The stability of thin-soft films · Ian Morrison© 2009 Not presented during Lecture 2. 22....

The stability of thin (soft) filmsThe stability of thin (soft) films

Lecture 2 – Thermodynamics and stabilityy

October 30, 2009

Ian Morrison© 2009

Review of Lecture 1

The 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

Ian Morrison© 2009 14

Josiah Willard Gibbs

JOSIAH WILLARD GIBBSBorn Feb. 11, 1839 – Died Apr. 22, 1903

PROFESSOR OF MATHEMATICAL PHYSICSIN YALE UNIVERSITY, 1871 - 1903

Ian Morrison© 2009

11-30-08

15

Gibbs’ interfacial phasepi idU TdS pdV dA dn= − +σ + μ∑

i iU TS pV A n= − +σ + μ∑Differential of total energy

Total energy

Bulk Phaseα

InterfacialPhase

Bulk Phaseβ

i ip μ∑0i iSdT Vdp Ad n d− + σ+ μ =∑0i iS dT V dp n dα α α α− + μ =∑

Total energy

Gibbs-Duhem equation

F h b lk h

Den

sity

∑0i iS dT V dp n dβ β β β− + μ =∑

i i id d dα βμ = μ = μ

For each bulk phase

At equilibrium

( ) ( ) ( ) 0i i i iS S S dT V V V dp Ad n n n dα β α β α β− − − − − + σ + − − μ =∑S S S Sσ α β= − − V V V Vσ α β= − − i i i in n n nσ α β= − −

0i iAd S dT V dp n dσ σ σσ+ − + μ =∑ Subtracting and renaming.

α σ β

Distancei ip μ∑

ii i i

nd d dσ

− σ = μ = Γ μ∑ ∑ -2 mol mii

nσ

Γ =

g g

Gibbs adsorption isotherm: Surface concentrations:


i i iAμ μ∑ ∑ i A

Gibbs’ surface phase2F C P= − +

p

nσ nσ

Gibbs adsorption isotherm: The surface excess:

For a 2-component system:

ii i i

nd d dA

− σ = μ = Γ μ∑ ∑ -2 mol mii

nA

Γ =

Bulk Phase Surface Bulk Phase Bulk Phase Surface Bulk Phasep y

1 1 2 2 with the Gibbs-Duhem relation

0

d d d

X d X d

− σ = Γ μ +Γ μ

+

Bulk Phaseα

SurfacePhase

Bulk Phaseβ

Den

sity

Bulk Phaseα

SurfacePhase

Bulk Phaseβ

Den

sity

Component 2

lid1 1 2 2

2 2

2

0

gives and the surface-excess concentration

G

G

X d X d

d d

X

μ + μ =

− σ = Γ μ

Γ = Γ Γ

D D

Component 1

solid

2 2 11

is independent of the dividing surface.X

Γ = Γ − Γ α β

Distance

α β

Distance

2 components, 2 components,


p2 phases

p1 phase

Interacting adsorption layers?Interacting adsorption layers?Bulk Phase

αInterfacial

PhaseBulk Phase

βBulk Phase

αSurfacePhase

Bulk Phaseβ

Bulk Phaseα

SurfacePhase

Bulk Phaseβ

Bulk Phaseα

SurfacePhase

Bulk Phaseβ

Den

sity

Den

sity

Den

sity

Component 1

Component 2

solid Den

sity

Component 1

Component 2

solid

α σ β

Distance

α β

Distance

α β

Distance

Component 1

2 phases, 2 components 1 phase, 2 components

α β

Distance

2 phases, 2 components 1 phase, 2 componentsBulk Phase

αBulk Phase

αSurfacePhase

SurfacePhase

Bulk Phaseβ

sity

Dens

Component 2

h

22 2 1 G X

XΓ = Γ − Γ

h is a new

Den

s sity

Component 1

solid solid

X1 and X2 are no longer constant. They depend on h.

1X h is a new degree of freedom


Distance

1 liquid phase, 2 components

Thermodynamics of thin films* - 1( )

1 2, , , ,

1

iT P N

GhA h σ σ

∂⎛ ⎞Π = − ⎜ ⎟∂⎝ ⎠ - **i i

idU TdS pdV dn G U TS pVμ= − + ⋅ = +∑

y

1 1 2 21

plates platesr

i i j j j ji j j

dG SdT VdP dN A d A dh d d dμ σ σ σ σ=

= − + + + Ψ − Π Ψ = Ψ +Ψ∑ ∑ ∑

( ) ( )

1

1 2 1 2, , , , , , , , , ,

i j j

i i hh

G T P N h A T P N k dk Gσ σ σ σ∞

=∞= Π +∫ is the free energy of Gibbs’ model including the adsorption at surfaces.

hG =∞

( )1 1 2 2 1 1 2 21 1

r

i i i ii i

dG d G dN d d SdT VdP N d A d d A dhμ σ σ μ σ σ= =

⎛ ⎞′ = − −Ψ −Ψ = − + + + Ψ + Ψ − Π⎜ ⎟⎝ ⎠

∑ ∑adsorption at surfaces.

As Gibbs did, we shall do, subtract the uniform properties of the bulk:

kk

sk k k

k k

V V

S S S S V s

=

= − = −

∑

∑ ∑

( )1 1 2 21

s si i

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

,si i k i k i k

k kN N N N V n= − = −∑ ∑


1,i r=

* Parallel plates.**Rowlinson and Widom, Appendix 1 - Thermodynamics

Thermodynamics of thin films - 2( )1 1 2 2

1,0s s

i ii r

dG S dT N d A d d A dh=

′ = − − − Ψ + Ψ − Π =∑ μ σ σ

y

Since the LHS is a total differential, so is the RHS, so …

1 2 1 2, , , , , , ,

1

i i

si

i h

NA h μ μ

μΨ Ψ Ψ Ψ

⎛ ⎞⎛ ⎞∂ ∂Π= ⎜ ⎟⎜ ⎟∂ ∂⎝ ⎠ ⎝ ⎠… …

1 2, , ,

1

i

si

ih

N dA h μ

μ∞

Ψ Ψ

⎛ ⎞⎛ ⎞∂⎜ ⎟Π = ⎜ ⎟⎜ ⎟∂⎝ ⎠⎝ ⎠∫

…

is a ‘modified” Gibbs equation.

ii i i

nd d dA

σ⎛ ⎞− σ = μ = Γ μ⎜ ⎟⎝ ⎠

∑ ∑⎝ ⎠ ⎝ ⎠

⎛ ⎞ 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 = ⎜ ⎟∂⎝ ⎠

∫…


Thermodynamics of thin films - 3

( )1 1 2 21,

0s si i

i rdG S dT N d A d d A dh

=

′ = − − − Ψ + Ψ − Π =∑ μ σ σ

y

1 sS⎛ ⎞∂ ∂Π⎛ ⎞

( )1 21 2

, , , ,, , ,

1

1ii

s

h

s s

SA h T

S S dh

Ψ ΨΨ Ψ

∞

⎛ ⎞∂ ∂Π⎛ ⎞=⎜ ⎟ ⎜ ⎟∂ ∂⎝ ⎠⎝ ⎠

∂Π⎛ ⎞⎜ ⎟∫

…… μμ

i th t d( ),

s sh

hh

S S dhA T=∞− = ⎜ ⎟∂⎝ ⎠∫

…is the excess entropy due to disjoining pressure.

If we know ( )1 2, , , , ih TΠ = Π Ψ Ψ μ

We have ( ) ( )1 2 1 2, , , , , , , , , ,i i hh

G T P N h A T P N k dk Gσ σ σ σ∞

=∞= Π +∫

So all the thermodynamic functions can be calculated.The rest is statistical mechanics soft matter scaling etc

sk k

k

S S V s= +∑


The rest is statistical mechanics, soft matter scaling, etc…

Stability of films between parallel plates( )

1 2, , , ,

1

iT P N

GhA h σ σ

∂Π = −

∂

y p p

( ) ( )0 0d h d P PΠ −

= =

2

2 0Gh

∂>

∂ Unstable:dh dh

( )

( )

1

0

GhA h

hh

∂Π = −

∂∂Π

<∂

Assuming all other thermodynamic values RTS.Stable whenh∂

ssu g a o e e ody a c a ues S

If the change in thickness is not by a reversible path, then the (non-equilibrium) ( ) ( )h h∂Π ∂Π

<

S ab e e

p , ( q )state is at a higher free energy so that: non eq eq

h h−

<∂ ∂

Some unstable films will be stable to fast e g Gibbs Marangoni stability a increasewill be stable to fast perturbations:

e.g. Gibbs-Marangoni stability – a increase in liquid surface tension with expansion because surfactant adsorption is slow.

Ian Morrison© 2009 22Not presented during Lecture 2.

Stability of films between convex bodies( ) ( ) ( ) ( ) ( ) ( )1 21 2

1 2

222 sins p s s r r

y y y

R RR RF y R h dh F y h dh F y h dhR R

πππ∞ ∞ ∞

− − −≅ Π ≅ Π ≅ Π+ Ω∫ ∫ ∫

y

The change in force with distance is: ( )F g y∂= − Π

∂g ( )g y

y∂

Stability of the equilibrium is: Hence:0dF< ( ) 0yΠ >e ce0

dy< ( ) 0yΠ >

If the change in thickness is t b ibl th th dF dF dFnot by a reversible path, then

the (non-equilibrium) state is at a higher force so that:

non eq eqdF dF

dy dy− < ( )non eq

dFg y

dy− > − Πor

e.g. If the collision of charged particles is more rapid than the time for the surface charge to equilibrate, the dispersion may appear to be more stable than it really is.

But interactions between convex surfaces are more


But, interactions between convex surfaces are more likely to be at equilibrium than parallel plates.

Not presented during Lecture 2.

Components of disjoining pressurep j g p

What overlaps?

El i fi ld f h d fl i f l• Electromagnetic fields from the random, quantum fluctuations of electrons in all the three phases. (Dispersion forces.)• Static electric fields from charged surfaces.• Interactions between molecules or polymers adsorbed on the surface.Interactions between molecules or polymers adsorbed on the surface.• Layered structuring of solvent molecules, solute molecules or dispersed phases.

Ian Morrison© 2009 24Not presented during Lecture 2.

Experimental evidence

Derjaguin, B.V.; Rabinovich, Y.I.; Churaev, N.V. “Direct measurement of molecular forces” Nature, 272, 313 – 318, 1978.

p

r

First direct measurements were reported in 1951. First

Quartz, platinum, gold in air. Glass threads in water. Glass threads in water.

pEnglish report in 1954.

( )1 21 22u F r rπ= ( )1 22u F r rπ

Rabinovich, Ya.I.; Derjaguin, B.V.; Churaev, N.V. Direct measurements of long-range

f f i d li id di Ad


surface forces in gas and liquid media. Adv. Colloid Interface Sci., 16, 63 – 78, 1982.


Polymolecular adsorbed layers?y y

Initial

Water on mica

“These distances are much larger than Debye lengths. For example: 0.1 N LiCl on polished diamond was 75 nm at 900 d / 2 Th D b l th h ld b

After collapse

Initial

dynes/cm2. The Debye length should be about 1 nm. Nor is it van der Waals forces because of the extreme sensitivity to the dissolved components.”

D j i B K k M L b d R f l l

p

Derjaguin, B.; Kussakov, M.; Lebedva. Range of molecular action of surfaces and polymolecular solvate (adsorbed) layers. C.R. Acad. Soc. URSS, 1939, 23(7), 671-673.

300 nm


300 nm


“Polywater”y

Rabinovich, Y.I.; Derjaguin, B.V. Interaction of hydrophobized filaments

Ian Morrison© 2009

j g y pin aqueous-electrolyte solutions. Colloids Surf., 30, 243 – 251, 1988.

27Not presented during Lecture 2.

Polywater – “Ice-nine”yIce-nine is a fictional material conceived by writer Kurt Vonnegut in his novel Cat's Cradle. It is supposed to be a more stable polymorph of water than common ice (Ice Ih) which instead of melting at 0° Celsius (32° Fahrenheit) melts at 45 8°C (114 4°F) When ice nine comes intomelting at 0 Celsius (32 Fahrenheit), melts at 45.8 C (114.4 F). When ice-nine comes into contact with liquid water below 45.8°C (which is thus effectively supercooled), it acts as a seed crystal, and causes the solidification (freezing) of the entire body of water which quickly crystallizes as ice-nine. A global catastrophe involving freezing the Earth's oceans by simple contact with ice-nine is used as a plot device in Vonnegut's novel. (wikipedia.com)

“Future observers will chuckle quietly over Derjaguin’s discomfiture…scientific history will see Derjaguin as a great physical chemist who has dominated theory and experiment in surface and colloid science for fifty years... Let the last words on polywater be those of Derjaguin: ‘It was not a matter of belief, it was a matter of performing better experiments.’”

Pethica B A Book review: Polywater by F Franks MITPethica, B.A. Book review: Polywater by F. Franks, MIT Press: 1981. J. Colloid Interface Sci., 21, 607, 1982.


Boris Vladimirovich Derjaguin, 1902 – 1994.


Jacob’s boxIsraelachvili, J. Intermolecular and surface forces, 2nd ed.; Academic Press: Amsterdam; 1985.

Surface forces apparatus: TwoSurface forces apparatus: Two mica surfaces, atomically smooth, accuracy to 1 Å. At the 10 Å scale fluid properties, such as relaxation times, can be 1010 times greater than in the bulk.

Disjoining pressure is only mentioned twice in this (terrific) book, once with negative Hamaker constants and once with electrostatic repulsion )


electrostatic repulsion.)


The stability of thin-soft films · Ian Morrison© 2009 Not presented during Lecture 2. 22....

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