next up previous
Next: Metastable Superheated Crystals Up: Electrostatic Interactions Previous: The DLVO Theory

Isolated Colloidal Pairs

Figure 2: Left: Colloidal interactions are measured in a hermetically sealed glass sample container, shown in cross-section. Two spheres are selected with optical tweezers and alternately trapped and released, their motions being recorded at 1/30 sec intervals as shown in the sequence of images. Spheres appear brighter when trapped because of light backscattered from the optical tweezers. Right: Interaction potentials [21] for pairs of polystyrene sulfate spheres in deionized water at $ T = 25\ensuremath{^\circ\mathrm{C}}\xspace $. Curves are labelled by the spheres' radii. Solid lines are nonlinear least squares fits to Eq. (9). Dashed lines are fits to the Sogami-Ise theory, Eq. (13).
\includegraphics[width=2in]{figures/apparatus} \includegraphics[height=2.8in]{figures/ur}

Since its development, the DLVO theory has profoundly influenced the study of macroionic systems. Testing its predictions directly through measurements on pairs of spheres has become possible with the recent development of experimental techniques capable of resolving colloids' delicate interactions without disturbing them. These fall into three categories: (1) measurements based on the equilibrium structure of low density suspensions [vondermassen94,kepler94,carbajaltinoco96], (2) measurements based on the nonequilibrium trajectories [crocker94,crocker96a,dinsmore96,larsen97] of spheres positioned and released by optical tweezers [28] and (3) measurements based on the dynamics of optically trapped spheres [20,29].

Methods (1) and (2) take advantage of the Boltzmann relationship

$\displaystyle \lim_{\bar \rho \rightarrow 0} g(r) = \exp \left[ - \beta U(r) \right]$ (10)

between the spheres' pair potential and their equilibrium pair correlation function, which, for an ergodic sample and $ r \ne 0$ can be evaluated as

$\displaystyle g(r) = \left< \frac{1}{\bar \rho^2 A} \int_A \rho( \vec x - \vec r, t ) \rho( \vec x, t) \, d \vec x \right>$ (11)

where $ \bar \rho$ is the mean concentration of spheres in area $ A$ and the angle brackets indicate an average over both time and angles. The two approaches differ in how they measure $ g(r)$, but agree in their principal result: isolated pairs of spheres, far from walls and unconfined by their neighbors, repel each other as predicted by Eq. (9).

Nonequilibrium optical tweezer measurements, such as the examples in Fig. 2.2 use a pair of optical traps to position two spheres at reproducible separations in a microscope's focal plane. Extinguishing the traps eliminates any perturbing influence of the intense optical field and frees the particles to move under random thermal forces and their mutual interaction. Their motions are captured in individual video fields at $ \tau = 1/30$ sec intervals and digitized. Each pair of consecutive images, such as the examples in Fig. 2.2, provides one discrete sampling of the probability $ P(x,\tau\vert x',0)$ that two spheres initially separated by $ x'$ will have moved to $ x$ a time $ \tau$ later. Repeatedly trapping and releasing the spheres over a range of initial separations enables us to sample $ P(x,\tau\vert x',0)$ uniformly. This probability density is the kernel of the master equation [30] describing how the nonequilibrium pair distribution function, $ \varrho(x,t)$, evolves in time:

$\displaystyle \varrho( x, t+\tau ) = \int_0^\infty P(x,\tau\vert x',0) \, \varrho(x',t) \, dx'.$ (12)

The domain of integration is conveniently limited in practice by the core repulsion which suppresses $ \varrho(x',t)$ at small $ x'$, and by the finite range of the interaction which renders $ P(x,\tau\vert x',0)$ diagonal and $ \varrho(x',t)$ independent of $ x'$ for large $ x'$. Consequently, Eq. (12) may be discretized and solved as an eigenvalue problem for the equilibrium distribution $ \lim_{t \rightarrow \infty} \varrho(x,t) \equiv \lim_{\bar \rho \rightarrow 0} g(x)$, whose logarithm is proportional to the pair potential through Eq. (10).

Data from Ref. [25] for three sizes of anionic polystyrene sulfate spheres dispersed in deionized water are reproduced in Fig. 2.2. Solid curves passing through the data points result from nonlinear least squares fits to Eq. (9) for the spheres' effective charges and the electrolyte's screening length. As expected [alexander84,lowen92,lowen93,gisler94,belloni98], the effective charges, ranging from $ Z = 6000$ for $ a = 0.327~\ensuremath{\mathrm{\mu m}}\xspace $ up to 22,800 for $ a = 0.765~\ensuremath{\mathrm{\mu m}}\xspace $, are one or two orders of magnitude smaller than the spheres' titratable charges [2,21]. The screening length of $ \kappa^{-1} = 280 \pm 10~\mathrm{nm}$ is comparable to the spheres' diameters and corresponds to a total ionic strength around $ 10^{-6}~\mathrm{M}$, a reasonable value for deionized water at $ T = 25\ensuremath{^\circ\mathrm{C}}\xspace $. These numbers will be useful for comparison with results in Sec. 5. The apparent success of the screened-Coulomb functional form does not validate the Debye-Hückel approximation, however, since the exact theory including all ion correlations has the same leading-order behavior [31].

Comparable results were obtained by Vondermassen et al. [22] from measurements on optical cross-sections of dilute bulk suspensions at low ionic strength. Sugimoto et al. [20] studied pairs of spheres at higher ionic strength trapped in optical tweezers and were able to measure the van der Waals contribution. In all cases, the measured pair potentials agree at least semi-quantitatively with predictions of the DLVO theory.

The observed pair repulsions at low ionic strength pose a challenge to theories predicting long-ranged pair-wise attractions. For example, Sogami and Ise [32,33] proposed that the colloidal pair potential can develop an attractive tail in the grand canonical ensemble when the simple ions' number is allowed to vary:

$\displaystyle U_{\mathrm SI} (r) = \frac{Z^2e^2}{\epsilon} \, \left[ 1 + \kappa a \coth \kappa a - \frac{\kappa r}{2} \right] \, \frac{\exp(-\kappa r)}{r}.$ (13)

This controversial theory has been quoted widely as a possible explanation for anomalous colloidal phenomena. However, Eq. (13) fails to describe the long-range pair repulsions in Fig. 2.2, and so cannot be expected to describe many-sphere behavior [21] through superposition.

next up previous
Next: Metastable Superheated Crystals Up: Electrostatic Interactions Previous: The DLVO Theory
David G. Grier 2001-01-16