# Sintering of Two Viscoelastic Particles: A Computational Approach
## 6 Results
### Effect of the Rheology  
Keeping the initial geometry constant, we assess the effect of the rheological model on the flow behavior of the system. We use a geometry of two equal particles \(R=1\) with initial contact radius \(y_{\rm n}=0.4\) and round off parameter \(R_{\rm n}=R\times(y_{\rm n}/R)^{3}\). We set the zero-shear-rate viscosity \(\eta_{0}=1\), solvent viscosity \(\eta_{\rm s}=0.01\) and surface tension \(\gamma=1\). First, the Deborah number is changed by varying the relaxation time \(\lambda=[0.01,\,0.1,\,0.5,\,1]\), resulting in Deborah numbers \({\rm De}=[0.01,\,0.1,\,0.5,\,1]\), respectively. The results are shown in Figure 10, where the contact radius \(y\) is depicted in time \(t\), using the Giesekus constitutive model with \(\alpha=0.1\). With increasing Deborah number, the initial increase in contact radius between \(t=0\) and \(t=0.05\) gets larger. By keeping the viscosity constant and varying the relaxation time, the modulus is changed \(G=\eta_{\rm p}/\lambda=[100,\,10,\,2,\,1]\), respectively. Initially, \(\frac{\gamma}{c}=0\), and from this, it follows that \(\mathbf{c}=\mathbf{B}\) and  
Figure 8: The evolution of the curvature \(\kappa\) in time \(t\) of two equal spheres with initial contact radius \(y_{\rm n}=0.4\) and different round off parameter \(R_{\rm n}=[R\times(y_{\rm n}/R)^{3},\,(R/2)\times(y_{\rm n}/R)^{3},\,(R/4) \times(y_{\rm n}/R)^{3}]\), using the Newtonian constitutive equation. Figure 9: The contour plot of \(r=g(z)\) at time \(t=[0,\,0.08,\,0.2]\) with initial contact radius \(y_{\rm n}=0.4\) and round off parameter \(R_{\rm n}=R\times(y_{\rm n}/R)^{3}\), using the Newtonian constitutive equation (**a**), and a zoom of the neck region (**b**). \(\mathbf{\tau}=G(\mathbf{B}-\mathbf{I})\), where \(\mathbf{B}\) is the Finger tensor. Since \(\mathbf{\tau}\) scales with \(\gamma/R\), which is kept constant, the Finger tensor \(\mathbf{B}\) has to increase for decreasing modulus \(G\). Consequently, the initial deformation increases for increasing Deborah number as shown in Figure 10. The deformation is not completely instantaneous, because \(\eta_{\text{s}}>0\). From \(t=0.05\) onwards, the shape transition gets slower for increasing Deborah number. From simulations, it follows that the same behavior holds for the XPP constitutive model. In Figure 11, the curvature \(\kappa\) at point \((0,\,y)\) at the surface of the neck is shown in time \(t\) for different Deborah numbers \(\text{De}=[0.01,\,0.1,\,0.5,\,1]\), using the Giesekus model with \(\alpha=0.1\). As is shown for the Newtonian constitutive behavior, the curvature \(\kappa\) at point \((0,\,y)\) increases with respect to the initial geometry until it reaches a maximum value and subsequently decreases for all different values of the Deborah number \(\text{De}=[0.01,\,0.1,\,0.5,\,1]\), as well. The maximum value of the curvature \(\kappa\) and the time \(t\) of the maximum depend on the value of the Deborah number De. Figure 11: The curvature \(\kappa\) at point \((0,\,y)\) at the surface of the neck in time \(t\) for different Deborah numbers \(\text{De}=[0.01,\,0.1,\,0.5,\,1]\), using the Giesekus model with \(\alpha=0.1\). The result of the Newtonian behavior is included, as well. Figure 10: The evolution of the contact radius \(y\) in time \(t\) for different Deborah numbers \(\text{De}=[0.01,\,0.1,\,0.5,\,1]\), using the Giesekus model with \(\alpha=0.1\). The result of the Newtonian behavior is included, as well. Furthermore, we assess the conformation tensor \(c\), which is a measure for the polymeric strain in the system and plays an important role in the crystallization kinetics of semicrystalline polymers like PA12. The dynamic evolution of the trace of the conformation tensor \(\mathrm{tr}(c)\), using the Giesekus constitutive model with \(\alpha=0.1\) and \(\mathrm{De}=1\), is shown in Figure 12. For visualization, the color bar ranges from three to eight, but the values locally exceed these numbers.