Strain-induced self-assembly of crystallites in elastomers

We propose a model of strain-induced crystallization in elastomers, combining Flory's original work with a description of crystallization adopted from the theory of micellar solutions. The experimentally observed hysteresis appears in the model due to a continuous, unidirectional change of the free energy difference between straightened polymer sections which occur isolated and those which are aggregated. The model yields good qualitative and even semiquantitative agreement with measurements of crystallization in natural rubber at variable cross-link density, strain amplitude, and temperature. The attendant description of the stress hysteresis is less good but still qualitatively correct.

Figure 1

Snapshot from a molecular dynamics computer simulation of cross-linked cis-polyisoprene (PI) in the undeformed state and in a stretched state $\lambda \approx 2.5$ [26]. The system consists of 20 PI chains consisting of 250 monomers each and containing 500 disulfide cross-links. Selected polymer chains are highlighted in color.

Figure 2

(a) Polymer chain containing $n$ Kuhn segments of length $a$ between two cross-links. Left: Unstrained with end-to-end vector $\vec{R}$. Right: Strained with end-to-end vector ${\vec{R}}^{\prime }$. The vector $\vec{c}$ indicates a straight portion of the chain containing $n_c$ Kuhn segments. (b) Straightened sections of the chains (red) bundle together into $m$-aggregates or crystallites. They are in equilibrium with 1-aggregates. The latter are often referred to as monomers. But monomer here has a different meaning and therefore we use the term 1-aggregate instead.

Figure 3

(a) Fraction of Kuhn segments in the “straightened” portion of a chain, $1-x_o$, vs strain, $\lambda$. (b) Force, $f$, divided by temperature, $T$, on a segment vs strain. Solid line, Eq. (9); dashed line, $\lambda -1/{\lambda }^2$. Parameter values: $n=36$, $\theta =0.19$, and $p$ according to (8).

Figure 4

(a) $x_1$ (dashed lines) and $x_m$ (solid lines) vs $\lambda$ for the parameter set discussed in the text. Here ${\alpha }_{\to }=1.15$ and ${\alpha }_{\leftarrow }=1.65$. The breaks in the slopes of $x_1$ signal the CAS, i.e., the onset of crystallite formation for the respective $\alpha$, and thus the increase of the attendant $x_m$. The red dots are experimental crystallite fractions taken from Fig. 2(b) in Ref. [7]. The experimental segment length between two cross-links is 238, which means that $n=59$. Red arrows indicate strain increase and strain relaxation, respectively. (b) Stress vs strain corresponding to the two solid lines in the upper panel. Specifically, we plot $f/T$ as given by Eq. (9). The quantity $x_o$ is calculated from $x_1$ and $x_m$ using Eq. (14). The red dots are experimental data from the above source [Fig. 2]. Red arrows again indicate strain increase and strain relaxation, respectively.

Figure 5

(a) Development of $\alpha$ during a strain cycle as indicated by the arrows. (b) Attendant $x_1$ (dashed lines) and $x_m$ (solid lines) vs strain, $\lambda$. (c) Stress vs strain corresponding to the two solid lines in panel (b). Specifically, we plot $f/T$ as given by Eq. (9). The quantity $x_o$ is calculated from $x_1$ and $x_m$ using Eq. (14). The symbols are data taken from Fig. 8 in Ref. [7]. Blue and red data denote mean segment lengths of 145 and 238 monomers, respectively. The colors of the data points and the colors of the theoretical curves do correspond.

Figure 6

(a) $\alpha$ vs $\lambda$ for different strain amplitudes. (b) Attendant $x_1$ (dashed lines) and $x_m$ (solid lines) vs $\lambda$. The experimental data (colored dots) are those of sample II ($N_c=238$ corresponding to $n=59$) in Fig. 5 of Ref. [14]. The colors of the data points and the theoretical curves do correspond. (c) Stress vs strain corresponding to the solid lines in panel (b). Specifically, we plot $f/T$ as given by Eq. (9). The quantity $x_o$ is calculated from $x_1$ and $x_m$ using Eq. (14). Only one experimental data set is shown for comparison.

Figure 7

(a) $x_m$ vs $\lambda$. The experimental data (colored dots) are taken from Fig. 3 in Ref. [20] ($N_c=238$; red, $20{}^{\circ }\mathrm{C}$; blue, 40 ${}^{\circ }\mathrm{C}$; black, $60{}^{\circ }\mathrm{C}$). The colors of the data points and the colors of the theoretical curves do correspond. (b) Theoretical force vs $\lambda$ corresponding to the solid lines in panel (b). Specifically, we plot $f/T$ as given by Eq. (9). The quantity $x_o$ is calculated from $x_1$ and $x_m$ using Eq. (14).