Surface Forces and Liquid Films (Continued)

Sofia University

Krassimir D. Danov

Department of Chemical Engineering, Faculty of Chemistry

Sofia University, Sofia, Bulgaria

Lecture at COST D43 School Fluids and Solid Interfaces

Sofia University, Sofia, Bulgaria

12 – 15 April, 2011

Oscillatory structural forces measured by colloid probe AFM.

(1) Van der Waals surface force: 3H

vw6

)()(

h

hAh

The Hamaker parameter, AH, depends on the film thickness, h, because of the

electromagnetic retardation effect [1,4]. The expression for AH reads [4]:

0222/322

222eP)0()0()0(

H d)~

2exp()21(

)~

21(

)(

)(

4

3 , zzh

z

zh

nn

nnhAAAA

ji

jiijiijiiji

νe = 3.0 x 1015 Hz – main electronic absorption frequency;

hP = 6.6 x 10– 34 J.s – Planck’s const;

c0 = 3.0 x 108 m/s – speed of light in a vacuum.0

e2/122 )(2~

c

hnnnh jij

(2) Electrostatic (Double Layer) Surface Force (General Approach)

0w

b2

2

d

d

x

Poisson equation in the film phase relates the electrostatic

potential, , to the bulk charge density, b [2,5,7]:

All ionic species in the bulk with concentrations, nj, follow the

Boltzmann distribution (constant electro-chemical potentials):

)exp(0 kT

qznn jjj

where q is the elementary charge, zj is the charge number, nj0 is the input concentration.

The bulk charge density, b is [2,5]: j

jjj

jjj kT

qzqnzqnz )exp(0b

The first integral of the Poisson-Boltzmann equation reads: const.)d

d(

220w

xp

where p is the local osmotic pressure j

jj kT

qznkTp )exp(0

In the case of symmetric films the electrostatic disjoining pressure (repulsion), el, is

defined as a difference between the pressure in the film midplane, pm, and that at large film

thicknesses, p0 [5]:

j

jj kT

qznkTpp ]1)[exp( m

00mel

Eq. (2.1)

Eq. (2.2)

(2) Electrostatic (Double Layer) Surface Force (General Approach)

hpp

s

m

2/1m

2/10w

)(

d)2(

For constant surface potential, s, s and h

are known and m is calculated from:

where ps is the osmotic pressure in the subsurface phase (at = s).

Charge regulation. In this case the surface charge density, s, relates the

surface potential through the condition of constant electro-chemical

potentials [6] and

The surface charge density, s, is calculated from the charge balance at the film surface:

ms0w

2

0w

s

2/ 2

d

dpp

xs

hx

For constant surface charge the system of equations, Eqs. (2.1), (2.3), and (2.4), is solved

numerically to obtain s and m.

Eq. (2.4)

Eq. (2.3)

)( sss

For example: For (1:1) surface active ion “1”

and counterion “2” with adsorptions 1 and 2

)( 21s q

Counterion binding Stern isotherm (KSt

– Stern constant) leads to the equation

)]/(exp[1

)]/(exp[

s20St

s20St

1

2

kTqnK

kTqnK

(3) Equilibrium Film Thicknesses, h0: Theory vs. Experiment [8]

c0el0vw0 )()()( Phhh Sodium dodecyl sulfate (SDS) -

NaC12H25SO4, CMC 8 mM

Cetyl-trimethylammonium bromide (CTAB) - (C16H33)N(CH3)3Br, CMC 0.9 mM

Cetyl-pyridinium chloride (CPC) - (C21H38NCl), CMC 1.0 mM

(3) Disjoining Pressure Isotherms: Theory vs. Experiment [21]

Sodium dodecyl sulfate (SDS)

Hexa-trimethylammonium bromide (HTAB)

Setup for measurement of disjoining pressure, (h), isotherms (Mysels-Jones

porous plate cell [9]).

(3) Disjoining Pressure Isotherms: Experiments – no Theory

For small concentration of ionic surfactants the DLVO theory cannot explain experimental

data.

(3) Colloidal – Probe AFM Measurements of Disjoining Pressure [10]

Force, F, in nN for 80 mM Brij 35.

Micelle volume fraction 0.257.

Force/Radius, F/R, in mN.m-1 for 133 mM Brij 35. Micelle volume fraction 0.401.

The aggregation number of micelles is 70.

The solid lines are drawn without adjustable parameters (formulas by

Trokhimchuk et al. [11]).

(4) Hydrodynamic Interaction in Thin Liquid Films [2,3]

Two immobile surfaces of a symmetric film with

thickness h(t,r) approach each others with

velocity U(t). Rf is the characteristic film radius.

Simplest version of the lubrication approximation (h<<Rf):

2/

2/2

0)(1 h

hr

zr dz

rU

zr

rr

t

hU

where: t is time; r and z are the

radial and vertical coordinates.

Continuity equation:

Momentum balance equation is simplified to: ),( and 2

2

c rtppr

p

zr

Simple solution:r

p

r

hUhz

r

pr

c

322

c 6 and )4(

8

1

Hydrodynamic

force, F: fff

003

3

c0

d)(2d6d)](),([2RRR

rrhrh

rUrrhrtpF

(h) is the disjoining pressure, which accounts for the molecular interactions in the film.

(4) Taylor vs. Reynolds regimes [2,3]

In the case of two spheres (Taylor) [12]:

Uh

RF

R

rhh c

0

22

0 2

3 and

For two disks (Reynolds) [13]: Uh

RF c3

4f

2

3

The life time can be defined as: in

cr

d)(

1h

h

hhU

where hin is the initial thickness and

hcr is the final critical film thickness.

In the case of buoyancy force: )ln(8

9

cr

inc

h

h

gR

where g is the gravity constant and

us the density difference.

The life time decreases with the increase of drop radii.

In the case of buoyancy force :

)11

(2

and 2

2in

2cr

5c2f

hhgRR

RF

The life time increases with the

increase of drop radii.

(4) Taylor vs. Reynolds regimes

Taylor

regime

Dickinson experiments for the life

time of small drops (β-casein, κ-

casein or lysozyme, 10–4 wt%

protein + 100 mM NaCl, pH=7) [14].

Our experiments for the life time of small and

large drops [15]

(4x10-4 wt% BSA + 150 mM NaCl, pH=6.4).

Strong dependence of the drops life time on the drop and film radii

for tangentially immobile film surfaces.

(4) Lubrication Approximation and Film Profile [2,16]

Two immobile surfaces of a symmetric film

with thickness h(t,r) approach each others.

The film profile changes with time and pm is

the pressure in the meniscus.

0)12

(1

0)(1

c

3

r

prh

rrt

h

zr

rrz

r Continuity equation:

Normal stress boundary condition: )()(2

20 h

r

hr

rrp

Rp

Simple solution: )4(8

1 22

c

hzr

pr

Film-profile-evolution equation

(stiff nonlinear problem):

f

00 d])(),([2

R

rrphrtpF

0)]}()(2

[12

{1

c

3

hr

hr

rrr

rh

rrt

h

The applied force is given by the expression:

(4) Study of Drainage and Stability of Small Foam Films Using AFM

Microscopy photographs of bubbles in the AFM with schematics of the two interacting bubbles and

the water film between them [17]:

(A) Side view of the bubble anchored on the tip of the cantilever. (B) Plan view of the custom-made cantilever with the hydrophobized circular anchor. (C) Side perspective of the bubble on the substrate. (D) Bottom view of the bubble showing the dark circular contact zone of radius, a (in focus) on the substrate and the bubble of radius, Rs. (E) Schematic of the bubble geometry.

Evolution of film profiles

and rim rupture effect.

(5) Interfacial Dynamics and Rheology – Complex Boundary Conditions

The velocities of both phases are

equal at liquid/liquid interface S:0v

S

The jump of bulk forces at S are compensated by the total surface forces:

sss 2 TnnTn HpS

where Ts is the surface

viscous stress tensor.

Marangoni

effect

Capillary

pressure

Surface viscosity

effect

For Newtonian interfaces (Boussinesq – Scriven law) [16]:

trssssssssshsssshdils )()(2 and 2):)(( vIIvDDIDIT

where: Is is the surface idem factor;

dil – surface dilatational viscosity;

sh – surface shear viscosity.

Rate of relative displacement

of surface points

(5) Lubrication Approximation for Complex Fluids in the Films [18]

The film phase contains one surfactant with

bulk concentration, c, adsorption, , and

interfacial tension, .

The larger bulk and surface diffusivities lead to larger surface velocity (mobility)!

r

phuz

hu

h

hr

c

22/

2/12

d1

Integrated-surfactant-mass-balance equation:

0)]22([1

)2( ssss

r

cDh

rDuhcur

rrhc

t

Continuity equation for mobile surfaces:

cs – the subsurface concentration, u – the surface velocity,

the mean velocity is defined as:

For slow processes the deviations of concentrations and adsorptions are small and

ch

rh

hDDu

a

as and

ln)

2( Adsorption length (known from

the adsorption isotherm)

0)]12

([1

c

3

r

phhur

rrt

h

(5) Lubrication Approximation for Complex Fluids in the Film [19]

The larger Gibbs elasticity and surface viscosity suppress the surface mobility!

Tangential stress boundary condition (s = dil+sh – total interfacial viscosity):

)]([)( s

2/d

2/c ru

rrrrrzz hz

zr

hz

r

Normal stress boundary condition closes the problem for film evolution in time:

For slow processes the Marangoni term has an explicit form and

viscous friction

(film phase)

viscous friction

(drop phase)

Marangoni

effect

Boussinesq

effect

ln and )]([

2

2)(

2 Gs

sa

Ga

2/d

ErurrrhDDh

uEh

rzr

ph

hz

zr

The Gibbs elasticity, EG, is known from the surface equation of state or from independent

rheological experiments.

)()(2

22

2/d0 h

r

hr

rrp

zRp

hz

z

(5) Role of Surfactant on the Drainage Rate of Thin Films [19]

characteristic surface

diffusion length

}1]1)1

11ln()1(

)1({[

2

/1

1 2rf

s2rf

s4rfss

Re

Nh

h

bN

h

bh

Nh

h

hhbU

U

In the case of surfactants for this geometry we have:

Two truncated spheres In the case of two spheres

(Taylor velocity):

2cc

s0Ta

3

)(2

R

FFhU

In the case of two plates

(Reynolds velocity):

4c

s30

Re3

)(2

R

FFhU

c0

22rf

aG

c

G

scs ,

3 ,

6

Rh

RN

hE

Db

E

Dh

bulk diffusivity

number

dimensionless film

radius

In the case of two spherical drops: In the case of emulsion

plane parallel films:

)/1( sRe hhbUU 1s

s

sTa }1)

1

11ln(]1)1({[

2

h

h

bb

h

h

h

hUU

(5) Inverse Systems – Surfactants in the Disperse Phase

where: c – the density of liquid in the film phase;

Fs – force arising from the disjoining pressure;

– characteristic thickness of the boundary layer

in the drop phase.

Surface active components

in the disperse phase

In this case the diffusion fluxes from the disperse phase are

large enough to suppress the Marangoni effect and [3,20]

0c

d3/1

s40cc

43d

Re

])(

108[

hFFh

R

U

U

Surfactant in the continuous phase:

0.1M lauryl alcohol (1); 2 mM C8H18O3S (2).Surfactant

in the disperse

phase (benzene

films): C8H18O3S 0

mM (1); 0.1 mM

(2); 2 mM (3).

Film life time diagram

Film life time

diagram

Basic References

1. J.N. Israelachvili, Intermolecular and Surface Forces, Academic Press, London, 1992.

2. K.D. Danov, Effect of surfactants on drop stability and thin film drainage, in: V. Starov,

I.B. Ivanov (Eds.), Fluid Mechanics of Surfactant and Polymer Solutions, Springer, New

York, 2004, pp. 1–38.

3. P.A. Kralchevsky, K.D. Danov, N.D. Denkov. Chemical physics of colloid systems and

Interfaces, Chapter 7 in Handbook of Surface and Colloid Chemistry", (Third Edition;

K. S. Birdi, Ed.). CRC Press, Boca Raton, 2008; pp. 197-377.

Additional References

4. W.B. Russel, D.A. Saville, W.R. Schowalter, Colloidal Dispersions, Cambridge Univ.

Press, Cambridge, 1989.

5. B.V. Derjaguin, N.V. Churaev, V.M. Muller, Surface Forces, Plenum Press: Consultants

Bureau, New York, 1987.

6. P.A. Kralchevsky, K.D. Danov, G. Broze, A. Mehreteab, Thermodynamics of ionic

surfactant adsorption with account for the counterion binding: effect of salts of various

valency, Langmuir 15(7) (1999) 2351–2365.

7. L.D. Landau, E.M. Lifshitz, Electrodynamics of Continuous Media, Butterworth-Heinemann,

Oxford, 2004.

8. K.D. Danov, E.S. Basheva, P.A. Kralchevsky, K.P. Ananthapadmanabhan, A. Lips, The

metastable states of foam films containing electrically charged micelles or particles:

Experiment and quantitative interpretation, Adv. Colloid Interface Sci. (2011) – in press.

9. Mysels, K. J.; Jones, M. N. Direct Measurement of the Variation of Double-Layer

Repulsion with Distance. Discuss. Faraday Soc. 42 (1966) 42-50.

10. N.C. Christov, K.D. Danov, Y. Zeng, P.A. Kralchevsky, R. von Klitzing, Oscillatory

structural forces due to nonionic surfactant micelles: data by colloidal-probe AFM vs.

theory, Langmuir 26(2) (2010) 915–923 .

11. A. Trokhymchuk, D. Henderson, A. Nikolov, D.T. Wasan, A Simple Calculation of

Structural and Depletion Forces for Fluids/Suspensions Confined in a Film,

Langmuir 17 (2001) 4940-4947.

12. In fact, this solution does not appear in any G.I. Taylor’s publications but in the article by

W. Hardy, I. Bircumshaw, Proc. R. Soc. London A 108 (1925) 1 it was published.

13. O. Reynolds, On the theory of lubrication, Phil. Trans. Roy. Soc. (Lond.) A177 (1886) 157234.

14. E. Dickinson, B.S. Murray, G. Stainsby, Coalescence stability of emulsion-sized droplets at a planar oil-water interface and the relationship to protein film surface rheology, J.

Chem. Soc. Faraday Trans. 84 (1988) 871883.

15. T. D. Gurkov, E. S. Basheva, Hydrodynamic behavior and stability of approaching

deformable drops, in: A. T. Hubbard (Ed.), Encyclopedia of Surface & Colloid Science,

Marcel Dekker, New York, 2002.

16. D.A. Edwards, H. Brenner, D.T. Wasan, Interfacial Transport Processes and Rheology,

Butterworth-Heinemann, Boston, 1991.

17. I.U. Vakarelski, R. Manica, X. Tang, S.Y. O’Shea, G.W. Stevens, F. Grieser, R.R.

Dagastine, D.Y.C. Chan, Dynamic interactions between microbubbles in water, PNAS 107

(2010) 11177.

18. I.B. Ivanov, D.S. Dimitrov, Thin film drainage, Chapter 7, in: I.B Ivanov (ed.), Thin Liquid

Films, M. Dekker, New York, 1988.

19. K.D. Danov, D.S. Valkovska, I.B. Ivanov, Effect of surfactants on the film drainage, J.

Colloid Interface Sci. 211 (1999) 291–303.

20. I.B. Ivanov, Effect of surface mobility on the dynamic behavior of thin liquid film, Pure

Appl. Chem. 52 (1980) 12411262..

21. K.D. Danov, E.S. Basheva, P.A. Kralchevsky, The Hydration Surface Force – an Effect

Due to the Discreteness and Finite Size of Surface Ions and Bound Counterions.

Curr. Opin. Colloid Interface Sci. (2011) – a manuscript in preparation.