# Difference between revisions of "Drops, menisci, and lenses"

## 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:$\sigma _{sv}-\left( \sigma _{lv}+\sigma _{sl} \right)$

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

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

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

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

## 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: $\tilde{P}=\int\limits_{0}^{z}{\rho g\left( e-\tilde{z} \right)d\tilde{z}}=\rho g\left( ez-\frac{z^{2}}{2} \right)$

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

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

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

## Menisci shapes

The ascending meniscus against a vertical wall (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.

$\frac{1}{R\left( z \right)}=-\frac{{\ddot{z}}}{\left[ 1+\dot{z}^{2} \right]^{3/2}}$

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

(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.

Along the vertical dotted line, the pressure varies as:$p\left( z \right)=p_{0}-\rho gz$

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

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

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}\;}[/itex] Meniscus height << capillary rise $h\left( \theta _{E}=0 \right)=\sqrt{2}\kappa ^{-1}$