Assessing nozzle flow dynamics in Fused Filament Fabrication through the parametric map $\alpha-\lambda$

Material extrusion, an additive manufacturing (AM) method, entails precisely dispensing material through a nozzle, following established standards [1]. Fused filament fabrication (FFF), introduced by S. Scott Crump in 1989 [2], currently stands as the predominant technology for material extrusion [3]. FFF allows for the rapid, cost-effective creation of intricate three-dimensional objects suitable for practical applications [4]. Moreover, post-processing techniques like sanding, chemical treatment, polishing, and metal plating, among others, can be employed to enhance surface quality [5, 6, 7].

The FFF procedure consists of introducing thermoplastic materials, usually in filament shape, from spools into a temperature-regulated liquefier, where they transform into liquid state. Subsequently, the molten material is extruded through a nozzle and guided by the print head onto a build plate in specific patterns to create a 3D object. Common quality issues in FFF systems encompass over-extrusion and under-extrusion, leading to the formation of blobs or holes. Another extrusion challenge, known as annular backflow, happens when the molten polymer reverses its flow along the ring-shaped region between the filament and the liquefier wall, escaping the heated area and cooling below the glass temperature, where solid/fluid transition occurs ($T_{g}$) [8]. These issues undoubtedly arise from the interplay of extrusion process parameters with the rheological properties of the molten filament [9]; however, state-of-the-art lacks fundamental research to understand the underlying mechanisms fully that allow for technical solutions to these issues.

The printing nozzle is at the heart of the FFF process, being a critical component responsible for precise material deposition [10, 11, 12]. The nozzle is a metal piece connecting the liquefier to the die through a contraction. It is well documented in the literature [13, 14, 15, 16, 17, 18, 19] that the flow of viscoelastic fluids through an axisymmetric contraction geometry can lead to elastic instabilities responsible for generating not only upstream vortices but also excess pressure drop. Furthermore, smaller characteristic length scales in the geometry intensify elastic effects. Previous efforts sought to replicate the polymer melt extrusion process within an FFF printer nozzle using microfluidic chips of a planar geometry with shear-thinning polymer solutions. This led to the construction of a Deborah-Reynolds flow map, revealing the production of growing upstream vortices under typical printing flow conditions in an FFF nozzle die [20]. Subsequently, the formation of vortices upstream of the tapered region in the nozzle, driven by elastic instabilities, was observed. This study was complemented in [21], where numerical analysis confirmed the presence of these vortices when molten Polycarbonate flowed through an axisymmetric nozzle geometry. It also allowed for the disentanglement of the relative significance of extension-induced and shear-induced elastic stresses in this extrusion process. It was concluded that within the upstream vortex, extensional elastic stresses predominated, but their importance diminished with higher extrusion velocities; moreover, they played a determinant role in the size of the upstream vortex structures. However, shear-induced normal stress differences may account for the excess pressure drop and a change in equilibrium height ($H^{*}$) in the backflow region, where the polymer melt flows upward between the solid filament and the liquefier wall. Recently, an experimental study with Polylactic Acid (PLA) [22] allowed the experimental determination of the nozzle pressure drop, which, in combination with traditional top-of-nozzle pressure measurements, permitted determining the pressure drop occurring in the liquefier and the calculation of equilibrium height in the backflow region based on experimental data exclusively. Results showed a non-linear increase in the pressure measured by the feeders, which was associated with shear-induced elastic instabilities occurring in the backflow region at the same extrusion velocity at which the maximum equilibrium height ($H^{*}$) occurred.

Most of the research papers available in the literature dealing with FFF are focused either on a single polymeric material, such as our previous works [21, 22], or on exclusively the viscous effect on flow dynamics [23, 24, 25], disregarding the viscoelastic intrinsic nature of the polymer melts affecting the stability of the flow through the nozzle at real printing speed [20]. In this work, we aim to go beyond the state-of-the-art by analyzing numerically how the non-linear rheological properties of different materials commonly used in FFF, such as PET-G, PET-CF, PLA, PC, ABS and PA6/66, can be reduced to limited numbers of parameters, i.e. Giesekus model parameters, and how this can help in sorting the materials and understanding the FFF nozzle flow dynamics; special attention will be given to the shear and extensional stress developed in the fluid, and how these are connected elastic instabilities, extra pressure drop and, ultimately, backflow issues.

The kinematics of the flow through the nozzle geometry is complex, i.e. at the wall, the polymer melt will undergo pure shear flow ($\dot{\gamma}=\frac{\partial u_{z}}{\partial r}$) and, at the centerline, the rate of deformation is purely extensional ($\dot{\epsilon}=\frac{\partial u_{z}}{\partial z}$). The complexity of the contraction flow is well represented by the flow-type parameter [57], defined as:

Figure 4 shows a composition of the contour plots corresponding to the gradient of flow type (left half) and the normal stress differences (right half) for all the materials considered in this study. Different patterns in the flow type can be observed depending on the material extruded; as expected, the flow type is mainly dominated by the shear flow (green), although important portions of the fluid domain, axisymmetric and with a dart shape, are pure extensional flow (red). As mentioned above, that complexity leads to a complex rate of deformation distribution, which, depending on the rheological properties of the fluid, results in a complex distribution of stress differences, depending on the material’s rheological properties.

Figure 5 shows the typical S-shaped curve for the upstream vortex size ($X_{r}$ [mm]) with a growing flow rate (Q [mm³/s]) for all materials, which tends asymptotically to the Moffatt vortex size at low flow rates and to a saturation size at high flow rates [14]. The combination of considerably high relaxation time and viscosity resulted in a faster growth rate of the upstream vortex size, as could be observed for Group 3 of polymers, whereas relatively low viscosities and relaxation time exhibited a lower rate of growth in upstream vortex structures (e.g. PET-G). The sequence observed in Figure 5 can be inferred from the product $\eta\cdot\lambda$ (Table 2), which is an estimation of the zero-shear first normal stress coefficient ($\Psi_{10}$ [48],  A); thus, the larger the first normal stress coefficient is, the more prone is the polymer to develop upstream vortices [16].

In FFF, the solid thermoplastic filament is melted in the liquefier and extruded through a nozzle. The rollers push down the solid filament, which works as a piston to impose a pressure-driven flow to the polymer melt. It has been previously reported [22] that the total pressure drop measured in the feeders can be decoupled as a sum of the pressure drop in the liquefier and the pressure drop in the nozzle, being the latter very close to the one predicted by the numerical model proposed here. In a cylindrical coordinate system, the conservation momentum equation for the steady-state Poiseuille flow of viscoelastic material through the nozzle gets reduced to the following ones, corresponding to $r$ and $z$-directions, respectively [38]:

By combining the data presented in Figure  6 with the viscosity curves depicted in Figure 3, one can observe that the larger the viscosity of the polymer, the larger the pressure drop. A stronger shear thinning behaviour (higher values in the $\alpha$ parameter) results in a bend of the $\Delta P-Q$ curve, whereas low values of $\alpha$ lead to elastic turbulence, with oscillating and larger pressure drops. This, together with the shapes of normal stress difference plots, allows for the understanding that a more squared/plug-shaped region in the center of the tapered region, along with the approximation of the vortex to the center of the channel (more pronounced in the PLA case (Figure 4)), leads to a higher magnitude of elastic turbulence/instabilities in the reentrant corner, a critical region of the flow.

By plotting the vortex size against the pressure drop for all materials (Figure 7), it is possible to attest that polymers with relatively low relaxation time ($\lambda<1$ s) are more sensitive, and the pressure drop increases drastically for a small increase in vortex size in its initial stages. This increase then becomes more gradual as the recirculation grows.

The shaded regions in Figures 5,6 and 7 represent the printing conditions when elastic turbulence is dominating. In that region, the pressure drop values and the vortices for this nozzle begin to oscillate and the shaded regions indicate the space between the minimum and maximum values recorded in the simulations, as in [22].

Although die-swell was not considered a matter of investigation in this work, it can be an important parameter in FFF, particularly on the fiber orientation and the resulting mechanical properties [66]. Die-swell is not affected by the presence of the upstream vortices [67], and it has been reported in the literature [68] that the swelling ratio decreases as the value of the $\alpha$ parameter increases. Therefore, according to Table 2, the larger die-swell would be expected for polymers belonging to Group 2, followed by ABS, PET-G, PET-CF and, finally, PLA.

This study has conducted a thorough numerical examination of upstream vortex formation in a standard Fused Filament Fabrication (FFF) nozzle during stable flow conditions. The investigation utilized the Giesekus constitutive model, incorporating data from steady simple shear experiments and transient extensional viscosity experiments conducted at different extension rates.

Significant contributions have been made to the body of knowledge previously derived from prior work [21, 20]. It has been demonstrated that even under steady flow conditions at extrusion velocities representative of practical scenarios, growing upstream vortices are influenced by elastic instabilities. These instabilities manifest as physical disturbances in pressure, leading to the phenomenon known as “elastic turbulence”.

Complex flow patterns induced within the nozzle, ranging from simple shear to extensional flow, have been critically examined. These flow phenomena affect the behaviour of molten polymer from the nozzle walls to the axis (as depicted in Figure 4). This intricate behaviour persists through the tapered region and further upstream into the liquefier, while the flow inside the die is primarily characterized by simple shear.

Two key observations have arisen from the analysis of flow types and variations in normal stress: Firstly, the vortex tip marks the culmination of elastic stresses induced by extensional flow, and secondly, the shape of the stress contour plot has a direct impact on the proximity of the upstream vortex to the nozzle centerline. These observations are further supported by the consistent appearance of rheological groups identified during the characterization. In the examination of the plots displaying normal stress surfaces, the intricate interplay between these stresses surrounding the contraction has been discerned. This relationship sheds light on the rheological properties of the fluid and contributes significantly to variations in pressure drop and the emergence of instabilities. This fluid characteristic appears to correlate with elevated excess pressure drop, potentially causing operational challenges during printing.

Furthermore, the sensitivity of various materials to the creation of recirculation vortices has been uncovered. A thorough comprehension of total pressure drop and upstream vortex formation is paramount for optimizing the FFF process. These factors directly influence extrusion rates, backflow tendencies, and overall print quality, as previously discussed [22]. Elevated pressure drops can give rise to complications such as filament deformation, irregular flow patterns, or even nozzle clogging. Additionally, this insight empowers us to anticipate potential challenges and develop strategies to mitigate them, thereby enhancing the efficiency and reliability of FFF-based 3D printing.

James and Roos recently demonstrated in their work [67] that it is feasible for a viscoelastic fluid to traverse a converging channel without inducing upstream vortices, resulting in an equivalent pressure drop across the contraction as that of a Newtonian fluid with the same viscosity. Under these circumstances, elasticity does not impact pressure drop within the nozzle, but influences die swell at the exit. Achieving this involves modifying the nozzle’s shape to facilitate the radial distribution of normal stress and pressure, prompting streamlines to migrate to the centerline, consequently reducing the shear rate and shear stress at the wall. Consequently, the energy expended on shear within the channel decreases, with the saved energy being redirected toward polymer-related viscous dissipation. That conclusion [67], together with the result obtained in Figure 7 inspired us and allowed us to envisage the possibility of having an optimal nozzle shape for each polymer rheology so that upstream vortices would be suppressed and, consequently, the pressure drop in the nozzle would be minimum, being dependent exclusively on the viscosity of the material.

This work showed that the tandem "nozzle shape - polymer rheology" affects the quality and performance of FFF, especially at high speeds. To prevent problems such as filament distortion, backflow and clogging, future research and development could focus on implementing Optimized Shape Design [69] to specific polymer rheologies. These improvements can help advance Fused Filament Fabrication technology and make 3D printing more efficient and reliable.

The authors thank Johan Versteegh for the fruitful discussion and selfless support. FJGR and TS acknowledge the financial support from Ultimaker B.V., and also LA/P/0045/2020 (ALiCE), UIDP/00532/2020 (CEFT) and the program Stimulus of Scientific Employment, Individual Support-2020.03203.CEECIND, funded by FEDER funds through COMPETE2020 – Programa Operacional Competitividade e Internacionalização (POCI); and by national funds (PIDDAC) through FCT/MCTES.