Langmuir 2005, 21, 7633-7637
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Articles Critical Concentration for Colloidal Crystallization Determined with Microliter Centrifuged Suspensions Toshimitsu Kanai,† Tsutomu Sawada,*,† Junpei Yamanaka,‡ and Kenji Kitamura† National Institute for Materials Science, 1-1 Namiki, Tsukuba, Ibaraki 305-0044, Japan, and Faculty of Pharmaceutical Sciences, Nagoya City University, 3-1 Tanabe, Mizuho, Nagoya, Aichi 467-8603, Japan Received May 2, 2005 We report an elegant method using centrifugal sedimentation for determining the critical particle concentration for colloidal crystallization. A small amount of a dilute suspension of monodispersed particles stored in a flat capillary cell was centrifuged to temporarily generate a nonequilibrium gradient of the particle concentration including a crystalline-noncrystalline phase boundary in the cell. In the recovering process after the centrifugation, the particle concentration of the crystalline phase at the boundary was found to always have the equilibrium value, although the global concentration distribution evolved with time. The critical concentration was determined based on spatially resolved spectrometry. The present method requires only one batch of a suspension of the order of microliters and is applicable up to high concentration regions near the closest packing without the effect of the particle aggregation.
Introduction Colloidal particles have been used in a broad range of applications in technology and material processing.1 Recently, colloidal crystals,2 which are three-dimensional periodic arrays of charged colloidal particle in dispersions, have received considerable attention for certain applications, especially as photonic crystals3 in the optical regime. A variety of functional particles have been prepared, e.g., magnetic materials and TiO2;4 hence, the fabrication of their colloidal crystals is expected to be the next stage. One of the most important techniques for colloidal crystallization is determining the critical particle con* To whom correspondence should be addressed. E-mail:
[email protected]. Telephone: +81-29-851-3354. Fax: +81-29-852-7449. † National Institute for Materials Science. ‡ Nagoya City University. (1) Evans, D. F.; Wennerstrom, H. A. The Colloidal Domain, Where Physics, Chemistry, Biology, and Technology Meet; VCH: New York, 1994. (2) (a) Pieranski, P. Contemp. Phys. 1983, 24, 25. (b) Ito, K.; Sumaru, K.; Ise, N. Phys. Rev. B 1992, 46, 3105. (c) Ise, N.; Smalley, M. V. Phys. Rev. B 1994, 50, 16722. (d) Ordering and Phase Transitions in Charged Colloids; Arora, A. K., Tata, B. V. R., Eds.; VCH: New York, 1996. (e) Gast, A. P.; Russel, W. B. Phys. Today 1998, 51, 24. (3) (a) Yablonovitch, E. Phys. Rev. Lett. 1987, 58, 2059. (b) John, S. Phys. Rev. Lett. 1987, 58, 2486. (c) Joannopoulos, J. D.; Meade, R. D.; Winn, J. N. Photonic Crystals; Princeton University Press: Princeton, NJ, 1995. (d) Sakoda, K. Optical Properties of Photonic Crystals; Springer-Verlag: Berlin, Germany, 2001. (e) Weitz, D. A.; Russel, W. B., Eds. MRS Bull. 2004, 29, 82. (f) Polman, A.; Wiltzius, P., Eds. MRS Bull. 2001, 26, 608. (g) Grier, D. G., Ed. MRS Bull. 1998, 23, 21. (4) (a) Li, Y. L.; Ishigaki, T. Chem. Mater. 2001, 13, 1577. (b) Kim, B.; Tripp, S. L.; Wei, A. J. Am. Chem. Soc. 2001, 123, 7955. (c) Xu, X.; Majetich, S. A.; Asher, S. A. J. Am. Chem. Soc. 2002, 124, 13864. (d) Feldmann, C. Adv. Funct. Mater. 2003, 13, 101. (e) Breen, M. L.; Dinsmore, A. D.; Pink, R. H.; Qadri, S. B.; Ratna, B. R. Langmuir 2001, 17, 903. (f) Velikov, K. P.; Moroz, A.; van Blaaderen, A. Appl. Phys. Lett. 2002, 80, 49. (g) Yin, Y.; Lu, Y.; Gates, B.; Xia, Y. Chem. Mater. 2001, 13, 1146. (h) Zhao, Y.; Sadtler, B.; Lin, M.; Hockerman, G. H.; Wei, A. Chem. Commun. 2004, 784. (i) Sugimoto, T. Adv. Colloid Interface Sci. 1987, 65.
centration (CPC) above which the particle system is in a state of the crystalline phase. The CPC can be a measure of the strength of the interparticle interaction and is a practical guide for designing crystallization conditions. For example, near the CPC, the colloidal suspension tends to form large single crystalline domains; from the viewpoint of stability, crystals easily melt under the shear stress. The conditions for optimized fabrication should be determined from these considerations. The CPC is conventionally determined using a homogeneous colloidal suspension under equilibrium,5 which requires a relatively large amount of suspension. In general, monodispersed particles are valuable especially in early developmental stages; therefore, their large consumption could be a practical bottleneck in the phase study of the colloidal crystallization. Previously, we briefly reported that the crystallinenoncrystalline boundary in centrifuged nonequilibrium colloidal suspensions can be regarded as being a state of the local phase equilibrium.6 On the basis of this phenomenon, the phase diagram of colloidal suspensions can be determined with an extremely small amount of suspension. In this paper, we make detailed comparisons between the present nonequilibrium method and the conventional equilibrium method. We also show that the present method makes it possible to determine the crystallization phase diagram over a wide particle concentration range up to a high concentration near the closest packing condition; in the conventional method, such an estimation is difficult because of aggregation. (5) (a) Hachisu, S.; Kobayashi, Y.; Kose, A. J. Colloid Interface Sci. 1973, 42, 342. (b) Takano, K.; Hachisu, S. J. Colloid Interface Sci. 1978, 66, 124. (c) Kremer, K.; Robbins, M. O.; Grest, G. S. Phys. Rev. Lett. 1986, 57, 2694. (d) Yamanaka, J.; Yoshida, H.; Koga, H.; Ise, N.; Hashimoto, T. Phys. Rev. Lett. 1998, 80, 5806. (e) Toyotama, A.; Sawada, T.; Yamanaka, J.; Kitamura, K. Langmuir 2003, 19, 3236. (6) Kanai, T.; Sawada, T.; Yamanaka, J.; Kitamura, K. J. Am. Chem. Soc. 2004, 126, 13210.
10.1021/la051177f CCC: $30.25 © 2005 American Chemical Society Published on Web 07/12/2005
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Figure 1. Photograph of the colloidal suspension with a salt concentration of 1 mM in a flat capillary cell (A) before and (B) after centrifugation at 2000 rpm for 20 h.
Experimental Procedures Nonequilibrium Method. Suspensions of uniform-sized polystyrene microspheres in water (Duke Scientific Corp.; fraction, 10%; diameter, 198 nm; standard deviation, 3%) with various salt (NaCl) concentrations were prepared as test samples. A small amount of suspension, which was in a state of the noncrystalline phase, was stored in a flat capillary cell (inner dimension, thickness, 0.1 mm; width, 9 mm; length, 50 mm), as shown in Figure 1A. The sample volume was 45 µL. The cell was hermetically sealed and then centrifuged at 2000 rpm (ca. 700 G) for 5-20 h. After centrifugation, an iridescent crystalline region formed at the bottom (indicated in the figure) of the cell and a milky white noncrystalline region remained above the crystalline one (Figure 1B). It is noted that the capillary gap is sufficiently thin to suppress the generation of convective flow in the suspension even when the cell is horizontalized. Spatially resolved reflectance spectra at an incidence normal to the capillary face were measured at a region including the crystalline-noncrystalline boundary using an imaging spectrometer (Kawasaki Steel Techno-research Corp., ImSpector) with an inplane spatial resolution of 100 µm. Because the colloidal crystal has the characteristic of directing its close-packed lattice plane toward the cell wall, the reflectance spectra for the crystalline region show a peak corresponding to the Bragg reflection from the close-packed plane. The particle concentration in the crystalline region including the CPC can be determined from the peak wavelength, as described below. Equilibrium Method. To investigate the characteristics of the present nonequilibrium method, the conventional equilibrium method was also performed to determine the CPC. Beginning with the same original suspensions (doped with salt) as in the nonequilibrium method, concentrated suspensions exhibiting iridescence were prepared by decanting extra water after mild centrifugation of the relatively dilute starting suspension. Then, 1 mL of the concentrated suspension was diluted step by step in a square quartz cell with a small amount of water (a few tens of microliters) of the same salt concentration until the iridescence disappeared. At each dilution step, the suspension was homogenized and equilibrated, after which the normal reflectance spectrum was measured using a fiber spectrometer (Soma Optics, Fastevert S-2600). The particle concentration of the crystalline phase was determined from the peak wavelength of the obtained reflectance spectrum as described below. Determination of the Particle Concentration from the Bragg Wavelength. The relation between the Bragg wavelength and particle concentration is described as follows. Here, we assume the crystal structure to be the cubic-close-packed
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Figure 2. (A) Particle concentration as a function of the Bragg wavelength. (B) Derivative of the particle concentration in terms of the Bragg wavelength. structure of spheres, i.e., face-centered cubic (fcc), which is the most common structure. This structure exists under the present conditions; however, when the particle concentration is very dilute and the particle surface charge is low, the crystal structure could possibly be body-centered cubic (bcc). Thus, the close-packed plane parallel to the cell wall is fcc (111), and the Bragg condition for normal incidence is written as
2ncd111 ) λ
(1)
where nc is the refractive index of the colloidal crystal, d111 is the interplanar spacing of (111) planes, and λ is the wavelength of light. The particle volume fraction φ of the fcc structure is derived from a geometrical consideration by using d111 and the particle diameter d
φ)
( )
2π d 9x3 d111
3
(2)
From eqs 1 and 2, we obtain
φ)
( )
2π 2ncd 9x3 λ
3
(3)
The volume-weighted average of the refractive indices of the dispersion medium ndm and particle np is known to provide a good approximation of nc,7 as follows:
nc ) ndm(1 - φ) + npφ
(4)
Combining eqs 3 and 4, the particle concentration φ can be calculated from the Bragg wavelength with the given particle diameter d. For the present condition (ndm ) 1.33, np ) 1.59, and d ) 198 nm), the particle concentration is calculated as a function of the Bragg wavelength and plotted in Figure 2A and the derivative, which gives uncertainty in φ because of an error in the wavelength measurement, is shown in Figure 2B. (7) (a) Hiltner, P. A.; Krieger, I. M. J. Phys. Chem. 1969, 73, 2386. (b) Mu¨ller, M.; Zentel, R.; Maka, T.; Romanov, S. G.; Torres, C. M. S. Adv. Mater. 2000, 12, 1499.
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Figure 4. (A) Distribution of the particle concentration in the centrifuged colloidal suspension along the centrifugal force direction, obtained 0 (b), 1 (4), 5 (2), and 10 (O) days after stopping the centrifugation. (B) Long-time stability of the particle concentration at the moving phase boundary.
Figure 3. (A) RGB-composed image around the crystalline boundary of the centrifuged colloidal suspension with a salt concentration of 1 mM after centrifugation at 2000 rpm. (B) Reflectance spectra at points a-h indicated in A. (C) Reflectance spectra of homogeneous colloidal suspensions in the dilution steps. The salt concentration is 1 mM.
Results Behavior of the Crystalline-Noncrystalline Boundary in the Centrifuged Suspension. Figure 3A shows an image, reconstructed from the spatially resolved reflectance spectra data, around the crystallinenoncrystalline boundary of the centrifuged suspension stored in a flat capillary cell (with a salt concentration of 1 mM), where the colored region is in a crystalline phase. The color contrast is caused by a difference in the Bragg wavelength. In Figure 3B, the spectra at the representative points indicated in A are shown. The peak in each spectrum, which is due to the Bragg reflection from fcc (111) planes parallel to the cell surface, continuously shifts to longer wavelengths while moving from the bottom side to the phase boundary (point a to point f) and disappears
in the disordered region (points g and h). An analogous behavior in the peak shift was observed during the stepby-step dilution of the suspension in the equilibrium method. Figure 3C shows the normal reflectance spectra for homogeneous suspensions at the dilution steps, where the reflection peak exhibits a red shift as the particle concentration decreases, and it finally disappears. In the equilibrium method, the peak wavelength just before the extinction should yield the CPC. A comparison of parts B and C of Figure 3 suggests the possibility that a single experiment during which the capillary cell consumes only a few tens of microliters of the suspension can replace a set of dilution experiments that involve the consumption of milliliters of the suspension. However, this is surprising because the centrifuged suspension in the capillary cell is neither in a global equilibrium nor in a steady state and the boundary position moves with time. Figure 4A shows spatial profiles of the particle concentration in the crystalline region 0, 1, 5, and 10 days after stopping the centrifugation near the phase boundary of the centrifuged sample (a salt concentration of 1 mM) along the centrifugal force direction; the particle concentration was determined from the Bragg wavelength read from the spatially resolved spectrum data. Here, each profile can be regarded as a snapshot because the movement speed of the boundary is low enough (less than 2 mm/day) to be negligible for the spectrum measurement. A remarkable characteristic of these profiles is that the particle concentration just below the crystalline boundary does not vary although the global shape changes with time. The long time stability of the concentration at the
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tration for the hard-sphere system8 (φ ) 0.545), thus suggesting that the present method is applicable for a system in which the electrostatic interparticle interaction is almost perfectly shielded. Discussion
Figure 5. Correlation between the particle concentration at the nonequilibrium phase boundary and the CPC, as determined by the equilibrium method for suspensions with the indicated salt concentrations.
Figure 6. Plot of the critical concentration versus the salt concentration determined by the nonequilibrium method.
boundary is demonstrated in Figure 4B. An identical behavior was generally observed under other centrifugation conditions, i.e., different rotation speeds and durations (results not shown). This strongly suggests that the phase equilibrium is locally attained at the phase boundary despite its unsteady position; this was confirmed by comparing the CPCs obtained from a centrifuged suspension (nonequilibrium method) and homogeneous suspension (equilibrium method), as explained below. CPCs Determined by the Nonequilibrium Method. Figure 5 shows the correlation between the particle concentration at the nonequilibrium phase boundary and the CPC, as determined by the equilibrium method for suspensions with the indicated salt concentrations. Because all of the data points are considered to lie on the 45° solid line within the errors, it can be concluded that the nonequilibrium method yields the equilibrium CPCs. The present method is particularly advantageous for measurement in high particle concentration conditions. Because in the conventional method, the measurement proceeds from a concentrated condition to a dilute one and requires a relatively long waiting time for equilibration, particle aggregation is a serious obstacle especially when the CPC is high. In contrast, in the present method, the measurement process starts from a dilute suspension, and therefore, the probability of particle aggregation can be significantly reduced. This allows us to study the phase of the crystallization in a very high concentration region. Figure 6 demonstrates the applicability of the present nonequilibrium method. The CPC saturated at approximately 0.6, which is close to an Alder transition concen-
The essence of the present method is that a large gradient, in the particle concentration, from the crystalline to noncrystalline phase is intentionally generated in a flat capillary cell by centrifugation, and the particle concentration of the crystalline phase at the transient boundary is determined from spectroscopic measurement. This method is significantly based on characteristics specific to colloidal crystals, i.e., the ability of the relatively large particle mass to generate the concentration gradient by centrifugal sedimentation within a realistic time frame and the slow particle diffusion that retains the transient coexisting state of the two phases during spectral measurement. Although atomic ions or molecules as solutes in ordinary solutions do not satisfy these conditions, submicrometer colloidal particles satisfy them. In addition, because the lattice constant of colloidal crystals is of the order of the wavelength of visible light, the particle concentration can be optically determined in situ by knowing the Bragg reflected wavelength. In the present measurement, imaging spectrometry was performed to determine the CPC. However, a more conventional instrument, such as a microscope spectrometer or a standard spectrometer with a mask, may be used to measure the local reflectance spectrum at the phase boundary. We considered only the case of the fcc structure described earlier. If a crystal assumes the bcc structure, the closepacked lattice plane is (110), which is parallel to the cell surface. Hence, eq 3 for the fcc structure is replaced by the following equation:
φ)
( )
π 2ncd 6x2 λ
3
(5)
This causes a somewhat lower particle concentration than in the case of the fcc structure (relative difference is approximately 8%). This difference would be insignificant in many practical cases. The inhomogeneous distribution of macroions such as charged colloidal particles yields an internal macroscopic electric field; therefore, the distribution of microions can also be inhomogeneous.9,10 This phenomenon is known as the Donnan equilibrium and can seriously affect the present system. If significant inhomogeneity in the salt concentration arises in the centrifuged suspension because of this mechanism, the particle concentration at the boundary will not agree with the CPC determined in the homogeneous system. As already seen in Figure 5, the quantitative agreement between the CPCs obtained using the present method and those obtained using the conventional equilibrium method is excellent. Therefore, we can neglect the Donnan effect at least under the present conditions. This should be because of the relatively highsalt concentration condition in the present experiments. According to van Roij,10 the effect of the inhomogeneous charge distribution in the sedimentation of charged colloids becomes less serious as the salt concentration (8) (a) Pusey, P. N.; van Megen, W. Nature 1986, 320, 340. (b) Hynninen, A.-P.; Dijkstra, M. Phys. Rev. E 2003, 68, 021407. (9) Rasa, M.; Philipse, A. Nature 2004, 429, 857. (10) van Roij, R. J. Phys.: Condens. Matter 2003, 15, S3569.
Critical Concentration for Colloidal Crystallization
increases. The present method should be cautiously used when the salt concentration is very low. Concluding Remarks We have succeeded in determining the CPC for colloidal crystallization over a wide particle concentration region using a centrifuged suspension. A large gradient in the particle concentration from the crystalline to noncrystalline region was intentionally formed in the flat capillary cell by the centrifugation. The distribution of the particle concentration in the crystalline region was examined by the Bragg reflection measurement. Although the position of the crystalline boundary developed with time after
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stopping the centrifugation, the particle concentration at the crystalline boundary did not vary and it was in accordance with the equilibrium particle concentration between the ordered and disordered phase. The present method requires only one batch of a small amount of centrifuged sample of the order of microliters without the need to prepare many samples with different concentrations; it is also applicable up to high concentration regions near the closest packing with a low possibility of particle aggregation. The sample volume in the present measurement is only 45 µL, and it can be further reduced by reducing the cell size, e.g., the cell width. LA051177F
