INTRODUCTION
In many aspects, colloidal particles suspended in a liquid behave like large idealized atoms that exhibit liquid, glass, and crystal phases similar to those observed in atomic systems.[1-6] Phase transitions in colloidal systems have been intensively studied over the last decade not only because of the theoretical interest for addressing fundamental questions about the nature of liquids, crystals, and glasses, but also for many practical applications of col-loids.[7-11] In recent years, colloidal dispersions have been used extensively for the fabrication of nanostruc-tured materials such as photonic crystals, catalysts, membranes, and ceramics, and for device applications.1-11-16-1 Current uses of colloidal particles as the building blocks of materials rely mostly on empirical approaches. Meanwhile, present knowledge on the structural and thermo-dynamic properties of colloidal dispersions is primarily based on an effective one-component model (OCM) where colloidal particles are represented by hard spheres and all remaining components in the dispersion, including solvent molecules, small ions, and polymers, are represented as a continuous medium.[17-23] Although the OCM approach is attractive because of its simplicity, application to practical systems is often limited by incomplete understanding of colloidal forces.[24-28]
Amid numerous conventional colloids, aqueous dispersions of poly-N-isopropylacrylamide (PNIPAM) microgel particles, first synthesized by Pelton and Chibante[29] in 1986, are of special interest for studying phase transitions and for the fabrication of colloid-based advanced materials.[30-36] Nearly monodispersed PNIPAM particles can now be routinely prepared in a wide range of colloidal sizes (50 nm up to 1 mm) and with a variety of physiochemical characterizations.[35,37] Because the size of PNIPAM particles is temperature-sensitive,[38] crystallization at different colloidal volume fractions can be conveniently measured by varying temperatures. By tuning the preparation conditions and the composition of the aqueous solution, the interaction potential between microgel particles can vary from star-polymer-like to hard-sphere-like potential for short-range repulsion, from electrostatically neutral to highly ionizable for long-range electrostatic interactions, and from essentially no attraction to strong attraction for van der Waals forces.[36,39] Furthermore, steric repulsion can be introduced by grafting polymers on the surface of PNIPAM particles.[40] The versatility in interaction potential makes PNIPAM microgel particles attractive for studying a broad variety of interesting phenomena in colloidal systems.[32,41-44] Although the practical values of PNIPAM particles have been long recognized, most previous studies on the phy-siochemical properties of PNIPAM dispersions have focused on particle preparations, swelling, rheology, and light (neutron) scattering measurements.1-35,36,45-47-1 Little work has been reported on the relationship between the temperature-dependent interparticle potential and the phase behavior of PNIPAM dispersions. Unlike that in a conventional colloidal system, the interparticle potential in aqueous dispersions of PNIPAM microgel particles is sensitive to temperature changes. Consequently, the phase diagram of PNIPAM dispersions may be noticeably different from those for ordinary colloids where, in most cases, the interparticle potential is essentially invariant with temperature.
In this topic, we report our recent investigations on the colloidal forces and phase behavior of neutral PNIPAM particles dispersed in pure water.[48-50] We will first discuss thermodynamic methods for the characterization of colloidal forces based on dynamic and static light scattering measurements. The analytical expression of colloidal forces allows us to construct a theoretical phase diagram that can be compared with that obtained from spectroscopic measurements. In particular, we will illustrate how the volume transition of PNIPAM particles affects the interaction potential and determines a novel phase diagram that has not been observed in conventional colloids. Finally, we discuss briefly the kinetics of crystallization in an aqueous dispersion of PNIPAM particles.
COLLOIDAL FORCES BETWEEN MICROGEL PARTICLES
Qualitatively, the pair potential between neutral microgel particles includes a short-range repulsion that is similar to the interaction between two polymer-coated surfaces, and a longer-ranged van der Waals-like attraction that arises from the difference in the Hamaker constants of the particle and the solvent.[51,52] To obtain a quantitative expression for the interparticle potential that covers a broad range of temperature, we assume that the interaction potential can be represented by a Sutherland-like function that includes a hard-sphere repulsion and a van der Waals attraction. A similar potential was used Senff and Richtering[43,53] for representing the phase behavior and rheological properties of PNIPAM microgel dispersions. Fig. 1 shows that the relative zero shear viscosity of PNIPAM dispersion can be approximately represented by that for hard spheres. Besides, as shown later, the kinetics of microgel crystallization also resembles that for hard spheres.
The variation of microgel diameter as a function of temperature is related to the swelling of microgel particles. The classical theory of gel swelling, proposed many years ago by Flory,[54] asserts uniform distributions of polymer segments and crosslinking points throughout the polymer network. However, recent experiments based on nuclear magnetic resonance (NMR) and light scattering investigations suggest the heterogeneous nature of PNI-PAM particles.[51,55] To take into account the heterogeneity, we use an empirical modification of the Flory-Rehner theory proposed by Hino and Prausnitz.[56] This theory has been applied successfully to describing the volume transitions of bulk PNIPAM gels. Because the physics for the volume transition is irrelevant to the particle size as long as the surface effect is unimportant, and because the effect of the osmotic pressure of dispersion on the swelling of an individual particle is neg-ligible,[42] we assume that the same thermodynamic model for bulk polymer gels is also applicable to micro-gel particles.
Fig. 1 Relative zero shear viscosity at different temperatures vs. the effective volume fraction. The line represents the master curve of hard-sphere suspensions.
According to the modified Flory-Rehner theory,[56] the polymer volume fraction within the particle f is determined from:
where m is the average number of segments between two neighboring crosslinking points in the microgel network, and f 0 is the polymer volume fraction in the reference state where the conformation of network chains is closest to that of unperturbed Gaussian chains. Approximately, f0 is equal to the volume fraction of polymers within the microgel particles at the condition of preparation. The Flory polymer-solvent energy parameter w is given empirically as a function of temperature T and polymer volume fraction f :[56]
At a given temperature, Eq. 1 can be solved to find the polymer volume fraction f from which the diameter of PNIPAM particles (s) is calculated:
where s0 is the particle diameter at the reference state. In this work, we fit the average chain length m and the volume fraction of polymer f 0 at the reference state to the diameters of microgel particles obtained from static and dynamic light scattering measurements. Because the osmotic pressure of microgel dispersion is small in comparison to the gel swelling pressure, the microgel particle concentration has a negligible effect on the chemical potential of water. As a result, the size of microgel particles should be insensitive to the overall concentration of colloidal dispersions.
The van der Waals attraction beyond the hard-sphere diameter may be represented by a power law potential:
where H is the Hamaker constant. We assume that n = 8 in considering that the range of attraction between colloidal particles (relative to the particle size) is shorter than that between atomic molecules. The calculated results are not sensitive to a small change in n if the Hamaker constant is obtained by fitting to osmotic second virial coefficients from static light scattering experiments. Because the interparticle attraction arises from the dispersion forces between polymeric segments from different particles, the Hamaker constant of microgel particles is approximately given by:[57]
where pm represents the number density of polymeric groups within each particle. The proportionality constant in Eq. 5 is independent of temperature and polymeric group density pm. Following Eqs. 4 and 5, we obtain the attractive potential because of the van der Waals forces:
where kA is a dimensionless constant, and T0 is the reference temperature that is introduced for the purpose of dimensionality. In Eq. 6, the parameters T0, s0, and kA are temperature-independent and they can be obtained by fitting to osmotic second virial coefficients from static light scattering measurements. An osmotic second virial coefficient in a colloidal dispersion is defined as:
where r stands for the center-to-center distance between colloidal particles, k is the Boltzmann constant, and u(r) is the interparticle potential.
The particle size and the osmotic second virial coefficients of microgel dispersions can be conveniently measured using static and dynamic light scattering. To correlate the radius of PNIPAM gel particles as a function of temperature, we combine the radius of gyration Rg with the hydrodynamic radius RH from static and dynamic light scattering measurements, respectively. By assuming that
Fig. 2 Radius of PNIPAM particles vs. temperature. The points are averages from dynamic and static light scattering measurements with the error bars showing the differences between the two. The line is calculated from Eqs. 1-3.
Fig. 2 Radius of PNIPAM particles vs. temperature. The points are averages from dynamic and static light scattering measurements with the error bars showing the differences between the two. The line is calculated from Eqs. 1-3. the microgel particles are homogeneous spheres, we define the effective radius:
This effective radius is introduced to represent the excluded volume of microgel particles as required in phase equilibrium calculations. Fig. 2 presents the radius of microgel particles near the volume transition temperature (approximately at 34.3°C). The error bars give the difference between the hydrodynamic radius and that calculated from the radius of gyration. The solid line is calculated by using the modified Flory-Rehner theory (Eqs. 1-3). In the calculation of the particle radius, the polymer fraction at the condition of preparation f 0 = 0.0884, the average number of segments between two neighboring crosslinking points m = 34, and the particle radius at the preparation condition R0= 125.8 nm are obtained by fitting to the experimental data. These model parameters are in good agreement with experiments.
Fig. 3 presents the reduced osmotic second virial coefficients (B2/#HS) for PNIPAM microgel particles dispersed in water. Here the open circles are data points from static light scattering measurements and the line is calculated using Eqs. 6 and 7, with the molecular weight MW= 1.73 x 107 g/mol and the proportionality constant kA = 6.43 x 10"5 obtained by fitting to the experimental data. The hard-sphere second virial coefficient is related to the particle diameter s by BHS = (2p/3)s3. A positive osmotic second virial coefficient means that the overall interparticle potential is repulsive; otherwise, it is attractive. Fig. 3 indicates that below the volume transition temperature, the PNIPAM particles behave essentially like hard spheres. In this case, the microgel particles are in the swollen state and they contain up to 97% of water by volume. As a result, the van der Waals attraction between colloidal particles is negligible because of the close match in the Hamaker constants of the particle and the water. The reduced osmotic second virial coefficient exhibits a sharp change at the volume transition temperature, beyond which it turns to negative, indicating an increase in the van der Waals attraction as the particles collapse. Fig. 3 suggests that with temperature-dependent size and energy parameters, the Sutherland-type function captures the essential features of the interaction potential between microgel particles.
Fig. 3 The reduced osmotic second virial coefficients (B2/BHS) for PNIPAM particles dispersed in pure water. Points are from static light scattering and the line is calculated from Eq. 7 normalized by the second virial coefficient BHS = (2p/3)s3 for the corresponding hard spheres.
Because the number density of polymeric segments within each particle is closely related to the particle size, we expect that the attraction between microgel particles is highly temperature-sensitive. Fig. 4 shows the reduced energy parameter e/(kT) near the volume transition temperature as obtained by correlation with the osmotic second virial coefficients from experiments. Experimental values for this parameter are not included because it is difficult to make a direct (or indirect) measurement of the reduced energy parameter. But the credibility of calculated results is implied in the comparison of osmotic second virial coefficients, as shown in Fig. 3. Remarkably, the energy parameter increases by over six orders of magnitude as temperature changes from 24°C to 36°C, with the sharpest increase at the volume transition temperature of 34°C.
