Computational indentation in weakly cross-linked polymer networks

Abstract

Crosslinking is one of the important pathways to tune properties of polymers for desired applications. Crosslinking of polymers ranges from weakly crosslinked elastomers to highly crosslinked epoxies. We present a computational study of indentation in weakly crosslinked polymer (WCP) networks. Indentation study offers a significant advantage by providing a direct relationship between a material’s local structure and its mechanical properties, while bulk mechanical testing is unable to do it. Although the complex network structure plays a crucial role in determining the mechanical characteristics of weakly cross-linked polymer (WCP) networks, indentation studies in these systems have received less attention. We explore the mechanical properties of a weakly crosslinked polymer network (WCP) using two indenters of different sizes. We establish a relationship between force-depth response, force-relaxation and local bond breaking in WCP network. This will help us to optimize the design of crosslinked polymer materials for desired applications and also guide future experiments.

Appendix A

1.1 A.1 Bond distribution

The network formation framework presented in Sect. 2.3 produces a reasonably uniform sample; see supplementary Fig. 9. The bond distribution is relatively homogeneous, except near the two interfaces, where the confinement walls induced depletion zone.

Fig. 9

The distribution of bond density $f (z^*)$ is in the z direction. The data are shown for every second ($x_\textrm{f}=0.50$), fourth ($x_{\textrm{f}}=0.25$), fifth ($x_{\textrm{f}}=0.20$) and every tenth ($x_{\textrm{f}} = 0.10$)

1.2 A.2 Effect of velocity

In order to investigate the effect of indentation velocity, we performed our simulations at $v=0.005\sigma /\tau$ and $v=0.05\sigma /\tau$, and the response is shown in Fig. 10. It can be depicted that the $F-d$ response of $v=0.05\sigma /\tau$ has an insignificant difference compared to $v=0.005\sigma /\tau$ force response, see Fig. 10. At the same time lower indentation velocities needed high computational power. Therefore we have chosen $v=0.05\sigma /\tau$ in our study.

Fig. 10

Force F as a function of the indentation depth d. The data are shown for a tetra-functional network with fraction monomers $x_{\textrm{f}}$ with the indenters $r_{\textrm{ind}}=10.0\sigma$ and $r_{\textrm{ind}}=5.0\sigma$, and velocities $v=0.005\sigma / \tau$ and $v=0.05\sigma /\tau$

1.3 A.3 Loading and unloading mechanics

In Fig. 7, loading and unloading mechanics have shown for a fraction of monomers $x_{\textrm{f}}=0.5$ and $x_{\textrm{f}}=0.25$. In this section, the mechanics of the fraction of monomers $x_{\textrm{f}}=0.2$ and $x_{\textrm{f}}=0.1$ are shown (Fig. 11).

Fig. 11

Force F verses indentation depth d, the results are shown for the indenter $r_{\textrm{ind}}=5.0\sigma$ and $r_\textrm{ind}=10\sigma$ of a tetra-functional network with the fraction of a monomers $x_{\textrm{f}}=0.2$ and $x_{\textrm{f}}=0.1$, and the indentation is done at a velocity $v=0.05\sigma /\tau$. These data are the remainder of Fig. 7

1.4 A.4 Effect of indenter radius

In order to see the effect of the indenter radius on the $F-d$ response, We have chosen two indenters of different sizes ((Fig. 12). The maximum force F for the indenter radius $r_{\textrm{ind}}=10.0\sigma$ is approximately five times higher than the maximum force F of the indenter radius $r_{\textrm{ind}}=5.0\sigma$, during the loading condition. A large force drops $\Delta F$ observed at indentation depth $d\approx 29\sigma$ is because large numbers of bonds are broken and are clearly seen in Fig. 6 .

Fig. 12

Force F as a function of the indentation depth d. Results are shown for the indenter $r_{\textrm{ind}}=5.0\sigma$ and $r_{\textrm{ind}}=10.0\sigma$, Data are shown for a tetra-functional network with the fraction of a monomer $x_{\textrm{f}}=0.5$, and the indentation is performed at a constant loading velocity $v=0.05\sigma /\tau$

1.5 A.5 Effect of fraction of monomers

The fraction of monomers $x_{\textrm{f}}$ is the parameter from which we can construct a range of weakly cross-linked polymers (WCP). As the $x_{\textrm{f}}$ increases the $F_{\textrm{max}}$ increases, which means $x_{\textrm{f}}$ is making stiff the sample, and it can be seen in Fig. 13.

Fig. 13

Force F as a function of the indentation depth d. Parts (a) and b show the data for the indenter $r_\textrm{ind}=5.0\sigma$ and $r_{\textrm{ind}}=10.0\sigma$, The data are shown for a tetrafunctional network with the fraction of monomers $x_\textrm{f}=0.5$, $x_{\textrm{f}}=0.25$ $x_{\textrm{f}}=0.2$ and $x_{\textrm{f}}=0.1$