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Langmuir 1999, 15, 2684-2702
Transitions between Ordered and Disordered Phases and Their Coexistence in Dilute Ionic Colloidal Dispersions Hiroshi Yoshida,†,‡ Junpei Yamanaka,*,†,§ Tadanori Koga,†,| Tsuyoshi Koga,†,⊥ Norio Ise,X and Takeji Hashimoto†,∇ Hashimoto Polymer Phasing Project, ERATO, Japan Science and Technology Corporation, 15 Morimoto, Shimogamo, Sakyo, Kyoto 606-0805, Japan, Central Laboratory, Rengo Co., Ltd., 186-1-4, Ohhiraki, Fukushima, Osaka 553-0007, Japan, and Department of Polymer Chemistry, Graduate School of Engineering, Kyoto University, Sakyo, Kyoto 606-8501, Japan Received September 23, 1998. In Final Form: January 21, 1999 Order-disorder transitions in dilute aqueous dispersions of ionic colloids were studied experimentally in colloidal silica and polymer latex systems. A three-dimensional phase diagram of the silica system was determined by treating the particle density, the effective surface charge number, and the ionic strength as variables. On the basis of the phase diagram, the structures of the dispersions were studied by means of ultrasmall-angle X-ray scattering and confocal laser scanning microscopy. The results clearly revealed the presence of a reentrant order-disorder transition with increasing charge number, and a wide biphasic region where ordered and disordered phases coexist. The nearest interparticle distance 2Dexp was almost equal to that calculated assuming a homogeneous particle distribution, 2D0, in the ordered single-phase region, while 2Dexp was smaller than 2D0 under the biphasic condition. Further studies concerning the crystallization process of the dispersion in the biphasic region demonstrated the development of disordered liquidlike regions inside the space-filling crystalline grains which was induced by lattice contraction. The results suggest that the commonly held view based on a Yukawa-type potential, in which only repulsive interparticle interactions are considered, cannot explain the phase behavior of the dilute ionic colloids.
I. Introduction Ionic colloidal particles dispersed in media having high dielectric constants, such as aqueous dispersions of ionic polymer latex and colloidal silica particles, are stabilized via a long range electrostatic interparticle interaction.1 It has been well-known that these systems undergo an order-disorder phase transition (ODT) with respect to the spatial distribution of the particles.2-10 In the ordered phase, particles are arranged in body-centered-cubic (bcc) or face-centered-cubic (fcc) lattices, while in the disordered * To whom correspondence should be addressed. † Japan Science and Technology Corporation. ‡ Present address: Hitachi Research Lab., Hitachi Ltd., 7-1-1 Omika, Hitachi, Ibaraki 319-1292, Japan. § Present address: Faculty of Pharmaceutical Sciences, Nagoya City University, 1-3 Tanabe-dori, Mizuho, Nagoya 467-8603, Japan. | Present address: Department of Chemistry, State University of New York at Stony Brook, Stony Brook, New York 11794-3400. ⊥ Present address: Department of Polymer Chemistry, Graduate School of Engineering, Kyoto University, Sakyo, Kyoto 606-8501, Japan. X Rengo Co., Ltd. ∇ Kyoto University. (1) For a recent review article, see: Ordering and Phase Transitions in Charged Colloids; Arora, A. K., Tata, B. V. R., Eds.; VCH Publisher: New York, 1996. (2) Hachisu, S.; Kobayashi, Y.; Kose, A. J. Colloid Interface Sci. 1973, 42, 2. (3) Yoshiyama, T.; Sogami, I. Langmuir 1987, 3, 854. (4) Monovoukas, Y.; Gast, A. P. J. Colloid Interface Sci. 1989, 128, 533. (5) Palberg, T.; Mo¨nch, W.; Schwarz, J.; Leiderer, P. J. Chem. Phys. 1995, 102, 5082. (6) Palberg, T.; Mo¨nch, W.; Bitzer, F.; Piazza, R.; Bellini, T. Phys. Rev. Lett. 1995, 74, 4555. (7) Wu¨rth, M.; Schwarz, J.; Culis, F.; Leiderer, P.; Palberg, T. Phys. Rev. E 1995, 52, 6415. (8) Yamanaka, J.; Koga, T.; Ise, N.; Hashimoto, T. Phys. Rev. E 1996, 53, R4317. (9) Yamanaka, J.; Yoshida, H.; Koga, T.; Ise, N.; Hashimoto, T. Phys. Rev. Lett. 1998, 80, 5806. (10) Monovoukas, Y.; Gast, A. P. Phase Trans. 1990, 21, 183.
phase, colloidal systems take liquidlike or gaslike orders. Figure 1 shows typical recent examples of confocal laser scanning micrographs for an aqueous dispersion of ionic polymer latex particles in its ordered state. The lattice spacing of the ordered array depends on the particle concentration and the interparticle interactions, and typically lies in the range 0.1-1 µm.1 Thus, due to Bragg diffraction of visible light, the ordered dispersions are iridescent, as demonstrated in Figure 2. The ordered colloidal particles are known to form higher order grain (crystallite) structures, as can be seen in Figure 1a.10 These grain structures have a close correspondence with those for metallic alloy systems. For over two decades, colloidal systems have been attracting substantial interest as fascinating models for studying ODTs because of their relatively large spatial and temporal scales and also because it is possible to alter the magnitude of the interparticle interaction easily. When sufficiently dilute conditions are concerned, one can safely assume that the driving force for the ODT is purely enthalpic. Consequently, ODTs in ionic colloidal systems are governed by several parameters which determine the magnitude of the electrostatic interaction, such as the effective surface charge number Ze, the particle number concentration np, and the salt concentration Cs. A variety of studies have been reported concerning, for example, the phase behavior,1-10 dispersion structures,2,3,10-13 and dynamics7,13-17 of ODTs as a function of these parameters. (11) Gast, A. P.; Monovoukas, Y. Nature 1991, 351, 553. (12) Yoshida, H.; Ito, K.; Ise, N. Phys. Rev. B 1991, 44, 435. (13) Yoshida, H.; Yamanaka, J.; Koga, T.; Ise, N.; Hashimoto, T. Langmuir 1998, 14, 569. (14) Yoshida, H.; Ito, K.; Ise, N. J. Chem. Soc., Faraday Trans. 1991, 87, 371. (15) Dhont, J. K. G.; Smith, C.; Lekkerkerker, H. N. W. J. Colloid Interface Sci. 1995, 98, 6. (16) Lo¨wen, H. in Ordering and Phase Transitions in Charged Colloids; Arora, A. K., Tata, B. V. R., Eds.; VCH Publisher: New York, 1996; Chapter 8.
10.1021/la981316b CCC: $18.00 © 1999 American Chemical Society Published on Web 03/19/1999
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Figure 1. Confocal laser scanning micrographs showing a typical ordered state of an aqueous dispersion of ionic polystyrene-based latex particles at (a) low and (b) high magnifications. Micrograph b was obtained by zooming into the region indicated by the white square in micrograph a. The white striplike patterns observed in part a are a consequence of interference of the laser beam due to the ordered array of the colloidal particles. Differences in the appearances of individual grains and ordered arrays observed in parts a and b reflect variations of the orientation of the ordered lattice with respect to the focal plane. Sample: SS-10. Diameter: 0.12 µm. Volume fraction of the particles φ ) 2.5 × 10-3 (np ) 2.8 µm-3). Salt concentration Cs ) (1-2) × 10-6 M. Micrographs were taken using an Ar laser (wavelength 488 nm) and a 40× objective. See section II for experimental details concerning the sample and confocal laser scanning microscope. The distance between the focal plane and the inner surface of the cover slip (cell wall) was Zfocal ) 120 µm.
However, there still remain several fundamental questions including the nature of the screened electrostatic interaction which governs ODTs in ionic colloidal systems. In deionized colloids, the ODT was found to take place at a very low particle volume fraction φ. Furthermore, it has been revealed that the ordered phase coexists with a disordered phase in a biphasic region between the ordered and disordered single-phase regions.2,3,11,13,17-19 Hachisu et al. were the first to find the coexistence of ordered and disordered regions by microscopy.2 Yoshiyama and Sogami observed the structural inhomogeneity by an independent technique, a Lang method.3 To account for these findings, two contradictory explanations have been advanced. One is an “effective” hard sphere model, which is essentially based on a Yukawa-type long range repulsive interaction. In this assumption, the particle size is enlarged by adding the thickness of the surrounding counterion clouds to the bare radius.19-21 By adapting the enlarged particle size, the effective φ values at which the coexistence and ODT occur can be raised to those predicted by Alder et al. for a hard sphere system.24 Therefore, it has been claimed (17) Ito, K.; Nakamura, H.; Yoshida, H.; Ise, N. J. Am. Chem. Soc. 1988, 110, 6955. (18) Kose, A.; Hachisu, S. J. Colloid Interface Sci. 1974, 46, 460. (19) Hachisu, S.; Kobayashi, Y. J. Colloid Interface Sci. 1974, 46, 470. (20) Wadati, M.; Toda, M. J. Phys. Soc. Jpn. 1972, 32, 1147. (21) The effective hard sphere model has been reported to successfully explain some of the experimental results (see ref 11 for an example). However, a major drawback of this model lies in the vagueness of its physical background. In addition, it has been suggested that it contradicts the historically important interpretation by Perrin of sedimentation equilibria.22,23 (22) Perrin, J. Les Atomes; Libraire Felix Alcan: Paris, 1913. (23) Ise, N.; Yoshida, H. Acc. Chem. Res. 1996, 29, 3. (24) Alder, B. J.; Wainwright, T. E. J. Chem. Phys. 1959, 31, 459.
Figure 2. Aqueous dispersion of SI-80P silica particles containing crystallites, which show iridescence due to Bragg diffraction of visible light. Effective surface charge number Ze ) 510, salt concentration Cs ) 2 × 10-6 M, and particle number density np ) 16 µm-3 (φ ) 0.01).
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that ODT and the coexistence can be possibly elucidated within the framework of purely repulsive interaction. Another explanation that is much more straightforward is to assume a long range electrostatic attractive interaction between the particles in addition to the widely accepted Yukawa-type repulsive interaction. The existence of such a long range attraction has recently been demonstrated experimentally1,3,13,14,17,25-35 and also studied theoretically.36 Here, we discuss our recent results concerning ODT in dilute ionic colloidal dispersions, which we believe provide further experimental evidence to add to ongoing discussions on the nature of the interparticle interaction. The present study is based on our recent work9 on a threedimensional phase diagram of the ODT as a function of Ze, np, and Cs, whose details will be reported herein. To examine the influence of Ze, we employed aqueous dispersions of colloidal silica particles. It is known that the surface of silica particles is covered by weakly acidic silanol groups (Si-OH).45 Thus, by adding Brønsted bases to the dispersion, the degree of dissociation of the silanol groups, and hence Ze, can be continuously tuned. Utilizing this feature, we systematically studied the influence of Ze on ODT and found that the disordered state became ordered and thereafter reentered into the disordered state, with increasing Ze. The presence of the reentrant ODT was further generalized using a series of polystyrenebased latex particles possessing various Ze values. On the basis of the three-dimensional phase diagram, we have studied dispersion structures by means of an X-ray scattering method and also by direct observation utilizing confocal laser scanning microscopy (CLSM).46 (25) Dosho, S.; et al. Langmuir 1993, 9, 394. (26) Konishi, T.; Ise, N.; Matsuoka, H.; Yamaoka, H.; Sogami, I.; Yoshiyama, T. Phys. Rev. B 1995, 51, 3914. (27) Yoshida H.; Ito, K.; Ise, N. J. Chem. Soc., Faraday Trans. 1991, 87, 371. (28) Ito, K.; Nakamura, H.; Yoshida, H.; Ise, N. J. Am. Chem. Soc. 1988, 110, 6955. (29) Yoshida, H.; Ise, N.; Hashimoto, T. J. Chem. Phys. 1995, 103, 10146. (30) Ito, K.; Yoshida, H.; Ise, N. Science 1994, 263, 66. (31) Crocker, J. C.; Grier, D. G. Phy. Rev. Lett. 1996, 76, 3862. (32) Larsen, A. E.; Grier, D. G. Nature 1997, 385, 230. (33) Tata, B. V. R.; Arora, A. K. Phys. Rev. Lett. 1992, 69, 3778. (34) Tata, B. V. R.; Arora, A. K. Phys. Rev. Lett. 1994, 72, 786. (35) Muramoto, T.; Ito, K.; Kitano, H. J. Am. Chem. Soc. 1997, 119, 3592. (36) There have been several theoretical attempts to formulate a long range attraction in colloidal dispersions. Attention should be paid to the hydrophobic force predicted by Ninham et al.37 and the work on the charge regulation primitive model by Spalla and Belloni.38 Recently, Wasan et al. calculated the effective pair interaction potential by solving the Ornstein-Zernike equation with the mean spherical approximation closure and also found the existence of an attractive minimum in the pair interaction potential.39 In the framework of the “primitive” model, which treats a colloidal dispersion as a mixture of charged particles in a dielectric continuum, and within the linearized Poisson-Boltzmann treatment, Sogami predicts the existence of a long range attractive interaction.40 Overbeek’s criticism of this theory41 was shown to be incorrect, since it violates the Gibbs-Duhem equation.42 For a detailed discussion, see a monograph by Schmitz43 and his recent paper,44 in which the existence of the attractive minimum based on the pairwise “Gibbsian” free energy is proposed. (37) Dispersion Forces; Mahanty, J., Ninham, B., Eds.; Academic Press: New York, 1976. (38) Spalla, O.; Belloni, L. Phys. Rev. Lett. 1995, 74, 2515. (39) Chu, X.; Wasan, D. T. J. Colloid Interface. Sci. 1996, 184, 268. (40) Sogami, I. S.; Ise, N. J. Chem. Phys. 1984, 81, 6320. (41) Overbeek, J. Th. G. J. Chem. Phys. 1987, 87, 4406. (42) Smalley, M. V. Mol. Phys. 1990, 71, 1251. (43) Schmitz, K. S. Macroions in Solution and Colloidal Suspension; VCH Publishers: New York, 1993; p 126. (44) Schmitz, K. S. Langmuir 1996, 12, 3828. (45) For a review article, see: Iler, R. K. The Chemistry of Silica; Wiley: New York, 1979; Chapters 3 and 6. (46) Confocal Microscopy; Wilson, T., Ed.; Academic Press: London, 1990.
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The former method provides us with statistically averaged information from a macroscopic region, while the microscopy gives us insight into the local structural characteristics of the dispersions. Preliminary results have already been reported in our previous studies.8,9,13 In the present paper, we report systematic and more detailed accounts including the influences of Ze, np, and Cs on the dispersion structures and also on their time evolutions. Light scattering is one of the commonly applied scattering methods to study systems having length scales of microns to submicrons. However, because colloids are often highly turbid, light scattering is not straightforwardly applicable. A contrast matching technique which adjusts refractive index differences between the dispersion medium and the particles may overcome this limitation, but it is difficult to apply the technique to aqueous dispersions. Therefore, in the present study, we used an ultrasmallangle X-ray scattering (USAXS) method utilizing a BonseHart camera.47 USAXS can detect electron density fluctuations up to several microns and is suitable for studying colloidal systems. Because silica particles possess high electron density contrasts in their aqueous dispersions, precise determination of the dispersion structures is possible by applying this technique. Microscopy is widely applied for studying colloidal systems.2 However, conventional microscopic observation is limited to a region close to a cover slip (container wall), where the glass-dispersion interface might seriously influence the dispersion properties. To eliminate such a wall effect, we applied CLSM and successfully observed both individual particles and their higher order organizations simultaneously in the internal region of the dispersions (up to 200 µm inside). Especially we focused our attention on the location and the dispersion structures of a biphasic region, where the ordered and disordered phases coexist, using this technique. The organization of the present paper is as follows. Experimental details are described in section II. Determination of the effective surface charge number of the particles is mentioned in section III.1. Section III.2 presents a phase diagram of a ODT and the USAXS results utilizing the silica systems. The coexistence of the ordered and disordered phases is discussed in section III.3 on the basis of USAXS and CLSM studies for the silica dispersions as well as ionic polymer latex systems. Finally, conclusions are made in section IV. II. Experimental Section II.1. Colloidal Particles. The colloidal silica particles, Seahoster KE-P10W and Cataloid SI-80P, were purchased from Nippon Shokubai Co., Ltd. (Osaka, Japan) and Catalyst & Chemicals Co., Ltd. (Tokyo Japan), respectively, in the form of aqueous dispersions stabilized under alkaline conditions. The polystyrene-based latex particles, SS-10 and SS-16, were synthesized in our laboratory by copolymerization of styrene and styrenesulfonate following a standard emulsifier-free emulsion polymerization method.48 Polystyrene-based latex N100 was purchased from Sekisui Chemical Co., Ltd. (Osaka, Japan). The diameters of the particles used are compiled in Table 1. They were determined as follows. The diameters of the silica particles, which have sufficient contrast in an aqueous dispersion for X-ray measurements, were measured by the USAXS method described in section II.3. The scattering profiles of the dispersions were obtained at Cs ) 5.0 × 10-4 M, for which electrostatic interparticle interactions are negligibly small. The volume fraction of the samples was φ ) 3 × 10-2 (np ) 37 and 48 µm-3 for KE-P10W and SI-80P, respectively49). The scattering profiles (47) Bonse, U.; Hart, M. Z. Phys. 1966, 189, 151. (48) Chonde, Y.; Kridger, I. M. J. Appl. Polym. Sci. 1981, 26, 1819. (49) φ ) np × (4/3)πR3, where R is a particle radius (in µm).
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Table 1. Properties of the Colloidal Particles Used. sample silica KE-P10W SI-80P latex SS-10 SS-16 N100
diametera (µm) surface charge avg stand dev group 0.12 0.11 0.12 0.11 0.12
0.006 0.006 0.002 0.002 0.003
silanol silanol sulfonic acid sulfonic acid sulfonic acid
Zab
Zeb
200 510 4 400 720 10 300 1100 13 800 1300
a Diameters of silica particles are determined by USAXS, while those of latex particles are determined by TEM and DLS. b For protonated particles in deionized condition (Cs ) 2 × 10-6 M).
were analyzed by fitting to the theoretical form factor for an isolated particle.50 For the latex particles, transmission electron microscopy (TEM) and dynamic light scattering (DLS) methods were applied to determine the particle diameters. TEM observations were performed by using a JEM-2000FXZ model (JEOL, Japan) operated at 200 kV for dried samples. For DLS measurements, an ALV-5000e system (ALV GmbH, Germany) was employed to measure autocorrelation functions of sufficiently dilute dispersions at φ ) 1 × 10-5 (np ) 0.01 µm-3) and Cs ) 5.0 × 10-4 M. The autocorrelation functions thus obtained were analyzed by the CONTIN algorithm to determine particle diameters and their deviations. The diameters determined by these methods resulted in identical values within experimental errors. Purification of the dispersions was performed thoroughly as follows. First the dispersions were dialyzed using cellulose tubes (pore size ) 2.4 nm) against purified water for about 30 days. The completeness of the dialysis was judged by conductivity measurements. Then the dispersions were further washed by ultrafiltration using an ultrafiltration cell (Model 8400, Amicon Inc., Beverly, MA) and 0.05 µm pore size membranes (Type VM, Millipore, Bedford, MA). The conductivity and UV absorption of the filtrate were traced to determine the end point of the procedure. After these processes, ion-exchange resin beads (AG501-X8 (D), Bio-Rad Laboratories, Hercules, CA) were introduced into the dispersions and the dispersions were kept standing for at least a week. The specific gravity (F) of the particles, which was determined by a picnometer method, was 2.24 and 2.23 g/cm3 for KE-10PW51 and SI-80P silica particles, respectively. The F values of the polystyrene latex particles were 1.05 g/cm3 for all samples. The particle concentrations of the dispersions were determined by a drying-out method using these F values. Dissolution of the silica into monosilicates had been confirmed to be negligible under the present experimental conditions.51 The water used for the experiments was obtained with a Milli-Q system (Type XQ or SP-TOC, Millipore, Bedford, MA) and had a conductivity of 0.4-0.6 µS/cm. The concentration of ionic impurities in the water was estimated to be (1-2) × 10-6 M. Aqueous solutions of NaOH were prepared using analytical grade NaOH (Merck, Darmstadt, Germany) just prior to use, and their concentrations were determined by conductometric titration with standard HCl aqueous solutions. II.2. Potentiometric and Conductometric Measurements. The electrical conductivity was measured by employing a conductivity meter (DS-12, Horiba, Japan) and two kinds of conductance cells, each of them having a pair of parallel platinum electrodes (cell constants: 1.00 and 1.28 cm-1). The temperature was controlled at 25.00 ( 0.05 °C. pH measurements were performed by using a Horiba F-13 pH meter at room temperature, and the observed pH values were corrected to those at 25 °C. Samples for the measurements were prepared under an Ar atmosphere to minimize contamination by carbon dioxide. To avoid ionic impurities from glass walls, plastic bottles and apparatuses were used. II.3. Ultrasmall-Angle X-ray Measurements. A USAXS apparatus employing a Bonse-Hart camera47,52 was applied to (50) Konishi, T.; Yamahara, E.; Ise, N. Langmuir 1996, 12, 2608. (51) Yamanaka, J.; Ise, N.; Miyoshi, H.; Yamaguchi, T. Phys. Rev. E 1995, 51, 1276. (52) Hart, M.; Koga, T.; Takano, Y. J. Appl. Crystallogr. 1995, 28, 568.
determine the structures of the dispersions. A Cu KR line beam with a wavelength of λ ) 0.1542 nm was used as the X-ray source. Further details of the apparatus used have been presented in a previous paper.53 A sample dispersion was introduced into a quartz capillary having an inner diameter of 2 mm and a thickness of 0.02 mm. After Ar gas had been introduced, the capillary was doubly sealed using plastic films and chemical paste. Unless otherwise stated, the background ionic strength of the dispersion in the capillary was determined to be (2-3) × 10-6 M by control conductometric measurements. The capillary cell was held horizontally with a specially designed cell holder for the measurement. The temperature was controlled at 25.0 ( 0.1 °C. All scattering profiles obtained were corrected for absorption by the sample and background scattering. The smearing effect due to the height and width of the incident X-ray beam was also corrected, except for the profiles showing sharp peaks of sixfold symmetry. II.4. Confocal Laser Scanning Microscopy. An inverted type confocal laser scanning microscope (LSM 410, Carl-Zeiss, Oberkochen, Germany) was used with two water immersiontype objectives having different magnifications: a 40× objective (numerical aperture (NA) 1.2, C-Apochromat, Carl-Zeiss) and a 63× objective (NA 1.2, C-Apochromat, Carl-Zeiss). As light sources, we employed Ar lasers having wavelengths of λ ) 488 and 364 nm. Specially designed quartz and/or polystyrene cells (10 mm in diameter and 15 mm in height, with the bottom made of a cover slip) were used for the observations. The sample dispersion was filtered using a cellulose acetate membrane filter (pore size ) 0.45 µm, DISMIC 25CS, Advante Toyo Co., Ltd., Tokyo, Japan) and then introduced into the cell. For the samples free of NaOH, 30 mg of mixed bed ion-exchange resin beads (AG501-X8 (D), Bio-Rad Laboratories) was also introduced into the cell to maintain the deionized condition. For the colloidal silica samples prepared with additions of NaOH, anion-exchange resin beads (AG 1-X8, Bio-Rad Laboratories) were used instead. The resin beads were kept 3 mm away from the cell bottom using a nylon mesh screen. The ionic strengths of the dispersions in the cell with ion-exchange resin beads were determined to be (1-2) × 10-6 M by control conductometric measurements for both cases. The sample cell was then placed on the stage of the confocal laser scanning microscope, and observations were performed in reflection mode at 24 ( 1 °C. All CLSM observations were performed in internal regions of the samples to rule out any possible influence of the cell wall (cover slip). The distance between the focal plane and the inner surface of the cell wall, Zfocal, was 200 µm. It should be noted that the sizes of the colloidal particles used for the experiments were smaller than the resolution limit of the microscope. Thus, the particles imaged by CLSM correspond to the projections of point spread functions of the particles on the focal plane, which correctly demonstrate the positions (centers of gravity) of the individual particles but not their sizes (see p 25 in ref 46).
III. Results and Discussion III.1. Effective Charge Number. Counterions of colloidal particles are partly condensed near the particle surface due to a strong electrostatic field, which reduces the net surface charge of the particles. Therefore, the effective charge number of the particle Ze is quite different from the analytical (bare) charge number Za. When particles are dispersed in liquids, it is Ze not Za that determines the interaction and, consequently, the state of the dispersion. In the present study, the surface charge numbers of the silica dispersions were altered by partly neutralizing their surface silanol groups by adding NaOH to the system. The Za and Ze values for the partially neutralized silica particles were determined as follows. It has been reported that the isoelectric point of silica is at a pH of about 2.45 At higher pH values, the surface bears negative charges due to dissociation of the weakly acidic silanol groups, (53) Koga, T.; Hart, M.; Hashimoto, T. J. Appl. Crystallogr. 1996, 29, 318.
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at 25 °C in water54) and the particles. κb is the background conductivity due to the water and a small amount of excess NaOH. κb was calculated as the sum of the conductivities of the ionic impurities in the water and the excess NaOH estimated from the pH values. Cc is the concentration of counterions, which is related to Ze by
Ze ) 10-15NACc/np
Figure 3. Potentiometric (a) and conductometric (b) titration curves for salt-free (Cs ) 2 × 10-6 M) aqueous dispersions of KE-P10W (filled circles) and SI-80P (open circles) silica particles at φ ) 0.02 (np ) 24 and 32 µm-3 for KE-P10W and SI-80P, respectively). The [NaOH] regions indicated by solid and dashed arrows in part b correspond to 6 < pH < 8 for SI-80P and KEP10W dispersions, respectively.
and the charge number increases with increasing pH.45 Figure 3a shows potentiometric titration curves for saltfree aqueous dispersions of KE-P10W and SI-80P particles by NaOH solutions at φ ) 0.02 (np ) 24 and 32 µm-3, respectively). In the absence of NaOH, both dispersions were weakly acidic, which was attributable to dissociation of the silanol groups. As the neutralization proceeded, the pH values increased monotonically. We note that the counterions of the silica particles are hydronium (H3O+) ions (hereafter designated simply as H+) in the absence of NaOH and that they are gradually replaced by Na+ by addition of NaOH. For example, at pH ) 6, the [NaOH] value for the KE-P10W dispersion was 5.0 × 10-5 M, while [H+] was 1 × 10-6 M. Thus, under this condition, most of the counterions were Na+. At sufficiently large [NaOH]’s, added NaOH might partly exist in the dispersion medium in excess. However, even at pH ) 8 ([NaOH] ) 4.5 × 10-4 M for the KE-P10W dispersion), the concentration of excess NaOH was approximately 1 × 10-6 M. Thus, at pH < 8, added NaOH was almost completely used up in neutralizing the silanol groups. From the considerations mentioned above, we can conclude that, in the pH range of 6 < pH < 8, most of the counterions are Na+ ions and that the concentration of excess NaOH is negligibly small. Thus, in this pH region, we can safely assume that Za is simply determined by the ratio of [NaOH] to np in the dispersion, which leads to a linear relationship between Za and [NaOH] as
Za ) 10-15NA[NaOH]/np
(1)
where NA is Avogadro’s number and np is given in inverse cubic micrometers. The electrical conductivity of the silica dispersion κ is given by
κ ) 10-3(λcCc + λpnp) + κb
(2)
where λc and λp are the equivalent conductivities of the counterions (349.82 and 50.10 S cm/mol for H+ and Na+,
(3)
Figure 3b shows conductometric titration curves for KEP10W and SI-80P silica dispersions under the same φ and Cs conditions as those for Figure 3a. Both curves show minima at small [NaOH] values. As we have reported before,55 the presence of the minimum can be reasonably explained in terms of (1) the occurrence of counterion exchange between H+ and Na+, which reduces the (averaged) value of λc, and (2) an increase in the degree of dissociation, with which Cc and λp increase. After passing through a minimum, κ increased with increasing [NaOH]. For 6 < pH < 8 (shown by arrows in Figure 3b), the plots were slightly concave upward. As will be described below, this behavior is attributable to counterion condensation. The Cc and Ze values were estimated by comparing the conductivities of silica dispersions having potassium and sodium ions as counterions (hereafter designated as Kand Na-types), assuming that the degrees of dissociation, and thus Cc and Ze, are the same for both cases. A similar treatment was adapted by Vink to determine the charge number of sodium carboxymethylcelluloses in their aqueous solutions.56 It was reported that the Ze values for Kand Na-type silica particles agreed to within 10%, as determined by applying potentiometry.57 From the expressions for κ (eq 2) for these two types of dispersions, we have
κNa-κK ) 10-3(λNa-λK)Cc
(4)
where κNa and κK are the conductivities of Na- and K-type silica dispersions and λNa and λK are the equivalent conductivities of Na+ and K+ ()73.5 S cm/mol), respectively. The measured κNa and κK values for SI-80P silica dispersions at four np values are shown in Figure 4. The Za values were 3800 for all np values examined. By applying eqs 3 and 4, we calculated Cc and Ze from κNa and κK. The transport number of the particles tp (≡λpnp/(λcCc + λpnp)) did not vary significantly with np in the range np ) 16-48 µm-3 (φ ) 0.01-0.03) and was 0.50 ( 0.02 for the Na-type silica. We assumed that the tp value is independent of Za. In Figure 5a, the Ze values thus obtained are plotted against the Za value calculated from [NaOH] using eq 1. It is clearly seen that Ze was always smaller than Za and more significantly different from Za at larger Za’s. This trend is reasonably attributable to counterion condensation, which is expected to be more pronounced at higher surface potentials. We have reported earlier that the relation between the effective and analytical charge densities for ionic polymer latex systems obeyed a power law from experimental data for 16 kinds of latex particles having diameters of 0.08-1.0 µm.55 A double-logarithmic plot of Ze versus Za for the presently examined silica systems displayed in Figure 5b showed good linearity, which implies that the power law holds also for the silica systems. Table 2 shows the coefficients C1 and C2 for the (54) The values at infinite dilution were used in the light of the small values of the counterion concentration Cc. (55) Yamanaka, J.; Hayashi, Y.; Ise, N.; Yamaguchi, T. Phys. Rev. E 1997, 55, 3028. (56) Vink, H. Makromol. Chem. 1982, 183, 2273. (57) Abendroth, R. P. J. Colloid Interface Sci. 1970, 34, 591.
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Figure 4. Electrical conductivities κ versus particle concentration np plot for salt-free dispersions of SI-80P having potassium ions (a) and sodium ions (b) as counterions. Za ) 3800.
Figure 5. Relationships between Ze and Za for salt-free dispersions of the two kinds of silica in linear (a) and double logarithmic (b) scales. φ ) 0.02 (np ) 24 and 32 µm-3 for KEP10W and SI-80P, respectively). Table 2. Coefficients C1 and C2 for the Empirical Relationship between Ze and Za for Silica and Latex Dispersions (Eq 5) sample silica
KE-P10W SI-80P
latexa
C1
C2
0.51 0.52 0.54
1.23 1.16 1.39
a
C1 and C2 values for the latex spheres were determined using the data in ref 55.
empirical relationship
log Ze ) C1 log Za + C2
(5)
Figure 6. Three-dimensional phase diagram for ODT of the silica dispersions as a function of Ze, np, and Cs. The orderdisorder phase boundary is shown by rectangles, which correspond to a region between the smallest Cs for the disordered state and the largest Cs for the ordered state at given Ze and np values. The region whose Cs is smaller than that at the phase boundary is in the ordered phase. The dashed lines are drawn as guides for the eye. The order-disorder coexistence regime is included in the “ordered phase” on this diagram. Sample: KE-P10W.
for SI-80P and KE-P10W silica and the latex systems obtained under salt-free conditions. The values of C1 and C2 were not very different from each other for these three samples. The values of Ze in the absence of NaOH, which were calculated from the measured conductivities and tp values obtained above by using eqs 2 and 3, were 200 and 510 for KE-P10W and SI-80P, as shown in Table 1. Our previous study showed that Ze determined by conductometry at Cs ) 5.0 × 10-4 M agreed with that under the salt-free condition within experimental errors.51 Since we are concerned with Cs less than 1 × 10-4 M in the present study, we assumed that the C1 and C2 values determined here for the salt-free system are valid at any Cs values examined in the present study.58 The latex particles used had strongly acidic sulfonic acid groups on their surfaces. The Za values of the particles were determined by conductometric titration using aqueous NaOH solutions. The values of Ze (with H+ as counterions) were estimated by performing conductivity measurements on the deionized dispersions and by using eqs 2 and 3. In this case the tp values of the particles were assumed to be 0.11 ( 0.02.59 The charge numbers thus obtained are listed in Table 1. III.2. Order-Disorder Transition. III.2.a. Phase Diagram. The ODTs of the silica dispersions were examined at various Ze, np, and Cs conditions. The charge number was varied by adding NaOH in the range of pH < 8. The largest Za we studied was 16 000 (analytical charge density, σa ≡ eZa/4πR2 ) 6 µC/cm2; where e is the electronic charge and R is the particle radius), which gave a Ze of 2400 (effective charge density, σe ≡ eZe/4πR2 ) 0.8 µC/cm2). Cs was estimated from the sum of concentrations of added NaCl, ionic impurities in the water used (2 × 10-6 M), and excess NaOH. (58) At much larger Cs values, the Ze of silica is known to increase with Cs. See ref 45. (59) Ito, K.; Ise, N.; Okubo, T. J. Chem. Phys. 1985, 82, 5732.
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Figure 7. Cross sections of the three-dimensional order-disorder phase boundary for the KE-P10W system presented in Figure 6. (a) Phase boundaries at constant np values of 24, 37, and 49 µm-3. The regions below the phase boundaries, which are shown by rectangles, are ordered phases. (b) Order-disorder phase boundary at a constant Ze of 510. The boundary determined for the SI-80P system (open symbols) is presented as well as that for the KE-P10W system (filled symbols). (c) Cross section of the phase boundary at Cs ) 2 × 10-6 M. The region below the phase boundary is the disordered phase. For all of the diagrams, order-disorder coexistence regimes are included in the “ordered phase”.
Figure 6 is the phase diagram for KE-P10W dispersions as a function of Ze, np, and Cs. It was constructed by observing iridescence due to Bragg diffraction of visible light from the ordered states. The order-disorder phase boundary is shown by rectangles, which correspond to a region between the smallest Cs for the disordered state and the largest Cs for the ordered state at given Ze and np values. Parts a-c of Figure 7 demonstrate cross sections of the three-dimensional phase boundary at constant np, Ze, and Cs, respectively. We note that an order-disorder coexistence regime, if any, is included in the “ordered phase” on the diagrams. With increasing np at fixed Ze, Cs at the order-disorder boundary increased monotonically, which is consistent with the trend reported in the previous studies.1,4,5,18-20 On the other hand, characteristic behavior was observed for the Ze dependence. At small Ze, a disordered phase existed even under salt-free conditions (Cs ) 2 × 10-6 M), at all np values examined. On increasing Ze, ordering took place and then the phase boundary shifted toward higher Cs, as expected. However, with a further increase in Ze, a maximum or a plateau region was observed at around Ze ) 1000-1500 (σe ) 0.4-0.5 µC/cm2), after which the Cs value at the boundary decreased. In other words, there existed a reentrant disordered state in the high-Ze region. The existence of the reentrant ODT was further confirmed by USAXS measurements of the silica dispersions and also by direct CLSM observations using latex dispersions, as will be described in sections III.2.b and III.3. Detailed discussions of the reentrant ODT, including a comparison with a numerical simulation study based on the Yukawa potential UY,60,61 have been reported in our previous paper.9 Several authors4,6 have reported that the phase boundary predicted by the Yukawa potential shows a close agreement with the observed ones, when coupled with a renormalized charge Z*, proposed by Alexander et al.62 It should be noted, however, that these conclusions were drawn on the basis of experimental results at relatively small Ze values. The phase diagram presented in Figure 6 was indeed close to the boundary calculated (60) Kremer, K.; Robbins, M. O.; Grest, G. S. Phys. Rev. Lett. 1986, 57, 2694. (61) Robbins, M. O.; Kremer K.; Grest, G. S. J. Chem. Phys. 1988, 88, 3286. (62) Alexander, S.; Chaikin, P. M.; Grant, P.; Morales, G. J.; Pincus, P. J. Chem. Phys. 1984, 80, 5776.
with UY and Z*, in the small-Ze region (approximately Ze < 1000). However, with further increase in Ze, the predicted boundary tended to level off due to a saturation of Z* and did not demonstrate the reentrant ODT. Thus, in our opinion, arguments in terms of the Yukawa potential and the charge renormalization concept are not applicable to highly charged colloidal systems.9 When the observed Ze (Figure 5) was used instead of Z*, the phase boundary predicted from UY exhibited a maximum. This is because the screening effect is more pronounced at higher Ze values resulted from an increase of the counterion concentration Cc and thus an increase in the ionic strength of the dispersion with Ze. The predicted maximum, however, lies at Ze = 2500, which was much higher than the observed one (Ze = 1000). In the framework of the attraction assumption, Tata et al. have performed Monte Carlo simulation studies63,64 with the Sogami potential.40 Although their simulations were performed under salt-free conditions, they found the occurrence of the reentrant transition with Ze in qualitative agreement with the present results. We also point out that spontaneous formation of void structures, where dense and less dense disordered states coexist, is observed in the reentrant disordered region at low np.29 In Figure 7b, the phase boundary for the SI-80P dispersion at the same Ze ()510) is also shown for comparison. It is in close agreement with that for KEP10W. It should be noted that the former had H+ as counterions, while most of counterions for the latter were Na+. Although a slight difference between their particle radii might affect the transition point, the fairly good agreement between their phase boundaries appears to suggest that the effect of counterion species (H+ and Na+) on the ordering phenomena may not be pronounced. III.2.b.USAXS Studies. The phase diagram was further examined by applying the USAXS method. In Figures 8 and 9, typical scattering profiles for KE-P10W silica dispersions at np ) 37 µm-3 are demonstrated. The profiles are presented as the scattering intensity in arbitrary units versus scattering vector q, which is defined as
q ) 4π sin(θ/2)/λ
(6)
where θ is the scattering angle and λ is the wavelength of the X-ray beam. Figure 8 shows the scattering profiles at Cs ) 1.0 × 10-5 M and at three Ze values. In the
Transitions between Ordered and Disordered Phases
Figure 8. Smeared USAXS profiles showing the reentrant ODT for KE-P10W silica dispersions with varying Ze: (a) Ze ) 200; (b) Ze ) 910; (c) Ze ) 1840. Cs ) 1.0 × 10-5 M and np ) 37 µm-3. The integers beside the arrows show the relative positions of the diffraction peaks with respect to the first peak position.
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Figure 10. USAXS profile showing a powderlike pattern for the ordered KE-P10W silica dispersion. Ze ) 570, Cs ) (2-3) × 10-6 M, and np ) 37 µm-3. The profile was corrected for smearing effects due to the X-ray beam shape and the USAXS optics. The relative positions of the diffraction peaks with respect to the first peak position are indicated by the sides of the arrows.
Figure 11. Experimental conditions where disordered (squares), bcc lattice with sixfold symmetry (circles), and bcc powderlike patterns (triangles) were observed by USAXS measurements of KE-P10W colloidal silica dispersions. np ) 37 µm-3. The relevant order-disorder phase boundary is also shown by gray rectangles. The dashed line is drawn as a guide for the eye. Figure 9. Smeared USAXS profiles showing a sixfold symmetry for the ordered KE-P10W silica dispersions at three rotation angles ω around the capillary axis. Ze ) 910, Cs ) 1.0 × 10-5 M, and np ) 37 µm-3.
disordered region (Ze ) 200, profile a) and the reentrant disordered region (Ze )1840, profile c), the USAXS profile showed a couple of broad peaks, which are typical for a liquidlike structure, in addition to the form factor of an isolated sphere. On the other hand, sharp peaks were observed in the ordered region (Ze ) 910, profile b). The relative peak positions of the sharp peaks with respect to the first peak position were integers from 1 to 4. This suggests that these peaks can be ascribed to the first to the fourth orders of Bragg diffraction. When measure(63) Tata, B. V. R.; Ise, N. Phys. Rev. B 1996, 54, 6050. (64) The reentrant disordered state in the high-Ze region was attributed to a glass structure. See ref 65 for a detailed discussion.
ments were performed by rotating the sample with respect to the capillary axis, similar scattering profiles were observed at each multiple angle of ω ) 60°, as demonstrated in Figure 9. This result implies that the structure had a sixfold symmetry. Therefore, as extensively discussed by Konishi et al.,26 this scattering profile can be attributed to diffraction from the (110) plane of a “single” body-centered-cubic (bcc) lattice, which was maintained parallel to the capillary axis with the [11h 1] direction being vertically upward. As shown in Figure 1, ordered colloidal dispersions consisted of grains (crystallites). The present USAXS results suggest that the sizes of the ordered grains were as large as the width of the incident X-ray beam (approximately 1 mm) under these experimental conditions. Figure 10 demonstrates the USAXS profile of the ordered structure formed at Cs ) (2-3) × 10-6 M, np ) 37 µm-3, and Ze ) 570. Again, several diffraction peaks were
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Figure 12. Confocal laser scanning micrographs showing time evolution of the grain structures in a KE-P10W colloidal silica dispersion. np ) 37 µm-3, [NaOH] ) 0.61 × 10-4 M, Ze ) 570, Cs ) (1-2) × 10-6 M. At t ) 0, the dispersion was homogenized by shear melting. Each micrograph was taken at different lateral positions of the sample at a fixed axial position of Zfocal ) 200 µm. Micrographs were taken using a λ ) 488 nm Ar laser and a 40× (NA ) 1.2) objective. The scale bar superimposed on part a applies to all other micrographs.
observed. However, in this case, the relative positions of the peaks could be represented by square roots of integers from 1 to 5, and the profile did not markedly change with rotations of the sample with respect to the capillary axis. Thus, the profile was ascribable to a powderlike diffraction pattern of a bcc lattice. The first to the fifth diffraction peaks are due to diffraction from the (110), (200), (211), (220), and (310) planes of the bcc lattice. These findings suggest that small ordered grains are randomly oriented in the dispersion under these conditions. In Figure 11, experimental conditions where the three kinds of scattering profiles described above (disordered, bcc lattice with sixfold symmetry, and bcc powderlike pattern) were observed are compiled. The relevant phase boundary for np ) 37 µm-3 determined above is also shown. Clearly, the variation of the USAXS profiles shows a good
correspondence with the phase diagram determined by observing iridescence. It should be recalled that large grains were formed near the order-disorder phase boundary with the lattice plane being oriented parallel to the cell wall, while randomly oriented small grains were observed under off-boundary conditions. The observed difference in the grain orientations appears to be due to a variation in the nucleation and growth mechanisms with changing experimental conditions: At conditions near the phase boundary, where the nucleation rates are small, the crystallization appears to take place mainly at the cell wall, where the required nucleation energy is smaller than that in the bulk (inhomogeneous nucleation). In this case, the arrangement of the lattice planes is strongly regulated by the wall effect,
Transitions between Ordered and Disordered Phases
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Figure 13. Confocal laser scanning micrographs showing an “anomaly” grain growth process in a KE-P10W colloidal silica dispersion observed at a fixed position of the sample. np ) 49 µm-3, [NaOH] ) 0.81 × 10-4 M, Ze ) 570, Cs ) (1-2) × 10-6 M. At t ) 0, the dispersion was homogenized by shear melting. Micrographs were taken using a λ ) 488 nm Ar laser and a 40× (NA ) 1.2) objective at Zfocal ) 200 µm. The scale bar superimposed on part a applies to all other micrographs.
resulting in grain orientations parallel to the cell wall. On the other hand, under conditions far from the phase boundary, where the nucleation rates are larger, the grains are developed by a homogeneous nucleation mechanism throughout the dispersion. The size of the grains may increase due to the grain growth mechanism (see section III.2.c). However, if compared at a time when the crystal structures started to be space-filling, the grain size is larger when the number of the nuclei is smaller. Thus, it is to be expected that the grain size increases when the phase boundary is approached, which is in accord with the observed trend. This variation of the nucleation and growth mechanisms with the experimental conditions is consistent with the currently accepted view on those for atomic and molecular systems.66
III.2.c. Grain Growth. The average size of the ordered grains increased with time following a grain growth mechanism. Confocal laser scanning micrographs showing the time evolution process of the grain structure are demonstrated in Figure 12 (sample: KE-P10W, Ze ) 570, np ) 37 µm-3, Cs ) (1-2) × 10-6 M). The difference in grayness of individual ordered regions reflects variations in their orientations with respect to the focal plane.10 As will be discussed in section III.3.b (Figure 16), this experimental condition lies in the ordered one-phase region, and thus a space-filling ordered structure was (65) Tata, B. V. R.; Yamahara, E.; Rajamani, P. V.; Ise, N. Phys. Rev. Lett. 1997, 78, 2660. (66) Gunton, J. D.; Miguel, M. San; Sahni, P. S. In Phase Separations and Critical Phenomena; Domb, C., Lewobitz, J. L., Eds.; Academic Press: New York, 1973; Vol. 8.
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Table 3. Influence of Surface Charge Number on USAXS Profiles and 2Dexp Values in Colloidal Silica Dispersionsa [NaOH] Cs (×10-6 M) (×10-4 M) 2-3
10
a
0 0.5 0.6 1.0 1.5 2.0 0 1.0 1.5 6.0
Za
USAXS profile
Ze
200 820 520 980 570 1600 740 2500 910 3300 1060 200 1600 740 2500 910 9800 1840
disorder bcc (sixfold) bcc (powder) bcc (powder) bcc (powder) bcc (powder) disorder bcc (powder) bcc (sixfold) disorder
2Dexp 2D0 (µm) (µm) 0.23 0.27 0.28 0.30 0.33 0.30 0.23 0.31 0.33 0.28
0.30 0.33 0.33 0.33 0.33 0.33 0.30 0.33 0.33 0.30
Sample: KE-P10W. np ) 37 µm-3.
Table 4. Influence of Salt Concentration Cs on USAXS Profiles and 2Dexp Values of Colloidal Silica Dispersionsa Cs (×10-6 M)
USAXS profile
2Dexp (µm)
2D0 (µm)
2-3 5 8 10 17 20 30 42 52
bcc (powder) bcc (powder) bcc (powder) bcc (sixfold) disorder disorder disorder disorder disorder
0.30 0.29 0.29 0.30 0.24 0.24 0.23 0.22 0.22
0.33 0.33 0.33 0.33 0.30 0.30 0.30 0.30 0.30
a Sample: KE-P10W. [NaOH] ) 1.0 × 10-4 M, Z ) 1600, Z ) a e 740, and np ) 37 µm-3.
Table 5. Influence of Particle Number Density np on USAXS Profiles and 2Dexp Values of Colloidal Silica Dispersionsa np (µm-3)
USAXS profile
2Dexp (µm)
2D0 (µm)
29 40 48 56 72 80
bcc (powder) bcc (powder) bcc (powder) bcc (powder) bcc (powder) bcc (powder)
0.33 0.30 0.28 0.28 0.25 0.24
0.36 0.32 0.31 0.29 0.26 0.25
a
Sample: SI-80P. Ze ) 510, and Cs ) (2-3) × 10-6 M.
was observed only under low Cs (e2 × 10-6 M) and high np (g49 µm-3) conditions. A similar grain growth process, characterized by the sudden rapid growth of a few grains only, has been reported for metallic systems and is referred to as “secondary recrystallization”.68 It has been reported (1) that the secondary recrystallization takes place above a welldefined minimum temperature and (2) that the size of the secondary grains becomes larger when the minimum temperature is approached.68 Although we could not examine the presence of the “minimum temperature” for the colloidal systems, the fact that this unusual growth process was detected under conditions far from the boundary seems to be reasonable in light of the trend in the secondary grain size for the metallic system. III.3. Coexistence of Ordered and Disordered Phases. III.3.a. Interparticle Distance. The nearest neighbor distance between particles arranged in a bcc lattice, 2Dexp, was determined from the positions of the Bragg peaks in the USAXS profiles by the following relations:
dhkl ) 2πi/qihkl
(7)
aexp ) dhkl(h2 + k2 + l2)1/2
(8)
2Dexp ) (x3/2)aexp
(9)
where h, k, and l are the Miller indices, qihkl is the peak position in the scattering profile due to the i-th order diffraction from an (hkl) plane, dhkl is the distance between (hkl) planes, and aexp is the lattice constant. The average interparticle distance 2D0 was also calculated by the following relations assuming that the dispersion is homogeneously occupied by a perfect bcc lattice.
a0 ) (2/np)1/3
(10)
2D0 ) (x3/2)a0
(11)
where a0 is the lattice constant of a volume-filling bcc lattice containing no lattice defects. For the disordered states, 2Dexp and 2D0 are defined as
formed. A quantitative discussion of the growth law of the grains will be reported separately.67 When compared at the same evolution time, the average grain size was larger under experimental conditions closer to the order-disorder phase boundary, even in the case without the wall effects. For example, when np was reduced while keeping Cs and Ze constant (cf. Figure 6), the observed grain size became larger, as will be demonstrated in section III.3.b (Figure 15b-d). Similar trends for the grain size were also confirmed for the dependence on Ze and Cs. Under “deep quenched” conditions far from the ODT boundary, we sometimes observed that small numbers of extraordinarily large grains grew very rapidly, while the rest of the grains remained small and were eventually swallowed by the large grains. Figure 13 shows CLSM images exemplifying this growth process (sample: KEP10W, Ze ) 570, np ) 49 µm-3, Cs ) (1-2) × 10-6 M). By taking optical sections of the dispersion using CLSM, we found that the large grains started to grow in the internal regions of the dispersion but not in the regions near the cover slip. Therefore, in this case, the large grains were not formed as a consequence of inhomogeneous nucleation at the cell wall. We note that this unusual grain growth
where qm is the scattering vector at the first peak position in the USAXS profiles. Tables 3-5 show the influences of Ze, Cs, and np on 2Dexp, respectively. It is clearly seen from these tables that the 2Dexp values for the ordered states are often smaller than the corresponding 2D0. These results can be interpreted as implying that the ordered regions do not fill the total volume of the dispersion but coexist with disordered regions (liquid and/or gas) having a smaller np than that in the former. We note that the inequality relationship 2Dexp < 2D0 has been reported for ordered dispersions of various dilute ionic colloids under low-ionicstrength conditions.3,8,9,12-14,17,25,26,28,31,32,34,69,70 In the ordered states, 2Dexp increased with increasing Ze from 520 to 910, at Cs ) (2-3) × 10-6 M, and eventually had an identical value to 2D0 (Table 3). This appears to be attributable to a decrease in the volume of the coexisting
(67) Koga, T.; Yoshioka, M.; Yamanaka, J.; Yoshida, H.; Hashimoto, T. In preparation.
(68) Physical Metallurgy, 4th ed.; Cahn, R. W., Haasen, P., Eds.; Elsevier Science: Netherlands, 1996; Vol. 3, Chapter 28.
2Dexp ≡ 2π/qm
(12)
2D0 ≡ (1/np)1/3
(13)
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Figure 14. Variation of the internal structures of the deionized dilute aqueous colloidal dispersions with the effective surface charge number Ze shown by confocal laser scanning micrographs. np ) 2 µm-3 and Cs ) (1-2) × 10-6 M. (a) SI-80P colloidal silica dispersion at [NaOH] ) 0 M; (b) SS-10, (c) SS-16, and (d) N100 polystyrene latex dispersions. The distance of the focal plane from the inside surface of the cover slip was Zfocal ) 200 µm. Micrographs were taken using a λ ) 488 nm Ar laser and a 40× (NA ) 1.2) objective.
disordered phase with Ze. In other words, a one-phase (volume-filling) ordered state is favored under conditions far from the ODT boundary. The decrease of 2Dexp with further increasing Ze to 1060 seems to be reasonable, since the reentrant ODT boundary is approached (cf. Figure 6). A similar behavior of 2Dexp against Ze was observed for Cs ) 10 × 10-6 M (Table 3). In the disordered states, the 2Dexp values were always smaller than those in the ordered states. This may be partly due to a difference between the definitions of 2Dexp in the ordered and disordered regions. However, we note that when particles are distributed homogeneously with liquidlike (or gaslike) spatial distributions, 2Dexp is generally smaller than 2D0, depending on the magnitude of the interparticle interaction. In fact, 2Dexp may be as
small as the particle diameter (0.12 µm in the present case) in the hard sphere limit.71 For the present system, the influence of Cs on 2Dexp in the ordered state was not marked, as shown in Table 4. In the disordered state at Cs g 17 × 10-6 M, 2Dexp decreased with increasing Cs, which is reasonably attributable to reduced electrostatic interaction, as described above. Matsuoka et al.69,70 have performed detailed USAXS measurements on the Cs dependence of 2Dexp, employing several kinds of latex dispersions (diameter ) 0.12-0.26 µm, Za ) 8000-78000). They discovered that 2Dexp (69) Matsuoka, H.; Harada, T.; Yamaoka, H. Langmuir 1994, 10, 4423. (70) Matsuoka, H.; Harada, T.; Kago, K.; Yamaoka, H. Langmuir 1996, 12, 5588. (71) For a review article, see: Egelstaff, P. A. An Introduction to the Liquid State; Academic Press: New York, 1967.
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Figure 15. Variation of the internal structures of the dilute aqueous silica dispersions with particle number density np shown by confocal laser scanning micrographs. Ze ) 510, and Cs ) (1-2) × 10-6 M. Samples: SI-80P colloidal silica dispersions. The distance of the focal plane from the inside surface of the cover slip was Zfocal ) 200 µm. Micrographs were taken using a λ ) 488 nm Ar laser and a 40× objective.
increased with Cs in the ordered state. The maximum value of 2Dexp was equal to 2D0, which was observed at the ODT point. With a further increase in Cs, the system crossed the ODT boundary. In the disordered state, 2Dexp decreased monotonically with increasing Cs, as detected also in the present study. This trend was observed for all the latex particles they employed. The dispersions Matsuoka et al. examined had a face-centered-cubic lattice (fcc) structure in their ordered states, while the present silica systems had bcc lattice structures. Furthermore, the Cs value at the ODT point for the latter was smaller than that of the former. These differences suggest that the interparticle interactions in the present systems are weaker than those employed in their experiments. Thus, it seems plausible that the variation of 2Dexp might not be successfully detected in the present study. Table 5 presents the influence of np on 2Dexp in the ordered state. When np was increased, 2Dexp approached 2D0. This is reasonable, since the ordered one-phase
structure is expected to be favored against the orderdisorder coexistence under higher np conditions. III.3.b. Direct Observations of Order-Disorder Coexistence. The USAXS results presented above suggested that the ordered phase coexisted with a less dense disordered phase at conditions close to the ODT boundary, while a space-filling ordered structure was formed at conditions away from the boundary. However, it is a formidable task to clarify detailed coexistence structures solely by the scattering method without any assumptions. Thus, it is desirable to confirm the existence of the order-disorder coexistence by independent experiments. In the present study, this was achieved by CLSM observations, as will be discussed in the following sections. Figure 14 shows the variation of CLSM images with Ze for silica and latex dispersions at np ) 2 µm-3 and Cs ) (1-2) × 10-6 M. The micrographs were taken in internal regions of the dispersions (distance from the cover slip, Zfocal ) 200 µm), ruling out possible influences of the
Transitions between Ordered and Disordered Phases
Figure 16. Phase diagram of the ODT for deionized ionic colloidal dispersions determined by direct observation utilizing a confocal laser scanning microscope at constant Cs ) (1-2) × 10-6 M; circles, space-filling ordered; triangles, order-disorder coexistence; times signs, disordered. Data points at Ze ) 200, 510, 720, 1100, and 1300 were obtained by using KE-P10W, SI-80P, SS-10, SS-16, and N100, respectively. For the data points at Ze ) 570, KE-P10W dispersions were employed with addition of NaOH.
container walls on the observed structures. At Ze ) 510, the dispersion was homogeneous and disordered (Figure 14a). The particles showed an extensive Brownian motion. However, as displayed in Figure 14b, the internal structure of the dispersion changed to order-disorder coexistence at Ze ) 720. A zoomed-in micrograph (Figure 14b inset) clearly shows that the ordered lattice coexisted with a disordered region. However, individual particles in the disordered region were smeared out and could not be seen due to their Brownian motion. The interface between the ordered grains and the disordered region is rounded. This is typical for the interface between disordered and ordered regions, as it minimizes the surface free energy of the grains by minimizing their interfacial area. As Ze further increased to 1100, the structure became ordered throughout the dispersion and then changed to disordered at Ze ) 1300 (Figure 14c and d). Figure 15 shows CLSM images of the internal structures of the dispersions at four np values at Ze ) 510 and Cs ) (1-2) × 10-6 M. As np increased from 2 to 80 µm-3, the dispersion structure changed from disordered (Figure 15a) to order-disorder coexistence (Figure 15b and c) and then to an ordered one-phase structure (Figure 15d). It is important to note that the order-disorder coexistence structure observed here is somewhat unexpected. As can be seen in Figure 15b and c, the disordered regions exist not only in the interstitial regions between the crystalline grains but also inside the grains.13 Further discussion of this structure is given in section III.3.c. By means of the direct CLSM observations for the dispersions at various np and Ze values, a phase diagram of the system was constructed and is presented in Figure 16. It is clearly seen in the figure that np at the ODT point decreased and then, after passing through a minimum, increased with increasing Ze. This tendency almost quantitatively agrees with the results obtained by observing iridescence (Figures 6 and 7) and by applying USAXS, which clearly confirms the presence of the reentrant ODT with increasing Ze. In Figure 16, a biphasic region determined by CLSM observations, where ordered and disordered regions coexist, is also displayed. The biphasic region was considerably wider than expected, suggesting that the coexistence might be difficult to
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understand solely by assuming the repulsive interaction. At Ze ) 510, the dispersion entered into the biphasic region at np ≈ 7 µm-3 and then into the ordered single phase at np ≈ 100 µm-3. Furthermore, as has been shown by the USAXS measurements, the particle densities in the ordered phases are characteristically larger than those in the disordered phases. As stated in the Introduction, the coexistence of ordered and disordered phases in dilute colloidal dispersions has often been claimed to be explicable by an “effective” hard sphere model. In this case, the origin of the coexistence is basically entropic. However, the numerical simulation72 and experimental73 results obtained with purely repulsive interparticle interaction suggest that the biphasic region may be relatively narrow and that the particle density difference between the ordered and disordered phases may be smaller compared to those determined in the present experiment. Therefore, although we cannot provide further quantitative discussion, the present results seem to be very naturally explained by assuming the presence of a net interparticle attraction.74 III.3.c. “Swiss Cheese”-like Structure.13 As briefly mentioned in the previous section, the morphologies of the order-disorder coexistence structures observed for the colloidal silica dispersions at Ze ) 510 were somewhat unexpected. Namely, we found that the disordered regions existed not only in the interstitial regions between the crystalline grains but also inside the grains. According solely to a conventional nucleation and growth process,75 it is difficult to account for the formation of such a structure. In the present section, we will consider this dispersion structure further. In Figure 17, we present typical confocal laser scanning micrographs showing the order-disorder coexistence structure observed in the internal region of an SI-80P dispersion at Ze ) 510, np ) 29 µm-3, and Cs ) (1-2) × 10-6 M. These micrographs were taken at the evolution time t ) 1 day after homogenizing the dispersion by shear-melting a preexisting structure. A zoomed-in micrograph (Figure 17b), clearly shows that ordered lattices were coexisting with the disordered regions. As stated previously, the difference in grayness of individual ordered regions seen in Figure 17a reflects variations in their orientations with respect to the focal plane.10 Figure 18 presents a three-dimensional feature of the structure observed by CLSM for the dispersion at a slightly smaller np value of 24 µm-3. Cross sections of the three-dimensionally reconstructed CLSM image parallel (Figure 18a) and perpendicular (Figure 18b and c) to the focal plane doubtlessly display that spherical disordered regions exist in the internal region of “potato”-like ordered grains. If we consider the disordered regions in the ordered grains as holes, the ordered grains have structures resembling those of a cheese with holes (“Swiss cheese”). Determination of the lattice structure of the ordered grain was performed by USAXS for the same dispersion used for the CLSM observation presented in Figure 17 at t ) 1 day. The level of Cs was adjusted to that of the CLSM observation by introducing mixed-bed ion-exchange resin beads into the USAXS cell. The scattering profile showed a typical powderlike diffraction pattern of a bcc lattice, (72) Hoover, W. G.; Ree, F. H. J. Chem. Phys. 1967, 47, 4873; 1968, 49, 3609. (73) Pusey, P. N.; van Megen, W. Nature 1986, 320, 340. (74) Polydispersities in particle sizes and surface charge numbers may also affect the ODT and the coexistence of the ordered and disordered phases. Further studies are in progress to evaluate this effect. (75) Aastuen, D. J. W.; Clark, N. A.; Cotter, L. A.; Ackerson, B. J. Phys. Rev. Lett. 1986, 57, 1733.
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Figure 17. Confocal laser scanning micrographs showing the order-disorder coexistence in a SI-80P silica colloid dispersion at (a) low and (b) high magnifications at t ) 1 day. A schematic illustration distinguishing the ordered and disordered phases in micrograph a is also presented for clarity (inset in part A). Cs ) (1-2) × 10-6 M, np ) 29 µm-3, and Ze ) 510. The distance of the focal plane from the inside surface of the cover slip was Zfocal ) 200 µm. Micrograph (a) was taken using a λ ) 488 nm Ar laser and a 40× objective while micrograph (b) was obtained by zooming into the region indicated by the white square in micrograph (a) by employing a λ ) 364 nm Ar laser and a 63× objective.
which was similar to the one presented in Figure 10. The 2Dexp value determined from it was 0.33 µm, while 2D0 was 0.36 µm, as shown in Tables 5 and 6. Thus, it can be concluded that the local particle density of the ordered grain is higher than that of the coexisting disordered phase, as was expected. To investigate the crystallization process of the “Swiss cheese”-like structure, further CLSM observations were performed, and the results are presented in Figure 19. These CLSM images show the time evolution of the internal structure of the dispersion under the same conditions as those employed in Figure 17 (np ) 29 µm-3). The dispersion was homogenized to the disordered structure by shear melting at t ) 0. This state can be regarded as equivalent to a supercooled metallic alloy melt. In
