1. DLVO theory: General
DLVO (Derjaguin, Landau, Verwey, Overbeek)
♦ Electric Double Layer begins to interfere
- electrostatic repulsion becomes significant
♦ Van der Waals Attraction
In order to agglomerate, two particles on a collision
course must have sufficient kinetic energy due to
their velocity and mass to “jump over” the energy
barrier
Steric Stabilization
- adsorption of polymer on particle surface prevent
the particles from coming close enough for van der
Waals attraction to cause flocculation
For flocculation
- mechanical bridging by long chain polymer
enables flocculation in spite of the electrostatic
forces that would normally make them repel each
other.
http://www.zeta-meter.com/
Advanced Electronic Ceramics I (2004)
DLVO theory: General
Potential Energy Curve
Φtot = ΦR + ΦA
Repulsion Born repulsion for atoms
Double layer for colloids
Attraction x-6 Van Der Waals for atoms
D-2 for plates
R/D for spheres
Intermolecular force
1) Strong Bonding ionic bonding
covalent bonding
metallic bonding
2) Weak Bonding Van Der Waals Bonding
1. Debye (permanent dipole-induced dipole)
2. Keesom (permanent dipole-permanent dipole)
3. London (induced dipole-induced dipole)
Advanced Electronic Ceramics I (2004)

2. Van Der Waals Bonding for atoms
1. Debye (permanent dipole-induced dipole)
α1, α2: polarizability
µ1: permanent dipole
2. Keesom (permanent dipole-permanent dipole) moment
µ2: induced dipole moment
x: distance from dipole
3. London (induced dipole-induced dipole)
ΦVDWA = -βx-6 β: various interaction parameters(Jm6)
Advanced Electronic Ceramics I (2004)
Van Der Waals Attraction for plates
As δ → ∞
δ δ
D
ΦR = [64nokTγo2/(κ)] exp (-κD)
where γo = [exp(Zeϕo/2kT)-1]/[exp(Zeϕo/2kT)+1]
(assumption D >> κ-1)
ρNA/M : number of molecule per cubic centimeter
(M= molecular weight)
A: Hamaker constant (energy unit): Typical range : 10-20 ~ 10-19 J
- a materials constant that depends on the dielectric properties of two
materials and the intervening medium
Advanced Electronic Ceramics I (2004)

3. Van Der Waals Attraction for spheres
As R >> s
2R 2R
s
ΦR = [64πRnokTγo2/(κ2)] exp (-κs)
where γo = [exp(Zeϕo/2kT)-1]/[exp(Zeϕo/2kT)+1]
assumption: D>> κ-1
Advanced Electronic Ceramics I (2004)
Van der Waals Attraction and Surface Tension
- The difficulties of calculating β due to the lack of the information
about the polarizability, permanent dipole orientation, chemical
homogeneity of the surface
- Evaluation of Hamaker constant via surface tension
L
WLL
WLL: work of cohesion
L
L
Advanced Electronic Ceramics I (2004)

4. Van der Waals Attraction and Surface Tension
WLL = 2γL = ΦD=∞ - ΦD=do Do: intermolecular spacing
γd: dispersion component of surface
2γL = A/(12πdo 2)
A=24πγLdo2 tension
when additional interaction besides London forces operates between
the molecule
A = 4πγddo2/(1.2)
The estimation of Hamaker constant via the direct measurement of
VDW forces as a function of separation using the displacement of
sensitive spring and also from capacitance type measurement
is not easy due to the external vibration and surface roughness
Advanced Electronic Ceramics I (2004)
Hamaker constant 1
When the materials interact across a
liquid, their Hamaker constants
decreases but remains high.
Advanced Electronic Ceramics I (2004)

5. Hamaker constant 2
Flocculation
occur
+ +
2 1 2 1 2 2 1 1
particle solvent
Change in potential energy in above reaction
∆Φ = Φ11 + Φ22 -2Φ12
ΦA∝ A
A212 = A11 + A22 -2A12 (A12 = (A11A22)1/2 : geometric mixing rule)
∴ A212 = (A111/2 - A221/2)2
1. Effective Hamaker constant A212 always >0
( identical particles exert a net attraction due to van der Waals forces
in a medium as well as under vacuum)
2. Embedding a particles in a medium generally diminishes the VDWA.
3. No interaction at A11=A22
- can be used to evaluate the A11 and A22
Advanced Electronic Ceramics I (2004)
Repulsive and Attractive Potentials
Both mode of interaction become weaker
as the separation becomes larger.
At sufficiently large spacing the particles
exert no influence each other.
For spherical particle
r
2R 2R
s
Advanced Electronic Ceramics I (2004)

6. Repulsive and Attractive Potentials
Kinetic of
flocculation offer
some clues as to the
height of the
maximum
Metastable:
possessing a degree
of kinetic stability
eventhough it lacks
thermodynamic
stability
Advanced Electronic Ceramics I (2004)
DLVO: Hamaker constant
For plates
Φtot = [64nokTγo2/(κ)] exp (-κd) -A/(12πd2)
where γo = [exp(Zeϕo/2kT)-1]/[exp(Zeϕo/2kT)+1]
(assumption D >> κ-1)
A212↑ → VDWA [-A/(12πd2)]↓
Advanced Electronic Ceramics I (2004)

7. DLVO: ϕo
For plates
Φtot = [64nokTγo2/(κ)] exp (-κD) -A/(12πd2)
where γo = [exp(Zeϕo/2kT)-1]/[exp(Zeϕo/2kT)+1]
(assumption D >> κ-1)
ϕo ↑ → γo ≈ 1 → ΦR ↑
sensitivity of the ΦR to the ϕo values
decreases as ϕo values increases.
For some system, ϕo values is adjustable
by varying the concentration of potential
determining ions. (remember Nernst
equation)
Advanced Electronic Ceramics I (2004)
DLVO: κ
For spheres
Φtot = [64πRnokTγo2/(κ2)] exp (-κs)
-AR/(12s)
where γo = [exp(Zeϕo/2kT)-1]
[exp(Zeϕo/2kT)+1]
assumption: D>> κ-1
κ ↑ → ΦR ↓
Advanced Electronic Ceramics I (2004)

8. DLVO and CFC
Advanced Electronic Ceramics I (2004)
DLVO : summary
Φtot = [64πRnokTγo2/(κ2)] exp (-κs) -AR/(12s)
For spheres:
Φtot = [64nokTγo2/(κ)] exp (-κD) -A/(12πd2)
For plates:
where γo = [exp(Zeϕo/2kT)-1]/[exp(Zeϕo/2kT)+1]
(assumption D >> κ-1)
1. The higher the potential at the surface of particle(ϕo) - and therefore
throughout the double layer - the larger repulsion(ΦR) between the
particles will be.
2. The lower concentration of indifferent electrolyte, the longer is the
distance from the surface before the repulsion drops significantly.(κ)
3. The larger Hamaker constant(A), the larger is the attraction between
macroscopic bodies.(ΦA)
Advanced Electronic Ceramics I (2004)