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Bentonite Erosion and Colloid Transport Mats Jansson 1

The DLVO theory

Electro-Osmotic




The Stern Layer

The Stern layer

•Ions are firmly attached(immobilized) to colloid surface

•The potential drops from thesurface potential almost linearlythrough the Stern layer




The Diffuse Double Layer

The Stern layer

The Diffuse layer

The DiffuseDouble Layer

P o t e n t i a l

Distance from Colloid

Surface Potential

Stern Layer

Zeta PotentialDiffuse Layer

Zeta Potential(Low Concentration)




The Repulsive Energy

x

ϕ

ϕ (x)

ϕ ( ∞ )=0

d

Distance between Stern layers





x

ϕ

ϕ (x)

ϕ ( ∞ )=0


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x

ϕ

ϕ (x)

ϕ ( ∞ )=0

d/2




ϕ (x)

d/2




ϕ (x)

d/2

Excess of total ion concentration at d/2=> Osmotic pressure

Δp = pd/2 - p∞= RT [c+

d/2 + c-d/2 - 2c∞]

The osmotic pressure becomes

Δp = RT 2c∞ (cosh (z ψd/2) – 1)

(ψd/2 = F ϕd/2 / (RT)

Together with the Bolzmann equation

(F = Faradays constant)

RT

Fz-

i,xi,

xi

eccϕ

∞=




ϕ (x)

d/2

Δp = RT 2c∞ (cosh (z ψd/2) – 1)

(ψd/2 = F ϕd/2 / (RT)

Cut the system in two at x=d/2. Look at the point x=d/2

As the electrical field strength is zero,the electrical force exerted on either half of the system is zero

⇒ The remaining force is due to the osmotic pressure difference(described by the equation below)

The repulsive force is electro-osmotic in nature, not electrostatic in the proper sense!




Δp = RT 2c∞ (cosh (z ψd/2) – 1)

The repulsive force can be calculated if ψd/2 is known

Evaluating ψd/2 is very difficult

Approximation ψd/2 « 1 is useful (fulfilled when d is suff. large)

ψd/2 can be obtained from superposition the potentials of the isolated plates

and approximateψd/2 = (8/z) γ0 e-κd/2

γ0 = surface charge density

ψd/2 = (8/z) γ0 e-κd/2

lengthDebye1

,

22

== ∑ ∞

κ ε κ

RT

c z F i i




Δp = RT 2c∞ (cosh (z ψd/2) – 1)

ψd/2 = (8/z) γ0 e-κd/2

Sinceψd/2 « 1

cosh (x) = 1 + x2/2

Δp = 64 RT c∞ γ02 e-κd

The electro-osmotic energy per surface area is then

ωel = - ∞∫d Δp dx

ωel

= 64 RT c∞

γ0

2 (1/κ) e-κd



The Attractive Energy

The most important interaction energy besides the electro-osmoticis the van der Waals energy (dispersion energy)

For our system the van der Waals energy per surface area is

ωvdW = - A / (12 π d2

)

A is the Hamaker constant (typically about 10-20 J)A is very difficult to determine experimentally



DLVO theory

Derjaguin-Landau-Verwey-Oberbeek

ω = 64 RT c∞ γ02 (1/κ) e-κd – A / (12π d2)

ω =ωel +ωvdW

Repulsion (electro-osmotic)

Attraction (van der Waal)



DLVO theory

ω = 64 RT c∞ γ02 (1/κ) e-κd – A / (12π d2)

Increase c∞

222

2

0 1264

22

d

A

ec z F

RT

c RT

d RT

c z F

π

ε

γ ω ε

−=

∑−

∞∞

∞

∑

=> Repulsion decreases



Repulsion c∞ varied

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 2 4 6 8 10

c=1

c=0.1

c=0.01

222

20

12

64

22

d

Ae

c z F

RT c RT

d RT

c z F

π

ε γ ω

ε −=∑

−

∞

∞

∞

∑

d ce

c

c y ∞−

∞

∞=




Secondary minimum

Secondary minimum

If the surface area of the particle is largeEnergy in secondary minimum > kT

=> Causes a stabilization (reversible coagulation)




DLVO summary

• Electro-osmotic repulsion

• van der Waal attraction(difficult to obtain a correct Hamaker constant)

• At long and short distances vdW > Electro-osmotic

• Electro-osmotic barrier

• When surface are is large reversible coagulation mayoccur at secondary minimum




Extended DLVO theory

• Hydration forces

• Hydrophobic forces

• Oscillatory forces

• Membrane fluctuations

• Water structure forces

When the DLVO theory fails to explain experimental results,an extra term is often added, such as

The ”theory” is then called the Extended DLVO theory

d d h h d




DLVO extended with hydration

DLVO theory fails to describe systems withvery hydrophobic or very hydrophilic particles

or in other words

DLVO theory is only applicable for lyophobic colloidswith advancing contact angle Θa between 15º and 64º

(Θa for Na-montmorillonite = 17°)

Small Θ Large Θ

DLVO d d i h h d i




DLVO extended with hydration

Small Θ Large Θ

The DLVO expression can be extendedwith a hydration component:

ω =ωel +ωvdW + ωH

ωH = a/2 (C1 e-d/D1 + C2 e-d/D2)

C1, C2 constants, D1, D2 decay lengths

E t d d DLVO DLVO




Extended DLVO vs DLVO

Silica particles (r = 6 nm) in 0.01M NaCl

S h u

k un

C h en

T h e s i s

2 0 0

7

DLVO th D b k




DLVO theory: Drawbacks

•Lifschitz theory of attractive forces

•Ion fluctuation forces

•Charge regulations in the double layer

•Specific ion effects

The DLVO theory does not account for

DLVO th D b k




DLVO theory: Drawbacks

•Proper description of vdW force

•Surface charge density

•Surface potentials

•Debye length, etc.

Even so, an extended term is often needed to

explain the behaviour of a system

The DLVO theory has a numberof (adjustable) parameters

”Forces can vary in magnitude by a factor of 50 or moreby simply changing the counter-ion from e.g. bromide to acetate” Boström et al Phys Rev Lett 87, (16) 8103 (2001) (click here to read)

Pashley et al J Phys Chem 90 1637 (1986)

Concluding remarks

http://www.kemi.kth.se/nuchem/colloid/bostrom.pdf

http://www.kemi.kth.se/nuchem/colloid/bostrom.pdf




Concluding remarks

• The DLVO theory has been established since 1945

It can often describe trends and tendencies, but

• In many articles I found the DLVO theory fails to

quantitatively explain/(predict) experimental results

(without parameter fitting or extended theories)

"... DLVO-theory is completely inadequate (to put it gently)

in almost every system so far investigated".

Christiansen J Dispersion Sci Technol. 9: 171 (1988)