Surface Forces and Liquid Films (Continued)
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.
Sofia University
(1) Van der Waals surface force:
AH (h)
 vw (h)  
6 h 3
The Hamaker parameter, AH, depends on the film thickness, h, because of the
electromagnetic retardation effect [1,4]. The expression for AH reads [4]:
AH  Aiji( 0)  Aiji( 0) , Aiji( 0)
~
2
2 2 
~
3hP e (ni  n j )
(1  2h z )

exp( 2h z ) d z
2
2 3/ 2 
2 2
4 (ni  n j ) 0 (1  2 z )
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.
~
 h
h  2 n j (ni2  n 2j )1 / 2 e
c0
(2) Electrostatic (Double Layer) Surface Force (General Approach)
b
d2

2
 w 0
dx
z j q
n j  n j 0 exp( 
)
kT
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):
where q is the elementary charge, zj is the charge number, nj0 is the input concentration.
The bulk charge density, b is [2,5]:
 b   z j qn j   z j qn j 0 exp( 
j
j
The first integral of the Poisson-Boltzmann equation reads:
p
 w  0 d
(
z j q
kT
)
) 2  const.
2 dx
z j q
p  kT  n j 0 exp( 
) Eq. (2.1)
kT
j
where p is the local osmotic pressure
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]:
 el  pm  p0  kT  n j 0 [exp( 
j
z j q m
kT
)  1]
Eq. (2.2)
(2) Electrostatic (Double Layer) Surface Force (General Approach)
For constant surface potential, s, s and
h are known and m is calculated from:
(2 w 0 )1 / 2
s

m
d
 h
1/ 2
( p  pm )
Eq. (2.3)
The surface charge density, s, is calculated from the charge balance at the film surface:
d
dx
x h / 2
s
 s2


 ps  pm
 w 0
2 w  0
Eq. (2.4)
where ps is the osmotic pressure in the subsurface phase (at  = s).
For constant surface charge the system of equations, Eqs. (2.1), (2.3), and (2.4), is solved
numerically to obtain s and m.
Charge regulation. In this case the surface charge density, s, relates the
surface potential through the condition of constant electro-chemical
potentials [6] and
For example: For (1:1) surface active ion “1”
and counterion “2” with adsorptions G1 and G2
s  q(G1  G2 )
s  s ( s )
Counterion binding Stern isotherm (KSt
– Stern constant) leads to the equation
K St n20 exp[ q s /( kT )]
G2

G1 1  KSt n20 exp[ q s /( kT )]
(3) Equilibrium Film Thicknesses, h0: Theory vs. Experiment [8]
(h0 )   vw (h0 )   el (h0 )  Pc
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]
Setup for measurement of disjoining
pressure, (h), isotherms (Mysels-Jones
porous plate cell [9]).
Sodium dodecyl sulfate (SDS)
Hexa-trimethylammonium bromide (HTAB)
(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.
h
U 
t
where: t is time; r and z are the
radial and vertical coordinates.
Simplest version of the lubrication approximation (h<<Rf):
1 

r
Continuity equation:
(r r )  z  0  U    r dz
r r
z
2 h / 2
2

 p
Momentum balance equation is simplified to:
c 2r 
and p  p(t , r )
r
z
3
1 p
h p
2
2
Simple solution:  
(
4
z

h
)
and
U


r
8 c r
6r c r
h/2
Hydrodynamic
force, F:
Rf
Rf
R
f
r3
F  2  [ p (t , r )   (h)]r d r  6cU  3 d r  2   (h)r d r
0
0 h
0
(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]:
r2
3 R 2
h  h0 
and F 
cU
R
2h0
hin
The life time can be defined as:
1

dh
U ( h)
h
where hin is the initial thickness and
hcr is the final critical film thickness.
cr
In the case of buoyancy force:
9 c
hin

ln( )
8 gR
hcr
where g is the gravity constant and
 us the density difference.
The life time decreases with the increase of drop radii.
For two disks (Reynolds) [13]:
3 Rf4
F
cU
3
2h
In the case of buoyancy force :
The life time increases with the
increase of drop radii.
c 5
2 2
1
1
F 
Rf and  
gR  ( 2  2 )
R
2
hcr hin
(4) Taylor vs. Reynolds regimes
Taylor
regime
Dickinson experiments for the life
Our experiments for the life time of small and
time of small drops (-casein,  -
large drops [15]
casein or lysozyme, 10–4 wt%
protein + 100 mM NaCl, pH=7) [14].
(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.
Simple solution:
1 p
r 
(4 z 2  h 2 )
8 c r
Continuity equation:
1 
 z
h 1  rh 3 p
(r r ) 
0 

(
)0
r r
z
t r r 12 c r
Normal stress boundary condition:
Film-profile-evolution equation
(stiff nonlinear problem):
2
  h
p0 
 p
( r )   ( h)
R
2r r r
h 1  rh 3    h

{
[
(r )   (h)]}  0
t r r 12 c r 2r r r
Rf
The applied force is given by the expression:
F  2  [ p(t , r )   (h)  p0 ]r d r
0
(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:
 vS 0
The jump of bulk forces at S are compensated by the total surface forces:
  pn  T  n  S 
2 H n
 s
 s  Ts
where Ts is the surface
Capillary
Marangoni
Surface viscosity
viscous stress tensor.
pressure
effect
effect
For Newtonian interfaces (Boussinesq – Scriven law) [16]:
Ts  (dil sh )(Is : Ds )Is  2sh Ds and 2 Ds  (s vs )  Is  Is  (s vs ) tr
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, G, and
interfacial tension, .
Integrated-surfactant-mass-balance equation:
cs

1 
G
(2G  hcs ) 
[r (2Gu  hcsu  2 Ds
 Dh
)]  0
t
r r
r
r
cs – the subsurface concentration, u – the surface velocity,
the mean velocity is defined as:
1
h 2 p
u
r d z  u 

h h / 2
12 c r
h/2
For slow processes the deviations of concentrations and adsorptions are small and
u  ( Ds  D
h  ln G
G
)
and ha 
2ha
r
c
Adsorption length (known from
the adsorption isotherm)
The larger bulk and surface diffusivities lead to larger surface velocity (mobility)!
Continuity equation for mobile surfaces:
h 1 
h 3 p

[r (hu 
)]  0
t r r
12 c r
(5) Lubrication Approximation for Complex Fluids in the Film [19]
Tangential stress boundary condition (ms = mdil+msh – total interfacial viscosity):
r
c
z
r  z
  s 
 d (

)

 [
(ru )]
z
r z h / 2 r r r r
z h / 2
viscous friction
viscous friction
Marangoni
Boussinesq
(film phase)
(drop phase)
effect
effect
For slow processes the Marangoni term has an explicit form and
2ha EGu
h p
 
  

 d ( r  z )

 [ s
(ru )] and EG  
2 r
z
r z h / 2
2ha Ds  hD r r r
 ln G
The Gibbs elasticity, EG, is known from the surface equation of state or from
independent rheological experiments.
The larger Gibbs elasticity and surface viscosity suppress the surface mobility!
Normal stress boundary condition closes the problem for film evolution in time:
p0 
2

 2d z
R
z
 p
z h / 2
 
2r r
(r
h
)   ( h)
r
(5) Role of Surfactant on the Drainage Rate of Thin Films [19]
Two truncated spheres
In the case of two spheres
In the case of two plates
(Taylor velocity):
(Reynolds velocity):
2h ( F  Fs )
UT a  0
3c Rc2
U Re
2h03 ( F  Fs )

3c R 4
In the case of surfactants for this geometry we have:
6 D
hs  c s
EG
characteristic surface
3 D
b c
EG ha
,
,
N rf2
R2

h0 Rc
bulk diffusivity dimensionless film
diffusion length
number
radius
U Re
1
2h
h(1  b)
1 hs
2
2


{[
(
1

N
)
ln(
1

)

1
]

N
rf
rf  1}
4
U
1  b  hs / h hs N rf
hs
1 b h
In the case of two spherical drops:
In the case of emulsion
plane parallel films:
hs h
1 hs
U  U T a {[ (1  b)  1] ln( 1 
)  1}1
2h hs
1 b h
U  U Re (1  b  hs / h)
(5) Inverse Systems – Surfactants 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]
108d3 R 4 1 / 3 dd
U
[
] 
4
U Re
c h0
cc h0 ( F  Fs )
where: c – the density of liquid in the film phase;
Surface active components
in the disperse phase
Fs – force arising from the disjoining pressure;
d – characteristic thickness of the boundary layer
in the drop phase.
Surfactant in the continuous phase:
Surfactant
in the disperse
Film life time
phase (benzene
diagram
films): C8H18O3S
0 mM (1); 0.1 mM
(2); 2 mM (3).
0.1M lauryl alcohol (1); 2 mM C8H18O3S (2).
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, ButterworthHeinemann, 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)
157234.
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) 871883.
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) 12411262.
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.