View source for Drops, menisci, and lenses

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== Thickness of a large drop ==
Small drops are spherical segments.  Large drops are flattened.
[[Image: DeGennes_Fig_2-4.gif|thumb| 400px | center | 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>






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== 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.
[[Image: DeGennes_Fig_2-4.gif|thumb| 400px | center | 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>








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== A brief discription of Menisci ==

'''Meniscus''', plural: ''menisci'', from the [[Greek language|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 [[convex set|concave or convex]].  A convex meniscus occurs when the molecules have a stronger attraction to each other than to the container.<ref> Moore, John W.; Stanitski, Conrad L.; Jurs, Peter C.  Chemistry: The Molecular Science.  Belmont, CA: Brooks/Cole, 2005.  290.</ref>  This may be seen between [[Mercury (element)|mercury]] and [[glass]] in barometers.<ref> Moore, John W.; Stanitski, Conrad L.; Jurs, Peter C.  Chemistry: The Molecular Science.  Belmont, CA: Brooks/Cole, 2005.  290.</ref> 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 [[plant]]s.  Honey, water, milk etc. 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 of magnitude 2πrdσ.  Mercury etc. have an upper convex meniscus. 

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 [[parallax 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.
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== 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.

[[Image: Ascending_meniscus_1.png |thumb| 800px | left| ]] <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.

[[Image: Pressure_in_a_vertical_meniscus.png|thumb| 400px | left| ]] 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>






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== Meniscus on a fiber ==
[[Image: De_Gennes_Fig_2-14.gif |thumb| 400px | center | 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: <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>







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== Meniscus wetting - AFM tips ==
The end of an atomic force microscope probe has to be of molecular dimensions for best sensitivity:
[[Image: De_Gennes_Fig_2-16.gif|thumb| 400px | center | de Gennes, 2004, Fig. 2-16]]

They are made by immersion in an etching solution:
[[Image: De_Gennes_Fig_2-15.gif|thumb| 400px | center | de Gennes, 2004, Fig. 2-15]]


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.








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