Lecture 10: Capillary Forces
In the last lecture…
Surface and interfacial energy
Surface tension
Calculating Surface energy of materials
Adhesion energy of simple liquids
Wetting interactions
Contact angles
LV  4N A 



8 N A  3M 
A

24Do2
2
3
In this lecture…
1) Capillary pressure in liquids
2) Capillary rise
3) Synthesis of nanostructures
Further reading
Gases, Liquids and Solids and other states of
matter., 3rd Ed. D. Tabor, Cambridge University Press,
p280-290, 1993
Intermolecular and surface forces, J. Israelachvili,
p330-334
Pressure difference across a liquid
vapour interface
In the last lecture we
considered the energy
associated with creating flat
surfaces. What happens if a
surface between two liquids
becomes curved?
Is there a pressure
difference across the
interface?
Capillary pressure
There is a pressure difference
across a curved liquid interface
which acts to try to reduce the
area of the interface. This is
called the Capillary Pressure
1
1 
P     
 R1 R2 
 is surface energy/tension (Jm-2)
R1 and R2 are principal radii of
curvature in two orthogonal
directions
Problem
Calculate the capillary pressure inside
a spherical air bubble of radius 10mm
in water =72 mJm-2
Capillary Rise
When a fine capillary is
placed inside a liquid, a
curved liquid meniscus
forms.
The resulting pressure
drop across the interface
causes the fluid to be drawn
up inside the capillary
This is referred to as
capillary rise
See OHP
P2 > P1 (= Patmos)
Pressure difference between
surface of reservoir and P2
forces fluid up the column
P1
P2
h
Height of fluid in a capillary
The height of fluid (h) in a capillary of radius, R, is determined
by balancing the capillary pressure drop across the
meniscus with the pressure change required to draw the
fluid up the capillary (See OHP)
For a fluid of density, , and
surface tension, 
h
2
h
gR
Problem
An anodised aluminum oxide membrane contains
a regular array of pores with a radius of 500 nm
which penetrate throughout the entire thickness of
the membrane. The membrane is placed on top of
a molten polymer liquid which has a surface
energy of =50mJm-2 and density of 1100kgm-3 in
such a way that the pores fill under capillary
action
Obtain an estimate for the maximum thickness of
membrane which can be filled in this way
Filling of Nanopores
The phenomenon of capillary rise can be used to
manufacture nanoscale structures.
Porous membranes are placed on the surface of liquids and
allowed to fill under capillary action.
Porous AlOx membranes
500nm
Xiang et al,
Macromolecules,
37(15), p5660 2004
Summary of key concepts
Capillary pressure due to a
curved liquid interface
1
1 
P     
 R1 R2 
Capillary pressure is
2
responsible for phenomenon h 
gR
of capillary rise
This method can be used to
make some interesting
nanoscale structures
h