# Drops, menisci, and lenses

## Contents

## Thickness of a large drop

Small drops are spherical segments. Large drops are flattened. The relative size of the radius of drop, in the absence of a surface, to the capillary length determines the extent to which the drop spreads across the surface. For the case of small droplets (r << <math>\kappa</math>), only capillary forces influence the drop. These drops will be spherical in shape and form a contact angle with the surface that depends on the spreading parameter. Large droplets (r<< <math>\kappa</math>) are much more interesting. At this size, the droplet is large compared to the capillary length and gravity will play an important role. In this case (as seen in the figure below), the drop will form a pancake on the surface. The thickness of this pancake may be calculated by balancing surface forces with hydrostatic pressure.

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.

Real-Lab Example:

I actually did this sort of experiment for a microdrop experiment that I was working on at the University of Illinois, Urbana-Champaign.

The spirit of the project, done in Dr. Martin Gruebel's lab, was to find a way to characterize a single protein (a variation on GFP, green-fluorescent protein) folding and unfolding. One of the interesting parts of this GFP is that the fluorophore remains intact during folding and unfolding, but fluorescence is quenched upon entry of water into the beta barrel that the fluorophore is residing in. This makes it clear to tell when the protein has unfolded.

In order to study the folding and unfolding characteristics, Dr. Gruebele and his graduate students designed a cube lens, into which a micron size drop of protein solution would be driven and suspended in a fixed optical trap. This drop would be small enough and the original solution at a low enough concentration that there would statistically be only one GFP present in the trap. The protein would then be excited, and the fluorescence monitored. The cubic lens allows there to be very low loss given the low number of photons coming off of the individual protein. This is a very valuable way of finding and studying the halfway point of the protein in folding kinetics; you can gain a lot from looking at the pH and ionic strength conditions that cause half of the protein solution to be folded and the other half unfolded, dynamically.

Enter the microdrop generator:

As always, the goal of a part of my project in the lab was create something cheap, reproducible, and easy to make; the caveat being that it also needed to be able to make drops approximately 10 microns in diameter, which is somewhat unheard of. After a bit of indecision of which way to go, we settled on a modification of the way AFM tips are made and the way inkjet printers work: basically, I tried using HF to etch the capillary tube to a fine point and attaching a peizoelectric driver to force the droplet out of the tube. This driving effect, where the drop is pushed out and then pulled up quickly, would be a really neat force effect to study--ideas on what is acting on the drop throughout? What would change with different fluid properties?

I really wish that I had seen the de Gennes picture of this before starting my project, as it would had led to a bit more credibility to my theory that the glass really would be etched down to a point if it were just suspended in the acid...

So: the production. Taking a borosilicate capillary tube, the graduate student that I was working with drew the tube under heat to a very fine point, creating a short tube of about 400 microns in length that narrowed down from about 50 microns to a closed point over the length. I then took this tube and suspended it in an HF solution that was adjusted to give a gradual etch. To make sure that the acid didn't shoot up the fine capillary tube once the micron sized hole opened (which, I assure you, happened to me the first time I did this, and it was quite unpleasant), I applied a very slight opposing pressure using filtered nitrogen; filtered because one of the main frustrations with mircodroplet generation is clogging due to particulate build up in the nozzle.

Do it work? Sort of, although the evenness that is shown in the de Gennes picture is not exactly what I got under the microscope. I feel like this could be due to a number of things, most likely glass weakening near the pulled tip and micro-cracks from rapid cooling that could have been infiltrated by the HF. The etching was progressing well, though I was quite freaked out by the bone dissolving power of HF, and a little wary of working with it. The graduate student continuing the project was hoping to test out the drop size from the tips created as I was leaving in May this year.

This book: http://books.google.com/books?id=JKZ8MHbAlEgC&printsec=frontcover has a great bit of information on the physical limitations of making micron sized drops, and a lot on the tricky soft matter aspects of it, including what to do about Newtonian, Shear Thinning, and Shear Thickening fluids, which I found quite interesting. --BPappas 16:35, 14 October 2008 (UTC)

## Another Example: Liquid Optics

In the October 2008 issue of *Nature Photonics,* Carlos Lopez and Amir Hirsa describe a novel type of lens that uses drops of liquid water to focus light. This has been accomplished before using electro-wetting, but these researchers used sound waves to oscillate the shape of the droplet. The water was confined by a millimeter-sized cylinder drilled inside a Teflon plate. The cylinder was overfilled so that a droplet bulged outwards from the opening, but did not spread along the hydrophobic Telfon. Unlike some of the cases considered above, surface tension dominated over gravitational forces, so that the droplets were optically-smooth, spherical segments. By vibrating the Teflon plate, the researchers could cause the bubble to resonate and change its focal plane between 4 and 22 mm away from the Teflon surface.

Why is this useful? Traditionally, it has been relatively difficult to auto-focus a camera, which leads to a delay before recording an image. However, using a novel algorithm, the researchers are able to find an image that is sharply in focus in less than the time it takes the bubble to make a single oscillation (10 ms) Moreover, by continuously varying the focal plane, it may be possible to enable three-dimensional imaging.

For a nice summary of the work, see "Liquid optics: Oscillating lenses focus fast" by Claudiu Stan in *Nature Photonics* 2, 595 - 596 (2008).