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Kinetic Analyses of the Colloidal Crystallization of Silica Spheres As Studied by Reflection Spectroscopy
JOURNAL OF COLLOID AND INTERFACE SCIENCE ARTICLE NO.
192, 490–496 (1997)
CS975028
Kinetic Analyses of the Colloidal Crystallization of Silica Spheres As Studied by Reflection Spectroscopy Tsuneo Okubo 1 and Shinji Okada Department of Applied Chemistry, Gifu University, Gifu 501-11, Japan Received April 11, 1997; accepted June 2, 1997
Reflection spectroscopy is used in kinetic analyses of the nucleation and growth processes of colloidal crystals of silica spheres (110 nm in diameter) in exhaustively deionized aqueous suspensions. Sphere concentrations range from 0.001 to 0.0025 in volume fraction ( f ) for the nucleation and 0.0014 to 0.0036 for the crystallization process, respectively. Induction periods are from 1 to 500 s and become longer with decreasing sphere concentration. Nucleation rates are 10 03 to 10 3 mm03 s 01 and increase sharply as sphere concentration increases. The crystallization process has been observed through the sharpening and the increase of intensity in the reflection peaks for the suspension in a test tube, which stands still after being inverted. Crystal growth rates £ range from 2 to 15 mm/s and decrease linearly as the reciprocal sphere concentration increases. Crystal growth rates represented by the number of unit cells u also increase as f increases, ranging from 2 to 23 unit cells/ s. The importance of electrostatic intersphere repulsion through electrical double layers and of cooperative fluctuation of colloidal spheres in crystallization processes is supported. q 1997 Academic Press Key Words: colloidal crystal; crystal growth kinetics; reflection spectrum; nucleation; silica spheres.
ties. When the sample suspensions are carefully deionized with resins for more than three weeks, the fc-value becomes quite low, in the range from 0.0001 to 0.0003, as we have reported recently (30). We have observed the process of colloidal crystallization in exhaustively deionized and highly diluted aqueous suspensions through the sharpening of the reflection peak in reflection spectroscopy (15, 19). The crystallization process was unexpectedly fast. In a previous paper (31), we studied colloidal crystallization kinetically by reflection spectroscopy and dynamic light-scattering techniques. Nucleation rates were 5.6 1 10 04 to 3.2 1 10 03 mm03 s 01 at sphere concentrations from volume fraction ( f ) 0.0006 to 0.001. Crystal growth rates ranged from 2.9 to 20.7 mm/s and increased substantially as the sphere concentration increased. However, reflection spectroscopy measurements have been performed for only one sample of silica spheres at volume fraction 0.001. In this report, colloidal crystallization kinetics has been studied in much more detail and systematically in the range of sphere concentrations from 0.0015 to 0.0036 by measuring reflection spectra in the course of crystallization.
1. INTRODUCTION
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 occur sharply. The mechanism of colloidal crystal growth has been studied by several researchers hitherto for the two group systems of colloidal crystals, diluted and deionized aqueous suspensions (14–20) and concentrated suspensions in the refractive index matched organic solvents (21–29). We must pay close attention to the complete deionization of the suspension, especially for aqueous systems. The critical sphere concentration for melting is quite sensitive to the degree of deionization and increases sharply for the suspensions contaminated with ionic impuri1 To whom correspondences should be addressed at Department of Applied Chemistry, Faculty of Engineering, Gifu University, Yanagido 1-1, Gifu 501-11, Japan.
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2.1. Materials Colloidal silica spheres of CS-91 were a gift from Catalysts & Chemicals Ind. Co. (Tokyo). Diameter (d 0 ), standard deviation ( d ) from the mean diameter, and polydispersity index ( d /d 0 ) were 110 nm, 4.5 nm, and 0.041, respectively. The values of d 0 and d were determined from an electron microscope. Charge density of the spheres was determined to be 0.48 mC/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 two years before use, since the newly produced silica spheres always released a considerable number of alkali ions from
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FIG. 2. Intensity of the reflection peaks in the course of crystallization of CS-91 spheres at 237C. s: f Å 0.00143; X: 0.00179; n: 0.00262.
FIG. 1. Reflection spectra in the course of crystallization of CS-91 spheres at 247C. f Å 0.00179, 17 days after suspension preparation with the resins. Curve 1: 5 s after mixing; 2: 15 s; 3: 20 s; 4: 25 s; 5: 600 s.
the sphere surfaces for a long 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. Close-Up Photography
q is the scattering vector, and Dq denotes ql 0 qs , with ql and qs referring to the largest and smallest wavelengths corresponding to the half-width of the reflection peak. It should be noted here that Eq. [1] is derivable by assuming the powder-like random distribution of a large number of colloidal crystals in the reflection volume. However, this assumption is difficult to hold for colloidal crystals, since the most dense planes of the colloidal single crystals are apt to orient parallel to the cell plane. On the other hand, the peak intensity, I, of the reflection spectra is approximated by (34) I } Ncryst L 3 } L 3 ,
[2]
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
Photographs of colloidal crystals in a test tube were made with a Canon EOS10 camera, macrolens (EF50 mm, f Å 2.5), and life-size converter EF. Velvia film (Fuji chrome, RVP135, ISO Å 50) was used. The light source was a pocket-type flashlight (BF-775, Xenon 4, National). 2.3. Reflection Spectroscopy The reflection spectra at an incident angle of 907 were recorded on a multichannel photodetector, MCPD-110B, Otsuka Electronics, Hirakata, Osaka, connected to a Y-type optical-fiber cable. The size of the colloidal single crystals from the homogeneous nucleation, L, may be estimated by Scherrer’s equation from the half-width of the reflection spectra (32–34) as L } nDq 01
[1]
where n is the number of the order in the Bragg equation.
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FIG. 3. Intensity of the reflection peaks in the course of crystallization of CS-91 spheres at 237C. s: f Å 0.00154; X: 0.00214; n: 0.00227; h: 0.00238; l: 0.00357.
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FIG. 4. Induction period for CS-91 spheres at 237C. s: this work; X: previous work.
process, being equal to the total number of nuclei which were formed in the whole course of crystallization. 3. RESULTS
Typical reflection spectra in the course of crystallization are shown in Fig. 1 at f Å 0.00179. The spectra were taken every 5 s. The reflection peak became sharp with time in the course of crystallization, which corresponds to the increase in crystal size. The background intensity decreased slightly in the course of crystallization, though this is not clear in the figure, where the backgrounds of the spectra were adjusted at the same level. This decrease is ascribed to the diminition in the multiple scattering of the suspension. Figures 2 and 3 show 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, espe-
FIG. 5. Cube root of the reflection peak intensity in the course of crystallization of CS-91 spheres at 237C. s: f Å 0.00143; X: 0.00179; n: 0.00262.
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FIG. 6. Cube root of the reflection peak intensity in the course of crystallization of CS-91 spheres at 237C. s: f Å 0.00154; X: 0.00214; n: 0.00227; h: 0.00238; l: 0.00357.
cially at low sphere concentration. The induction period (ti ) was observed clearly at low concentrations of spheres and determined from the intersection of the slope line and the time axis. ti increased sharply as sphere concentration decreased. This parameter will be discussed later in Section 4.1. The crystal growth rate, £, in mm/s was evaluated from (1) the cube-root plot of the reflection intensity using Eq. [2] and (2) the half-width of the reflection peak using Eq. [1], respectively. In these calculations, the final mean sizes of colloidal crystals formed were estimated with the naked eye and with a ruler. Crystal size increased rather sharply as sphere concentration decreased and largest at a bit higher sphere concentration than the critical sphere concentration of melting, fc . Figure 4 shows the induction periods. Figures 5 and 6 show the typical examples of the cube root of intensity against time plots. Furthermore, Fig. 7 shows the size vs time plots evaluated using Eq. [2] and the average size of crystals.
FIG. 7. Crystal size estimated from the half-width method in the course of crystallization of CS-91 spheres at 237C. s: f Å 0.00143; X: 0.00179; n: 0.00262.
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0.00154, 0.00167, 0.00179, 0.00214, 0.00227, and 0.00262, respectively. The l-values are given by l Å 0.904d 0f 01 / 3 .
[6]
It should be mentioned here that the size of single crystals is polydisperse, as observed hitherto (33). Therefore, the clear-cut separation of the nucleation step from the crystallization process will be difficult, and the nucleation reaction may remain even in the crystal growth period. Furthermore, the Ostwald ripening of the crystals, i.e., further growth of large crystals and disappearance of small ones, has not been observed at all for our colloidal crystallization systems. FIG. 8. Nucleation rates for CS-91 spheres at 237C. s: this work, X: previous work (31).
4.2. Crystal Growth Process According to Wilson (35) and Frenkel (36), the crystal growth rate of a crystal, £, is given by
4. DISCUSSION
£ Å £`[1 0 exp( 0 Dm/kBT )].
4.1. Nucleation Process Most kinetic measurements on colloidal crystallization, including this work, have observed an induction period after which the crystal growth starts, especially in the diluted suspensions. This observation supports that the kinetics of colloidal crystallization is explainable by the classical diffusive crystallization theory including nucleation and crystal growth processes. The number of nuclei that germinate per unit time, the nucleation rate, £n , is given by £n Å Nn /ti ,
[3]
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 the cubic lattice by q
Nc Å 2L 3 /l 3
[4]
where L and l are the mean size of single crystals formed and the closest intersphere distance, respectively. Total number of colloidal spheres (NT ) in a unit volume is f /[(4/ 3) p(d 0 /2) 3 ]. Then £n is given by q
£n Å NT /Ncti Å fl 3 /[(4 2/3) p(d 0 /2) 3 L 3ti ].
[5]
£n-values thus estimated are shown in Fig. 8. The mean size of single crystals formed, L, was estimated by closeup photography to be 1000, 1200, 570, 500, 460, 420, 200, 120 and 120 mm at f Å 0.00056, 0.000955, 0.00143,
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[7]
Here, Dm is the chemical potential difference between the crystal and liquid phases and is given by Dm Å mcryst 0 mliq .
[8]
Dm ú 0 means that crystallization will proceed. We should note that Dm/kBT Å ln( f / fc ) à s,
[9]
where s is the relative supersaturation given by ( f 0 fc )/ fc . From Eqs. [7] and [9] is derived £ Å £`[1 0 exp( 0 s )].
[10]
Equation [10] is further simplified as £ Å £` 0 £`fc / f.
[11]
Figure 9 shows £ thus evaluated vs 1/ f plots. The data are rather scattered but linearity is good. £` and fc are 15.3 mm/ s and 0.0014, respectively. Degree of deionization of the sample suspensions in this work is clearly low compared with that reported previously, 0.00055(31). Uncertainty of £` in Fig. 9 is rather high, and it is safely concluded that £` of this work is certainly higher than in the previous work, i.e., 38 mm/s. It should be noted here that the lattice spacing of colloidal crystals is not constant but increases rather sharply as sphere concentration decreases, because the most important intersphere interaction in the colloidal crystals is the electrostatic ‘‘repulsion.’’ Thus, the crystallization rate will be represented much more appropriately by the increase of the unit
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FIG. 9. Crystal growth rates, £` as a function of reciprocal sphere concentration estimated from the cube-root method ( s ) and half-width method (X) for CS-91 spheres at 237C.
cells per time, u Å £ /l, rather than the actual length per time, £, where l is a mean intersphere distance determined uniquely as a function of sphere concentration by Eq. [6]. Thus u Å u` 0 u`fc / f.
[12]
Fig. 10 displays the plots against 1/ f of u-values evaluated in this work and also in previous work. The linearity between u and 1/ f was good, and the u` and fc values of this work were evaluated to be 24 unit cells/s and 0.0014 in volume fraction, respectively. Those in previous work were 44 unit
cells/s and 0.00055 in volume fraction, respectively. The large difference in fc-values between the two works is ascribed to the difference in degree of deionization of the colloidal suspensions. In this work, all samples were prepared in test tubes with ion-exchange resins. However, the measurements were made within several hours after suspension preparation. Clearly, the degree of deionization of the suspension in this work is lower than in previous work, where measurements were made about 2 wk after suspension preparation with ion-exchange resins. We should note here that the kinetic parameters of the colloidal crystallization are highly sensitive to the degree of deionization of suspensions.
FIG. 10. Crystal growth rates, u, as a function of reciprocal sphere concentration estimated from the cube-root ( s ) and half-width (X) methods in this work and from DLS ( l ), half-width ( m ), and cube-root ( j ) methods in previous work on CS-91 spheres.
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We assume here that Eq. [6] is applicable to the colloidal system and the maximum growth rate, £` , is determined by the diffusion of single spheres near the crystal surface. Then £` Å 4D0 / j.
[13]
Here j is the mean diffusion length (path) and D0 is the diffusion coefficient of spheres in the crystallization suspension when j and D0 are assumed to be the mean intersphere distance (l, 1.18 mm in this work) and the diffusion coefficient of spheres in the gas-like distribution is calculated by the Stokes–Einstein equation (4.36 1 10 012 m2 /s). Then £` will be given by £` Å 4kBT/3phd 0l
[14]
where d 0 is the sphere diameter. £` is calculated to be 14.8 mm/s, which agrees surprisingly well with the observation, 15.3 mm/s. We should note here that D0 should be the diffusion coefficient in the supersaturated liquids, for example, 1 1 10 012 m2 /s in the previous work on dynamic light-scattering. Furthermore, the mean diffusion length ( j ) is clearly overestimated when j à l is taken, since the colloidal spheres in supersaturated liquids must move in a cooperative manner by the electrostatic repulsive forces. In this sense, colloidal crystal systems are quite similar to fused metal systems. Slight movement of the effectively enlarged spheres, including electrical double layers, will be enough to allow crystallization. Thus, the agreement of £` between the calculation and the observation will be explained by the balancing of two factors in the decrease of D0 and j. 4.3. Importance of Intersphere Repulsion and Synchronous Fluctuation of Colloidal Spheres in Crystallization The critical concentration of melting ( fc ) of colloidal crystals in the exhaustively deionized state is very low, around 0.0002 in volume fraction in most cases, as described in the Introduction. However, the effective concentration of spheres including the expanded electrical double layers is substantially high, close to volume fraction 0.74, corresponding to close packing. Each sphere in the colloidal crystals interacts very strongly and in a cooperative manner through electrical double layers by electrostatic repulsion. Recently, Schatzel and Ackerson (27) proposed the importance of dynamic phase transition by density fluctuation in colloidal crystallization. It should be noted that the synchronous fluctuation of colloidal spheres in crystal cages plays an important role in colloidal crystallization. The importance of the dynamic phase transition has been reported for other crystals such as metals (38) and polymers (39). Many researchers including Hachisu et al. have clarified the importance of electrostatic repulsive forces between
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spheres in colloidal crystal systems hitherto (40, 41). Intersphere repulsion should also exist in the course of crystallization including nucleation and growth. The electric double layers for colloidal crystals are more expanded than those of colloidal liquids (42, 43). This means that the microscopic intersphere repulsion force in the colloidal crystals is weak, though very slightly, compared with that of colloidal liquids. Therefore, there should be apparent intersphere attraction in the course of nucleation and growth processes in the phase transition from liquids to crystals. In other words, the slight difference in the repulsive forces in the crystals and the liquids leads to apparent attraction between spheres in the nucleus regions. This apparent attraction in the colloidal crystallization processes must be one of the main reasons the kinetic and morphological properties of colloidal crystals are so similar to those of other crystal systems such as metals, protein crystals, and minerals. Among the many crystal systems, crystallization of fused metals will be one of the best systems that colloidal crystals mimic. ACKNOWLEDGMENTS This work was supported by a grant-in-aid from Japan Space Utilization Promotion Center. 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.
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.). 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. Bunsenges. 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 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, UK, 1989. 13. Sood, A. K., Solid State Phys. 45, 2 (1991). 14. Ackerson, B. J., and Clark, N. A., Phys. Rev. Lett. 46, 123 (1981). 15. Okubo, T., J. Chem. Soc. Faraday Trans. 1 84, 1163 (1988). 16. Aastuen, D. J. W., Clark, N. A., Swindal, J. C., and Muzny, C. D., Phase Transitions 21, 139 (1990). 17. Lowen, H., Palberg, T., and Simon, R., Phys. Rev. Lett. 70, 1557 (1993). 18. Grier, D. G., and Murray, C. A., J. Chem. Phys. 100, 9088 (1994). 19. Okubo, T., ‘‘Macro-ion Characterization. From Dilute Solutions to
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22. 23. 24. 25. 26. 27. 28. 29. 30.
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Complex Fluids,’’ ACS Symposium Series, Vol. 548, Chap. 28. Am. Chem. Soc., Washington, DC, 1994. Wurth, N., Schwarz, J., Culis, F., Leidener, and Palberg, T., Phys. Rev. E. 52, 6415 (1995). Pusey, P. N., and van Megen, W., in ‘‘Complex and Supramolecular Fluids’’ (Safran, S. A., and Clark, N. A., Eds.), pp. 673–698. Wiley– Interscience, New York, 1987. Davis, K. E., and Russel, W. B., Ceram. Trans. B 1, 693 (1988). Russel, W. B., Phase Transitions 21, 127 (1990). Harkless, C. R., Singh, M. A., Nagler, S. E., Stephenson, G. B., and Jordan-Sweet, J. L., Phys. Rev. Lett. 64, 2285 (1990). Dhont, J. K. G., Smits, C., and Lekkerkerker, H. N. W., J. Colloid Interface Sci. 152, 386 (1992). Schatzel, K., and Ackerson, B. J., Phys. Rev. Lett. 68, 337 (1992). Schatzel, K., and Ackerson, B. J., Phys. Rev. E 48, 3766 (1993). Verhaegh, N. A. M., and van Blaaderen, A., Langmuir 10, 1427 (1994). Butler, S., and Harrowell, P., Phys. Rev. E 52, 6424 (1995). Okubo, T., Langmuir 10, 1695 (1994).
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31. Okubo, T., Okada, S., and Tsuchida, A., J. Colloid Interface Sci. 189, 337 (1997). 32. James, R. W., in ‘‘The Optical Principles of the Diffraction of X-rays. The Crystal State’’ (L. Bragg, Ed.), Vol. II. Cornell Univ. Press, Ithaca, NY, 1965. 33. Okubo, T., J. Chem. Soc. Faraday Trans. 1 82, 3163 (1986). 34. Dhont, J. K. G., Smits, C., and Lekkerkerker, H. N. W., J. Colloid Interface Sci. 152, 386 (1992). 35. Wilson, H. A., Phil. Mag. 50, 238 (1900). 36. Frenkel, J., Phys. Z. Sowjetunion 1, 498 (1932). 37. Hartman, P., Ed., ‘‘Crystal Growth: An Introduction.’’ North-Holland, Amsterdam, 1973. 38. Fujita, H., in ‘‘Proceedings, 4th Asia–Pacific Conference, Workshop on Electron Microscopy, 1988,’’ p. 215. 39. Imai, M., Mori, K., Mizukami, T., Kaji, K., and Kanaya, T., Polymer 33, 4451 (1992). 40. Okubo, T., Prog. Polym. Sci. 18, 481 (1993). 41. Okubo, T., Colloids Surf. A 109, 77 (1996). 42. Arora, A. K., Tata, B. V. R., Sood, A. K., and Kasavana, R., Phys. Rev. Lett. 60, 2438 (1988). 43. Okubo, T., J. Chem. Soc. Faraday Trans. 86, 2871 (1990).
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CS975028
Kinetic Analyses of the Colloidal Crystallization of Silica Spheres As Studied by Reflection Spectroscopy Tsuneo Okubo 1 and Shinji Okada Department of Applied Chemistry, Gifu University, Gifu 501-11, Japan Received April 11, 1997; accepted June 2, 1997
Reflection spectroscopy is used in kinetic analyses of the nucleation and growth processes of colloidal crystals of silica spheres (110 nm in diameter) in exhaustively deionized aqueous suspensions. Sphere concentrations range from 0.001 to 0.0025 in volume fraction ( f ) for the nucleation and 0.0014 to 0.0036 for the crystallization process, respectively. Induction periods are from 1 to 500 s and become longer with decreasing sphere concentration. Nucleation rates are 10 03 to 10 3 mm03 s 01 and increase sharply as sphere concentration increases. The crystallization process has been observed through the sharpening and the increase of intensity in the reflection peaks for the suspension in a test tube, which stands still after being inverted. Crystal growth rates £ range from 2 to 15 mm/s and decrease linearly as the reciprocal sphere concentration increases. Crystal growth rates represented by the number of unit cells u also increase as f increases, ranging from 2 to 23 unit cells/ s. The importance of electrostatic intersphere repulsion through electrical double layers and of cooperative fluctuation of colloidal spheres in crystallization processes is supported. q 1997 Academic Press Key Words: colloidal crystal; crystal growth kinetics; reflection spectrum; nucleation; silica spheres.
ties. When the sample suspensions are carefully deionized with resins for more than three weeks, the fc-value becomes quite low, in the range from 0.0001 to 0.0003, as we have reported recently (30). We have observed the process of colloidal crystallization in exhaustively deionized and highly diluted aqueous suspensions through the sharpening of the reflection peak in reflection spectroscopy (15, 19). The crystallization process was unexpectedly fast. In a previous paper (31), we studied colloidal crystallization kinetically by reflection spectroscopy and dynamic light-scattering techniques. Nucleation rates were 5.6 1 10 04 to 3.2 1 10 03 mm03 s 01 at sphere concentrations from volume fraction ( f ) 0.0006 to 0.001. Crystal growth rates ranged from 2.9 to 20.7 mm/s and increased substantially as the sphere concentration increased. However, reflection spectroscopy measurements have been performed for only one sample of silica spheres at volume fraction 0.001. In this report, colloidal crystallization kinetics has been studied in much more detail and systematically in the range of sphere concentrations from 0.0015 to 0.0036 by measuring reflection spectra in the course of crystallization.
1. INTRODUCTION
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 occur sharply. The mechanism of colloidal crystal growth has been studied by several researchers hitherto for the two group systems of colloidal crystals, diluted and deionized aqueous suspensions (14–20) and concentrated suspensions in the refractive index matched organic solvents (21–29). We must pay close attention to the complete deionization of the suspension, especially for aqueous systems. The critical sphere concentration for melting is quite sensitive to the degree of deionization and increases sharply for the suspensions contaminated with ionic impuri1 To whom correspondences should be addressed at Department of Applied Chemistry, Faculty of Engineering, Gifu University, Yanagido 1-1, Gifu 501-11, Japan.
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2.1. Materials Colloidal silica spheres of CS-91 were a gift from Catalysts & Chemicals Ind. Co. (Tokyo). Diameter (d 0 ), standard deviation ( d ) from the mean diameter, and polydispersity index ( d /d 0 ) were 110 nm, 4.5 nm, and 0.041, respectively. The values of d 0 and d were determined from an electron microscope. Charge density of the spheres was determined to be 0.48 mC/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 two years before use, since the newly produced silica spheres always released a considerable number of alkali ions from
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0021-9797/97 $25.00 Copyright q 1997 by Academic Press All rights of reproduction in any form reserved.
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FIG. 2. Intensity of the reflection peaks in the course of crystallization of CS-91 spheres at 237C. s: f Å 0.00143; X: 0.00179; n: 0.00262.
FIG. 1. Reflection spectra in the course of crystallization of CS-91 spheres at 247C. f Å 0.00179, 17 days after suspension preparation with the resins. Curve 1: 5 s after mixing; 2: 15 s; 3: 20 s; 4: 25 s; 5: 600 s.
the sphere surfaces for a long 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. Close-Up Photography
q is the scattering vector, and Dq denotes ql 0 qs , with ql and qs referring to the largest and smallest wavelengths corresponding to the half-width of the reflection peak. It should be noted here that Eq. [1] is derivable by assuming the powder-like random distribution of a large number of colloidal crystals in the reflection volume. However, this assumption is difficult to hold for colloidal crystals, since the most dense planes of the colloidal single crystals are apt to orient parallel to the cell plane. On the other hand, the peak intensity, I, of the reflection spectra is approximated by (34) I } Ncryst L 3 } L 3 ,
[2]
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
Photographs of colloidal crystals in a test tube were made with a Canon EOS10 camera, macrolens (EF50 mm, f Å 2.5), and life-size converter EF. Velvia film (Fuji chrome, RVP135, ISO Å 50) was used. The light source was a pocket-type flashlight (BF-775, Xenon 4, National). 2.3. Reflection Spectroscopy The reflection spectra at an incident angle of 907 were recorded on a multichannel photodetector, MCPD-110B, Otsuka Electronics, Hirakata, Osaka, connected to a Y-type optical-fiber cable. The size of the colloidal single crystals from the homogeneous nucleation, L, may be estimated by Scherrer’s equation from the half-width of the reflection spectra (32–34) as L } nDq 01
[1]
where n is the number of the order in the Bragg equation.
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FIG. 3. Intensity of the reflection peaks in the course of crystallization of CS-91 spheres at 237C. s: f Å 0.00154; X: 0.00214; n: 0.00227; h: 0.00238; l: 0.00357.
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FIG. 4. Induction period for CS-91 spheres at 237C. s: this work; X: previous work.
process, being equal to the total number of nuclei which were formed in the whole course of crystallization. 3. RESULTS
Typical reflection spectra in the course of crystallization are shown in Fig. 1 at f Å 0.00179. The spectra were taken every 5 s. The reflection peak became sharp with time in the course of crystallization, which corresponds to the increase in crystal size. The background intensity decreased slightly in the course of crystallization, though this is not clear in the figure, where the backgrounds of the spectra were adjusted at the same level. This decrease is ascribed to the diminition in the multiple scattering of the suspension. Figures 2 and 3 show 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, espe-
FIG. 5. Cube root of the reflection peak intensity in the course of crystallization of CS-91 spheres at 237C. s: f Å 0.00143; X: 0.00179; n: 0.00262.
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FIG. 6. Cube root of the reflection peak intensity in the course of crystallization of CS-91 spheres at 237C. s: f Å 0.00154; X: 0.00214; n: 0.00227; h: 0.00238; l: 0.00357.
cially at low sphere concentration. The induction period (ti ) was observed clearly at low concentrations of spheres and determined from the intersection of the slope line and the time axis. ti increased sharply as sphere concentration decreased. This parameter will be discussed later in Section 4.1. The crystal growth rate, £, in mm/s was evaluated from (1) the cube-root plot of the reflection intensity using Eq. [2] and (2) the half-width of the reflection peak using Eq. [1], respectively. In these calculations, the final mean sizes of colloidal crystals formed were estimated with the naked eye and with a ruler. Crystal size increased rather sharply as sphere concentration decreased and largest at a bit higher sphere concentration than the critical sphere concentration of melting, fc . Figure 4 shows the induction periods. Figures 5 and 6 show the typical examples of the cube root of intensity against time plots. Furthermore, Fig. 7 shows the size vs time plots evaluated using Eq. [2] and the average size of crystals.
FIG. 7. Crystal size estimated from the half-width method in the course of crystallization of CS-91 spheres at 237C. s: f Å 0.00143; X: 0.00179; n: 0.00262.
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0.00154, 0.00167, 0.00179, 0.00214, 0.00227, and 0.00262, respectively. The l-values are given by l Å 0.904d 0f 01 / 3 .
[6]
It should be mentioned here that the size of single crystals is polydisperse, as observed hitherto (33). Therefore, the clear-cut separation of the nucleation step from the crystallization process will be difficult, and the nucleation reaction may remain even in the crystal growth period. Furthermore, the Ostwald ripening of the crystals, i.e., further growth of large crystals and disappearance of small ones, has not been observed at all for our colloidal crystallization systems. FIG. 8. Nucleation rates for CS-91 spheres at 237C. s: this work, X: previous work (31).
4.2. Crystal Growth Process According to Wilson (35) and Frenkel (36), the crystal growth rate of a crystal, £, is given by
4. DISCUSSION
£ Å £`[1 0 exp( 0 Dm/kBT )].
4.1. Nucleation Process Most kinetic measurements on colloidal crystallization, including this work, have observed an induction period after which the crystal growth starts, especially in the diluted suspensions. This observation supports that the kinetics of colloidal crystallization is explainable by the classical diffusive crystallization theory including nucleation and crystal growth processes. The number of nuclei that germinate per unit time, the nucleation rate, £n , is given by £n Å Nn /ti ,
[3]
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 the cubic lattice by q
Nc Å 2L 3 /l 3
[4]
where L and l are the mean size of single crystals formed and the closest intersphere distance, respectively. Total number of colloidal spheres (NT ) in a unit volume is f /[(4/ 3) p(d 0 /2) 3 ]. Then £n is given by q
£n Å NT /Ncti Å fl 3 /[(4 2/3) p(d 0 /2) 3 L 3ti ].
[5]
£n-values thus estimated are shown in Fig. 8. The mean size of single crystals formed, L, was estimated by closeup photography to be 1000, 1200, 570, 500, 460, 420, 200, 120 and 120 mm at f Å 0.00056, 0.000955, 0.00143,
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[7]
Here, Dm is the chemical potential difference between the crystal and liquid phases and is given by Dm Å mcryst 0 mliq .
[8]
Dm ú 0 means that crystallization will proceed. We should note that Dm/kBT Å ln( f / fc ) à s,
[9]
where s is the relative supersaturation given by ( f 0 fc )/ fc . From Eqs. [7] and [9] is derived £ Å £`[1 0 exp( 0 s )].
[10]
Equation [10] is further simplified as £ Å £` 0 £`fc / f.
[11]
Figure 9 shows £ thus evaluated vs 1/ f plots. The data are rather scattered but linearity is good. £` and fc are 15.3 mm/ s and 0.0014, respectively. Degree of deionization of the sample suspensions in this work is clearly low compared with that reported previously, 0.00055(31). Uncertainty of £` in Fig. 9 is rather high, and it is safely concluded that £` of this work is certainly higher than in the previous work, i.e., 38 mm/s. It should be noted here that the lattice spacing of colloidal crystals is not constant but increases rather sharply as sphere concentration decreases, because the most important intersphere interaction in the colloidal crystals is the electrostatic ‘‘repulsion.’’ Thus, the crystallization rate will be represented much more appropriately by the increase of the unit
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FIG. 9. Crystal growth rates, £` as a function of reciprocal sphere concentration estimated from the cube-root method ( s ) and half-width method (X) for CS-91 spheres at 237C.
cells per time, u Å £ /l, rather than the actual length per time, £, where l is a mean intersphere distance determined uniquely as a function of sphere concentration by Eq. [6]. Thus u Å u` 0 u`fc / f.
[12]
Fig. 10 displays the plots against 1/ f of u-values evaluated in this work and also in previous work. The linearity between u and 1/ f was good, and the u` and fc values of this work were evaluated to be 24 unit cells/s and 0.0014 in volume fraction, respectively. Those in previous work were 44 unit
cells/s and 0.00055 in volume fraction, respectively. The large difference in fc-values between the two works is ascribed to the difference in degree of deionization of the colloidal suspensions. In this work, all samples were prepared in test tubes with ion-exchange resins. However, the measurements were made within several hours after suspension preparation. Clearly, the degree of deionization of the suspension in this work is lower than in previous work, where measurements were made about 2 wk after suspension preparation with ion-exchange resins. We should note here that the kinetic parameters of the colloidal crystallization are highly sensitive to the degree of deionization of suspensions.
FIG. 10. Crystal growth rates, u, as a function of reciprocal sphere concentration estimated from the cube-root ( s ) and half-width (X) methods in this work and from DLS ( l ), half-width ( m ), and cube-root ( j ) methods in previous work on CS-91 spheres.
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We assume here that Eq. [6] is applicable to the colloidal system and the maximum growth rate, £` , is determined by the diffusion of single spheres near the crystal surface. Then £` Å 4D0 / j.
[13]
Here j is the mean diffusion length (path) and D0 is the diffusion coefficient of spheres in the crystallization suspension when j and D0 are assumed to be the mean intersphere distance (l, 1.18 mm in this work) and the diffusion coefficient of spheres in the gas-like distribution is calculated by the Stokes–Einstein equation (4.36 1 10 012 m2 /s). Then £` will be given by £` Å 4kBT/3phd 0l
[14]
where d 0 is the sphere diameter. £` is calculated to be 14.8 mm/s, which agrees surprisingly well with the observation, 15.3 mm/s. We should note here that D0 should be the diffusion coefficient in the supersaturated liquids, for example, 1 1 10 012 m2 /s in the previous work on dynamic light-scattering. Furthermore, the mean diffusion length ( j ) is clearly overestimated when j à l is taken, since the colloidal spheres in supersaturated liquids must move in a cooperative manner by the electrostatic repulsive forces. In this sense, colloidal crystal systems are quite similar to fused metal systems. Slight movement of the effectively enlarged spheres, including electrical double layers, will be enough to allow crystallization. Thus, the agreement of £` between the calculation and the observation will be explained by the balancing of two factors in the decrease of D0 and j. 4.3. Importance of Intersphere Repulsion and Synchronous Fluctuation of Colloidal Spheres in Crystallization The critical concentration of melting ( fc ) of colloidal crystals in the exhaustively deionized state is very low, around 0.0002 in volume fraction in most cases, as described in the Introduction. However, the effective concentration of spheres including the expanded electrical double layers is substantially high, close to volume fraction 0.74, corresponding to close packing. Each sphere in the colloidal crystals interacts very strongly and in a cooperative manner through electrical double layers by electrostatic repulsion. Recently, Schatzel and Ackerson (27) proposed the importance of dynamic phase transition by density fluctuation in colloidal crystallization. It should be noted that the synchronous fluctuation of colloidal spheres in crystal cages plays an important role in colloidal crystallization. The importance of the dynamic phase transition has been reported for other crystals such as metals (38) and polymers (39). Many researchers including Hachisu et al. have clarified the importance of electrostatic repulsive forces between
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spheres in colloidal crystal systems hitherto (40, 41). Intersphere repulsion should also exist in the course of crystallization including nucleation and growth. The electric double layers for colloidal crystals are more expanded than those of colloidal liquids (42, 43). This means that the microscopic intersphere repulsion force in the colloidal crystals is weak, though very slightly, compared with that of colloidal liquids. Therefore, there should be apparent intersphere attraction in the course of nucleation and growth processes in the phase transition from liquids to crystals. In other words, the slight difference in the repulsive forces in the crystals and the liquids leads to apparent attraction between spheres in the nucleus regions. This apparent attraction in the colloidal crystallization processes must be one of the main reasons the kinetic and morphological properties of colloidal crystals are so similar to those of other crystal systems such as metals, protein crystals, and minerals. Among the many crystal systems, crystallization of fused metals will be one of the best systems that colloidal crystals mimic. ACKNOWLEDGMENTS This work was supported by a grant-in-aid from Japan Space Utilization Promotion Center. 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.
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