Surface tension

Oct 2013

Dispersed liquids in nature

Artificiallly dispersed liquids

http://www-math.mit.edu/~bush/gallery.htmlhttp://lmlm6-62.univ-lille1.fr/lml/perso/pbrunet/page_perso.html

http://www.dolomite-microfluidics.com/products/system-solutions

There is an extra energy associated with a surface

Molecules on the surface are less bounded than molecules in the bulk.

There is therefore an energy associated with a free surface.

More rigorously: there is a free energy associated with an interface.

This energy is the surface tension g

Solids have a surface tension too

More about this later

How many molecules are on the surface of a 10-nm droplet ?

The molar volume of liquid nitrogen is 34.6 cm³/mol, which converts to a molecule « diameter » of 4.8 Ǻ

Consider a drop of diameter D = 100 Ǻ

The total volume is 4/3*p*(D/2)3, and that of the surface layer is 4/3*p*[ (D/2)3 – (D/2-t)3]

A droplet of 10 nm in diameter has therfore 25 % of the molecules on the surface!

t ~ 4.8 Ǻ

D = 100 Ǻ

The surface tension can be thought of as a force

http://www.funsci.com/fun3_en/exper2/exper2.htm

F

A 

The energy per unit area is gIt can often be thought of as a force

A didactic setup(Cf. Wilhelmy plate)

http://www.funsci.com/fun3_en/exper2/exper2.htm

Values of g for some usual liquids

Substance Surface tension (mN/m)

Water (10°C) 74.2

Water (25°C) 72.0

Water (50°C) 67.9

Mercury (25°C) 485.5

Acetone (25°C) 23.5

Ethanol (25°C) 23.2

Formamide (25°C) 57.0

Nitrogen (77°K) 8.85

Values from Butt, Graf & Kappl, Physics and Chemistry of Interfaces, Wiley, 2006

A rough estimation of g from the energy of vaporization

Cyclohexane @ 25°CUevap = 30.5 kJ/mol r=773 kg/m³M=84.16 g/mol

Assuming a cubic structure (6-coordinated), Uevap converts to 5.08 kJ/mol for each bond.Each molecule on the surface has one missing bond.All is left to determine is the area of each molecule on the surface.This can be estimated as a² with NA a³ r = M, i.e. a = 0.565 nm

The final estimate is about 26 mJ/m²(the experimental value is ~ 25 mJ/m²)

Stefan’s law

From K.S. Birdi, Surface and Colloid Chemistry, Principles and Applications

2010, CRC Press

Jozef Stefan 1835-1893

There is an entropic contribution to sin addition to the energetic contribution

T

P

Tc

gas

liquid

solid

𝜎=𝜎0 (1−𝑇 /𝑇𝑐 )𝑛 with n ~ 1

The surface tension vanishes at the critical temperature

𝑑𝑈=𝑇𝑑𝑆+𝜎 𝑑𝐴𝑑𝐹=−𝑆𝑑𝑇+𝜎 𝑑𝐴with𝐹=𝑈 −𝑇𝑆

The corresponding Maxwell’s relation reads

( 𝜕𝑆𝜕 𝐴 )𝑇

=−( 𝜕𝜎𝜕𝑇 )𝐴

¿−𝑑𝜎𝑑𝑇

≥0

Surfactants

Surfactant classification according to the composition of

their head: nonionic, anionic, cationic, amphoteric.

Effect of surfactants on surface tension

Critical micellar concentration

Minimal surfaces

Costa’s minimal surface (1982)

Minimal surfaces in materials sciences

Micron, Volume 38, July 2007, Pages 478–485

Young-Laplace equation

R

Pext

Pin

g g

The mechanical equilibrium of a half-droplet requires

 

For a more complex interface, the relation is

 with R1 & R2 being the principal radii of curvature

Laplace

Nous devons donc envisager l’état présent de l’univers comme l’effet de son état antérieur et comme la cause de celui qui va suivre. Une intelligence qui, pour un instant donné, connaîtrait toutes les forces dont la nature est animée, et la situation respective des êtres qui la composent, si d’ailleurs elle était assez vaste pour soumettre ces données à l’Analyse, embrasserait dans la même formule les mouvements des plus grands corps de l’univers et ceux du plus léger atome : rien ne serait incertain pour elle et l’avenir, comme le passé serait présent à ses yeux.  (Laplace, Essai philosophique sur les probabilités, 1795)

Géomètre de première catégorie, Laplace n’a pas tardé à se montrer un administrateur plus que médiocre ; de son premier travail nous avons immédiatement compris que nous nous étions trompés. Laplace ne traitait aucune question d’un bon point de vue : il cherchait des subtilités de partout, il avait seulement des idées problématiques et enfin il portait l’esprit de l’infiniment petit jusque dans l’administration. (Napoléon Bonaparte, à propos du passage de Laplace au ministère de l’intérieur)

Pierre-Simon de Laplace(1749-1827)

Thomas Young

Thomas Young 1773-1829

• Optics: the double-slit interference experiments• Mechanics: Young’s modulus• Wetting phenomena: Young-Laplace & Young-Dupré relations• Vision and color theory: Young-Helmholtz, 3-dimensional color space• Physiology: pulse speed• Linguistics: he introduced the idea of « Indo-European languages »• Egyptology: he decyphered the hieroglyphs (before Champollion)• Music: Young temperament

What happens now?

A small and a large droplets lie on a fiber wetted by a thin liquid film.

How does the system evolve?

Laplace equation applies to solid particles as well

Lattice Contraction and Surface Stress of fcc NanocrystalsJ. Phys. Chem. B, 2001, 105 (27), pp 6275–6277

Ostwald ripening

Wilhelm Ostwald1853-1932

http://iopscience.iop.org/1367-2630/7/1/040/media/movie1_SD.mpg

New Journal of Physics Volume 7 2005D G A L Aarts et al 2005 New J. Phys. 7 40 doi:10.1088/1367-2630/7/1/040

Interfacial dynamics in demixing systems with ultralow interfacial tension

What is the pressure in a soap bubble ?

« Amusons nous sur la terre comme sur l'onde,Malheureux celui qui se fait un nom

Richesses, Honneurs, Faux éclat de ce monde Tout n’est que boules de savon. »

Taken from the Nobel lecture of P.-G. de Gennes Rev. Modern Phys. 64 (1992) 645

And in an anti-bubble?

http://www.youtube.com/watch?v=6r_8Pp9WkF0

Some menisci configurations

   

1 2

Think about the pressure difference between 1 and 2

Plateau-Rayleigh instability

Plateau-Rayleigh instabilities in materials science

Appl. Phys. Lett. 85, 5337 (2004)

Nano Lett., 2007, 7 (1), pp 183–187

Cu nanowires

PMMA in pores

Joseph Plateau

Joseph Plateau1801-1883

Doctoral thesis at the University of Liège (1829)Professor of experimental physics in Ghent University (1835)

The phenakistiscope(“phenakizein” means “to deceive”)

Plateau’s laws

John Strutt (1842-1919-3rd baron Rayleigh

Second Cavendish professor of physics at Cambridge (following Maxwell)

• Dynamic soaring (flying of birds)• Rayleigh scattering (why is the sky blue)• Rayleigh waves (acoustics and mechanics)• Rayleigh-Jeans law (blackbody radiation)• Discovery of Argon with Ramsay (Nobel Prize for physics 1904)• Etc.

Other types of capillary instabilities

Soft Matter, 2008,4, 1403-1413

Rayleigh-Taylor instabilityhttp://www.physicscentral.com/explore/

pictures/cup.cfm

Sessile drop or bubble

New J. Phys. 5 (2003) 59

Tate’s law

𝑚𝑔=2𝜋 𝑅𝜎

is a rough approximation.

More accurate measurements are obtained by analyzing the shape of droplet using Laplace’s law.Can you do that?

The spinning droplet

Optics and Lasers in EngineeringVolume 46, Issue 12, December 2008, Pages

893–899

for small values of s(mostly intefacial tensions)

Capillary forces

What is the force between two spheres?

A more accurate analysis of forces involved in g measurement

Surfactants in nature

Pulmonary surfactants

What can you tell about the pressure in a liquid from the shape of a rising bubble?

http://people.rit.edu/andpph/exhibit-bubbles.html