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DYNAMIC DROP TENSIOMETER
The measurement of the dilatational rheological
properties of the interfaces

 

I/ INTERFACIAL RHEOLOGY

 

a) Shear and dilatational rheology

Rheological interfacial parameters can be defined for :

  • shearing deformation

  • Area compression/expansion

Shear

Dilatation

 

 

b) Rheological interfacial parameters

Surface dilatational modulus in compression and expansion is defined by :

 

E = dg / (dA/A) = dg / dln(A)

g is the interfacial tension

A is the area of the interface

 

 

In oscillatory experiments E is a complex quantity :

E = I E I.cos (q) + i. I E I sin(q)

q = phase angle

 

 

 

Elastic component :

E' = I E I.cos(q)

 

Viscous component :

E'' = I E I.sin(q) / w

 

with w =2.p.f

f = frequency

 

II/ PRINCIPLE OF MEASUREMENTS

 

a) The Dynamic Drop Tensiometer (DDT)

Two equations are used :

1) Laplace -Young equation :

 

DP is the pressure difference at the interface

g is the interfacial tension

R and R' are the radii of curvature

 

 

2) Forces equilibrium througt every horizontal plane

 

 

Dp : Pressure difference at the interface

g : Interfacial tension

V : Volume under the horizontal plane

rh et rl : Densities of the two fluids

g : Gravity

 

The shape of the drop depend on only one parameter w :

 

 

Shape factor

or

Bond number

 

Dr : Difference of densities of the two fluids

g : Gravity

b = 1/r (r is the apex radius of curvature).

c : Capillary constant

 

 

b) Surface rheology measurements

In oscillatory experiments E considered as a transfert function :

 

 

At a given frequency E is a complex number :

 

FT(dg) = E.FT(dln(A))

 

FT means Fourier transform

We can write :

 

 

E : Modulus

q : Phase angle