Critical Thickness of Rupture of Foam Films in Relation to the Acting Surface Forces

measurements: critical thickness of film ruptureair

airair

water waterhydrophobic attraction

very low surfactantconcentration

P. Kralchevsky, K. Danov, J. Angarska, B. Dimitrova, K. P. Ananth, Alex Lips

University of Sofia (BG), University of Shumen (BG), Unilever Research (USA)

Basic Idea: The air is “hydrophobic phase” ⇒ Hydrophobic attraction could be present in foam films ⇒ It would affect the critical thickness of film rupture!

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(Invited Lecture at the Conference “EUFOAM”, Marne la Vallée, France, July 5-8, 2004)

Film radius (µm)

100 200 300 400 500

Crit

ical

thic

knes

s (n

m)

25

30

35

40

45

0.43 mM SDS + 0.25 M NaCl

Comparison of theory and experiment: No adjustable parameters!

Foam Films:

Relatively high SDS concentration;

High NaClconcentration – to suppress the double layer repulsion;

Van der Waals forcesin retarded regimetaken into account.

Experiment: E. Manev, S. Sazdanova, D.T. Wasan, J. Colloid Interf. Sci. 97 (1984) 591.

Theory: D.Valkovska, K. Danov, I. B. Ivanov, Adv. Colloid Interface Sci. 96 (2002) 101.

h

SDS = Sodium Dodecyl Sulfate

2

Theory: D.Valkovska, K. Danov, I. B. Ivanov, Adv. Colloid Interface Sci. 96 (2002) 101:

Assumptions:

1. Simultaneous film drainage and growth of interfacial perturbations;

2. The critical wave has amplitude = h/2;

3. Unbounded waves: Hankel transformation: ∫∞

=0

0 )(),(),( kdkkrJtkAtr ξζ

),(

)()exp( 0

khYhV

krJt

=

∝

ω

ωζ

ω < 0 ⇒ the wave will decay (stable film);

ω > 0 ⇒ the wave will grow (unstable film);

ω = 0 ⇒ transition from stability to instability.

film

air

air

h

z

ζ ∝ ( )J kr0λ

3

Theory of Critical Thickness: Valkovska, Danov, Ivanov (2002) (continued)

0-5.0

-2.5

5.0

2.5

0.0

2 10864

R=50 mµ

hst=38 nm

hcr=30 nm

h tr=35 nm

kcr

h =400 nmstable film

Dimensionless wave number, k

Dim

ensi

onle

ss fu

nctio

n, Y

unstable film

),(

)()exp( 0

khYhV

krJt

=

∝

ω

ωζ

At h = hst – the first unstable wave appears (“stability” thickness);

At h = hcr – the film breaks (“critical” thickness); the respective wave number is kcr

At h = htr – the “critical wave” becomes unstable (“transitional” thickness).

Y > 0 ⇒ growth

Y < 0 ⇒ decay

h

4

Critical Thickness, Basic Equations: Valkovska, Danov, Ivanov (2002)

∫∫ Π−Π′

=Π−

tr

cr

tr

cr c

3

c

6

3cr

2

2cr h

h

h

hdh

Phdh

Ph

hRk σ

3cr

2

2cr

3tr

trc4tr

3cr

tr 2)]([

24)(

hRkh

hPhh

hσ

+Π−=Π′

)32

exp()( tr

cr c

3

3cr

2cr

tr4/1

B

2tr

cr ∫ Π−Π′

−=h

hdh

Ph

hk

hTkh

hσ

Three equations for determining hcr, kcr and htr

Numerical solution: Bisection method

Single roots: convenient computations

Input Quantities:

R – Film radius; Pc – Capillary pressure; σ – Surface tension; T – temperature

Disjoining pressure: Π(h) = Πvw(h) + Πhb(h) (van der Waals + hydrophobic)

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Physical Origin of the Hydrophobic Surface Force

Two kinds of hydrophobic surface forces:

(1) Due to gaseous capillary bridges (cavitation) between

the hydrophobic surfaces

(2) Due to hydrogen-bond-propagated ordering of water molecules in the vicinity of

hydrophobic surfaces

Ordered H2O in the film, Disordered H2O in the bulk ⇒ Gain of Entropy !

Eriksson, J. C. et al. J. Chem. Soc. Faraday Trans. 2 1989, 85, 163.

gas water

gas bridge betweentwo hydrophobic surfaces

water

hydrophobicinterface

bulk

Yushchenko et al., J. Colloid Interface Sci. 96

(1983) 307

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Hydrophobic Surface Force of the Second Kind

Hydrophobic Disjoining Pressure in a water film:

h – film thickness;

B – strength of the hydrophobic attraction; depends on the interfacial

hydrophobicity;

λ – decaylength; bulk property: depends on the solution conditions.

Eriksson, J. C. et al. J. Chem. Soc. Faraday Trans. 2 1989, 85, 163.

Example: Structuring of water molecules in the vicinity of paraffin-water interface, including the paraffin tails of surfactants:

Standard free energy of surfactant adsorption:

oad

oad

oad STHG ∆−∆=∆

Enthalpy: | | = 1−7 kJ/mol,

Entropy term: = 23−37 kJ/mol

oadH∆oadST∆

)2/(sinh1

4)( 2hb λπλ h

Bh −=Π

water

surfactant molecule

air

waterair

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Experiment with Foam Films

0.3 M NaCl

0.1 M NaCl

(Experimental cell of Scheludko & Exerowa)

Film radius R = 155 µm; Solutions: 10 µM SDS

0.3 M NaCl the film looks dark gray in reflected light just before it ruptures.

0.1 M NaCl in primary film, formation of black spots, corresponding to a secondary film, is seen.

Air

8

At 0.3 M NaCl – the electrostatic effects are suppressed: working concentration!

No equilibrium films – only the critical thickness of film rupture is measured.

dzz

zhzhnnnnhhAji

ji ∫∞

+−+

+

−=

0222/322

222eP

)21()~2exp()~21(

)()(

43)(

πν

3vw 6)()(hhAh

π−=Π

Πvw – only dispersion interactionwith account for the electro-magnetic retardation effect (Russel et al):

At 0.1 M NaCl – indications about existence of primary films and Πel are seen.

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Experimental Results and Theoretical Fits

Film radius, R (µm)

60 80 100 120 140 160

Crit

ical

thic

knes

s, h

cr (

nm)

25

30

35

40SDS + 0.3 M NaCl 0.5 µM SDS

1 µM SDS

10 µM SDS

10 µM SDS for Πhb = 0

1 µM SDS for Πhb = 0

Disjoining pressure:

Π(h) = Πvw(h) + Πhb(h)

Eriksson et al.:

λ - decay length, independent of the surface hydrophobicity;

B – characterizes the strength of the hydrophobic interaction, depends on surface hydrophobicity.

)2/(sinh1

4)( 2hb λπλ h

Bh −=ΠEriksson formula:

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Parameter Values

System λ (nm) B (mJ/m2)

0.5 µM SDS + 0.3 M NaCl 15.85 6.56 × 10−4

1.0 µM SDS + 0.3 M NaCl 15.85 4.71 × 10−4

10 µM SDS + 0.3 M NaCl 15.85 3.34 × 10−5

DDOA-covered mica* 15.8 0.6

F-surfactant-covered mica* 15.8 0.9

*Data by Eriksson et al. DDOA = dimethyl-dioctadecyl-ammonium bromide; λ is constant; B decreases with the rise of SDS concentration

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Key: The adsorption of SDS and Na+ render the surface hydrophilic.

The adsorptions can be determined from fit of surface-tension data:

SDS Concentration (mM)

0.01 0.1 1 10

Surf

ace

tens

ion,

σ

(mN

/m)

20

30

40

50

60

70

80

0.0 mM0.5 mM0.8 mM1.0 mM2.5 mM4.0 mM5.0 mM8.0 mM10 mM20 mM115 mM

NaCl

van der Waals adsorption isothermData: Tajima et al. (1970-1971)

Parameters of the best fit:

Adsorption energy:

ESDS = 12.5 kT

ENa = 1.6 kT

Area per SDSmolecule:

0.30 nm2 (cross-section of the SO4 – group)

The curves for all NaCl concentrations are fitted simultaneously

[Kolev, Danov, Kralchevsky et al., Langmuir 18 (2002) 9106–9109]12

CSDS(µM)

Γ1 / Γ∞(%)

Γ2 / Γ∞(%)

EG(mN/m)

0.5 4.2 % 1.0 % 0.560

1.0 6.9 % 2.1 % 0.921

10.0 33.2 % 19.4 % 6.15

Results from the fit of the SDS surface tension isotherms:

Γ1 – surfactant (DS–) adsorption;

Γ2 – counterion (Na+) adsorption;

Γ∞ – maximal adsorption in a closely-packed surfactant adsorption monolayer;

EG – surface dilatational (Gibbs) elasticity.

air

air

Na

Na

air

water waterhydrophobic attraction

surfactant

13

(0.3 M NaCl)

Foam filmsSDS + 0.3 M NaCl

Inverse ionic adsorption, (Γ0 + Γ1 + Γ2)−1 (m2/µmol)

0.0 0.5 1.0 1.5 2.0 2.5 3.0

Para

met

er B

(10

−4 m

J/m

2 )

-2

0

2

4

6

8

10

Slope:2.74 ± 0.16 (10−13 J.mol.m−4)

Plot of B vs. (Γ0 + Γ1 + Γ2)−1;

Γ0, Γ1 and Γ2 are the adsorptions of OH−, DS− and Na+ ions

(the adsorbed ions render the air-water interface hydrophilic)

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Conclusions

1. The van der Waals attraction, alone, is insufficient to explain the results, especially for the lower SDS concentrations, 0.5 and 1 µM.

2. If the difference is attributed to the hydrophobic attraction, then a very good agreement between theory and experiment is achieved.

3. From the best fit, we determine decaylength λ ≈ 15.8 nm, which coincides with results by other authors for hydrophobized mica surfaces.

4. The strength of the hydrophobic interaction, B, is found to be inversely proportional to the surface density of the adsorbed ions.

5. With the decrease of SDS adsorption, greater areas of bare hydrophobic air-water interface are uncovered ⇒ hydrophobic attraction becomes stronger.

Details can be found in: Langmuir 20 (2004) 1799.

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