# 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0 is the liquid being withdrawn from the surface and (b) F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.)

## Oscillation Modes of a Triple Line

The triple line can make oscillations similar to a vibrating string. To do this, we have to account for the following:

1) Fringe elasticity of the line contact

2) Viscous friction

When <math>\theta_d</math> is small we can write the energy equation as:

<math>E=\frac{1}{2} \gamma \theta^2_E \vert q \vert u^2_q </math>

where q is a wavevector, <math>u_q</math> is amplitude, and <math>\theta_E</math> angle shown in the previous image. We can therefore write the force as:

<math>F=-\frac{dE}{du_q}=-\gamma \theta^2_E \vert q \vert u_q</math>

If we balance this force with the viscous force we get:

<math>-\gamma \theta^2_E \vert q \vert u_q=-\frac{3\nu l}{\theta_e}\frac{du_q}{dt}</math>

Solving the equation, we obtain the exponential relaxation with characteristic frequency of:

<math>\nu=\frac{1}{\tau_q}=c \vert q \vert </math> where c is a constant.

Essentially, at first the defect looks like a sharp "Gaussian-like" curve that gets smeared out in time. At any point in time t, only the region of <math>\pm ct</math> around the point zero is relaxed (see image below).

(de Gennes p. 147)

## 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.

Lab-on-a-chip biomedical/diagnostic applications are a new and ambitious research field. To test for the presence of a particular gene or protein in a blood/saliva sample, the DNA fragment or peptide -respectively- has to first be separated from the rest of the particles that are present in the sample. The way this is traditionally done in the lab is by an electrophoresis process: different peptides have different charge-to-mass ratios and thus travel at different speeds in the presence of an electric field. This has to be reproduced within the lab-on-a-chip environment for a drop of reagent. In addition though, the drop has to be separated into two parts: a) the useful peptide-containing part that will be analyzed further in the chip and b) the useless junk-peptides part. Both processes, the electrophoresis and the drop separation, rely on electric fields. In fact, the electrophoresis requires a low-level electric field, while the drop-splitting requires a higher electric field. The concept is illustrated in the picture:

In practice, there are a few barriers to overcome in order to combine electrophoresis and electro-wetting on a chip. Some of the problems involved are: the cross-talk between the two electric fields and internal liquid flow during droplet separation which mixes back the particles. In order to overcome these, a more aggressive separation technique is implemented. Magnetic beads coated with a specific antibody sequester the particles of interest. The magnetic beads can then be confined to a single electrode by the presence of a concentrated magnetic field:

This info and lovely images were extracted from *Fair,R.B., Digital Microfluidics:is a true lab-on-a-chip possible?, Microfluid Nanofluid 3, 245-281 (2007)*.

## 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