Short Fiber Suspensions

Fiber-reinforced polymeric composites are extensively manufactured by various polymer melt processes such as extrusion, injection, and compression molding. Generally, in a suspension of fiber, the flow field alters the orientation of the fibers; at the same time, the presence of fibers (and their mean orientation) changes the stress response of the suspension.

Fiber Orientation Prediction

The continuum approach for predicting the orientation of rigid short fibers with uniform volume fraction in a suspension [149] uses an orientation tensor to define the mean orientation. The orientation tensors are different moments of the fiber orientation probability distribution function $ψ$ . They provide a computationally effective approach for evaluating the fiber orientations in the complex flow field. The orientation tensors are defined as:

Figure 1. EQUATION_DISPLAY

\begin{matrix} {\begin{matrix} a \end{matrix}}_{2} & = & \int_{0}^{2 π} ⅆ φ \int_{0}^{π} p p ψ (θ, φ, t) \sin θ d θ \\ A_{4} & = & \int_{0}^{2 π} ⅆ φ \int_{0}^{π} p p p p ψ (θ, φ, t) \sin θ d θ \end{matrix}

(749)

where:

$θ$ and $φ$ are the polar and azimuthal angles in spherical coordinates.
$p$ is the orientation vector for a single fiber. $p p$ and $p p p p$ are second- and fourth-order tensors, dyadic products of $p$ .

In spherical coordinates, the single fiber orientation vector is written as:

p = (\begin{matrix} \sin θ \cos φ \\ \sin θ \sin φ \\ \cos φ \end{matrix})

The orientation tensors must satisfy the following symmetric and normalization properties:

\begin{matrix} a_{i j} & \begin{matrix} = \end{matrix} & a_{j i} \\ A_{i j k l} = A_{i j l k} & = & A_{j i k l} = A_{k l i j} \\ a_{i i} & = & 1 \\ A_{i j k k} & = & a_{i j} \end{matrix}

Additionally, the probability distribution function $ψ$ must satisfy the Fokker-Planck (population balance) equation. The Fokker-Planck equation can be written as:

Figure 2. EQUATION_DISPLAY

\frac{D ψ}{D t} = - \nabla\cdot (ψ \dot{p} - \nabla (D_{r} ψ))

(750)

where $D_{r}$ is the rotary diffusivity of the short fibers in the viscous fluid.

To derive an evolutionary equation for $a_{2}$ , the Fokker-Planck equation is multiplied by $p p$ [149] and then integrated both side of equations over a unit sphere. From this, the evolutionary equation for $a_{2}$ can be written as:

Figure 3. EQUATION_DISPLAY

\frac{D a_{2}}{D t} = α [- (Ω \cdot a_{2} - a_{2} \cdot a_{2}) + λ (D \cdot a_{2} + a_{2} \cdot D - 2 A_{4} : D) + 2 \dot{γ} C_{I} (I - 3 a_{2})]

(751)

where:

$D / D t$ is the material derivative.
$D$ and $Ω$ are the strain rate and vorticity tensors.
$\dot{γ}$ is the shear rate, $\dot{γ} = 2 \sqrt{{I I}_{D}}$ , where ${I I}_{D}$ is the second invariant of the train rate tensor.
$α$ is the slip coefficient of the model.
$C_{I}$ is an empirical parameter described by the Bay [153] and Phan-Thien [203] models. $\dot{γ} C_{I} = D_{r}$ .
In the Bay model:
Figure 4. EQUATION_DISPLAY

$C_{I} = 0.0184 \exp (- 0.7148 ϕ r)$
(752)

In the Phan-Thien model:

Figure 5. EQUATION_DISPLAY

$C_{I} = 0.03 [1 - \exp (- 0.0224 ϕ r)]$
(753)

$r$ is the aspect ratio of the fiber.

It is evident from Eqn. (751) that $a_{2}$ is a function of $A_{4}$ , which presents a closure dilemma. To circumvent the need for the fourth-order orientation tensor $A_{4}$ , this tensor is approximated by the lower-order orientation tensor. Chung and Kwon [160] proposed the Invariant-Based Optimal Fitting (IBOF) closure approximation which relates $A_{4}$ to $a_{2}$ and the unit tensor according to:

A_{i j k l} = β_{1} S (δ_{i j} δ_{k l}) + β_{2} S (δ_{i j} a_{k l}) + β_{3} S (a_{i j} a_{k l}) + β_{4} S (δ_{i j} a_{k m} a_{m l}) + β_{5} S (a_{i j} a_{k m} a_{m l}) + β_{5} S (a_{i m} a_{m j} a_{k n} a_{n l})

where $S$ indicates the symmetric part of its argument defined as:

S (T_{i j k l}) = \frac{1}{24} (T_{i j k l} + T_{j i k l} + T_{i j l k} + T_{j i l k} + T_{k l i j} + T_{l k i j} + T_{k l j i} + T_{l k j i} + T_{i k j l} + T_{k i j l} + T_{i k l j} + T_{k i l j} + T_{j l i k} + T_{l j i k} + T_{j l k i} + T_{l j k i} + T_{i l j k} + T_{l i j k} + T_{i l k j} + T_{l i k j} + T_{j k i l} + T_{k j i l} + T_{j k l i} + T_{k j l i})

—that is, with the subscripts of the stress tensor $T$ taken in all possible orders.

The six set of coefficient $β_{i}$ are functions of second and third invariants of $a_{2}$ .

Fiber Suspension Rheology

The presence of the fibers has an extra contribution on stress. It can be shown [179] that the bulk stress contribution due to presence of a spheroidal fibers in a Newtonian fluid is governed by:

Figure 6. EQUATION_DISPLAY

T_{suspension} = 2 μ_{s} D + T_{fiber}

(754)

where:

Figure 7. EQUATION_DISPLAY

T_{fiber} = 2 μ_{s} ϕ [C_{1} D + C_{2} A_{4} : D]

(755)

and:

$ϕ$ is the volume fraction of the fibers in the suspension.
$μ_{s}$ is the solvent viscosity.
$A_{4}$ is determined using the IBOF closure approximation.
$C_{1}$ and $C_{2}$ are two constants, determined by any of five models.

The models available for determining $C_{1}$ and $C_{2}$ are:

Lipscomb et al.: This model is suited to dilute fiber suspensions [191].
Figure 8. EQUATION_DISPLAY

$\begin{matrix} C_{1} = 2 \\ C_{2} = \frac{r^{2}}{2 \ln r} \end{matrix}$
(756)
Batchelor: This model is suited to dilute fiber suspensions [152].
Figure 9. EQUATION_DISPLAY

$\begin{matrix} C_{1} = 0 \\ C_{2} = \frac{2 r^{2}}{3 \ln 2 r} \end{matrix}$
(757)
Shaqfeh & Fredrickson: This model is suited to dilute fiber suspensions [209].
Figure 10. EQUATION_DISPLAY

$\begin{matrix} C_{1} = 0 \\ C_{2} = \frac{4 r^{2}}{3 \ln (\frac{1}{ϕ})} [1 - \frac{\ln (\ln (\frac{1}{ϕ}))}{\ln (\frac{1}{ϕ})} + \frac{0.6634}{\ln (\frac{1}{ϕ})}] \end{matrix}$
(758)
Phan-Thien & Graham: This model is suited to semi-concentrated fiber suspensions [203].
Figure 11. EQUATION_DISPLAY

$\begin{matrix} C_{1} = 0 \\ C_{2} = \frac{r^{2} (2 - \frac{ϕ}{0.53 - 0.013 r})}{2 (\ln (2 r) - 1.5) {(1 - \frac{ϕ}{0.53 - 0.013 r})}^{2}} \end{matrix}$
(759)
Dinh & Armstrong: This model is suited to semi-concentrated fiber suspensions [164].
Figure 12. EQUATION_DISPLAY

$\begin{matrix} C_{1} = 0 \\ C_{2} = \frac{r^{2}}{3 \ln (\frac{π}{2 ϕ r})} \end{matrix}$
(760)

The suspension is in the dilute regime when $ϕ < \frac{1}{r^{2}}$ . The suspension is in the semi-concentrated regime when $\frac{1}{r^{2}} < ϕ < \frac{1}{r}$ .