Effects of contact angles

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Spreading coefficient

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

This is often called the “initial” spreading coefficient.

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.

Initial and final contact angles

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

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

where $\theta _{e}\ge \theta$

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

$\pi _{e}$ 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:

$S=\sigma _{s}-\left( \sigma _{sl}+\sigma _{lv} \right)$

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

$S=\sigma _{lv}\left( \cos \theta +1 \right)$

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:

$S_{e}=\sigma _{lv}\left( \cos \theta _{e}+1 \right)+\pi _{e}$

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:

de Gennes, 2004, Fig. 2.17

$I=\sigma _{sv}-\sigma _{sl}$

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

$S=\sigma _{sv}-\sigma _{sl}-\sigma _{lv}$

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 $\cos \theta _{e}$ for a series of liquids was plotted against their surface tension:

Zisman, ACS Symp. Ser. 43, 1, 1964.

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 $\sigma _{c}$ (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.

Morrison, Fig. 10.3

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

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

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:

$r\sigma _{sv}=r\sigma _{lv}\text{cos}\varphi +\sigma _{sl}$

Combining with the Young-Dupré equation gives:

$\text{cos}\varphi =r\cos \theta _{e}$

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

$\text{cos}\theta =\sum{f_{i}\cos \theta _{i}}$

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

$\text{cos}\theta =f_{1}\cos \theta _{1}+f_{2}\cos \theta _{2}$

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

$\text{cos}\theta =f_{1}\cos \theta _{1}-f_{2}$

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.

Woodward, J.T.; Schwartz, D.K. Dewetting modes of surfactant solution as a function of the spreading coefficient., Langmuir, 13, 6873-6876, 1997.