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Correlated dynamics of weakly charged silica spheres at an air-water interface

Optical microscopy and multi-particle tracking are used to investigate the spatially correlated motion of weakly charged silica spheres at an air-water interface for different area fraction $n$ occupied by the particles. When the area fraction is very small, e.g. $n=0.03$, the correlation function along the line joining the centers of particles $D_{rr}$ decays with inter-particle distance $R$ as $1/R^{0.86\pm0.02}$, and the function perpendicular to this line $D_{θθ}$ decays with $R$ as $1/R^{1.45\pm0.03}$, which differs from the results of [Phys. Rev. Lett. 97, 176001 (2006)] with low surface viscosity (where $D_{rr}\propto 1/R$, $D_{θθ}\propto 1/R^2$). We argue that the differences arise from the Coulomb interaction between particles. The Coulomb interaction enhances the correlated motion of particles. Experimental results show that with the increase of $n$, the decay rate of $D_{rr}$ and $D_{θθ}$ with $R$ decreases and the cross-correlation enhances for the Coulomb interaction increases. The Coulomb interaction between colloidal particles should serve as an effective surface viscoelastical role in our system. With the scaled separation $\frac R {d}(\frac {η_{w}d} {η_{es,2p}})^{3/2}$, the correlated motions for various values of $n$ and different particles can be scaled onto a single master curve, where $d$ is particles' diameter, $η_{w}$ is the viscosity of the water, and $η_{es,2p}$ is the effective surface viscosity whose measurements agree well with that of one-particle surface viscosity $η_{es,1p}$. The effective surface viscosity $η_{es,2p}$ as a function of the area fraction $n$ for different silica spheres is presented.