  Recent experiments (Guan et al. 2016a,b) showed many interesting phenomena on
dynamic contact angle hysteresis while there is still a lack of complete
theoretical interpretation. In this work, we study the time averaging of the
apparent advancing and receding contact angles on surfaces with periodic
chemical patterns. We first derive a Cox-type boundary condition for the
apparent dynamic contact angle on homogeneous surfaces using Onsager
variational principle. Based on this condition, we propose a reduced model for
some typical moving contact line problems on chemically inhomogeneous surfaces
in two dimensions. Multiscale expansion and averaging techniques are employed
to approximate the model for asymptotically small chemical patterns. We obtain
a quantitative formula for the averaged dynamic contact angles. It gives
explicitly how the advancing and receding contact angles depend on the velocity
and the chemical inhomogeneity of the substrate. The formula is a
coarse-graining version of the Cox-type boundary condition on inhomogeneous
surfaces. Numerical simulations are presented to validate the analytical
results. The numerical results also show that the formula characterizes very
well the complicated behaviour of dynamic contact angle hysteresis observed in
the experiments.
