## 3 Fused Filament Fabrication
### Rheological Behavior of Thermoplastics in FFF  
Rheology is the study of the deformation and flow properties of materials when subjected to external stress. In the FFF process, understanding the rheological behavior of molten thermoplastics within the heated die is essential for controlling process parameters and determining the quality of the printed part. In FFF, the filament feedstock is initially pulled by pinch rollers into the filament feed tube or directly into the heat break in the hot end. In this configuration, the filament acts as a continuous piston that pushes the molten material through the heated channel and the threaded nozzle onto the build plate [24]. The pressure required to extrude the melt is influenced by both the rheological behavior of the material and the geometric parameters of the print head. A failure in filament feeding can occur when the extrusion pressure exceeds the filament's ability to support the load, leading to buckling. In this context, the viscosity of the molten polymer determines the resistance to flow, while the compressive elastic modulus of the solid filament determines its ability to carry the compressive load [25]. The primary geometric parameters of the hot end include the inlet diameter (D1), corresponding to the filament size; the outlet diameter (D2) of the nozzle; the convergence angle (\(\beta\)); and the lengths of the extruder nozzle inlet (L1) and outlet (L2) channels [26]. These geometric factors influence the pressure decay along the flow path and therefore the extrusion force required. As illustrated in Figure 6, the molten polymer flows through four regions in the extrusion system. In the initial region (I), flow is pressure-driven within the hot end channel. At the center, both shear stress (\(\tau\)) and shear rate (\(\dot{\gamma}\)) are minimal, while the axial velocity is maximal (\(vz\)). At the tube walls, the situation reverses--shear stress and shear rate reach their maximum values, while the velocity drops to zero [27]. This condition creates a shear gradient across the cross-section, where molecular orientation varies with the degree of shear. Molecules near the wall become increasingly aligned, while those at the center remain more disordered [28]. The mechanical behavior of this flow can be described by basic rheological equations that relate to stress and deformation under flow conditions. Shear stress (\(\tau\)) is defined as the force applied (F) divided by the cross-sectional area (A) of the channel, expressed as: \(\tau=\text{F}\ /\ \text{A}\). Shear rate (\(\dot{\gamma}\)), on the other hand, is defined as the velocity gradient across the radius of the flow channel, where \(d\underline{v}z\) is the variation of axial velocity across the tube diameter and \(d\underline{v}\) is the distance from the center of the tube, yielding: \(\dot{\gamma}=d\underline{v}\underline{d}\underline{d}\underline{v}\). Figure 5: Schematic representation of key physical phenomena in FFF (self-developed). The polymer melt rheological behavior depends on the applied shear rate. At exceptionally low shear rates, typically below \(10^{1}\) s\({}^{-1}\), thermoplastics is considered a Newtonian fluid, exhibiting constant viscosity. In most polymer processing conditions, including FFF, shear rates fall between \(10^{1}\) and \(10^{5}\) s\({}^{-1}\), where the fluid exhibits non-Newtonian, shear-thinning (pseudoplastic) behaviors. At even higher shear rates, viscosity may plateau again, restoring Newtonian behavior [28]. Figure 7 illustrates this relationship on a logarithmic scale. For FFF, shear rates within the nozzle are typically in the range of 100-200 s\({}^{-1}\)[25]. Specifically for PLA, pseudoplastic behavior has been observed between 1 and \(10^{3}\) s\({}^{-1}\)[29]. This non-Newtonian behavior is described by the Power Law (Ostwald-de Waele) model, where k is the consistency index and n is the flow behavior index: \(\tau=\text{k}\cdot\hat{\gamma}^{n}\). Mathematical models for simulating flow in FFF have used this equation [26].