압출유동을 겪은 섬유현탁액의 Skin/Core 배향 구조

이성재, 이승종*

수원대학교 고분자공학과, 서울대학교 화학공학과*

Skin/Core Orientation Structure in Extruded Parts of Fiber Suspensions

Seong Jae Lee and Seung Jong Lee*

Department of Polymer Engineering, The University of Suwon, Suwon 445-743

Department of Chemical Engineering, Seoul National University, Seoul 151-742*

Introduction

Skin and core orientation structure in the extruded part of short fiber reinforced composites has great influence on the mechanical and physical properties of product, which originally results from the flow field and the orientation of fibers during processings. In fact, the orientation distribution of fibers in a final product is extremely complex. Some locations can retain the random orientation distribution of fibers, while others may have highly aligned state of fibers. In order to predict the mechanical and physical properties of the extruded product, therefore, it is important to understand the flow field and the orientation distribution of fibers during the extrusion process. In this study, the three-dimensional extrudate swell flows of fiber suspensions are investigated by using the finite element method for the coupled system of the flow and orientation fields, where major concerns are focused on the degree of swelling and the orientation distribution of fibers. The highly oriented skin and the randomly oriented core of the extruded part are predicted from the coupled solution of flow and orientation fields.

Modeling and Governing Equations

The steady extrusion flow under isothermal condition is governed by the continuity, momentum, constitutive, and fiber orientation equations. Inertial effect is negligible because of the highly viscous nature of polymeric materials and the creeping nature of the flow, and body force term is assumed to be negligible compared to the rheological florce. The continuity and momentum equations for incompressible flow are given by

[eqn: DEL CDOT bold v` = 0]	(1)

[eqn: - TRIANGLED p `+ TRIANGLED CDOT bold tau ` = 0]	(2)

The general continuum model approach for formulating the constitutive equation is based on the extensive theories of the stress field around the ellipsoid with any translation and rotation [1]. This model for dilute and semiconcentrated fiber suspensions leads to following form of the constitutive equation,

[eqn: bold tau = 2mu bold D `+ 2mu phi & left{C_1 bold D ``:< bold pppp > + C_2 `(bold D CDOT< bold pp > + <bold pp > CDOT bold D ) #~&`````+ C_3 bold D `+F< bold pp > D_r `right}] (3)

In the last term of Eq. (3), [eqn: D_r`] is the rotary diffusivity due to Brownian motion, whose term is usually neglected for suspensions of large fibers. The stress shape coefficients [eqn: C_1], [eqn: C_2] and [eqn: C_3] for various limiting cases can be found in Hinch and Leal[2]. For large fiber aspect ratio, the constitutive model expressed as Eq. (3) may reduce to

[eqn: bold tau ` = 2 mu bold D + mu alpha bold D ``:<bold pppp >]  (4)

[eqn: alpha` = phi` r^2 over {ln2r -1.5}]  (5)

because the coefficients [eqn: C_2] and [eqn: C_3] are much smaller than [eqn: C_1].

The equation of motion for a single particle can be expressed as follows,

[eqn: bold {dot p } = - bold W CDOT bold p `+ lambda ( bold D CDOT bold p `- bold D` : bold ppp ) - D_r 1 over PSI {PARTIAL PSI} over {PARTIAL bold p }]	(6)

where [eqn: lambda] is a parameter related to the shape of the particle, i.e., [eqn: (r^2 - 1 ) / (r^2 +1)]. Folgar and Tucker [3] have suggested this type of a mathematical model as a model

for concentrated suspensions of non-Brownian large fibers, and replaced [eqn: D_r`] by

[eqn: C_I dot {gamma````}]. Here, [eqn: C_I `] is a phenomenological interaction coefficient which models the

effects of interactions between fibers.

Substituting Eq. (6) into the continuity equation for the distribution function, known as the Fokker-Plank equation

[eqn: {D PSI} over Dt = - PARTIAL over {partial bold p } CDOT ( PSI bold {dot p} `)]	(7)

and integrating [eqn: bold pp `] with [eqn: PSI `] over all possible directions in configuration space gives the equation of change for orientation tensors, that is, the orientation equation.

[eqn: bold v CDOT DEL bold a_2 =&-bold W CDOT bold a_2 + bold a_2 CDOT bold W ``+ lambda(bold D CDOT bold a_2 + bold a_2 CDOT bold D `- 2bold D``: bold a_4 `) #~&+2C_I dot {gamma````} ( bold I - nu bold a_2 `)] (8)

Here, the tensors [eqn: bold a_2`] and [eqn: bold a_4`] stand for [eqn: < bold pp `>] and [eqn: < bold pppp `>], respectively.

Numerical Methods

Extrudate swell flows of fiber suspensions have free surfaces which are not known a priori and inherent nonlinearities which mainly come from the constitutive equation coupled with the orientation equation. As a numerical technique suitable for deforming domains in various flow problems, the standard Galerkin finite element method is imployed to solve the three-dimensional flow and orientation fields, Eqs. (1), (2), (4) and (8). The velocity components and the orientation tensor components are approximated in terms of the 27-node triquadratic shape function, and the pressure is approximated in terms of the 8-node trilinear shape function in the hexahedron elements. The integration scheme used here is 27-point Gaussian quadrature [4].

The resulting nonlinear algebraic system of equations must be solved by iterative techniques. At the die inlet prior to solving the system, the fully developed velocity profile is imposed, which is also numerically generated by solving the corresponding two-dimensional(y-z plane), fully developed flow problem. After every nonlinear iteration based on the latest velocity information, the free surface is calculated separately by using both the pathline approach and the mapping method. This procedure is repeated until a certain convergence is obtained at a given value of the suspensity intensity [eqn: alpha`] and then the value of [eqn: alpha`] is increased gradually to obtain the solutions at higher [eqn: alpha`] of interest. Flowsheet of the numerical procedure for the extrudate swell problem of fiber suspension is schematically shown in Fig. 1.

Fig. 1 Numerical procedure for the extrudate swell flows of fiber suspensions.

Results and Discussion

Finite element mesh, which is comprised of 160 hexahedron elements, is used for the extrudate swell of fiber suspensions from square die. The geometry is chosen such that the total length of the system is 8cm, the die exit is at x=3cm, and the die half-width and half-height is 1cm, respectively. The convergence is obtained within about 10 iterations for the Newtonian flow without fibers, but it requires more iterations as [eqn: alpha `] becomes higher value or [eqn: C_I`] becomes lower value.

In Fig. 2, the predicted extrudate swell and the orientation distributions of fibers with the random fiber input are shown at the outer surface and the cross section of far downstream, at the center symmetric plane, and at the x-z quarter plane. The initial random fibers along the centerline pass through and exit the die, and move to the downstream direction of flow, gradually producing the perpendicular orientation distributions. As a result, the perpendicular orientation distributions to the streamline direction occur along the centerline as the flow exit the die due to the effects of the fiber interaction and the diverging flow.

The orientation tensor components along the z-direction of the x-z center symmetric line at the cross section of far downstream are shown in Fig. 3. In the case of the random fiber input, the skin/core orientation structure appears clearly, especially for the lower fiber interaction. However, in the case of the aligned fiber input, the central and surface regions have higher alignments whereas the mid-region shows relatively random states. The different structures resulted from random and aligned fiber inputs seem the characteristics observed only in short dies. For better prediction of the realistic skin/core orientation structure, it is necessary to introduce the idea of the fiber interaction coefficient as a function of average interfiber spacing and fiber orientation state [5].

References

1. Batchelor, G. K., J. Fluid Mech., 41, 545 (1970).

2. Hinch, E. J. and L. G. Leal, J. Fluid Mech., 52, 683 (1972).

3. Folgar, F. and C. L. Tucker, J. Reinf. Plast. Compos., 3, 98 (1984).

4. Lee, S. J. and S. J. Lee, Korean J. Rheol., 2(2), 35 (1990).

5. Park, K. S., S. J. Lee and S. J. Lee, Korean J. Rheol., 8, 49 (1996).

Fig. 2 Perspective view of three-dimensional finite element mesh and fiber orientation distributions at [eqn: alpha ``]=1, [eqn: C_I``]=0.01 and [eqn: r``]=100 with random fiber input: (a) at surface and cross section, (b) at center symmetric plane, (c) at x-z quarter plane.

Fig. 3 Orientation tensor components along z-direction of x-z symmetric line at cross section of far downstream:[eqn: alpha ``]=1, [eqn: C_I``]=0.01 and [eqn: r``]=100 with random fiber input.