# Effects of contact angles

## Contents

- 1 Spreading coefficient
- 2 Initial and final contact angles
- 3 Spreading pressure in the spreading coefficient
- 4 Capillary rise
- 5 Zisman's rule
- 6 Critical surface tensions
- 7 Jamin effect
- 8 Relation to Decompression Sickness ("The Bends")
- 9 Contact Angle Hysteresis
- 10 Surface heterogeneity
- 11 Test of Cassie model

## Spreading coefficient

The spreading coefficient is: <math>S=\sigma _{s}-\left( \sigma _{sl}+\sigma _{lv} \right)</math>

If the spreading coefficient is positive, the liquid “spreads” on the solid (b); if the spreading coefficient is negative, the liquid “partially spreads” on the solid (a). If the contact angle is less than 90o, the liquid is said to “wet” the solid; if the contact angle is greater than 90o, the liquid is said to “not wet” the solid.

This is often called the “initial” spreading coefficient because in a fluid-fluid interface, the interfacial tension between the two fluids is measured at the instant the interface is formed, before any mutual solubilization of the two liquids take place.

In a fluid-fluid interface, spreading behavior can be very different before and after mutual solubilization. One example which illustrates the difference between initial and final behavior of spreading is hexanol on water.

The Initial S of hexanol and water :

<math>\begin{align}

& S_{init}=\text{72}\text{.8 - (24}\text{.8+6}\text{.8)} \\ & \therefore S_{init}=41.2\frac{mJ}{m^{2}} \\

\end{align}</math>

However, after equilibrium, some hexanol dissolves in the water and thus reduce the surface tension of water from 72.8 to 28.5 <math>\frac{mJ}{m^{2}}</math>. This causes the S to become negative:

<math>\begin{align}

& S_{final}=\text{28}\text{.5 - (24}\text{.8+6}\text{.8)} \\ & \therefore S_{final}=-3.0\frac{mJ}{m^{2}} \\

\end{align}</math>

## Initial and final contact angles

Based on a force balance Young and Dupré (independently) asserted: <math>\sigma _{s}=\sigma _{lv}\text{cos}\theta +\sigma _{sl}</math>

If vapor is adsorbed on the solid (a spreading pressure is created) and <math>\sigma _{sv}=\sigma _{lv}\text{cos}\theta _{e}+\sigma _{sl}</math>

where <math>\theta _{e}\ge \theta </math>

The change in energy of the solid with adsorption can be expressed as: <math>\sigma _{sv}=\sigma _{s}-\pi _{e}</math>

<math>\pi _{e}</math> is called the spreading pressure.

## Spreading pressure in the spreading coefficient

The “initial” spreading coefficient described above refers to spreading before any vapor adsorption on the solid. It can be combined with the Young and Dupré equation to give:

<math>S=\sigma _{s}-\left( \sigma _{sl}+\sigma _{lv} \right)</math>

It can be combined with the Young and Dupré equation to give:

<math>S=\sigma _{lv}\left( \cos \theta +1 \right)</math>

This definition clearly implies that the spreading coefficient cannot be greater than twice the surface tension of the liquid no matter the solid! This is clearly wrong.

However, including the spreading pressure in the calculation of the spreading coefficient produces:

<math>S_{e}=\sigma _{lv}\left( \cos \theta _{e}+1 \right)+\pi _{e}</math>

Derjaguin introduced the more general concept of a “disjoining” pressure to account for these kinds of corrections in all of capillarity.

## Capillary rise

The driving force for capillary rise is the replacing of a high energy solid/vapor interface with a lower energy (generally) solid/liquid interface:

<math>I=\sigma _{sv}-\sigma _{sl}</math>

The spreading of a liquid across a solid has a smaller driving force:

<math>S=\sigma _{sv}-\sigma _{sl}-\sigma _{lv}</math>

Therefore wicking is more common than spreading.

## Zisman's rule

In the 1950’s and 60’s Zisman found empirically that the wettability of solid surfaces could be ranked if <math>\cos \theta _{e}</math> for a series of liquids was plotted against their surface tension:

The data are extrapolated to where the cosine is one and that surface tension taken as the “critical” surface tension.

## Critical surface tensions

Solid | Nylon | PVC | PE | PVF2 | PVF4 |

<math>\sigma _{c}</math> (mN/m) | 46 | 39 | 31 | 28 | 18 |

The scatter in the data led to a more careful modeling of the interaction between the solid and the liquid, led by the work done by F.M. Fowkes at Lehigh.

The general idea is that the interaction of a solid and a liquid is not a single dimensional quantity but contains “components” such as acid-base and dispersion forces.

Fowkes, F.M. Dispersion force contributions to surface and interfacial tensions, contact angles, and heats of immersion. ACS Symp. Ser. 43, 99 – 11, 1964.

## Jamin effect

In 1860 Jamin noticed that an ordinary cylindrical capillary tube filled with a chain of alternate air and water bubbles is able to sustain a finite pressure. We now know that this is a consequence of a difference between the advancing and receding contact angles leading to pressure differences.

If neither the advancing nor the receding contact angles differ from bubble to bubble then:

<math>P=\frac{2n\sigma _{lv}\left( \cos \theta _{r}-\cos \theta _{a} \right)}{r}+P_{0}</math>

## Relation to Decompression Sickness ("The Bends")

The Jamin Effect contributes to the prevalence of Decompression Sickness (DCS) or otherwise known as 'the bends'. DCS is commonly seen in the following circumstances:

* A scuba diva ascends quickly without taking proper decompression steps * Cabin pressure in an aircraft fails * Divers fly shortly after diving.

As most people know, the main cause is the bubbles that form in the capillaries. The pressure changes cause inert gases like nitrogen to form gas bubbles. When the body is exposed to decreased pressures, nitrogen dissolved in tissues and fluids in the body comes out of solution and forms bubbles. The natural blood flow and movement of the body creates micronuclei which serve as seeds to the bubbles during ascent, allowing nitrogen to diffuse into them. The nonwettable surface of the inside of blood vessels furthers the ability of the bubbles to form. The result is the existence of large bubbles that can stop a diver's heart.

The bubbles from in the capillaries and normally would be filtered out when they reach the alveoli in the lungs. However, the vast majority of cases of DCS are caused by bubbles that can't circulate. This is thought to be due in part to the Jamin Effect described above. The pockets between the bubbles act as a plug that can support a small level of pressure making it difficult for the heart to push the blood along. This mechanism is a primary cause of neurolofic DCI.

For more information check out: http://www.nationmaster.com/encyclopedia/Decompression-sickness and http://www.dtmag.com/Stories/Dive%20Psychology/09-00-2feature.htm.

## Contact Angle Hysteresis

The real surfaces that we deal in experiments are heterogeneous and exhibits some roughness or surface variations. A liquid drop resting on such surface might be in a metastable equilibrium instead of a stable equilibrium as mostly discussed.

On an ideally smooth and homogeneous surface, the theoretical equilibrium contact angle is <math>\theta _{y}</math> or Young's angle which corresponds to the lowest energy state for a system.

The equilibrium contact angle on a rough and heterogeneous surface is known as <math>\theta _{w}</math> and <math>\theta _{c}</math> respectively. Although these angles correspond to the lowest energy state, the angles found in experiments are often different. The contact angle of such system is found to be in a metastable state, in which the advancing and receding angles are different.

One example of such case is a liquid drop holding a steady contact angle on a solid surface. In an ideal system where the surface is smooth and homogeneous, the addition of a small volume of liquid to the drop will cause the drop front to advance. The same goes for the removal of liquid from the drop, which should cause the drop front to recede. However, in most real systems, these are not the case. The addition or removal of liquid in a real system often cause the droplet to increase or decrease in height without any surface front movement. The contact angle thus increases or decreases with each volume changes. When enough liquid is added or removed, the surface front will suddenly advance or recede.

## Surface heterogeneity

Two empirical equations have been proposed for heterogeneous surfaces:

**Wenzel equation**: surface roughness increases the contact area between the liquid and the solid so the Young-Dupré equation is modified to give:

<math>r\sigma _{sv}=r\sigma _{lv}\text{cos}\varphi +\sigma _{sl}</math>

Combining with the Young-Dupré equation gives:

<math>\text{cos}\varphi =r\cos \theta _{e}</math>

**Cassie equation**: the surface chemistry is not uniform but contains kinds of “patches” each with a different contact angle.

<math>\text{cos}\theta =\sum{f_{i}\cos \theta _{i}}</math>

A common assumption is that the surface has just two kinds of “patches”:

<math>\text{cos}\theta =f_{1}\cos \theta _{1}+f_{2}\cos \theta _{2}</math>

The most interesting case is if one “patch” is a hole so the contact angle is 2π

<math>\text{cos}\theta =f_{1}\cos \theta _{1}-f_{2}</math>

**Superhydrophobic surfaces**

Theory of superhydrophobic surfaces incorporates both Wenzel and Cassie models:

The surface is superhydrophobic as a result of both surface roughness (Wenzel Model) and inhomogeneity (air and solid interfaces: Cassie Model).

## Test of Cassie model

The cosine of the static contact angle of water on various subsaturated monolayers plotted versus the surface coverage measured directly using the atomic force microscope.