A Novel Approach to the Normal Force in Shear of Nematic Liquid Crystals in the Cone-and-Plate Rheometer

Abstract

We reexamine the interpretation of the normal force measured with a cone-and-plate rheometer for nematic liquid crystals. We point out the fact that the quite widespread belief that the normal force is directly related to the well-known rheological function 𝒩₁ (i.e., the first normal stress difference) fails for these complex fluids. After a brief presentation of the theoretical bases leading to the general expression of the normal force, this new approach is applied to nematic liquid crystals within the framework of Leslie-Ericksen theory. In order to avoid heavy numerical computations the full Leslie-Ericksen equations describing the three-dimensional flow within the cone-and-plate cell are reduced, thanks to reasonable approximations, to an effective one-dimensional problem (i.e., a system of partial differential equations with spatial derivatives with respect to one single coordinate). Results for low molecular weight liquid crystals are presented and discussed. The main feature evidenced in this work is the notable difference between the normal force calculated in this way and the 𝒩₁ function, in particular the normal force function becomes negative for high shear rate while 𝒩₁ stays positive.

Keywords:

• cone-and-plate
• Leslie-Ericksen
• nematic
• normal force
• shear

Acknowledgments

This work was partly supported by “Fundação para a Ciência e a Tecnologia” through a research grant to A. Véron.

Notes

¹ r: radius; θ: polar angle; φ: azimuthal angle.

²The Leslie viscosities satisfy the Parodi relation α₂ + α₃ = α₆ − α₅.

³In practice, the angle between the cone and the plate is very small, in the order of 10⁻³ rad.

⁴The case j = 3 is given by Eq. (I-3) because of axial symmetry.

⁵This choice is complete with an initial homogeneous condition exhibiting axial symmetry.

⁶From Eq. (15) we have and .

⁷The mathematical foundations leading to Eq. (IV-1) may be found in [10].