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 15), 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 ( ϕ = 1 %, 5%, and 10%) and Weissenberg number (by varying the Hencky strain rate, ϵ ˙ H = 0.01 1 s 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 ( ϕ D / 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.

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 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 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- α-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.

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.

The orientation of a rodlike particle can be described using two angles: θ ( t ) and ψ ( t ), defined in Fig. 1. In uniaxial extension along the z-axis, θ ( t ) is the alignment angle. If θ is close to zero, the particle is aligned with the extension direction. The second angle, ψ ( t ), is the angle between the y-axis and the projection of the rodlike particle on the x y-plane. The orientation vector, p, can be written using these angles,
p = p x e x + p y e y + p z e z ,
(1)
with
p x = sin ( ψ ) sin ( θ ) ,
(2)
p y = cos ( ψ ) sin ( θ ) ,
(3)
p z = cos ( θ ) .
(4)
FIG. 1.

(a) Schematics of the relevant angles to describe fiber orientation, (b) graphical representation of a uniaxial extensional flow field, according to u = z ϵ ˙ H e z y 1 2 ϵ ˙ H e y x 1 2 ϵ ˙ H e x.

FIG. 1.

(a) Schematics of the relevant angles to describe fiber orientation, (b) graphical representation of a uniaxial extensional flow field, according to u = z ϵ ˙ H e z y 1 2 ϵ ˙ H e y x 1 2 ϵ ˙ H e x.

Close modal
Moreover, e x, e y, and e z are unit vectors along the principal axes. The rate of change of p, is described by Jeffery’s equation and is given by [24]
d p d t = W p + β ( p D p D p p ) ,
(5)
where β is a constant called the particle shape factor, which is related to the geometry and aspect ratio of the rodlike particle, according to β = ( r e 2 1 ) / ( r e 2 + 1 ), with r e = L / D for a prolate ellipse and r e = 1.14 ( L / D ) 0.844 for a cylinder [25]. For large aspect ratios, β approaches unity. Moreover, D is the rate-of-deformation tensor and W is the vorticity tensor,
D = 1 2 ( u + ( u ) T ) ,
(6)
W = 1 2 ( u ( u ) T ) ,
(7)
where u is the velocity vector. For uniaxial extension along the z-direction, these tensors are W = 0, and D = ϵ ˙ H ( e z e z 1 2 e x e x 1 2 e y e y ), where ϵ ˙ H is the Hencky strain rate. In Eq. (5), rigid body rotation is captured by the first term on the right-hand side, whereas the effect of fluid deformation is given by the next term. The last term in Eq. (5) is a vector parallel to p. Subtracting this term from the middle term ( p D) ensures that the magnitude of p remains unity.
After substitution of Eqs. (1)–(4) in Eq. (5), the resulting three equations can be solved for the unknown angles. After simplification, the differential equations for both angles are [23]
d ψ d t = 0 with θ 0 and θ π ,
(8)
d θ d t = 3 4 β ϵ ˙ H sin ( 2 θ ) .
(9)
These equations can be integrated using the initial conditions ψ ( t = 0 ) = ψ 0 and θ ( t = 0 ) = θ 0 to yield
ψ = ψ 0 with θ 0 and θ π ,
(10)
θ = arctan ( exp [ 3 2 ϵ ˙ H β t ] tan ( θ 0 ) ) .
(11)

The alignment angle, θ, is independent of ψ due to radial symmetry around the z-axis. Hence, ψ 0 is only required to uniquely initialize the orientation of the particle. Figure 2 shows the orientation trajectories for different initial values of θ with β = 1. It can be observed that the orientation rate decreases when the initial alignment angle approaches 0 ° or 90 °. At initial alignment angles of exactly 0 ° and 90 °, 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.

FIG. 2.

Particle orientation trajectories for large aspect ratio and different initial values of the alignment angle θ, according to Jeffery’s equation [Eq. (11)] for uniaxial extension.

FIG. 2.

Particle orientation trajectories for large aspect ratio and different initial values of the alignment angle θ, according to Jeffery’s equation [Eq. (11)] for uniaxial extension.

Close modal
This single particle model is extended to a multiparticle model by solving Eq. (11) for multiple particles with randomized initial orientation. Every time step the particle orientation tensor, P, can be computed by averaging the dyadic product between the particle unit vectors, according to [26],
P = 1 N i = 1 N p i p i ,
(12)
where N is the number of particles in the multiparticle model and p i is the orientation vector for particle i. The anisotropy parameter, ξ m, is a measure for the alignment of the particles in the bulk, which compares the average orientation component in the x- and y-directions with the average orientation component in the z-direction, as follows:
ξ m = 1 ν x 2 + ν y 2 ν z 1 P x x 2 + P y y 2 P z z ,
(13)
where ν x , ν y, and ν z are the eigenvalues of the particle orientation tensor. For extension along the z-axis with initial random particle orientation, all off-diagonal components of P at every time step are almost zero. Therefore, the diagonal components of P equal the eigenvalues of this tensor. Initially, for a random fiber orientation, ν x = ν y = ν z = 1 / 3, and the anisotropy parameter becomes ξ m = 0.41. Due to the lack of interaction between particles, in the multiparticle model, it is possible to obtain full orientation of particles in the flow direction. In this case, ν x = ν y = 0, ν z = 1 and the anisotropy parameter becomes ξ m = 1. In case all fibers align perpendicular to the stretch direction, ν z ν x and ν z ν y and the anisotropy parameter becomes ξ m = . By systematically increasing the number of particles in the multiparticle model, it was determined that at least 1000 particles are required to obtain a reproducible average anisotropy. For this amount of particles, the anisotropy of the particles in the bulk as a function of strain is investigated for different particle aspect ratios, see Fig. 3. The figure shows that for particle aspect ratios larger than 7, the orientation kinetics are independent of the particle aspect ratio.
FIG. 3.

Anisotropy [Eqs. (10)–(13)] of ellipsoidal particles as a function of linear strain for different particle aspect ratios in uniaxial extension. Note that for isotropic orientation, ξ m , 0 = 0.41.

FIG. 3.

Anisotropy [Eqs. (10)–(13)] of ellipsoidal particles as a function of linear strain for different particle aspect ratios in uniaxial extension. Note that for isotropic orientation, ξ m , 0 = 0.41.

Close modal

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.

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.

For this study, low density polyethylene, LDPE ( M w = 153 344 g mol 1, M n = 12 810 g mol 1, see Sec. S1 in the supplementary material [40]) and linear low density polyethylene, LLDPE ( M w = 148 059 g mol 1, M n = 36 230 g mol 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 1.51, 20 °C) closely matches the refractive index of glass ( n 1.52). 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  μm and a volume averaged length of 111  μm. 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.

First polyethylene and glass fibers are mechanically mixed to obtain a homogeneously mixed extrudate. The mixing is performed at 230 °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 °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 × 2 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 × width) of approximately 16 × 10 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.

The rheology of the polymer matrix is characterized using small amplitude oscillatory shear (SAOS) measurements with an ARES melts LS 818 007 equipped with a 25 mm diameter plate-plate geometry and a nitrogen flushed convection oven. First, a strain sweep is conducted to determine the linear regime. Subsequently, frequency sweeps in the range of 0.01 to 100 rad  s 1 are performed in this regime (10% strain) at temperatures ranging from 140 °C to 240 °C. Using the same apparatus and geometry, the shear stress growth coefficient is measured, as follows [32]:
η s + = M 2 π R geo 3 ( 3 + d ln M d ln γ ˙ R ) 1 γ ˙ R ,
(14)
where η s + is the shear stress growth coefficient, which will be referred to as the transient shear viscosity in the remainder of the paper, M is the torque, R geo the radius of the plate-plate geometry, and γ ˙ R the shear rate in the material at the edge of the sample. In a plate-plate geometry the shear rate is a function of the radius, and therefore, the correction term, d ln M / d ln γ ˙ R, is required to obtain a correct value of the transient shear viscosity in shear thinning polymer melts. Each measurement is performed on a new sample at a temperature of 150 °C, which is determined using an external thermocouple. The data for the first normal stress growth function are presented in Fig. S2 in the supplementary material [40].
CT-scans were performed using a Nanotom 160 NF (General Electric/Phoenix) with a voltage of 60 kV, a current of 310  μA and a voxel size of about 1.3  μm. Analysis of the scans was performed in GeoDict [33], which allows for identifying the start- and end coordinates in 3D space of individual fibers. The fiber orientation tensor and the anisotropy parameter of the fibers in the CT-scans are computed using an approach similar to Eqs. (12)–(13):
ξ CT = 1 P x x , CT 2 + P y y , CT 2 P z z , CT ,
(15)
where ξ CT is the anisotropy of the fibers in the bulk as determined by the CT-scan.

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 ( η = 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  μm are visible ( q-range from 1.9 × 10 4 to 1.05 × 10 3 nm 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.

FIG. 4.

Schematic representation of the SER-SALS setup. From right to left, the most relevant components of the setup are a Neon-Helium laser (1), a Sentmanat extensional measurement device (2), a convection oven (3) with an oil-filled cuvette (4), a detector plate (5), and a camera (6).

FIG. 4.

Schematic representation of the SER-SALS setup. From right to left, the most relevant components of the setup are a Neon-Helium laser (1), a Sentmanat extensional measurement device (2), a convection oven (3) with an oil-filled cuvette (4), a detector plate (5), and a camera (6).

Close modal

1. Extensional rheology

The transient uniaxial extensional viscosity is determined using the Sentmanat extensional measurement device and is computed as [31]
η E + = 1 ε ˙ H 1 A 0 exp ( ε ˙ H t ) Δ ρ 2 / 3 M 2 r D ,
(16)
where ε ˙ H is the applied Hencky strain rate, A 0 is the initial cross-sectional area of the sample, t is the experimental time, M is the measured torque, r D is the drum radius of the SER, and Δ ρ is a correction factor for the changing dimensions of the sample going from the solid state to the molten state, for which the preset value of 1.2 given by the manufacturer is used.

For all extensional experiments, the sample temperature, determined using an external thermocouple, is set to 150 °C to ensure consistency with the shear rheometer. After lowering the sample in silicon oil, a torque of 15  μNm 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

The level of anisotropy or the alignment of the glass fibers in the sample is determined by the eccentricity of the scattering pattern. At 100 frames per second, a camera captures the scattering patterns in time. The background image is subtracted from each individual image to remove noise and the intensities originating solely from the primary laser beam. Next, the beam stop and the suspending rope of the beam stop are removed from the image, resulting in a scattering pattern suitable for analysis, see Fig. 5(a). Then, a contour plot is made, based on intensity levels corresponding to fixed q-values along the vertical direction center line through the beam stop (azimuthal coordinate of 0 rad), which results in Fig. 5(b). Subsequently, ellipses are fitted through the points of the contour plot, see Fig. 5(c). In general, ellipses can be described using a matrix notation. The eigenvalues of this matrix will be referred to as the eigenvalues of the ellipse. A two-dimensional ellipse, see Fig. 5(d), has two eigenvalues, ν y = ( a / 2 ) 2 and ν z = ( b / 2 ) 2, where a and b are the long and short axis of the ellipse, respectively. Since the 2D scattering pattern only yields two eigenvalues, the anisotropy of the fibers according to the SALS experiment, ξ SALS, is determined by the ratio of both eigenvalues, similar to Sec. II,
ξ SALS = ( 1 ν z ν y ) .
(17)
FIG. 5.

Typical image analysis for the obtained scattering patterns: (a) scattering pattern after background subtraction and beamstop removal, (b) intensity isocontour plots, (c) ellipses describing the intensity isocontours, and (d) the eigenvalues of the outer ellipse.

FIG. 5.

Typical image analysis for the obtained scattering patterns: (a) scattering pattern after background subtraction and beamstop removal, (b) intensity isocontour plots, (c) ellipses describing the intensity isocontours, and (d) the eigenvalues of the outer ellipse.

Close modal

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 ° 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.

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 1.41 , 20 °C). An additional benefit is that silicon oil will not chemically interact with polyethylene. Also, the viscosity of the oil should be low ( η oil 10 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 ( ε ˙ 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 1. To completely remove the lubrication layer at a strain rate of 1  s 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 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.

FIG. 6.

Sample width as a function of time for LLDPE with a Hencky strain rate, ε ˙ H, of 1  s 1. (a) Initial geometry of the sample including the definition of the width, (b) data for cuvette filled with 1 Pa s silicon oil, and (c) with 10 mPa s silicon oil. For (c), p300 sanded Kapton tape is applied to the drum of the SER. The markers are the experimentally obtained data, the solid lines are the predictions for uniaxial extension for a SER, w 0 exp ( ε ˙ H t / 2 ) (constant sample length between the sample contacts and the drum), and the dashed lines correspond to the prediction for uniaxial extension when only the ends of the sample are fixed to the drum. The different colors indicate repeat measurements.

FIG. 6.

Sample width as a function of time for LLDPE with a Hencky strain rate, ε ˙ H, of 1  s 1. (a) Initial geometry of the sample including the definition of the width, (b) data for cuvette filled with 1 Pa s silicon oil, and (c) with 10 mPa s silicon oil. For (c), p300 sanded Kapton tape is applied to the drum of the SER. The markers are the experimentally obtained data, the solid lines are the predictions for uniaxial extension for a SER, w 0 exp ( ε ˙ H t / 2 ) (constant sample length between the sample contacts and the drum), and the dashed lines correspond to the prediction for uniaxial extension when only the ends of the sample are fixed to the drum. The different colors indicate repeat measurements.

Close modal
FIG. 7.

Transient uniaxial extensional viscosity of LLDPE with a Hencky strain rate of 1.0  s 1 measured in different environments. The solid black line represents the linear viscoelastic response of LLDPE.

FIG. 7.

Transient uniaxial extensional viscosity of LLDPE with a Hencky strain rate of 1.0  s 1 measured in different environments. The solid black line represents the linear viscoelastic response of LLDPE.

Close modal

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.

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.

1. Rheology of the polymer matrix

Small amplitude oscillatory shear (SAOS) experiments are performed to characterize the linear viscoelastic (LVE) response of the matrix materials. Time–temperature–superposition (TTS) is used to construct a master curve of the storage and loss moduli, where the vertical shift factor b T = 1 and the horizontal shift factor a T follows an Arrhenius-like equation [35],
a T = exp [ Δ H R ( 1 T 1 T ref ) ] ,
(18)
where Δ H is the activation energy for flow in J mol 1, R = 8.314 J K 1 mol 1 is the ideal gas constant, T is the temperature in K, and T ref = 423 K is the reference temperature. The master curves are captured using seven Maxwell modes for LDPE and five Maxwell modes for LLDPE, as shown by the black lines in Fig. 8. The model parameters are summarized in Table I. From the Maxwell model fit, the linear viscoelastic behavior is obtained. In shear flow,
η S LVE + = i = 1 N η 0 , i [ 1 exp ( t λ i ) ] ,
(19)
where η S LVE + is the transient linear viscoelastic viscosity, N is the number of modes used to describe the dynamic frequency sweep, η 0 , i is the zero-shear viscosity for the i th Maxwell mode, and λ i is the relaxation time for the i th Maxwell mode.
FIG. 8.

Master curve at 150  °C for the storage ( G ) and loss moduli ( G ) as a function of the time-temperature shifted frequency for (a) LDPE and (b) LLDPE at representative temperatures. The black lines are fits with the multimode Maxwell model. The used shift factors are shown in the insets.

FIG. 8.

Master curve at 150  °C for the storage ( G ) and loss moduli ( G ) as a function of the time-temperature shifted frequency for (a) LDPE and (b) LLDPE at representative temperatures. The black lines are fits with the multimode Maxwell model. The used shift factors are shown in the insets.

Close modal
TABLE I.

Arrhenius and Maxwel model parameters describing the linear viscoelastic response of LDPE and LLDPE at a temperature of 150 °C.

LDPELLDPE
ΔH (kJ mol−150 27.5 
η 0 1 (Pa s), λ1 (s) 9.29 × 102, 1.0 × 10−2 3.97 × 103, 1.15 × 10−3 
η 0 2 (Pa s), λ2 (s) 3.11 × 103, 8.51 × 10−2 7.11 × 103, 6.35 × 10−2 
η 0 3 (Pa s), λ3 (s) 8.91 × 103, 5.18 × 10−1 5.33 × 103, 3.18 × 10−1 
η 0 4 (Pa s), λ4 (s) 2.18 × 104, 3.02 × 100 1.61 × 103, 2.06 × 100 
η 0 5 (Pa s), λ5 (s) 3.82 × 104, 1.65 × 101 5.44 × 102, 1.70 × 101 
η 0 6 (Pa s), λ6 (s) 3.76 × 104, 7.53 × 101 — 
η 0 7 (Pa s), λ7 (s) 2.43 × 104, 2.74 × 102 — 
LDPELLDPE
ΔH (kJ mol−150 27.5 
η 0 1 (Pa s), λ1 (s) 9.29 × 102, 1.0 × 10−2 3.97 × 103, 1.15 × 10−3 
η 0 2 (Pa s), λ2 (s) 3.11 × 103, 8.51 × 10−2 7.11 × 103, 6.35 × 10−2 
η 0 3 (Pa s), λ3 (s) 8.91 × 103, 5.18 × 10−1 5.33 × 103, 3.18 × 10−1 
η 0 4 (Pa s), λ4 (s) 2.18 × 104, 3.02 × 100 1.61 × 103, 2.06 × 100 
η 0 5 (Pa s), λ5 (s) 3.82 × 104, 1.65 × 101 5.44 × 102, 1.70 × 101 
η 0 6 (Pa s), λ6 (s) 3.76 × 104, 7.53 × 101 — 
η 0 7 (Pa s), λ7 (s) 2.43 × 104, 2.74 × 102 — 

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  °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 η E LVE + ( t ) = 3 η S LVE + ( t ). For LLDPE, the linear viscoelastic response slightly underestimates the experimental transient uniaxial extensional viscosity.

FIG. 9.

Transient shear viscosities at different shear rates measured at a temperature of 150  °C for (a) LDPE and (b) LLDPE. The black solid line represents the linear viscoelastic behavior, as predicted from the SAOS data.

FIG. 9.

Transient shear viscosities at different shear rates measured at a temperature of 150  °C for (a) LDPE and (b) LLDPE. The black solid line represents the linear viscoelastic behavior, as predicted from the SAOS data.

Close modal
FIG. 10.

Transient uniaxial extensional viscosities at different Hencky strain rates measured at a temperature of 150  °C for (a) LDPE and (b) LLDPE. The black solid line represents the linear viscoelastic behavior, as predicted from the SAOS data.

FIG. 10.

Transient uniaxial extensional viscosities at different Hencky strain rates measured at a temperature of 150  °C for (a) LDPE and (b) LLDPE. The black solid line represents the linear viscoelastic behavior, as predicted from the SAOS data.

Close modal
The material properties describing the viscoelastic behavior can be used to derive the Weissenberg number, which is a characteristic dimensionless number relating the elastic to the viscous stresses in the experiment
Wi = λ c ε ˙ H ,
(20)
where λ c is a characteristic timescale of the material. Typically, for the study of crystallization kinetics, the longest relaxation time is important [36]. However, in the melt state, the viscosity-averaged relaxation time gives a better representative time scale, which is defined as
λ c = i = 1 N η 0 , i λ i i = 1 N η 0 , i .
(21)

For LDPE and LLDPE, the average relaxation times are, respectively, 53 and 0.62 s at a temperature of 150  °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 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.

FIG. 11.

(a) Anisotropy of the scattering pattern as a function of linear strain for LLDPE with a fiber volume fraction of 1%, extended with a Hencky strain rate of 0.01  s 1, (b), (c), (d), and (e) show pictures of the scattering pattern corresponding to linear strains of 0, 0.5, 1.8, and 5.8, (f)–(i) show the corresponding isocontour plots at different intensities including the fitted ellipses. The anisotropies corresponding to the second largest (red) ellipses ( ξ SALS) are 0.10, 0.63, 0.90, and 0.98 for (f)–(i), respectively.

FIG. 11.

(a) Anisotropy of the scattering pattern as a function of linear strain for LLDPE with a fiber volume fraction of 1%, extended with a Hencky strain rate of 0.01  s 1, (b), (c), (d), and (e) show pictures of the scattering pattern corresponding to linear strains of 0, 0.5, 1.8, and 5.8, (f)–(i) show the corresponding isocontour plots at different intensities including the fitted ellipses. The anisotropies corresponding to the second largest (red) ellipses ( ξ SALS) are 0.10, 0.63, 0.90, and 0.98 for (f)–(i), respectively.

Close modal

First, nonextended samples are cut into a size of 1 × 1 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 y y- and z z 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 ± 0.017 and P x x , CT , 0 = 0.008 ± 0.0013, meaning that fibers are initially oriented in the z y-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 ± 0.1215). 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 y z-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.

FIG. 12.

3D representation of CT-scanned samples. (a) Initial orientation of LDPE with fiber volume fraction of 1%. The resulting anisotropy ξ CT = 0.2, where P x x , CT , 0 = 0.25, (b) initial orientation of LDPE with fiber volume fraction of 5%. The resulting anisotropy ξ CT = 0.1, where P x x , CT , 0 = 0.014, and (c) initial orientation of LDPE with fiber volume fraction of 10%. The resulting anisotropy ξ CT = 0.24, where P x x , CT , 0 = 0.008.

FIG. 12.

3D representation of CT-scanned samples. (a) Initial orientation of LDPE with fiber volume fraction of 1%. The resulting anisotropy ξ CT = 0.2, where P x x , CT , 0 = 0.25, (b) initial orientation of LDPE with fiber volume fraction of 5%. The resulting anisotropy ξ CT = 0.1, where P x x , CT , 0 = 0.014, and (c) initial orientation of LDPE with fiber volume fraction of 10%. The resulting anisotropy ξ CT = 0.24, where P x x , CT , 0 = 0.008.

Close modal
Figure 11(a) clearly shows that the degree of anisotropy levels off at a strain of about 4–5. Since this was the case for all experiments performed, the level of anisotropy obtained with the SALS method is compared to the level of anisotropy measured using a CT scan after extension until this strain. First, a SER-SALS measurement is performed with a Hencky strain rate of 0.01  s 1 until a linear strain of 5. Then, the strain rate is set to zero for 50 s to ensure internal stress relaxation of the material, and afterward, the anisotropy ξ SALS is determined. Stress relaxation after extension with these low Hencky strain rates did not influence the scattering pattern. Next, the sample is removed from the setup, cooled down fast, and cut into a size of 3 × 4 mm 2 at the location of the laser beam. Subsequently, a CT-scan is performed to determine ξ CT. The visualization in Geodict of a 1 × 1 mm 2 piece of a stretched CT-scanned sample is shown for different fiber volume fractions in Fig. 13. The figure shows that the fibers display an orientational distribution around the director, which points in the extension direction. The ratio between the anisotropy of the fibers in the bulk according to the CT-scan and the anisotropy of the fibers in the bulk according to the SALS experiment for the same sample is defined as the correction factor, C,
C = ξ CT ξ SALS .
(22)
FIG. 13.

3D representation of CT-scanned samples. (a) LDPE with fiber volume fraction of 1%. The sample is extended using a Hencky strain rate of 0.01  s 1 to a linear strain of 5. The resulting anisotropy ξ CT = 0.97, (b) LDPE with fiber volume fraction of 5%. The sample is extended using a Hencky strain rate of 0.01  s 1 to a linear strain of 5. The resulting anisotropy ξ CT = 0.97, and (c) LDPE with fiber volume fraction of 10%. The sample is extended using a Hencky strain rate of 0.01  s 1 to a linear strain of 3. The resulting anisotropy ξ CT = 0.94.

FIG. 13.

3D representation of CT-scanned samples. (a) LDPE with fiber volume fraction of 1%. The sample is extended using a Hencky strain rate of 0.01  s 1 to a linear strain of 5. The resulting anisotropy ξ CT = 0.97, (b) LDPE with fiber volume fraction of 5%. The sample is extended using a Hencky strain rate of 0.01  s 1 to a linear strain of 5. The resulting anisotropy ξ CT = 0.97, and (c) LDPE with fiber volume fraction of 10%. The sample is extended using a Hencky strain rate of 0.01  s 1 to a linear strain of 3. The resulting anisotropy ξ CT = 0.94.

Close modal
The correction factor is, in general, dependent on q-value and fiber volume fraction. Since the q-value is fixed in this study, the correction factor is only computed for different volume fractions, see Table II, where three repetitions are performed to compute the error. The anisotropy determined using SALS highly agrees with the anisotropy determined using the CT-scan. For reference purposes, the table also includes the corresponding sample thicknesses, h. Finally, C is used to correct the anisotropy determined with the SALS measurement technique to the more accurate anisotropy value obtained by the CT-scan:
ξ e = ξ SALS C .
(23)
TABLE II.

Ratio between 3D anisotropy from x-ray tomography and 2D anisotropy from SALS for different fiber volume fractions.

1% Fibers5% Fibers10% 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% Fibers5% Fibers10% 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, ξ e, will be considered.

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 β = 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 y z-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 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.

FIG. 14.

Anisotropy, ξ e, for Weissenberg numbers between 0.53 and 53, as a function of linear strain for (a) LDPE with fiber volume fraction of 1%, (b) LDPE with fiber volume fraction of 5%, and (c) LDPE with fiber volume fraction of 10%. The solid black line corresponds to Jeffery’s prediction for uniaxial extension, ξ m, [Eqs. (10)–(13)], with P x x , 0 as determined from average CT-scans. Filled symbols indicate averages with error bars indicating the standard deviation. Open symbols indicate individual experimental data points.

FIG. 14.

Anisotropy, ξ e, for Weissenberg numbers between 0.53 and 53, as a function of linear strain for (a) LDPE with fiber volume fraction of 1%, (b) LDPE with fiber volume fraction of 5%, and (c) LDPE with fiber volume fraction of 10%. The solid black line corresponds to Jeffery’s prediction for uniaxial extension, ξ m, [Eqs. (10)–(13)], with P x x , 0 as determined from average CT-scans. Filled symbols indicate averages with error bars indicating the standard deviation. Open symbols indicate individual experimental data points.

Close modal
FIG. 15.

Anisotropy, ξ e, for Weissenberg numbers between 0.0062 and 0.62, as a function of linear strain for (a) LLDPE with fiber volume fraction of 1%, (b) LLDPE with fiber volume fraction of 5%, and (c) LLDPE with fiber volume fraction of 10%. The solid black line corresponds to Jeffery’s prediction for uniaxial extension, ξ m, [Eqs. (10)–(13)], with P x x , 0 as determined from average CT-scans. Filled symbols indicate averages with error bars indicating the standard deviation. Open symbols indicate individual experimental data points.

FIG. 15.

Anisotropy, ξ e, for Weissenberg numbers between 0.0062 and 0.62, as a function of linear strain for (a) LLDPE with fiber volume fraction of 1%, (b) LLDPE with fiber volume fraction of 5%, and (c) LLDPE with fiber volume fraction of 10%. The solid black line corresponds to Jeffery’s prediction for uniaxial extension, ξ m, [Eqs. (10)–(13)], with P x x , 0 as determined from average CT-scans. Filled symbols indicate averages with error bars indicating the standard deviation. Open symbols indicate individual experimental data points.

Close modal

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 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 0.5 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.

FIG. 16.

Comparison between anisotropy as a function of linear strain for LDPE with fiber volume fraction of 5% and initial thickness of 0.2 mm and (a) LDPE with fiber volume fraction of 5% and initial thickness of 0.4 mm and (b) LDPE with fiber volume fraction of 10% and thickness of 0.2 mm. The solid black line corresponds to Jeffery’s prediction for uniaxial extension, ξ m, [Eqs. (10)–(13)], with P x x , 0 as determined from average CT-scans. The error bars indicate the standard deviation.

FIG. 16.

Comparison between anisotropy as a function of linear strain for LDPE with fiber volume fraction of 5% and initial thickness of 0.2 mm and (a) LDPE with fiber volume fraction of 5% and initial thickness of 0.4 mm and (b) LDPE with fiber volume fraction of 10% and thickness of 0.2 mm. The solid black line corresponds to Jeffery’s prediction for uniaxial extension, ξ m, [Eqs. (10)–(13)], with P x x , 0 as determined from average CT-scans. The error bars indicate the standard deviation.

Close modal

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 1 and Kobayashi et al. considered Hencky strain rates between 0.003 and 0.3  s 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, ϕ = 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

In this section, the rheology of the fiber composites is investigated. The uniaxial extensional viscosity of the composites is measured simultaneously with the fiber orientation and is compared to the Batchelor expression for rodlike particles perfectly aligned in the flow direction [39]:
η E B ( t ) = η E pure ( t ) ( 1 + 4 ϕ r e 2 9 ln ( π / ϕ ) ) ,
(24)
where η E pure is the transient uniaxial extensional viscosity of the matrix, r e is the aspect ratio ( L / D) and ϕ the fiber volume fraction. Figures 17(a) and 17(b) show the transient uniaxial extensional viscosity for a Hencky strain rate of 1.0  s 1 for LDPE and LLDPE composites, respectively. The experimentally obtained average transient uniaxial extensional viscosities are indicated with markers and the solid lines indicate the Batchelor prediction at that fiber volume fraction. The transient uniaxial extensional viscosity for a Hencky strain rate of 0.1  s 1 is shown for LDPE and LLDPE composites in Figs. 17(c) and 17(d). The rheological data for LLDPE at high strains shows a decrease in viscosity, which implies local necking or failure of the sample. Unfortunately, for the thin fiber-filled samples, no transient extensional viscosity data could be obtained at a Hencky strain rate of 0.01 s 1 due to limitations in the accuracy of the setup.
FIG. 17.

Transient uniaxial extensional viscosity as a function of time during the in situ SALS measurements. (a) LDPE with ε ˙ H = 1 s 1, (b) LLDPE with ε ˙ H = 1 s 1, (c) LDPE with ε ˙ H = 0.1 s 1, and (d) LLDPE with ε ˙ H = 0.1 s 1. The solid lines correspond to the Batchelor prediction [Eq. (24)]. The error bars indicate the standard deviation.

FIG. 17.

Transient uniaxial extensional viscosity as a function of time during the in situ SALS measurements. (a) LDPE with ε ˙ H = 1 s 1, (b) LLDPE with ε ˙ H = 1 s 1, (c) LDPE with ε ˙ H = 0.1 s 1, and (d) LLDPE with ε ˙ H = 0.1 s 1. The solid lines correspond to the Batchelor prediction [Eq. (24)]. The error bars indicate the standard deviation.

Close modal

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- α-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 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.

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 ( ϕ D / 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.

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.

The authors have no conflicts to disclose.

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

1.
Laurencin
,
T.
,
L.
Orgéas
,
P. J. J.
Dumont
,
S. R.
du Roscoat
,
P.
Laure
,
S.
Le Corre
,
L.
Silva
,
R.
Mokso
, and
M.
Terrien
, “
3D real-time and in situ characterisation of fibre kinematics in dilute non-newtonian fibre suspensions during confined and lubricated compression flow
,”
Compos. Sci. Technol.
134
,
258
266
(
2016
).
2.
Simon
,
S. A.
,
A.
Bechara Senior
, and
T.
Osswald
, “
Experimental validation of a direct fiber model for orientation prediction
,”
J. Compos. Sci.
4
(
2
),
59
(
2020
).
3.
Thi
,
T. B. N.
,
M.
Morioka
,
A.
Yokoyama
,
S.
Hamanaka
,
K.
Yamashita
, and
C.
Nonomura
, “
Measurement of fiber orientation distribution in injection-molded short-glass-fiber composites using x-ray computed tomography
,”
J. Mater. Process. Technol.
219
,
1
9
(
2015
).
4.
McGee
,
S. H.
, and
R. L.
McCullough
, “
Characterization of fiber orientation in short-fiber composites
,”
J. Appl. Phys.
55
(
5
),
1394
1403
(
1984
).
5.
Pignon
,
F.
,
A.
Magnin
,
J. M.
Piau
,
G.
Belina
, and
P.
Panine
, “
Structure and orientation dynamics of sepiolite fibers–poly (ethylene oxide) aqueous suspensions under extensional and shear flow, probed by in situ SAXS
,”
Rheol. Acta
48
,
563
578
(
2009
).
6.
Pujari
,
S.
,
S. S.
Rahatekar
,
J. W.
Gilman
,
K. K.
Koziol
,
A. H.
Windle
, and
W. R.
Burghardt
, “
Orientation dynamics in multiwalled carbon nanotube dispersions under shear flow
,”
J. Chem. Phys.
130
(
21
),
214903
(
2009
).
7.
Pujari
,
S.
,
L.
Dougherty
,
C.
Mobuchon
,
P. J.
Carreau
,
M. C.
Heuzey
, and
W. R.
Burghardt
, “
X-ray scattering measurements of particle orientation in a sheared polymer/clay dispersion
,”
Rheol. Acta
50
,
3
16
(
2011
).
8.
Pujari
,
S.
,
S.
Rahatekar
,
J. W.
Gilman
,
K. K.
Koziol
,
A. H.
Windle
, and
W. R.
Burghardt
, “
Shear-induced anisotropy of concentrated multiwalled carbon nanotube suspensions using x-ray scattering
,”
J. Rheol.
55
(
5
),
1033
1058
(
2011
).
9.
Dykes
,
L. M. C.
,
J. M.
Torkelson
,
W. R.
Burghardt
, and
R.
Krishnamoorti
, “
Shear-induced orientation in polymer/clay dispersions via in situ x-ray scattering
,”
Polymer
51
(
21
),
4916
4927
(
2010
).
10.
Scirocco
,
R.
,
J.
Vermant
, and
J.
Mewis
, “
Effect of the viscoelasticity of the suspending fluid on structure formation in suspensions
,”
J. Non-Newtonian Fluid Mech.
117
(
2-3
),
183
192
(
2004
).
11.
Pasquino
,
R.
,
D.
Panariello
, and
N.
Grizzuti
, “
Migration and alignment of spherical particles in sheared viscoelastic suspensions. A quantitative determination of the flow-induced self-assembly kinetics
,”
J. Colloid Interface Sci.
394
,
49
54
(
2013
).
12.
Vermant
,
J.
,
P.
Van Puyvelde
,
P.
Moldenaers
,
J.
Mewis
, and
G. G.
Fuller
, “
Anisotropy and orientation of the microstructure in viscous emulsions during shear flow
,”
Langmuir
14
(
7
),
1612
1617
(
1998
).
13.
Mohammad Mehdipour
,
N.
,
N.
Reddy
,
R. J.
Shor
, and
G.
Natale
, “
Orientation dynamics of anisotropic and polydisperse colloidal suspensions
,”
Phys. Fluids
34
(
8
),
083317
(
2022
).
14.
Girasole
,
T.
,
J. N.
Le Toulouzan
,
J.
Mroczka
, and
D.
Wysoczanski
, “
Fiber orientation and concentration analysis by light scattering: Experimental setup and diagnosis
,”
Rev. Sci. Instrum.
68
(
7
),
2805
2811
(
1997
).
15.
Girasole
,
T.
,
H.
Bultynck
,
G.
Gouesbet
,
G.
Gréhan
,
F.
Le Meur
,
J.
Le Toulouzan
,
J.
Mroczka
,
K. F.
Ren
,
C.
Rozé
, and
D.
Wysoczanski
, “
Cylindrical fibre orientation analysis by light scattering. part 1: Numerical aspects
,”
Part. Part. Syst. Charact.
14
(
4
),
163
174
(
1997
).
16.
Girasole
,
T.
,
G.
Gouesbet
,
G.
Gréhan
,
J.
Le Toulouzan
,
J.
Mroczka
,
K. F.
Ren
, and
D.
Wysoczanski
, “
Cylindrical fibre orientation analysis by light scattering. part 2: Experimental aspects
,”
Part. Part. Syst. Charact.
14
(
5
),
211
218
(
1997
).
17.
Wagner
,
A. H.
,
D. M.
Kalyon
,
R.
Yazici
, and
T. J.
Fiske
, “
Uniaxial extensional flow behavior of a glass fiber-filled engineering plastic
,”
J. Reinf. Plast. Compos.
22
(
4
),
327
337
(
2003
).
18.
Zhang
,
W.
,
S.
Komoto
,
H.
Yamane
,
M.
Takahashi
, and
J. L.
White
, “
Biaxial extensional flow behavior and fiber dispersion and orientation in short glass fiber filled polypropylene melts
,”
J. Soc. Rheol. Jpn.
29
(
3
),
111
120
(
2001
).
19.
Kobayashi
,
M.
,
T.
Takahashi
,
J.
Takimoto
, and
K.
Koyama
, “
Flow-induced whisker orientation and viscosity for molten composite systems in a uniaxial elongational flow field
,”
Polymer
36
(
20
),
3927
3933
(
1995
).
20.
Férec
,
J.
,
M. C.
Heuzey
,
J.
Pérez-González
,
L.
Vargas
,
G.
Ausias
, and
P. J.
Carreau
, “
Investigation of the rheological properties of short glass fiber-filled polypropylene in extensional flow
,”
Rheol. Acta
48
(
1
),
59
72
(
2009
).
21.
Wang
,
J.
,
W.
Yu
,
C.
Zhou
,
Y.
Guo
,
W.
Zoetelief
, and
P.
Steeman
, “
Elongational rheology of glass fiber-filled polymer composites
,”
Rheol. Acta
55
,
833
845
(
2016
).
22.
Jeffery
,
G. B.
, “
The motion of ellipsoidal particles immersed in a viscous fluid
,”
Proc. R. Soc. A
102
,
161
179
(
1922
).
23.
Takserman-Krozer
,
R.
, and
A.
Ziabicki
, “
Behavior of polymer solutions in a velocity field with parallel gradient. I. Orientation of rigid ellipsoids in a dilute solution
,”
J. Polym. Sci., Part A: Gener. Pap.
1
(
1
),
491
506
(
1963
).
24.
Tucker
,
C. L.
,
Fundamentals of Fiber Orientation Description, Measurement and Prediction
(
Hanser Publishers
,
Munich
,
2023
).
25.
Harris
,
J. B.
, and
J. F. T.
Pittman
, “
Equivalent ellipsoidal axis ratios of slender rod-like particles
,”
J. Colloid Interface Sci.
50
,
280
282
(
1975
).
26.
Advani
,
S. G.
, and
C. L.
Tucker
, “
The use of tensors to describe and predict fiber orientation in short fiber composites
,”
J. Rheol.
31
,
751
784
(
1987
).
27.
Wang
,
J.
,
J. F.
O’Gara
, and
C. L.
Tucker
, “
An objective model for slow orientation kinetics in concentrated fiber suspensions: Theory and rheological evidence
,”
J. Rheol.
52
(
5
),
1179
1200
(
2008
).
28.
Eberle
,
A. P. R.
,
G. M.
Vélez-García
,
D. G.
Baird
, and
P.
Wapperom
, “
Fiber orientation kinetics of a concentrated short glass fiber suspension in startup of simple shear flow
,”
J. Non-Newtonian Fluid Mech.
165
(
3-4
),
110
119
(
2010
).
29.
Sepehr
,
M.
,
G.
Ausias
, and
P. J.
Carreau
, “
Rheological properties of short fiber filled polypropylene in transient shear flow
,”
J. Non-Newtonian Fluid Mech.
123
(
1
),
19
32
(
2004
).
30.
d’Avino
,
G.
,
M. A.
Hulsen
,
F.
Greco
, and
P. L.
Maffettone
, “
Bistability and metabistability scenario in the dynamics of an ellipsoidal particle in a sheared viscoelastic fluid
,”
Phys. Rev. E
89
(
4
),
043006
(
2014
).
31.
Sentmanat
,
M.
,
B. N.
Wang
, and
G. H.
McKinley
, “
Measuring the transient extensional rheology of polyethylene melts using the ser universal testing platform
,”
J. Rheol.
49
(
3
),
585
606
(
2005
).
32.
Macosko
,
C. W.
,
Rheology: Principles, Measurements, and Applications
(
Wiley-VCH
,
New York
,
1994
).
33.
Math2Market GmbH, Germany, Geodict simulation software release 2022.
34.
Portale
,
G.
,
E. M.
Troisi
,
G. W. M.
Peters
, and
W.
Bras
, “
Real-time fast structuring of polymers using synchrotron WAXD/SAXS techniques
,”
Polymer
277
,
127
165
(
2015
).
35.
Morrison
,
F. A.
,
Understanding Rheology
(
Oxford University Press
,
New York
,
2001
).
36.
Housmans
,
J. W.
,
G. W. M.
Peters
, and
H.
Meijer
, “
Flow-induced crystallization of propylene/ethylene random copolymers
,”
J. Therm. Anal. Calorim.
98
(
3
),
693
705
(
2009
).
37.
Advani
,
S. G.
, and
C. L.
Tucker
, “
Orientation behavior of fibers in concentrated suspensions
,”
J. Reinf. Plast. Compos.
3
,
98
119
(
1984
).
38.
Férec
,
J.
,
G.
Ausias
,
M. C.
Heuzey
, and
P. J.
Carreau
, “
Modeling fiber interactions in semiconcentrated fiber suspensions
,”
J. Rheol.
53
(
1
),
49
72
(
2009
).
39.
Batchelor
,
G. K.
, “
The stress generated in a non-dilute suspension of elongated particles by pure straining motion
,”
J. Fluid Mech.
46
(
4
),
813
829
(
1971
).
40.
See the supplementary material online for GPC measurements, fiber length distribution data, and first normal stress growth function data.

Supplementary Material