We reconcile two scaling laws that have been proposed in the literature for
the slip length associated with a moving contact line in diffuse interface
models, by demonstrating each to apply in a different regime of the ratio of
the microscopic interfacial width $l$ and the macroscopic diffusive length
$l_D= (M\eta)^{1/2}$, where $\eta$ is the fluid viscosity and $M$ the mobility
governing intermolecular diffusion. For small $l_D/l$ we find a diffuse
interface regime in which the slip length scales as $\xi \sim(l_Dl)^{1/2}$. For
larger $l_D/l>1$ we find a sharp interface regime in which the slip length
depends only on the diffusive length, $\xi \sim l_D \sim (M\eta)^{1/2}$, and
therefore only on the macroscopic variables $\eta$ and $M$, independent of the
microscopic interfacial width $l$. We also give evidence that modifying the
microscopic interfacial terms in the model's free energy functional appears to
affect the value of the slip length only the diffuse interface regime,
consistent with the slip length depending only on macroscopic variables in the
sharp interface regime. Finally, we demonstrate the dependence of the dynamic
contact angle on the capillary number to be in excellent agreement with the
theoretical prediction of \cite{Cox1986}, provided we allow the slip length to
be rescaled by a dimensionless prefactor. This prefactor appears to converge to
unity in the sharp interface limit, but is smaller in the diffuse interface
limit. The excellent agreement of results obtained using three independent
numerical methods, across several decades of the relevant dimensionless
variables, demonstrates our findings to be free of numerical artifacts.Comment: 24 pages, 5 figure