the number of metastable crystals formed are so huge and their size becomes quite small. Then, the very long period of time will be required for the crystals being matched with other crystals and cell wall. Ostwald ripening is also highly plausible to occur in the slow process, though direct observation has not been successful until Wong et al. (35) observed ripening for colloidal crystallization processes. ACKNOWLEDGMENTS Drs. M. Komatsu and M. Hirai of Catalysts & Chemicals Ind. Co. (Tokyo and Kitakyusyu) are deeply thanked for providing the silica samples. Professor Akira Tsuchida of Gifu University is acknowledged for his valuable comments. The Ministry of Education, Science, Sports and Culture is thanked for Grants-inAid for Scientific Research on Priority Area (A) (11167241) and for Scientific Research (B) (11450367). T.O. thanks deeply the late Professor Emeritus Sei Hachisu for his continual encouragements and comments on our work on colloidal crystals.
REFERENCES 1. Vanderhoff, W., van de Hul, H. J., Tausk, R. J. M., and Overbeek, J. Th. G., in “Clean Surfaces: Their Preparation and Characterization for Interfacial Studies” (G. Goldfinger, Ed.). p. 15. Dekker, New York, 1970. 2. Hiltner, P. A., Papir, Y. S., and Krieger, I. M., J. Phys. Chem. 75, 1881 (1971). 3. Kose, A., Ozaki, M., Takano, K., Kobayashi, Y., and Hachisu, S., J. Colloid Interface Sci. 44, 330 (1973). 4. Williams, R., Crandall, R. S., and Wojtowicz, P. J., Phys. Rev. Lett. 37, 348 (1976). 5. Mitaku, S., Ohtsuki, T., Enari, K., Kishimoto, A., and Okano, K., Jpn. J. Appl. Phys. 17, 305 (1978). 6. Lindsay, H. M., and Chaikin, P. M., J. Chem. Phys. 76, 3774 (1982). 7. Pieranski, P., Contemp. Phys. 24, 25 (1983). 8. Ottewill, R. H., Ber. Bunsen-Ges. Phys. Chem. 89, 517 (1985). 9. Aastuen, D. J. W., Clark, N. A., Cotter, L. K., and Ackerson, B. J., Phys. Rev. Lett. 57, 1733 (1986). 10. Pusey, P. N., and van Megen, W., Nature (London) 320, 340 (1986). 11. Okubo, T., Acc. Chem. Res. 21, 281 (1988). 12. Russel, W. B., Saville, D. A., and Schowalter, W. R., “Colloidal Dispersions.” Cambridge Univ. Press, Cambridge, 1989.
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13. Stevens, M. J., Falk, M. L., and Robins, M. O., J. Chem. Phys. 104, 5209 (1996). 14. Okubo, T., J. Chem. Soc. Faraday Trans. 1 84, 1163 (1988). 15. Okubo, T., “Macro-ion Characterization. From Dilute Solutions to Complex Fluids,” ACS Symp. Ser. 548, Chap. 28. Am. Chem. Soc., Washington, DC, 1994. 16. Okubo, T., Okada, S., and Tsuchida, A., J. Colloid Interface Sci. 189, 337 (1997). 17. Okubo, T., and Okada, S., J. Colloid Interface Sci. 192, 490 (1997). 18. Okubo, T., and Ishiki, H., J. Colloid Interface Sci. 211, 151 (1999). 19. Okubo, T., Tsuchida, A., and Kato, T., Colloid Polym. Sci. 277, 191 (1999). 20. Okubo, T., and Okada, S., J. Colloid Interface Sci. 204, 198 (1998). 21. Okubo, T., Tsuchida, A., Okuda, T., Fujitsuna, K., Ishikawa, M., Morita, T., and Tada, T., Colloids Surf. A 153, 515 (1999). 22. Okubo, T., Tsuchida, A., Takahashi, S., Taguchi, K., and Ishikawa, M., Colloid Polym. Sci. 278, 202 (2000). 23. Okubo, T., Tsuchida, A., Yoshimi, H., Taguchi, K., Kiriyama, K., and Takyo, E., Colloid Polym. Sci., in press.
24. Dhont, J. K. G., Smits, C., and Lekkerkerker, H. N. W., J. Colloid Interface Sci. 152, 386 (1992). 25. Harland, J. L., and van Megen, W., Phys. Rev. E 55, 3054 (1997). 26. Vrij, A., Penders, M. H. G. M., Rouw, P. W., de Kruif, C. G., Dhont, J. K. G., Smits, C., and Lekkerkerker, H. N. W., Faraday Discuss. Chem. Soc. 90, 31 (1990). 27. Frenkel, J., Phys. Z. Sowjetunion 1, 498 (1932). 28. Hartman, P., ed., “Crystal Growth: An Introduction,” North-Holland, Amsterdam, 1973. 29. Okubo, T., J. Chem. Soc. Faraday Trans. 82, 3163 (1986). 30. Okubo, T., Langmuir 10, 1695 (1994). 31. Okubo, T., Prog. Polym. Sci. 18, 481 (1993). 32. Schatzel, K., and Ackerson, B. J., Phys. Rev. Lett. 68, 337 (1992). 33. Fujita, H., “Proc. 4th Asia-Pacific Conf., Workshop on Electron Microscopy,” p. 215. 1988. 34. Imai, M., Mori, K., Mizukami, T., Kaji, K., and Kanaya, T., Polymer 33, 4451 (1992). 35. Wong, E. M., Bonevich, J. E., and Searson, P. C., J. Phys. Chem. B 102, 7770 (1998).
Kinetic Analyses of Colloidal Crystallization in a Wide Range of Sphere Concentrations as Studied by Reflection Spectroscopy Tsuneo Okubo1 and Hisanori Ishiki Department of Applied Chemistry and Graduate School of Materials Science, Gifu University, Yanagido 1-1, Gifu 501-1193, Japan Received January 24, 2000; accepted April 24, 2000
The nucleation and growth rates in the colloidal crystallization of silica spheres (103 nm in diameter) from 0.006 to 0.04 in volume fraction (φ) have been measured by reflection spectroscopy. Kinetics of the crystallization has been discussed in a wide sphere concentration range (from φ = 0.0005 to φ = 0.04) using the data of this work and the previous work (110 nm in diameter) in exhaustively deionized aqueous suspensions. The induction period for nucleation decreases sharply as the sphere concentration increases. The nucleation rate increases substantially from 1 × 10−3 to 1 × 107 mm−3 s−1 when φ increases from 0.0005 to 0.04. The crystal growth process consists of the fast growing step toward metastable crystals (rate v 1 ) and slow growth accompanied with the reorientation toward stable ones (rate v 2 ). The v 1 values increase first from 5 to 20 µm/s and then turn back to 5 µm/s after passing a maximum. v 1 above φ = 0.01 remains at 5 µm/s and is insensitive to sphere concentration. The slow step is observed in the high-sphere concentrations only, and v 2 decreases sharply from 3 µm/s to 0.7 nm/s when sphere concentration increases from 0.004 to 0.04 in volume fraction. Importance of the electrostatic intersphere repulsion by overlapping of the electrical double layers and the cooperative and synchronized fluctuation of colloidal spheres in the crystallization processes are supported strongly. °C 2000 Academic Press Key Words: colloidal crystal; nucleation rate; crystal growth rate; reflection spectroscopy; colloidal silica spheres.
1. INTRODUCTION
Recently, keen attention has been paid to the structural and dynamic properties of colloidal crystals (1–13). Suspensions showing crystal-like structures are ideal for model studies of crystals. Furthermore, phase transition phenomena of crystallization or melting takes place sharply. Several researchers have studied the mechanism of colloidal crystallization for two groups of colloidal crystals: (a) diluted and deionized aqueous suspensions, and (b) concentrated suspensions in the refractive index matched organic solvents. The former are very convenient models for both soft- and hard-sphere systems depending on the ionic concentrations of the suspension, i.e., soft crystals in the
1
To whom correspondence should be addressed.
completely deionized state and hard crystals in the presence of a rather large amount of sodium chloride, for example. Colloidal crystallization takes place for monodispersed colloidal particles in suspension. Many researchers have clarified that the colloidal crystals are formed by Brownian movement of colloidal particles and the interparticle electrostatic repulsion (1–13). In other words, colloidal particles form crystal-like distribution automatically with the help of Brownian movement of the particles for maximizing packing density and/or minimizing dead space. The lattice spacing of colloidal crystals changes by the particle concentration exclusively, which supports strongly the importance of the interparticle repulsion induced by Brownian movement of particles themselves and also the electrostatic interparticle repulsive forces. We have studied the nucleation and crystallization processes in the exhaustively deionized and highly diluted aqueous suspensions from the sharpening of the reflection peak (14–17). The crystallization process was unexpectedly fast. In the previous papers (16, 17), nucleation rates in aqueous media increased drastically as sphere concentration increased. Crystal growth rates also increased substantially as the sphere concentration increased, but only in a narrow concentrstion range. Furthermore, nucleation and crystallization rates were very sensitive to the degree of deionization of the aqueous suspensions. Recently, we further studied the kinetics of colloidal crystallization in a sinusoidal electric field (18), in the presence of salt (19), and in alcoholic organic solvents (20). Microgravity experiments using aircraft have been made for the kinetic analyses of colloidal crystallization (1–23). Recently, several researchers reported that crystallization rates for the hard-sphere system increase in diluted concentrations and then decrease after passing a maximum as the sphere concentration increases (24–26). In this paper, colloidal crystallization rates in the soft-sphere colloidal systems have been investigated in detail over a very wide range of sphere concentrations (from 0.0005 to 0.05 in volume fraction) to compare the kinetic data for the soft-sphere and hard-sphere systems. The crystallization rates of the former are, however, too fast to be followed by the conventional inverted mixing method. Thus, in the present work stopped-flowtype reflection spectroscopy has been applied for the kinetic analyses.
151
0021-9797/00 $35.00
C 2000 by Academic Press Copyright ° All rights of reproduction in any form reserved.
152
OKUBO AND ISHIKI
2. MATERIALS AND METHODS
2.1. Materials Colloidal silica spheres of CS-82 were a gift from Catalyst & Chemicals Ind. Co. (Tokyo). The diameter (d0 ), standard deviation (δ) from the mean diameter, and polydispersity index (δ/d0 ) were 103 nm, 13.2 nm, and 0.13, respectively. The values of d0 and δ were determined by electron microscopy. Charge density of the spheres was determined to be 0.38 µC/cm2 for strongly acidic groups by conductometric titration with a Horiba model DS-14 conductivity meter. The sphere sample was carefully purified several times using an ultrafiltration cell (model 202, membrane: Diaflo-XM300, Amicon Co.). Then, the sample was treated on a mixed bed of cation- and anion-exchange resins [Bio-Rad, AG501-X8 (D), 20–50 mesh] more than 4 years before use since the newly produced silica spheres always released a considerable amount of alkali ions from the sphere surfaces for a long period of time. Water used for the purification and for suspension preparation was purified by a Milli-Q reagent grade system (Milli-RO5 plus and Milli-Q plus, Millipore Co., Bedford, MA). 2.2. Reflection Spectroscopy The cell system used, which was the same as that reported previously (21), consists of a quartz observation cell (40 × 10 × 2 mm), a column of cation- and anion-exchange resins (Bio-Rad), and a peristaltic pump (Masterflex 7524-10, Illinois) as shown in Fig. 1. The pump circulates the colloidal suspensions first to the resin column and then to the observation cell to deionize the suspension continuously. The flow rate was usually 3 ml/min. The rate was increased to 9 ml/min before the measurements were made. A light beam from a halogen lamp (Hayashi LA-150SX, Tokyo) hits the cell wall through a Y-type optical fiber cable and the reflection spectra at the angle of 180◦ are taken on a photonic multichannel analyzer (PMA-50, Hamamatsu Photonics, Hamamatsu). When the colloidal suspension is passed into the narrow observation cell, the crystals are melted away by the shear flow. After the flow was stopped nucleation and crystallization started. These processes were followed from the growth of the Bragg reflection peaks after a certain induc-
tion time. In our experimental conditions the suspension in the flow cell (2-mm path length) was transparent. Thus, the reflection spectroscopy in our experiments affords information on the crystal growth processes over the whole depth of the suspension in the cell. 2.3. Determination of the Nucleation and Crystal Growth Rates Most kinetic measurements on colloidal crystallization including this work have observed an induction period, after which the crystal growth starts especially in diluted suspensions. This observation supports the fact that the kinetics of colloidal crystallization is explainable by the classical diffusive crystallization theory including nucleation and crystal growth processes (17, 27, 28). The number of nuclei that germinate per unit time, the nucleation rate, vn , is given by vn = Nn /ti ,
[1]
where Nn is the total number of nuclei which are formed during the nucleation process and ti is the induction period. Here, we assume that the number of nuclei equals the number of single crystals formed. The number of sphere particles per mean size of single crystal, Nc , will be given for a cubic lattice by Nc =
√
2L 3 /l 3 ,
[2]
where L and l are the mean size of single crystals formed and the nearest-neighbor intersphere distance, respectively. The total number of colloidal spheres (NT ) in a unit volume is φ/[(4/3)π(d0 /2)3 ]. Then, vn is given by √ vn = NT /Nc ti = φl 3 /[(4 2/3)π (d0 /2)3 L 3 ti ].
[3]
The size of the colloidal single crystals from the homogeneous nucleation, L is estimated from the peak intensity of the reflection spectra (24), I ∝ Ncryst L 3 ∝ L 3 ,
[4]
where Ncryst is the number of single crystals in the reflecting volume, which is directly proportional to the number concentration of crystals in the final stages of the crystallization process, being equal to the total number of nuclei formed in the whole course of crystallization. The final size of the crystals formed was determined independently using a CCD microscope (type VH-5910, Keyence, Osaka). The first crystal growth rate of a metastable crystal, v1 is given by (16) v1 = v1∞ [1 − exp(−σ )]. FIG. 1.
Schematic representation of the apparatus.
[5]
Here, σ is the relative supersaturation given by (φ − φc )/φc . φc
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COLLOIDAL CRYSTALLIZATION KINETICS
is the critical concentration of melting. v1∞ is the maximum crystal growth rate. Equation [5] is further simplified as v1 = v1∞ − v1∞ φc /φ.
[6]
Equation [6] means that the growth rate should decrease linearly as the reciprocal sphere concentration increases and was supported experimentally, but only in a very low-sphere concentration range (16). 3. RESULTS AND DISCUSSION
3.1. Reflection Spectroscopy Typical reflection spectra in the course of crystallization are shown in Fig. 2 at φ = 0.02. The reflection peak became sharp with time in the course of crystallization, which corresponds to an increase in crystal size. The background intensity decreased in the course of crystallization, which is ascribed to the diminution in the multiple scattering of the suspension in the course of crystallization. Figure 3 shows the peak shifts in the course of crystallization for the same sphere samples shown in Fig. 2. Assignment of the crystal lattice structure, whether the lattices are face-centeredcubic or body-centered-cubic, could not be made since the spectra gave the single peaks only. Clearly, the peak wavelength, λ, decreased in the course of crystallization. This peak shift reflects the fact that metastable and loose crystals are formed in the first crystallization step and then the crystals become stable and compact ones. Figure 4 shows the λp values observed at the final stage of crystallization in this work. The nearest-neighbor intersphere distance, l, is determined from λp using Eq. [7] in an aqueous suspension at 25◦ C (29).
FIG. 3. Change in the reflection peak wavelengh in the course of crystallization of CS-82 spheres at 25◦ C. s, φ = 0.004; X, 0.008; 1, 0.02.
tance, l0 , on the other hand, is obtained from Eq. [8] (29), l0 = 0.904d0 φ −1/3 ,
[8]
where d0 is the diameter of the spheres, 103 nm in this work. The solid curve in the figure shows the l0 values thus calculated. Excellent agreement between l and l0 demonstrates that the lattice spacing of the colloidal crystal changes by the sphere concentration exclusively and, furthermore, the crystals are formed by the intersphere repulsion induced by Brownian movement of spheres themselves and also by the electrostatic intersphere repulsive forces. 3.2. Nucleation Process
The calculated values of the nearest-neigher intersphere dis-
Figure 5 shows the induction periods (ti ) determined in this work from the intersection of the slope line and the time axis in the reflection peak intensity vs t plots (see Figs. 7 and 8, for example). ti increased sharply as the sphere concentration decreased. The nucleation rate, vn was estimated using Eqs. [1]–[3] and observed ti . In Fig. 6 the vn values obtained in the previous work (16, 17) are also cited by triangles and crosses with the values in this work denoted by open circles, respectively.
FIG. 2. Reflection spectra in the course of crystallization of CS-82 spheres at 25◦ C. φ = 0.02, Curve 1,0 s after stopped flow; 2, 4.0 s; 3, 8.0 s; 4, 12.0 s; 5, 16.0 s; 6, 19.4 s.
FIG. 4. Reflection peak wavelength of colloidal crystals formed finally of CS-82 spheres as a function of sphere concentration at 25◦ C. s, observed; -------, calculated.
l = 0.460λp
[7]
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FIG. 5. Induction period of colloidal crystallization of CS-82 spheres at 25◦ C. Large circles are the mean of the repeated experimental values shown by small circles.
The nucleation rate increased substantially as the sphere concentration increased. As is clear in Fig. 6, vn increased by 1010 fold when φ increased by 102 -fold! Thus, an increase in sphere concentration is followed by an increase in vn and number of nuclei (Nn ) and resulted in a decrease in the size of single crystals formed as discussed by Dhont et al. (24). It should be mentioned here that the size of single crystals, which is observed with the naked eyes, is polydisperse as has often been observed (30). Therefore, the clear-cut separation of the nucleation step from the crystallization process will be difficult, and the nucleation reaction may remain operative, even in the crystal growth period. 3.3. Crystal Growth Process Figure 7 shows typical examples of the time dependencies of the corrected peak intensities, which were obtained from the peak intensity observed subtracted by the background intensity. Intensity vs time curves show S-shaped characteristics, especially at low-sphere concentrations. The crystal growth rates, v1 and v2 , were evaluated from the cubed-root plot of the peak intensity using Eq. [4]. Typical plots are shown in Fig. 8.
FIG. 6. Nucleation rates of CS-82 (s: this work) and CS-91 (X, Ref. (25); 1, Ref. (24)) spheres plotted as a function of sphere concentration at 25◦ C.
FIG. 7. Reflection peak intensity in the course of crystallization of CS-82 spheres at 25◦ C. ———, φ = 0.004; ·········, 0.008; ------, 0.02.
Before calculation of the first and second growth rates, v1 and v2 , the apparent growth rates, v10 and v20 , were determined as the reciprocal periods where the initial linear lines intersect two horizontal lines, giving initial and final intensities, 1/(tf − ti ), where ti is the induction time. In the calculation of v1 and v2 , the final mean size of single crystals must be known. CCD camera and close-up photography were used to measure size, L, and the results are shown in Fig. 9. L was largest when the sphere concentration was slightly higher than the critical concentration of melting and decreased sharply when the sphere concentration increased. This dependency of L with φ has been observed often before [11, 15, 30, 31], though the sphere concentration where the size becomes largest depends strongly on the degree of deionization of the suspensions. Figure 10 shows the v1 values thus evaluated from v10 multiplied by L. Again, the experimental errors are rather large, as is usual for the kinetic parameters of soft colloidal crystals, which is quite sensitive to the degree of deionization. However, it is clear that v1 increases first from 5 to 20 µm/s and then decreases back to 5 µm/s, passing a maximum. The v1 values above φ = 0.01 remain at 5 µm/s and are quite insensitive to sphere concentration. It should be mentioned here that the v1
FIG. 8. Cube roots of the reflection peak intensity in the course of crystallization of CS-82 spheres at 25◦ C. s, φ = 0.02.
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FIG. 9. Size of crystals formed finally of CS-82 (s, this work) and CS-91 (X, Ref. (17); 1, Ref. (16) spheres plotted as a function of sphere concentration at 25◦ C.
values in the diluted suspensions decrease linearly as the reciprocal concentration of spheres increases (16). A clear-cut explanation for the crystallization rate first going up and then going down as the sphere concentration increases is not obtained yet. However, the main course will be the dynamic nature of phase transition; i.e., spheres in the colloidal lattices interact with each other strongly and further in a cooperative manner through the electrical double layers. This dynamic phase transition is different from the classical one, which is explained by the translational diffusion of monomers. It should be noted here that the importance of the dynamic phase transition has been reported for colloidal crystals (32) and other crystals such as metals (33) and polymers (34). Figure 11 shows the plots of log v2 versus φ, which demonstrates that the slow step of crystallization decreases very sharply as the sphere concentration increases. v2 decreases from 3 µm/s to 0.7 nm/s when φ increases from 0.004 to 0.04. The second slow crystallization step is ascribed to the reorientation of the metastable crystals formed in the first step toward the stable ones matched with the other crystals mediated by the grain bound-
FIG. 10. First-step growth rates, v1 of CS-82 (s, this work) and CS-91 (X, Ref. (17) 1, Ref. (16); u, Ref. (18)) spheres plotted as a function of sphere concentration at 25◦ C.
FIG. 11. Second-step growth rates, v2 , of CS-82 spheres plotted as a function of sphere concentration at 25◦ C.
aries and the cell wall. When the sphere concentration is high, the number of metastable crystals formed are so huge and their size becomes quite small. Then, the very long period of time will be required for the crystals being matched with other crystals and cell wall. Ostwald ripening is also highly plausible to occur in the slow process, though direct observation has not been successful until Wong et al. (35) observed ripening for colloidal crystallization processes. ACKNOWLEDGMENTS Drs. M. Komatsu and M. Hirai of Catalysts & Chemicals Ind. Co. (Tokyo and Kitakyusyu) are deeply thanked for providing the silica samples. Professor Akira Tsuchida of Gifu University is acknowledged for his valuable comments. The Ministry of Education, Science, Sports and Culture is thanked for Grants-inAid for Scientific Research on Priority Area (A) (11167241) and for Scientific Research (B) (11450367). T.O. thanks deeply the late Professor Emeritus Sei Hachisu for his continual encouragements and comments on our work on colloidal crystals.
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