Macroscopic Internal Variables and Mesoscopic Theory: A Comparison considering Liquid Crystals

Internal and mesoscopic variables differ from each other fundamentally: both are state space variables, but mesoscopic variables are additional equipped with a distribution function introducing a statistical item into consideration which is missing in connection with internal variables. Thus, the alignment tensor of liquid crystal theory can be introduced as an internal variable or as one generated by a mesoscopic background using the microscopic director as mesoscopic variable. Because the mesoscopic variable is part of the state space, the corresponding balance equations change into mesoscopic balances, and additionally an evolution equation of the mesoscopic distribution function appears. The flexibility of the mesoscopic concept is not only demonstrated for liquid crystals, but is also discussed for dipolar media and flexible fibers.

Characterization and Alignment of Carbon Nanofibers under Shear Force in Microchannel

This work presents a novel method to align CNFs by using shear forces in microchannels. Effect of two different microchannel sizes (1 mm × 0.1 mm and 1 mm × 0.2 mm) on CNFs alignment is investigated. SEM images of CNFs preform display significant alignment by both microchannels, which can be interpreted using a second-order alignment tensor and a manual angle meter. In the second-order alignment tensor description, an elongated ellipse can signify high degree of alignment in the direction of the major axis. When the microchannel size is 1 mm × 0.2 mm, the lengths of major and minor axes of the ellipse are 0.982 to 0.018. An angle meter manually shows that 85% of the CNFs are aligned in the direction between 60° and 90° when the microchannel is 1 mm × 0.2 mm. Both methods can demonstrate that better alignment of CNFs can be obtained using the 1 mm × 0.2 mm microchannel.

Continuum Dislocation Dynamics Based on the Second Order Alignment Tensor

ABSTRACTIn the current paper we present a continuum theory of dislocations based on the second-order alignment tensor in conjunction with the classical dislocation density tensor (Kröner-Nye-tensor) and a scalar dislocation curvature measure. The second-order alignment tensor is a symmetric second order tensor characterizing the orientation distribution of dislocations in elliptic form. It is closely connected to total densities of screw and edge dislocations introduced in the literature. The scalar dislocation curvature density is a conserved quantity the integral of which represents the total number of dislocations in the system. The presented evolution equations of these dislocation density measures partly parallel earlier developed theories based on screw-edge decompositions but handle line length changes and segment reorientation consistently. We demonstrate that the presented equations allow predicting the evolution of a single dislocation loop in a non-trivial velocity field.

The helicity and vorticity of liquid-crystal flows

We present explicit expressions of the helicity conservation in nematic liquid-crystal flows, for both the Ericksen–Leslie and Landau–de Gennes theories. This is done by using a minimal coupling argument that leads to an Euler-like equation for a modified vorticity involving both velocity and structure fields (e.g. director and alignment tensor). This equation for the modified vorticity shares many relevant properties with ideal fluid dynamics, and it allows for vortex-filament configurations, as well as point vortices, in two dimensions. We extend all these results to particles of arbitrary shape by considering systems with fully broken rotational symmetry.