The dispersion-force contributions to thin-soft films ... · The Lifshitz constant is a double...

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

Lecture 4 – Dispersion-force contributionsNovember 13, 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 2Lecture 4 - Dispersion-forces

Review of Lecture 2( )

1 2, , , ,

1

iT P N

GhA h σ σ

∂Π = −

∂

( )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 = ⎜ ⎟∂⎝ ⎠

∫…

Ian Morrison© 2009 3

much excess is on either plate!

Lecture 4 - Dispersion-forces

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


( )02 2 cosh coshs m

zenσκ

= Φ − Φ

P.N. Lebedev (1894) Light and forces( ) g

Hertz's interpretation of light oscillations as electromagnetic processes, hides another problem, the processes that take place in a molecular vibrator when it emits light energy into the surrounding space.

This problem leads us into the field of spectral analysis it brings us toThis problem leads us into the field of spectral analysis it brings us to, the field of molecular forces.

Having accepted the standpoint of the electromagnetic theory of light, we have to conclude that ponderomotive forces must exist between two light-emitting molecules as between two vibrators in which electromagnetic oscillations have been generated.

A nonlinear force that a charged particle experiences in an inhomogeneous oscillating electromagnetic field. (e.wikipedia.org/wiki/Ponderomotive_force) Lebedev, P.N. WiedemannAnn 52 621 (1894) collected works Moscow (1913) p 56

22

24peF Em

= − ∇ω

Ian Morrison© 2009 Lecture 4 - Dispersion-forces 5

Ann., 52, 621 (1894); collected works, Moscow (1913), p. 56

Hamaker’s model (1937) ( )

The intermolecular attraction is due t L d (di i ) i

63U r −= − Λ 2

AGΔ =−to London (dispersion) energies: 11 112U r= Λ 212 Hπ

M l l i

rMolecules in

particle 1 Molecules in particle 2

H

Ian Morrison© 2009 Lecture 4 - Dispersion-forces

H

6

The physicsp y

Planck (1900) Energy levels of standing, oscillatory EM waves in a vacuous cavity with conductingwaves in a vacuous cavity with conducting walls. The temperature derivative gave heat capacity.

Casimir (1948) Energy levels of standing, oscillatory EM waves across a vacuum between two conducting flat plates. The distance derivative gave the force per unit area

Lifshitz (1954)

derivative gave the force per unit area.

Energy levels of standing, oscillatory EM waves across any medium between anywaves across any medium between any two flat plates. The distance derivative gave the force per unit area.


Electromagnetic fluctuationsg

“Empty space” can be considered as a turmoil of electromagnetic waves of all f i d l thfrequencies and wavelengths.

Absorption frequencies: those at which charges naturally dance or those where charge polarization quells the fluctuations and stills the spacewhere charge polarization quells the fluctuations and stills the space between the surfaces.

Retardation occurs when the time it takes for perturbations to be sensed b b th ti l b i t t


by both particles becomes important.

The modern view – “fluctuation-dissipation”p

• Electromagnetic fluctuations abound – quantum and thermal

• Producing spontaneous, transient electric and magnetic fields

• Which have the strongest effects at absorption frequencies.

• Net forces are coordinated interactions averaged over time• Net forces are coordinated interactions, averaged over time.

• Calculate “dispersion” forces from optical spectra.


Data for dielectric functions

The absorption(or reflection) spectrumThe absorption(or reflection) spectrum is measured. Sometimes a single peak in the UV and an average IR is sufficient.

The dielectric spectrum is calculated from the absorption spectrum. The only additional information needed isonly additional information needed is the static dielectric constant.


Model dielectric functions( )niε ξ

Dielectric permittivity – how a material responds to an electric fieldImaginary frequency – field varies exponentially rather than sinusoidallyg y q y p y y

Optical data: ( ) 2 ref absnε ω κ′′ =( )( )

2 2

2 2

1Reflectivity

1ref abs

normal

ref abs

n

n

κ

κ

− +=

+ +

( ) ( )2 2

0

21i dωε ω

ε ξ ωπ ω ξ

∞ ′′= +

+∫Kramers-Kronig relation:

( )ref abs

Why imaginary frequencies – the functions are smooth. Dielectric constants vary wildly with real frequency (near peaks).

For a system of dipole and resonant-damped oscillators:

( ) 2 211

j j

j jj j j

d fi

gε ξ

ξτ ω ξ ξ= + +

+ + +∑ ∑


Sample spectral parameters( ) 2 21

1j j

j jj j j

d fi

gε ξ

ξτ ω ξ ξ= + +

+ + +∑ ∑

p p p

PolystyreneFour‐term fit UV frequencies only

6.35 14.6 0.65

14.0 96.9 5.0

11.0 44.4 3.5

20.1 136.9 11.5

GoldFour‐term fit To absorption data

‐ 9.7 3.21

Parsegian, 2006, pp 2682.9 4.95 0.67

4.0 41.55 2.22

8.9 207.76 8.50

Four‐term fit To absorption data

‐ 40.11 ‐

3.87 59.61 2.62

8.37 122.55 6.41

23.46 1031.19 27.57

Four‐term fit To absorption data

‐ 53.0 1.8

3.0 5.0 0.8

Ian Morrison© 2009

4.8 104.0 4.4

12Lecture 4 - Dispersion-forces

Lifshitz theory 123123 212

AGHπ

Δ =−y

( )3 ´ R lkTA l∞

Δ Δ ×∑The Lifshitz constant is a double summation of products of dielectric functions:

12 Hπ

( )123 12 320

Rel2 n

nA l

=

= Δ Δ ×∑of products of dielectric functions:

The prime on the summation indicates that the n = 0 term is given half weight.

( ) ( ) ( ) ( )1 2 3 2dn n n ni i i iε ξ − ε ξ ε ξ − ε ξΔ Δ

The dielectric functions are differences in dielectric

the n 0 term is given half weight.

( ) ( )( ) ( )

( ) ( )( ) ( )

1 2 3 212 32

1 2 3 2

and n n n n

n n n ni i i iξ ξ ξ ξ

Δ = Δ =ε ξ + ε ξ ε ξ + ε ξ

differences in dielectric constants over a series of imaginary frequencies:

24n

kTnhπ

ξ =The sampling frequencies are:


Sampling (Matsubara) frequenciesp g ( ) q

The zero frequency term is the static dielectric constant.

22

n

n

kTnkT n

ξ ≡ π

πξ ≡

Frequencies are evenly spaced such that photon energies, from quantum mechanics, of each frequency is a multiple of thermal energy kT, from n

( ) 142.41 10n roomradT ns

ξ ≡ ×

thermodynamics.

The first sampling frequency corresponds to an IR frequency

11

2 7.82c mπλ = = μ

ξ

corresponds to an IR frequency.

A few more through the IR and visible (excitations are thermal)

Most sampling frequencies are in the UV and x-ray regions (excitations are quantum mechanical)


are quantum mechanical)

Lifshitz coefficients4.05 roomkT zJ=

Ian Morrison© 2009 15

Parsegian, 2006, pp 19, 64, 265

Lecture 4 - Dispersion-forces

The first results – Derjaguin et al

Derjaguin, B.V.; Churaev, N.V.; Muller, V.M. Surface forces; Consultants Bureau: New York; 1987.

j g

First instrument direct measurements of molecular forces between macroscopic b di (1951)

Quartz-quartz interactions in vacuum and in air. Lines are for Lifshitz theory (1954)


bodies (1951). for Lifshitz theory (1954)

Calculation vs measurement

Force - separation for TiO at the PZCGenerally:

Force - separation for TiO2 at the PZC

50

100

( ) 2 2 1+

1 1

ir uvn

n n

c ciε ξ = +⎛ ⎞ ⎛ ⎞ξ ξ

+ +⎜ ⎟ ⎜ ⎟

F/R

(μN

/m)

-50

0

direction ε(0) ωIR(rad/s) CIR ωUV(rad/s) CUV

1 1ir uv

+ +⎜ ⎟ ⎜ ⎟ω ω⎝ ⎠ ⎝ ⎠

10 0 10 20 30 40 50 60-200

-150

-100perpendicular 86 1 x 1014 80 7.49 x 1015 4.77 parallel 170 1 x 1014 163 7.24 x 1015 6.01

Separation (nm)

-10 0 10 20 30 40 50 60

Larson, I.; et alJACS, 1993, 115,11885-11890.


Other geometriesg


Archimedean bouyancyy y

For pair-wise interactions:

12 12 12 12 12ab cc ac cbΣ = Σ +Σ −Σ −Σ

in modern notation, . . Hamaker constants:

acb ab cc ac cb

i eA A A A A= + − −when :

2aca aa cc ac

a bA A A A

== + −

Derjaguin, B. (with Obuchov, E.) Ultramicrometric analysis of solvate layers and elementary expansion effects. Acta Phys.-Chim., 1936, 5(1), 1-22.


Hamaker geometriesParsegian, V.A. van der Waals forces; Cambridge University Press: New York; 2006.

gH

212A

lπ−

32A R⎛ ⎞

( ) ( )2 22

1 2 112 2

Al l a l aπ

⎡ ⎤− − +⎢ ⎥

+ +⎢ ⎥⎣ ⎦

29A R

z⎛ ⎞− ⎜ ⎟⎝ ⎠

2212

A Llπ

−

1/2

3/224A R

l−

1 23 A AA

( )1 2

1 2

16

R RAR R l

−+

1 2

8 rzπ−

A R−

6A R

l−

616A R

6 l

1 2A AA−


69 z− 42 zπ

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