# Drops, menisci, and lenses

## Contents

## Thickness of a large drop

Small drops are spherical segments. Large drops are flattened.

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.

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>

## A brief discription of menisci

**Meniscus**, plural:

*menisci*, from the Greek for "crescent", is a curve in the surface of a liquid and is produced in response to the surface of the container or another object. It can be either concave or convex. A convex meniscus occurs when the molecules have a stronger attraction to each other than to the container. This may be seen between mercury and glass in barometers. Conversely, a concave meniscus occurs when the molecules of the liquid attract those of the container. This can be seen between water and glass. Capillary action acts on concave menisci to pull the liquid up, and on convex menisci to pull the liquid down. This phenomenon is important in transpirational pull in plants. Honey, water, milk all have a lower meniscus. When a tube of a narrow bore, often called a capillary tube, is dipped into a liquid and the liquid “wets” the tube (with zero contact angle), the liquid surface inside the tube forms a concave meniscus, which is a virtually spherical surface having the same radius,

*r*, as the inside of the tube. The tube experiences a downward force.

When reading a scale on the side of a container filled with liquid, the meniscus must be taken into account in order to obtain a precise measurement. Manufacturers take the meniscus into account and calibrate their measurement marks relative to the resulting meniscus. The measurement is taken with the meniscus at eye level to eliminate error, and at the central point of the curve of the meniscus, i.e. the top of the meniscus, in the unusual case of a liquid like mercury, or more usually, the bottom of the meniscus in water and most other liquids.

## 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.

<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.

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

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: <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>

## Meniscus wetting - AFM tips

The end of an atomic force microscope probe has to be of molecular dimensions for best sensitivity:

They are made by immersion in an etching solution:

The mechanism that produces the sharp tip can be explained by the variation in meniscus height as the etched dip becomes smaller and smaller by considering the meniscus height as a function of radius: <math>h\approx b\ln \left( \frac{2\kappa ^{-1}}{b} \right)</math>

Typical capillary lengths are mm’s, so the menisci are also few mm’s. The meniscus height is a few times the fiber radius or a few 10’s mm. Therefore the meniscus is much wider than high.

During the etching of an AFM tip, as the end gets narrower, the meniscus drops, and so on, producing a fine point at the very last.