Princen-Kiss model

Another important rheological property of dry foams and highly concentrated emulsions is G, the shear modulus. Princen and Kiss [57] demonstrated that this property was dependent on < >, the volume fraction of the system. Previously, Stamenovic et al. [58] and, much earlier, Derjaguin and coworker [59], had derived an expression for the shear modulus of foams of volume fraction very close to unity. The value was found to depend on the surface tension of the liquid phase (in foams), for the particular case of (Jja 1. However, Princen demonstrated that the values of G obtained were overestimated by a factor of two. This error was attributed to the model used by Stamenovic and coworker, which failed to maintain the equilibrium condition that three films always meet at angles of 120° during deformation. 

As it is seen from Eq. (8.23), the volumetric equilibrium model gives a value of the yield stress considerably smaller than that for a two-dimensional system (Eqs. (8.16) - (8.18)). On the basis of experiments with emulsions Princen and Kiss [13] have derived the following expression about the yield stress... 

The G -0 curves can be analysed using the model suggested by Princen and Kiss... 

The validity of Princen s theory for concentrated water-in-oil emulsions was also investigated by Ponton et al. (2001), using the droplet size distribution determined by laser diffractometry based on the Mie theory model. Comparing the surface-volume diameter and the mass fractions of emulsions depicted an increase in the particle size with the volume fraction reduction. They showed that their experimental data (as obtained by oscillatory measurements and droplet-size distribution) corroborated the expression of the elastic shear modulus for the two-dimensional model proposed by Princen and Kiss (1986). In this model, G is proportional to (a/Rsv) l v ( l v- l c) where a is the interfacial tension, Rsv is the volume-surface radius (as obtained by laser diffractometry), and Oy and Oc are the volume fraction and the critical volume fraction, respectively (Ponton et al. 2001). The latter was found to be 0.714 experimentally, which is close to the value obtained by Princen ( 0.712) (Ponton et al. 2001). 

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