Palierne equation

Palierne equation connects the dynamic modulus of emulsions with the dynamic modulus of the two phases, size of the droplets and the interphase surface tension. The equation can also be used for suspensions of viscoelastic solid particles in viscoelastic fluids. The equation is named after French rheologist Jean-François Palierne, who proposed the equation in 1991.
For the dilute emulsions Palierne equation looks like:
where formula_2 is the dynamic modulus of the emulsion, formula_3 is the dynamic modulus of the continuous phase (matrix), formula_4 is the volume fraction of the disperse phase and the formula_5 is given as
where formula_7 is the dynamic modulus of the disperse phase, formula_8 is the surface tension between the phases and formula_9 is the radius of the droplets.
For the suspension of solid particles the value of formula_5 is given as
The Palierne equation is usually extended for the finite volume concentrations of the disperse phase formula_4 as: