An investigation is made into the moving contact line dynamics of
a Cahn–Hilliard–van der Waals (CHW) diffuse mean-field interface.
The interface separates two incompressible viscous fluids and can evolve
either through convection or through
diffusion driven by chemical potential gradients. The purpose of this paper is to show
how the CHW moving contact line compares to the classical sharp interface contact
line. It therefore discusses the asymptotics of the CHW contact line velocity and
chemical potential fields as the interface thickness ε and the mobility κ both go to
zero. The CHW and classical velocity fields have the same outer behaviour but can
have very different inner behaviours and physics. In the CHW model, wall–liquid
bonds are broken by chemical potential gradients instead of by shear and change of
material at the wall is accomplished by diffusion rather than convection. The result
is, mathematically at least, that the CHW moving contact line can exist even with
no-slip conditions for the velocity. The relevance and realism or lack thereof of this
is considered through the course of the paper.The two contacting fluids are assumed to be Newtonian and, to a first approximation,
to obey the no-slip condition. The analysis is linear. For simplicity most of
the analysis and results are for a 90° contact angle and for the fluids having equal
dynamic viscosity μ and mobility κ. There are two regions of flow. To leading order
the outer-region velocity field is the same as for sharp interfaces (flow field
independent of r) while the chemical potential behaves like
r−ξ, ξ = π/2/max{θeq,
π − θeq}, θeq being the equilibrium contact
angle. An exception to this occurs for θeq = 90°,
when the chemical potential behaves like ln r/r. The diffusive and viscous contact line
singularities implied by these outer solutions are resolved in the inner region through
chemical diffusion. The length scale of the inner region is about 10√μκ – typically
about 0.5–5 nm. Diffusive fluxes in this region are O(1). These counterbalance the
effects of the velocity, which, because of the assumed no-slip boundary condition,
fluxes material through the interface in a narrow boundary layer next to the wall.The asymptotic analysis is supplemented by both linearized and nonlinear finite
difference calculations. These are made at two scales, experimental and nanoscale.
The first set is done to show CHW interface behaviour and to test the qualitative
applicability of the CHW model and its asymptotic theory to practical computations
of experimental scale, nonlinear, low capillary number flows. The nanoscale calculations
are carried out with realistic interface thicknesses and diffusivities and with
various assumed levels of shear-induced slip. These are discussed in an attempt to
evaluate the physical relevance of the CHW diffusive model. The various asymptotic
and numerical results together indicate a potential usefullness for the CHW model
for calculating and modelling wetting and dewetting flows.
Stick-slip Motion of Moving Contact Line on Chemically Patterned Surfaces