Numerical Simulation of Molding Hele-Shaw Flow of Polymeric Liquid Crystals

Abstract

To develop a general-purpose program for predicting the molding flow of polymeric liquid crystals, we present a basic model and its computational procedure. The flow is modeled by the Transversely Isotropic Fluid theory, which is equivalent to the Leslie-Ericksen equations in the high viscosity limit. In the modeling, the Hele-Shaw approximation is applied to reduce computational power. A finite difference technique is used to solve the governing equations, except for the angular momentum equation, which is solved by a streamline integration method. Two molds with thin and simple shape cavities are selected to evaluate the model. The computational results for the locations of the flow front, and for the distributions of the temperature and the molecular orientation show that the model successfully predicts a smooth molding process and that the molecular orientation direction depends strongly on the position in the gap direction. Since alignment of molecules is disordered by the occurrence of tumbling behavior, which depends on the fluid temperature and shear strain, the mold wall temperature and the gate position are important for effective molding.Therefore, the theoretical and numerical studies of PLCs are strongly expected.There exist two representative theories for the flow of liquid crystals; one is the Leslie-Ericksen (L-E) theory (10)(11) and the other is the Doi theory (12)(13) . Although the L-E theory is mainly applied to low molar mass liquid crystals and the Doi theory to PLCs, there is also a case where the L-E theory was used for PLC flows (14) . The Doi theory can explain the occurrence of the negative first normal stress difference and the peculiar behaviors of the director, but a huge computation task is required, because, in the theory, an orientation state is expressed in terms of an orientation distribution function (15)(16) . Thus, a reduced Doi theory, in which the orientation distribution function is approximated with a second-order tensor, has commonly been employed (17)-(24) . However, even this simpler theory is still too complicated for use as the basic equations of general-purpose simulation programs. On the other hand, since it is constructed with a vector, the L-E theory is comparatively easy to treat and is appropriate for general-purpose programs, although rigorous expression of the orientation state is inadequate.Our final goal is to develop a general-purpose simulation program which is able to predict the thin film molding flow of PLCs. In the present study, we propose a model in which the Transversely Isotropic Fluid (TIF) theory (25)(26) , obtained by simplifying the L-E theory, is used as a constitutive equation, and show some computation results. This study deals with only monodomain PLCs, because there is no theory that can satisfactorily represent the polydomain structure, and also, the polydomain structure is destroyed in a mold where a comparatively large velocity gradient is generated.

Fully decoupled unconditionally stable Crank–Nicolson leapfrog numerical methods for the Cahn–Hilliard–Darcy system

“…A prominent instability in this setting is the so-called Saffman-Taylor instability which occurs in the displacement a viscous fluid by a less viscous one in porous materials and exhibits fingering patterns-viscous fingering [28,39]. Such phenomena has important applications in oil recovery, infiltration, and many other fields including tumor growth in biomechanics [44], crystal solidification [48], electrowetting [37] and polymer liquid crystal techniques [8]. Due to the instability and the resulting topological changes of the interface, classical sharp interface models such as the Muskat problem could be ill-posed.…”

“…FIGURE8 Snapshots of phase variable for viscous fingering with respect to different viscosity ratios. (a) 𝜈 1 ∶ 𝜈 2 = 1 ∶ 5, t = 0.2, 0.4, 0.6, 0.9, 1.2, (b) 𝜈 1 ∶ 𝜈 2 = 1 ∶ 10, t = 0.2, 0.4, 0.6, 0.9, 1.2, (c) 𝜈 1 ∶ 𝜈 2 = 1 ∶ 20, t = 0.2, 0.4, 0.6, 0.9, 1.2.…”