surface tension

1. 98
Fdx= gdA = gLdx
Surface Energy
• It is an experimental observation that
liquids tend to draw up into spherical
drops.
• A sphere is the geometric form which
has the smallest surface area for a
given volume.
• Thus it is clear that the surface of the
liquid must have a higher energy than
the bulk.
• This energy is known as the free
surface energy g, with units of Jm-2
and typical values of 30-100 mJm-2.
• Sometimes the unit is given
equivalently as Nm-1, particularly
when quoted as a surface tension.
• The free surface energy is equivalent
to a line tension acting in all
directions parallel to the surface.
• We can use a virtual work argument
to show this for a force F acting on
an area dA and moving through a
distance dx:
• Thus g = F/L
• And the surface energy is equivalent
to a line tension per unit length ie a
surface tension.
F
L
dx

2. 99
Why is there a Surface Tension?
• At the simplest level, we can ascribe the existence of surface tension to the
reduction in bonds for molecules at the liquid surface.
• Formally it is the additional free energy per unit area required to remove
molecules from the bulk to create the surface.
• Denoted by
i
i
i n
P
T
n
V
T
n
V
S A
G
A
F
A
U
,
,
,
,
,
,








=








=








=
g

3. 100
Continuum Approach
• Assuming a Van der Waals interaction, then the
force can be worked out by summing pairwise
interactions between the two surfaces
where A is the Hamaker constant, typically
~10-20J (See QS for proof of this).
• Most easily a could be taken as half the
average intermolecular distance, but this leads
to values systematically too small: this picture
has only dealt with a static case, and more
sophisticated analysis is required to get
agreement.
h
• F(h) is the force/unit area between
the two halves of the liquid


=
a
dh
h
F )
(
2
1
g
3
6
)
(
h
A
h
F

=
2
24 a
A

g =
• Imagine cutting a volume
of liquid and pulling it apart
a is some cut-off distance.

4. 101
Bubbles and Droplets
• Therefore
• And the pressure is higher inside the
drop.
• For a bubble (as in a soap bubble) which
is an air-filled film, there are two
surfaces and
• For a more general (non-spherical) drop,
with two principal radii of curvature R1
and R2
• Imagine expanding a droplet from
radius R to R+dR, with a corresponding
increase in surface area DA
• Work done is Dp4R2dR, where Dp is
the difference in internal and external
pressure, ie is the pressure driving the
expansion.
• This must balance the work done in
expanding the interface gDA=g8RdR
R+dR
R
R
p
g
2
=
D
R
p
g
4
=
D









=
D
2
1
1
1
R
R
p g

5. 102
Pierre-Simon Laplace
1749-1827
• The Laplace disjoining pressure
is one of the less familiar contributions of this French
scientist.
• You will previously have come across
–Laplace’s equation
–Laplace transforms
• He was particularly interested in ‘Celestial Mechanics’
• And he had to survive the French Revolution and
Napoleon!
• He briefly (for 6 weeks!) served as Minister for the
Interior, but was deemed by Napoleon a ‘mediocre
administrator’ despite his scientific fame.









=
D
2
1
1
1
R
R
p g
0
2
=
 V

6. 103
Measuring Surface Tension I
• Capillary Rise Experiments
• As the liquid rises up the tube, wetting
the side of the tube under the action of
hydrostatic pressure, we must have a
balance of forces at equilibrium arising
from the pressure difference across the
meniscus and the drop in atmospheric
pressure over capillary rise distance h.
• For a capillary of uniform radius R,
then R1=R2 and this can be rearranged
to give
• r is the density of the fluid
• h is the meniscus rise.
• This method works well for low
viscosity, simple liquids.
• It assumes 'complete wetting' as we will
see later (so the effect of contact angle
is ignored).
gh
R
R
r
g =









2
1
1
1
h
2
ghR
r
g =

7. 104
Interfacial Tension between Two Liquids
• There will also be an interfacial
tension between two liquids in
contact.
• The more different the liquids (e.g
in polarity) the larger this will be.
• This is the driving force for bulk
phase separation in immiscible
fluids such as oil and water.
Total surface surface energy low
energy high due as interfacial area
to large surface low
area
• If liquids are to be made more
miscible, adding a molecule which
sits at the interface and reduces the
interfacial tension will be
necessary.
• In the case of oil/water mixtures
such a molecule will typically
consist of two parts, one of which
is hydrophobic and the other
hydrophilic. i.e we are talking
about amphiphiles or surfactants.
• Such molecules are said to be
interfacially active.

8. 105
Surfactants
• Recall these have a polar head and a
hydrophobic tail, and are similar to
the lipids which turn up in cell
membranes.
• Preferentially they will adsorb to an
air-water (or water-oil) interface.
• Let S-1 be the monolayer surface
density (S is the area per polar
head).
• The molecules can be compressed at
the surface of a Langmuir trough.
• Surface pressure P to compress a
monolayer given by
• Piston is in equilibrium when
• This yields the form of the equilibrium
curve relating surface area S to applied
surface pressure P:
T
F






S


-
=
S
P )
(
P
-
=
S o
g
g )
(

9. 106
Reduction in Surface Tension due to Surfactant
PG de Gennes, F Brochard-Wyart and D Quéré
• Chemical potential of neutral surfactant at the
air-water surface where concentration is not
dilute (F(S) is free energy/ molecule in the
bulk):
• And (at constant T)
• Therefore
where mo
surf is the free energy of a single
surfactant molecule at the surface.
• In equilibrium, the chemical potentials for the
bulk and surface are equal at the interface .
• Thus if c is bulk surfactant concentration,
we can obtain a relation between S, c and g
(mo
w is the standard chemical potential of
bulk pure water and is > mo
surf since the
surfactant is not completely immersed) .
• In equilibrium (for a dilute solution of a
non-ionic surfactant)
• This permits the area S of the molecule to
be determined.
)
(
.
)
( S
P
S

S
= F
surf
m
P
S
= d
d surf .
m

P
P
S

=
0
.d
o
surf
surf m
m
T
B
B
B
surf
B
o
W
vol
c
T
k
or
d
d
c
dc
T
k
or
d
c
T
k
c
T
k








-
=
S
S
S
-
=
P
S

=
P
S

D
=

=

=

P
ln
1
1
)
(
.
.
.
ln
ln
0
g
g
m
m
m
m

10. 107
Concentration Dependence of Surface Tension
• Typically there is a substantial
reduction in surface tension up to some
plateau
• The surface tension decreases to the
point where there is an energetically
more favourable arrangement of
molecules.
• This is when micelles start to form.
• As we have seen, this occurs when
msurf >mmicelle.and we reach the cmc
(recall section on self-assembly).
• Thereafter more micelles form in
preference to more surface
adsorption.
• The concentration at which this
occurs is known as the critical
micelle concentration (cmc).
• The micelles themselves may
migrate to the interface.
• In principle for oil-water interfaces
(as opposed to free surfaces) it is
possible for the interfacial tension
to fall to zero.

11. 108
Foams
• Surfactants play a key role in
stabilising foams, such as beer
froth or whipped cream.
• They sit at the surface of the liquid
film between the gas bubbles and
lower the surface energy there.
• They also slow the drainage and
rupture of the liquid film.
• This allows the foam to survive for
longer.
• The films between the bubbles are
known as Plateau Borders.
• In many everyday products, getting
the drainage right is hugely
important for its function – and this
comes back to surfactants.
Joseph Plateau 1801-83
who did most of his
work on soap when he
was blind.

12. 109
Contact Angles
• When a liquid is deposited on a
surface, it may not form a
continuous film – wetting – but
instead may break up into droplets.
• The shape of the droplets is
defined by the relative surface
energies.
• This is equally true for a solid drop
forming from its melt on a surface.
• S denotes Solid, V vapour and L liquid
respectively
• Balancing forces (recall surface energy 
surface tension/unit length) at the contact
line, where the solid, liquid and vapour
phases meet
• (gSV denotes surface energy between
substrate and vapour etc)
• This is known as Young's equation.
• Condition for complete wetting is that there
is no real solution for q.
gLV
q
g
gSL
SV
Liquid
Saturated
Vapour
Solid
q
g
g
g cos
LV
SL
SV 
=

13. 110
Thomas Young (1773-1829)
• His epitaph states
"...a man alike eminent in almost every
department of human learning.“
• Through his medical practice he got interested in
the human eye.
• This led him to study optics, and led to his
famous Young’s slits experiments.
• Discovered the cause of astigmatism.
• Postulated how the receptors in the eye perceive
colour.
• And he also managed to find time to give a
formal meaning to energy: he assigned the term
energy to the quantity mv2 and defined work done
as (force x distance), proportional to energy.
• Plus he derived the equation for surface tension,
and defined Young’s modulus!

14. 111
Wetting and Spreading
• The spreading parameter S is given
by
• This coefficient determines
whether a droplet forms, or the
surface is completely wetted.
• It is a measure of the difference in
surface energy between the
substrate dry and wet.
• If S>0, the liquid spreads
completely to cover the surface and
lower its surface energy: q is zero.
• If S<0, partial wetting is said to
occur, with a finite contact angle.
• If the contact angle is 180o
, the
liquid forms a complete sphere and
the liquid is non-wetting.
• In general, liquids will spread on
highly polarizable substrates such
as metal and glass.
• They may or may not on plastics –
if the liquid is less polarizable than
the substrate it will.
)
( LV
SL
SV
S g
g
g 
-
=
Wetting layer
Non-wetting

15. 112
Measuring Surface Tension II
• Measuring the contact angle is
obviously a good way of
determining the surface energy of a
liquid on a particular substrate, if
the other surface energy terms are
known.
• A goniometer can be used to
measure the angle accurately, or
photographs taken on which
measurements are made.
• However, there are experimental
difficulties to take into account.
• Roughness: If the surface is rough,
then the local contact angle and the
macroscopic (measured) contact
angle will differ. This is a very
hard problem to deal with, both
theoretically and experimentally.
• If the droplet grows, the advancing
and receding angles will differ due
to hysteresis effects. These are also
not well understood. In general the
advancing angle is used, although
sometimes both are quoted.
• Surface cleanliness is a major
issue. Finger grease, for instance,
can completely change the
measurements.
Molten polymer on substrate
q

16. 113
Capillarity with Finite Contact Angle
• If the meniscus has the same radius
R for each of its radii of curvature,
then
• This equation implies either that we
can determine the surface energy, if
we measure q, but it is also a
convenient geometry with which to
measure q.
• If the contact angle is finite, then
we must modify our earlier
analysis.
h
q
Contact
angle
R
g
h
q
g
r
cos
2
=

17. 114
• As we can see from the previous
image, accuracy using optical
microscopes can be limited due to
refraction at the interface.
• Using Environmental SEM offers a
potentially more accurate route.
• Can look at the effect of different
surfaces too
• And in principle could watch
advancing and receding droplets,
and measure hysteretic effects
Experimental Challenges
Optical
ESEM
Water droplets on a cellulose textile
fibre.
Droplets adopt a so-called unduloidal
shape on cylindrical fibres.
A smaller contact angle is seen
for water droplets on the polar
glass substrate (lower)
compared with the polystyrene
(upper) surface.

18. 115
Further Optical Techniques – using
Interference to probe ‘Droplet’ Shape
• Similar approaches
have been developed
to study cell adhesion.
• Here a model
phospholipid vesicle is
examined.
• Thermal fluctuations
mean the shape is far
from spherical.
• Interference fringes can
be used to monitor drop
shape/thickness, and the
way in which the droplet
spreads.
• The dynamics of
spreading can thus be
followed in real time

19. 116
Hydrophobic and Hydrophilic Surfaces
Hydrophobic surfaces
• Leaves, duck feathers etc are designed so
that water rapidly forms droplets and rolls
off them
('water off a duck's back').
• Aircraft are sprayed with a hydrophobic
liquid so that a continuous film of water
does not form which can transform into
solid ice during flight, substantially
increasing weight.
• Teflon frying pans and saucepans are used
to prevent most things – not just water –
sticking. Made from polytetrafluorine
ethylene (PTFE).
Hydrophilic surfaces
• Contact lenses must be made out of
materials that favour wetting, and
prevent the lens adhering to the
cornea.
• In many industrial processes such
as paper coating, wetting must be
achieved very fast to cope with the
speed of the process (m's per
second).
• Likewise with adhesives need a
continuous film to form to give
good adhesive strength.
Hydrophobic surfaces are obviously ones that repel water whereas hydrophilic ones
are covered with a wetting layer.
Different applications have different requirements.

20. 117
How does Soap Work?
• Soap – sodium and potassium salts of fatty
acids traditionally – have long been used.
• However because they react with Ca2+ and
Mg2+ ions to form scum, modern detergents
use different chemistry, but follow the same
physical principles.
• The molecule must wet the substrate (fabric
etc) so that it comes into contact with the
surface and the dirt.
• If the contaminant is an oily fluid, the
molecule must reduce the surface angle.
• It removes the oil by a 'rollup' mechanism.
• The dirt is then solubilised, and can then be
removed mechanically.

21. 118
Colloids
• Colloids are systems in which one of the systems (at least) has dimensions
of ~1mm or less.
• Thus many aspects of nanotechnology are essentially colloidal.
• Examples:
Solid in liquid such as Indian Ink or sunscreen
Suspension
Liquid in Liquid such as mayonnaise or salad dressing
Emulsion
Gas in Liquid such as beer or soap foam
Foam
Gas in Solid such as bath sponge or ice cream
Sponge

22. 119
Stability of Colloids
• Colloids of necessity have a lot of
surface area, since the inclusions are so
small.
• We would therefore expect a move
towards aggregation/complete phase
separation to minimise surface energy.
• Why doesn't this happen?
• It does if things don't go right!
• The trick is to introduce long range
repulsions to overcome the short range
van der Waals attraction.
• Van der Waal's attraction  1/r6
where r is separation.
Johannes van der Waals
1837-1923
Nobel Prize in Physics
1910
• It arises from interactions between
dipoles in the two surfaces.
• Energy U due to dipole moment p in
local field EL is
• Where
And p is the induced dipole due to the field
EL so that p  EL so that
3
4
2
r
o

p
EL =
6
2
1
.
r
U
U



 L
L E
E
p
L
E
p.
2
1

23. 120
Interactions between Colloidal Particles
(Recap from 1st section)
• Van der Waals force (also known as
the dispersion force) leads to a long
range attraction.
• At very short distances there is a
hard core repulsion
• So why don’t colloidal particles
always stick together?
• Two main routes to prevent
aggregation in the ‘primary
minimum’, by introducing a
repulsion force.
1. By introduction of an
electrostatic repulsion.
2. By steric repulsion.
• Both these routes are of great
practical importance.
• Colloids turn up in many
situations: ink, ferrofluids, milk,
clay, blood….
• We will return to stabilisation
methods later.
Interaction
energy
distance
Hard core
Repulsion
Van der
Waals
attraction

24. 121
Interactions between Surfaces
• This form of the potential is correct
for pairwise interactions between
two dipoles.
• Between macroscopic surfaces one
must sum over all appropriate such
pairs, and hence this depends on
the geometry of the objects.
• Also, a correction needs to be made
at larger distances, since the force
between the two dipoles are said to
be 'retarded'; basically this means
they are out of phase.
• At large distances then U  1/r7
Experiments carried out at the
Cavendish by Tabor and Winterton
Configuration: crossed mica cylinders
Measure separation via interferometry.
Use springs of different stiffness.
Determine when attraction sufficient to
Cause surfaces to jump together into contact.

25. 122
Results from Surfaces Force Apparatus
• A is Hamaker's constant;
theoretical fits plotted.
• Change in slope indicates transition
from 1/r6 to 1/r7 behaviour.
• Confirms theoretical ideas.
• Subsequent modifications to the
apparatus permitted actual force
measurements to be made.
Distance at which jump
into contact occurs
Tabor and Winterton, Proc Roy Soc 1969

26. 123
• Consider a (positively) charged
surface in an aqueous, ion-containing
environment.
+ - + -
+ - - +
+ - + - +
+ - - + -
+ - + -
Double no field in bulk
Layer of solution – diffuse layer
Charged Colloids in Aqueous Environments
• So to stabilise colloids, long range
repulsive forces need to be
introduced to overcome the short
range dipolar attraction.
• This is often achieved by charge
stabilisation.
• This is not the only possible route,
there is a second common
mechanism known as steric
stabilisation, which involves
polymers (we will return to this
later).
• Counter ions line up to form a double layer.
• Co-ions are repelled.
• Thermal effects cause the double layer to be
somewhat diffuse.

27. 124
Potential and Ion Distributions near the Surface
We will construct a model to
evaluate these parameters exactly
assuming:
• Surfaces are perfectly flat with a
uniform charge s.
• In the diffuse region, charges are
point-like and obey the Boltzmann
distribution.
• The influence of the solvent is
limited to its relative permittivity ,
which is assumed constant within
the diffuse region.
• The electrolyte is assumed to be
symmetrical (e.g 1:1) with charge
number z.
Counter-ions
Co-ions
distance x
no
Ion distribution must have this form, and hence
potential must also fall away from the surface.
 o
Potential

x

28. 125
Debye-Hückel Theory
• The potential at the surface is o, and
at a distance x from surface it is (x).
• Assume the surface is positively
charged.
• Bulk ion concentration no. Then
• Hence the local charge density r is
given by
• The charge obeys Poisson's
equation (in 1D, since we are only
interested in the distance away
from the surface).
• Apply boundary conditions:
 = o at x = 0 and
 = 0 and d/dx = 0 at x = 
• This gives solution of the form
(you are not expected to be able to
prove this!)







 
=







 -
=
-

T
k
x
ze
n
x
n
T
k
x
ze
n
x
n
B
o
B
o
)
(
exp
)
(
)
(
exp
)
(










-
=









-
-
=
-
= -

T
k
ze
zen
T
k
ze
T
k
ze
zen
n
n
ze
B
o
B
B
o

r


r
r
sinh
2
exp
exp
)
(








=
-
=
T
k
ze
zen
dx
d
B
o
o
o



r

sinh
2
2
2








-
-
-

=
x
x
ze
T
kB

g

g

exp
1
exp
1
ln
2

29. 126
Debye-Hückel Theory cont
• In this expression
• And
• In the Debye-Hückel
approximation
• Valid for o  50mV
• In this case we can expand the
exponential in g to give
• Hence
• This shows the potential has an
exponential fall off with distance
and 1/ is the distance over which
the potential falls by a factor of e.
• 1/ is the screening length
 DH no and  z
1
2
exp
1
2
exp







-






=
T
k
ze
T
k
ze
B
o
B
o


g
2
/
1
2
2
2






=
T
k
z
n
e
B
o
o


1

T
k
ze
B
o

T
k
ze
B
o
4

g 
x
o 

 -
= exp

30. 127
Debye-Hückel Theory cont
For a monovalent ion, typical values of 1/
o can be related to surface charge density
so within DH approximation.
The surface potential therefore depends on
both the surface charge density and the
ionic composition of the medium.
Approximation breaks down if o too high.
Concn of monovalent ion 1/
0.1M 1nm
0.001M 10nm
o
o
o dx 

r
s ~
0


-
=
There are a series of further refinements
possible, but this is good enough to help
us understand colloidal stability.
Exact and
Debye-Hückel
approximation for
o=75mV and o =
25mV

31. 128
Charge Stabilisation of Colloids
• Consider two charged surfaces each at
potential o separated by distance h in
an electrolyte.
m
o o
• If you bring the surfaces together
exactly what happens depends on
whether interaction occurs at constant
surface charge, surface potential or
something intermediate. In practice
equilibrium may not be maintained
anyhow, and situation difficult to
analyse exactly.
• Work within DH approximation.
• As surfaces approach each other the local
concentration of ions in the gap increases.
• This gives rise to an osmotic pressure of
liquid trying to enter the gap to reduce ion
concentration.
• Osmotic pressure difference P at midpoint
• Note it is the number of ions not their
charge which matters.
• This expression can be expanded within
the DH approximation.
• The osmotic repulsion is the source of the
force which acts to overcome the van der
Waals attraction and keep the surfaces
apart.






-
-

=

=
2
exp
exp
ions
of
no
excess
T
k
ze
T
k
ze
Tn
k
T
k
B
m
B
m
o
B
B





32. 129
Charge Stabilisation of Colloids cont
• Expand the expression for .
• Now m is the sum of two separate
potentials at h/2
• So at the midpoint
• The case for spherical particles can
be similarly treated and results in a
similar form for the osmotic
pressure ie P  exp-h.
(although now the particle radius R
also appears in the expression).
• The total expression for the net
force between two spherical
particles becomes
attraction repulsion
• A is Hamaker's constant, and C is a
constant.
2
2
2
)
2
2
1
1
2
1
1
(








=
-







 -

-




















=
T
k
ze
T
k
n
T
k
ze
T
k
ze
T
k
ze
T
k
ze
T
k
n
B
m
B
o
B
m
B
m
B
m
B
m
B
o







2
/
exp
2 h
o
m 

 -
=
h
T
k
ze
T
k
n o
B
B
o 

 -








= exp
4 2
2
h
C
h
AR
F 
-
-
= exp
12 2

33. 130
Net Potential
• The net potential strongly depends on the
ionic strength and surface potential.
• If the ionic strength is too high, screening
is so effective that short range attraction
takes over and coagulation occurs.
• Otherwise particles may sit in secondary
minimum in a stable aggregate or even
crystal.
• 250nm particles in no salt (left) when
colloidal crystal forms, or high salt, when a
disordered fractal aggregate forms.
Primary
minimum Secondary minimum

34. 131
Packing and Excluded Volume
• Thus packing can be either regular or
random, depending on circumstances.
• Phase transitions can be observed in
colloids as a function of concentration,
and different structures can coexist.
• Why should random packing sometimes
be of lower free energy than crystals?
• The answer lies in the concept of
excluded volume, and is similar to the
argument for the existence of the
hydrophobic force.
• To understand this consider a hard
sphere colloid, analogous to a hard
sphere gas.
• For an ideal gas
where a is a
constant.
But if the atoms have finite volume b, the
volume accessible is reduced to V-Nb
Crudely
at low volume fractions
Thus per atom
• The finite size of the atoms gives rise to a
repulsive term- the atoms cannot overlap.
• For colloids as well there is a similar
effective excluded volume.
• The good packing in the crystal means
there is more space for the atoms/colloids to
explore thereby increasing entropy, despite the
long range order.






=
N
aV
k
S B
ideal ln
b
V
N
k
S
V
bN
k
S
N
Nb
V
a
k
S
B
ideal
B
ideal
B
-






-

=





 -
=
~
1
ln
)
(
ln
b
V
N
T
k
F
F B
ideal 






=

35. 132
Colloidal Crystals
• Sometimes you don't want the colloid
to be stabilised!
• Well-ordered colloidal crystals can
form, with the same symmetries as for
atomic crystals.
• The optical properties of colloidal
crystals form the basis for opals, in
which aggregates of silica are
dispersed in 5-10% of water.
• The local differences in packing give
rise to optical effects giving precious
opals their distinctive colours.
• Synthetic opals have much more
regular packing.
• More generally they can be used as
model systems, e.g as macroscopic
hard sphere fluids to help physicists
understand the nature of interactions.
• Polystyrene beads of diameter
~700nm

36. 133
Photonic Crystals
• Photonic crystals possess
microstructures where the refractive
index is modulated periodically in
space.
• They are a large scale analogue of
atomic crystals
Dispersion curves along different
symmetry directions.
• They can have a photonic band gap
• This means, in principle, that in all
directions at a certain wavelength the
light is stopped from propagating, and
hence is confined.
• But to achieve this need very perfect
(i.e defect free) structures.
• Colloidal crystals are thought to be one
way of achieving this, and also allow
tuning of properties by changing size
of the particles.