Model for the shear modulus

In previous publications the shear modulus for the multilamellar phases was considered to be the result of the interactions of hard sphere particles [46-48]. In this picture each charged multilamellar vesicle is treated as a hard sphere. The theoretical treatment of the samples would then be similar to latex systems. The modulus of the systems depends on the chainlength of the surfactants that are used for the preparation of the systems if all other parameters like charge density, salinity, and concentration of surfactants and cosurfactants are kept constant. It can be argued that the differences of the moduli result from a change of the particle density of the vesicles. But these values are not known exactly. Systems with different chainlengths have similar conductivities which suggests that the particle density is also similar and therefore not responsible for the different shear moduli. 

Furthermore the birefringence looks the same, too. If the particle density of the vesicles decreases and if the mean size of the vesicles increases then the birefringence should increase. But this is not the case. The different moduli must therefore have a different origin. We propose, consequently, a different model for the explanation of the magnitude of the shear moduli. For the treatment of multilamellar vesicle phases and Lq phases this model was proposed by E. v. d. 

Linden [49]. To our knowledge the theory has not yet been applied to experimental results. E. v. d. Linden assumes that multilamellar vesicles (droplets) are deformed in shear flow from a spherical to an elliptical shape. Turning into the deformed state the energy of closed shells is shifted because their curvature as well as their interlamellar distance D are changed. Due to the interaction of the bilayers, expressed by the bulk compression modulus B, the inner shells are deformed and the total deformation energy of the lamellar droplet gets minimized. Assuming that the volume of a droplet is not modified by the deformation, the surface A must increase. One can define an effective surface tension (Tef[=ElAA. E. v. d. Linden obtains  

bulk compression modulus and bending constant, depend on the charge density of the bilayers and the shielding of the charges with excess salt. 

This means the theory of E. v. d. Linden results in a calculation of the geometrical average of the compression and bending energy k per unit volume. 

---