The demand for fiber-filled polymers has witnessed a significant upswing in recent years. A comprehensive understanding of the local fiber orientation is imperative to accurately predict the mechanical properties of fiber-filled products. In this study, we experimentally investigated the fiber orientation kinetics in uniaxial extensional flows. For this, we equipped a rheometer with a Sentmanat extensional measurement device and with an optical train that allows us to measure the fiber orientation *in situ* during uniaxial extension using small angle light scattering. We investigated an experimental system with glass fibers for the suspended phase ( $L/D=8\u221215$), and for the matrix either low density polyethylene, which shows strain hardening in extension, or linear low density polyethylene, which shows no strain hardening. For these two polymer matrices, the fiber orientation kinetics were investigated as a function of fiber volume fraction ( $\varphi =1%$, 5%, and 10%) and Weissenberg number (by varying the Hencky strain rate, $ \u03f5 \u02d9 H=0.01\u22121 s \u2212 1$). We found that all these parameters did not influence the fiber orientation kinetics in uniaxial extension and that these kinetics can be described by a multiparticle model, based on Jeffery’s equation for single particles. Our results show that, in uniaxial extension, fiber orientation is solely determined by the applied strain and that, up to the concentrated regime ( $\varphi \u2248D/L$), fiber-fiber interactions do not influence the fiber orientation. The extensional stress growth coefficient of these composites, which is measured simultaneously with the orientation, shows high agreement with Batchelor’s equation for rodlike suspensions.

## I. INTRODUCTION

In the last decades, the interest in fiber-reinforced polymer materials for industrial applications has significantly increased. Products manufactured from these composites have, among others improved stiffness, strength, heat resistance, and are lightweight. However, these improvements depend on the local orientation of the fibers. Therefore, it is of interest to develop an understanding of the relation between polymer flow during processing and fiber orientation. During processing, polymer melts are usually subjected to a combination of shear and extensional flows. For injection molding, the material is predominantly subjected to shear flow near the walls, while extensional flow dominates in the flow front and in converging or diverging flow situations. For an extrusion or 3D printing process, the polymer mostly experiences shear flow, while for blow molding, fiber spinning, and thermoforming, extensional flow fields are strongly present.

In general, *in situ* orientation of a dispersed phase during flow can be measured using different techniques. The most common techniques include microscopy, (synchrotron) microcomputed tomography (micro-CT) [1–3], small angle x-ray scattering (SAXS) [4–9], and small angle light scattering (SALS) [10–13]. A synchrotron micro-CT approach was used by Laurencin *et al.* [1] to study the orientation of polyvinylidene fluoride fibers in a hydrocarbon gel matrix during compression flow. Recently, a SALS approach was used by Mehdipour *et al.* [13] to study the anisotropy of micrometer sized rodlike titanium dioxide particles in shear flow. SAXS is used to study the microstructure of smaller, nanoscale sized particles, e.g., Pujari *et al.* [6–8] used SAXS to study the orientation kinetics of carbon nanotubes in shear flow. Usually, a micro-CT approach results in an accurate 3D description of the microstructure and thus the orientation, but a full CT-scan can take a long time, which makes it difficult to study high rates occurring in processing conditions. On the other hand, scattering techniques can reach high acquisition rates, but they only provide an average and a 2D projection of the structure, which can be interpreted in different manners to obtain a value for the anisotropy. Anisotropy can be computed using an orientation distribution approach, wherein the scattering intensity at a certain $q$-value is weighted by a spherical harmonic with a maximum in the flow direction, as was done by Scirocco *et al.* [10]. Alternatively, the second-moment image intensity tensor can be computed and the difference in eigenvalues of this tensor can be used as a measure for anisotropy, as was done by Vermant *et al.* [12]. These different methods yield different values for the anisotropy. Therefore, it is difficult to compare these interpretable experimental scattering results with existing orientation models. Additionally, the so-far presented literature using light scattering techniques only investigated shear flow. However, in the present study, uniaxial extensional flow is considered. *In situ* fiber orientation measurements using SALS, are to the knowledge of the authors, never done in uniaxial extension, although Girasole *et al.* [14–16] demonstrated that it is possible to characterize fiber orientation of short and long glass fibers *ex situ* with SALS.

In uniaxial extension, Wagner *et al.* [17] determined the strain at which most fibers are aligned in the stretch direction. They considered a glass fiber—polyamide 6 composite with a fiber aspect ratio ( $L/D$) of 50 and measured the fiber orientation *ex situ* using x-ray microradiography. They argued that most fibers are aligned in the stretch direction at a Hencky strain of 2 (linear strain of 6). Yet, they only considered a single Weissenberg number. The effect of the Weissenberg number on the fiber orientation kinetics was considered by Zhang *et al.* [18] and Kobayashi *et al.* [19], who found contradictory results. Zhang *et al.* [18] argued that the fiber orientation kinetics decreased for increasing strain rates, whereas Kobayashi *et al.* [19] argued that the fiber orientation kinetics are independent of the strain rate. The former authors measured the fiber orientation *ex situ* for a glass fiber—polypropylene composite using scanning electron microscopy for Hencky strain rates in the range of 0.0046–0.46 $ s \u2212 1$, whereas the latter authors measured the orientation *in situ* for a potassium-titanate whisker-polystyrene composite using x-ray diffraction for Hencky strain rates in the range of 0.0038–0.3 $ s \u2212 1$. The aspect ratio of their whiskers ( $L/D$) was distributed between 20 and 100. Additionally, Kobayashi *et al.* investigated the influence of whisker volume fraction and argued that up to a whisker volume fraction of 10%, the orientation kinetics are independent of volume fraction. However, it still remains unclear how the rheology of the matrix influences the fiber orientation kinetics at different Weissenberg numbers and volume fractions.

The fiber orientation also influences the rheological properties of the composite as shown by Ferec *et al.* [20], who considered polypropylene glass fiber composites, and Wang *et al.* [21], who considered polyethylene-co- $\alpha $-octene (POE) and long chain branching POE glass fiber composites. They both investigated the extensional stress growth coefficient as a function of initial fiber orientation and both showed a large difference in viscosity between samples with fibers initially oriented in the stretch direction and samples with fibers initially oriented perpendicular to the stretch direction. However, contradictory results were obtained when both authors considered the strain hardening behavior of the fiber-filled composites. Ferec *et al.* [20] argued that the fibers induce strain hardening, while Wang *et al.* [21] demonstrated a decrease of strain hardening due to the presence of fibers.

Much remains unclear about the combined effects of the polymer matrix’s Weissenberg number, strain hardening, and fiber volume fraction on the fiber orientation kinetics in uniaxial extensional flow and how the resulting fiber orientation influences the rheological properties. In this work, these effects will be investigated for non-Brownian fiber suspensions using an in-house developed setup that allows to simultaneously measure the fiber orientation *in situ* using SALS and the rheological properties of the composite. The experimentally obtained anisotropy of the fibers in the bulk as a function of strain will be compared to a model, based on the Jeffery equation, that describes the anisotropy of multiple noninteracting fibers in the bulk. Through this comparative analysis, we seek to gain deeper insights into the interplay between strain-induced fiber orientation and the rheological behavior of the composite. This knowledge will contribute to a more comprehensive understanding of the mechanical properties and performance of fiber-filled polymers, enabling more precise predictions and further advancements in their design and manufacturing processes.

## II. MODELING FIBER ORIENTATION

In this section, a simple model is introduced that describes the evolution of the anisotropy of a polymer composite containing noninteracting rodlike particles in uniaxial extensional flow. The model is based on Jeffery’s equation [22]. In 1922, Jeffery derived an analytical equation that describes the flow-induced orientation dynamics of an ellipsoidal particle in a dilute suspension in shear flow. Jeffery neglected inertia in a Newtonian fluid and thus considered Stokes flow. Later, in 1963, Takserman-Krozer and Ziabicki [23] used Jeffery’s approach to find a similar equation for uniaxial extensional flow.

**p**, is described by Jeffery’s equation and is given by [24]

The alignment angle, $\theta $, is independent of $\psi $ due to radial symmetry around the $z$-axis. Hence, $ \psi 0$ is only required to uniquely initialize the orientation of the particle. Figure 2 shows the orientation trajectories for different initial values of $\theta $ with $\beta =1$. It can be observed that the orientation rate decreases when the initial alignment angle approaches $ 0 \xb0$ or $ 90 \xb0$. At initial alignment angles of exactly $ 0 \xb0$ and $ 90 \xb0$, the orientation does not change at all. For all other angles, the fiber orientation increases monotonically to the stretch direction, which contrasts with Jeffery orbits for shear flow [24], wherein fibers tumble.

Alternatively, Folgar–Tucker models can be used to predict the fiber orientation kinetics [26]. These models are also based on Jeffery’s equation but include an additional diffusive term to take fiber-fiber interactions into account. Moreover, they require a closure relation, while the multiparticle model does not. Rheological experiments in shear flow showed that these models over predicted the rate of fiber orientation [27–29]. Therefore, the Folgar–Tucker models are extended with additional fitting parameters to slow down the orientation kinetics. One of such models is called the reduced strain closure model [27]. Additionally, direct numerical simulations can be used to model the fiber orientation [30]. This method is more accurate because fiber-fiber interactions are described by the hydrodynamic forces in the fluid, but it is computationally more expensive. In the present work, comparison of the experimental results with the model for noninteracting particles is used to determine the contribution of such interactions.

## III. MATERIALS AND METHODS

This section explains the materials used, the sample preparation, and the setup for measuring both the fiber orientation and the rheological properties during uniaxial extensional flow.

### A. Materials

For this study, low density polyethylene, LDPE ( $ M w=153344 g mol \u2212 1$, $ M n=12810 g mol \u2212 1$, see Sec. S1 in the supplementary material [40]) and linear low density polyethylene, LLDPE ( $ M w=148059 g mol \u2212 1$, $ M n=36230 g mol \u2212 1$, see Sec. S1 in the supplementary material [40]) are used as matrix materials. The extensional stress growth coefficient, which will be referred to as transient uniaxial extensional viscosity in the remainder of the paper, of LDPE shows strain hardening and that of LLDPE shows no strain hardening [31]. The refractive index of LDPE and LLDPE above the glass transition temperature ( $n\u22481.51$, $20 \xb0$C) closely matches the refractive index of glass ( $n\u22481.52$). Hence, to limit sample turbidity, glass fibers are chosen for the suspended phase. The glass fibers (EPH80M-10A) are kindly provided by Nippon Electric Glass and have a diameter of 10 $\mu $m and a volume averaged length of 111 $\mu $m. The fiber length distribution in the fiber composites, as determined via x-ray tomography, is shown in Fig. S1 in the supplementary material [40]. The fibers are amino-silane-coated to ensure better adhesion to the polymer matrix. For this system the minimum Péclet number, which is the ratio between convection effects and diffusion effects, is of order $ 10 8$, meaning that Brownian forces do not play a role.

### B. Sample preparation

First polyethylene and glass fibers are mechanically mixed to obtain a homogeneously mixed extrudate. The mixing is performed at $230 \xb0$C with a rotational speed of 50 rpm for 8 min using a DSM Micro 15cc twin screw compounder. After 7 min, the rotational speed is lowered to 10 rpm to minimize the number of air bubbles in the extrudate.

Next, the extrudate is cut into small granules and is compression molded at $160 \xb0$C into thin sheets. To maximally randomize the orientation of the fibers in the film, the thin sheets are cut into smaller sheets of approximately $2\xd72 mm$ and compression molded again. During this step, the top and bottom surfaces are protected by a smooth Kapton film to increase the surface smoothness of the resulting thin sheet, which is needed to perform scattering experiments at room temperature. Finally, the thin sheets are cut into samples with dimensions (length $\xd7$ width) of approximately $16\xd710$ mm. During both compression molding cycles, the following pressure-time profile is used: during the first 3 min, no pressure is applied, during the next 3 min, a pressure of 20 kN is applied, and finally, for 5 min, a pressure of 40 kN is applied.

Fiber volume fractions of 1%, 5%, and 10% are made (2.6%, 13%, and 26% fiber weight fraction, respectively). The fiber volume fraction constrains the thickness of the samples because small angle light scattering is used to determine fiber orientation. This technique will only detect noise if the laser beam encounters too many fibers. Therefore, the sample thickness is adjusted according to the fiber volume fraction. The used volume fractions correspond to sample thicknesses of 1.33, 0.44, and 0.22 mm, respectively.

### C. Shear rheology

### D. *Ex situ* X-ray tomography

### E. *In situ* SER-SALS setup

An in-house setup is built to determine the fiber orientation *in situ* during uniaxial extensional flows using SALS. The setup is schematically shown in Fig. 4. From right to left, the setup consists of a Neon-Helium laser with a wavelength of 632.8 nm, an Anton Paar MCR 502 rheometer, a white detector plate, and a Hamamatsu C14400 camera. The Anton Paar rheometer is equipped with a Sentmanat Extensional Rheology (SER) 2 measurement system and a Convection Temperature control Device (CTD) 300/GL. The CTD 300/GL is a special oven with optically flat glass windows that are perpendicular to the laser source. In the past this oven was used for x-ray measurements [34]. In the oven, a quartz cuvette, filled with silicon oil ( $\eta =10$ mPa s), is placed and the samples are immersed in the oil during stretching. This is required to prevent unwanted scattering due to surface roughness generated during stretching of the fiber-filled samples. The distance from the sample to the detector plate is approximately 40 cm so that characteristic distances between 3 and 16 $\mu $m are visible ( $q$-range from $1.9\xd7 10 \u2212 4$ to $1.05\xd7 10 \u2212 3 nm \u2212 1$), which is about the order of the fibers’ diameter. Furthermore, the laser spot has a diameter of about 1mm, which means that characteristic distances and thus the average orientation information of minimal 2000 fibers are initially visible in the scattering pattern.

#### 1. Extensional rheology

For all extensional experiments, the sample temperature, determined using an external thermocouple, is set to $150 \xb0$C to ensure consistency with the shear rheometer. After lowering the sample in silicon oil, a torque of 15 $\mu $Nm is applied for 80 s to prevent sagging during heating of the sample. Then, the desired strain rate is applied. At a maximum Hencky strain of 4 (a linear strain of about 50), the stretching is stopped. Finally, the sample is lifted out of the oven and an image of the background of the detector is made. Per parameter set, three repeat measurements are performed.

#### 2. Image analysis in SER-SALS

In the remainder of this paper, the eccentricity will be computed from the second largest ellipse as this provides an optimal trade-off between reducing interference from the beam stop region as well as the edge of the detector. This manner of computing the anisotropy in the system enables to compare the results to the anisotropy obtained from the multiparticle model [Eq. (13)] since here the anisotropy in the system is also defined as the ratio between eigenvalues. Note that the fraction in Eq. (17) is inverted with respect to the definition in Eq. (13) since the scattering pattern is rotated with $ 90 \xb0$ as compared to the real system. Furthermore, the multiparticle model uses the 3D orientation, whereas here only a 2D projection is considered. This difference will further be discussed in Sec. IV A 2.

### F. Effect of silicon oil on the applied uniaxial extension

During the experiment, the fibers in the sample rotate and cause surface roughness. As a result, the laser light will scatter on the surface of the sample, resulting in unwanted contributions to the scattering pattern. To prevent this, the experiments are performed in a quartz cuvette filled with silicon oil. The oil covers the samples’ surface and has a similar refractive index as the matrix polymer ( $n\u22481.41,20 \xb0$C). An additional benefit is that silicon oil will not chemically interact with polyethylene. Also, the viscosity of the oil should be low ( $ \eta oil\u224810$ mPa s) because high viscous oil results in slip between the drums of the SER and the sample, as will be discussed in the next paragraph.

Evidence of the slip is shown in Fig. 6(b). The markers are the width, $w$, obtained by tracking a grid drawn on the sample using an optical camera during a uniaxial extensional experiment in a cuvette filled with silicon oil having a viscosity of 1 Pa s [see Fig. 6(a)]. The solid lines are the expected decrease in width during uniaxial extension according to $w= w 0 exp ( \u2212 \epsilon \u02d9 H t / 2 )$, with $ w 0$ being the initial width of the sample. In the experiment, the width decreases slower than expected because a lubrication layer of silicon oil remained between the new piece of sample being wound on the drum and the drum itself, resulting in a changing contact-contact distance. The dashed line in Fig. 6(b) corresponds to the sample thinning when only the sample ends would be fixed to the drum and the full sample would be extended. Since the drums rotate at a constant speed, this results in an extension rate that decreases with time. As can be seen in Fig. 6(b), the sample thinning is consistent with the second scenario. The data indicate that no lubrication layer between the drums of the SER and the sample is present when 10 mPa s silicon oil is used as cuvette fluid for the Hencky strain rates between 0.01 and 0.1 $ s \u2212 1$. To completely remove the lubrication layer at a strain rate of 1 $ s \u2212 1$, p300 sanded Kapton tape was stuck to the drums of the SER. For this situation, the experimentally tracked width during an extensional experiment is shown in Fig. 6(c) (markers). The solid line represents the expected sample thinning. The transient uniaxial extensional viscosity measured at a Hencky strain rate of 1.0 $ s \u2212 1$ in a cuvette filled with each of the silicon oils is compared to the transient uniaxial extensional viscosity in air and is shown in Fig. 7 for LLDPE. The transient uniaxial extensional viscosity of LLDPE measured in a cuvette filled with 1 mPa s silicon oil shows a nonrealistic strain hardening behavior. This results from the underestimation of the cross-sectional area when assuming absence of slip whereas in reality the oil lubrication layer causes only the sample ends to stick to the drums. The latter is not present for a cuvette filled with 10 mPa s silicon oil. Hence, the transient uniaxial extensional viscosity measured in a cuvette filled with 10 mPa s silicon oil shows agreement with the transient uniaxial extensional viscosity measured in air. The fluctuations in viscosity at early times in Fig. 7 originate from start-up effects in the stress-controlled rheometer.

The width is tracked at different strain rates for the highest and lowest sample aspect ratios ( $w/h$). For all experimental settings, the width decreased as expected, meaning that the applied strain rate is equal to the local strain rate at the location of the laser beam and that perfect uniaxial extension is occurring. As a result, the transient uniaxial extensional viscosities in air and in silicon oil (of 10 mPa s) match.

## IV. RESULTS AND DISCUSSION

In order to measure the fiber orientation kinetics as a function of matrix properties, first characterization experiments are performed, which include the rheological characterization of the composite’s matrix to determine the characteristic internal relaxation times and validation of the anisotropy of the fibers obtained in the SALS experiments with the anisotropy of the fibers obtained from a CT-scan. Afterward, in Sec. IV B, the fiber orientation kinetics is investigated as well as the corresponding rheology of the fiber composites.

### A. Characterization and verification

#### 1. Rheology of the polymer matrix

. | LDPE . | LLDPE . |
---|---|---|

ΔH (kJ mol^{−1}) | 50 | 27.5 |

$ \eta 0 1$ (Pa s), λ_{1} (s) | 9.29 × 10^{2}, 1.0 × 10^{−2} | 3.97 × 10^{3}, 1.15 × 10^{−3} |

$ \eta 0 2$ (Pa s), λ_{2} (s) | 3.11 × 10^{3}, 8.51 × 10^{−2} | 7.11 × 10^{3}, 6.35 × 10^{−2} |

$ \eta 0 3$ (Pa s), λ_{3} (s) | 8.91 × 10^{3}, 5.18 × 10^{−1} | 5.33 × 10^{3}, 3.18 × 10^{−1} |

$ \eta 0 4$ (Pa s), λ_{4} (s) | 2.18 × 10^{4}, 3.02 × 10^{0} | 1.61 × 10^{3}, 2.06 × 10^{0} |

$ \eta 0 5$ (Pa s), λ_{5} (s) | 3.82 × 10^{4}, 1.65 × 10^{1} | 5.44 × 10^{2}, 1.70 × 10^{1} |

$ \eta 0 6$ (Pa s), λ_{6} (s) | 3.76 × 10^{4}, 7.53 × 10^{1} | — |

$ \eta 0 7$ (Pa s), λ_{7} (s) | 2.43 × 10^{4}, 2.74 × 10^{2} | — |

. | LDPE . | LLDPE . |
---|---|---|

ΔH (kJ mol^{−1}) | 50 | 27.5 |

$ \eta 0 1$ (Pa s), λ_{1} (s) | 9.29 × 10^{2}, 1.0 × 10^{−2} | 3.97 × 10^{3}, 1.15 × 10^{−3} |

$ \eta 0 2$ (Pa s), λ_{2} (s) | 3.11 × 10^{3}, 8.51 × 10^{−2} | 7.11 × 10^{3}, 6.35 × 10^{−2} |

$ \eta 0 3$ (Pa s), λ_{3} (s) | 8.91 × 10^{3}, 5.18 × 10^{−1} | 5.33 × 10^{3}, 3.18 × 10^{−1} |

$ \eta 0 4$ (Pa s), λ_{4} (s) | 2.18 × 10^{4}, 3.02 × 10^{0} | 1.61 × 10^{3}, 2.06 × 10^{0} |

$ \eta 0 5$ (Pa s), λ_{5} (s) | 3.82 × 10^{4}, 1.65 × 10^{1} | 5.44 × 10^{2}, 1.70 × 10^{1} |

$ \eta 0 6$ (Pa s), λ_{6} (s) | 3.76 × 10^{4}, 7.53 × 10^{1} | — |

$ \eta 0 7$ (Pa s), λ_{7} (s) | 2.43 × 10^{4}, 2.74 × 10^{2} | — |

Figures 9 and 10 show, respectively, the transient shear viscosities and the transient uniaxial extensional viscosities for LDPE and LLDPE at different rates for a temperature of 150 $ \xb0$C. The black solid line in Fig. 9 represents the linear viscoelastic behavior. The black solid line in Fig. 10 represents the linear viscoelastic behavior of the transient uniaxial extensional viscosity computed using $ \eta E LVE +(t)=3 \eta S LVE +(t)$. For LLDPE, the linear viscoelastic response slightly underestimates the experimental transient uniaxial extensional viscosity.

For LDPE and LLDPE, the average relaxation times are, respectively, 53 and 0.62 s at a temperature of 150 $ \xb0$C. The resulting range of considered Weissenberg numbers in this study is $ Wi=0.53$–53 for LDPE and $ Wi=0.0062$–0.62 for LLDPE using Hencky strain rates between 0.01 and 1 $ s \u2212 1$. In case the longest relaxation time is used as characteristic timescale, the Weissenberg number ranged between 2.74 and 274 and 0.17 and 17 for LDPE and LLDPE, respectively. The effect of strain hardening of the matrix on fiber orientation can be investigated by comparing the response in LDPE and LLDPE. The transient uniaxial extensional viscosities of the matrix clearly show strain hardening behavior for LDPE and negligible strain hardening for LLDPE, as the comparison between Figs. 10(a) and 10(b) shows.

#### 2. Comparison of *in situ* SALS with *ex situ* CT-scans

During a SER-SALS experiment, the scattering pattern changes over time. Initially, at a strain equal to zero, fibers are randomly oriented in the bulk, resulting in circular scattering patterns. For increasing strains, fibers become more aligned with the extension direction and the resulting scattering pattern elongates to a vertical ellipse, as shown in Fig. 11. Since the ellipse remains oriented vertically, the orientation angle of the ellipse, indicating the average fiber orientation direction, does not change and is not considered in this work. In this section, the orientation resulting from the scattering patterns is compared to the orientation resulting from CT-scans.

First, nonextended samples are cut into a size of $1\xd71 mm 2$ at the location of the laser beam, which is only a small part of the sample, and CT-scans are made from these samples. A visualization in Geodict of nonextended CT-scanned samples is shown in Fig. 12 for different fiber volume fractions. For uniaxial extensional flow, all principal directions of orientation align with the coordinate axes. All the off-diagonal elements of the orientation tensor are thus zero [24]. The three nonzero elements of the fiber orientation tensor of the compression molded samples are determined, and especially the $yy$- and $zz$ tensor components differ per location and sample. The three repeat measurements show that the average component in the $x$ direction of the fiber orientation tensor for the thin samples with 5 and 10% fiber volume fraction is initially almost zero, respectively, $ P x x , CT , 0=0.02\xb10.017$ and $ P x x , CT , 0=0.008\xb10.0013$, meaning that fibers are initially oriented in the $zy$-plane. The thick samples with 1% fiber volume fraction can have a small component in the $x$ direction of the fiber orientation tensor ( $ P x x , CT , 0=0.13\xb10.1215$). To compare experimental data with the results of the multiparticle model, the initial particle orientation distribution in the model is chosen such that it yields a $ P x x$ value that is similar to $ P x x , CT , 0$, while the particle orientation in the $yz$-plane is taken isotropic ( $ P y y= P z z$). It should be noted that the scattering pattern only gives the ratio $ P y y/ P z z$, whereas the model includes $ P x x$. However, since $ P x x$ is small, and descending in time, it does not influence the model data significantly.

. | 1% Fibers . | 5% Fibers . | 10% Fibers . |
---|---|---|---|

h | 1.4 mm | 0.4 mm | 0.2 mm |

C | 1.00 ± 0.02 | 1.015 ± 0.01 | 1.05 ± 0.04 |

. | 1% Fibers . | 5% Fibers . | 10% Fibers . |
---|---|---|---|

h | 1.4 mm | 0.4 mm | 0.2 mm |

C | 1.00 ± 0.02 | 1.015 ± 0.01 | 1.05 ± 0.04 |

The values of $C$ are almost equal to one, implying that hardly any correction is applied. In the remainder of this paper, the corrected anisotropy, $ \xi e$, will be considered.

### B. Fiber orientation results

#### 1. Fiber orientation during extension

In this section, the fiber orientation is investigated during uniaxial extensional flow where the effect of three different parameters i.e., the Weissenberg number (characterized by the relaxation time, which is a linear viscoelastic parameter), the presence of strain hardening (a nonlinear viscoelastic effect), and the fiber volume fraction is considered. The fiber orientation kinetics were determined as described in Sec. III E. The results are shown in Figs. 14 and 15 for LDPE and LLDPE, respectively. The solid markers in these graphs show the average anisotropy of the fibers in the bulk as a function of linear strain for different combinations of matrix material (LDPE or LLDPE) and fiber volume fraction (1%, 5%, or 10%). Per graph three decades of Weissenberg numbers are considered. Every graph also includes the prediction of the multiparticle model presented in Sec. II, where $\beta =1$ is chosen because the glass fibers in the experiments have an aspect ratio between 8 and 15. The initial orientation component in the thickness direction, $ P x x , 0$, in this model is chosen such that it agrees with the average CT-scans as explained in Sec. IV A 2, while the initial orientation in the $yz$-plane remains isotropic, $ P y y , 0= P z z , 0$. The model and the experiments are compared in this manner because in both cases particle alignment is computed by correlating eigenvalues. Initially, the anisotropy of the fibers in the samples ranged between $\u2212$1.0 and 0.5. The corresponding initial strain of these samples is corrected so that the initial anisotropy agrees with the model prediction. As a result, at low strain, data for repeat measurements is not always available. Therefore, the individual experimental data points are shown with open markers, without averaging. Overall, this experimental study resulted in three key observations.

The first key result is that for all combinations of matrix material and volume fraction, the Weissenberg number does not influence the fiber orientation kinetics, as the graphs in Figs. 14 and 15 show. In Fig. 14, the anisotropy of the fibers is determined for LDPE with Hencky strain rates between 0.01 and 1 $ s \u2212 1$, thus Weissenberg numbers between 0.53 and 53. For every considered volume fraction, the spread in the experimental data for repeat measurements matches the spread for different Hencky strain rates, implying that the anisotropies of the fibers in the bulk for different Weissenberg numbers overlap. The same can be concluded for LLDPE, the results of which are shown in Fig. 15 where Weissenberg numbers between 0.0062 and 0.62 are considered.

The second key result is that matrix rheology does not affect the fiber orientation, at least up to a fiber volume fraction of 10%. The matrix rheology in uniaxial extensional flow is significantly different for LDPE compared to LLDPE. LDPE shows strain hardening and LLDPE does not, even at the same $ Wi$ number ( $ Wi\u22480.5$ for LLDPE at the highest strain rate and LDPE at the lowest strain rate), as Fig. 10 shows. The effect of this matrix behavior on the fiber orientation can be investigated by comparing the corresponding panels of Figs. 14 and 15. At identical fiber volume fractions, the anisotropy of the fibers in the bulk follows the same kinetics.

The third key result is that an increase in fiber volume fraction results in a minor increase in fiber orientation rate, as can be most clearly observed around a strain of 1–2. This effect can be seen by comparing the individual graphs of Figs. 14 and 15 for LDPE and LLDPE, respectively. This minor effect could be the result of confinement, as for increasing fiber volume fraction, the thickness of the samples decreases. To further investigate confinement effects, LDPE samples are prepared with a fiber volume fraction of 5% and a thickness of 0.2 mm. The fiber orientation kinetics of these samples are determined and compared to the fiber orientation kinetics of LDPE with 5% fiber volume fraction and thickness of 0.4 mm [Fig. 16(a)] and compared to the fiber orientation kinetics of LDPE with 10% fiber volume fraction and thickness of 0.2 mm [Fig. 16(b)]. As the anisotropy of the fibers in the bulk better agrees for samples having the same thickness but a different volume fraction, the minor acceleration of orientation for thinner samples can be attributed to confinement effects.

The fiber orientation kinetics for all considered experiments show high agreement with the model prediction. The model is based on Jeffery’s equation, which is derived for a single particle in a Newtonian fluid. Since these fiber orientation kinetics match the fiber orientation kinetics for different viscoelastic matrix properties, the fiber orientation is only determined by the applied external flow, and not by the Weissenberg number or the rheology of the matrix. Hence, the fiber orientation is purely determined by the strain reached, independent of the strain rate. The negligible effect of the Weissenberg number on the fiber orientation kinetics was also seen by both Wagner *et al.* [17] and Kobayashi *et al.* [19]. Via respective *ex situ* and *in situ* characterizations, Wagner *et al.* considered Hencky strain rates between 0.062 and 0.53 $ s \u2212 1$ and Kobayashi *et al.* considered Hencky strain rates between 0.003 and 0.3 $ s \u2212 1$. Thus, our results generalize the observations of Kobayashi *et al.* [19] and Wagner *et al.* [17] to materials with a wide range of rheological properties and Weissenberg numbers.

The experimental system considers fiber volume fractions up to 10% with a volume averaged aspect ratio of 11, implying that the system reaches the concentrated regime, $\varphi =D/L$, where fibers generally should interact [37,38]. Yet no influence of fiber-fiber interactions on the fiber orientation kinetics is observed, since similar kinetics are obtained for different volume fractions and these match Jeffery’s equation, which also does not consider fiber-fiber interactions. The results obtained in this study are in agreement with the experiments conducted by Kobayashi *et al.* [19], who did not see significant differences in whisker orientation for volume fractions 5% and 10% in uniaxial extension with whisker aspect ratio ( $L/D$) of 20–100. These results show that for uniaxial extension up to a fiber volume fraction of 10%, the additional diffusive and strain reduction terms in the Folgar–Tucker models are not required.

#### 2. Rheology of the fiber composites

From Fig. 17, two key results can be obtained. First, for the strain hardening LDPE matrix, increasing the volume fraction of fibers decreases the amount of strain hardening in the composite and for the non strain hardening LLDPE matrix, strain hardening remains absent in the composites. This can be seen by comparing the experimentally obtained transient uniaxial extensional viscosity of the composites with that of Batchelor’s prediction. The strain hardening index resulting from the latter remains constant as a function of the fiber volume fraction, whereas the data are clearly lower at higher strains and fiber volume fractions. These results qualitatively agree with the results obtained by Wang *et al.* [21], who considered the strain hardening behavior of glass fiber-filled long chain branching polyethylene-co- $\alpha $-octene (LCB POE) (strain hardening matrix) and glass fiber-filled linear POE. The decrease of the strain hardening behavior of the composite for a strain hardening matrix also agrees with the results obtained by Kobayashi *et al.* [19]. The results presented here are however different from the results obtained by Ferec *et al.* [20], who could reach higher strains without local material failure, and found that the presence of fibers increases the strain hardening behavior of the composite for a non-strain hardening matrix.

The second key result is that, overall, Batchelor’s prediction describes the transient uniaxial extensional rheology rather well, which is remarkable since this model assumes perfect alignment of the fibers in the stretch direction. According to the results obtained in the previous section, full alignment of the fibers in the sample is reached at a linear strain of about 4–5, which is reached in 1.7–1.8 s at a Hencky strain rate of 1.0 $ s \u2212 1$. The initial fiber orientation influences the uniaxial extensional viscosity of the composites, as investigated by Ferec *et al.* [20] and Wang *et al.* [21]. However, both found that the transient uniaxial extensional viscosity of composites with fibers oriented randomly and fibers oriented parallel to the stretch direction closely agrees, which can be the reason why Batchelor’s expression matches the experimentally found viscosity.

## V. CONCLUSIONS

In this study, the fiber orientation kinetics of polymer fiber composites was investigated *in situ* during uniaxial extensional flow. First, a multiparticle model, based on Jeffery’s equation, was derived which describes the anisotropy of rodlike particles in the bulk for this flow type.

Experimentally, a setup was built to measure the fiber orientation, *in situ*, using small angle light scattering. Using a Sentmanat extensional measurement device on a rotational rheometer, uniaxial extension was applied on polyethylene-glass-fiber composites that were submerged in a cuvette filled with silicon oil. The oil was required to prevent scattering from surface artifacts occurring during the experiment. Moreover, low viscous oil is essential to prevent slip between the drums and the sample.

The influence of Weissenberg number on the fiber orientation was investigated by varying the applied Hencky strain rate. For all considered systems, the Weissenberg number was varied over three decades and did not affect the fiber orientation. Additionally, the experimental systems were chosen to isolate the influence of strain hardening behavior of the matrix on the fiber orientation. For all considered fiber volume fractions, the strain hardening behavior of the matrix did not significantly influence the fiber orientation. Finally, the influence of fiber volume fraction on the fiber orientation was investigated. Experimental systems up to a fiber volume fraction of 10% were considered and no significant influence of fiber volume fraction on the fiber orientation kinetics was observed. For all considered cases, similar fiber orientation kinetics were obtained, which overlap with the multiparticle model. The multiparticle model does not include fiber-fiber interactions. This implies that for uniaxial extension, up to the concentrated regime ( $\varphi \u2248D/L$), fiber-fiber interactions are negligible and fiber orientation is determined solely by the strain. This also implies that the diffusive term and the strain reduction term in the Folgar–Tucker models is not required for the regime considered in this study.

The simultaneously measured transient uniaxial extensional viscosity can be well predicted by the Batchelor expression for oriented fibers. For a strain hardening matrix, the amount of strain hardening in the composite decreases for increasing fiber volume fractions, while the amount of strain hardening in the composite is not influenced by fiber volume fraction for composites with a non strain hardening matrix. An increasing fiber volume fraction also increased the zero extensional viscosity.

## ACKNOWLEDGMENTS

This research forms part of the research programme of the Dutch Polymer Institute (DPI), Project No. 840, Engineering the rheology ANd processinG-induced structural anisotropy of poLymEr composites with non-Brownian fibrous particles (ANGLE). The authors thank Nippon Electric Glass NL for supplying the short glass fibers, and Rafael Sablong of PTG Eindhoven for the GPC measurements.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.

## REFERENCES

*Fundamentals of Fiber Orientation Description, Measurement and Prediction*

*Rheology: Principles, Measurements, and Applications*