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Interaction Measurements

Most physical properties of dense colloidal suspensions are either determined or modified by the form of the interaction potential between the individual colloidal spheres. While present models for the pair potential have been used to achieve qualitative agreement between theoretical and observed phase diagrams and rheological properties, detailed measurements are needed to secure quantitative agreement and resolve questions of interpretation surrounding a variety of unexplained experimental phenomena. For example, the dependence of the pairwise interaction on temperature, volume fraction and boundary conditions are still being investigated. In practical applications, pairwise interaction measurements provide detailed information regarding the state of charge and local chemical environment of particles in suspension.

The effective pairwise interaction potential tex2html_wrap_inline1127 is encoded in the equilibrium distribution of particles in a suspension through the relation




is the two-particle correlation function and angle brackets indicate an average over angles. Static snapshots in principal can yield measurements of tex2html_wrap_inline1127 provided that the suspension's concentration is low enough to avoid many-body contributions[7]. This restriction was circumvented by Kepler and Fraden [6] who augmented their imaging measurements of g(r) with molecular dynamics simulations to correct for many-body effects. Direct imaging avoids the problems encountered in efforts to invert light scattering data to extract pair potentials [23], since detector noise contributes little to the error in estimating g(r). Geometrically confining colloid to a single layer to avoid complications from the poor depth resolution of video microscopy, however, introduces wall-mediated interactions [6].

As with the measurements of diffusivity described above, we use blinking optical tweezers to facilitate measurements of the pair interaction in unconfined colloid. This technique allows us to collect all the necessary data with one pair of spheres located far from the sample container's glass walls and far from other particles. We align a pair of traps in the focal plane along a line parallel to the video lines and separated by no less than ten times the typical distance a free sphere can diffuse in 1/60 sec. Under these conditions, out-of-plane diffusion is sufficiently small that the three-dimensional center-to-center separation can be approximated by its projection into the plane.

Rather than measuring the equilibrium pair distribution g(r) directly, we use optical tweezers to create reproducible initial configurations away from which the pair of particles diffuse. By considering the dynamics of interacting Brownian particles, we can extract the equilibrium pair correlation function from a collection of such trajectories. Since the inertial damping time for the system is much smaller than the time scales on which we measure dynamics, we can approximate the master equation for the dynamical pair correlation function g(r,t) by


where tex2html_wrap_inline1139 is a Markovian propagator [24]. The steady-state solution to eqn. (22) is the equilibrium pair distribution function, g(r). After spatially discretizing eqn. (22) the discrete pair correlation function tex2html_wrap_inline1143 is the nontrivial eigenvector of a system of linear equations:


where tex2html_wrap_inline837 is the transition probability matrix for a pair of particles initially separated by distance tex2html_wrap_inline1147 to be separated by distance tex2html_wrap_inline1149 1/30 sec later and corresponds to tex2html_wrap_inline1151 .

In practice, we build up tex2html_wrap_inline837 by binning pair trajectory data according to the initial and final center-to-center separations in each time step. Each row of the counting data is normalized independently to conserve probability in eqn. (23). A typical example of tex2html_wrap_inline837 measured in this manner is shown in Fig. 7(a). Details for calculating such matrices are published elsewhere [5]. With a chopper wheel set to turn off the tweezers for six fields out of every twenty, we typically are able to videotape 15,000 fields of data in less than half an hour. To collect data at a range of different pair separations, we adjust the spacing between the traps every few minutes by slightly rotating a gimbal-mounted mirror in the laser projection optics. We programmed our frame grabber to selectively digitize only those frames with the traps off by monitoring each image's background brightness, which decreases noticeably when the laser is chopped. Once tex2html_wrap_inline837 has been calculated from the trajectory data, g(r) can be calculated by solving eqn. (22). Its logarithm is then an estimate of the interaction potential in units of the thermal energy tex2html_wrap_inline1161 . The pair potential for the propagator matrix shown in Fig. 7(a) appears in Fig. 7(b).

The method we have outlined for measuring the effective pair interaction potential rests on very few assumptions regarding the nature of the interaction. We require, for example, non-potential interactions to be negligible to ensure history independence of the propagator. Interactions lacking spherical symmetry such as those between Brownian dipoles or charged ellipsoids also would require a more sophisticated analysis. With these caveats, however, we are free to interpret our results for charged spherical latices within the conventional DLVO theory [25] for colloidal interactions.

The electrostatic part of the DLVO potential resembles a Yukawa interaction


where tex2html_wrap_inline1163 is the effective charge for spheres of radius tex2html_wrap_inline1115 , tex2html_wrap_inline1167 is the Debye-Hückel screening length and tex2html_wrap_inline811 is the dielectric coefficient of the suspending fluid. The screening length depends on the concentration n of z-valent counterions and determines the range of the interaction. The charge renormalization theory of Alexander et al. [26] suggests that a sphere's effective charge scales with its radius tex2html_wrap_inline1115


where tex2html_wrap_inline1177 is known as the Bjerrum length, and C is a constant predicted [26] to be about 10. The solid line in Fig. 7(b) is a fit of eqns. (24) and (25) to our measured interaction potential. The extracted screening length tex2html_wrap_inline1181 nm corresponds to an electrolyte concentration of about tex2html_wrap_inline1183 M, which presumably is due to dissolved airborne CO tex2html_wrap_inline1185 . The fit value of tex2html_wrap_inline1187 is consistent with the values predicted by molecular dynamics simulations [27] (C=7) and neutron scattering measurements on micelles [28] (C=6).

We chose to fit our data to eqn. (24) as a test of the charge renormalization theory of constant surface charge. Fitting to other forms would allow estimation of quantities such as the zeta potential. Consequently, this technique might be considered complementary to electrophoretic surface potential measurements.

Our prototype interaction measurement system reports a reliable interaction curve complete with fits for quantities of interest in roughly 10 hours. Most of this time is spent acquiring and analyzing video frames under software control. Performing more of these operations with specialized yet readily available hardware would reduce the processing time to under 1 hour. This is comparable to the time required for more traditional colloidal characterization measurements. Unlike conventional techniques, moreover, digital video microscopy coupled with blinking optical tweezers can measure the interactions between a particular pair of particles rather than averaging over a sample.

Next: Summary Up: Dynamical Measurements Previous: Blinking Optical Tweezers

David G. Grier
Mon Mar 11 23:01:27 CST 1996