# Dynamics of forced wetting

## Contents

## Dynamics of the triple line

A force exerted on the triple line (a) F<0 is the liquid being withdrawn from the surface and (b) F>0 is the liquid being forced across the surface.

The dynamic contact angle is less than the equilibrium angle when the fluid is pulled back (or the substrate pulled forward); and the dynamic contact angle is greater than the equilibrium angle when the fluid is pushed forward (or the substrate pulled back).

(de Gennes, 2004, 00 139f.)

## Application of wetting: Lab-on-a-chip

The principles of forced wetting, in this case caused by an electric field, could enable "digital microfluidics." In this technique, discrete droplets of fluid are individually manipulated like the charges in an integrated circuit. The droplets are sandwiched between two hydrophobic plates, which provide an insulating layer in front of an array of electrodes (see below):

Dr. Richard Fair at Duke University is one of the leaders in this emerging field and much more information, including some fantastic videos, can be found on his website.

## Observations of the triple line

Original work from: Hoffman, R. *JCIS*, *50*, 228, **1975**.

The experimental observation and data analysis: θD as a function of Ca=ηV/σ for various silicone oils in a glass tube.

The data show that the velocity of the triple line scales with dynamic contact angle as: <math>V\propto \theta _{D}^{3}</math>

AN INCREDIBLE RESULT!!

## Forced Wetting

Vertical extraction of a plate from a pool of liquid.

F < 0 is extraction of a plate from a liquid.

- At low pull rates the triple line remains at a fixed height, that is, it moves with V = - Vp relative to the plate.
- At higher pull rates, the triple line moves with a finite thickness. This is called forced wetting.

The observations on vertical extraction is summarized on this graph:

As the pulling rate increases, the dynamic contact angle decreases from the equilibrium contact angle. When a critical extraction velocity is reached, the triple line has no stable position and a thick film is pulled from the pool of liquid.

## Mechanical model of forced wetting

Forced flow of a spreading liquid. The velocity V of the line is the average of the velocity profile in the figure.

Assume a perfect wedge-shaped liquid border with: <math>\tan \theta \approx \theta \text{ is a constant}</math>

The velocity profile goes from 0 on the substrate to 1.5V at the liquid surface.* Note that the upper surface is moving faster that the liquid. The top surface moves to the triple line and then pins to the solid surface. The motion is like a tank’s tread.

The velocity goes from 0 at *z* = 0 to 1.5V at *z* = θDx: <math>\frac{dv}{dz}\approx \frac{1.5V}{\theta _{D}x}</math>

- This is an exercise left for the reader on p.110!

## Viscous dissipation model of forced wetting

The energy dissipated by viscous flow (per unit length of the triple line in the y-direction) is

<math>T\dot{S}=\int\limits_{0}^{\infty }{dx}\int\limits_{0}^{\theta _{D}x}{\eta \left( \frac{dV}{dz} \right)}^{2}dz</math>

The integrations proceed normally except for the prefactor which de Gennes adds at the end (and I do not understand.)

<math>\begin{align}

& \text{ }\approx \int\limits_{0}^{\infty }{dx}\int\limits_{0}^{\theta _{D}x}{\eta \left( \frac{V}{\theta _{D}x} \right)}^{2}dz \\ & \text{ }\approx \frac{\eta V^{2}}{\theta _{D}^{2}}\int\limits_{0}^{\infty }{\frac{\theta _{D}x}{x^{2}}dx\text{ }} \\ & \text{ }=\frac{3\eta V^{2}}{\theta _{D}}\int\limits_{0}^{\infty }{\frac{dx}{x}}\text{ } \\

\end{align}</math>

The final integral diverges in both limits. Not good for the average physicist.

The singularities at the limits are removed (by de Gennes et al) by integrating from a molecular distance of closest approach to the size of the meniscus slip length, L.*

<math>\int\limits_{0}^{\infty }{\frac{dx}{x}}\approx \int\limits_{a}^{L}{\frac{dx}{x}}=\ln \left( \frac{L}{a} \right)\equiv l\text{ }</math>

L is a parameter, having the dimensions of length, which is approximately the length of the leading meniscus before the angle reaches the equilibrium value and is seen experimentally. Ref. 2 & 3. p. 150. The numerical (experimental) value of varies from 15 to 20.*

gives

<math>\text{ }T\dot{S}=\frac{3\eta l}{\theta _{D}}V^{2}\text{ }</math>

The force of traction pulling the liquid toward the dry region is:

<math>F\left( \theta _{D} \right)=\sigma _{so}-\sigma _{sl}-\sigma \cos \theta _{D}</math>

Remember that <math>F\left( \theta _{E} \right)=\sigma _{so}-\sigma _{sl}-\sigma \cos \theta _{E}=0</math>

<math>\therefore F\left( \theta _{D} \right)=\sigma \left( \cos \theta _{E}-\cos \theta _{D} \right)</math>

(No normal person would have thought of these two steps.)

<math>\begin{align}

& F\left( \theta _{D} \right)\simeq \sigma \left( \left( 1-\frac{\theta _{E}^{2}}{2} \right)-\left( 1-\frac{\theta _{D}^{2}}{2} \right) \right) \\ & F\left( \theta _{D} \right)\simeq \sigma \left( \frac{\theta _{D}^{2}}{2}-\frac{\theta _{E}^{2}}{2} \right) \\

\end{align}</math>

The rate of energy generation is FV, where V is the velocity of the triple line.

<math>F\left( \theta _{D} \right)V=\frac{V\sigma }{2}\left( \theta _{D}^{2}-\theta _{E}^{2} \right)</math>

The energy balance is: <math>T\dot{S}=FV</math>

<math>\frac{3\eta l}{\theta _{D}}V^{2}\text{ =}\frac{V}{2}\sigma \left( \theta _{D}^{2}-\theta _{E}^{2} \right)</math>

<math>V\text{ =}\frac{V^{*}}{6l}\theta _{D}\left( \theta _{D}^{2}-\theta _{E}^{2} \right)</math>

<math>\text{where }V^{*}=\frac{\sigma }{\eta }</math> Is a recurring parameter in the capillarity of viscous materials and has a value typically near 30 m/s.

The velocity of the triple line does vary as the third power of the dynamic contact angle as found from experiment. The energy dissipation is high in a narrow, moving wedge.

## Chemical model of forced wetting

Model the wetting process as a molecule of liquid approaches the triple line and binds on the substrate. If the contact angle retains the value θD, then the force per unit length remains as before:

<math>F\left( \theta _{D} \right)=\sigma _{so}-\sigma _{sl}-\sigma \cos \theta _{D}</math>

The energy liberated is <math>Fa^2</math>, where a is the molecular length. Assume an energy barrier, *U*, this leads to a characteristic jump frequency (adsorption and desorption):

<math>\frac{1}{\tau }=\frac{1}{\tau _{0}}\left\{ \exp \left( \frac{-U+{}^{Fa^{2}}\!\!\diagup\!\!{}_{2}\;}{k_{B}T} \right)-\exp \left( \frac{-U-{}^{Fa^{2}}\!\!\diagup\!\!{}_{2}\;}{k_{B}T} \right) \right\}</math>

where τ0 is a microscopic time on the order of <math>10^{-11}</math>s.

If the forces are weak, then <math>\frac{1}{\tau }\approx \frac{1}{\tau _{0}}\exp \left( \frac{-U}{k_{B}T} \right)\frac{Fa^{2}}{k_{B}T}</math>

We can define a velocity as <math>V=\frac{a}{\tau }</math> so that <math>V\approx V_{0}\exp \left( \frac{-U}{k_{B}T} \right)\frac{Fa^{2}}{k_{B}T}</math>

or <math>FV\simeq \frac{k_{B}T}{V_{0}a^{2}}\exp \left( \frac{U}{k_{B}T} \right)V^{2}\approx \eta _{1}V^{2}</math>

where <math>\eta _{1}=\frac{k_{B}T}{V_{0}a^{2}}\exp \left( \frac{U}{k_{B}T} \right)</math>

Interesting, but some of the parameters do not seem physical (yet). And no clear preference for mechanical or molecular models (yet).

From: de Gennes, 2004, p. 145