## 3 Fused Filament Fabrication
### Rheological Behavior of Thermoplastics in FFF
A value of n = 1 indicates Newtonian behavior, while n < 1 characterizes pseudoplastic fluids [30]. The lower the value of n, the more pseudoplastic the material, resulting in a flatter flow front. The velocity profile shown in Figure 7 corresponds to a Newtonian fluid [31]. After region I, the molten polymer enters the convergent channel (region II), where the flow becomes elongational due to the change in cross-section. This behavior is analogous to the inlet region of a traditional extrusion die. In region III, the flow again becomes pressure-driven, but at increased velocity due to the smaller diameter. Upon exiting the nozzle (region IV), the flow profile does not present speed variation along the extrudate diameter, as the polymer enters at ambient pressure and undergoes relaxation, leading to an increase in  
Figure 6: Schematic illustration of the molten polymer flow profile and nozzle geometry, showing regions I-IV along the extrusion path. Based on Schramm [27]. Figure 7: Rheological behavior of molten thermoplastics, showing the variation of viscosity (\(\upmu\)) with shear rate (\(\dot{\gamma}\)). Based on Manrich [28]. diameter--a viscoelastic phenomenon known as extrudate swell (D3/D2 ratio) [27]. It is considered that the molten polymer undergoes an axisymmetric contraction flowing through the nozzle [32]. The cylindrical volume in the region of the inlet nozzle is elongated and its diameter is reduced; after it passes the convergent channel, it increases again at the nozzle outlet, reducing its length [27]. Molten polymers show a dependence on the speed, intensity, and time at which stress is applied, i.e., it is dependent on their thermo-mechanical history. They can partially return to their original shape after being stretched because their long molecular chains line up when stressed and then relax back to a random state when the stress is gone [31]. This behavior generates normal stress components when the polymer is out of the nozzle, which causes swell [28]. Whether in fluid or solid state, this behavior is called viscoelastic [33], as mathematical theories are applied in both cases [34]. The extent of extrudate swell depends on the thermo-mechanical history of the melt. The longer the residence time inside the nozzle, the greater the relaxation of elongational deformation and the smaller the residual contribution to swell. However, shear-induced elastic deformation within the nozzle still contributes to swelling. To mitigate this effect, it is recommended to reduce the extrusion speed, increase the processing temperature (at constant shear rate), increase the length-to-diameter ratio (L2/D2), and reduce the diameter ratio (D1/D2) [28; 34]. Other elastic effects during flow include melt fracture, shark skin, and draw resonance. These phenomena are caused by flow instabilities arising from elastic turbulence at the nozzle inlet, especially at high deformation rates in elongational flow, where stress levels exceed the cohesive strength of the polymer [31; 35]. Since viscosity is temperature-dependent, extrusion behavior is strongly influenced by thermal conditions. Excessively elevated temperatures can lead to discharge fluctuation and thermal degradation, while low temperatures increase viscosity and pressure demand, potentially resulting in melt fracture [31]. The average level of molecular orientation and residual stress in the printed part depends on three main factors: I) the shear-induced orientation during deposition; II) the relaxation time of the polymer chains (typically between 0.1 and 0.5 s for certain polymers); and III) the effective cooling time between processing temperature (Tp) and solidification temperature (Ts). This cooling time is determined by the size and geometry of the part, the polymer's specific heat, and the temperature intervals Tp-Ts and Tp-Tr, where Tr is room temperature [34]. The filament softens and melts in the heater block, which contains a machined flow channel. The melt quantity depends on the heat flux and material feed rate [24; 26].