Charge renormalization in nominally apolar colloidal dispersions

We present high-resolution measurements of the pair interactions between dielectric spheres dispersed in a fluid medium with a low dielectric constant. Despite the absence of charge control agents or added organic salts, these measurements reveal strong and long-ranged repulsions consistent with substantial charges on the particles whose interactions are screened by trace concentrations of mobile ions in solution. The dependence of the estimated charge on the particles' radii is consistent with charge renormalization theory and, thus, offers insights into the charging mechanism in this interesting class of model systems. The measurement technique, based on optical-tweezer manipulation and artifact-free particle tracking, makes use of optimal statistical methods to reduce measurement errors to the femtonewton frontier while covering an extremely wide range of interaction energies.

Figure 1

(a) Schematic of a two-particle blinking optical trapping experiment with the traps on. (b) Typical bright-field image of 2.3-$\mu \mathrm{m}$-diameter PMMA spheres in CXB-dodecane overlain with circles marking the measured sphere positions. (c) Single-blink trajectory of 2.3-$\mu \mathrm{m}$ spheres with an initial particle separation of 4.$8\mu \mathrm{m}$. The particles move 1.$6\mu \mathrm{m}$ within the 86-ms blink period.

Figure 2

Kernel density estimates of (a) the probability distribution function $\rho \left(r\right)$, (b) the velocity $v\left(r\right)$, (c) the diffusivity $D\left(r\right)$, and (d) the force $F\left(r\right)$, all as functions of particle separation $r$ for 2.3-$\mu \mathrm{m}$-diameter spheres. (e) Pair potentials $U\left(r\right)$ in units of the thermal energy scale for spheres of various sizes obtained by numerically integrating measured pair forces. (f) Selected data from (e) replotted to emphasize the screened-Coulomb form. Dashed curves are best fits to the pure Coulomb repulsion. Shaded regions are uncertainties for the relevant quantities.

Figure 3

Effective charge number $Z^{*}\left(a_p\right)$ of PMMA spheres suspended in CXB-odecane with respect to particle size $a_p$. The solid curve is the limiting effective charge due to counterion condensation from Eq. (16). The dashed horizontal line indicates the mean effective surface potential of $5.2k_{B}T$ and the shaded region indicates the uncertainty in this value, $\pm 0.4k_{B}T$.