Figure 1

(Color online) (a) Illustration of the experimental setup, in which the force $F$ at separation $D$ between the solid surfaces of the two negatively charged colloids is measured by using optical tweezers. One colloid is fixed by the tip of a micropipette (blue) and the other in an optical trap established by a photonic potential (brown). (b) Microscope image of the two colloids under study. (c) Illustration of the image analysis algorithm to determine the geometrical center of the two colloids.Reuse & Permissions

Figure 2

(Color online) Force $F$ vs separation $D$ for one single pair of colloids in aqueous solution of varying salt and salt concentrations from experiments (symbols) and the DLVO theory with fitted values of the parameters $Z$ and $R$ (curves). (a) $\mathrm{NaCl}$ at $0.3\mathrm{mM}$ (black squares), $0.55\mathrm{mM}$ (red circles), $1\mathrm{mM}$ (green triangles), $2\mathrm{mM}$ (blue nablas), $4\mathrm{mM}$ (cyan diamonds), $10\mathrm{mM}$ (magenta left triangles), and $30\mathrm{mM}$ (orange right triangles). To ensure a full reproducibility of the exchange of the medium and to exclude hysteresis effects due to possible adsorption effects on the colloids the sample cell was flushed again with $0.3\mathrm{mM}\mathrm{NaCl}$ (gold diamonds). (b) ${\mathrm{CaCl}}_2$ at $0.15\mathrm{mM}$ (black open squares), $0.3\mathrm{mM}$ (red open circles); $0.5\mathrm{mM}$ (green open triangles), $1\mathrm{mM}$ (blue open nablas), $1.5\mathrm{mM}$ (cyan open diamonds), $3\mathrm{mM}$ (magenta open left triangles), $0.15\mathrm{mM}$ (orange open right triangles), and finally $0.3\mathrm{mM}\mathrm{NaCl}$ (gold open diamonds). (c) ${\mathrm{LaCl}}_3$ at $3\mathrm{\mu }\mathrm{M}$ (black red solid square), $10\mathrm{\mu }\mathrm{M}$ (black green solid circle), $30\mathrm{\mu }\mathrm{M}$ (black blue solid triangle), $100\mathrm{\mu }\mathrm{M}$ (black cyan solid nablas), $3\mathrm{\mu }\mathrm{M}$ (black magenta solid diamonds), and $0.3\mathrm{mM}\mathrm{NaCl}$ solution (black gold solid diamonds). Some indicative error crosses are given.Reuse & Permissions

Figure 3

Contour plots of the root-mean-square deviation ${\sigma }^F$ between experimental and predicted [Eq. (1)] force-separation dependences determined at different colloidal charge $Z$ and radius $R$ for (a) $\mathrm{NaCl}$, (b) ${\mathrm{CaCl}}_2$, and (c) ${\mathrm{LaCl}}_3$ salt solutions. The locations of the optimal values of $Z$ and $R$ are given (symbols), and corresponding values of ${\sigma }^F$ are (a) ${\sigma }_{\min }^F=1.28\mathrm{pN}$, (b) ${\sigma }_{\min }^F=1.10\mathrm{pN}$, and (c) ${\sigma }_{\min }^F=0.40\mathrm{pN}$. The three contour curves represent ${\sigma }^F={\sigma }_{\min }^F+0.25\mathrm{pN}$, ${\sigma }_{\min }^F+0.5\mathrm{pN}$, and ${\sigma }_{\min }^F+0.75\mathrm{pN}$, respectively.Reuse & Permissions

Figure 4

(Color online) Calculated force $F$ vs separation $D$ for one single pair of colloids in aqueous solution of varying salt and salt concentrations for (a) monovalent counterions at $c_{\mathrm{salt}}=0.3$, 0.55, 1, 2, 4, 10, and $30\mathrm{mM}$ (right to left) with $Z=218000$ and $R=1134.6\mathrm{nm}$ and (b) divalent counterions at $c_{\mathrm{salt}}=0.15$, 0.3, 0.5, 1, 1.5, and $3\mathrm{mM}$ (right to left) with $Z=137000$ and $R=1131.9\mathrm{nm}$ using the DLVO theory (solid curves) and the PB theory (dotted curves).Reuse & Permissions

Figure 5

(Color online) Experimental (symbols) and predicted [Eq. (1), curves with parameters from Table ] surface separation at forces of $2\mathrm{pN}$ (left ordinate) and $4\mathrm{pN}$ (right ordinate) vs (a) $\mathrm{NaCl}$ concentration, (b) ${\mathrm{CaCl}}_2$ concentration, and (c) ${\mathrm{LaCl}}_3$ concentration for colloid pairs 1 (black squares), 2 (green triangles), and 3 (red circles).Reuse & Permissions