Modeling the Rheology and Orientation Distribution of Short Glass Fibers Suspended in Polymeric Fluids: Simple Shear Flow

In this paper we present a constitutive relation for predicting the rheology of short glass fibers suspended in a polymeric matrix. The performance of the model is assessed through its ability to predict the steady-state and transient shear rheology as well as qualitatively predict the fiber orientation distribution of a short glass fiber (0.5 mm, L/D < 30) filled polypropylene. In this approach the total extra stress is equal to the sum of the contributions from the fibers (modified Doi theory), the polymer and the rod-polymer interaction (multi-mode viscoelastic constitutive relation). Introduction Adding high modulus and strength fibers to thermoplastics can significantly increase the mechanical properties of the neat matrix, especially when the fibers are aligned in the direction of mechanical interest [1]. Therefore it is desirable to be able to predict the fiber orientation as a function of processing conditions to optimize mold design to maximize mechanical properties of the final part [2]. The overall goal of our research is to be able to accurately predict the fiber orientation in injection molded parts using a finite element analysis. The goal of this paper is to present a constitutive relation for modeling the suspension rheology and predicting the fiber orientation distribution in simple shear flows. The presence of fibers can significantly affect the rheology of the neat matrix, especially at the high fiber concentrations, and aspect ratio of industrial interest. The composite rheology is thought to be influenced by the fiber orientation distribution, concentration, and aspect ratio of the fiber, the viscoelastic nature of the suspending medium, and the degree of fiber-matrix interaction [3]. Due to the brief nature of this paper, our analysis and discussion will be limited to the general effect fiber and its orientation distribution has on the steady-state and transient rheology of polymer melts and the model’s ability to qualitatively predict the rheological behavior. Preceding the introduction of modeling glass fiber suspensions we believe it is pertinent to make a few general comments on the rheological behavior of fiber suspensions as it will aid in the explanation of the model development. The first is regarding the steady-state rheology and the second regarding the transient rheology. Generally speaking, the steady-state viscosity vs. shear rate curve is similar in nature to what one would expect from a neat polymer. When the steady-state rheology of a suspension is compared to its neat counterpart it typically has an enhanced Newtonian plateau and can exhibit a shear thinning behavior at lower shear rates than the neat resin. At high shear rates the viscosity curves typically merge. In some cases, typically at very high fiber loading, the suspensions can exhibit yield-like behavior [4]. Point being, the steady-state viscosity can be predicted with a number of shear rate dependent empiricisms, i.e. Carreau-Yasuda model. Conversely, the transient shear rheology of fiber suspensions is typically easily distinguishable from that of a neat resin. For example, when a sample with an isotropic fiber orientation is subjected to a stress growth upon inception of steady shear flow test, the sample will exhibit a large stress overshoot in both the shear stress and the normal stress differences. This is believed to be a result of the fiber aligning itself in the principle flow direction. Once aligned the stresses reach a steady-state [3]. Hence, the transient rheological behavior is coupled with the fiber orientation and being able to model the evolution of orientation is imperative to correctly predicting the rheology. Nearly all theoretical work on modeling the flow of fiber suspensions starts with the work of Jeffery [5] who investigated the motion of a single elliptically shaped particle in a Newtonian suspending medium. Effort has been made in extending the idea of a single particle to that of a distribution of fiber orientations in a suspension [6, 7]. Further work has been done to extend the theory into more concentrated fiber regimes where hydrodynamic interaction becomes an increasing factor [8, 9]. However, these theories are all based on Newtonian suspensions and, therefore, are, in general, incapable of capturing the non-Newtonian behavior of polymeric suspensions. With respect to modeling the fiber orientation in an injection molded part, the majority of work has been accomplished by using an approach that decouples the fiber orientation with the flow field. Hence the rheology of the suspension is taken as that of a generalized Newtonian fluid to predict the flow field and then a modified Jefferey’s equation is used to post calculate the fiber orientation [10]. To capture both the effects of the fiber and the nonNewtonian suspending medium in an approach where the flow is coupled with the fiber orientation we propose an additive scheme, where the total extra stress is equal to a sum of contributions from the fiber, the suspending medium and the interaction between the fiber and the suspending medium. In the model, the contribution of the fiber is calculated using a special form of the Doi theory for concentrated rigid rod molecules. As a note, because the Doi theory was developed for rigid rods we will synonymously use the term rods to refer to glass fibers in our real system. The contribution from the suspending medium is captured using a viscoelastic constitutive relation, and the interaction between the fiber and the polymer is captured by expanding the viscoelastic constitutive relation into its multi-mode form and fitting the long relaxation times of the suspension, which is believed to be influenced by the presence of the fiber. Theory We begin with the simple framework that the total extra is equal to the sum of the contribution to the stress tensor from the rods, the matrix, and the interaction between the rods and the matrix as follows: n interactio matrix rods total τ τ τ τ + + = (1) Rod contribution: The starting point for the development of the contribution of the rods to the extra stress is Doi’s molecular theory for mono-disperse rod-like molecules suspended in a Newtonian fluid. The theory begins with the dilute solution case where a rod is free to rotate and translate without interacting with other rods. It was then extended to concentrated systems which spontaneously become anisotropic after a critical concentration without the presence of any external fields due to excluded volume effects [11]. The Doi theory for rod-like molecules consists of two components. The first, calculating the rod orientation distribution and its evolution under external forces. The second, post calculating the stress tensor which is a function of the rod orientation. In both, the quadratic closure approximation is used. The rod orientation within the system is characterized by the deviatoric form of the orientation order parameter tensor ( S ), and is defined as: δ 3 1 − = u u S (2) where u is a unit vector parallel to the axis of a rod, δ is the unit tensor, and the brackets • represent the ensemble average over the distribution function. In simple shear flow the time evolution of S is equal to the contributions from Brownian motion, ( ) S F , plus the contribution from the macroscopic flow field, ( ) S v G , ∇ : ( ) ( ) S v G S F t S , ∇ + = ∂ ∂ (3) The Brownian motion contribution is prevalent in the case of rod-like molecules or in the case where the rods are on the length scale where the effect of Brownian motion is a contributing factor and is defined by: