Introduction

In microfluidics an increasing surface area to volume ratio of liquids means that surface effects are of greater significance.

The correct description of the physics at solid-liquid interfaces then becomes imperative to the success of any attempt to model this class of flows.

How do variations in the wettability of a substrate affect the flow of an adjacent liquid?

A no-slip condition predicts no effect.

Molecular dynamics simulations suggest that this is not the case (Priezjev et al 05, Qian et al 05), see

Fig. 1

.

Fig. 1.

Snapshot of a molecular dynamics simulation showing flow driven by a shear over a patterned surface. Red indicates a hydrophobic regions while blue represents hydrophilic regions.

Note variations in density near the surface indicate a normal component of the flow.

Aim

Examine the case of a plane-parallel shear flow that encounters a change in solid substrate (

Fig. 2)

. Here we consider the case where solid 1 is more hydrophilic.

This is carried out in a continuum framework using the interface formation model.

Fig. 2. Sketch of the problem

Viscous flow over a chemically patterned surface J.E. Sprittles

and Y.D. Shikhmurzaev School of Mathematics, University of Birmingham, Edgbaston, B15 2TT

Interface formation model

 The no-slip boundary condition is generalised to allow for situations in which the classical fluid mechanics approach breaks down (e.g. Shikhmurzaev 93 and Shikhmurzaev 05).

 The interface is treated as a system in its own right, interacting with the bulk via mass, energy and momentum fluxes.

 In the continuum approximation the interaction across a layer of finite width (of the order of nanometres), caused by an asymmetry of intermolecular forces at the interface, is described by a set of equations to be applied at the surface.  The state of the interface is described by properties such as surface tension, surface density and surface velocity.

 The concept of wettability naturally fits into the model without any ad-hoc alterations. Specifically the equilibrium surface density is higher for a more hydrophilic solid-liquid interface.

Problem Formulation

Consider the steady flow of an incompressible, viscous, Newtonian fluid over a stationary flat solid surface, driven by a shear in the far field.

The bulk flow is described by the Navier-Stokes equations.

The boundary conditions to be applied at the solid liquid interface are provided by the interface formation model.

For a given liquid, a solid’s wettability is defined by the equilibrium contact angle θ, which a liquid-gas free surface would form with that solid.

Solid 1 (2) is characterised by a contact angle θ 1 (θ 2 ) The resulting set of equations are solved numerically using the finite element method.

Results

It was found that variations in wettability disturb the flow of adjacent liquid

(

Fig. 3)

.

Notice the non-zero normal velocity on the liquid facing side of each solid-liquid interface.

Conclusions

When fluid particles forming the interface are driven by the outer flow from a hydrophilic region into a hydrophobic one, the surface interacts with the bulk in order to attain its new equilibrium state.

Notably this creates a normal component to the flow.

This effect is qualitatively in agreement with molecular dynamics simulations and is here realised in a continuum framework.

Fig. 3. Streamlines of the flow.

How is the magnitude of the effect dependent on the solids chosen?

Slip on the surface is caused by shear stress and Marangoni effects.

Figs. 4 and 5

show the velocity components for three different combinations of solids where, in each case, the first solid is more hydrophilic.

Fig. 4. Tangential velocity on the surface

Curve 1: θ 1 = 10 o Curve 2: θ 1 = 60 o Curve 3: θ 1 = 10 o θ 2 = 60 o θ 2 = 110 o θ 2 = 110 o There is a flux out of the interface due to the surface density being above its equilibrium value.

Fig. 5. Normal velocity on the surface.

Our results suggest that for a given liquid the normal flux per unit time J, is related to the wettability of the solids by

J

 cos  1  cos  2 This may be considered as a measure of the effect that a patterned surface has on an adjacent flow.

For the case when solid 1 is more hydrophobic, there is a flux into the interface.

Literature cited

N.V. Priezjev, A.A. Darhuber and S.M. Troian. Slip behaviour in liquid films on surfaces of patterned wettability: Comparison between continuum and molecular dynamics simulations.

Phys. Rev. E

71:041608, 2005 T. Qian, X. Wang and P. Sheng. Hydrodynamic boundary condition at chemically patterned surfaces: A continuum deduction from molecular dynamics.

Phys. Rev. E

, 72:022501, 2005 Y.D. Shikhmurzaev. The moving contact line on a smooth solid surface.

Int. J. Multiphase Flow

, 19:589, 1993. Y. D. Shikhmurzaev. Macroscopic rupture of free liquid films.

C.R. Mecanique

, 333:205, 2005.

Acknowledgments

The author acknowledges the financial support of Kodak and the EPSRC.

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.