Dynamic mechanisms for shear-dependent apparent slip on hydrophobic surfaces, E. Lauga and M. P. Brenner, Phys. Rev. E (2003)

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Wiki Entry by Robin Kirkpatrick, AP 225, Fall 2011

Introduction

The no-slip boundary condition is a result of microscopic surface roughness that results in viscous dissipation of the fluid so that it is rough stationary near the surface, and is independent of the material properties of the solid and liquid. However, experimentally, it is observed that the 'degree of no slip' is shear dependent.

The degree of slip can be described by the slip length, <math> \lambda </math> which is the distance from the surface that the velocity extrapolates to zero. <math>\lambda</math> is typically computed via <math>\lambda = U_s/\gamma^'</math>, with <math>U_s</math> being the slip velocity and <math>\gamma^'</math> being the strain rate at the surface. However, very large slip lengths have been observed experimentally and have been shown to increase with strain rate. This assumption results in an overestimation of the slip length, and is independent of interfacial shear, which is inconsistent with experimental observation.

One explanation for this phenomena is that nanobubbles exist on hydrophobic surfaces, allowing for a "hydrophobic attraction" resulting in a large no-slip distance. The authors note that, at a gas-liquid interface, one expects there to be a nonzero velocity, and that the slip length is only a function of the ratio of the viscosities in the fluid (<math> \eta1 </math>) and gas phases ((<math> \eta2 </math>) ) and is independent of shear.

In this article, the authors ask whether or not a bubble layer at each surface (of a sphere oscillating near a surface) explains the experimental observations of slip experiments. The authors calculate the dynamic response of bubbles in the presence of an oscillatory shear. They propose that in this "squeeze flow experiment", bubble size changes as a result of pressure changes in the fluid, diffusion of gas across the interface, and compression and expansion of gas in the bubble. This "leaky mattress effect" results in reduction of the size of the bubbles which results in an apparent slip at the boundary, as a smaller volume of fluid is required to exit the gap.


Theory

Oscillatory squeeze flow experimenets involve a sphere of radius a oscillating with velocity Vs in a liquid with viscosity <math>\eta</math> at a surface D away from a planar surface. Both surfaces are thought to be covered with microscopic gas bubbles with contact angles <math>\theta </math> and radii of curvature of Ro. They assume that each surface is covered with bubbles with a surface density <math> \phi </math>.

RLKMB1.jpg

For brevity, this summary will not go through the entire derivation, rather it will highlight the assumptions made in each step.

1. The total force on a bubble is a result of an elastic force due to pressure fluctuations inside the bubble, Fe that acts on an area <math>\psi S</math>, and a viscous force due to pressure fluctuations that act on the area <math>(1- \phi S)</math>. Thus the total time dependent force on a bubble is <math> F(t) = (1 - \phi)F_h + \phi F_b</math>

2. At small frequency oscillations, the gas is isothermal, and the gas in the bubbles obey the ideal gas law p(t)h(t)/m(t) = constant, where p is the pressure in the bubble, h is the height of the bubble, and m is the mass of a single bubble.

3. It is assumed that there are small changes in h, p, and m, which allows linearization about the equillibrium points.

4. Pressure fluctuations in the gas and bubble are proportional, therefore bubble and hydrodynamic forces are proportional. <math>F_b = \alpha F_h</math>

5. In the experiment, the sphere's distance is from the surface is of the form <math> D(t) = dsin(\omega t)</math>, so the velocity is <math>V_s (t) = d/dt(D(t))</math>.

6. The bubble loses mass due to vertical diffusion across the bubble boundary and radial diffusion across the apparatus. The former is computed via Henry's law, and the latter is computed by computing the flux integral, assuming a spherical bubble. We thus arrive at a condition for h (the height of the bubble) of

<math>\frac{dh}{dt} = -k_1 F_b - k_2 \frac{dF_b}{dt}</math>

<math>k_1 = \frac{b \kappa (R_o)^2 I(\theta) c_0}{\rho _0 p_0 S L_r}</math>

<math>k_2 = \frac{c_0 h_0}{c_{inf}p_0S}(1 + \frac{c_{inf}(D-2h_0)}{\rho_0 h_0})</math>

<math>\delta = 12 \pi \eta a^2 f_{slip} /D </math>

<math> f_{slip} = \frac{D}{3 \lambda} [(1 + \frac{D}{6 \lambda}) ln(1 + \frac{6 \lambda} {D} -1]</math>

where <math> c_{inf} </math> is the far field gas concentration, S is the area of the sphere, <math> \kappa </math> is the gas transfer ration, f_slip is the factor that the vicsous force is decreased due to having fluid slip, and p_0 and h_0 are the pressure and heights that we expand about in the linearization (ie the equillibrium values).

8. The final solution is

<math>F(t)/F_{lub} = (f_{slip} \frac{(1- \phi + \alpha \phi)(1-\delta \alpha k_1)}{(1 + \delta \alpha k_1)^2 + (\delta \omega \alpha k_2)^2}) (cos( \omega t) + \frac{(\delta \omega \alpha k_2}{1 + \delta \alpha k_1 }sin(\omega t)) </math>


9. in this limit of D -> inf, <math> \alpha </math> ->1 so that the out of phase ratio of the peak force is given by

<math>\frac{f^*}{f_{slip}} = \frac{1}{1 + (\delta k_1 + \frac{(\omega \delta k_2)^2}{1 + \delta k_1})}</math>


and the in-phase ratio is given by

<math>g^*= f^* \frac{ \delta \omega k_2}{ 1 + \delta k_1}</math>

Interestingly, the out-of phase component allows us to predict the effect elasticity of the bubble interface.

Therefore, the apparant slip length depends on the size of the system, increases with fluid viscosity, is shear dependent (due to frequncy dependence) which leads to increase in apparent slip for increasing shear.

The authors compared these numerical results with previsouly published experiemental data for this sqeueeze flow epxeriment.

Figure 3 shows one set of experimental results compared with their dynamic model. 3a, 3b, and 3c correspond to the authors using different parameters in the model for a single data set. Briefly, the authors fit the data to their model in a least squares sense allowing theta to vary with the other parameters corresponding to the measured system parameters. Their results were found to match fairly well with experiment for large surface coverage. They find differences in contact angle from data fitting to the macroscopically measured contact angle, which they attribute to electrical effects and intermolecular forces of bubbles on surfaces. The authors note that for bubble coverage to be responsible for apparant slip, very high surface coverage is required.

RLKMB3.jpg


Discussion

The authors present a "leaking mattress" model which is a result of the dynamic deformation due to pressure fluctuations and mass transfer at the bubble boundary. Their results are consitent with the observations that the decrease in viscous force increases with the viscosity of the fluid and the size of the sphere.

This paper highlight the importance of considering all relevant transport processes when considering hydrodynamic coupling problems. In this case, the presence of bubbles on surfaces results in an apparant 'slip' effect which is due to the "leaking mattress" effect which becomes more important as the viscosity and sphere size increase. It is still an open question when and how these bubble layers arise. However, the nature of the results for this 'slip' problem in the presence of bubble layer and a 'macroscopic' noslip problem are inherently different.

Keywords

No-Slip Boundary Condition, Diffusion