Lecture 11
Colloidal interactions I
In the last lecture…
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
Capillary action can be used to
manufacture novel nanoscale
structures
h
In this lecture…
1)What are Colloids and why are they important?
2) Colloidal stability
3) Interactions between charged surfaces in
solutions
4) Osmotic pressure between two surfaces
Further reading
Intermolecular and Surface Forces
J. Israelachvili, Chapter 12 p213-259
Colloidal Dispersions: suspensions,
emulsions and foams,
I. D. Morrison and S. Ross, Wiley p316-379
Colloids and nanoparticles
Colloids are mixtures
where small particles (<1
mm in diameter) of one
substance are suspended
in a second medium or
continous phase
Continuous phase (e.g. water)
They are not solutions!
The material is not
completely dispersed, but
exists as discrete particles
Colloidal particle
Why are colloids important?
Colloidal interactions govern the properties of nanoscale
systems in liquid environments
Gold colloids
Food
Science
Ferrofluids
Quantum dots (CdSe)
Biomolecules
Key challenge in manufacturing
colloidal dispersions
When materials are dispersed in
particulate form they have a very
large surface area.
Their natural tendency is to stick
together or flocculate to form an
aggregate (more on aggregation
in a later lecture)
AR
Fsphere/ sphere  
3
12D
We have already seen that
dispersion forces result in a
mutual attraction between
particles which can drive the
flocculation of even very
dilute systems
How do we stop things sticking
together?
We need to introduce a short range repulsion force to keep
particles apart.
In water, this is often done by decorating the surface of the
particles with charged groups and adding a small amount of
salt
add water
add salt
But the reason why the particles repel is not
purely electrostatic …
Charged surfaces in electrolytes
The added salt is often referred to
as an electrolyte
In practice the electrolyte
concentration is much higher than
the concentration of ions due to
dissociated groups
Some of the charged ions from
the salt form layers at the particle
surfaces, ensuring that charge
neutrality is maintained
So if everything is charge neutral, how do the particles
repel one another?
Repulsive forces between charged
surfaces in an electrolyte
When two charged surfaces
are brought close together the
counter-ion concentration
between the surfaces is larger
than outside this region.
This results in an osmotic
pressure, P, which pushes the
surfaces apart and tries to
restore a uniform concentration
(see OHP)
P  (n  n  2no )k BT
Where n+ and n- are number densities of
positive and negative ions in gap
How do we calculate n+ and n-?
The number of counter ions of each charge is determined
by Boltzmann statistics
 E 

n  no exp  
 k BT 
Where no is the
number density in
bulk solution
The energies of the charged counter ions with valence z
are given by
E   zeV (x)
Where V(x) is the electrostatic
potential at a position x in the
solution
Osmotic pressure between charged
surfaces
Inserting the previous result into our expression for the
osmotic pressure gives (see OHP)

 zeV ( x)  
  1
P  (n  n  2no )k BT  2no k BT  cosh 

k
T
B




In the limit of small surface potentials
this reduces to
2 2
zeV ( x)
 1
k BT
z e
2
V (x)
P  no
k BT
So if we know V(x) we can calculate the
osmotic pressure!
What form of V(x) should we use?
The true form of the electrical
potential between colloids is
complicated.
A good approximation to V(x) is the
form for the potential for an isolated
charged surface in an electrolyte.
To obtain this we must first solve the
Poisson equation
d 2V

 2 
dx
o
where  is the charge density (Cm-3)
and o is the permittivity of free space
Note: This equation is a differential form of Gauss’s law!
(see F31CO1 + F32ON1 notes). Derivation not examinable
Potential near a planar charged
surface I
The total charge density in the gap
between the two surfaces is the
sum of densities due to positive
and negative counterions
If counterions have a valence of ± z
then
assuming that n+ and n- obey
Boltzmann statistics, as before we
have
(see OHP)
    
    zen
    zen
  zen  n 
 E 

n  no exp  
 k BT 
Potential near a planar charged
surface II
Combining the above results gives a modified version of the
Poisson equation
 zeV ( x) 
 zeV ( x)  
d 2V zeno 
 
  exp 
 
 2 
exp

dx
o   k BT 
 k BT  
In the limit of small potentials
2
zeV ( x)
 1
k BT
2 2
this reduces to
d V 2 z e no

V ( x)
2
dx
o k BT
The Debye screening length
d 2V 2 z 2 e 2 no

V ( x)
2
dx
o k BT
Assuming that the potential
decays to zero for infinite x,
this equation has a solution
which is a simple
exponential decay
V ( x)  Vo exp( x)
Where 1/ is the Debye
screening length
 o k BT 

 
2 2 
  2no z e 
1
1/ is the distance over which electrostatic
interactions are screened in an electrolyte and
Vo is the potential at the surface
1
2
Problem 1
Two surfaces are charged to a surface
potential of 10mV and are suspended in water
(=80) which also has a monovalent salt
dissolved in it.
Calculate the Debye screening length
between the surfaces at room temperature if
the salt concentration is
a)1mM
b)100mM
Osmotic pressure between surfaces
(revisited)
Recall from a previous slide that
the osmotic pressure due to
counterions has the form
z 2e 2
2
V (x)
P  no
k BT
If we insert our exponential form
for the decay of the potential this
gives
2 2
2
o
z eV
P  no
exp( D)
k BT
where D=2x is the separation between the surfaces
Total pressure between surfaces
The total pressure between two charged surfaces in an
electrolyte is
Ptot  P  Pdispersion
where Pdispersion is the pressure due to attractive dispersion
interactions
2 2
2
o
z eV
A
Ptot  no
exp( D) 
3
k BT
6
D



osmotic pressure
attractivedispersion
forces
This is a simplified form of the Derjaguin, Landau, Verwey,
Overbeek (DLVO) theory of colloidal stability
Pressure (Force) vs separation
Long range attractive dispersion forces (negative)
Short range double layer repulsion forces (positive)
The effects of adding electrolyte
As more salt is added, electrostatic effects are more
strongly screened → eventually attractive dispersion forces
dominate and surfaces will stick together
Summary of key concepts
Colloidal stability is important in many
areas of nanotechnology
It can be achieved by charging
surfaces
The pressure exerted on charged surfaces in an electrolyte
is determined by a balance between the osmotic pressure
due to counter-ions and the dispersion pressure
z 2 e 2Vo2
A
Ptot  no
exp( D) 
3
k BT
6

D


osmotic pressure
attractivedispersion
forces