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Langmuir 1996, 12, 5588-5594
Exact Evaluation of the Salt Concentration Dependence of Interparticle Distance in Colloidal Crystals by Ultra-Small-Angle X-ray Scattering. 2. The Universality of the Maximum in the Interparticle Distance-Salt Concentration Relationship Hideki Matsuoka,* Tamotsu Harada, Keitaro Kago, and Hitoshi Yamaoka Department of Polymer Chemistry, Kyoto University, Kyoto 606-01, Japan Received October 23, 1995. In Final Form: August 23, 1996X We have investigated the salt concentration dependence of the interparticle distance in colloidal crystals in dispersion by the ultra-small-angle X-ray scattering (USAXS) technique. USAXS gives us the internal, global, and general information for the structure of colloidal crystals. The nearest neighbor interparticle distance (2Dexp) increased at low salt concentration with increasing ionic strength in the dispersion. After passing through a maximum, 2Dexp decreased. The maximum value of 2Dexp was nearly equal to the calculated distance (2D0), assuming uniform distribution of the particles in the dispersion. This quite unique tendency, which we reported in our previous letter (Langmuir 1994, 10, 4423), was found to be universal by systematic USAXS experiments for various kinds of latices which were different in size and charged state. The salt concentration at the maximum of the 2Dexp vs [salt] profile was different for each latex particle, but it was found that this position is scaled by κa (κ-1, Debye length; a, particle radius): the condition of κa ) 1.3 is the maximum point for all latices studied. The theories proposed for colloidal interaction to date cannot explain this curious phenomenon. These observations by USAXS seem to be predicting some possibility of transition in structure and in interaction at this maximum position.
Introduction The formation of ordered structures, or colloidal crystals, in ionic colloidal dispersions at low ionic strength is a hot topic in colloid science.1 The formation of such structures had been anticipated from the observation of iridescence in dispersions of latex particles: The iridescence is caused by the Bragg diffraction of visible light by an ordered array of latex particles, as shown by Hachisu et al., who used an ultramicroscope to observe the structure directly.2 To date, the phenomenon has attracted keen attention because of its uniqueness not only as a colloidal phenomena but also as a model for atomic systems and metals. In a concentrated dispersion, the structure formation is easily understood in terms of an excluded volume effect. However, the interesting thing is that the ordered structure formation can be observed in rather dilute dispersions. From this fact and that the addition of the salt destroys the ordered structure, the origin of the structure formation has been thought to be due to electrostatic forces. The widely accepted concept was an explanation by DerjaguinLandau-Verwey-Overbeek (DLVO) theory,3 that is by an electrostatic repulsion force between the charged particles. However, there have been many reports by Ise and co-workers1 on interesting phenomena which cannot be simply explained by an electrostatic repulsion force. These include the observation of a two-state structure,4 void structures,5 etc. by ultramicroscope and laser scanning (confocal) microscope (LSM) techniques. From these results, the existence of an electrostatic attractive force
in addition to the electrostatic repulsion has been proposed by Ise et al.1 Theoretical considerations on the attraction have been reported by Sogami et al.6,7 and further developed by Sogami-Shinohara-Smalley.8 They cause controversial discussions in the field.9 Although microscopic observation (direct observation, photographs, video imagery) is very impressive since it provides direct information for the naked eye, one should pay attention to the fact that the information obtained by these techniques is very limited local information. In addition, the field of vision is limited to the region very close to the cell (normally glass) surface. The low number of particles sampled in such data is also always a problem. Hence, it is thought to be very difficult to prove the universality of the information obtained by microscopic observation. One may say that this limitation is partially, but not completely, circumvented by the LSM technique. The same situation applies to the reflection spectra technique. On the other hand, the scattering technique gives us global information. But the light scattering (LS) technique cannot be applied to turbid colloidal systems. For latex particles made of polystyrene or poly(methyl methacrylate) in water, there is a strong possibility that the scattering does not satisfy the Rayleigh condition10 but should be treated as Mie scattering.11 The condition is limited by the refractive index and the size of the particle relative to the wavelength.12 Since there is a possibility to apparently satisfy the Rayleigh condition for relatively small latex particles,13,14 some structural studies for latex
* To whom correspondence should be addressed. X Abstract published in Advance ACS Abstracts, November 1, 1996.
(6) Sogami, I. S. Phys. Lett. A 1983, 96, 199. (7) Sogami, I. S.; Ise, N. J. Chem. Phys. 1984, 81, 6320. (8) Sogami, I. S.; Shinohara, T.; Smalley, M. V. Mol. Phys. 1991, 74, 599. Sogami, I. S.; Shinohara, T.; Smalley, M. V. Mol. Phys. 1992, 76, 1. (9) Levine, S.; Hall, D. G. Langmuir 1992, 8, 1090. Ettelaie, R. Langmuir 1993, 9, 1888. Overbeek, J. Th. G. Mol. Phys. 1993, 80, 685. Smalley, M. V. Langmuir 1995, 11, 1813. Smalley, M. V.; Sogami, I. S. Mol. Phys., in press. (10) Rayleigh, R. Philos. Mag. 1871, 41, 107, 447. (11) Mie, G. Ann. Phys. 1908, 25, 377. (12) van de Hulst, H. C. Light Scattering by Small Particles; Dover: New York, 1957.
(1) For a convenient review, see: Dosho, S.; et al. Langmuir 1993, 9, 394. (2) Kose, A.; Ozaki, M.; Takano, K.; Kobayashi, K.; Hachisu, S. J. Colloid Interface Sci. 1973, 44, 330. (3) Verwey, E. J. W.; Overbeek, J. Th. G. Theory of the Stability of Lyophobic Colloids; Elsevier: Amsterdam, 1948. Derjaguin, B. V.; Landau, L. Acta Physicochim. 1941, 14, 633. (4) Ise, N.; Okubo, T. Acc. Chem. Res. 1980, 13, 303. (5) Ito, K.; Yoshida, Y.; Ise, N. Chem. Lett. 1992, 2081. The top cover of Langmuir 1993, 9 (2).
S0743-7463(95)00916-4 CCC: $12.00
© 1996 American Chemical Society
Interparticle Distance-Salt Concentration Relationship
dispersions by LS have been reported.15 Recently, many attempts have also been made to estimate the static structure factor S(q) by dynamic light scattering (DLS),16-18 but similar problems are encountered as for LS. The limitation cannot be removed if one uses visible laser light as the observation tool for a turbid system such as a colloidal dispersion. If one use X-rays as the light source of the scattering measurement, the effect of the turbidity becomes no problem because of the penetrating power of X-rays. However, because of the short wavelength of X-rays, the small-angle resolution needs to be dramatically higher than that of conventional small-angle scattering in order to study colloidal systems whose dimensions are of the order of several hundred or thousand angstroms. In these circumstances, we have constructed an ultrasmall-angle X-ray scattering (USAXS) instrument, following the principle of Bonse and Hart.19 The optical principle of USAXS is very different from the conventional one,19 and its small-angle resolution extends the spatial range of observation up to about 8 µm, which is suitable for the structural study of colloidal systems.20 We have shown that the USAXS technique can give us useful and general information about the internal structure of turbid colloidal systems and opaque polymer materials.21-25 Since it is clear that the driving force for the formation of the ordered structure or colloidal crystal is electrostatic, the structure is quite sensitive to the salt concentration (i.e. the ionic strength) of the dispersion. Under high-salt conditions, the ordered structure cannot be maintained. The salt concentration dependence of the nearest neighbor interparticle distance (2Dexp) has been measured by microscopy, and the result was a monotonic decrease of 2Dexp with increasing salt concentration.26 The theoretical predictions gave the same tendency.7 However, we definitely thought that it was necessary to check this point for the internal structure, which can be studied only by USAXS experiments, and then a maximum was observed in the 2Dexp vs salt concentration curve, as briefly described in our previous letter.27 In this study, we have applied USAXS measurements to many latex systems which were different in size and in electric charge, to confirm the universality of our previous observation.27 It was found that a maximum was observed for all the systems studied and that its position is scaled by κa (κ-1, Debye length; a, the radius of the latex particle). Some consideration has been given to the USAXS profiles before and after the transition (i.e. maximum) position in order to find clues to the origin of this curious phenomenon. (13) Kerker, M.; Farone, W. A.; Matijevic, E. J. Opt. Soc. Am. 1963, 53, 758. (14) Kerker, M. The Scattering of Light and Other Electromagnetic Radiation; Academic: London, 1969. (15) For example: Hertl, W.; Versmold, H.; Wittig, U. Ber. BunsenGes. Phys. Chem. 1984, 88, 1063. (16) Brown, J. C.; Pusey, P. N.; Goodwin, J. W.; Ottewill, R. H. J. Phys. A: Math. Gen. 1975, 35, 1448. (17) Gruener, F.; Lehmann, W. J. Phys. A: Math. Gen. 1980, 13, 2155. (18) Pusey, P. N.; Tough, R. J. A. In Dynamic Light Scattering; Pecora, R., Ed.; Plenum: New York, 1985; Chapter 4. (19) Bonse, U.; Hart, M. Z. Phys. 1966, 189, 151. (20) Matsuoka, H.; Kakigami, K.; Ise, N. The Rigaku J. 1991, 8, 21. (21) Matsuoka, H.; Kakigami, K.; Ise, N.; Kobayashi, Y.; Machitani, Y.; Kikuchi, T.; Kato, T. Proc. Natl. Acad. Sci. USA 1991, 88, 6618. (22) Matsuoka, H.; Kakigami, K.; Ise, N. Proc. Jpn. Acad. 1991, 67B, 170. (23) Matsuoka, H.; Nakatani, Y.; Yamakawa, M.; Ise, N. J. Phys. IV 1993, 3, 451, Colloque C8. (24) Matsuoka, H.; Ise, N. Chemtract 1993, 4, 53. (25) Matsuoka, H.; Ise, N. Adv. Polym. Sci. 1994, 114, 187. (26) Ise, N.; Ito, K.; Okubo, T.; Dosho, S. Sogami, I. J. Am. Chem. Soc. 1985, 107, 8074. (27) Matsuoka, H.; Harada, T.; Yamaoka, H. Langmuir 1994, 10, 4423.
Langmuir, Vol. 12, No. 23, 1996 5589 Table 1. Properties of Latices sample name
diameter (Å)
polydispersity (%)
surface charge number
MSC-14 MS-27 MS-28 MS-30
1200 1890 2200 2600
5 5 4 4
8 000 22 000 31 000 78 000
Experimental Section USAXS Apparatus. The USAXS apparatus used in this study was a Bonse-Hart type from Rigaku Corporation (Tokyo, Japan). This instrument can detect the scattering from structures on the length scale from 300 Å to 8 µm. The details of the apparatus are fully described elsewhere.20 Synthesis and Characterization of the Latices. The sample latices were prepared in our laboratory. Methyl methacrylate was used as the monomer, potassium oxydisulfate as the initiator, and the potassium salt of p-styrenesulfonic acid as the stabilizer. No emulsifier was used in the preparations. The polymerizations were performed at 70 °C for 30∼40 h, and impurities in the dispersions were completely removed by ultrafiltration. In order to remove ionic impurities, an ionexchange resin was added to the dispersions, separated from the particles by a nylon mesh. Due to the deionization process, iridescence was observed in all the dispersions caused by the strong interaction between particles. In the present study, four latices with different properties were used. The characteristics of the latices used are summarized in Table 1. To determine the radius of the samples, USAXS measurements were performed on the dried latices. The radius28 and polydispersity29 of the samples were estimated by fitting a calculated theoretical curve which takes the polydispersity into account to the measured data.24,29 The surface charge numbers of the samples were determined by conductometric titration. Sample Preparation and USAXS Measurement. For the preparation of sample dispersions, we took the utmost care. Samples at various salt concentrations were prepared by adding a suitable amount of Milli-Q water and dilute salt (NaCl) solution to the stock solutions. The sample container was then rotated for several hours. After keeping it undisturbed overnight, the sample was put into a glass capillary (diameter, 2.0 mm) by Teflon tube and syringe. After 24 h in the capillary, the measurement of the sample was started. A typical USAXS measurement was performed with an angle step of 2 s of an arc and accumulation time of 30 s and took about 5 h. The scattering pattern was measured for both sides of the direct beam; hence, the original data had a symmetrical shape. From the symmetry, the exact point of zero scattering angle was determined. The background scattering, i.e. the scattering from water, in the capillary was also measured and subtracted from the scattering from the sample dispersion with the correction of the absorption factor. The absorption factor was determined from the ratio of the direct beam intensities after passing through the samples whose profiles were measured separately with a smaller angle step (0.1 s). The data thus obtained were desmeared with the assumption of an infinitely thin width and infinitely long beam geometry.30 In addition, continuous measurements were performed to observe the change in the ordered structure in the dispersion from the time when the sample was put into the glass capillary. Measurement under Deionized Conditions. The ordered structure in the fully deionized dispersion is very important for studying the dependence of the interparticle distance on the ionic strength in the dispersion. This is because there is a possibility that a small quantity of ionic impurities is contained in the sample with a nominal salt concentration of [NaCl] ) 0 M. We therefore also prepared a sample with an ion-exchange resin coexisting in the capillary and determined the time evolution of the structure. The ion-exchange resin was set at the bottom part of the capillary so that the X-ray beam did not hit the resins. (28) Guinier, A.; Fournet, G. Small-angle Scattering of X-rays; Wiley: New York, 1955. (29) Hashimoto, T.; Fujimura, M.; Kawai, H. Macromolecules 1980, 13, 1660. (30) Grady, B. P.; Matsuoka, H.; Ise, N.; Nakatani, Y.; Cooper, S. L. Macromolecules 1993, 26, 4064.
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Data Treatment. The scattering vector q is defined by
q ) 4π sin(θ)/λ
(1)
where 2θ is the scattering angle and λ the wavelength of the X-rays (1.5406 Å, Cu KR1). The scattering intensity I(q) for a spherical particle is expressed as31
I(q) ) KP(q) S(q)
(2)
where P(q) is the particle form factor, S(q) is the interparticle structure factor, and K is a constant. P(q) for a monodisperse sphere is given by28
P(q) )
[
]
3{sin(qR) - qR cos(qR)} (qR)3
2
(3)
The value of S(q) in the high-q region can be taken to be unity. Hence, S(q) can be calculated by fitting I(q) with P(q) in the high-q region, by taking polydispersity into account. Although, strictly speaking, eq 2 is not perfectly correct even for spherical particles if they are polydisperse (f 2 * f 2 ) P(q), f is the amplitude of the form factor), the polydispersity of the samples used here was low enough to make eq 2 effectively correct. S(q) is a function which depends only on the spatial distribution of the center of gravity of the particles. Therefore, the interparticle distance in the ordered structure can be exactly calculated from S(q). In addition, the comparison of the intensity of the first peak for samples with various salt concentrations gives information about the degree of ordering in the structures that the particles form in the dispersion. Calculation of Debye Length. The Debye Length κ-1 was calculated by the method by Sumaru et al.,32 i.e. by eq 16 in ref 32
κ2 ) 4πλB(n+0 + n-0)
Figure 1. USAXS curves of MS30 latex dispersed in water at various salt concentrations. [Latex] ) 4.3 vol %. The ordinate has been shifted by 2 decades to avoid superimposing the data. The solid line (at the bottom) is the theoretical curve for an isolated sphere of diameter 2600 Å with polydispersity 4% which was used for the function P(q).
(4)
where λB is the Bjerrum length and n+0 and n-0 are the activities of the simple cation and anion, respectively, which are obtained by solving the nonlinear Poisson-Boltzmann equation
[
]
1 d 2 dφ(r) r ) -4πλB[n+(r) - n-(r)] ) dr r2 dr -4πλB[n+0e-φ(r) - n-0eφ(r)] (5) with the boundary conditions
φ(r0) ) 0
(6)
ZaλB dφ | )- 2 dr r)a a
(7)
Results and Discussion Figure 1 shows the USAXS profiles for latex (MS-30) dispersions under various conditions of added salt. Clear Bragg peaks were observed at low ionic strengths, indicating the formation of an ordered structure, or colloidal crystals, in the dispersion. The behavior here is quite similar to that of the MS-27 latex dispersions reported in our previous letter.27 The relative peak positions are compatible with those of the so-called powder pattern for a face-centered cubic lattice (x3:x4:x8:x11: x12).33,34 Hence, it appears that there are many small (31) Hayter, J. B.; Penfold, J. Colloid Polym. Sci. 1983, 26, 1022. (32) Sumaru, K.; Yamaoka, H.; Ito, K. Ber. Bunsen-Ges. Phys. Chem. 1996, 100, 1176. (33) Cullity, D. B. Elements of X-ray Diffraction, 2nd ed.; AddisonWesley: Reading, PA, 1977. (34) Warren, B. E. X-ray Diffraction; Addison-Wesley: Reading, PA, 1969.
Figure 2. Interparticle structure factor S(q) for an MS-30 latex dispersion ([latex] ) 4.3 vol %) obtained from the USAXS profiles shown in Figure 1. The ordinate has been shifted for each data set.
colloidal crystals in the dispersion and that they are oriented randomly.35 This situation means that there is no significant wall effect under this condition.63 The Bragg peaks became lower and broader with increasing salt concentration, indicating the decrease in the degree of order in the ordered structure caused by the shielding effect of the salt. The previously observed curious behavior that the first-order peak position initially moves toward smaller angles with increasing salt and then, after passing through a maximum, moves to higher angles is also clearly observable for the MS-30 latex. Figure 2 shows the functions S(q) evaluated from the profiles in Figure 1. S(q) reflects the spatial distribution of the center of gravity of the latex particles only. The curious change of the peak position becomes much clearer, (35) If the colloidal crystal grows and becomes single-crystal-like, additional peaks appear on the powder-like diffraction pattern. The positions of the additional peaks depend on the orientation of the sample. Matsuoka, H.; et al. In preparation.
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Figure 3. Salt concentration dependence of the peak height of the function S(q) in Figure 2.
Figure 4. Salt concentration dependence of the nearest neighbor interparticle distance (2Dexp). 2D0 (6730 Å) is the distance calculated by assuming that the particles distributed uniformly in the dispersion.
and the peak intensity, which reflects the degree of order, decreases monotonically with increasing salt concentration, as shown by Figure 3. The degree of order certainly corresponds to the salt concentration. Figure 4 shows the 2Dexp values as a function of the salt concentration. In this plot, a clear maximum is observed, as was the case for the MS-27 latices reported previously. The salt concentration of the maximum is 5 × 10-6 M, which is different from that for the MS-27 latex probably due to the different size and charge density of the particles. However, as will be shown later, the maximum position can be well scaled by a simple parameter. The USAXS profiles shown in Figure 2 were measured 2 days after sample preparation for all salt concentrations. Since the S(q)max value is simply related to salt concentration, as described above, it appears that all the USAXS profiles reflect equilibrium structures. However, it is reasonable to think that the time evolution of the structure (the rate of the colloidal crystal growth) might depend on the amount of added salt, so it was necessary to check whether or not the structure observed by USAXS really reflects the equilibrium structure for each salt condition. We checked this point by measuring the time evolution
Figure 5. Time evolution of the function S(q) obtained for the MS-27 latex dispersion ([latex] ) 3.8 vol %) at various salt concentrations: [salt] ) (a) 0 M; (b) 10-5 M; (c) 7 × 10-5 M.
of the USAXS profile for the MS-27 latex under different salt conditions. Figure 5 shows the USAXS profiles at 11 ∼ 43 h after sample preparation at salt concentrations of 0, 10-5, and 7 × 10-5 M. At each salt concentration, all the USAXS curves showed exactly the same profiles. This result strongly suggests that the structures we have detected by USAXS 2 days after the sample preparation are equilibrium structures. The curious maximum in the 2Dexp vs salt concentration plot is not due to a nonequilibrium nature of the structures. The salt concentration at the maximum of 2Dexp was different for each latex system. For example, it was about 10-5 M for the MS-27 latex, 0.5 × 10-5 M for MS-30, 10-5 M for MS-28, and 1.6 × 10-5 M for MSC-14. In our previous paper,27 the maximum salt concentration corresponded to 25% of the counterion number density. In Figure 6, the 2Dexp values normalized by 2D0 have been plotted against κa. Using this normalization, the maximum position
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Figure 6. Relationship between 2Dexp and κa. The 2Dexp values have been normalized by the appropriate 2D0 value.
Figure 8. Comparison of the data for MS-27 with two theories.
with radius (a + κ-1),36 i.e.
2Dexp ) 2(a + κ-1) and Sogami theory (eq 42 in ref 7),
Rmin ) [C + 1 + {(C + 1)(C + 3)}1/2]/κ
Figure 7. Change of 2Dexp under the deionized process. The 2Dexp values have been normalized to the 2D0 values. Measurement time means the time of coexistence with the ionexchange resin.
comes at κa ) 1.3 for all the latex dispersions studied here. This quite new finding strongly supports the conclusion that our curious observation in the 2Dexp vs salt concentration curves is not an artifact but a universal phenomenon. To strongly reconfirm the maximum behavior in 2Dexp vs salt concentration, we measured 2Dexp as a function of time for the system coexisting with ion-exchange resin. This experiment was performed because results in the very low κa region (say κa < 0.7 in Figure 6) were not available due to difficulties in the sample preparation, especially in the control of the salt concentration. In this system, the ion-exchange process is the rate-determining step: the change in 2Dexp with time corresponds to lowering of the salt concentration by the action of the resin not to crystal growth under constant (very low) salt conditions. The 2Dexp value decreased with time, which means that 2Dexp decreases with decreasing salt concentration (Figure 7). This behavior corresponds to the behavior for κa < 0.7 in Figure 6. The maximum of 2Dexp as a function of salt concentration was thereby further confirmed. We compared our observations with theoretical predictions. An example, for the MS-27 latex, is shown in Figure 8. An effective hard sphere model, pure repulsive theory in which the latex particle behaves like a hard sphere
where C ) (κam coth κam + κan coth κan)/2, were considered. The predictions by both of these theories show a monotonic decrease of 2Dexp with increasing salt concentration: they do not predict the existence of the maximum. The slopes of the plots at higher salt concentrations are very similar to that of our USAXS result, but the absolute values are greatly different: the USAXS result is much larger than the theoretical predictions. This situation may mean that the behavior of 2Dexp at relatively high salt concentrations (κa > 1.3) could be qualitatively explained by these theories if some additional factor were taken into account. One possibility for this factor may be the so-called saltfractionation effect.37 By this effect, the salt concentrations in the ordered regions and the disordered regions are different (lower in the ordered and higher in the disordered regions). If such an effect exists, observation of the two-state structure and void structures can be explained. On this point, Smalley et al. will give a detailed theoretical treatment.38 As 2Dexp reaches a maximum value at κa ) 1.3, we thought that there may be a possibility of some kind of transition at this transition point. So, we compared the USAXS profiles themselves before and after this transition point. Parts a and b of Figure 9 show USAXS profiles for κa < 1.3 and κa > 1.3, respectively, for three latex samples. At first sight, the profiles in Figure 9a look like the scattering pattern from a solid crystal because many (Bragg) peaks are observed at specific scattering angles. Even though the degree of order is not so high as in a real solid crystal, it is obvious that these profiles cannot be reproduced by the simple theory for liquid structure39,40 or colloidal theory based on liquid theory.41,42 On the other (36) Okubo, T. Acc. Chem. Res. 1988, 21, 281. (37) Smalley, M. V. Langmuir 1994, 10, 2884. (38) Smalley, M. V.; Schartl, W.; Hashimoto, T. Langmuir 1996, 12, 2340. (39) Hansen, J. P.; McDonald, I. R. Theory of Simple Liquid, 2nd ed.; Academic: San Diego, CA, 1986. (40) Zernike, F.; Prins, J. A. Phys. Z. 1927, 41, 184. (41) Hayter, J. B.; Penfold, J. Mol. Phys. 1981, 42, 109.
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Figure 10. S(qmax) as a function of κa.
Figure 9. Comparison of USAXS profiles (a) before and (b) after the transition point.
hand, the profiles in Figure 9b do look like those for liquid structure:39 the peak is very broad and weak and shows oscillatory behavior, not peaks at specific angles. This situation reminds us of the possibility of solid-liquid transition described by the Lindemann rule.43 The Lindemann rule and the Hansen-Verlet criterion predict a solid-liquid transition at S(q)max ) 2.85 for hard spheres interacting via Lennard-Jones potential.44 In our case, the latex particles are not hard spheres and there is a strong electrostatic interaction, so it is clear that a completely similar treatment to that given by the Lindemann rule is not applicable to our system. However, there remains a large possibility that such a kind of transition does occur at the transition point. In Figure 10, S(q)max values have been plotted as a function of κa. Although the data are scattered, the tendency of S(q)max to decrease with increasing κa is clearly seen and the S(q)max value at the critical point (i.e. κa ) 1.3) is about 3, which is very close to Lindemann’s prediction and the Hansen-Verlet criterion. As discussed so far, there seems to be some transition point at κa ) 1.3. At least, there is a structural transition of ordered structure of colloidal particles between solidlike and liquid-like at this point. There should be some (42) Hansen, J. P.; Hayter, J. B. Mol. Phys. 1982, 46, 651. (43) Ziman, J. M. Models of Disorder; Cambridge University Press: Cambridge, U.K., 1979. (44) Hansen, J. P.; Verlet, L. Phys. Rev. 1969, 184, 151.
origin causing this structural transition, which may be a transition of interparticle interaction and/or may be that of particle property. Although this origin is still a mystery for us, some possibilities can be pointed out now. (i) By very simple consideration, an existence of some kind of electrostatic attractive interaction below κa ) 1.3 can be predicted. Since 2Dexp decreased with decreasing salt below κa ) 1.3, it may be natural to think that there is an electrostatic attraction in this region. The decreasing tendency of 2Dexp with decreasing salt can be easily explained by a less electrostatic shielding effect. (ii) The salt fractionation proposed by Smalley38 et al., as described already, is a possibility, as is (iii) some change of particle property at this point, for example, the effective charge density. For possibility i, the origin of the electrostatic attraction is a problem: it should be a new type of attractive force. However, it is quite interesting to note the fact that 2Dexp is equal to 2D0 at κa ) 1.3: With decreasing salt concentration, 2Dexp increases, as is predicted by both Sogami and DLVO, and reaches 2D0 at κa ) 1.3. 2D0 is the maximum interparticle distance which is physically possible. Sogami and DLVO predict a further increase of 2Dexp with decreasing salt, but this is physically impossible. In this sense, the region below κa ) 1.3 is a very special situation: the colloidal particles consisting of the particle itself and a counterion cloud are, in a sense, overlapping, unexpectedly. It is not unrealistic that there is a new type of interaction which has never been considered in such a special situation. One possibility may be the role of counterions. The counterion cloud is forced to be overlapped below κa ) 1.3; the situation that the counterions are shared by some colloidal particles may be achieved, which is already pointed out by Schmitz et al.45 This situation is quite similar to those of metal bonding and covalent bonding of atomic systems. There is a possibility that a very rapid motion of counterions between colloidal particles is an origin of attraction, since such dynamic properties have never been considered in the electrostatic interaction theories. By possibility ii, the increasing tendency of 2Dexp below κa ) 1.3 with increasing salt concentration can be interpreted as a decreasing tendency of salt concentration in ordered regions with increasing average salt concentration in the whole system.38 This unique tendency is predicted to occur below κa ) 0.82 by Smalley theory.38 The slight difference of κa value at the transition point is acceptable, since the method of calculation of κ is different. The position of the Sogami potential well shifts toward larger distance with decreasing salt concentration. However, an interpretation of the decreasing tendency of S(q)max with increasing average salt concentration may be a problem for this (45) Schmitz, K. S.; Lu, M.; Gauntt, J. J. Chem. Phys. 1983, 78, 5059.
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interpretation. For possibility iii, the change of effective charge density on the particle surface cannot be denied although we have believed that this is constant at all the salt concentrations studied here. If the salt ions are adsorbed on the colloidal particle surface, leading to a decrease of the effective charge below κa ) 1.3, the increase of 2Dexp with increasing salt concentration can be explained by Sogami theory, since the position of the secondary minimum of the Sogami potential shifts to larger distance with decreasing surface charge.46 For this interpretation, the explanation of the universal tendency at the transition point, i.e. κa ) 1.3 and 2Dexp ) 2D0, is a problem. In addition, very recently, Yamanaka et al. observed a similar tendency to ours for a colloidal silica dispersion under the condition of constant surface charge, although it is a preliminary result.47 Another possible interpretation may be by the concept of effective temperature. For the region of κa < 1.3, the structure is a solid-like lattice. The point κa ∼ 1.3 seems to correspond to melting. Hence, the increase of ionic strength implies the increase of effective temperature, which results in the expansion of the lattice, i.e. an increase of 2Dexp in this region. It is thought to also be necessary to pay attention to the concept of a lattice for the interpretation of this phenomenon. To confirm this, a detection of the thermal vibration of the lattice point is needed, which should be our further work. A detailed analysis based on these considerations and comparison with results of computer simulations48 will be our next target. For the quantitative estimation of the structure, especially in the region of the solid-like state, the three-dimensional paracrystal theory proposed by Hosemann49 and extended by one of us50 may be useful. Using this analysis, the degree of distortion of the crystal structure, the g-factor, can be estimated. The g-factor should be a useful parameter in addition to S(q)max, and such an analysis is already in progress.51 To check a transition in the interaction force, direct measurements, such as those provided by the atomic force microscope (AFM) and surface force apparatus (SFA), which have already been applied to colloidal systems,52,53 may be powerful. However, great care should have to be taken in controlling the effect of ionic impurities in such measurements. For the estimation of the interaction potential itself, the total internal reflection microscopy (or evanescent wave light scattering microscopy)54,55 might prove interesting. Recently, the role of the counterions in electrical interactions has attracted considerable attention (not in (46) Tata, B. V. Private communication. (47) Yamanaka, J.; Koga, T.; Ise, N.; Hashimoto, T. Proceedings of 48th Annual Meeting on Colloid and Surface Chemistry; Chemical Society of Japan: Sapporo, 1995; p 40. (48) Sood, A. K. In Solid State Physics; Ehrenreich, H., Turnbull, D., Eds.; Academic Press: New York, 1991; Vol. 45, p 1. Tata, B. V.; Arora, A. K.; Valsakumar, M. C. Phys. Rev. E 1993, 47, 3404. (49) Hosemann, R.; Bagchi, S. N. Direct Analysis of Diffraction by Matter; North Holland: Amsterdam, 1966. (50) Matsuoka, H.; et al. Phys. Rev. B 1987, 36, 1754. Matsuoka, H.; et al. Phys. Rev. B 1990, 41, 3854. (51) Matsuoka, H.; Harada, T.; Kago, K.; Yamaoka, H. Polym. Prepr. Jpn. 1995, 44, 2959. Matsuoka, H.; Harada, T.; Kago, K.; Yamaoka, H. Proceedings of 48th Annual Meeting on Colloid and Surface Chemistry; Chemical Society of Japan: Sapporo, 1995; p 576. Matsuoka, H.; Harada, T.; Kago, K.; Yamaoka, H. In preparation. (52) Li, Y. Q.; Tao, N. J.; Pan, J.; Garcia, A. A.; Lindsay, S. M. Langmuir 1993, 9, 637. (53) Mizukami, M.; Kurihara, K. Polym. Prepr. Jpn. 1995, 44, 298. (54) Prieve, D. C.; Frej, N. A. Langmuir 1990, 6, 396. (55) Tanimoto, S.; Matsuoka, H.; Yamaoka, H. Colloid Polym. Sci. 1995, 273, 1201.
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the sense of the effective hard sphere model).45,56 There is a strong possibility that the so-called counterion-sharing effect plays an important role in colloidal interactions, as described above. In this sense, the very recent attempt to determine the counterion cloud (counterion distribution) by scattering techniques should become important.57,58 On the other hand, the possibility of another kind of attractive force proposed by Ninham et al.59 by the van der Waals approach should also be taken into account. It may be interesting to note the similarity of our curious finding to the famous finding by Lin, Lee, and Schurr on the ordinary-extraordinary (O-E) transition.60 The O-E transition is a dramatic change in the diffusion coefficient in polyelectrolyte solutions with decreasing salt. Our finding is also a curious behavior at low ionic strength. It has also been reported that there is some curious behavior in the diffusion of colloidal particles61 and viscosity behavior62 at κa ) 1. Analysis of any similarities or dissimilarities between these observations and our finding may be very interesting. Conclusion The curious phenomenon of a maximum in the 2Dexpsalt concentration relationship has been found to be universal by this systematic USAXS study for colloidal particles of different properties. The maximum position is scaled by κa and found to be κa ) 1.3. This quite unique finding predicts a necessity to consider an additional factor, which has never been taken into account, for formation of colloidal crystals and interparticle interaction between colloidal particles. From the structural point of view, the transition point, κa ) 1.3, seems to be a transition point of solid-like and liquid-like structure of colloidal crystals. For the origin of this transition, a new type of attractive interaction, salt fractionation, and change of surface charge can be predicted. Much more detailed analyses, such as a quantitative estimation of the solid-liquid transition and for the origin of the transition, are now in progress in our laboratory. Acknowledgment. This work was supported by grants-in-aid of the Ministry of Education, Science and Culture (Nos. 07805087 and 08226226). The work was also partly supported by the Ogasawara Foundation for the Promotion of Science & Engineering, to whom our sincere gratitude is due. We express our deep thanks to Dr. Martin V. Smalley (currently at the Polymer Phasing Project, Kyoto) for useful comments and discussions and for his kind help with the preparation of the manuscript. Useful comments were received from reviewers, to whom our sincere thanks are due. LA950916X (56) Ray, J.; Manning, G. S. Langmuir 1994, 10, 2450. Yoshino, S. Polym. Prepr. Jpn. 1994, 43, 1454. (57) Liu, Y. C.; Ku, C. Y.; LoNostro, P.; Chen, S. H. Phys. Rev. E 1995, 51, 4598. (58) Sumaru, K.; Matsuoka, H.; Yamaoka, H.; Wignall, G. D. Phys. Rev. E 1996, 53, 1744. (59) Ninham, B. W.; Parsegian, U. A. In Dispersion Forces; Mahanty, J., Ninham, B. W., Eds.; Academic: New York, 1976; Chapter 7. (60) Lin, S. C.; Lee, W. I.; Schurr, J. M. Biopolymers 1978, 17, 1041. (61) Schumacher, G. A.; van de Ven, T.; Faraday Discuss. Chem. Soc. 1987, 83, 75. (62) Yamanaka, J.; Ise, N.; Miyoshi, H.; Yamaguchi, T. Phys. Rev. E 1995, 51, 1276. (63) Under the condition that the colloidal crystal forms a very large single crystal compared to the capillary size, the wall effect becomes significant and the crystal has some special orientation to the capillary wall (see: Konishi; et al. Phys. Rev. B 1995, 51, 3914).
