  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 $D_{\theta\theta}$ 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 $D_{rr}\propto 1/R$, $D_{\theta\theta}\propto 1/R^2$). 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 $D_{\theta\theta}$ 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 $\frac R {d}(\frac {\eta_{w}d}
{\eta_{es,2p}})^{3/2}$, the correlated motions for various values of $n$ and
different particles can be scaled onto a single master curve, where $d$ is
particles' diameter, $\eta_{w}$ is the viscosity of the water, and
$\eta_{es,2p}$ is the effective surface viscosity whose measurements agree well
with that of one-particle surface viscosity $\eta_{es,1p}$. The effective
surface viscosity $\eta_{es,2p}$ as a function of the area fraction $n$ for
different silica spheres is presented.
