Orientation Texture Formation during Plane Poiseuille Flow of a Nematic Polymer ( 네마틱 고분자의 평판 포아셀 흐름에서의 배향 조직 형성)

한원희*, 이재욱1

경북 영주시 풍기읍 동양대학교 화학공학과*

서울 마포구 서강대학교 화학공학과1

Corresponding Author 이메일: whhan@phenix.dyu.ac.kr

Won Hee HAN*, Jae Wook Lee1

Department of Chemical Engineering, Dong Yang University, YoungJou, Kyungbook*, Department of Chemical Engineering, Sogang University, Seoul1

Corresponding Author e-mail: whhan@phenix.dyu.ac.kr

We give a nonlinear numerical analysis of orientation texture formation of a nematic polymer (polybenzyl glutamate) during Plane Poiseuille flow. Leslie-Ericksen equation with finite element method has been used to obtain the time dependent orientation texture evolutions during Plane Poiseuille flow. Bounding surfaces have homeotropic anchoring condition in which average molecular orientation is perpendicular to the surface. Inlet and outlet have zero surface couple condition. As the flow rate increases orientation texture changes from uniform orientation to alternatively twisting orientation along the flow direction, which is similar to the shear flow case. Further flow rate increase results in herring-bone type orientation texture formation which is unlike the tubular orientation walls during shear flow.

INTRODUCTION

Nematic liquid crystals are classfied into two types according to their orientation aligning behavior during shear flow; one is flow aligning type and the other non-aligning type (De Gennes[1]). For the flow aligning type, director orientation, or the local average molecular orientation, becomes spatially uniform during shear flow making some small angle between the stream line and the director orientation. Usually, low molar mass nematics show flow aligning behavior while high molar mass nematics show flow non-aligning behavior. One important characterisitcs of non-aligning type nematics is the director escaping from the shear plane spanned by the shear flow axis and the thickness axis. Han[2] showed that many pattern formation phenomena in shear flow of nematics have their origin in the non-aligning behavior in which no orientation angle can diminish the viscous torque of director.

The present study show orientation texture formation during plane Poiseuille flow, which is similar to the shear flow case in that the director orientation cannot be constrained within the shear plane as the flow rate increases. The plane poiseuille flow of a nematic polymer (polybenzyl glutamate) give rise to some amazing herringbone-like texture not seen in the shear flow computations as will be introduced below.

THEORY

We use the Leslie-Ericksen (LE) continuum theory for computing the orientation profile evolution during plane Poiseuille flow. The LE theory consists of the linear momentum

balance and the torque balance of the unit length vector n=(nx,ny,nz), director, representing local average orientation. The details of LE equation are found in De Gennes[1]. Therefore, we are not going to elaborate on the LE equation details. The orientation behavior during flow is controlled by the Ericksen number E, a characteristic ratio between the viscous and elastic torques given as E=□1DU/(K11K22K33)1/3 where □1, U and D(=3.5X10-4m) denote the twist viscosity, maximum center velocity and the gap thickness respectively. We use the material constant data of Srajer, et. al.[3] for the computation.

We use the Galerkin finite element method and a fully implicit time integration method with 60 by 60 quadrilateral bilinear elements to obtain the director field, n=(nx(x,y,t), ny(x,y,t), nz(x,y,t)) under simple plane poiseuille flow kinematics v=(D(h-y)y□(t),0,0) where □ is the Heavyside step function. At t=0, a uniform homeotropic director field is randomly perturbed with a maximum perturbation □ni2=10-4, i=x,y,z. The fixed homeotropic anchoring condition is used as n=(0,1,0) at y=0 and D. To minimize the inlet and outlet end effects of finite length computational domain, a large aspect ratio (L/D) of 16 was used. At the inlet and outlet, vanishing contact torque condition convenient for implementing the finite element method was used.

RESULTS AND DISCUSSIONS

The critical Ericksen number Ec is defined as the Ericksen number at which director orientation can no longer constrained within the shear plane spanned by the flow (x) axis and the thickness (y) axis. For the present plane Poiseuille flow case, Ec is about 80,000. Figure 1 shows z-component of the director along the centerline (y/D=0.5) at different time for E=81700. As the time goes by, z-component of the director becomes nonzero meaning that the directors escape the x-y plane. Since the escape can be of either left or right alternating director escape along the flow direction is obtained. Outside the centerline, directors are mostly within the x-y plane and pointing parallel to the flow direction.

(a)

(b)

Figuer 4. Density plots of nz component of director. Flow occurs from left to right in horizontal direction. The aspect ratio (L/D) is 16. Black region represents nz=-1, white region nz=+1, medium gray region nz=0. Other values of nz is represented with shades in such a way that region of larger values are lighter.

Figure 2 shows the z-component director evolution along the centerline for higher Ericsen number. Now we have much more complex and short period alternation of director escape sense. For even higher Ericksen number, z-component of the director shows similary complex pattern evolution as shown in Figure 3. However, as shown in Figure 4(a) and (b) density plots, where black means nz=-1, white nz=+1, medium gray nz=0, outside the centerline the steady state orientation texture (a) corresponding to Figure 2 has alternating director escapes occuring only near the centerline while the orientation texture (b) corresponding to Figure 3 has director escapes outside the center region. Figure 4(b) resembles herringbone. Unlike the shear flow case shown in [4,5], pairs of tubular orientation walls are joined in the center stem-like orientation wall and the tubular walls consist of black and white regions. This computed amazing herringbone texture is waiting for the due experimental validation. A further extensive study will bring more light on the fundamental understanding of the computed phenomena.

REFERENCES

[1] P. G. De Gennes, The physics of liquid crystals, (Clarendon Press, Oxford 1974).

[2] W. H. Han, Ph.D. Dissertation, McGill University (1995).

[3] G. Srajer, S. Fraden, and R. B. Meyer, Phys. Rev. A, 39, 4828-4832 (1989)

[4] W. H. Han and A. D. Rey, Macromolecules, 28, 8401-8405 (1995).

[5] W. H. Han, J. H. Cho, J. K. Jeon, S. G. Noh, and S. T. Koh, Mol. Cryst. Liq. Cryst., 366, 929-936 (2001).