Unidirectional Crystallization of Charged Colloidal Silica Due to the

Masakatsu Yonese,† Kensaku Ito,§ Tsutomu Sawada,| Fumio Uchida,‡ and Yoshimasa Ohki. ⊥. Faculty of Pharmaceutical Sciences, Nagoya City UniVers...
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Langmuir 2007, 23, 7510-7517

Unidirectional Crystallization of Charged Colloidal Silica Due to the Diffusion of a Base Masako Murai,† Hiroshi Yamada,†,‡ Junpei Yamanaka,*,† Sachiko Onda,† Masakatsu Yonese,† Kensaku Ito,§ Tsutomu Sawada,| Fumio Uchida,‡ and Yoshimasa Ohki⊥ Faculty of Pharmaceutical Sciences, Nagoya City UniVersity, 3-1 Tanabe, Mizuho, Nagoya, Aichi 467-8603, Japan, Fuji Chemicals Co., Ltd., 1-35-1 Deyashiki-Nishi, Hirakata, Osaka 573-0003, Japan, Department of Science and Engineering, UniVersity of Toyama, 3190 Gofuku, Toyama 930-8555, Japan, National Institute for Materials Science, 1-1 Namiki, Tsukuba, Ibaraki 305-0044, Japan, and Japan Aerospace Exploration Agency, 2-1-1 Sengen, Tsukuba, Ibaraki 305-8505, Japan ReceiVed March 14, 2007. In Final Form: April 19, 2007 Dilute aqueous dispersions of charged colloidal silica (particle volume fraction ) ∼0.03-0.04, particle diameter ) 110 nm) exhibit unidirectional crystal growth due to the diffusion of a weak base, pyridine (Py). Similar diffusioncrystallization is enabled by a salt of a weak acid and a strong base, sodium hydrogen carbonate (NaHCO3). The resulting crystals consist of columnar (or cubic) crystal grains with a maximum height of a few centimeters and a maximum width of 1 cm. The crystal growth process is attributed to a combination of (i) the diffusion of Py or NaHCO3 accompanied by a charging reaction of the silica particles and (ii) the charge-induced crystallization of the silica colloids. Theoretical growth curves based on the reaction-diffusion model for the case of Py were in good agreement with the observed curves. We also report the immobilization of the resulting large crystals by using a polymer hydrogel matrix.

I. Introduction Charged colloidal particles dispersed in aqueous media are stabilized due to long-range electrostatic interparticle interaction. When the interaction is weak, their spatial distribution is disordered. With increasing interaction magnitude, the charged colloids undergo a phase transition to the ordered “crystal” state, where the particles are regularly arranged.1 Thus far, the crystallizations of charged colloids have been studied extensively as models to examine phase transition in general.2 Furthermore, in recent years, colloidal crystals have attracted considerable attention as photonic crystals,3 since their Bragg wavelengths usually lie in the optical regime. In particular, recent progress in the gel immobilization technique4 has increased the applicability of charged colloidal crystals. We have reported that silica colloids undergo charge-induced crystallization upon the addition of a base.5 The silica particles are slightly charged in their aqueous dispersions due to the self* To whom correspondence should be addressed. E-mail: yamanaka@ phar.nagoya-cu.ac.jp. † Nagoya City University. ‡ Fuji Chemicals Co., Ltd. § University of Toyama. | National Institute for Materials Science. ⊥ Japan Aerospace Exploration Agency. (1) (a) Pieranski, P. Contemp. Phys. 1983, 24, 25. (b) Russel, W. B.; Saville, D. A.; Schowalter, W. R. Colloidal Dispersions; Cambridge University Press: New York, 1989. (c) Sood, A. K. In Solid State Physics; Ehrenreich, H., Turnbull, D., Eds.; Academic Press: New York, 1991. (d) Arora, A. K., Tata, B. V. R., Eds. Ordering and Phase Transition in Charged Colloids; VCH: New York, 1996. (e) Ise, N.; Sogami, I. S. Structure Formation in Solution; Springer: Berlin, 2005. (2) (a) Anderson, V. J.; Lekkerkerker, H. N. W. Nature 2002, 416, 811. (b) Yethiraj, A.; van Blaaderen, A. Nature 2003, 421, 513. (3) Joannopoulos, J.; Meade, R.; Winn, J. Photonic Crystals; Princeton University Press: Princeton, NJ, 1995. (4) (a) Kamenetzky, E. A.; Magliocco, L. G.; Panzer, H. P. Science 1994, 263, 207. (b) Holtz, J. H.; Asher, S. A. Nature 1997, 389, 829. (c) Asher, S. A.; Holtz, J. H.; Weissman, J. MRS Bull. 1998, 23, 44. (d) Iwayama, Y.; Yamanaka, J.; Takiguchi, Y.; Takasaka, M.; Ito, K.; Shinohara, T.; Sawada, T.; Yonese, M. Langmuir 2003, 19, 977. (e) Toyotama, A.; Kanai, T.; Sawada, T.; Yamanaka, J.; Ito, K.; Kitamura, K. Langmuir 2005, 21, 10268. (f) Toyotama, A.; Sawada, T.; Yamanaka, J.; Kitamura, K. Langmuir 2006, 22, 1952.

dissociations of silanol groups on their surfaces (tSisOH T tSisO- + H+).6 Since silanols are weak acids, the charge number, Z, of the silica particle increases with the addition of a base such as sodium hydroxide (tSisOH + Na+OH- f tSisO- Na+ + H2O). At moderately large values of Z, silica colloids take crystal states.5 Based on these findings, in our previous brief paper, we examined the directed crystal growth of silica colloids due to the diffusion of pyridine (Py).7 Py is a weak base with pKa ) 5.42 at 25 °C (py + H2O T pyH+ + OH-, where py and pyH+ denote the undissociated Py and pyridinium ions, respectively). Figure 1 shows the experimental setup used and the proposed model of crystal growth. Py molecules are diffused into colloidal silica from a reservoir of an aqueous Py solution through a semipermeable membrane. Upon diffusion, a part of Py provides charges on the silica surface (tSisOH + py f tSisO- + pyH+), with the pyH+ ions being the counterions. The other part of Py remains undissociated in the medium under our experimental conditions (pH < 8). The crystal region is extended by the diffusion of these mobile Py molecules in the medium. In the same study, we discussed the crystal growth mechanism based on growth curves obtained under limited conditions. In the present study, diffusion-crystallization is investigated in greater detail to verify the above-mentioned mechanism. First, we report the acid-base reaction in the Py/silica system and the crystallization phase diagram. Subsequently, we systematically examine the growth curves at various Py concentrations by using two types of semipermeable membranes (polymer hydrogel and poly(tetrafluoroethylene) membranes). The influence of the salt (NaCl) concentration is also studied. We demonstrate that the (5) (a) Yamanaka, J.; Koga, T.; Ise, N.; Hashimoto, T. Phys. ReV. E 1996, 53, R4314. (b) Yamanaka, J.; Yoshida, H.; Koga, T.; Ise, N.; Hashimoto, T. Phys. ReV. Lett. 1998, 80, 5806. (c) Yoshida, H.; Yamanaka, J.; Koga, T.; Koga, T.; Ise, N.; Hashimoto, T. Langmuir 1999, 15, 2684. (6) Iler, R. K. The Chemistry of Silica; Wiley: New York, 1979. (7) Yamanaka, J.; Murai, M.; Iwayama, Y.; Yonese, M.; Ito, K.; Sawada, T. J. Am. Chem. Soc. 2004, 126, 7156.

10.1021/la700754s CCC: $37.00 © 2007 American Chemical Society Published on Web 05/27/2007

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Figure 3. Illustration of two types of cells equipped with (a) a polymer hydrogel membrane and (b) a PTFE membrane. Figure 1. Schematic diagram of the experimental setup and the proposed mechanism of unidirectional crystal growth of silica colloids by the diffusion of pyridine. py and pyH+ denote undissociated and ionized pyridine, respectively.

Figure 2. Illustration of the chemical species present in the NaHCO3/ silica systems.

observed growth can be explained by the theoretical growth curves based on the above-mentioned model. In this paper we also report diffusion-crystallization by sodium hydrogen carbonate (NaHCO3), which is a salt of a weak acid and a strong base; we have briefly mentioned this in our previous papers.7,8 The pKa value of the silanol groups is as low as 6-7 in the absence of a base.6 Since this value is comparable to that for the primary dissociation of carbonic acid (H2CO3 T H+ + HCO3-, pKa ) 6.35), the Z value of the silica particles should increase because of the addition of NaHCO3 (tSisOH + Na+ + HCO3- f tSisO- + Na+ + H2CO3). The chemical species present in the NaHCO3/silica system are illustrated in Figure 2 (produced H2CO3 molecules undergo partial dissociation, H2CO3 T H+ + HCO3-). Thus, additions of NaHCO3 may also cause charge-induced crystallization. Furthermore, the diffusion of mobile NaHCO3 in the medium would result in unidirectional crystal growth. However, mobile NaHCO3 (and H2CO3 T H+ + HCO3-) increases the ionic strength of the medium I; this is in contrast with the case of Py, where the mobile Py molecules in the medium are undissociated and have little influence on the I value. Since interparticle interaction is weaker at a higher I values, crystallization may not be preferable at very large NaHCO3 concentrations. In the present study, we systematically examine the crystallization conditions and crystal growth curves. We show that charge-induced crystallization occurs when the NaHCO3 concentration is adequately small; diffusion-crystallization is observed also at low NaHCO3 concentrations. Usually, the maximum size of colloidal crystals is in the millimeter range. Since the application of colloidal crystals is greatly restricted due to their size, various methods9 have recently been devised to fabricate large single crystals. They include shear annealing10 and colloidal epitaxy,11 whereby welloriented colloidal crystals with a large area have been obtained. (8) Wakabayashi, N.; Yamanaka, J.; Murai, M.; Ito, K.; Sawada, T.; Yonese, M. Langmuir 2006, 22, 7936. (9) (a) van Blaaderen, A. MRS Bull. 2004, 29, 85. (b) Sullivan, M.; Zhao, K.; Harrison, C.; Austin, R. H.; Megens, M.; Hollingsworth, A.; Russel, W. B.; Cheng, Z.; Mason, T.; Chaikin, P. M. J. Phys.: Condens. Matter 2003, 15, S11. (10) (a) Clark, N. A.; Hurd, A. J.; Ackerson, B. J. Nature 1979, 281, 57. (b) Stipp, A.; Biehl, R.; Preis, T.; Liu, J.; Fontecha, A. B.; Scho¨pe, H. J.; Palberg, T. J. Phys.: Condens. Matter 2004, 16, S3885. (c) Kanai, T.; Sawada, T.; Toyotama, A.; Kitamura, K. AdV. Funct. Mater. 2005, 15, 25. (11) (a) van Blaaderen, A.; Wiltzius, P. Nature 1997, 385, 321. (b) Hoogenboom, J. P.; Yethiraj, A.; van Langen-Suurling, A. K.; Romijn, J.; van Blaaderen, A. Phys. ReV. Lett. 2002, 89, 256104.

As we have reported eariler,8 crystals obtained by the present method have a maximum length of a few centimeters and a maximum cross-sectional area of 1 × 1 cm2. We also describe the immobilization of these large crystals by using a polymer gel matrix. The organization of this paper is as follows. The experimental details are provided in section II. We then describe the crystallizations in the Py/silica and NaHCO3/silica systems in sections III and IV. The gel immobilization of the crystals is reported in section V. The conclusions of this study are described in section VI. II. Experimental Section II.A. Materials. Colloidal silica was purchased from Nippon Shokubai Co., Ltd. (Osaka, Japan). It was dialyzed in a cellulose tube (pore size ) 2.4 nm) against purified water for more than 30 days. The completeness of the dialysis was determined from the electrical conductivity of the water. After the addition of a mixed bed of cation- and anion-exchange resin beads (AG 501-X8 (D), Bio-Rad Laboratories, Hercules, CA), the sample was left standing for at least 1 week for further deionization. The particle diameter estimated by the dynamic light scattering method was 110 nm. By electrical conductivity measurements, the Z value of the purified silica particles was determined to be 170 (surface charge density ) 0.07 µC/cm2). The specific gravity of silica, F, estimated by the pycnometer method was 2.12. The particle volume fraction, φ, of the sample was determined by the drying-out method using the F value.12 All the experiments for the Py/silica and NaHCO3/silica systems were performed at φ ) 0.034 and 0.039, respectively. Water was purified by using a Milli-Q Simpli-Lab system (Millipore, Billerica, MA), and its electrical conductivity was 0.4-0.6 µS/cm. For sample preparations, a polystyrene or Teflon apparatus was used instead of glassware to avoid the elution of ionic impurities from the container wall. II.B. Methods. II.B.1. pH Titration. Titrations of the silica colloids were perfomed by using a type F-14 pH meter (Horiba Co., Ltd., Kyoto, Japan) for 15 mL of the silica colloid without the addition of salt. The temperature was controlled at 25 ( 0.05 °C. The titration experiments were repeated at least three times to obtain average values. II.B.2. Phase Diagram. The crystallization phase diagram was determined by detecting the Bragg diffraction from the crystal state 10 min after the base and/or salt was added to the sample. II.B.3. Unidirectional Crystal Growth. The growth experiments were conducted using the experimental setup shown in Figure 1. We used poly(methyl methacrylate) cells (1 cm × 1 cm × 4.5 cm, wall thickness ) 1 mm) with semipermeable membranes at their bottoms. Two types of membranes were used, as illustrated in Figure 3. The polymer hydrogel membrane (Figure 3a) was prepared at the bottom of the cell by the method described in section II.B.4 and was brought into contact with the reservoir through a circular hole of 6 mm diameter; the gel membranes having thicknesses of 4 and 6 mm were used for the experiments using Py and NaHCO3, respectively. A poly(tetrafluoroethylene) (PTFE) membrane (thickness ) 0.1 mm, (12) In ref 7, we use the F value of silica purchased from the same producers, that is, F ) 2.24, which yields φ ) 0.032 for the present sample. The φ values should be corrected to φ ) 0.034, as described herein.

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Figure 4. pH titration curves of colloidal silica using NaOH (open circles) and Py (filled circles); S and C values for [Py] ) 0.8 mM are represented graphically. Advantec Co., Ltd., Tokyo, Japan) was attached to an open cell bottom (Figure 3b). The effective diffusion constants of Py in the membranes (Dm), as determined by permeability measurements at 25 °C, were 0.60D0 and 0.38D0 (D0 is the diffusion constant of Py in water) for the gel and PTFE membranes, respectively. The entire experimental system was maintained in an air-conditioned incubator controlled at 25 ( 1 °C. II.B.4. Gel Membrane and Gelation of the Crystal. Poly(Nmethylolacrylamide) hydrogels were used for preparing the gel membranes described in section II.B.3 and for immobilizing the crystals. The reaction solutions for the gelations were typically composed of 0.67 M N-methylolacrylamide (gel monomer), 10 mM N,N′-methylene bisacrylamide (cross-linker), and 0.1 mg/mL 2,2′azobis[2-methyl-N-(2-hydroxyethyl)-propionamide] (photoinduced radical-polymerization initiator). They were deionized with ionexchange resin beads and then bubbled with argon gas to remove dissolved oxygen and carbon dioxide, which would inhibit radical polymerization. To fabricate the gel-immobilized crystals, the abovementioned gelation reagents were dissolved in both the silica colloid and the base solution in the reservoir (volume ) 120 mL). The gelations were performed under UV illumination using a 400 W Hg lamp for ∼20 min.

III. Crystal Growth Due to the Diffusion of Py III.A. Electrostatic Adsorption of Py onto Silica Particles. We first examine the acid-base equilibrium between our silica particles and Py in a homogeneous system. All the crystallization experiments described in the present paper were performed under the condition that the pH of the silica colloid is C†, respectively, where C† is the threshold concentration. The set of coefficients in eqs 1 and 2 is determined by curve fitting such that the values of both S and dS/dC of the two fitting curves are continuous at C ) C†. The best results are obtained for k ) 2.13 × 104 M-1, Γ ) 8.45 × 10-5 M, R ) 1.77 × 10-3 M, n ) 0.371, and C† ) 80 µM (fitting residuals ) 3% and 6% on average for C e C† and C > C†, respectively).15 The fitted results for the two C regions are shown in Figure 5b and c by the red and blue solid curves; the dashed vertical lines correspond to C ) C†. In the following sections, we represent the crystallization conditions in terms of S, which is more directly related to the experimental results than Z. Z is related to S as follows: at S ) 0, the present silica has a small charge Z ) Z0 () 170) due to the self-dissociation of the silanols. For Z . Z0, Z is approximately equal to the number of adsorbed base molecules per particle, that is, Z ) NAS/(103n), where NA is Avogadro’s constant and n is the number density of the particles (n ) φ/(4πap3), where ap is the particle radius). In the present case, Z ) 12.3S, when S is represented in micromoles.16 III.B. Crystallization Phase Diagram. Charge-induced crystallization of homogeneous silica colloids was examined under various base (NaOH and Py) and salt (NaCl) concentrations. Figure 6 shows the phase diagram defined by S and [NaCl]. The open symbols in Figure 6 represent the phase boundaries determined using NaOH with different concentrations of NaCl at a given [NaOH]. Here, we assume that S ) [NaOH] because C ∼ 0, as described in section III.A. In the absence of NaOH, the colloid was disordered, while it underwent charge-induced crystallization with an increase in S. The phase boundary shifts toward higher values of [NaCl] with increasing S, as expected from the increase in the interaction magnitude. The filled symbols in Figure 6 represent the phase boundaries obtained by the addition of Py, as determined by changing [Py] at given (15) In our analysis in ref 7, we applied the Langmuir-type adsorption isotherm for the entire [Py] region, with k ) 8.8 × 103 M-1 and Γ ) 1.15 × 10-4 M, based on preliminary data of the S-C relation. This yields S/[Py] ∼ 40% for [Py] ) 1 mM and a significantly slower theoretical growth than the present one. (16) We note that the effective charge number Ze is less than Z due to counterion condensation. We have obtained an empirical relationship between Ze and Z for silica colloids (ref 5).

Figure 7. Overviews of colloidal silica crystals formed by Py diffusion obtained using the (a and b) 4-mm-thick gel membranes and (c and d) 0.1-mm-thick PTFE membranes. (a) and (c) show the growth process for [Py]0 ) 1 mM, while (b) and (d) show crystals obtained for [Py]0 ) 10 mM at t ) 23 h. [NaCl] ) 0 M in all cases.

concentrations of NaCl; in this case, the S values were calculated from [Py] based on the S-C relationship determined in section III.A. In the upper abscissa of Figure 6, several values of [Py] are shown for reference. The phase boundaries for the two bases were in close agreement, which supports the validity of the S-C relation. III.C. Unidirectional Crystal Growth. Unidirectional diffusion-crystallization was systematically examined. Figure 7 shows overviews of the crystals obtained by using the (a and b) gel and (c and d) PTFE membranes, without the addition of extraneous salt. Parts a and c of Figure 7 represent the growth processes for the concentration of Py in the reservoir, [Py]0, equal to 1 mM at three evolution times, t. In parts b and d of Figure 7, the samples at [Py]0 ) 10 mM and t ) 23 h are shown for comparison. The crystal regions exhibited the diffraction color, while the disordered regions were opaque. The growth rate was larger for higher values of [Py], and the thinner PTFE membrane produced faster growth than the gel membrane. The crystal grains were columnar or cubic, and their sizes varied depending on both [Py]0 and the type of membrane; large crystals were obtained during slow growth. As shown in Figure 7a, the cross-sectional size of the crystal grain was sometimes the same as that of the sample cell (1 cm × 1 cm). This is significantly larger than the crystal grain size in homogeneous colloids, which is ∼2 mm at the maximum. The diffaction wavelength, λm, of the crystal exhibited a red shift with the crystal height. For example, at t ) 23 h (crystal height ) ∼2 cm), the λm values at the bottom and top of a crystal obtained at [Py]0 ) 1 mM using the PTFE membrane cell were 550 and 690 nm, respectively. The φ values determined from the λm values were ∼0.045 and ∼0.02, respectively. It appears that this nonuniformity in φ was due to the gravitational compression of the crystal and/or a migration of the particles under a gradient of chemical potential. Further studies are in progress in this regard. Figure 8 shows the observed crystal height, h, plotted against t for [Py]0 ) 100, 10, and 1 mM (shown by symbols of different colors) obtained by using the gel membrane in the absence of

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Figure 8. Plot of observed crystal heights h against t at three values of [Py]0 for the gel membrane in the absence of salt (symbols). The solid and dashed curves show the theoretical heights h′ for [NaCl] ) 0 M and 5 µM, respectively.

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Figure 11. f(C)-C curve (solid curve). f values at the high and low C limits are shown by dashed horizontal lines.

We have observed similar diffusion-crystallizations for other weak bases, such as trimethylpyridine. A systematic study on a relation between their pKa values and the crystallization behavior is underway. III.D. Theoretical Growth Curves. III.D.1. Diffusion Equation. Here, we examine the theoretical crystal growth curves based on the adsorption-diffusion model shown in Figure 1. We consider one-dimensional diffusion in the x direction. Without adsorption, the diffusion equation for C ) C(x, t) is given by

∂C ∂2C )D 2 ∂t ∂x

Figure 9. Observed and theoretical growth for the PTFE membrane at four values of [Py]0.

(3)

where D is the diffusion constant of Py. Equation 3 is modified for the adsorption-diffusion as follows. In the pH titration experiments (section III.A), we observe that the equilibrium between S and C is achieved in a few minutes; this period is significantly less than the time scale for the diffusioncrystallization (section III.C). Therefore, we assume an instantaneous adsorption equilibrium. Thus, we can apply the equilibrium S-C relation (section III.A) to the diffusion process. The adsorption-diffusion equation17 for C ) C(x, t) and S ) S(x, t) is given by

∂2C ∂S ∂C )D 2 ∂t ∂t ∂x

(4)

By using an identical relation, ∂S/∂t ) (∂S/∂C)(∂C/∂t), eq 4 is represented as

∂C ∂2C ) f(C)D 2 ∂t ∂x Figure 10. Plot of observed crystal heights h (symbols) against t at three values of [NaCl] for [Py]0 ) 1 mM using the gel membrane. The curves show the theoretical heights h′ calculated for each value of [NaCl].

salt. The data points in the single growth experiments are represented by the same symbols (circle, traingle, or square); the data from our previous report7 are represented by open symbols in Figure 8. Figure 9 shows the results for the PTFE membrane for [Py]0 ) 100, 10, 1, and 0.1 mM without salt. The influence of salt on the growth was examined by adding NaCl at the same concentration both in the silica colloid and in the reservoir. Growth curves at three values of [NaCl] ([Py]0 ) 1 mM; gel membrane) are shown in Figure 10, which reveal a clear reduction in the growth rate with [NaCl]. A comparison between the observed and theoretical growths is described below.

(5)

where

f(C) )

1 1+

∂S ∂C

(6)

That is, eq 4 is reduced to a diffusion equation with a variable diffusion coefficient, f(C) D. For instantaneous equilibrium, we can replace ∂S/∂C in eq 6 with dS/dC at equilibrium. Figure 11 shows the f(C)-C plot obtained by using the fitted S-C curve determined in section III.A. f(C) varies with C from 0.357 (C ) 0) to 1 (C f ∞). At the high C limit, eq 5 approaches the diffusion equation without adsorption, that is, eq 3. (17) Crank, J. The Mathematics of Diffusion, 2nd ed.; Oxford University Press: New York, 1975; Chapter 14.

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We numerically solved the diffusion equations by a finitedifference method to determine C(x, t) for the geometry of the experimental setups (Figure 3). The reservoir/membrane boundary was denoted as x ) 0. We assumed that C(0, t) is constant at [Py]0, since the volume of the reservoir is significantly larger (>100 times) than that of the cell. In the membrane region, we used eq 3 with D ) Dm (see section II.B.3 in the Methods). In the colloid region, eqs 5 and 6 were applied after correcting the D value for the obstruction effect due to the colloidal particles; we applied the expression derived by Jefferson et al.18 for diffusion in a composite medium where nonpermeable spheres are arranged on a regular crystalline lattice in the medium:

D π )1D0 4(1 + 2R)2

(7)

R ) 0.403φ-1/3 - 0.5

(8)

where

Here, D0 is the diffusion coefficient of Py in water. We used D0 at an infinite dilution () 8.75 × 10-6 cm2/s at 25 °C)19 for all values of C because of the small C dependence of D0 in the [Py] region considered.20 In dilute colloids, the obstruction effect did not appear to be significantly influenced by the particle arrangements. Therefore, we applied eqs 7 and 8 for the diffusions in both the crystal and disordered regions. Since the diffusing species is a noncharged Py molecule, the effect of particle charges on the diffusion process can be safely ruled out. At φ ) 0.034, D/D0 ) 0.873 and D ) 7.64 × 10-6 cm2/s. III.D.2. Concentration Profiles and Growth CurVes. Figure 12a shows the time evolution of the concentration profile of C calculated for [Py]0 ) 1 mM and the gel membrane cell geometry (0 < x e 0.1 cm, cell wall with a hole; 0.1 cm < x e 0.5 cm, gel membrane; 0.5 cm < x e 4 cm, silica colloid). The discontinuity of C observed at the gel/colloid boundary (x ) xB ) 0.5 cm) is due to the adsorption of Py in the colloid region. By applying the S-C relationship to C(x, t) in the colloid region, we evaluate S(x, t), as shown in Figure 12b. Based on the phase diagram (Figure 6), we define the critical value of S for the crystallization, S/, at the given values of [NaCl]. Hereafter, we use S/ ) 20, 35, 55, and 70 µM at [NaCl] ) 0, 5, 10, and 15 µM, respectively. By assuming that crystallization occurs instantaneously when the condition S(x, t) ) S/ is satisfied, we determine the location of the crystal front, x ) x/, at a given value of t. In Figure 12b, x/ for S/ ) 20 µM at t ) 10 h is shown as an example. Thus, the theoretical crystal height h′ is obtained as h′ ) x/ - xB. Finally, by estimating h′ at various values of t, we obtain the theoretical crystal growth curves. III.D.3. Comparison with the ObserVed Growth. Here, we compare the theoretical growth curves with the observed ones. As described in section III.C, the φ values in the colloids are slightly nonuniform during the growth process. However, as discussed in the Supporting Information, nonuniformity in φ has little effect on the theorectial growth. Thus, we use the growth curves obtained at the initial φ value () 0.034). In Figure 8, the h′-t curves calculated for three values of [Py]0 are represented by solid curves. A higher value of [Py]0 produces faster growth, as expected from the diffusion-controlled growth mechanism. The theoretical growth curves are in good (18) Jefferson, T. B.; Witzell, O. W.; Sibbett, W. L. Ind. Eng. Chem. 1958, 50, 1589. (19) Brun, B.; Salvinien, J. J. Chim. Phys. Phys.sChim. Biol. 1967, 64, 1319. (20) D0 at C ) 100 mM is ∼0.4% less than that at the infinite dilution (ref 19).

Figure 12. Time evolutions of the profiles for (a) C and (b) S at [Py]0 ) 1 mM in the case of the gel membrane cell. Values of t shown in (a) also apply for (b). The solid vertical lines in (a) and (b) indicate the membrane/colloid boundary (x ) 0.5 cm). In (b), the location of the crystal front (x ) x/) is shown when S/ ) 20 µM at t ) 10 h.

Figure 13. Crystallization phase diagram obtained by the addition of NaOH or NaHCO3 at various NaCl concentrations. Open and filled symbols denote the phase boundaries for NaOH and NaHCO3, respectively. Gray-colored symbols are applicable to both NaOH and NaHCO3.

agreement with the observed curves. At [Py]0 ) 1 mM, the value of h is considerably less than that of h′. This might be due to the presence of trace ionic impurities, which have a more significant effect in the case of slower growth. The growth curves at [NaCl] ) 5 µM (S/ ) 35 µM) for all values of [Py]0 are shown by the dashed curves in Figure 8. The growth curves are also calculated for the PTFE membrane, as shown in Figure 9; the solid and dashed curves are obtained for [NaCl] ) 0 and 5 µM, respectively. They are in good correspondence with the observed ones over a wide [Py]0 region. The theoretical growths at three values of [NaCl] (Figure 10) also show close agreement with the experimental ones. The above-mentioned good correspondence between theory and experiment under various conditions suggests that the present unidirectional growth can be explained in terms of the model shown in Figure 1.

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Figure 14. Unidirectional crystal growth due to the diffusion of NaHCO3. (a) Images of the crystals obtained for [NaHCO3]0 ) 30 and 90 µM at evolution time t ) 48 h. (b) Crystal growth process for [NaHCO3]0 ) 60 µM at four t values.

IV. Crystal Growth Due to the Diffusion of NaHCO3 In this section, we describe the diffusion-crytallization process by NaHCO3, which is a salt of a weak acid and a strong base. In the NaHCO3/silica system, free NaHCO3 may be present in the medium at equilibrium (see Figure 2), which increases the ionic strength. This is in contrast with the case of Py, where free Py molecules in the medium are undissociated and have little influence on the I value. IV.A. Crystallization Phase Diagram. Figure 13 shows a comparison of the crystallization phase diagrams in homogeneous silica colloids obtained by additions of NaOH and NaHCO3 at various NaCl concentrations. The phase boundaries for NaOH and NaHCO3 are denoted by open and filled symbols, respectively; the gray-colored symbols are applicable to both NaOH and NaHCO3. The addition of NaHCO3 caused re-entrant melting of the crystal at high NaHCO3 (∼100 µM at [NaCl] ) 0 M) concentrations; that is, the crystal region for the NaHCO3/silica system was restricted to the low NaHCO3 concentration regime. This can be explained by the increases in both Z and I with NaHCO3 concentration. IV.B. Unidirectional Crystal Growth. On the basis of the phase diagram, we examined the unidirectional crystallization. Figure 14a shows the images of the crystals formed at different NaHCO3 concentrations in the reservoir ([NaHCO3]0 ) 30 and 90 µM) at t ) 48 h. The crystal growth processes at [NaHCO3]0 ) 60 µM for four t values are shown in Figure 14b. As seen in Figure 14, the crystals formed by NaHCO3 diffusion had a maximum size of a few centimeters in height and 1 cm in width; the crystal size does not vary considerably with [NaHCO3]0. In Figure 15, the plots of h against t for [NaHCO3]0 ) 30, 60, and 90 µM are denoted by red, blue, and green symbols, respectively. The data points and bars represent the averaged values for at least three measurements and the standard deviations, respectively. The growth was faster at higher [NaHCO3]0 values, as was also observed in the case of Py diffusion. At [NaHCO3]0 ) 120 and 150 µM, directed crystallization was also observed; however, the crystals melted over time from the bottom at t > 100 h. This appears to be reasonable, as the reentrant melting occurred at [NaHCO3] ∼ 100 µM in the homogeneous phase diagram (Figure 13). Such melting was not observed for [NaHCO3]0 e 90 µM, at least not within a period of 1 month. As seen above, the conditions for unidirectional growth (without remelting) are restricted to the low-[NaHCO3]0 regime. Consequently, the crystal growth rates, R, are inevitably small. In the case of Py diffusion where the diffusing species is a noncharged Py molecule, directed growth was observed over a much wider range of [Py]0 values (0.1-100 mM, section III.C). The fastest growth was observed at [Py]0 ) 100 mM (R ) a few

Figure 15. Plots of the observed crystal height h against time t for unidirectional crystal growth at [NaHCO3]0 ) 90, 60, and 30 µM (red, blue, and green symbols, respectively). The bars represent the standard deviations. The solid and dashed curves represent the theoretical height h′.

centimeters per day). With a decrease in [Py]0, the R value markedly reduced, and the crystal size simultaneously increased. At [Py]0 ) 1 mM, nearly centimeter-sized crystals were formed at R ∼ 1 cm/day. As seen in Figure 15, the largest R value that could be attained by NaHCO3 diffusion (∼0.8 cm/day) is still smaller than this value. This inherently slow growth explains the large crystal size formed by NaHCO3 diffusion. In the case of Py diffusion, the crystallization process could be explained by the adsorption-diffusion equation for Py on the basis of the adsorption characterstics in a homogeneous system (section III.D). However, this relatively simple model is not applicable here because of the following reasons. The reaction between silanols and NaHCO3 produces H2CO3 (H2CO3 T H+ + HCO3-), which also undergoes diffusion. Therefore, the conservation relationship in the homogeneous system, [Na+] ) [HCO3-] + [H2CO3], is not locally satisfied in the diffusion process. Detailed formalization of the diffusion process is in progress, based on the multicomponent reaction-diffusion of the mobile species in silica colloids in combination with the local acid-base equilibrium. Here, we simply compare the observed crystal growth curves with the hypothetical ones calculated by assuming that the NaHCO3 molecules undergo diffusion without any chemical reactions. We numerically solved the diffusion equation (eq 3) by using the D value of free NaHCO3 in water at an infinite dilution (D0 ) 1.25 × 10-5 cm2/s at 25 °C) after correcting for the obstruction effect due to the colloidal particles (eqs 5 and 6). Based on the phase diagram (Figure 13), we assumed that crystallization takes place at [NaHCO3] ) 15 µM. The solid curves in Figure 15 are the calculated growths for the three [NaHCO3]0 values. The theoretical curves were

Unidirectional Crystallization of Colloidal Silica

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Figure 16. Gel-immobilized large crystals obtained at (a) [Py]0 ) 1 mM and t ) 20 h and at (b) [NaHCO3]0 ) 60 µM and t ) 5 days. (c) Part of the large crystal cut from the sample shown in (b) with side view images taken for two different incident light angles. (d) Cross sections of the gelled crystals cut at the locations indicated by the line shown in (c); the images were taken from two directions.

comparable to the experimental ones at [NaHCO3]0 ) 90 and 60 µM, suggesting a small effect of the chemical reaction on the diffusion process at high [NaHCO3]0 values. On the other hand, at [NaHCO3]0 ) 30 µM, the calculated growth was somewhat faster than the theoretical one; the dashed curve in Figure 15 shows the calculated growth for an apparent D0 value of 1.9 × 10-5 cm2/s, which exhibits a significantly closer agreement with the observed one. Preliminary multicomponent reactiondiffusion analysis shows that the codiffusion of NaHCO3 with H+HCO3- and H2CO3, which have larger diffusion coefficients (2.10 × 10-5 and 1.94 × 10-5 cm2/s at 25 °C, respectively) than NaHCO3, enables faster transport of Na+ ions than the case of one-component NaHCO3 diffusion, particlularly at low [NaHCO3]0 values. This is in accordance with the above-mentioned result.

V. Immobilization of Crystals by a Polymer Gel Matrix The large crystals obtained in this study may be applicable as photonic materials. Gel immobilization of the colloidal crystals has been studied extensively, as it is a considerably important technique for device applications.4 In many studies, polyacrylamide gels have been utilized to immobilize colloidal crystals. However, they are not suitable for the large crystals obtained in this study because the acrylamide monomer decomposes in aqueous solution to produce small amounts of ionic impurities, resulting in the melting of the large crystals. However, we were able to immobilize the large crystals by using N-methylolacrylamide as the gel monomer, which could be satisfactorily deionized by using the ion-exchange method. An image of such a large gelled crystal obtained by using Py diffuion is shown in Figure 16a ([Py]0 ) 1 mM and t ) 24 h without salt). Figure 16b shows the gel-immobilized crystals obtained by NaHCO3 diffusion ([NaHCO3]0 ) 60 µM and t ) 5 days), containing a large portion of the crystal (volume ) 1.3 cm3). From this original sample, the large crystal was cut out. Figure 16c shows the images of its side plane, which is illuminated at two different incident light angles. This large gelled crystal was further cut at the location indicated by the vertical line in Figure 16c. Figure 16d displays images of its edge plane taken from two different angles. As seen in parts c and d of Figure 16, both the side and cross-sectional

planes of the gelled crystal exhibited considerably uniform Bragg diffraction colors at specific wavelengths, depending on the diffraction geometries. This indicates that the crystals obtained here have a single-crystalline nature.

VI. Conclusions In this paper, we described the unidirectional crystal growth of charged colloidal silica driven by the diffusion of bases. The theoretical growth curves for the Py/silica system were calculated based on the adsorption-diffusion equations by using the adsorption relation and the crystallization phase diagram, and they showed good agreement with the observed growths. This suggests that the unidirectional crystallization is due to (i) the adsorption-diffusion of Py, resulting in the charging of the silica particles, and (ii) charge-induced crystallization. Directed growth was also observed due to the diffusion of NaHCO3 at low concentrations. Colloidal crystals with sizes in the cubiccentimeter range were obtained by the present method, which could be immobilized in the polymer hydrogel matrix. We expect that the large gelled crystals obtained here will be useful as photonic materials. Acknowledgment. The authors sincerely thank Dr. Yoshihiro Takiguchi, Hamamatsu Photonics K.K., for his suggestions on the applicability of colloidal crystals as photonic materials. Thanks are also due to Mr. Yoichi Oba, Interface Technical Laboratories, for his helpful suggestions regarding the gel monomer. J.Y. expresses sincere thanks to Dr. Tsuyoshi Koga, Kyoto University, for his advice concerning the numerical calculations. We are grateful to Mr. Masanobu Hayashi, Nagoya City University, for performing the transport measurements for the PTFE membrane. This study was supported by a Grant-in-Aid from the Ministry of Education, Science and Culture, Japan. A part of this study was performed as the “Three-Dimensional Photonic-Crystal (3DPC) Project” of the Japan Aerospace Exploration Agency (JAXA). Supporting Information Available: Theoretical crystal growth curves for Py diffusion at various values of φ. This material is available free of charge via the Internet at http://pubs.acs.org. LA700754S