Computational indentation in highly cross-linked polymer networks

Indentation is a common experimental technique to study the mechanics of polymeric materials. The main advantage of using indentation is this provides a direct correlation between the microstructure and the small-scale mechanical response, which is otherwise difficult within the standard tensile testing. The majority of studies have investigated hydrogels, microgels, elastomers, and even soft biomaterials. However, a less investigated system is the indentation in highly cross-linked polymer (HCP) networks, where the complex network structure plays a key role in dictating their physical properties. In this work, we investigate the structure-property relationship in HCP networks using the computational indentation of a generic model. We establish a correlation between the local bond breaking, network rearrangement, and small-scale mechanics. The results are compared with the elastic-plastic deformation model. HCPs harden upon indentation.

Figure 1

Interaction potentials between the bonded monomers $u_{\mathrm{b}}\left(r\right)$ as a function of intermonomer distance $r$. The quartic parameters are taken from the literature [55]. The equilibrium bond length for these models is around ${\ell }_{\mathrm{b}}=0.97\sigma$ represented by the vertical line.

Figure 2

Curing percentages $C$ during the formation of total number of bonds ${\mathcal{N}}_{\mathrm{b}}$ with time $t$ for two network functionalities $n$. The data are shown for the trifunctional ($n=3$) and the tetrafunctional ($n=4$) systems. Network cure is performed via FENE bonds; see Eq. (2). Note that for the representation purpose we have shown results only for the initial time of $5\tau$, while the total simulation time is $5\times {10}^3\tau$.

Figure 3

Force $F$ and the percentage of broken bonds $\mathcal{B}$ as a function of the indentation depth $d$. (a, b) Data for the indenter radius $R=15.0\sigma$ (a) and $5.0\sigma$ (b). The color codes of the data sets are consistent with their corresponding $y\mathrm{axes}$. Data are shown for an indentation velocity of $0.005\sigma /\tau$ and for a tetrafunctional network. Vertical lines highlight the major drops in $F$ and the corresponding $\mathcal{B}$. Initial depletion zones below $d<10.0\sigma$ (a) and $d<5.0\sigma$ (b) are because the lowest part of the indentation tips are certain distance away from the network surface. Arrow indicates at the $d$ values where the bond breaking starts. The insets show different pictorial representations of the indentation tips entering the sample at different $d$. The top and the bottom panels show the cases just before and after the bond breaking in the samples, respectively.

Figure 4

Distribution of the force drops $P\left(\mathrm{\Delta }F\right)$ obtained from Fig. 3. $\mathrm{\Delta }F$ is calculated at the onset of bond breaking in the tetrafunctional networks. The data are shown for an indenter radius $R=5\sigma$. The lines are power-law fits, $▵F^{-1.0}$ (for ${10}^{-4}<▵F<{10}^{-3}$) and $▵F^{-1.8}$ (for $▵F>{10}^{-3}$). We have also added the data corresponding to the unloading cycle. The data are averaged over three different simulation runs, as described by Fig. 13 in Appendix pp6.

Figure 5

Same as Fig. 3, however, during both loading and unloading cycles. Arrows indicate the corresponding loading and unloading curves. [(a), (b)] Data for the indenter radius $R=15.0\sigma$ (a) and $R=5.0\sigma$ (b). For clarity of presentation, we have shown only four data sets in (a). More data are presented in Fig. 9 in Appendix pp2.

Figure 6

Same as Fig. 3, but for two different network functionalities. The inset also includes the percentage of bond breaking $\mathcal{B}$ for the trifunctional network. For this network, bond breaking starts at around a depth of $d\simeq 21.0\sigma$. Data are shown for an indenter radius $R=5.0\sigma$.

Figure 7

The first maximum at the force drop $\tilde{F}$ and corresponding depth $\tilde{d}$ as a function of the percentage of the network cure $\mathcal{C}$. The data are shown for two different network functionalities and for an indenter radius $R=5.0\sigma$. Note that for the calculation of $\tilde{d}$, the initial depletion zones in the force-extension curves are subtracted.

Figure 8

Bond density $f\left(z^{*}\right)$ along the $z\mathrm{direction}$ of the sample, i.e., the direction along which the indentation is applied. The data are shown for a tetrafunctional network.

Figure 9

Force $F$ as a function indentation depth $d$ during three different unloading cycles starting at different $d$. Data are shown for the tetrafunctional network and for an indenter radius $R=15.0\sigma$. The unloading is performed at a constant unloading velocity $v=0.005\sigma /\tau$. During the loading cycle, bond breaking starts only around $d\simeq 18.0\sigma$. Therefore, the unloading data starting at $d\simeq 14.0\sigma$ represent the elastic deformation. Note the data here are an extension of those in Fig. 5.

Figure 10

Main panel shows the temporal relaxations of the depth $d$, and the inset presents relaxation of force $F$. The data are shown for the indenter radius $R=15.0\sigma$ and for a tetrafunctional network.

Figure 11

Force $F$ as a function indentation depth $d$ for two network functionalities: the trifunctional ($n=3$) and the tetrafunctional ($n=4$) networks. Data are shown for an indenter radius $R=15.0\sigma$. The indentation is performed at a constant velocity $v=0.005\sigma /\tau$.

Figure 12

Force $F$ as a function indentation depth $d$ for two network functionalities: the trifunctional (a) and the tetrafunctional (b) networks. Data are shown for three curing percentages $\mathcal{C}$. Note that for the clarity of presentation, we have shown only the smoothed data.

Figure 13

Force $F$ as a function indentation depth $d$ for the trifunctional (a) and the tetrafunctional (b) networks. Data are shown for an indenter radius $R=5.0\sigma$ and for three sets. The indentation is performed at a constant velocity $v=0.005\sigma /\tau$.