The stability of thin (soft) films

Lecture 5 – Stability of thin filmsLong range forces*Long range forcesNovember 20, 2009

e.g. Dispersion, electrostatic, polymeric

Review of Lecture 1Review of Lecture 1The disjoining pressure is a jump in pressure at the boundary It does

P

pressure at the boundary. It does not vary between the plates.

( ) 1

T P N

GhA h∂

Π = −∂

h

P1 2, , , , iT P NA h σ σ∂ P

Force between 2 spheres: ( ) ( )1 2

1 2

2s s

R RF y h dhR Rπ ∞

− ≅ Π+ ∫

( ) ( )G hh

h∂

Π = −∂

1

2

G(h)

Curve 1 – StableCurve 2 – Metastable

1 2 yR R+

( )( )

00

0s sl lv

hG S

G

σ σ σ∂

= − − =

∞ =

2

3

h0Curve 3 - Unstable

2Lecture 5 - Stability of thin filmsIan Morrison ©2009

Review of Lecture 2( )

1 2, , , ,

1

iT P N

GhA h σ σ

∂Π = −

∂Review of Lecture 2

( )1 1 2 2s s

i idG S dT dN A d d A dhμ σ σ= − + + Ψ +Ψ − Π∑

( )1 1 2 21

s si i

i

dG S dT N d A d d A dh′ = − − − Ψ + Ψ − Π∑ μ σ σ

1,i r=

1,i r=

is a ‘modified” nσ⎛ ⎞1 s

iN dμ∞ ⎛ ⎞∂

Π = ⎜ ⎟∫ is a modified Gibbs equation.

ii i i

nd d dA

⎛ ⎞− σ = μ = Γ μ⎜ ⎟⎝ ⎠

∑ ∑1 2, , ,i

ih

dA h μ

μΨ Ψ

Π = ⎜ ⎟∂⎝ ⎠∫

…

is the excess adsorption due to disjoining pressure. Note that we do not know how much excess is on either plate!,

si

iih h

N A dhA μ

∞ ⎛ ⎞∂Π−Γ = ⎜ ⎟∂⎝ ⎠

∫…

3

much excess is on either plate!

Lecture 5 - Stability of thin filmsIan Morrison ©2009

Review of Lecture 3

( ) ( )2 2

2 h oh E EεΠ = − 2 0

0

2

lim 4 sinh2

4li

hnkT

nkTσ π

Ψ

→

Ψ⎛ ⎞Π = ⎜ ⎟⎝ ⎠

Π( )20

4limh

nkTh

σ πκ→

Π =

0h dΦ Φ

( )

0

2 2 cosh coshm m

h dκ

Φ

Φ=

Φ − Φ∫

F th h f f t t t t ti l Φ i t tFor the approach of surfaces at constant potential, Φ0 is constant.

For the approach of surfaces at constant surface charge density Φ0 is:

( )2 2 h hzenΦ Φ

4Lecture 5 - Stability of thin films

( )02 2 cosh coshs m

zenσκ

= Φ − Φ

Ian Morrison ©2009

Review of Lecture 4123

123 212AG

HπΔ =− ( )123 12 32

0

3 ´ Rel2 n

n

kTA l∞

=

= Δ Δ ×∑

Review of Lecture 4Lifshitz (1954) – The distance derivative of the standing, oscillatory EM waves between two flat plates is the force per area.p p

24n

kTnhπ

ξ =( ) 2 ref absnε ω κ′′ =

Convert optical data to dielectric data: The Matsubara frequencies:

( ) ( )( ) ( )

1 212

n ni ii i

ε ξ − ε ξΔ =

ξ ξ

( )( )

2 2

2 2

1Reflectivity

1ref abs

normal

ref abs

n

n

κ

κ

− +=

+ +

The dielectric differences:

( ) ( )( ) ( )( ) ( )

121 2

3 232

3 2

n n

n n

n n

i i

i ii i

ε ξ + ε ξ

ε ξ − ε ξΔ =

ε ξ + ε ξ

( ) ( )2 2

0

21i dωε ω

ε ξ ωπ ω ξ

∞ ′′= +

+∫

( )123 12 323 ´ RelnkTA l

∞

= Δ Δ ×∑( ) 2 211

j jn

d fi

gε ξ

ξ τ ω ξ ξ= + +

+ + +∑ ∑

Fit dielectric data to empirical equation:The Lifshitz constant:

Lecture 5 - Stability of thin films 5Ian Morrison ©2009

( )123 12 3202 n

n=∑1j jn j j j n ngξ τ ω ξ ξ+ + +

Energy of thin liquid films( )

1 2, , , ,

1

iT P N

GeA h σ σ

∂Π = −

∂From lecture 1.

Energy of thin liquid films

( )Energy P eσ σ= + + ( )2

02as 0

sl lv

s

P em

Energye

σ σ

σ

= + +

→ → 02

2as

s

sl lv

mEnergye

mσ σ→∞ → +

m

( )( )

00 s sl lvP Sσ σ σ= − − =Range of P(e):Between surfaces: 1/e2

( ) 0P ∞ =For polymers: size of coilsElectrostatics: 10’s nm in water

100’s nm in oil

6Ian Morrison ©2009 Lecture 5 - Stability of thin films

*de Gennes, P-G; Brochard-Wyart, F.; Quéré, D.Capillarity and wetting phenomena: Drops, bubbles, pearls, waves.

Springer: New York; 2004.

Energy at constant volumeEnergy at constant volume

( )EnergyE P eσ σ= = + + ( )l lE P e Aσ σ= + +⎡ ⎤⎣ ⎦or( )2 sl lvE P em

σ σ= = + +

Differentiating gives:

( )sl lvE P e Aσ σ+ +⎡ ⎤⎣ ⎦

( ) ( )sl lv

dP edE P e dA A de

deσ σ

⎛ ⎞= + + +⎡ ⎤ ⎜ ⎟⎣ ⎦

⎝ ⎠

or

Consider changes at A

( ) sl lv

de

P e dA A deσ σ

⎣ ⎦⎝ ⎠

= + + − Π⎡ ⎤⎣ ⎦

Consider changes at constant volume = Ae 0 or A eAde edA

dA de+ = = −

The surface tension of the film is the dE⎛ ⎞change in energy with area: film

Ae constant

dEdA

σ=

⎛ ⎞= ⎜ ⎟⎝ ⎠

( ) ( )fil l l P e e eσ σ σ= + + + Π

Lecture 5 - Stability of thin films 7

( ) ( )film sl lv P e e eσ σ σ+ + + Π

Ian Morrison ©2009

Stability of thin filmsdGBWQ, Fig. 4.3

Stability of thin films

( ) ( )film sl lv P e e eσ σ σ= + + + ΠThe energy of a film at constant volume is:

The differential with respect to thickness is:

constant volume is:

( ) ( ) ( ) ( )2 2

2 20 0filmd d P e d P ee e e e

de de deσ

= + −Π +Π + =

A film is stable (locally)only if:

( )2

2 0d P e

de>

(c) represents the splitting of the film in (a) into two thickness of partial areas andpartial areas, α1 and α2

( ) ( ) ( )1 1 2 2

1 2 1P e P e P eα α

α α+ <

+ =

Lecture 5 - Stability of thin films 8

1 2

This is the energy at constant Ae!Ian Morrison ©2009

Total wettingdGBWQ, Fig. 4.4

Total wetting

0 0s lv slS σ σ σ= − − >0s lv sl

Films greater than ec are stable.

Lecture 5 - Stability of thin films 9Ian Morrison ©2009

Thickness of a large dropThickness of a large drop.Small drops are spherical segments. Large drops are flattened.

To “spread” the drop requires an force per unit length:

( )The hydrostatic pressure integrated over the depth of the drop is a force per unit

( )sv lv slσ σ σ− +

( ) 21e

P g e z dz geρ ρ= − =∫the depth of the drop is a force per unit length pushing to “spread” the drop:

( )0 2

P g e z dz geρ ρ∫

At equilibrium the sum of the two is zero: ( ) 21 02sv lv sl geσ σ σ ρ− + + =

Substituting the Young-Dupré equation: ( ) 21 1 cos2lv e geσ θ ρ− =

Re-arranging gives: 1 2 sin2ee θκ − ⎛ ⎞= ⎜ ⎟

⎝ ⎠

Lecture 5 - Stability of thin films 10

2⎜ ⎟⎝ ⎠ dGBWQ, Fig. 2.4

Ian Morrison ©2009

Partial wettingdGBWQ, Fig. 4.4

Partial wetting

0S 0

0

0cos

s lv sl

s lv sl

S σ σ σσ σ θ σ= − − <

= +

Lecture 5 - Stability of thin films 11Ian Morrison ©2009

Pseudo‐partial wettingdGBWQ, Fig. 4.4

Pseudo partial wetting

0 0cos

s lv slS σ σ σσ π σ σ θ σ= − − <

− = = +0 coss e sv lv slσ π σ σ θ σ= = +

σLp

σπ θ

p0

σSLπe

σSV

θ

0

lnp

e RT d pπ = Γ∫

Gibbs’ adsorption isotherm:

Lecture 5 - Stability of thin films 12Ian Morrison ©2009

0e p∫

Wetting, gas adsorption, and contact angles( ) ( )film sl lv P e e eσ σ σ= + + + Π

σSL

σL

πe

σSV

θ

p0

If the liquid has a finite contact angle

cossv lv slσ σ θ σ= +

σSV If the liquid has a finite contact angle, then gas adsorption is finite at p0.

( )( )

lp

RT dΓ

Γ Γ∫( )filmdP

σ

Ian Morrison ©2009 Lecture 5 - Stability of thin films 13

( )0

lnRT d pπ Γ = Γ∫( )film eP ede

= −

Wetting – contact angles – gas adsorption

( ) ( )film sl lv P e e eσ σ σ= + + + Π ( ) ( )121 212 lv cAP e e e

eσ

π= = =

g g g p

σSL

σL

πe θ

p0

σSLe

σSV

θ

From a thermodynamic argument by Adamson:

From a thermodynamic argument by deGennes et al:

( ) ( )( ) ( )

cos

cos 1sl lv m m m sl lv

m lv

P e e e

P e

σ σ σ σ θ

σ θ

+ + + Π = +

= −( )cos 1lvArea σ θ= −

Ian Morrison ©2009 Lecture 5 - Stability of thin films 14

The contact angle on small particles!

Interactions on treated surfaces( ) ( )2 22

1 2 112 2

Al l a l aπ

⎡ ⎤− − +⎢ ⎥

+ +⎢ ⎥⎣ ⎦Slabs of thickness a, separation l:

Interactions on treated surfacesSilane treatment of glass

(a) Water spreads on clean glass – thick films are stable.

(b) On thinly silanized glass, thick films or water are

G ld fil l i

thick films or water are stable, thin films of water are unstable.

Gold films on plastic

(a) Thin films of water are unstable on plasticunstable on plastic.

(b) Thin films of water are stable on gold-coated plastic.

Lecture 5 - Stability of thin films 15dGBWQ, p. 98.Ian Morrison ©2009