Drops, menisci, and lenses

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Thickness of a large drop

Small drops are spherical segments. Large drops are flattened.

de Gennes, 2004, Fig. 2.4

To “spread” the drop requires an force per unit length:<math>\sigma _{sv}-\left( \sigma _{lv}+\sigma _{sl} \right)</math>

The hydrostatic pressure integrated over the depth of the drop is a force per unit length pushing to “spread” the drop: <math>\tilde{P}=\int\limits_{0}^{e}{\rho g\left( e-\tilde{z} \right)d\tilde{z}}=\frac{1}{2}\rho ge^{2}</math>

At equilibrium the sum of the two is zero: <math>\sigma _{sv}-\left( \sigma _{lv}+\sigma _{sl} \right)+\frac{1}{2}\rho ge^{2}=0</math>

Substituting the Young-Dupré equation: <math>\sigma _{lv}\left( 1-\cos \theta _{e} \right)=\frac{1}{2}\rho ge^{2}</math>

Re-arranging gives: <math>\text{ }e=2\kappa ^{-1}\sin \left( \frac{\theta _{e}}{2} \right)</math>

Profile of a large drop

The general ideas used to calculate the thickness are used again to calculate the profile. With a couple of modifications.

de Gennes, 2004, Fig. 2.4

The limits on the integration are changed: <math>\tilde{P}=\int\limits_{0}^{z}{\rho g\left( e-\tilde{z} \right)d\tilde{z}}=\rho g\left( ez-\frac{z^{2}}{2} \right)</math>

The “spreading” force per unit length is now: Where q<math>\theta </math> is the angle marked in the diagram: <math>\sigma _{sv}-\left( \sigma _{lv}\cos \theta +\sigma _{sl} \right)</math>

“It can be shown” from the diagram that: <math>\cos \theta =\sqrt{1+\dot{z}^{2}}</math>

This results in a differential equation for the shape consistent to the algebraic equation for the drop thickness: <math>\sigma _{lv}\left( \sqrt{1+\dot{z}^{2}}-\cos \theta _{e} \right)=\frac{1}{2}\rho g\left( 2ez-z^{2} \right)</math>

Menisci against a wall

The ascending meniscus against a vertical wall is shown below. (de Gennes, 2004, pp. 45f). The Laplace equation, shown on the diagram, is a differential equation that describes the shape of the meniscus. The curvature increases with height.

Ascending meniscus 1.png
<math>\frac{1}{R\left( z \right)}=-\frac{{\ddot{z}}}{\left[ 1+\dot{z}^{2} \right]^{3/2}}</math>

Substituting the curvature into the Laplace equation and integrating twice gives: <math>x-x_{0}=\kappa ^{-1}\cosh ^{-1}\left( \frac{2\kappa ^{-1}}{z} \right)-2\kappa ^{-1}\left( 1-\frac{z^{2}}{4\kappa ^{-2}} \right)^{1/2}</math>

(Where x0 makes z = h at x = 0) A correct, but not illuminating, result.

de Gennes provides a more illuminating derivation by considering the equilibrium of forces.

Pressure in a vertical meniscus.png
Along the vertical dotted line, the pressure varies as: <math>p\left( z \right)=p_{0}-\rho gz</math>

This produces a total horizontal force on the line: <math>\tilde{p}=\int\limits_{0}^{z}{\rho gzdz}=\frac{1}{2}\rho gz^{2}</math>

The balance is of hydrostatic force, the horizontal component of surface tension, and the surface tension of the liquid surface (z = 0) gives: <math>\frac{1}{2}\rho gz^{2}+\sigma \sin \theta =\sigma </math>

Evalutating at z = h where the angle is the contact angle and re-arranging gives: <math>h=\sqrt{2}\kappa ^{-1}\left( 1-\sin \theta _{E} \right)^{{1}/{2}\;}</math> Meniscus height << capillary rise <math>h\left( \theta _{E}=0 \right)=\sqrt{2}\kappa ^{-1}</math>

Meniscus on a fiber

de Gennes, 2004, Fig. 2.14

As usual, the meniscus obeys: <math>p_{0}+\sigma \left( \frac{1}{R_{1}}+\frac{1}{R_{2}} \right)=p_{0}-\rho gz</math>

Since the meniscus height is small, the hydrostatic term is small and the film has no curvature! <math>\left( \frac{1}{R_{1}}+\frac{1}{R_{2}} \right)=0</math>

Assuming <math>\theta =0</math> for simplicity, the profile is a catenary curve:

for simplicity, the profile is a catenary curve: <math>r\left( z \right)=b\cosh \left( \frac{z-h}{b} \right)</math>

Dropping the hydrostatic term left this equation with the wrong limit: <math>r\left( 0 \right)=\kappa ^{-1}</math>,

Assuming that the meniscus is lost at the capillary length: <math>\kappa ^{-1}=b\cosh \left( \frac{-h}{b} \right)</math>

Hence: <math>h\approx b\ln \left( \frac{2\kappa ^{-1}}{b} \right)</math>