Experimental demonstration

We demonstrate our procedure by measuring the well-understood electrostatic interactions between micrometer-scale charge-stabilized colloidal silica spheres dispersed in deionized water. In this case, the electrostatic pair potential for two spheres of radius $a$ each carrying effective charge $Z^\ast$ (47,19,46) has the form (48)

$$\beta u(r) = {Z^\ast}^2 \, \lambda_B \, \frac{\exp(2\kappa a)}{(1 + \kappa a)^2} \, \frac{\exp(-\kappa r)}{r},$$ (8)

where $\lambda_B = e^2 / (4 \pi \epsilon k_B T)$ is the Bjerrum length in a medium of dielectric constant $\epsilon$ , and $\kappa^{-1}$ is the Debye-Hückel screening length, given by $\kappa^2 = 4 \pi \lambda_B n$ in a concentration $n$ of monovalent ions. Previous measurements (2,17,22,19) have confirmed that Eq. (8) accurately describes the interactions between pairs of highly charged colloidal spheres provided they are kept far enough away from charged surfaces (13,14,5,20,19) or other spheres (16,15).

Figure 3: (Color online) Measured pair potential for two 1.5  $\unit{\mu m}$ diameter colloidal silica spheres obtain with Eqs. (4) through (7) (circles) and by subtracting off single-particle optical contributions (diamonds). Inset: data replotted to facilitate comparison with Eq. (8).

We performed measurements on two silica spheres of nominal radius $a = 1.53~\ensuremath{\unit{\mu m}}\xspace$ (Bangs Labs 5303) dispersed in a 40  $\unit{\mu m}$ thick layer of water between a glass microscope slide and a #1.5 cover slip. The edges of the coverslip were sealed to the surface of the slide with Norland Type 63 UV-cured adhesive to prevent evaporation. The glass surfaces were cleaned by oxygen plasma etching before assembly.

A holographic line trap $L = 8~\ensuremath{\unit{\mu m}}\xspace$ long was focused near the midplane of the sample volume, far enough from the bounding surfaces to minimize their influence on the spheres' interactions. The line was designed to come to best focus uniformly in the microscope's focal plane, to have uniform phase along its length and a Gaussian intensity profile (1). Figure 2 shows the measured potential energy profile, which differs from the design by roughly 20%. Such variations would pose challenges if an accurate profile were required for our analysis. Because none of the spurious local potential energy wells is deep enough to trap a particle against thermal forces, however, deviations from the designed profile do not affect our measurement.

The curvature of the line's potential energy well was adjusted to bring the particles into proximity while still allowing them freedom of motion. Three half-hour data sets were obtained at laser powers of $\alpha = 0.4~\unit{W}$ , 0.6 W and 0.8 W. The overall efficiency of our optical train is roughly 5%, taking into account the theoretical efficiency of the line-forming shape-phase hologram (1). The total power projected onto each sphere is of the order of 3 mW, which is comparable to conditions in conventional point-like optical tweezers.

Thermal forces cause particles to wander away from the projected line. Equation (4) can be generalized to incorporate averages over the extra dimensions, with the appropriate redefinition of the inter-particle separation $r$ . The additional computational effort required for multi-dimensional integrals would be burdensome, however. Instead, we pruned the data set to include only those measurements with single-particle axial excursions smaller than 200 nm. After this, just 2,000 statistically independent measurements of the particles' positions were retained for each laser power. These were analyzed according Eqs. (4), (5) and (7) to obtain estimates for the intrinsic pair potential, which are plotted as points in Fig. 3. Even with such limited statistics, the results, plotted as circles in Fig. 3, are consistent with an energy resolution of $\pm 0.5~k_B T$ and a spatial resolution of $\pm 20~\unit{nm}$ , roughly twice the estimated uncertainty in the individual particles' positions.

The inset to Fig. 3 shows the same data plotted for easy comparison with the prediction of Eq. (8). The observed linear trend is consistent with the anticipated screened Coulomb repulsion, and thus with previous measurements on similar colloidal particles under similar conditions (17,22,19). The best-fit slope of this plot suggests a Debye-Hückel screening length of $\kappa^{-1} = 32 \pm 10~\unit{nm}$ which is consistent with a $n = 180~\unit{\mu M}$ total concentration of monovalent ions.

Based on the dissociation of terminal silanol groups with an estimated surface coverage of $6~\unit{nm^{-2}}$ , the silica particles' effective charge number is anticipated (3) to be no larger than $Z^\ast \leq 6500$ and is known to be reduced by the presence of a neighboring sphere. The generalized (19) charge renormalization (46) result,

$$Z^\ast = \frac{e \zeta_0}{k_B T} \, \frac{a}{\lambda_B} \, \left( 1 + \kappa a \right),$$ (9)

relates the effective charge number to the effective surface potential, $\zeta_0$ . Taking $e\zeta_0 = 112~\unit{meV}$ yields $Z^\ast = 5500$ . The solid curve in Fig. 3 is the prediction of Eq. (8) for these values.

Figure 4: (Color online) Position-averaged light-induced pair potential $v_2(r)$ estimated from the data in Fig. 3.

If we assume that there are no light-induced interactions, we can use the calibrated line profile to compute $u(r)$ from $P_\alpha(\vec{x},\vec{x}+\vec{r})$ directly. The results are plotted as diamonds in Fig. 3. The difference between $u(r)$ computed in this way and that obtained from Eq. (4) is the one-dimensional projection of

$$\int P_1(\vec{x}) \, P_1(\vec{x}+\vec{r}) \, v_2(\vec{x},\vec{x}+\vec{r}) \, d\vec{x} \, d\Omega_r \approx v_2(\vec{r}),$$ (10)

which provides at least a rough estimate for the light-induced interaction. This is consistent with a short-ranged exponential repulsion with a decay length of 80 nm, which is plotted in Fig. 4 as a dashed line. Such a repulsion is consistent with previous reports (34) of optically induced interactions. The peaks in Fig. 4 may be ascribed to power-dependent changes in the functional form of $v_1(\vec{x})$ . These discrepancies can be seen in Fig. 2 and are likely to explain the peaks in the estimate for $u(r)$ at $r = 1.66~\ensuremath{\unit{\mu m}}\xspace$ and 1.79  $\unit{\mu m}$ in Fig. 3.

Although results obtained with Eqs. (4) through (7) are subject to artifacts due to power-dependent changes in $v_1(\vec{x})$ , ignoring $v_2(\vec{x},\vec{x}+\vec{r})$ leads to subtle systematic errors. In particular, the result for $u(r)$ obtained by applying single-particle calibrations overestimates the repulsive force at small separations. The principal consequence for the present system would be to systematically overestimate the particles' effective charge number.

Even without these exigencies, the agreement between experiment and theory in this model system demonstrates that the protocol described above can be applied with reasonable confidence to systems whose underlying interactions are less well understood.