Lecture 6

Surface Forces II:Surface Forces II:Double Layer Force,

DLVO theory and beyondDLVO theory and beyond

The electrostatic double layer force• When electric double layers of two surfaces overlap

a force will arise.

++

far away

00

+ ++

0 0close to each other

• we expect this force to be a repulsive one as the p psurface charge is not completely screened.

Equation for DL-forceG f ( ) f f• Gibbs free energy (per unit area) of two infinitely separated double layers:

0

00

2g dϕ

σ ϕ∞ ′′= − ∫0

• if the surface are at a distance x from each other both surface charge and potential depend on x.

0 ( )

( ) 2 ( )x

g x x dϕ

σ ϕ ′′= ∫

both surface charge and potential depend on x.

00

( ) 2 ( )g x x dσ ϕ= − ∫

• the interaction energythe interaction energy( ) ( )w x g g x g∞= Δ = −

• the interaction force: d gdxΔ

Π = −

Equation for DL-force• Poisson-Boltzmann equation:

2e e dϕ ϕ−⎛ ⎞ 20 0 2 0

e ekT kT dec e e

dx

ϕ ϕ ϕεε⎛ ⎞

− − =⎜ ⎟⎝ ⎠

• after first integration:

2⎛ ⎞ ⎛ ⎞2

00 2

e ekT kT dc kT e e P

dx

ϕ ϕ εε ϕ−⎛ ⎞ ⎛ ⎞− − =⎜ ⎟ ⎜ ⎟⎝ ⎠⎝ ⎠

Equation for DL-force• The disjoining pressure:

20

0 02 2 2

e ekT kT dP kTc c kT e e

d

ϕ ϕ εε ϕξ

−⎛ ⎞ ⎛ ⎞Π = − = − − −⎜ ⎟ ⎜ ⎟

⎝ ⎠⎝ ⎠ 2 dξ⎝ ⎠⎝ ⎠

Electrostatic interaction between identical surfaces

• As at the equilibrium the pressure is the same q peverywhere, we calculate it in the middle of the gap:

⎛ ⎞0 2

m me ekT kTc kT e eϕ ϕ−⎛ ⎞

Π = − −⎜ ⎟⎝ ⎠

0dϕ

⎝ ⎠• for low potentials:

0; mdxϕ ϕ ϕ= =

22 20 0

22m mc e kT kTkT

εεϕ ϕλ

Π = = 22m mDkT λ

kTif we have and expression for the potential we can

02

02D

kTc e

εελ =for the potential we can calculate the pressure

Electrostatic interaction between identical surfaces

• for a slight overlap of DLs we can assume

2 ( / 2)m xϕ ϕ′=

2εε

• for low potentials

/2002

2Dx

D

e λεε ϕλ

−Π =

2/0 02( ) ( ) D

xx

D

w x x dx e λεε ϕλ

−

∞

′ ′= − Π = −∫D∞

• for high potentials

/2 00( ) 64 tanh 4

DxD

ew x c kT ekT

λϕλ −⎛ ⎞= ⎜ ⎟⎝ ⎠

The DLVO theory• The theory explains why dispersions aggregate when

salt concentration is increasedsalt concentration is increased

Derjaguin-Landau-Verwey-Overbeek

Derjaguin B V and Landau L D 1941 “Theory of the stability of strongly chargedDerjaguin B.V. and Landau L.D., 1941. Theory of the stability of strongly charged lyophobic sols and the adhesion of strongly charged particles in solutions of electrolytes”. Acta Physicochimica URSS 14, 633-662.

Verwey E.J.W. and Overbeek J.Th.G., 1948. “Theory of the Stability of Lyophobic Colloids”. Elsevier, New York

The DLVO theory• Coagulation of dispersed particles is explained by the

interplay of two forces: van der Waals and double-layer force (DLVO forces)

( )2

exp 2 1e kT Aϕ⎛ ⎞( )( ) ( )

00 2

0

exp 2 1( ) 64 exp

exp 2 1 12H

D D

e kT Aw x c kT xe kT xϕ

λ λϕ π

⎛ ⎞−= − −⎜ ⎟⎜ ⎟+⎝ ⎠

Non-DLVO forces

• Born repulsionp• origin in same interatomic

potential as van der Waals• repulsion from overlap of

electron shells• very short range force• very short range force• prevents contact between

attached particlesp• provides finite depth to

primary minimum

Non-DLVO forces• Solvation forces:

02( ) cos x xxf x f eπ −

⎛ ⎞= ⎜ ⎟ 00

0

( ) cosf x f ed

= ⎜ ⎟⎝ ⎠

Experimentally measured force between a SiN tip and mica in 1-propanol

Non-DLVO forces

– Hydration• “vicinal” water structure• repulsive force• corresponds to layer-by-

layer removal of vicinal water

• effect of ionic strength, tication

– Li+, Na+ > K+ > Cs+

Non-DLVO forces

– Hydrophobic interaction• attractive force• exponentially decaying• stronger than vdWaals forces• stronger than vdWaals forces• usually two component

assumed:– short-range (1-2nm) due to

change in water structure:– long-distance (up to 200nm) g ( p )

due to spontaneously formed gas bubbles

examples of hydrophobic interaction

Non-DLVO forces: Steric force

• A macromolecule in solution:• for a freely jointed segment chain model:

– Rms of the end-to-end distance: 0R l n=

l– radius of gyration: 0 6

l nR =

• Solvent influence – ideal solvent: interaction between the monomers is the same as with the

l tsolvent– good solvent: repulsive interaction between monomers, Rg increases– bad solvent: attraction between the monomers Rg decreasesbad solvent: attraction between the monomers, Rg decreases

Steric force• The force between two surface coated by a polymer is

determined by:– the nature of solvent– the way the polymer is deposited and its density

5 3 1 30L nl= Γ

21 gRΓ21 gRΓ

Steric force• Repulsive force caused by:

– Entropic factor (reduced configuration entropy)p ( g py)– osmotic pressure in the gap– intersegment force

• High density brush3944

3 2 02( )2

L xx kTx L

⎡ ⎤⎛ ⎞⎛ ⎞⎢ ⎥Π = Γ −⎜ ⎟⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦02x L⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦

• Low density brush2 2

2

2( ) 1 for 3 2g g

RkTx x Rx x

π⎛ ⎞ΓΠ = − ≤⎜ ⎟⎜ ⎟

⎝ ⎠2

( ) exp for 3 22 gg g

kT x xx x RR R

⎛ ⎞ΓΠ = − >⎜ ⎟⎜ ⎟

⎝ ⎠

Depletion forces• Attractive force between two particles in solution of

macromolecules will arise due to osmotic pressure macromolecules will arise due to osmotic pressure

• Asakura Oosawa theory:• Asakura-Oosawa theory:

( )20 0( ) 2 22( ) 0 2

pW D ckTR R D D R

W D D R

π⎧ ≈ − ≤⎪⎨⎪ >⎩ 0( ) 0 2W D D R⎪ = >⎩

• Alternatively could be explained due increase of total free volume available to macromoleculesfree volume available to macromolecules

Contact forces and adhesion• Hertz theory

– If two spheres having Young moduli E1, E2p g g 1, 2and Poisson ratios n1, n2 are pressed together:

effective radius*

3*

34Ra FE

=reduced Young modulus

effective radius 2

*

aR

δ =

– where:

reduced Young modulus

2 2* 1 2 1 2

*1 2 1 2

1 11R RRR R E E E

ν ν⋅ − −= = +

+1 2 1 2

Contact forces and adhesion• Johnson, Kendall and Roberts (JKR

theory)y)– additionally the work of cohesion is taken

into account

( )* 23 * * **

3 3 6 34Ra F WR WR F WRE

π π π⎛ ⎞= + + +⎜ ⎟⎝ ⎠⎝ ⎠

– so, a force is required to separate two solids

*3 ; 22adh s

F WR Wπ γ= =

• Derjagin, Muller, Toropov (DMT theory)– van der Waals forces are taken into account– van der Waals forces are taken into account

*4adh sF Rπγ=

Contact forces and adhesion

• Criterion:– if size of the neck is larger than

i t t i di t JKR th iinteratomic distance – JKR, otherwise DMT

/1/32 *

*2s

nRh

Eγ⎛ ⎞

≈ ⎜ ⎟⎝ ⎠

Problems

End of chapter problems:d o c ap e p ob e s• ch.6.8• ch.6.9