Crystallization Kinetics of Thermosensitive Colloids Probed by

It is found that the rates of the change in crystallinity, the average crystallite size, and the number density of crystallites at the most rapid stag...
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Langmuir 2004, 20, 8858-8864

Crystallization Kinetics of Thermosensitive Colloids Probed by Transmission Spectroscopy Shijun Tang and Zhibing Hu* Department of Physics, University of North Texas, Denton, Texas 76203

Zhengdong Cheng† Department of Physics and DEAS, Harvard University, Cambridge, Massachusetts 02138

Jianzhong Wu Department of Chemical and Environmental Engineering, University of California, Riverside, California 92521 Received March 26, 2004. In Final Form: June 14, 2004 The kinetics of crystallization of poly-N-isopropylacrylamide (PNIPAM) particles has been investigated using the UV-visible transmission spectroscopy. Since the particle size decreases with the increase in temperature, microgel dispersions of different volume fractions have been obtained by varying the temperature of a single sample. It is found that the rates of the change in crystallinity, the average crystallite size, and the number density of crystallites at the most rapid stage over a certain time interval at various temperatures can be described by the power-law relations. At 19 °C, the PNIPAM system behaves as a hard sphere system under microgravity. The hard sphere theory based on Monte Carlo simulation has been used as a reference point to compare with conventional hard spheres, soft spheres, and PNIPAM spheres.

Introduction Crystallization of colloidal systems provides not only the knowledge of complex fluids but also insights into the phase transitions in atomic systems.1-15 In recent years, colloidal crystals have been used extensively for the fabrication of nanostructured materials such as photonic crystals and membranes for device applications.16-19 Most * Corresponding author. E-mail: [email protected]. † Current address: Department of Chemical Engineering, Texas A&M University, College Station, TX 77843. E-mail: [email protected]. (1) Pusey, P. N. In Liquid, Freezing, and the Glass Transition; Les Houches, J. P., Levesque, H. D., Zinn-Justin, J., Eds.; Elsevier: Amsterdam, 1990. Pusey, P. N.; van Megen, W. Nature 1986, 320, 340. Anderson, V. J.; Lekkerkerker, H. N. W. Nature 2002, 416, 811-815. (2) Dixit, N. M.; Zukoski, C. F. Phys. Rev. E 2001, 64, 041604; 2002, 66, 051602. (3) Gasser, U.; Weeks, E. R.; Schofield, A.; Pusey, P. N.; Weitz, D. A. Science 2001, 292, 258. (4) van Megen W.; Underwood, S. M. Nature (London) 1993, 362, 616. (5) Harland, J. L.; Henderson, S. I.; Underwood, S. M.; van Megen, W. Phys. Rev. Lett. 1995, 75, 3572. Harland, J. L.; van Megen, W. Phys. Rev. E 1997, 55, 3054. (6) Okubo, T.; Tsuchida, A.; Kato, T. Colloid Polym. Sci. 1999, 277, 191. (7) Weeks, E. R.; Crocker, J. C.; Levitt, A. C.; Schofield, A.; Weitz, D. A. Science 2000, 287, 627. (8) Cheng, Z.; Zhu, J. X.; Russel, W. B.; Chaikin, P. M. Phys. Rev. Lett. 2000, 85, 1460. (9) Liu, J.; Weitz, D. A.; Ackerson, B. J. Phys. Rev. E 1993, 48, 1106. (10) Russel, W. B. Phase Transit. 1990, 21, 27. (11) Ackerson, B. J.; Schatzel, K. Phys. Rev. E 1995, 52, 6448. (12) Russel, W. B.; Chaikin, P. M.; Zhu, J. X.; Meyer, W. V.; Rogers, R. B. Langmuir 1997, 13, 3871. (13) Zhu, J.; Li, M.; Rogers, R.; Meyer, W.; Ottewill, R. H. STS-73 Space Shuttle Crew, Russel, W. B.; Chaikin, P. M. Nature (London) 1997, 387, 883. (14) Cheng, Z.; Chaikin, P. M.; Zhu, J.; Russel, W. B.; Meyer, W. V. Phys. Rev. Lett. 2002, 88, 015501. (15) Cheng, Z.; Zhu, J.; Russel, W. B.; Meyer, W. V.; Chaikin, Paul M. Appl. Opt. 2001, 40, 4148. (16) Xia, Y. N.; Gates, B.; Yin, Y.; Lu, Y. Adv. Mater. 2000, 12, 693. Fudouzi, H.; Xia, Y. N. Langmuir 2003, 19, 9653-9660.

previous works on the kinetics of colloidal crystallization have been focused on weakly charged or hard-sphere-like colloids using light scattering methods.1,5,14 The classical theory of nucleation and crystal growth has been adapted by Russel10 to hard-sphere-like colloids and extended and evaluated numerically by Ackerson and Scha¨tzel.11 However, large differences exist between theoretical calculations on hard spheres assisted by computer simulation20 and the experiments with model colloid systems. Meanwhile, the phase behavior of the aqueous dispersions of poly-N-isopropylacrylamide (PNIPAM) spheres has been intensively investigated.21-26 In particular, it was recently found that the volume transition of microgel particles affects the solvent-mediated interparticle forces and leads to a novel phase behavior.27 Here we report the first measurement of the crystallization kinetics in a PNIPAM microgel dispersion using UV-visible transmission spectroscopy. In contrast to (17) Chen, Y.; Ford, W. T.; Materer, N. F.; Teeters, D. Chem. Mater. 2001, 13, 2697-2704. (18) Takeoka, Y.; Watanabe, M. Adv. Mater. 2003, 15, 199. (19) Lee, Y. J.; Braun, P. V. Adv. Mater. 2003, 15, 563. (20) Auer. S.; Frenkel, D. Nature 2001, 409, 1020. Auer, S. Ph.D. thesis, The FOM Institute for Atomic and Molecular Physics 2002. (21) Weissman, J. M.; Sunkara, H. B.; Tse , A. S.; Asher, S. A. Science 1996, 274, 959. Holtz, J. H.; Asher, S. A. Nature 1997, 389, 829. Reese, C. E.; Mikhonin, A. V.; Kamenjicki, M.; Tikhonov, A.; Asher, S. A. J. Am. Chem. Soc. 2004, 126, 1493. (22) Senff, H.; Richtering, W. J. Chem. Phys. 1999, 111, 1705; Langmuir 1999, 15, 102. (23) Debord, J. D.; Lyon, L. A. J. Phys. Chem. B 2000, 104, 6327. Debord, S. B.; L. Lyon, A. J. Phys. Chem. B 2003, 107, 2927. Jones, C. D.; Lyon, L. A. J. Am. Chem. Soc. 2003, 125, 460. (24) Hu, Z. B.; Lu , X. H.; Gao, J.; Wang, C. J. Adv. Mater. 2000, 12, 1173. Hu, Z. B.; Lu, X. H.; Gao, J. Adv. Mater. 2001, 13, 1708. Gao, J.; Hu, Z. B. Langmuir 2002, 18, 1360. Hu, Z. B.; Huang, G. Angew. Chem., Int. Ed. 2003, 42, 4799. (25) Hellweg, T.; Dewhurst, C. D.; Bru¨ckner, E.; Kratz, K; Eimer, W. J. Colloid Polym. Sci. 2000, 278, 972. (26) Okubo, T.; Hase, H.; Kimura, H.; Kokufuta, E. Langmuir 2002, 18, 6783.

10.1021/la049203h CCC: $27.50 © 2004 American Chemical Society Published on Web 08/28/2004

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conventional colloidal particles such as silica and polystyrene, the PNIPAM microgel spheres investigated in this work contain up to 97 wt % water. Consequently, the density and the refractive index of the particles closely match up those of water, yielding a condition of minigravity (∼10-2 g) at room temperature. Furthermore, due to the temperature-responsive shrinkage of PNIPAM particles27,28 the attractive interparticle potential increases with temperature, resulting in phase behavior that deviates from the hard sphere systems at high temperatures. The kinetics of crystallization at different volume fractions can be conveniently measured by varying the temperature. This work is part of the current major thrust for developing systems where external responses (e.g., electric and magnetic fields) change the crystallization behavior. For example, it is reported that electric fields combined with patterned structures have been used to modulate the crystallization29 and Helseth and Fischer30 demonstrate that magnetic fields may be used to tune crystallization behavior by studying how a magnetic field may break up a crystal (and form chains in three dimensions). Experimental Section Synthesis of PNIPAM Microgels. We synthesized PNIPAM particles using precipitation polymerization.31 A 3.78 g portion of N-isopropylacrylamide monomer, 0.11 g of acrylic acid (AA) monomer, 0.0665 g of methylene-bis-acrylamide (BIS) as crosslinker, 0.106 g of sodium dodecyl sulfate (SDS) as surfactant, and 240 mL of deionized water were mixed in the flask. The solution was stirred at 300 rpm for 30 min under nitrogen. A 0.166 g portion of potassium persulfate (KPS) dissolved in 10 mL of deionized water was added to start the reaction. The reaction was carried out at 70 °C for 4 h. A small amount of poly(acrylic acid) was incorporated into the PNIPAM network with the molar ratio of the ionic group to NIPAM approximately equal to 2% at pH ) 4.3. In this experiment, all measurements were carried out at pH ) 4.3 so that the carboxyl groups were stabilized without causing additional repulsive interactions between particles. The PNIPAM particles can self-assemble in water by evaporating the solvent at a temperature higher than 34°C and then allowing the concentrated dispersion to reach an equilibrium state for 1 week. Dynamic Light Scattering Characterization. A laser light scattering (LLS) spectrometer (ALV Co., Germany) equipped with an ALV-5000 digital time correlator was used with a heliumneon laser (Uniphase 1145P, output power of 22 mW and wavelength of 632.8 nm) as the light source. The incident light was vertically polarized with respect to the scattering plane, and the light intensity was regulated with a beam attenuator (Newport M-925B). The scattered light was conducted through a single mode optical fiber leading to an active quenched avalanche photodiode (APD), the detector. In dynamic LLS, the intensity-intensity time correlation function G(2)(t,q) in the selfbeating mode was measured and can be expressed by32,33

G(2)(t,q) ) 〈I(t,q)I(0,q)〉 ) A(1 + β|g(1)(t,q)|2)

(1)

where t is the decay time, A is a measured baseline, β is the coherence factor, and g(1)(q,t) is the normalized first-order electric field time correlation function E(t,q). The value of β was estimated to be between 0.95 and 0.98, using eq 1 and at t f 0, g(1)(t,q) f 1. (27) Wu, J.; Zhou, B.; Hu, Z. B. Phys. Rev. Lett. 2003, 90, 048304. Wu, J.; Huang, G.; Hu, Z. B. Macromolecules 2003, 36, 440. (28) Shibayama, M.; Tanaka, T. Adv. Polym. Sci. 1993, 109, 1. (29) Golding, R. K.; Lewis, P. C.; Kumacheva, E.; Allard, M.; Sargent, E. H. Langmuir 2004, 20, 1414. (30) Helseth L. E.; Fischer, T. M. Phys. Rev. E 2003, 68, 051403. (31) Pelton, R. H.; Chibante, P. Colloids Surf. 1986, 20, 247. (32) Chu, B. Laser Light Scattering, 2nd ed.; Academic Press: New York, 1991. (33) Berne, B. J.; Pecora, R. Dynamic Light Scattering; Wiley: New York, 1976.

g(1)(q,t) is related to the line-width distribution G(Γ) by

g(1)(t,q) ) 〈E(t,q)E*(0,q)〉 )

∫

∞

0

G(Γ) e-Γt dΓ

(2)

G(Γ) can be obtained from the Laplace inversion of g(1)(q,t) using the CONTIN program developed by Provencher.34 g(1)(q,t) was also analyzed by a cumulant analysis to get the average line width 〈Γ〉 and the relative distribution width µ2/〈Γ〉2. The extrapolation of Γ/q2 to qf 0 led to the translational diffusion coefficient (D). Further, G(Γ) can be converted to the translational difffusion coefficient distribution G(D) and to the hydrodynamic radius distribution f(Rh) by using the Stokes-Einstein equation

Rh ) kBT/6πηD

(3)

where kB, T, and η are the Boltzmann constant, the absolute temperature, and the solvent viscosity, respectively. The dynamic light scattering experiments were performed at the scattering angle θ ) 90°. Using a smaller scattering angle could make the measurements more accurate, but it will not significantly affect the trends of the results. According to the dynamic light scattering measurements, the hydrodynamic radius of these particles was narrowly distributed. With increasing temperature, the PNIPAM particles undergo a continuous volume transition that collapse at about 34 °C, similar to that for a neutral or a weakly ionized bulk PNIPAM gel. In the temperature range from 19 to 25 °C, the hydrodynamic radius of microgel particles can be empirically described by Rh ) 250.5 - 1.014T, where the units for Rh and T are nanometers and °C, respectively. UV-Visible Spectroscopy Measurements. The polymer concentration of a dispersion was obtained by completely drying the dispersion at 60 °C and then weighing it. The turbidity (R) of the dispersions was measured as a function of the wavelength using a diode array UV-vis spectrometer (Agilent 8453) by calculating the ratio of the transmitted light intensity (It) to the incident intensity (I0) R ) -(1/d) ln(It/I0), where d is the thickness (1 cm) of the sampling cell.

Results and Discussion The crystallization of PNIPAM microgel dispersion can be readily observed by its appearance as shown in Figure 1. We started with a shear-melted dispersion of microgel particles with a polymer concentration equal to 15.2 g/L and the solution pH ) 4.3. The sample cell was first mounted into a sample holder to reach an equilibrium temperature and then was quickly taken out. After shaken quickly by hand, the cell was put back in the holder. Then, the entire crystallization process was quantitatively monitored by measuring the UV-vis transmission spectra. Because of the Bragg diffraction, the UV-vis spectrum exhibits a sharp attenuation peak in the crystalline phase. Above the freezing temperature 26 °C (corresponding to the freezing volume fraction), the peak disappears. Figure 2 shows typical turbidity profiles for the same aqueous dispersion of PNIPAM spheres (the polymer concentration is 15.2 g/L and the solution pH is 4.3) at 19 and 24 °C. At 19 °C (Figure 2a), a weak Bragg peak appears about 3 min after the shear melting is stopped. The turbidity increases with time and reaches a maximum at around 54 min. The same measurements were repeated at six other temperatures up to 25 °C, near the freezing temperature of the microgel dispersion (Tf ) 26 °C). We found that because of the changes in particle size, the kinetics of crystal nucleation depends strongly on tem(34) Provencher, S. W. Makromol. Chem. 1979, 180, 201.

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Figure 1. Progressive appearance of a PNIPAM microgel dispersion in crystallization at 22 °C. From left to right, the duration of crystallization is 0, 4, 11, 25, 89, and 152 min. Here the polymer concentration is 15.2 g/L and the solution pH is 4.3. At this temperature, the average hydrodynamic radius of the PNIPAM particles in a dilute water dispersion is 228 nm.

perature. For example, the nucleation rate at 24 °C (Figure 2b) is slower than that at 19 °C (Figure 2a). It is found that when the refractive indexes of the solvent and the particle are nearly matched (i.e., ns ∼ np), the attenuation of the transmission beam in the normal incidence is mainly due to the Bragg diffraction of light.35 For the Bragg peak intensity much smaller than that of the incident beam, the structure factor may be written as

S(λ,t) =

R(λ,t) SB′(λ,t) + Sdiff(λ,t) ) ) R(λ,0) Sfluid(λ,0) SB′(λ,t) + B(λ,t) ) SB(λ,t) + B(λ,t) (4) Sfluid(λ,0)

where SB(λ,t) describes the Bragg reflection of the crys(35) Astratov, V. N.; Adawi, A. W.; Fricker, S.; Skolnicket, M. S.; Whittaker, D. M.; Pusey, P. N. Phys. Rev. B 2002, 66, 165215. (36) Heymann, A.; Stipp, A.; Sinn, C.; Palberg, T. J. Colloid Interface Sci. 1998, 207, 119.

tallites, Sfluid(λ,0) arises from the fluid phase at t ) 0, and Sdiff(λ,t) represents the diffusive scattering from the fluid phase as well as grain boundaries of the crystalline phase.36 The structure factors SB(λ,t) of the developing crystals can be obtained by subtracting the background B(λ,t) from S(λ,t). The degree of crystallization X(t) is defined as the fraction of crystalline phase obtained from the area of the Bragg peak5,14

X(t) ) κ

∫∆λ SB(λ,t) dλ

(5)

where κ is a normalization constant. ∆λ is the wavelength range of the Bragg peak. The average linear crystal size (in units of the particle diameter 2R) is calculated from

L(t) ) πK/wq(t)R

(6)

where wq(t) is the width of the peak at half-maximum and

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Figure 2. The turbidities of the PNIPAM microgel dispersion measured from UV-vis transmission spectroscopy at various stages of nucleation: (a) T ) 19 °C; (b) T ) 24 °C. The Bragg peak arises with the formation of crystalline nucleates.

K ) 1.155 is the Scherrer constant for a crystal of cubic shape. The average number density of crystalline nucleates is then

Nc(t) ) X(t)/L3(t)

(7)

It has been shown by neutron scattering experiments that the crystal of PNIPAM dispersions exhibits a face-centered cubic (fcc) structure.21,25 The particle volume fraction φc of the crystalline phase is given by

φc(t) ) 0.0130[qm(t)R]3

Figure 3a shows X(t) at various temperatures. There are several stages during crystallization at each temperature. Following established procedures for hard sphere systems,5,14 in the most rapid kinetics stage the rate of crystallization is described with the power-law relation X(t) ∼ tµ, where the exponent µ increases from 0.21 at T ) 25 °C to 2.1 at T ) 19 °C. Figure 3b presents the average linear crystal size L(t). Eimer and co-workers observed that the radius of particles in the close-packed state is approximately the same as the radius of gyration in the dilute dispersion.25 So, Rg instead of Rh is used in eq 6 to calculate the linear crystal size. For PNIPAM particles, Rg/Rh ≈ 0.778.22,37 A similar procedure is applied to the calculation of φc(t). The variation of the crystallites’ size also follows a power-law relation L(t) ∼ tδ, with the exponent δ rising from 0.12 at 25 °C to 0.41 at 19 °C. Figure 3c shows the crystal number density as a function of time. The nucleation rate significantly increases as the sample is cooled from the freezing temperature of 26 °C to 19 °C. The rate of the crystal number density at the most rapid stage is described by Nc(t) ∼ tν. Following an initial fast increase, Nc(t) reaches a maximum and declines during further progress of crystallization (coarsening).5 Figure 3d shows φc(t) obtained from the Bragg peak positions using eq 8. The decrease of φc(t) versus time for the PNIPAM system originates from the decrease of osmotic pressure and is smaller than that for a hardsphere-like colloid.5 The colloidal crystal tends to be more compact for the same osmotic pressure. The suspension of colloidal PNIPAM at lower temperatures has the larger volume fraction, φ (i.e., φ > φm), due to the expansion of the microgel particles. Table 1 summarizes the exponents of the power-law relations for the PNIPAM system at 19 and 25 °C, including the results for colloidal hard spheres at a volume fraction of 0.552 in microgravity15 and 0.548 in gravity.5 Instrumental errors for the hydrodynamic radius measured by dynamic light scattering, turbidity, and wavelength measured by a UV-vis spectrophotometer are about (4 nm, (0.02 cm-1, and (0.5 nm, respectively. Considering error propagation in eqs 5-8, the error bars were estimated as shown in Figure 3. The errors in exponents of µ, δ, and ν are estimated to be 0.01, 0.07, and 0.26, respectively. Later, we also estimated and included error bars in our data in Figure 4. At 25 °C, however, the exponents are completely different. This is mainly due to the shrinkage of the PNIPAM particles as the temperature increases from 19 to 25 °C, resulting in a different volume fraction. In this case, there are no hard-sphere data available for comparison. For typical hard-sphere-like colloids, the gravitational length, defined as h ) kBT/(mg) where m is the buoyant mass of a particle and g is the acceleration due to gravity, is ∼30 µm on Earth.15,38 In contrast, the acceleration due to gravity and the gravitational length for the PNIPAM spheres in water are about 3 × 10-2 g and 600 µm, respectively. We observed that, as shown in Figure 1, sedimentation due to gravity is not significant in the aqueous dispersion of swollen PNIPAM microgel spheres. Although it is difficult to compare absolute power-law exponents in different colloidal systems under different conditions, the similarity as demonstrated in Table 1 between the PNIPAM dispersion at 19 °C and the hardsphere-like colloid in microgravity is striking.

(8)

where qm(t) is the wave vector corresponding to the peak position at time t.

(37) Wu, C.; Zhou, S. Q.; Auyeung, S. C. F.; Jiang, S. H. Angew. Makromol. Chem. 1996, 240, 123. (38) Biben, T.; Ohnesorge, R.; Lowen, H. Europhys. Lett. 1994, 28, 665.

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Figure 3. Crystallization kinetics for the PNIPAM microgel dispersion at various temperatures: (a) degree of crystallization X(t); (b) average linear crystal dimension L(t); (c) number density of crystals Nc(t); (d) volume fraction of the crystal phase φc(t). Time t is measured in seconds. Table 1. Summaries of the Exponents of the Power-Law Relations for the PNIPAM System at 19 and 25 °C, for Hard-Sphere-Like Colloids at a Volume Fraction of 0.552 in Microgravity (ref 15) and 0.548 in Gravity (ref 5)

hard spheres in gravity, ref 5 (φ ∼ 0.548) hard spheres in microgravity, ref 15 (φ ∼ 0.552) PNIPAM microgel spheres at 19 °C (φ ∼ 0.55) PNIPAM microgel spheres at 25 °C (φ ∼ 0.51)

PNIPAM microgel particles exhibit a volume transition at a temperature (≈34 °C) close to the low critical solution temperature (LCST) of the corresponding polymer solution. Below this temperature, the microgel particles are in the swollen state and contain up to 97 wt % water. In this case, both the gravity and London dispersion forces play a minor role because of the close matches of density and dielectric constant between the particles and aqueous

X(t) ∼ tµ µ

Nc(t) ∼ tν ν

L(t) ∼ tδ δ

4 2.2 2.1 0.2

3 1.0 0.9 0.05

0.5 0.47 0.41 0.12

solution. Specially, microgel particles are essentially hard spheres at 19 °C, which is far away from 34 °C. As shown in ref 27, we found that the van der Waals attraction between microgel particles can be safely neglected below the LCST by analyzing the osmotic second virial coefficients measured from static light scattering and that predicted from the thermodynamic models. However, near or above the LCST, the PNIPAM particles shrink and

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I ) ζ exp(-∆Gcrit/kBT)

(9)

where the critical Gibbs energy of nucleation is related to the difference in the chemical potentials of the fluid and the crystal ∆µ, the solution-solid interfacial tension γ, and the number density of the crystalline phase Fs

∆Gcrit )

16π γ3 3 [F ∆µ]2

(10)

s

Figure 4. The nucleation rates per unit volume (in units of σ-5D0 where σ is the particle diameter and D0 is the self-diffusion coefficient at infinite dilution) as a function of the particle volume fraction in PNIPAM microgel dispersion and in various hard-sphere systems. The solid line is calculated from CNT with the prefactor A ) 10-5.96 and the fluid-solid interfacial tension γσ2/kt ) 0.45 as the fitting parameters. The dashed line is the prediction of the CNT for monodispersed hard spheres using the parameters obtained from Monte Carlo simulation.20

strong van der Waals attraction arises due to the mismatch of the dielectric permittivity of the polymer and the surrounding aqueous solution. It is noted that surfactant may play an important role in crystallization. But in this study, the microgel dispersions were thoroughly dialyzed with deionized, distilled water and purified using an ultracentrifuge. The amount of residue surfactant in the dispersion may be negligible. In Figure 4, we compare the rate of nucleation per unit volume in the microgel dispersion investigated in this work with previous studies for colloidal spheres.5,20,39-41 Here we estimated that φ ) 0.55 with uncertainty about 0.01 for the PNIPAM system at 19 °C by two methods: (1) analysis of the fraction of crystalline phase X(t) as shown in Figure 3a, and (2) analysis of chemical potential and dynamic light scattering measurements. The volume fractions for other temperatures are calculated from the hydrodynamic radius. The detailed discussion will be published elsewhere. Also shown in Figure 4 are calculated results from the classical nucleation theory (CNT) using a semiempirical expression for the prefactor proposed by Harland et al.5 and an alternative expression by Auer and Frenkel.20 Other expressions for the experimentally determined kinetic prefactor have been proposed by Lekkerkerker et al.42 and by Palberg.43 According to CNT, the crystalnucleation rate per unit volume, I, depends exponentially on the Gibbs energy barrier in the formation of a critical nucleus, ∆Gcrit (39) Sinn, C.; Heymann, A.; Stipp, A.; Palberg, T. Prog. Colloid Polym. Sci. 2001, 118, 266. (40) Schatzel, K.; Ackerson, B. J. Phys. Rev. E 1993, 48, 3766. (41) Cheng, Z. Ph.D. Dissertation, Princeton University, 1998. (42) Lekkerkerker, H. N. W.; Dhont, J. K. G.; Verduin, H.; Smits, C.; van Duijneveldt, J. S. Physica A 1995, 213, 1829. (43) Palberg, T. J. Phys. Condens. Matter 1999, 11, R323-360.

Using the kinetic prefactor ζ ) Aφ5/3(1 - φ/0.58)2.6 proposed by Harland et al.,5 we find that CNT agrees reasonably well with experimental data with log(A) ) -5.96 and γσ2/kBT ) 0.45 as the fitting parameters. In these calculations, the chemical potentials of the fluid and solid phases are calculated from the CarnahanStarling equation of state44 and a modified cell model,27 respectively. It is noted that the value of γσ2/kBT is close to 0.5 for hard spheres.5 However, the value of log A is substantially smaller than that for the hard sphere system.5 This indicates that the rate of the colloidal nucleation in the PNIPAM system is smaller than that in the hard-sphere system around φm. The nucleation rate for the PNIPAM system is also slightly φ-dependent. It has been pointed out by Auer and Frenkel that in order to crystallize, colloidal fluids must be compressed beyond the freezing curve.20 Because the PNIPAM particles contain a large amount of water in their structures, they are soft in terms of low shear modulus, in contrast to conventional softness that was defined by the surface charges. It seems that the softness of the PNIAPM particles results in the slower kinetics. For a hard sphere system with φ g φm, nucleation occurs by an accelerated “burst” of nuclei,5 which seems not significant for the PNIPAM system. The prediction of CNT significantly deviates from the experimental data if the prefactor is calculated from20

ζ)

31/2 Flφs∆µ fnc 4π2 γ3/2 D0

(11)

with log(fnc/D0) ) 8827.2φl2 - 9346.3φl + 2475.6 and γ ) 3.6228φl - 1.1829 obtained from simulations for the nucleation of hard spheres.20 It is noted that with large sample dimensions of 10 mm × 10 mm × 44.5 mm, the surface effect may be not significant in this experiment. The detailed crystal form may be revealed using the neutron scattering method.25 In summary, we have determined the crystallization kinetics in an aqueous dispersion of PNIPAM microgel particles using a UV-vis transmission spectroscopy. Because the particle size decreases with the increase of temperature, different volume fractions have been obtained by varying the temperature of a single colloidal dispersion. Sedimentation due to gravity is insignificant for the swollen PNIPAM microgel spheres in water. The degree of crystallization X(t), average crystalline dimension L(t), number density of crystals Nc(t), and volume fraction φc(t) over a range of temperatures (19-25 °C) have been analyzed via the turbidity peak which arises from the Bragg scattering of the evolving crystals. It is found that the rates in X(t), L(t), and Nc(t) at the most rapid stage over a certain time interval can be described using the power law relations. Near 19 °C, the PNIPAM system behaves as a hard sphere system under micro(44) Carnahan, N. F.; Starling, K. E. J. Chem. Phys. 1969, 51, 635.

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gravity. Hard sphere theories have been used as a reference point to compare with conventional hard spheres, soft spheres, and PNIPAM spheres. The striking features that the nucleation rate is slightly φ-dependent and much smaller than the hard sphere system around φm may indicate that the softness (i.e., low shear modulus) of PNIPAM microgels plays an important role. This softness is very different from the conventional softness that is usually defined by the colloidal surface charges.

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Acknowledgment. We gratefully acknowledge the support from the National Science Foundation (DMR0102468) and the University of California Directed Research and Development. We thank Professor Daan Frenkel for generously sharing unpublished results and Mr. Bo Zhou for his technical assistance. LA049203H