5004
Langmuir 1998, 14, 5004-5010
Self-Diffusion in Concentrated Colloid Suspensions Studied by Digital Video Microscopy of Core-Shell Tracer Particles Andreas Kasper, Eckhard Bartsch, and Hans Sillescu* Institut fu¨ r Physikalische Chemie, Universita¨ t Mainz, Jakob-Welder-Weg 15, D-55099 Mainz, Germany Received October 6, 1997. In Final Form: May 26, 1998 Optical video microscopy and digital image processing have been used to study the self-diffusion of colloidal particles with a hard-sphere potential. The colloid particles consist of cross-linked polymers and are dispersed in a good solvent to avoid aggregation. To investigate single particle motion in highly concentrated dispersions, a host-tracer system, consisting of two different kinds of polymer particles, has been designed: the host particles are made of poly-t-butylacrylate (with ethanedioldiacrylate as crosslinker) and have the same refractive index as the employed solvent, 4-fluorotoluene. The tracer particles have a core-shell structure with a polystyrene core (cross-linked with m-diisopropenylbenzene) and a shell consisting of cross-linked poly-t-butylacrylate to match surface properties and interaction potential to those of the “invisible” particles. The motion of the strongly scattering core-shell particles (“tracer” particles) was observed by dark-field light microscopy. From the obtained particle trajectories, mean squared displacements, van Hove autocorrelation functions, and vector-vector correlation functions were calculated, yielding a direct real-space image of the “cage effect” at φ ) 0.52 and of the transition to a glassy state between φ ) 0.56 and φ ) 0.60, as expected for a hard sphere system. The extracted long-time self-diffusion coefficients Dself,long are fully consistent with a recent theoretical prediction using full manybody hydrodynamics at φ e 0.56 and a colloid glass transition at φg ) 0.583. However, even at φ ) 0.60, Dself,long seems to be still finite, possibly indicating the existence of long-time motion of colloidal particles even in the glassy state.
1. Introduction Brownian motion in concentrated colloid suspensions has mainly been studied by light scattering techniques.1 Using tracer particles in suspensions of index-matched host particles,2,3 colloids labeled with a photoreactive dye in applications of forced Rayleigh scattering,4,5 or fluorescence bleaching,6 tagged particle motion has been monitored in concentrated dispersions as well. With the recent development of video detection and digital image processing techniques, optical microscopy has attracted new interest as a means of studying structure and dynamics in colloidal dispersions. In experiments by Murray and co-workers,7-9 such techniques were applied to study the structure and dynamics of quasi-twodimensional and three-dimensional aqueous suspensions of charge-stabilized colloids. Other studies of similar systems concentrated on the determination of the particle interaction potentials from the particle trajectories of isolated particle pairs10,11 and from the pair distribution (1) Pusey, P. N. Colloidal Suspensions, In Liquids, Freezing and the Glass Transition, Les Houches Session L1; Levesque, D.; Hansen, J.-P.; Zinn-Justin, J., Eds.; Elsevier: Amsterdam, 1990. (2) van Megen, W.; Underwood, S. M. J. Chem. Phys. 1989, 91, 552. (3) van Megen, W.; Underwood, S. M.; Snook, I. J. Chem. Phys. 1986, 85, 4065. (4) Bartsch, E.; Frenz, V.; Mo¨ller, S.; Sillescu, H. Physica A 1993, 201, 363. (5) Renth, F.; Bartsch, E.; Kasper, A.; Kirsch, S.; Sto¨lken, S.; Sillescu, H.; Ko¨hler, W.; Scha¨fer, R. Prog. Colloid Polym. Sci. 1996, 100, 127. (6) Imhof, A.; van Blaaderen, A.; Maret, G.; Mellema, J.; Dhont, J. K. G. J. Chem. Phys. 1994, 100, 2170. (7) Murray, C. A.; Sprenger, W. O.; Wenk, R. A. Phys. Rev. B 1990, 42, 688. (8) Murray, C. A.; Van Winkle, D. H. Phys. Rev. Lett. 1987, 58, 1200. (9) Murray, C. A.; Grier, D. G. Annu. Rev. Phys. Chem. 1996, 47, 421. (10) Crocker, J. C.; Grier, D. G. J. Phys. Rev. Lett. 1994, 73, 352. (11) Crocker, J. C.; Grier, D. G. J. J. Colloid Interface Sci. 1996, 179, 298.
function g(r) for both quasi-two-dimensional12 and threedimensional samples.13 Kepler and Fraden14 used video microscopy to study the interactions between the colloid particles and the glass walls of a quasi-two-dimensional system. The dynamics of charge-stabilized colloid systems has already been investigated by several groups, including Bongers and Versmold who studied particle motion in a crystalline layer next to the sample wall,15 Verhaegh et al. who used confocal scanning laser microscopy to look at the properties of crystallizing and crystalline fluorescent silica spheres,16,17 and Ise et al. who observed ordering phenomena18 and the time evolution of void structures.19 Only very few video microscopy studies dealing with hardsphere colloid suspensions have been reported so far. Marcus et al. studied self-diffusion and phase transitions in confined quasi-two-dimensional colloid suspensions of sterically stabilized colloid spheres.20-22 The real-space structure of colloidal hard-sphere glasses has been investigated by van Blaaderen and Wiltzius who used confocal fluorescence microscopy.23 However, the dynamics of concentrated colloidal hard-sphere suspensions (up to the glass transition region) has to our knowledge not (12) Carbajal-Tinoco, M. D.; Castro-Roma´n, F.; Arauz-Lara, J. L. Phys. Rev. E 1996, 53, 3745. (13) Vondermassen, K.; Bongers, J.; Mueller, A.; Versmold, H. Langmuir 1994, 10, 1351. (14) Kepler, G. M.; Fraden, S. Langmuir 1994, 10, 2501. (15) Bongers, J.; Versmold, H. J. Chem. Phys. 1996, 104, 1519. (16) Verhaegh, N. A. M.; van Blaaderen, A. Langmuir 1994, 10, 1427. (17) Verhaegh, N. A. M.; van Duijneveldt, J. S.; van Blaaderen, A.; Lekkerkerker, H. N. W. J. Chem. Phys. 1995, 102, 1416. (18) Ise, N.; Matsuoka, H.; Ito, K.; Yoshida, H. Faraday Discuss. Chem. Soc. 1990, 90, 153. (19) Ito, K.; Yoshida, H.; Ise, N. Science 1994, 263, 66. (20) Marcus, A. H.; Lin, B.; Rice, S. A. Phys. Rev. E 1996, 53, 1765. (21) Marcus, A. H.; Rice, S. A. Phys. Rev. Lett. 1996, 77, 2577. (22) Marcus, A. H.; Rice, S. A. Phys. Rev. E 1997, 55, 637. (23) van Blaaderen, A.; Wiltzius, P. Science 1995, 270, 1177.
S0743-7463(97)01089-5 CCC: $15.00 © 1998 American Chemical Society Published on Web 08/07/1998
Colloid Self-Diffusion Studied by Video Microscopy
yet been studied by video microscopy. Here, a major interest results from the possibility of directly visualizing the processes that have been discussed in the context of glass dynamics of simple liquids; for examples, the cage effect, (possibly) cooperative hopping phenomena, or motions of dynamically correlated clusters (for recent reviews on the glass transition see refs 24-26). To date, these and alternative processes were inferred from the interpretation of scattering and relaxation experiments24,27-29 or have been discussed in the context of computer simulations.30 Optical microscopy on highly concentrated dispersions offers the possibility to check up on different suggested mechanisms of motion by directly following the particle trajectories in real space using colloidal particles as mesocopic model atoms. The colloid system we used for our studies consists of internally cross-linked polymer spheres in a good solvent, termed microgel or micronetwork colloids. The predominance of polymer-solvent interactions over polymerpolymer interactions prevents the aggregation of the colloidal particles. The polymer micronetworks take up solvent and increase in size. However, due to the crosslinks, the swelling process is limited and the particles keep their spherical shape. With this particular colloid system, hard-sphere behavior can be achieved by a high degree of internal cross-linking. In two previous reports, we demonstrated that a cross-link density of 1:10 (10 monomer units between cross-links) is sufficient to guarantee hard-sphere-like behavior, taking the static structure factor31 and the volume fraction dependence of the long-time self-diffusion coefficient Dself,long5 as criteria. Here we report on an optical microscopy study of a hosttracer system (Figure 1) that allows one to follow the trajectories of single colloidal particles up to high concentrations and thus to monitor single particle motion in the regime of the colloid glass transition (φg ≈ 0.58). The host particles consist of a polymer that has almost exactly the same refractive index as the solvent. Because of this very small contrast, these particles hardly scatter any light when they are illuminated. The second sort of particles, the tracers, have a core-shell structure. The core is made of a material with a large difference in refractive index in comparison with the solvent, whereas the shell is made of the same material as the host particles to match interaction potential and surface properties to those of the host particles. If the tracers have the same size as the host particles, it can be assumed that the few visible particles show exactly the same behavior in their dynamical properties as all the invisible particles, which is the basic requirement for identifying tracer diffusion with self-diffusion. Because the main interest of this work is on studying glass transition phenomena using colloids as model atoms, some size polydispersity (∼8%) had to be accepted to sufficiently suppress crystallization of the samples. (24) Go¨tze, W.; Sjo¨gren, L. Rep. Prog. Phys. 1992, 55, 241. (25) Cummins, H. Z.; Li, G.; Du, W. M.; Hernandez, J. Physica A 1994, 204, 169. (26) Disorder effects on relaxational processes. Glasses, Polymers, proteins; Richert, R.; Blumen, A., Eds; Springer-Verlag: Berlin, 1994. (27) Chamberlin, R. V.; Bo¨hmer, R.; Sanchez, E.; Angell, C. A. Phys. Rev. B 1992, 46, 787. (28) Cummins, H. Z.; Du, W. M.; Fuchs, M.; Go¨tze, W.; Hildebrand, S.; Latz, A.; Li, G.; Tao, N. J. Phys Rev. E 1993, 47, 4223. (29) Proceedings of the 2nd International Discussion Meeting on Relaxations in Complex Systems, Alicante, Ngai, K., Ed.; J. Non-Cryst. Solids 1994, 172-174. (30) Kob, W. In Annual Reviews of Computational Physics; Staufer, D., Ed.; World Scientific: Singapore, 1995; Vol 3. (31) Kasper, A.; Kirsch, S.; Renth, F.; Bartsch, E.; Sillescu, H. Prog. Colloid Polym. Sci. 1996, 100, 151.
Langmuir, Vol. 14, No. 18, 1998 5005
Figure 1. Sketch of a host-tracer system. The host particles, shown as open circles, have the same refractive index as the surrounding medium. The tracer particles have a shell of the same “invisible” material, but a core, drawn in black, made of a material with a large contrast in refractive index that allows for detection by dark-field microscopy.
The concentration of the host particles is varied from the highly dilute regime up to a volume fraction of φ ) 0.60, where the system is well in the glass transition region. The self-diffusion of the strongly scattering tracer particles is then studied by optical dark-field microscopy and digital image processing. From the observed particle trajectories, mean-squared displacements, the van Hove autocorrelation function, the non-Gaussian parameter R2,32 a vector correlation function (to quantify the cage effect) and the long-time self-diffusion coefficients are derived. 2. Experimental Details Sample Preparation and Characterization. The hostparticles were prepared by adapting one-step surfactant-free emulsion polymerization techniques from un-cross-linked lattices33,34 to the synthesis of acrylate microgel particles, with potassium peroxodisulfate as initiator and a temperature of 70 °C for a total reaction time of 48 h (see ref 35 for more details on the preparation of this type of particles). The ratio of the comonomers, tert-butylacrylate and ethanedioldiacrylate, was calculated to yield an average cross-link density of 1:10 (i.e., 10 monomer units between cross-links). For the tracer particles, a two-step polymerization process was used.35 In the first step, styrene and m-diisopropenylbenzene (cross-linker) were copolymerized in a way similar to the preparation of the host particles, using surfactant-free emulsion polymerization with potassium peroxide as initiator and a reaction temperature of 70 °C. In the second step, a mixture of tert-butylacrylate and ethanediolacrylate was added and polymerized under the same conditions. The presence of the crosslinked polystyrene seed particles prevented secondary nucleation and the acrylate copolymer formed a shell of ∼30 nm thickness around the polystyrene seeds. The polydispersities of both types of particles, as determined in water by static and dynamic light scattering,36 were ∼8%. Polydispersities between 6 and 10% are typical for the employed preparation technique. Because of the lack of stabilizing surfactant, the polymerization process reacts very sensitively to small fluctuations in the reaction conditions. So far, it is not (32) van Megen, W.; Underwood, S. J. Chem. Phys. 1988, 88, 7841. (33) Goodwin, J. W.; Hearn, J.; Ho, C. C.; Ottewill, R. H. Colloid Polym. Sci. 1974, 252, 464. (34) Goodwin, J. W.; Ottewill, R. H.; Pelton, J.; Vianello, G.; Yates, D. E. Br. Polym. J. 1978, 10, 173. (35) Kirsch, S.; Do¨rk, A.; Bartsch, E.; Sillescu, H.; Landfester, K.; Spiess, H. W.; Ma¨chtle, W. Submitted. (36) Bartsch, E.; Frenz, V.; Baschnagel, J.; Scha¨rtl, W.; Sillescu, H. J. Chem. Phys. 1997, 106, 3743.
5006 Langmuir, Vol. 14, No. 18, 1998
Figure 2. AFM picture of the core-shell tracer particles. The image shows an area of about 5 × 5 µm in size. The shell material is stable under AFM conditions. possible to achieve full control on size polydispersity. Because it is known that crystallization is slowed considerably (or even quenched) with size polydispersities >7%,1,37 we chose a batch with a size polydispersity of ∼8% and checked that crystallization does not occur on time scales relevant for the optical microscopy experiments.38 After the polymerization steps, both types of polymer particles were treated similarly. First, they were precipitated by adding methanol. Filtration and drying yielded a white polymer powder that was dissolved in tetrahydrofurane. The solution was then added in a dropwise manner to methanol to precipitate the polymer again. The solution-precipitation process was repeated again, and the resulting polymer was then dried in a vacuum oven at 40 °C. Finally, the dry powder was dissolved in 1,4dioxane and then freeze-dried. To determine the particle sizes, particle trajectories for dilute aqueous dispersions were recorded by video microscopy; from the mean-squared displacements, diffusion coefficients were determined (see “Data Evaluation” for details). Using the Stokes-Einstein equation, the hydrodynamic radii RH were calculated, yielding 250 nm for the tracer particles and 280 nm for the host micronetworks in aqueous dispersion. Within the experimental error, these results were in full agreement with data obtained by dynamic light scattering. Figure 2 shows a picture of the core-shell tracer particles that was obtained by atomic force microscopy (AFM), indicating the spherical shape and the narrow size distribution of the particles [see ref 31 for a transmission electron microscopy (TEM) picture of the tracer particles]. 4-Fluorotoluene (4-FT) was used as isorefractive solvent with good solvent properties for the polymer micronetwork particles. The swelling ratio S ) RH3 (in solvent)/RH3 (in water) relates the final hydrodynamic radius of the dissolved, swollen particles to their initial size (unswollen) in aqueous suspension. For the tracer particles, their swollen size was determined by video microscopy of a dilute 4-FT solution in the same manner as already described for the aqueous dispersions. The measured hydrodynamic radius RH was 455 nm, indicating a swelling ratio S of 6.0. Within the experimental error, this result was consistent with dynamic light scattering measurements. The host particles are invisible when they are dissolved in 4-fluorotoluene, due to (37) Morigouchi, I.; Kawasaki, K.; Kawakatsu, T. J. Phys. II (France) 1993, 3, 1179. (38) As a matter of fact, for the samples studied, no traces of crystallization were ever detected.
Kasper et al. the refractive index match, so dynamic light scattering does not yield reliable results for the hydrodynamic radius. Use of a different solvent (with a higher contrast) would lead to a different swelling ratio of the polymer micronetwork. Furthermore, determining the hydrodynamic radius with an experimental accuracy of, for example, 5% would still result in an error of 15% for the volume fraction. The best procedure to determine the volume fractions is to equate the measured volume fraction of freezing, φf, to the known hard-sphere value. Because we had to accept a polydispersity of 8% to monitor the slow glass transition dynamics, this procedure could not be applied.38 Therefore, the volume fractions and thus RH (4-FT) of the host particles were determined indirectly. Host-tracer samples of different concentrations were prepared (vide infra) and the particle trajectories of the tracers were monitored. From the mean-squared displacements, diffusion coefficients were calculated. The concentration-dependent diffusion coefficients (except for the data point at the highest concentration, φ ) 0.60) were then fitted to theoretical predictions for hard sphere colloids by Tokuyama and Oppenheim39 with the swelling ratio S, connected with the volume fraction φ of colloid particles, being the only fit parameter (see Figure 9 and the discussion there). The data yielded a result of S ) 4.0 for the host particles, which means that their hydrodynamic radius in solution was 445 nm. The experimental error is estimated to be of ∼5% for the volume fractions and, as a consequence, 2% for the hydrodynamic radius. Thus, within experimental error, tracer and host particles are of the same size. To prepare the samples for video microscopic investigation, different amounts of the freeze-dried host particles (F ) 0.97 g/cm3, the value for the un-cross-linked polymer, assuming that the density is uneffected by the cross-linker), together with a few micrograms of freeze-dried tracer particles (F ) 1.02 g/cm3), were dissolved in 4-fluorotoluene (F ) 1.00 g/cm3). Note that differences in density decrease when the particles take up solvent and swell. A volume swelling ratio of S ) 4.0 implies that 75% of the volume of the microgel particles is filled with solvent. Thus, the density of the swollen particle is practically identical to that of the solvent and sedimentation effects are negligible even for large particle sizes. The weight-fraction of tracer particles was chosen to be
