Dilatational rheology of complex fluid-fluid
interfaces
V.I. Kovalchuk
Institute of Biocolloid Chemistry
of National Academy of Sciences of Ukraine,
03142 Kiev, Ukraine
1
Dilatational rheology of complex fluid-fluid
interfaces
Scope
Diffusion and mixed relaxation kinetics in adsorption layers
Effect of equilibrium thermodynamic properties
Relaxation in mixed adsorption layers
Particles at interfaces
Dilatational rheology of thin liquid films
Summary and conclusions
V.I. Kovalchuk, Dilatational rheology – Lorentz Workshop, Leiden-2011
2
Interfacial rheology
Expansion /
Compression
Elasticity
Shear
Viscosity
V.I. Kovalchuk, Dilatational rheology – Lorentz Workshop, Leiden-2011
3
Surface dilational modulus
A
E - characterizes the response of the
surface tension  against relative surface
area change A:
Ei 
 i
ln Ai
A(t) can be arbitrary function of time
Ei  E r   i  Ei 
ln Ai  Fln At 
E r  - surface dalational elasticity
 t-surface dalational
i 
E iF
d  
viscosity
t
t    Et  t ln At dt 

where
V.I. Kovalchuk, Dilatational rheology – Lorentz Workshop, Leiden-2011
Et   F1Ei
4
Purely diffusion relaxation of adsorption layers
Diffusion in the bulk phase:
A
C
 2C
D 2
t
y
Boundary conditions:
C  C0
y  
1 dA 
C
 D
A dt
y
y0
Initial condition:
 2 
Ct   C t 



Additional conditions – surface tension
and adsorption isotherms:
   
V.I. Kovalchuk, Dilatational rheology – Lorentz Workshop, Leiden-2011
  CS 
5
Purely diffusion relaxation of adsorption layers
Diffusion in the bulk phase:
A
C
 2C
D 2
t
y
Boundary conditions:
C  C0
y  
1 dA 
C
 D
A dt
y
y0
Initial condition:
 2 
Ct   C t 



Additional conditions – surface tension
and adsorption isotherms:
   
V.I. Kovalchuk, Dilatational rheology – Lorentz Workshop, Leiden-2011
  CS 
6
Purely diffusion relaxation of adsorption layers
Diffusion in the bulk phase:
C
 2C
D 2
t
y
Dilational elasticity modulus:
E
d
d ln 
Boundary conditions:
C  C0
y  
1 dA 
C
 D
A dt
y
y0
1
dC
1  1  i  S
d
D
2
or:
1    i
E  E0
1  2  2 2
Initial condition:
 2 
Ct   C t 




Additional conditions – surface tension
and adsorption isotherms:
   
  CS 
where:
E0  d / d ln 
1/ 2
 dC  D 
   S  
 d  2 
J. Lucassen and M. van den Tempel, Chem. Eng. Sci., 27 (1972) 1283; J. Colloid Interface Sci., 41 (1972) 491
V.I. Kovalchuk, Dilatational rheology – Lorentz Workshop, Leiden-2011
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Lucassen – van den Tempel model:
E r  E0
1  D / 
1  2 D /   2D / 
E0  d / d ln 
Ei  E0
D / 
1  2 D /   2D / 
- limiting elasticity
2
 dC  D
D   S 
 d  2
- characteristic frequency of diffusion relaxation
Maxwell model:
2


E r  E0
2
1  
Ei  E0

2
1  
Kelvin-Voigt model:
E  EKV  iKV
EKV  const KV  const
V.I. Kovalchuk, Dilatational rheology – Lorentz Workshop, Leiden-2011
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Cole-Cole plot:
Lucassen –
van den Tempel
model:
Maxwell
model
0.5
2
Ei
Lucassenvan den Tempel
model
0.0
0.0
0.5
1.0
Er
-0.5
2
E0  
E0 
E02

 E r     Ei   
2  
2 
2

E 
E
O 0 , 0 
2 
 2
R
E0
2
Maxwell model:
2
-1.0
E0 
E02

2
 Er    Ei  
2 
4

 E0 
O ,0 
 2 
V.I. Kovalchuk, Dilatational rheology – Lorentz Workshop, Leiden-2011
R
E0
2
9
Microgravity experiments during the STS-107
space shuttle mission
0.013 mmol/l
0.019 mmol/l
0.026 mmol/l
0.038 mmol/l
0.064 mmol/l
0.115 mmol/l
0.22 mmol/l
0.42 mmol/l
60
r , mN/m
40
20
0
0.01
0.1
1
10
100
Frequency, Hz
Real part of complex surface dilatational modulus vs. frequency for different C12DMPO
concentrations.
V.I. Kovalchuk et al. / Journal of Colloid and Interface Science 280 (2004) 498–505
V.I. Kovalchuk, Dilatational rheology – Lorentz Workshop, Leiden-2011
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Microgravity experiments during the STS-107
space shuttle mission
0.013 mmol/l
0.019 mmol/l
0.026 mmol/l
0.038 mmol/l
0.064 mmol/l
0.115 mmol/l
0.22 mmol/l
0.42 mmol/l
15
i , mN/m
10
5
0
0.01
0.1
1
10
100
Frequency, Hz
Imaginary part of complex surface dilatational modulus vs frequency for different
C12DMPO concentrations.
V.I. Kovalchuk et al. / Journal of Colloid and Interface Science 280 (2004) 498–505
V.I. Kovalchuk, Dilatational rheology – Lorentz Workshop, Leiden-2011
11
Microgravity experiments during the STS-107
space shuttle mission
0.013 mmol/l
0.019 mmol/l
0.026 mmol/l
0.038 mmol/l
0.064 mmol/l
0.115 mmol/l
0.22 mmol/l
0.015
Ei, mN/m
0.010
0.005
0.000
0.00
0.02
0.04
0.06
Er, mN/m
Cole-Cole diagram for complex surface dilatational modulus for different C12DMPO
concentrations.
V.I. Kovalchuk, Dilatational rheology – Lorentz Workshop, Leiden-2011
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Diffusion from two adjacent phases
E  E0
A
D
1    i
1  2  2 2
2
 dC  D
D   S  ef
 d  2
D
Effective diffusion coefficient:
Def 

D  K D

2
Surfactant distribution coefficient:
C 0
K
C 0
V.I. Kovalchuk, Dilatational rheology – Lorentz Workshop, Leiden-2011
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Mixed adsorption kinetics
Diffusion in the bulk phase:
C
 2C
D 2
t
y
  CS 
1 dA 
C
 D
A dt
y
- no equilibrium at the interface

d


 k adCS 1    k des
dt

  
i  (1  i) / 2
E  E0
1  i  (1  i) / 2
Relaxation time:
  kdes  kadc /  
1
B.A. Noskov, Adv. Colloid Interface Sci., 69 (1996) 63.
V.I. Kovalchuk, Dilatational rheology – Lorentz Workshop, Leiden-2011
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Micellar solutions – Lucassen equation
disturbance time is much larger than the characteristic time of the “slow process”
- for ordinary surfactants of the order of milliseconds and characterizes the
change in the number of micelles

E  E0 1  (1  i)1  m
2  DM / D
1/ 2
1  m 
2

1 / 2 1
- the ratio of micelles to monomers diffusion coefficients
m
- the aggregation number
  C0  CK  / CK , CK = CMC (critical micelle concentration)
1/ 2
 dCS  D 

 
d


 2 
Effective diffusion coefficient:
Deff  D1  m1  m2  D
J. Lucassen, Faraday Discuss. Chem. Soc., 59 (1975) 76.
V.I. Kovalchuk, Dilatational rheology – Lorentz Workshop, Leiden-2011
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E0  d / d ln 
- limiting elasticity
2
 dC  D
D   S 
 d  2
- characteristic frequency of diffusion relaxation
Effect of equilibrium thermodynamic properties
Equilibrium surface tension and adsorption isotherms:
   
  CS 
V.I. Kovalchuk, Dilatational rheology – Lorentz Workshop, Leiden-2011
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Frumkin adsorption model

bc 

RT
ln1    a2
0


exp(2a)
1  
- surface pressure isotherm (equation of state)
- adsorption isotherm
 =  0
- the surface coverage
0
- the molar area
a
- the interaction constant
b
- the adsorption equilibrium constant
V.I. Kovalchuk, Dilatational rheology – Lorentz Workshop, Leiden-2011
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Intrinsic 2D monolayer compressibility
150
1E-03
C12DMPO
C12DMPO
Theory:
Experiment:
Experiment:
E0 
E0, mN/m
100
d
d ln 
E 2r  E i2
E r  Ei
1E-05
50
0
0.001
d E r  E i

dc
Ei
1E-04
d/dc, m
E0  
D
2
Theory:
d
dCS
1E-06
0.01
0.1
1
0.001
c, mmol/l
0.01
0.1
1
c, mmol/l
Frumkin model: Γ = const, Ω = 1/Γ = const
Intrinsic compressibility model: Ω = 1/Γ = Ω0(1 – εΠ)
(ε – intrinsic 2D monolayer compressibility)
V.I. Kovalchuk et al. / Journal of Colloid and Interface Science 280 (2004) 498–505
V.I. Kovalchuk, Dilatational rheology – Lorentz Workshop, Leiden-2011
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Intrinsic 2D monolayer compressibility
 = 0(1 – εΠ)
ε – intrinsic 2D monolayer compressibility,  – surface pressure
The area occupied by a molecule on the interface can continuously change
with the surface pressure.
V.I. Kovalchuk, R. Miller, V.B. Fainerman and G. Loglio / Advances in Colloid and Interface Science, 114-115 (2005) 303.
V.I. Kovalchuk, Dilatational rheology – Lorentz Workshop, Leiden-2011
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Intrinsic 2D monolayer compressibility
150
30
E0, mN/m
, mN/m
100
20
50
10
0
1E-4
1E-3
1E-2
1E-1
c, mmol/l
0
0.001
0.01
0.1
1
c, mmol/l
Frumkin model: Γ = const, Ω = 1/Γ = const
Intrinsic compressibility model: Ω = 1/Γ = Ω0(1 – εΠ)
The surface rheological characteristics are much more sensitive to the state and
interaction of molecules in the adsorption layer than equilibrium isotherms!
V.I. Kovalchuk et al. / Journal of Colloid and Interface Science 280 (2004) 498–505
V.I. Kovalchuk, Dilatational rheology – Lorentz Workshop, Leiden-2011
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Intrinsic 2D monolayer compressibility C14TAB adsorption
Dependence of C14TAB adsorption  on activity c*: neutron reflection data ( )
and calculations according to Frumkin and compressibility model.
V.I. Kovalchuk, R. Miller, V.B. Fainerman and G. Loglio / Advances in Colloid and Interface Science, 114-115 (2005) 303.
V.I. Kovalchuk, Dilatational rheology – Lorentz Workshop, Leiden-2011
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Reorientation model
Adsorbed molecules can
acquire two (or more)
orientation states with respect
to the interface.

 0
2
 ln1   S   S  0   a  S 
RT
 1

10



exp

2
a

S 
1 / 0



1  S 
0


 2

20
2aS 
bc 
exp 

 2 / 0
2 / 1  1  S 
 0

bc 
V.B. Fainerman, S.A. Zholob, E.H. Lucassen-Reynders and R. Miller, J. Colloid Interface Sci., 261 (2003) 180.
V.I. Kovalchuk, Dilatational rheology – Lorentz Workshop, Leiden-2011
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Protein Adsorption Layers

 0
 ln1  P   P 1  0 /  P   a P 2P
RT
bPj c P 
P Pj
1  P 
 j / P
exp  2a P  j / P P 
n
P   P P   i Pi
- the total surface coverage
i 1
i  1  i  10
with
- the molar area in state i ( 1  i  n )
min  1 and max  1  n  10
V.B. Fainerman, E.H. Lucassen-Reynders and R. Miller, Adv. Colloid Interface Sci., 106 (2003) 237.
V.I. Kovalchuk, Dilatational rheology – Lorentz Workshop, Leiden-2011
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Relaxation in mixed adsorption layers
V.I. Kovalchuk, Dilatational rheology – Lorentz Workshop, Leiden-2011
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Viscoelasticity of mixed adsorption layers

 i
i
2
i
(a11a 22  a12 a 21 ) 

a12
a11 

1
D2
D1D 2

 D1
2 
1   

E  
B   ln 1  

i
i
1
1     i
 

(a11a 22  a12 a 21 )
a 22 
a 21 
D2
2
B   ln 2    D1
D1D 2

1
where: B  1  i D1 a11  i D2 a 22  (i
D1D2 )  (a11a 22  a12a 21 )
This expression includes 6 parameters determined from
surface tension and adsorption isotherms:
  


  ln 1  2 ,
 

  ln 2


 1
and
a ij  (i c j )
ck  j
Jiang Q, Valentini JE, Chiew YC. J. Colloid Interface Sci. 174 (1995) 268.
V.I. Kovalchuk, Dilatational rheology – Lorentz Workshop, Leiden-2011
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Viscoelasticity of mixed adsorption layers
1
E
B0
1
B0

   
D2
D1 2

 1 
d 22 
d12  
i
i 1
  ln 1  2 


   
D1
D 2 1

 1 
d11 
d 21 
i
i 2
  ln 2  1 

where: B0  1  d11 D1 / i  d22 D2 / i  (1/ i) D1D2 (d11d22  d12d21 )
This expression also includes 6 parameters determined from
surface tension and adsorption isotherms:
  


  ln 1  2 ,
 

  ln 2


 1
and
d ij  c i / j 
k j
P. Joos, Dynamic Surface Phenomena, 1999
V.I. Kovalchuk, Dilatational rheology – Lorentz Workshop, Leiden-2011
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Dilational elasticity of mixed adsorption layers:
Mixture of C10DMPO and C14DMPO
30
7/200
5/200
20
|E| (mN/m)
3/200
3/500
10
2/500
5/2000
0
1E-02
1E-01
f (Hz)
1E+00
C14DMPO/C10DMPO concentrations in µmol/l
E.V. Aksenenko, V.I. Kovalchuk, V.B. Fainerman and R. Miller / J. Phys. Chem. C, 111 (2007) 14713
V.I. Kovalchuk, Dilatational rheology – Lorentz Workshop, Leiden-2011
27
Mixtures of proteins and surfactants
Protein/
non-ionic
surfactant
Protein/
ionic
surfactant
Cs. Kotsmar, et al. / Advances in Colloid and Interface Science 150 (2009) 41–54
V.I. Kovalchuk, Dilatational rheology – Lorentz Workshop, Leiden-2011
28
Equation of state for protein/non-ionic surfactant mixtures
*0

 ln(1  P  S )  P (1  0 / P )   P 2P  SS2  2 PS P S
RT
where:
P = PP - total surface coverage by protein molecules;
S = SS - surface coverage by surfactant molecules;
Adsorption isotherms:
P P1
b P1c P 
exp 2 P (1 / P )P  2 PS S 
1 / P
1  P  S 
S
bScS 
exp 2SS  2 PS P 
1  P  S 
These tree equations allow one to calculate the necessary
6 partial derivatives
E.V. Aksenenko et al. / Advances in Colloid and Interface Science 122 (2006) 57–66
V.I. Kovalchuk, Dilatational rheology – Lorentz Workshop, Leiden-2011
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Dilational rheology of mixed adsorption layers:
Mixture of C10DMPO and β-lactoglobulin
100
|E|, mN/m
80
0.02; 0.04; 0.1; 0.2; 0.4; 0.7
60
40
20
0
1E-03
1E-02
1E-01
1E+00
1E+01
1E+02
f, Hz
Dilational elasticity modulus |E| vs. frequency f at various C10DMPO concentrations (in mmol/l) in the
β-LG/C10DMPO mixtures. Experimental data from R. Miller et al., Tens. Surf. Deterg. 40 (2003) 256.
V.I. Kovalchuk et al., in Progress in Colloid and Interface Science, Vol.1, Brill, Leiden-Boston, 2009, p. 332-371.
V.I. Kovalchuk, Dilatational rheology – Lorentz Workshop, Leiden-2011
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Particles at interfaces
Position of a spherical particle at the water/air interface (top) and
modification of particles via adsorption of ionic surfactants (bottom).
R. Miller et al: Project Proposal for the Investigation of Particle-Stabilised Emulsions and Foams by Microgravity Experiments,
Microgravity sci. technol. XVIII-3/4 (2006) 104-107
V.I. Kovalchuk, Dilatational rheology – Lorentz Workshop, Leiden-2011
31
Particles at the interface

kT       
ln1        coh

 0   A   A 
A
- the available surface area per particle
0
- the molar area of solvent molecules

- the molar area of particles
coh
- the cohesion pressure
V.B. Fainerman, V.I. Kovalchuk, D.O. Grigoriev, M.E. Leser and R. Miller / NATO Science Series, Vol. 228, 2006, P. 79-90
V.I. Kovalchuk, Dilatational rheology – Lorentz Workshop, Leiden-2011
32
Surface pressure in a monolayer
of polymeric particles
Surface Pressure [mN/m]
50
40
30
20
10
0
0
10
20
30
40
50
Area [m2/g]
Dependence of surface pressure on the monolayer coverage for polymeric particles 113 nm in
diameter without dispersant (▲) and with dispersant (■). Experimental data according to E. Wolert
et al., Langmuir 17 (2001) 5671.
V.B. Fainerman, V.I. Kovalchuk, D.O. Grigoriev, M.E. Leser and R. Miller / NATO Science Series, Vol. 228, 2006, P. 79-90
V.I. Kovalchuk, Dilatational rheology – Lorentz Workshop, Leiden-2011
33
Particles at the interface
Apparent dilatation modulus of composite monolayers:
n
1
Xi

E 1 Ei
Xi is the surface fraction having the dilatational modulus Ei.
Corresponds to the case of particles which do not interact and do not
move but are characterized by a certain internal compressibility.
For incompressible particles:
E
ES
ES

XS 1  P
- excluded area effect
P is the surface coverage for particles, Es is the local elasticity of
interparticle space (e.g. covered by surfactants).
J. Lucassen, Colloids Surfaces, 65(1992) 139
V.I. Kovalchuk, Dilatational rheology – Lorentz Workshop, Leiden-2011
34
Particles at the interface
Alternatively, for a mixed particle-surfactant layers the surface elasticity can be
obtained by considering the surface pressure as a function of two variables:
S ,P 
where θP and θS is the surface coverage by particles and surfactant.
   d ln S    d ln  P
d
 
 
E
 
 
d ln A
  ln S  P d ln A   ln  P  S d ln A
For insoluble surfactant molecules and particles:
d ln S d ln  P

 1
d ln A d ln A
and
  
  



E  ES  E P  
 

  ln S  P   ln P S
R. Miller et al., Adv. Colloid Interface Sci., 128–130 (2006) 17–26
V.I. Kovalchuk, Dilatational rheology – Lorentz Workshop, Leiden-2011
35
Particles at the interface
For a mixed particle-surfactant layers described by the surface pressure isotherm:


kT

ln1  P  S    P  S (1  0 / S )  a S S2   coh (c S )
0
the partial elasticities are:

 d coh
 0 
S
2


ES  E 0 
 S 1 
 2a SS  

 S 
1  P  S
 d ln S
P P  S 
EP  E0
1  P  S
For fast oscillations:
For slow oscillations:
with
E0  kT / 0
E  ES  EP

E  E P  P ES
1  P
R. Miller et al., Adv. Colloid Interface Sci., 128–130 (2006) 17–26
V.I. Kovalchuk, Dilatational rheology – Lorentz Workshop, Leiden-2011
36
Dilatational rheology of thin liquid films
V.I. Kovalchuk, Dilatational rheology – Lorentz Workshop, Leiden-2011
37
Dilatational rheology of thin liquid films
F
F
F
F
Δγf
Δγf
Film elasticity:
d f
Ef 
d ln A
V.I. Kovalchuk, Dilatational rheology – Lorentz Workshop, Leiden-2011
γf = 2γ
Δγf ≠ 2Δγ
Ef ≠ 2E
38
Dilatational rheology of thin liquid films
300
C10-OH
250
Ef, mN/m
200
150
h0 = 10 m
0 = 42 mN/m
C8-OH
100
50
hcr
C6-OH
h0
0
1E-7
1E-6
1E-5
h, m
Film elasticity vs. thickness dependencies for normal alcohols: n-hexanol, n-octanol, n-decanol;
films with initial surface pressure 0 = 42 mN/m and initial thickness h0 = 10 m
V.I. Kovalchuk et al., in Progress in Colloid and Interface Science, Vol.1, Brill, Leiden-Boston, 2009, p.476-518.
V.I. Kovalchuk, Dilatational rheology – Lorentz Workshop, Leiden-2011
39
Film elasticity – time effect of disturbances
300
2E0
 dc D
 h i 

E f  2E 0 1 
tanh

 2 D 
 d i
1
|Ef|, Ei mN/m
200
1000 Hz
100
hcr
h0
100 Hz
10 Hz
0
1E-6
1E-5
EGibbs
h, m
Effect of characteristic disturbance time on film elasticity modulus E f (full lines) and its imaginary part
Ei (dotted lines). Frumkin isotherm with the parameters for C8 (octanol), h0 = 10 m, 0 = 42 mN/m.
V.I. Kovalchuk et al., in Progress in Colloid and Interface Science, Vol.1, Brill, Leiden-Boston, 2009, p.476-518.
V.I. Kovalchuk, Dilatational rheology – Lorentz Workshop, Leiden-2011
40
Summary and conclusions
Surface tension studies provide general information about the formation
of adsorption layers.
However, interfacial rheology gives more insight into the details of single
and mixed adsorption layers.
The study of interfacial rheological properties represents a versatile and
very sensitive experimental tool to investigate the adsorption layer
properties. This techniques requires, however, quantitative theories
combining interfacial dynamics and mass transfer aspects.
V.I. Kovalchuk, Dilatational rheology – Lorentz Workshop, Leiden-2011
41
Acknowledgements
Financial support by
Max-Planck-Institute of Colloids and Interfaces
and
COST D-43 Action
is gratefully acknowledged.
August 2008
V.I. Kovalchuk, Dilatational rheology – Lorentz Workshop, Leiden-2011
42