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

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(Equilibrium of a large drop)
(Shape of a large drop)
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[[Capillarity_and_wetting#Topics | Back to Topics.]]
 
[[Capillarity_and_wetting#Topics | Back to Topics.]]
  
== Shape of a large drop ==
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== Thickness of a large drop ==
 
Small drops are spherical segments.  Large drops are flattened.
 
Small drops are spherical segments.  Large drops are flattened.
 
[[Image: DeGennes_Fig_2-4.gif|thumb| 400px | center | de Gennes, 2004, Fig. 2.4]]
 
[[Image: DeGennes_Fig_2-4.gif|thumb| 400px | center | de Gennes, 2004, Fig. 2.4]]
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== Profile of a large drop ==
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The general ideas used to calculate the thickness are used again to calculate the profile. With a couple of modifications.
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[[Image: DeGennes_Fig_2-4.gif|thumb| 400px | center | de Gennes, 2004, Fig. 2.4]]
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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>
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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>
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“It can be shown” from the diagram that: <math>\cos \theta =\sqrt{1+\dot{z}^{2}}</math>
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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>

Revision as of 23:21, 27 September 2008

Back to Topics.

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>