Contributed by:
Surface energy, Surface tension, Continuum approach, Bubbles, and droplets, Capillarity, Colloids
1.
Surface Energy
• It is an experimental observation that • The free surface energy is equivalent
liquids tend to draw up into spherical to a line tension acting in all
drops. directions parallel to the surface.
• • We can use a virtual work argument
A sphere is the geometric form which
to show this for a force F acting on
has the smallest surface area for a an area dA and moving through a
given volume. distance dx:
• Thus it is clear that the surface of the
liquid must have a higher energy than
the bulk. L F
• This energy is known as the free
surface energy , with units of Jm-2
and typical values of 30-100 mJm-2.
dx
• Sometimes the unit is given
equivalently as Nm-1, particularly Fdx= dALdx
when quoted as a surface tension.
• Thus = F/L
• And the surface energy is equivalent
to a line tension per unit length ie a
surface tension.
1
2.
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
U F G
A S ,V ,ni A T ,V ,ni A T , P ,ni
2
3.
Continuum Approach
• Imagine cutting a volume • Assuming a Van der Waals interaction, then the
of liquid and pulling it apart force can be worked out by summing pairwise
interactions between the two surfaces
A
F ( h)
6 h3
where A is the Hamaker constant, typically ~10 -
h 20J (See QS for proof of this).
• F(h) is the force/unit area between A
the two halves of the liquid
24 a 2
•
1 Most easily a could be taken as half the average
F (h)dh intermolecular distance, but this leads to values
systematically too small: this picture has only
2 a dealt with a static case, and more sophisticated
analysis is required to get agreement.
a is some cut-off distance. 3
4.
Bubbles and Droplets
• Imagine expanding a droplet from
• Therefore
radius R to R+dR, with a corresponding
increase in surface area A 2
p
R
• And the pressure is higher inside the
R+dR drop.
R • For a bubble (as in a soap bubble) which
is an air-filled film, there are two
• Work done is p4R2dR, where p is surfaces and
the difference in internal and external 4
p
pressure, ie is the pressure driving the R
expansion.
• This must balance the work done in
• For a more general (non-spherical) drop,
expanding the interface A=8RdR
with two principal radii of curvature R1
and R2
1 1
p
R1 R2 4
5.
Pierre-Simon Laplace
1749-1827
• The Laplace disjoining pressure
1 1
p
R1 R2
is one of the less familiar contributions of this French
scientist.
• You will previously have come across
–Laplace’s equation
2V 0
–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.
5
6.
Measuring Surface Tension I
• Capillary Rise Experiments
1 1
gh
R1 R2
• For a capillary of uniform radius R,
h then R1=R2 and this can be rearranged
to give
ghR
2
• As the liquid rises up the tube, wetting • is the density of the fluid
the side of the tube under the action of • h is the meniscus rise.
hydrostatic pressure, we must have a
balance of forces at equilibrium arising • This method works well for low
from the pressure difference across the viscosity, simple liquids.
meniscus and the drop in atmospheric • It assumes 'complete wetting' as we
pressure over capillary rise distance h. will see later (so the effect of contact
angle is ignored).
6
7.
Interfacial Tension between Two Liquids
• There will also be an interfacial • If liquids are to be made more
tension between two liquids in miscible, adding a molecule which
contact. sits at the interface and reduces the
• The more different the liquids (e.g interfacial tension will be
in polarity) the larger this will be. necessary.
• This is the driving force for bulk • In the case of oil/water mixtures
phase separation in immiscible such a molecule will typically
fluids such as oil and water.
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.
Total surface surface energy low
energy high due as interfacial area
to large surface low
area 7
8.
Surfactants
• Recall these have a polar head and a • Surface pressure to compress a
hydrophobic tail, and are similar to monolayer given by
the lipids which turn up in cell
F
membranes. ()
T
• Preferentially they will adsorb to an
air-water (or water-oil) interface. • Piston is in equilibrium when
• Let be the monolayer surface
density ( is the area per polar () o
head).
• This yields the form of the equilibrium
• The molecules can be compressed at
curve relating surface area to applied
the surface of a Langmuir trough.
surface pressure :
8
9.
Reduction in Surface Tension due to Surfactant
PG de Gennes, F Brochard-Wyart and D Quéré
• Chemical potential of neutral surfactant at • Thus if c is bulk surfactant concentration,
the air-water surface where concentration is we can obtain a relation between , c and
not dilute (F() is free energy/ molecule in ow is the standard chemical potential
the bulk): of bulk pure water and is > o surf since the
surfactant is not completely immersed).
surf F () .() • In equilibrium (for a dilute solution of a
• And (at constant T) non-ionic surfactant)
d surf .d vol Wo k BT ln c surf
• Therefore k BT ln c .d
0
or
o
surf surf .d dc
k BT .d .d()
0 c
where osurf is the free energy of a single 1 1
or
surfactant molecule at the surface. k BT ln c T
• In equilibrium, the chemical potentials for
the bulk and surface are equal at the interface • This permits the area of the molecule to
. 9
be determined.
10.
Concentration Dependence of Surface Tension
• Typically there is a substantial
• As we have seen, this occurs when
reduction in surface tension up to some
surfmicelle.and we reach the cmc
plateau
(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 surface tension decreases to the • The micelles themselves may
point where there is an energetically migrate to the interface.
more favourable arrangement of • In principle for oil-water interfaces
molecules. (as opposed to free surfaces) it is
• This is when micelles start to form. possible for the interfacial tension
to fall to zero.
10
11.
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.
Joseph Plateau 1801-83
• This allows the foam to survive for
who did most of his
longer.
work on soap when he
was blind. • 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.
11
12.
Contact Angles
• S denotes Solid, V vapour and L liquid
respectively
Saturated
Vapour LV
SV
Liquid
Solid SL
• When a liquid is deposited on a • Balancing forces (recall surface energy
surface tension/unit length) at the contact
surface, it may not form a
line, where the solid, liquid and vapour
continuous film – wetting – but phases meet
instead may break up into droplets.
• The shape of the droplets is SV SL LV cos
defined by the relative surface
• (SV denotes surface energy between
energies.
substrate and vapour etc)
• This is equally true for a solid drop • This is known as Young's equation.
forming from its melt on a surface.
• Condition for complete wetting is that there
is no real solution for .
12
13.
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!
13
14.
Wetting and Spreading
• The spreading parameter S is given • If S<0, partial wetting is said to
by occur, with a finite contact angle.
S SV (SL LV ) • If the contact angle is 180o, the
liquid forms a complete sphere and
• This coefficient determines the liquid is non-wetting.
whether a droplet forms, or the
surface is completely wetted. Non-wetting
• It is a measure of the difference in
surface energy between the
substrate dry and wet.
• If S>0, the liquid spreads • In general, liquids will spread on
completely to cover the surface and highly polarizable substrates such
lower its surface energy: is zero. as metal and glass.
Wetting layer • They may or may not on plastics –
if the liquid is less polarizable than
the substrate it will.
14
15.
Measuring Surface Tension II
• Measuring the contact angle is • Roughness: If the surface is rough,
obviously a good way of then the local contact angle and the
determining the surface energy of a macroscopic (measured) contact
liquid on a particular substrate, if angle will differ. This is a very
the other surface energy terms are hard problem to deal with, both
known. theoretically and experimentally.
• A goniometer can be used to • If the droplet grows, the advancing
measure the angle accurately, or and receding angles will differ due
photographs taken on which to hysteresis effects. These are
measurements are made. also not well understood. In general
• However, there are experimental the advancing angle is used,
difficulties to take into account. although sometimes both are
quoted.
• Surface cleanliness is a major
Molten polymer on substrate issue. Finger grease, for instance,
can completely change the
measurements.
15
16.
Capillarity with Finite Contact Angle
• If the contact angle is finite, then • If the meniscus has the same radius
we must modify our earlier R for each of its radii of curvature,
analysis. then
2 cos
hg
R
h Contact
angle • This equation implies either that
we can determine the surface
energy, if we measure , but it is
also a convenient geometry with
which to measure .
16
17.
Experimental Challenges
ESEM • 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
A smaller contact angle is seen
Optical surfaces too
for water droplets on the polar
• And in principle could watch glass substrate (lower)
advancing and receding droplets, compared with the polystyrene
and measure hysteretic effects (upper) surface.
Water droplets on a cellulose textile
Droplets adopt a so-called unduloidal
17
shape on cylindrical fibres.
18.
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
• Interference fringes can mean the shape is far
be used to monitor drop from spherical.
shape/thickness, and the
way in which the droplet
spreads.
• The dynamics of
spreading can thus be
followed in real time
18
19.
Hydrophobic and Hydrophilic Surfaces
Hydrophobic surfaces are obviously ones that repel water whereas hydrophilic ones
are covered with a wetting layer.
Different applications have different requirements.
Hydrophobic surfaces Hydrophilic surfaces
• Leaves, duck feathers etc are designed so • Contact lenses must be made out of
that water rapidly forms droplets and rolls materials that favour wetting, and
off them prevent the lens adhering to the
cornea.
('water off a duck's back').
• In many industrial processes such
• Aircraft are sprayed with a hydrophobic as paper coating, wetting must be
liquid so that a continuous film of water
achieved very fast to cope with the
does not form which can transform into speed of the process (m's per
solid ice during flight, substantially second).
increasing weight.
• Likewise with adhesives need a
• Teflon frying pans and saucepans are used continuous film to form to give
to prevent most things – not just water – good adhesive strength.
sticking. Made from polytetrafluorine
ethylene (PTFE).
19
20.
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.
20
21.
Colloids
• Colloids are systems in which one of the systems (at least) has dimensions of ~1m 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
21
22.
Stability of Colloids
• Colloids of necessity have a lot of surface • It arises from interactions between
area, since the inclusions are so small. dipoles in the two surfaces.
• We would therefore expect a move
• Energy U due to dipole moment p in
towards aggregation/complete phase
separation to minimise surface energy. local field EL is 1
p. E L
• Why doesn't this happen? 2
• It does if things don't go right! • Where
• The trick is to introduce long range
repulsions to overcome the short range 2p
EL
van der Waals attraction. 4 o r 3
• Van der Waal's attraction 1/r6
where r is separation. And p is the induced dipole due to the field
EL so that p EL so that
Johannes van der Waals 2
U p.E L E L
Nobel Prize in Physics 1
U
1910 r6
22
23.
Interactions between Colloidal Particles
(Recap from 1st section)
• Two main routes to prevent
Hard core aggregation in the ‘primary
Repulsion minimum’, by introducing a
repulsion force.
distance
1. By introduction of an
Interaction energy electrostatic repulsion.
Van der
Waals 2. By steric repulsion.
• Both these routes are of great
• Van der Waals force (also known practical importance.
as the dispersion force) leads to a • Colloids turn up in many
long range attraction.
situations: ink, ferrofluids, milk,
• At very short distances there is a clay, blood….
hard core repulsion • We will return to stabilisation
• So why don’t colloidal particles methods later.
always stick together?
23
24.
Interactions between Surfaces
Experiments carried out at the
Cavendish by Tabor and Winterton
• 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 Configuration: crossed mica cylinders
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
Measure separation via interferometry.
Use springs of different stiffness.
Determine when attraction sufficient to
Cause surfaces to jump together into contact.
24
25.
Results from Surfaces Force Apparatus
Distance at which jump
into contact occurs
• 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.
Tabor and Winterton, Proc Roy Soc 1969
25
26.
Charged Colloids in Aqueous Environments
• So to stabilise colloids, long range • Consider a (positively) charged surface
repulsive forces need to be in an aqueous, ion-containing
introduced to overcome the short environment.
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 Double no field in bulk
stabilisation, which involves Layer of solution –
polymers (we will return to this diffuse layer
later).
• Counter ions line up to form a double layer.
• Co-ions are repelled.
• Thermal effects cause the double layer to be
somewhat diffuse. 26
27.
Potential and Ion Distributions near the Surface
Counter-ions We will construct a model to evaluate
these parameters exactly assuming:
distance x
no • Surfaces are perfectly flat with a
uniform charge .
Co-ions • In the diffuse region, charges are
point-like and obey the Boltzmann
distribution.
Ion distribution must have this form, and hence • The influence of the solvent is limited
potential must also fall away from the surface. to its relative permittivity , which is
assumed constant within the diffuse
o region.
• The electrolyte is assumed to be
symmetrical (e.g 1:1) with charge
Potential number z.
27
x
28.
Debye-Hückel Theory
• • The charge obeys Poisson's
The potential at the surface is o, and
equation (in 1D, since we are only
at a distance x from surface it is (x). interested in the distance away
• Assume the surface is positively from the surface).
charged.
d 2 2 zeno ze
• Bulk ion concentration no. Then 2
sinh
dx o o k BT
ze ( x)
n ( x) no exp
k BT • Apply boundary conditions:
ze ( x) o at x = 0 and
n ( x) no exp = 0 and ddx = 0 at x =
k BT
• This gives solution of the form
• Hence the local charge density is
given by 2k BT 1 exp x
ln
ze(n n ) ze 1 exp x
ze ze
zeno exp exp
k BT k BT
you are not expected to be able to
ze prove this!)
2 zeno sinh
k
B T 28
29.
Debye-Hückel Theory cont
• In this expression • In this case we can expand the
ze o exponential in to give
exp 1
2 k T
B
ze o ze o
exp 1
2k BT
4 k BT
• And 1/ 2
2e 2 no z 2 • Hence
k T
o exp x
o B
• This shows the potential has an
• exponential fall off with distance
In the Debye-Hückel
and 1/ is the distance over which
approximation
the potential falls by a factor of e.
ze o
1 • 1/ is the screening length
k BT DH no and z
• Valid for o 50mV
29
30.
Debye-Hückel Theory cont
For a monovalent ion, typical values of 1/
Exact and
Concn of monovalent ion 1/ Debye-Hückel
approximation for
0.1M 1nm o=75mV and o =
0.001M 10nm 25mV
can be related to surface charge
density o within DH approximation.
o dx ~ o o
0 There are a series of further refinements
possible, but this is good enough to help
us understand colloidal stability.
The surface potential therefore depends on
both the surface charge density and the
ionic composition of the medium.
Approximation breaks down if too high.
30
31.
Charge Stabilisation of Colloids
• Consider two charged surfaces each at • As surfaces approach each other the local
potential o separated by distance h in an concentration of ions in the gap increases.
electrolyte. • This gives rise to an osmotic pressure of
m liquid trying to enter the gap to reduce ion
o o concentration.
• Osmotic pressure difference at midpoint
k BT excess no of ions
ze m ze m
k BTno exp exp 2
k BT k BT
• If you bring the surfaces together exactly • Note it is the number of ions not their
what happens depends on whether
interaction occurs at constant surface charge which matters.
charge, surface potential or something • This expression can be expanded within the
intermediate. In practice equilibrium may DH approximation.
not be maintained anyhow, and situation • The osmotic repulsion is the source of the
difficult to analyse exactly.
force which acts to overcome the van der
• Work within DH approximation.
Waals attraction and keep the surfaces
apart. 31
32.
Charge Stabilisation of Colloids cont
• Expand the expression for . • The case for spherical particles can
be similarly treated and results in a
2
ze m 1 ze m similar form for the osmotic
no k BT (1 pressure ie exp-h.
k
B T 2 k T
B
2
(although now the particle radius R
ze m 1 ze m also appears in the expression).
1 2)
k BT 2 k BT
2 • The total expression for the net
ze m force between two spherical
no k BT
k BT particles becomes
• Now m is the sum of two separate
AR
potentials at h/2 F 2
C exp h
m 2 o exp h / 2 12h
• So at the midpoint attraction repulsion
• A is Hamaker's constant, and C is a
2 constant.
ze 2
4no k BT o exp h
k BT 32
33.
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.
Primary • Otherwise particles may sit in secondary
minimum Secondary minimum 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.
33
34.
Packing and Excluded Volume
• Thus packing can be either regular or But if the atoms have finite volume b, the
random, depending on circumstances. volume accessible is reduced to V-Nb
• Phase transitions can be observed in Crudely a (V Nb)
S k B ln
colloids as a function of concentration, N
and different structures can coexist. bN
S ideal k B ln 1
V
• Why should random packing sometimes
N
be of lower free energy than crystals? ~ S ideal k B b
V
• The answer lies in the concept of
excluded volume, and is similar to the at low volume fractions
argument for the existence of the Thus N per atom
hydrophobic force. F Fideal k BT b
V
• To understand this consider a hard
sphere colloid, analogous to a hard • The finite size of the atoms gives rise to a
sphere gas. repulsive term- the atoms cannot overlap.
• For an ideal gas • For colloids as well there is a similar
where a is a effective excluded volume.
aV • The good packing in the crystal means
Sideal k B ln constant.
N there is more space for the atoms/colloids to
explore thereby increasing entropy, despite the
long range order. 34
35.
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 35
36.
Photonic Crystals
• Photonic crystals possess • This means, in principle, that in all
microstructures where the directions at a certain wavelength
refractive index is modulated the light is stopped from
periodically in space. propagating, and hence is
• They are a large scale analogue of confined.
atomic crystals • 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.
Dispersion curves along different
symmetry directions.
• They can have a photonic band
gap
36
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