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How Homogeneous Are “Homogeneous Dispersions”? Counterion-Mediated Attraction between Like-Charged Species† Norio Ise,*,‡ Toshiki Konishi, and B. V. R. Tata§ Central Laboratory, Rengo Company, Ltd., 186-1-4, Ohhiraki, Fukushima, Osaka 553-0007, Japan Received August 24, 1998. In Final Form: December 8, 1998 In solutions or dispersions, solute distributions are considered to be more or less homogeneous and space-filling, particularly for concentrated ones. This is not experimentally the case, however, at low concentrations. Traverse photographs of a homogeneous dispersion of ionic latex particles (volume fraction φ ) 0.05) taken by a Lang camera show the coexistence of ordered domains of particles (as studied by Kossel line analysis) and disordered regions. The video imagery study indicates the presence of at least two diffusion modes for particles in a “homogeneous” dispersion (φ ) 0.02). The confocal laser scanning microscope (CLSM) study shows that negatively charged latex particles are positively adsorbed near likewise negatively charged glass interface. The ultra-small-angle X-ray scattering (USAXS) patterns of 4- or 6-fold symmetry are observed with five or four orders of Bragg diffraction from colloidal silica dispersions (φ ) 0.0376), suggesting the formation of a bcc single crystal with a lattice constant of 0.3 µm. The two-dimensional USAXS study of the colloidal silica dispersion (φ ) about 0.025) gives 22 scattering peaks below a scattering angle of 203′′, which uniquely prove that a single bcc crystal is formed, allowing us to accurately determine the lattice constant and direction of the crystal. The USAXS investigations again confirm the previously found fact that the closest interparticle distance was systematically smaller than the average distance expected from the overall particle concentration. For latices of poly(chlorostyrene-styrenesulfonate) copolymers, which allow concurrent scattering and microscopic studies, the inequality relation of the interparticle spacing is observed and the presence of void structures is visually confirmed at φ ) 0.03 or below. The re-entrant phase transition is found when the net charge density of particles is increased. The bcc-fcc transition, void formation, and the re-entrant behavior can be accounted for by the Monte Carlo simulation with the Sogami potential containing a short-range repulsion and a long-range attraction.
1. Introduction The electrostatic interaction between charged colloidal particles has been claimed to be purely repulsive according to the so-called DLVO (Derjaguin-Landau-VerweyOverbeek) theory.1 The DLVO theory is a mean field theory for one or two ionic particles immersed in simple electrolyte solutions, and strictly speaking, it should be valid and can be used only at the infinite dilution of particles. Intrinsically, therefore, this theory does not satisfy the Gibbs-Duhem (G-D) relationship, which is a thermodynamic requirement for multicomponent systems. It is only at infinite dilution of particles and/or for the infinitely small change of the chemical potential due to particles that the incompatibility with the G-D equation is permissible in practice. Paradoxical situations were pointed out to arise from ignoring or misusing the G-D equation in colloidal systems.2-5 However, the theory has often been claimed to explain various experimental observations carried out at finite particle concentrations. Recent analysis has shown that the agreements with some † Presented at Polyelectrolytes ’98, Inuyama, Japan, May 31June 3, 1998. ‡ Fax: +81-6-465-0220. E-mail:
[email protected]. § Present address: Indira Gandhi Centre for Atomic Research, Kalpakkam, TN 603 102, India.
(1) Derjaguin, B. V.; Landau, L. Acta Physiochem. 1941, 14, 633662. Verwey, E. J. W.; Overbeek, J. Th. G. Theory of the Stability of Lyophobic Colloids; Elsevier: Amsterdam, 1948. (2) Ise, N.; Matsuoka, H.; Ito, K.; Yoshida, H.; Yamanaka, J. Langmuir 1990, 6, 296-302, footnote 32. (3) Smalley, M. V. Mol. Phys. 1990, 71, 1251-1267. (4) Ise, N.; Yoshida, H. Acc. Chem. Res. 1996, 29, 3-5. (5) Schmitz, K. S. Acc. Chem. Res. 1996, 29, 7-11; Langmuir 1996, 12, 1407-1410.
experiments can be reached when not only the purely repulsive DLVO potential but also another potential, for example, the Sogami potential6 (containing short-range repulsion and long-range attraction) is used,7 indicating that the DLVO potential cannot be asserted to be the only correct one. Several paradoxes have been pointed out to emerge from the repulsion-only assumption, which is in line with the DLVO concept.4 Furthermore recent techniques provide experimental facts that are in contradiction with the DLVO theory and can be readily accepted if one assumes the contribution of an electrostatic attraction between particles and/or particles and interface. In this review, we survey such findings obtained for dispersions at low particle concentrations with the special reference to the homogeneity/inhomogeneity in the distribution of particles. Although we fully understand that, with the repulsion-only assumption, an inhomogeneous particle distribution was noticed in the simulation work by Alder et al.8 and the phase transition was demonstrated to take place at volume fractions (φ) of 0.50 or higher,9 the volume fraction where the order-disorder transition is observed for colloidal systems is below 0.05, about 1 order of magnitude lower than in the Alder transition, since only short distances from a central particle (where exclusively the repulsive portion of the total potential comes into (6) Sogami, I. S.; Ise, N. J. Chem. Phys. 1984, 81, 6320-6332. (7) Ise, N.; Ito, K.; Matsuoka, H.; Yoshida, H. In Ordering and Phase Transitions in Charged Colloids; Arora, A. K., Tata, B. V. R., Ed.; VCH Publishers: New York, 1996; p 139. (8) Alder, B. J.; Wainwright, T. W. J. Chem. Phys. 1957, 27, 12081209; 1959, 31, 459-466. (9) Hoover, W. G.; Ree, F. H. J. Chem. Phys. 1968, 49, 3609-3617.
10.1021/la981088l CCC: $18.00 © 1999 American Chemical Society Published on Web 03/19/1999
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Figure 1. (a) Traverse photograph by Ar beams recording the ordered and disordered regions of latex particles in a macroscopically homogeneous aqueous dispersion: particle diameter, 0.16 µm; analytical charge number, -3 × 104/particle; particle volume fraction φ, 0.050. The laser beams entering into the disordered regions are subject strongly to random scattering, diminishing light transmitted to the receiving film. The beams incident to the ordered domains are less dispersed and reach the film without sizable loss of intensity. (b) Backward Kossel image obtained by pinpointing the bright region in (a) with fine laser beams. The image indicates that the crystal has an fcc twin structure. Reproduced from ref 11 with the permission of the American Chemical Society.
question) matter at such high concentrations. It is highly plausible that, at such high concentrations, the contribution of putative attractive interaction cannot be detected. 2. Photographic Records of Inhomogeneous Distribution in Polymer Latex Dispersions: Two-State Structure By using the Lang method in X-ray topography,10 the global internal structure of aqueous dispersions of highly charged polymer latex particles was investigated by Yoshiyama and Sogami.11 After being kept motionless for several months, well-deionized, macroscopically “homogeneous” dispersions were introduced into quartz cuvettes, laser beams irradiated the dispersions, and traverse photographs were taken with the use of a Lang camera. An example is given in Figure 1a, which shows that there are at least two regions having different scattering powers in the dispersion. When fine laser beams were pinpointed on the bright regions, clear Kossel images of diffraction (for example, Figure 1b) were obtained. However no diffraction signal was gained from the dark regions. This implies that the bright and dark patterns in the traverse picture are direct photographic records of the coexistence of the ordered domains of particles and disordered regions in apparently homogeneous dispersions. This is what we call the two-state structure, that was first inferred to exist from the small-angle-X-ray scattering results of ionic polymer solutions12 and later observed by microscope for polymer latex dispersions.13 The Kossel image showed that the ordered domain had an fcc twin structure. Furthermore, the lattice constant (10) Lang, A. R. J. Appl. Phys. 1959, 30, 1748-1755. (11) Yoshiyama, T.; Sogami, I. S. Langmuir 1987, 3, 851-853. (12) Ise, N.; Okubo, T.; Yamamoto, K.; Kawai, H.; Hashimoto, T.; Fujimura, M.; Hiragi, Y. J. Am. Chem. Soc. 1980, 102, 7901-7906. (13) Ise, N.; Okubo, T.; Sugimura, M.; Ito, K.; Nolte, H. J. J. Chem. Phys. 1983, 78, 536-540.
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Figure 2. Trajectory of the latex particles (diameter, 0.3 µm; analytical charge density, -1.3 µC/cm2) in the ordered (lower part of the picture) and disordered (upper part) regions in a dispersion (φ ) 0.02). The particles (in a vertical plane) were photographed in 11/15 s by an Olympus microscope from the side of the dispersion cell and with a TV camera and recorded. The video image was transformed finally into a binary image by using an image data analyzer, and the coordinates of the particle centers were determined. This procedure was repeated for consecutive frames taken at an interval of 1/30 s. Then the centers in 11/15 s were regenerated in one new frame and connected by lines with the image data analyzer. The particle positions at the starting time were shown in green, and those after 11/15 s, in yellow. Note that some trajectories are without green and/or yellow ends. This implies that the particles show out-of-focusplane motion. In other words, the particles are not in a confined geometry. To avoid too complicated a picture, not all of the information in the time span was used. Reproduced from ref 15 with the permission of the American Chemical Society.
was determined to be 0.4697 µm. If the structure covered uniformly the dispersion volume, the lattice constant is expected to be 0.5413 µm for the dispersion of φ ) 0.05. Obviously contraction (13% in the lattice constant) took place during crystallization, as was observed previously for ionic polymer solutions and latex dispersions.12,13 It is to be noted that the contraction can hardly be noticed in concentrated dispersions, because it would be within the experimental uncertainty of the lattice constant. 3. Video Imagery Study of Structural Inhomogeneity in Polymer Latex Dispersions: Two-Diffusion Modes The experimental results obtained by video imagery were reviewed in detail previously.7,14 Thus here only one aspect most pertinent to the present review is mentioned, namely the thermal motion of particles in macroscopically homogeneous dispersions.15 Figure 2 shows the trajectories of latex particles in the ordered and disordered regions at φ ) 0.02. Clearly, the particles in the ordered region exhibit damped motion around the lattice points which is almost indiscernible in this figure in comparison with thermal motion in the disordered region. It is important to note that, while the dispersion appears macroscopically homogeneous, the solute particles have at least two diffusion (14) Dosho, S.; Ise, N.; Ito, K.; Iwai, S.; Kitano, H.; Matsuoka, H.; Nakamura, H.; Okumura, H.; Ono, T.; Sogami, I. S.; Ueno, Y.; Yoshida, H.; Yoshiyama, T. Langmuir 1993, 9, 394-411. (15) Ito, K.; Nakamura, H.; Yoshida, H.; Ise, N. J. Am. Chem. Soc. 1988, 110, 6955-6963.
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modes. As is well-known, this has been confirmed by dynamic light scattering for linear polyelectrolyte solutions.16 4. Positive Adsorption of Ionic Particles near Like-Charged Interface Another example of inhomogeneous distribution of particles is the positive adsorption of charged particles near like-charged interface. This was most unanticipated, since it contradicts with the double layer interaction theory in a straightforward manner. The phenomenon was first noticed and discussed for latex particles by using a confocal laser scanning microscope (CLSM).17 Thomas et al. found the same phenomenon for ionic micelle solutions by using the neutron reflection technique,18 and Ito et al. carried out a systematic investigation for latex dispersions,19 which showed that the positive adsorption of negatively charged latex particles near likewise negatively charged glass surface took place in the distance range of 5-50 µm from the glass surface at low ionic strengths and disappeared in the presence of 10-4 M NaCl. The salt concentration dependence indicated that the adsorption was caused by electrostatic interaction between the charged particles and the surface. By using mixtures of H2O and D2O as dispersant, the gravitational sedimentation was demonstrated not to be the driving force for the adsorption. In a more recent study,20 the adsorption was observed to be enhanced with increasing (negative) ζ potential of the surface. Clearly, the positive adsorption becomes more distinct with increasing magnitude of ζ potentials of the surface. The often advanced interpretation is that interparticle repulsion, which is intensified with increasing particle charges, pushes the particles toward the glass surface, resulting in positive adsorption. This interpretation is unacceptable, however, since the adsorption was observed to be enhanced with increasing (negative) ζ potential of the surface. The electrostatic attraction between the surface and particles is thus intensified with increasing charge number on the surface and particles. The higher the charge number, the stronger the attraction. Although this trend cannot be understood on the basis of the DLVO concept, it is easily accepted if we admit that the attraction is generated between like-charged particles and surface through the intermediary of counterions, since the higher charge number, the more the counterions, and hence the stronger the attraction. This is what the Sogami theory6 qualitatively predicts, as will be discussed below. 5. Ultra-Small-Angle X-ray Scattering of Colloidal (Single) Crystals 5.1. 4-Fold and 6-Fold Symmetries. The conventional small-angle X-ray scattering (SAXS) method is a useful tool to analyze the electron-density fluctuation in the order of 0.1 µm or below. Colloidal crystals typically have much larger dimensions so that this technique cannot be applied. Thus it is necessary to utilize an ultra-smallangle X-ray scattering (USAXS) method. By using the Bonse-Hart camera system21 with Ge or Si single crystals, Konishi et al. could attain the full widths at half-maximum of the rocking curve of 11.1 and 4.0 s of arc, respectively, (16) Schmitz, K. S. Macroions in Solution and Colloidal Suspension; VCH: New York, 1993. (17) See Figure 14 of a review article.14 (18) Lu, J. R.; Simister, E. A.; Thomas, R. K.; Penfold, J. J. Phys. Chem. 1993, 97, 13907-13913. (19) Ito, K.; Muramoto, T.; Kitano, H. J. Am. Chem. Soc. 1995, 117, 5005-5006; Proc. Jpn. Acad. 1996, 72B, 62-66. (20) Muramoto, T.; Ito, K.; Kitano, H. J. Am. Chem. Soc. 1997, 119, 3592-3595. (21) Bonse, U.; Hart, M. Z. Phys. 1966, 189, 151-162.
Figure 3. Ultra-small-angle X-ray (USAXS) scattering profile in counts per second as a function of the rotation angle of the second crystal (2θ˜ ): sample, colloidal silica particles (KE-P10W, Nippon Shokubai Co., Ltd., Osaka, Japan; particle diameter, 0.112 µm; standard deviation, 8%; net charge density, -0.06 µC/cm2 by conductivity measurement; analytical charge density by conductometric titration, -0.24 µC/cm2; particle volume fraction φ, 0.0376; temperature, 25 ( 1 °C. The particle size information was obtained by curve-fitting of the USAXS curves of dispersion sample at low concentration with the theoretical form factor for an isolated sphere (Konishi, T.; Yamahara, E.; Ise, N. Langmuir, 1996, 12, 2608-2610). The original silica aqueous dispersion was dialyzed against Milli-Q reagent grade water. The purified dispersion was introduced into a capillary (length ) 70 mm) of an inner diameter of 2 mm together with ion-exchange resin particles [AG501-X8(D), Bio-Rad Lab. Richmond, CA]. The upper part of the capillary (separated by a nylon mesh) and the lower part were filled with the resin particles. The middle part, where the X-ray beam hit, was free from the resin particles. The dispersion inside the capillary was iridescent. The USAXS apparatus consisted of an X-ray generator, Rotaflex RU-300 (60 kV-300 mA; target, Cu) and a Bonse-Hart camera with one set of Ge or Si single crystals as monochromator: upper curve, rotation angle of the capillary (ω) ) 0°; lower curve, ω ) 45°. The scattering profiles were obtained 4 days after the dispersion was introduced into the capillary. Taken from ref 23 with the permission of the American Chemical Society.
and studied the structure of (single) crystals of colloidal silica particles.22,23 The dispersions were extensively purified and then introduced into a glass or quartz capillary, in which crystallization was allowed to proceed. Figure 3 shows the scattering intensity measured for a silica dispersion in a vertically held capillary at various angles of (horizontal) rotation (2θ˜ ) of the second Ge crystal of the BonseHart camera, which is related to the true scattering angle 2θ by 2θ˜ ) 2θ cos φˆ with φˆ being the angle between the horizontal plane and the plane defined by the paths of the incident and scattered beams.23 The measurements were also done by varying the rotation angle ω of the sample capillary with respect to its axis, which was vertical. The top curve shows five orders of Bragg diffraction at (149n)′′ with n being an integer. When the capillary was rotated around its axis, the same curve was observed at ω ) (90m) ( 1° with m being an integer. Furthermore the dispersion showed similarly sharp peaks at diffraction angles of (109n)′′ when the capillary was rotated by (45 + 90m) ( 1° as shown by the lower curve. No peaks could be observed at ω * (45m)°. The peaks of the upper curve were ascribed to the (110) diffraction with φˆ ) 0 and those (22) Konishi, T.; Ise, N.; Matsuoka, H.; Yamaoka, H.; Sogami, I. S.; Yoshiyama, T. Phys. Rev. B 1995, 51, 3914-3917. (23) Konishi, T.; Ise, N. J. Am. Chem. Soc. 1995, 117, 8422-8424.
Attraction between Like-Charged Species
of the lower curve to the (020) reflections with φˆ ) 0 and also to the (011) and (011 h ) reflections with φˆ ) 45°. The observed ω- and 2θ˜ -dependencies of the profile can be explained by assuming a bcc structure with [001] direction being parallel to the capillary axis and with d110 ) 0.210 µm (from the Bragg relation, sin θ ) nλ/2d). Another interesting observation was that the same dispersion in another capillary displayed a 6-fold symmetry. Although the scattering profiles are not given here, the new symmetry corresponds to the bcc lattice having the same d110 maintained in the capillary with [11h 1] direction of the crystal being parallel to the capillary. It is important to note that we have never observed any other orientations besides the two discussed above for the silica system. Furthermore, with regard to the crystal size, it is pointed out, the Hosemann plot24 and application of the Scherrer equation25 were unsuccessful: the crystal was too large under the present conditions. 5.2. Contraction during Crystallization: Interparticle Attraction. From d110 obtained above, the lattice constant ac can be obtained to be 0.300 µm by using dhkl ) ac/(h2 + k2 + l2)1/2 for the cubic lattice. The closest interparticle distance 2Dexp [)(31/2/2)ac] was 0.260 µm, whereas the average interparticle distance 2D0 [)(31/2/ 2)(8π/3φ)1/3a, with the particle radius a] was 0.290 µm from the particle concentration φ (in vol %) for bcc symmetry. Obviously, 2Dexp < 2D0. This inequality relation was earlier reported by us on the small-angle X-ray profile on ionic polymer solutions.26 Our interpretation was and is correct but appears not to be convincing to some researchers because it was based on the single broad peak. Now with the several orders of diffraction in hand, we claim the existence of the two-state structure14,26 more strongly. The inequality relation implies that contraction took place during crystallization, or the crystal occupies not the whole dispersion volume but only 0.72 [)(2Dexp/ 2D0)3] of the total volume. The rest (0.28) must contain voids and/or free particles for the present case, although the voids were not visually confirmed. 5.3. (110) Planes Parallel to the Capillary Surface: Attraction between Particles and Surface. Whichever of the 6-fold or 4-fold symmetry is maintained, it is easily understood from the crystal geometry that the (110) planes must be parallel to, and in “contact” with, the capillary surface in agreement with the previous microscopic study of latex dispersions27 and Kossel line analysis.28 The present USAXS study negates the frequently advanced view that the most densely packed planes of the bcc structure are formed in “contact” with the container surface by gravitational sedimentation, since the capillary surface is kept vertical. We suggest that another factor that favors the (110) planes consisting of anionically charged colloidal silica particles in contact with (negatively charged) glass surface is the positive adsorption of anionic particles on the like-charged interface, which was discussed above. Because of this factor, the particle concentration near the surface can be higher than in the bulk, facilitating the crystallization at the interface. We note that the fact that a large (single) crystal grew with different symmetries in different capillaries can be accounted for in terms of the Ostwald ripening mecha(24) Hindeleh, A. M.; Hosemann, R. Polymer 1982, 23, 1101-1103. (25) See, for example: Cullity, B. D. Elements of X-ray Diffraction; Addison-Wesley: Reading: MA, 1978; Chapter 3. (26) Ise, N. Angew. Chem. 1986, 25, 323-334. (27) Ito, K.; Nakamura, H.; Ise, N. J. Chem. Phys. 1986, 85, 61436146 and references therein. (28) Yoshiyama, T.; Sogami, I. S.; Ise, N. Phys. Rev. Lett. 1984, 53, 2153-2156.
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Figure 4. Time evolution of voids observed by confocal laser scanning microscope with polystyrene-based latex (diameter, 0.120 µm; analytical charge density, -4.8 µC/cm2; net charge density, -0.48 µC/cm2): φ, 0.001; dispersant, H2O-D2O (densitymatched between solvent and particles); temperature, 23 °C. The observations were made (a) 0, (b) 15 days, and (c) 60 days after the dispersion cell was started to be kept standing without motion. The voids are shown as white objects. The black background shows the liquidlike structure. Each image was reconstructed from 40 horizontal cross sections of the dispersion photographs by CLSM by using the technique reported by H. Jinnai et al. (Macromolecules 1995, 27, 4782). The upper and lower planes of the reconstructed images were 67 and 3 µm away from the coverslip. The scale of the image (a) applies to all other images. Reproduced from ref 36 with the permission of American Institute of Physics.
nism,29 and the 6-fold symmetry seems to be more energetically stable than the 4-fold one as discussed recently.30 6. Recent Microscopic Study of Void Formation As mentioned above, 2Dexp < 2D0 held for various systems. Then, the mass balance requires that there must be simultaneously more dense (ordered) regions and less dense regions, for example voids containing no or few particles as an extreme case. Actually voids in dilute latex dispersions were photographed.31-35 Figure 4 is recent (29) Ostwald, Wil. Z. Phys. Chem. 1900, 34, 495-503. (30) Konishi, T.; Ise, N. Langmuir 1997, 13, 5007-5010.
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CLSM pictures which show how voids developed with time.36 Clearly voids can be unexpectedly large, for example as large as 50 × 200 × 50 µm3,14 and can be found deep inside the dispersions, for example in the distance range from 10 to 60 µm from the coverslip, ruling out the possibility that they were artifacts due to the interface effect by the “uncleaned” coverslip surface or due to particle confinement in a small space as adopted in a recent experiment.37 One of the simplest interpretations is that the voids are formed as a result of interparticle attraction. 7. Two-Dimensional Ultra-Small-Angle X-ray Scattering Study of Colloidal Single Crystal The USAXS apparatus discussed above has its advantage for turbid and/or liquid systems where other techniques such as SAXS, light scattering, and microscopy cannot be applied. However, it cannot conveniently be used for oriented systems because of the smearing effect22,23,38 in the Bonse-Hart camera in which X-rays are collimated only in one plane. Thus a modified version, which we call a two-dimensional ultra-small-angle X-ray scattering (2D-USAXS) apparatus,21,39 was constructed in our laboratory for structural analyses of oriented materials and applied for single crystals of colloidal silica particles in dispersions.40,41 Two sets of two channel-cut single crystals of Ge were used to collimate the X-ray beam in both the horizontal and vertical planes (Figure 5). Figure 6 shows contour plots of the 2D-USAXS intensities from a colloidal silica dispersion against χ and φs of the sample at a scattering vector of 4.02 × 10-3 Å-1 (2θ ) 203′′). Measurements were done also at 2θ ) 118 and 165′′. Twenty-two peaks were observed for 2θ e 203′′. From the observed values of 2θ, φs, and χ of the peaks, the scattering vector Q ) (Q cos χ cos φs, Q cos χ sin φs, Q sin χ) was calculated. Then by using Busing and Levy’s method42 without assuming a priori lattice symmetry, we used a 3 × 3 matrix, which is called UB, to connect the scattering vectors Qx, Qy, and Qz with the Miller indices of the diffraction planes as follows:
( ) () Qx
h
Qy ) UB k Qz l
(1)
Consequently we could determine the matrix UB to be
(
1.21
1.11
UB ) -0.59 0.73 0.93
-0.07 1.35
-0.98 0.93
)
× 105 cm-1
(2)
This implies that a single crystal of a bcc symmetry was formed with a lattice constant of 0.380 µm and the [11 h 1] direction is parallel to the capillary axis. (31) Ise, N. Proc. 19th Yamada Conference on Ordering and Organization in Ionic Solution; Ise, N., Sogami, I. S., Eds.; World Scientific: Singapore, 1988. (32) Kesavamoorthy, R.; Rajalakshmi, M.; Babu Rao, C. J. Phys.: Condens. Matter 1989, 1, 7149-7161. (33) Ito, K.; Yoshida, H.; Ise, N. Chem. Lett. 1992, 2081-2084. (34) Ito, K.; Yoshida, H.; Ise, N. Science 1994, 263, 66-68. (35) (a) Tata, B. V. R.; Yamahara, E.; Rajamani, P. V.; Ise, N. Phys. Rev. Lett. 1997, 78, 2660-2664. (b) Tata, B. V. R.; Ise, N. Phys. Rev. B 1996, 54, 6050-6053. (36) Yoshida, H.; Ise, N.; Hashimoto, T. J. Chem. Phys. 1995, 103, 10146-10151. (37) Crocker, J. C.; Grier, D. G. Phys. Rev. Lett. 1996, 77, 18971900. Larsen, A. E.; Grier, D. G. Nature 1997, 385, 230-233. (38) Guinier, A.; Fournet, G. Small-Angle X-ray Scattering; Wiley: New York, 1955.
Figure 5. Optical system of the two-dimensional ultra-smallangle X-ray scattering apparatus. The two sets of two Ge single crystals were cut parallel to the (111) planes. The mechanism of sample (capillary) rotation is as follows. The vertical ring (χ ring) can be rotated by ωs about the vertical axis. ωs ) 0 when the axis of the χ ring is parallel to the X-ray beam. To change the direction of the scattering vector, we rotated the sample by χ in the plane made by the χ ring and by φs about the sample axis shown in the figure. χ ) 0 when the sample axis is vertical, and φs ) 0 when the x axis of the coordinate system associated with the sample is in the plane of the χ ring. The y axis is normal to the plane of the χ ring when φs ) 0, and the z axis constitutes a right-handed rectangular coordinate system with the x and y axes. For changing the magnitude of the scattering vector, fixing ωC3 ) ωC4 ) 0, we rotate the third and fourth crystals together with the detector, which are surrounded by the dashed line in the figure, by 2θ about the vertical axis at the sample, while keeping the rotation angle ωs of the sample being equal to θ (bisecting position). By temporary removal of the first and fourth crystals, the apparatus is converted to a one-dimensional- (1D-) USAXS apparatus, in which the X-rays were not parallel in the vertical plane and then the detected intensity was smeared. Owing to this effect, it was difficult to study 1D-USAXS data from oriented systems to obtain the scattering intensity as a function of the scattering vector, while desmearing from disoriented samples is easy. Taken from ref 41 with the permission of the American Physical Society.
Conversely, by assuming a bcc single crystal with the lattice constant and direction mentioned above and eqs 1 and 2, we estimated the peak positions, and the results are indicated by circles. The excellent agreement between observed and calculated positions in Figure 6 is noteworthy. In light of this agreement, we positively claim that a large single crystal was formed in the capillary, although the concurrent presence of a small number of small crystals is feasible because free spaces are available in the capillary as a result of contraction during crystallization and also between the curved glass surface and the flat surface of the single crystal. Recently Vos et al.43 reported a structural analysis of polystyrene latex dispersion by using a USAXS technique. They reported many Bragg diffraction peaks from a concentrated water dispersion, which corresponded to an fcc single crystal of a lattice constant of 0.37 µm. The difference in the structural symmetry in their work and ours is due to their volume fraction of 0.56 being much higher than ours (below 0.1). The 2Dexp was found from their data to be 0.257 µm, which agrees with the 2D0 (0.265 µm) from the concentration. The agreement is reasonable (39) Pahl, R.; Bonse, U.; Pekala, R. W.; Kinney, J. H. J. Appl. Crystallogr. 1991, 24, 771-776. (40) Konishi, T.; Yamahara, E.; Furuta, T.; Ise, N. J. Appl. Crystallogr. 1997, 30, 854-856. (41) Konishi, T.; Ise, N. Phys. Rev. B 1998, 57, 2655-2658. (42) Busing, W. R.; Levy, H. A. Acta Crystallogr. 1967, 22, 457-464. (43) Vos, W. L.; Megens, M.; van Kats, C. M.; Bo¨secke, P. Langmuir, 1997, 13, 6004-6008.
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Figure 6. Contour plot of the 2D-USAXS intensity against χ and φs for a colloidal silica dispersion (about 2.5 vol %). Silica particles: the same as in Figure 3. The 2D-USAXS apparatus was composed of a rotating anode X-ray generator (Rotaflex RU-H3R; 60 kV-300 mV; target, Cu) and the Bonse-Hart optical systems with a Ge single crystal. The cross section of the incident X-ray was about 1 × 1 mm2. The full width at the half-maximum of the intensity profile against ωC3 and ωC4 was about 17′′, and the high collimation as a point focusing geometry was confirmed.40 The red circles indicate the calculated positions of the diffraction peaks considering a bcc lattice with a lattice constant of 0.380 µm and with [11 h 1] parallel to the capillary axis. The diffraction peaks were assigned to the diffraction planes of Miller indices given. Taken from ref 41 with the permission of the American Physical Society.
results reported by Robbins et al.46 based on the Yukawa potential and renormalized charge concept.47 Anyway, Figure 7 clearly shows that a subtle change in charge density causes drastic variation in the physical state of the colloidal systems. Unfortunately, the significance of the charge number was not adequately appreciated in previous arguments on colloidal behavior: It was usually treated as an adjustable parameter which was determined by curve-fitting with the theoretical framework (in most cases, with the DLVO-type theories).
Figure 7. Phase diagram or the liquid-solid-liquid transition of colloidal silica dispersions as functions of the net charge density σe, the particle volume fraction φ, and the salt concentration Cs. The solid-liquid boundary is shown by rectangles. The dashed curves are eye guides. Taken from ref 45 with the courtesy of J. Yamanaka and with the permission of the American Physical Society..
under such a high concentration, suggesting the one-state structure in contrast with the two-state structure (2Dexp < 2D0) in much lower concentrations. 8. Re-entrant Phase Equilibrium in Colloidal Dispersions by Varying Charge Number Recently Yamanaka et al. found out how to tune the charge number of silica particles by adjusting the quantity of NaOH to be added44 and studied the influence of the charge number on the solid-liquid-phase transition as functions of the charge density σe, salt concentration Cs, and volume fraction of particle φ.45 Figure 7 shows the phase diagram constructed mainly by iridescence observation. It is seen that, with increasing φ at fixed σe, the Cs at the boundary became larger monotonically as often reported. On increasing σe for φ g 0.02, however, the phase boundary first shifts toward higher Cs and a maximum appears at around σe ) 0.4-0.5 µC/cm2 with further increase, after which the Cs value at the boundary decreases. In other words, there exists a re-entrant liquid state at high σe. It was demonstrated that this re-entrant phase transition is not consistent with the simulation (44) Yamanaka, J.; Hayashi, Y.; Ise, N.; Yamaguchi, T. Phys. Rev. E 1997, 55, 3028-3036.
9. Charge Density and Interparticle Interaction That the charge number or density of colloidal particles was not given due attention in most of the previous works, is also demonstrated by the fact that the charge number obtained by the curve-fitting was not critically examined by independent methods. This would have had something to do with confidence in the DLVO theory. However, in light of various experimental observations in contradiction with this theory, this situation is rather unhappy, if it is aimed to scrutinize the pros and cons of the theory and the adopted experimental techniques (and hence the results obtained therefrom). Apart from this feature of the charge density, careful examination of some previous papers reveals an interesting trend, as seen from Table 1. This table shows the net charge number and density of particles employed by previous authors. In category A, in which only repulsion was claimed to be detected, the charge density was rather low, if the values obtained by curve-fitting are reliable. On the other hand, in category B, in which attraction and repulsion were detected, high charge density particles were used. The stronger attraction for higher charge density particles was observed by us13,48 and was pointed out to be in accord with the Feynman scheme,49 by which the intrinsic nature of the observed attraction could be understood clearly. Furthermore this tendency could be described by the Sogami theory6 but not by the DLVO (45) Yamanaka, J.; Yoshida, H.; Koga, T.; Ise, N.; Hashimoto, T. Phys. Rev. E. 1998, 80, 5806-5858. (46) Robbins, M. O.; Kremer. K.; Grest, G. S. J. Chem. Phys. 1988, 88, 3286. (47) Alexander, S.; Chaikin, P. M.; Grant, P.; Morales, G. J.; Pincus, P. J. Chem. Phys. 1984, 80, 5776. (48) Ise, N.; Okubo, T. Acc. Chem. Res. 1983, 13, 303-309. (49) Feynman, R. P.; Leighton, R. B.; Sands, M. The Feynman Lecture on Physics; Addison-Wesley: Reading, MA, 1963; Vol. 1, Chapter 2, pp 2-3.
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Table 1. Net Charge Density of Colloidal Particles author
net charges/ particle
net charge density (µC/cm2)
radius (Å)
Category A (Only Repulsion Detected or Claimed) Ackerson 360 0.038a 1000 Versmold 190 0.001b 3850 Grier 1991 0.02c (3.8c) 3200 Grier (2d) 3250 Grier 5964 0.07e 3250 Ito Kepler Yoshida Yamanaka Tata Tata
Category B (Attraction Detected) 65000 1.33f 515000 2g 530 0.23h 1400 0.5h 500 0.21i 1590 0.25 j
ref 50 51 52 53 37
implies that, at low and high concentrations of particles and salts and at low and high charge densities of particles, the potential minimum is too shallow and hence the attraction is very weak to be observed: At least qualitatively, the κ dependence of the potential minimum explains first what was mentioned above by the re-entrant phase transition and second why the attraction could not be detected by researchers in category A of Table 1. 10. Monte Carlo Simulation
2500 6350 530 600 550 900
54 55 56 45 57 35a
a Obtained by a curve-fitting to a model calculation by the authors. Estimated by a curve-fitting of experimental data with the DLVO theory. c Obtained by a curve-fitting of the measured potential with the DLVO potential for a Duke sample (N.5065A). This sample was also used in ref 37, in which a different σe value of 0.07 µC/cm2 was derived from an independent potential measurement with curvefitting with the DLVO potential. The value in the parentheses is estimated from the titratable, analytical charges reported (3.2 × 105 e/particle) by the authors. d Obtained from the titratable charges reported (1 e/10 nm2) by the authors. e Obtained from a curvefitting of the observed potential with the DLVO theory. f Derived from σa determined by the conductometric titration with the fraction of free counterions (f) obtained by the transference experiment assuming σe ) fσa, with f being 0.1 for the present condition. See: Ito, K.; Okubo, T.; Ise, N. J. Chem. Phys. 1985, 82, 5732. g Derived from a reported value of 0.1 e/nm2. Whether this value denotes σa or σe is not clear. If it refers to σa, σe would be about 0.16 µC/cm2, being still larger than those used in category A. h Determined by conductance measurements. i Obtained by curve-fitting of experimental data of vapor-liquid equilibrium with the Sogami theory. j Determined by conductance measurements. b
Figure 8. Position of the minimum of the Sogami effective pair potential and its depth as a function of κa. Taken from ref 6 with the permission of the American Institute of Physics.
potential (electrostatic repulsion plus van der Waals attraction). Figure 8 gives the position (Rmin) of the minimum of Sogami’s effective pair potential and its depth as a function of κa (κ, the extended Debye screening parameter; a, particle radius). The unique aspect is that the potential depth is zero at very low κa, becomes deeper with increasing κa, and passes through a minimum. This
The microscopic method provides local information in real space while the scattering techniques give average information on global structure in the Fourier space. They are supplementary. Concurrent studies by these two techniques are useful. For this purpose, poly(chlorostyrene-styrenesulfonate) (abbreviated as PCSS) latex particles were synthesized since they have strong electron density contrasts in water and can be made large enough to study under a microscope. Though the original paper should be consulted for detail,35a the USAXS study (carried out at latex volume fractions between 0.006 and 0.06) clearly confirmed 2Dexp < 2D0 also for the present case, and correspondingly, void structures were observed by CLSM. This result was further examined by constant volume Monte Carlo simulations35b using the Sogami potential6 which has the form Us(r) ) B[(A/r) - κ] exp(-κγ), where A ) 2 + κd coth(κd/2) and B ) 2[Ze sinh(κd/2)/κd]2/, κ2 ) 4πe2(npZ + Cs)/(kBT), d is the particle diameter, Ze is the particle charge, np is the particle concentration in cm-3, Cs is the salt concentration, T is the temperature (298 K), is the dielectric constant of water, and kB is the Boltzmann constant. The first term of Us(r) is repulsive whereas the second term is attractive. The position of the potential minimum Rm is given as Rm ) {A + [A(A + 4)]1/2}/2κ. The simulation was performed with 432 particles, and checks were done to test the equilibrium state and system size dependence. After equilibrium was reached, the pair correlation function, g(r), was calculated using the particle coordinates using procedures reported earlier58,59 and is shown in Figure 9a. The g(r) except at the lowest concentration showed the first peak and split second peak, in conformity with the USAXS results. Furthermore, the first peak appeared at smaller r than 2D0, as observed. Figure 9b gives the corresponding projection of the coordinates in the Monte Carlo cell, which clearly shows void structures. Figure 10 shows the effects of the net charge density σe and salt concentration Cs upon the structural ordering for dilute dispersions.60 For dispersions with σe ) 0.2 µC/cm2 and Cs ) 0, the simulation shows a homogeneous crystalline order (bcc) at φ ) 0.03. Upon an increase of the charge on the particles (Figure 10A), dispersions become inhomogeneous in the form of dense phase coexisting with voids, in the range studied. It seems that there exists a (50) Ackerson, B. J.; Clark, N. A. Phys. Rev. A 1984, 30, 906-918. (51) Vondermassen, K.; Bongers, J.; Mueller, A.; Versmold, H. Langmuir 1994, 10, 1351-1353. (52) Crocker, J. D.; Grier, D. C. Phys. Rev. Lett. 1994, 73, 352-355. (53) Larsen, A. E.; Grier, D. C. Phys. Rev. Lett. 1996, 76, 3862-3865. (54) Ito, K.; Nakamura, H.; Ise, N. J. Chem. Phys. 1986, 85, 61366142. (55) Kepler, G. M.; Fraden, S. Phys. Rev. Lett. 1994, 73, 356-359. (56) Yoshida, H.; Yamanaka, J.; Koga, T.; Ise, N.; Hashimoto, T. Langmuir 1998, 14, 569-574. (57) Tata, B.V.R.; Rajalskshmi, M.; Arora, A. K. Phys. Rev. Lett. 1992, 69, 3778-3781. (58) Tata, B. V. R.; Arora, A. K.; Valsakumar, M. C. Phys. Rev. E. 1993, 47, 3404-3411. (59) Tata, B. V. R.; Arora, A. K. J. Phys. Condens. Matter 1991, 3, 7983; 1992, 4, 7699; 1995, 7, 3817-3834. (60) Tata, B. V. R.; Ise, N. Phys. Rev. E. 1998, 58, 2237-2246.
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Figure 10. (A) Coordinate-averaged pair correlation function gc(r) vs r for dispersion for various charge density σe at a salt concentration Cs ) 0 and particle volume fraction φ ) 0.03. Curves a-c correspond to σe ) 0.2, 0.4, and 0.68 µC/cm2, respectively. The projections of the time-averaged particle coordinates in the Monte Carlo cell are shown in the insets. (B) gc(r) for various Cs for σe ) 0.35 µC/cm2 and φ ) 0.03. Curves a-c correspond to Cs ) 0, 2, and 15 µM, respectively. Taken from ref 60 with the permission of the American Physical Society.
Figure 9. (a) Pair correlation function g(r) obtained by Monte Carlo simulation using the Sogami potential at φ ) 6, 2.4, 1.2, and 0.6 vol %: particle diameter, 0.110 µm; net charge density, -0.25 µC/cm2. The top three curves were shifted vertically for the sake of clarity. The vertical line denotes the average spacing 2D0 obtained from the concentrations. (b) Projection of the particle coordinates in the Monte Carlo cell, showing void structure. The coordinates are normalized to the unit of 2D0. Taken from ref 35b with the permission of the American Physical Society.
critical charge density below which the dispersions stay homogeneously ordered. Figure 10B shows the influence of salt concentration on a dispersion whose charge density is just below the critical value. The dispersion, which is homogeneous and ordered at Cs ) 0, becomes inhomogeneous with increasing Cs in the range studied. To show clearly that the re-entrant phase equilibrium in terms of the charge density is at least qualitatively substantiated by the MC simulation with the Sogami potential, Figure 11 was constructed by combining the data in refs 35b and 60, where Smax is the first peak height of the structure factor obtained for φ ) 0.03, Cs ) 0, and d ) 0.110 µm. It is to be noted that the phase equilibrium (Figure 7) observed at Cs > 2 µM is qualitatively reproduced by the simulation, although quantitative comparison is yet to be established because of the difference in the salt concentrations.61 (61) The simulation at Cs ) 0 shows that the solid phase is spacefilling, whereas the experimental observation at Cs > 2 µM indicates the nonspace-filling nature of the solid phase.62 This appears to be due to the difference in the salt concentration. Although more detailed study is necessary and in progress, whether the phase is space-filling or not depends sensitively on the salt concentration, as seen from Figure 10B. (62) Yamanaka, J.; Koga, T.; Ise, N.; Hashimoto, T. Prog. Colloid Polym. Sci. 1997, 106, 270-273.
Figure 11. Structure parameter Smax obtained by the Monte Carlo simulation using the Sogami potential as a function of charge density of particles at φ ) 0.03, Cs ) 0, and d ) 0.110 µm.
Thus, it can be claimed that the Sogami potential provides at least qualitatively satisfactory agreement with the observation. We note that the DLVO potential cannot reproduce the inhomogeneous distribution of particles such as void formation in dilute dispersions. 11. Concluding Remarks The experimental facts obtained by recently developed techniques show that particle distribution in macroscopically homogeneous dispersions is not homogeneous; even fairly large voids are present. The qualitative agreement between these experiments and Monte Carlo simulation indicates that the Sogami potential is more realistic to explain such a situation than the DLVO potential. In other words, there is generated counterion-mediated attraction between like-charged particles, which was not considered in the DLVO theory. The most recent observation of positive adsorption of charged particles near the like-
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charged interface suggests that this concept should be extended from particle-particle systems to particleinterface systems. However, quantitative tests of the Sogami potential are not adequate enough; further critical examination is necessary and in progress of this potential and also of other theoretical approaches advanced by others.63 It is to be noted that the counterion-mediated attraction in question is of long range. For typical colloidal particles, the potential minimum appears at 0.5-1 µm from particle center. Direct measurements of the interparticle force by SFM and AFM techniques could cover only short distances (up to 0.1 µm in most cases), so that the attraction could not be detected so far. According to the optical trapping techniques employed by Grier and Sugimoto et al.,37,64 like-charged particles were reported to simply repel each other in conformity with the DLVO potential, unless confined near a glass wall. The reason they failed to detect the attraction is very clear to us: Grier’s samples were not highly charged, as shown in Table 1, and Sugimoto et al. chose high salt conditions and very short distances from particle surface, where the attraction is not measurably strong. It is hoped that longer distances and low salt conditions can be studied by technical improvement and well-characterized, highly charged particles be used in future work. Acknowledgment. The authors thank Professor I. Sogami, Department of Physics, Kyoto Sangyo University, Kyoto, Japan, and Drs. J. Yamanaka and Hiroshi Yoshida, ERATO, Japan Science and Technology Corp., for their constant discussion and help and Professor K. Ito, Department of Chemical and Biochemical Engineering, Toyama University, Toyama, Japan for his discussion and providing us materials before publication. B.V.R.T.’s stay in Osaka, Japan, was made possible by the Visiting Scientists Program of Rengo Co., Ltd. Note Added in Proof. In the foregoing article, counterion-mediated attraction between like-charged species has been discussed for colloidal particles. In light of the (63) Chu, X.; Wasan, D. T. J. Coll. Interface Sci. 1996, 184, 268-278. (64) Sugimoto, T.; Takahashi, T.; Itoh, H.; Sato, S.; Muramatsu, A. Langmuir 1997, 13, 5528-5530.
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fundamental nature of the attraction, a similar sort of attraction may be expected for other systems as well. An example was discussed by Yamanaka et al. as regards conformation of flexible linear polymers in solution (Yamanaka, J.; Matsuoka, H.; Kitano, H.; Hasegawa, H.; Ise, N. J. Am. Chem. Soc. 1990, 112, 587-592). It was believed that because of repulsive interaction between ionized groups inside a macroion, the macroion assumes a fully stretched conformation. Yamanaka et al. argued that a large number of counterions as well as ionized groups are present in the macroion domain, which corresponds to a highly concentrated ionic solution. In the widely accepted interpretation, these ionized groups were claimed to repel each other to result in a fully stretched conformation. The early justification for this was the exponent R of the Houwink-Mark-Sakurada equation being close to 2. Subsequent careful measurements of the intrinsic viscosity revealed the presence of a maximum in the reduced viscosity-concentration curve. After this maximum was taken into consideration, Yamanaka et al. obtained R values of 1.6-1.2 for polystyrenesulfonate samples, concluding that the macroion is not fully stretched. They ascribed this result to the formation of intramolecular multiple ions due to “counterion-mediated attraction between ionized groups”. This result from the viscosity measurements is in line with the light scattering data by Krause et al., who demonstrated that the observed angular dependence of the excess scattering intensity can be much better fitted by a single chain form factor assuming coillike chain conformation than the fully extended conformation for sodium polystyrenesulfonate in water at 3.46 × 10-4 g/L (Krause, R.; Maier, E. E.; M. Deggelmann, H.; Hagenbu¨chle, M.; Schulta, S. F.; Weber, R. Physica 1989, 160, 135). A recent computer simulation study showed that a “rodlike chain is a rarity” and “counterion condensation can dramatically shrink the polyelectrolyte chain”, though not for typically long chains (Stevens, M. J.; Kremer, K. J. Chem. Phys. 1995, 103, 1669-1690). More detailed experiments proving the multiple ion formation inside the macroion domain are awaited for. LA981088L
