Fast Solutions for the Fiber Orientation of Concentrated Suspensions of Short-fiber Composites Using the Exact Closure Method

The kinetics of the fiber orientation during processing of shortfiber composites governs both the processing characteristics and the cured part performance. The flow kinetics of the polymer melt dictates the fiber orientation kinetics, and in turn the underlying fiber orientation dictates the bulk flow characteristics. It is beyond computational comprehension to model the equation of motion of the full fiber orientation probability distribution function. Instead, typical industrial simulations rely on the computationally efficient equation of motion of the second-order orientation tensor (also known as the second-order moment of the orientation distribution function) to model the characteristics of the fiber orientation within a polymer suspension. Unfortunately, typical implementation forms of any order orientation tensor equation of motion requires the next higher, even ordered, orientation tensor, thus necessitating a closure of the higher order expression. The recently developed Fast Exact Closure avoids the classical closure problem by solving a set of related secondorder tensor equations of motion, and yields the exact solution for pure Jeffery’s motion as the diffusion goes to zero. Typical closures are obtained through a fitting process, and are often obtained by fitting for orientation states obtained from solutions of the full orientation distribution function, thus tying the closure to the flows from which it was fit. With the recent understandings of the limitations of the Folgar and Tucker (1984) model of fiber interactions during processing, it has become clear the importance of developing a closure that is independent of any choice of fitting data. The Fast Exact Closure presents an alternative in that it is constructed independent of any fitting process. Results demonstrate that when diffusion exists, the solution is not only physical, but solutions for flows experiencing Folgar-Tucker diffusion are shown to exhibit an equal to or greater accuracy than solutions relying on closures developed via a curve fitting approach. INTRODUCTION With the increasing industrial demand for high strength, low weight, rapidly produced parts, understanding the final part performance for short fiber injection molded composites from the underlying microstructure is becoming increasingly important. The kinetics of the fiber orientation during processing of a short-fiber composite governs both the processing characteristics and the cured part performance [1]. The flow kinetics of the polymer melt dictates the fiber orientation kinetics, and in turn the underlying fiber orientation dictates the bulk flow characteristics [2,3]. The Folgar-Tucker model for fiber interactions within the suspension has been accepted industrially for several decades [4], but with recent advances in part repeatability the limitation of their model has been observed [5]. Recent models [6-9] do not follow the same form as the Folgar-Tucker model and thus orientation prediction methods that are fit to the Folgar-Tucker model results may be invalid. Industrial simulations of the FolgarTucker model are beyond computational comprehension and thus the equation of motion for the moments of the distribution is traditionally solved [1]. Unfortunately the equation of motion for the moments of the distribution requires knowledge of the next higher ordered moment. Thus requiring some form of a closure to approximate the higher order moment in terms of the lower order moment. Advani and Tucker’s [1] hybrid closure has been used extensively in industrial simulations but has been shown to be inadequate in representing accurately the fiber orientation distribution function [10]. With the advent of the othotropic closure by Cintra and Tucker, high accuracy closures became available. Unfortunately, nearly all of the orthotropic closures were formed through a fitting process by taking information available from known flow solutions based on the Folgar and Tucker model of diffusion [12-17]. Thus it is unclear if any of these fitted closures could be, with high confidence, employed within the new diffusion models for fiber collision. It is important to point out, that the orthotropic closures of Wetzel [18] and VerWeyst [19] were constructed based on distributions formed through the elliptic integral form