# Dynamics of total wetting

## Dynamics of total wetting

Total wetting is the unforced motion of liquid across a surface.

Tanner was the first to explore the dynamics of total wetting (1979) and proposed: <math>\theta _{D}\left( t \right)\propto t^{-{}^{3}\!\!\diagup\!\!{}_{10}\;}</math>

Which is called “Tanner’s law”.

The surprising part of this observation is that the dynamic contact does not depend on the magnitude of the spreading coefficient. This may be related to a much early study by Hardy (1919) showing a precursor film spreading well ahead of the observable dynamic contact line.

Careful observation shows a precursor film that can be detected in advance of a spreading liquid. With the naked eye an observer can see the drop spread but not a precursor film.

Motion of a precursor film indicates that a great force must be at work (the film is so thin).

To analyze the rate of spreading of the drop we compare the force of the precursor film on the drop: <math>\sigma _{sl}+\sigma _{l}</math> since the precursor film is so thin to the retarding force of the drop with dynamic contact angle, θD. <math>-\sigma _{sl}-\sigma _{l}\cos \theta _{D}</math>

The net force is: <math>\tilde{F}\left( \theta _{D} \right)=\sigma _{l}-\sigma _{l}\cos \theta _{D}\cong \sigma _{l}\frac{\theta _{D}^{2}}{2}</math>

which we can use instead of: <math>F\left( \theta _{D} \right)\simeq \sigma \left( \frac{\theta _{D}^{2}}{2}-\frac{\theta _{E}^{2}}{2} \right)</math>

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

Now becomes: <math>V\text{ =}\frac{V^{*}}{6l}\theta _{D}^{3}</math>

Remember that: <math>V\text{ }\equiv \frac{dR}{dt}</math>\

The volume of the spherical cap is <math>\Omega \text{ =}\frac{\pi }{4}R^{3}\theta _{D}</math> (I get one sixth not one fourth.)

Taking the total differential (which equals 0) and re-arranging gives: <math>\frac{1}{\theta _{D}}\frac{d\theta _{D}}{dt}=-\frac{3}{R}\frac{dR}{dt}</math>

Substituting the velocity for the time derivative of the radius gives: <math>\frac{d\theta _{D}}{dt}\sim -\frac{V^{*}}{R}\theta _{D}^{4}</math> Dropping the constants.

Since the volume is constant: <math>\text{R}\sim L\theta _{D}^{{-1}/{3}\;}</math>

Substituting <math>\text{R}\sim L\theta _{D}^{{-1}/{3}\;}</math> into <math>\frac{d\theta _{D}}{dt}\sim -\frac{V^{*}}{R}\theta _{D}^{4}</math>

Gives <math>\frac{d\theta _{D}}{dt}\sim -\frac{V^{*}}{L}\theta _{D}^{{13}/{3}\;}</math>

Which, after integrating gives: <math>\theta _{D}^{{-10}/{3}\;}\sim \frac{V^{*}t}{L}</math>

or <math>\theta _{D}\sim \left( \frac{L}{V^{*}t} \right)^{3/10}</math>

In agreement with both Tanner and Marmur!

Substituting <math>\text{R}\sim L\theta _{D}^{{-1}/{3}\;}</math>

A very slow growth rate for the drop – as expected: <math>\text{R}\sim L\left( \frac{V^{*}t}{L} \right)^{1/10}</math>