Tensile strength of rubber described via the formation and rupture of load-bearing polymer chains

A theoretical picture describing the tensile strength ${\sigma }_T$ of elastomers is developed. ${\sigma }_T$ is composed of three factors: (1) the tensile strength of individual polymer load-bearing chains (LBCs) according to Eyring's theory, (2) an occupation number of LBC states using Fermi statistics, and (3) an excluded volume factor reducing the number of possible LBCs due to the presence of filler particles or crosslinks between polymers. This description is compared to experimental tensile strengths of carbon black (N339)-filled EPDM (Keltan 4450) as well as to other experiments in the literature studying the effects of temperature, filler concentration, and particle size as well as crosslink density.

Figure 1

(a) Cartoon of filler particles (shaded circles) inside an elastomer close to its breaking stress. The red line illustrate polymer LBCs or rather chain segments. The green lines are chains which do not carry much load. Dots indicate anchoring sites on the particle surface (black dots) or chemical or physical crosslinks (red dots). (b) Snapshot from an atomistic computer simulation studying the interaction between two silica particles tied to a surrounding polyisoprene matrix via short silanes (taken from Ref. [24]). Notice the void space appearing between the particles as they are pulled apart keeping one particle stationary while the other experiences a force acting on its center of mass.

Figure 2

Master curves obtained for his data points by Plagge [15] in comparison to Eq. (8) (solid line) for ${\epsilon }_o/R=2.0$ and $\mathrm{\Omega }=0$.

Figure 3

Temperature dependence of the tensile strength (hollow circles) in the plateau regime obtained by Plagge. Here $s={\sigma }_{T,o}\upsilon /\left(R{T}_o\right)$ and $t=T/T_o$. The solid circles follow from the hollow circles after subtracting the respective tensile strength for $\varphi =0$. The error bars, except in the case of the one shown explicitly, are comparable to the size of the circles. Dashed line: solution of Eq. (4) for $e=17.5$ ($E_b=40$ kJ/mol), $c=13.2$, and $q=4$; solid line: solution of Eq. (4) for $e=35$ ($E_b=80$ kJ/mol), $c=25.5$, and $q=9$; dotted line: solution of Eq. (4) for $e=66$ ($E_b=150$ kJ/mol), $c=46.5$, and $q=18$.

Figure 4

The symbols depict measured tensile strengths ${\sigma }_T$ vs filler volume fraction $\varphi$ at a series of temperatures $T$ (gray: $90{}^{\circ }\mathrm{C}$, red: $60{}^{\circ }\mathrm{C}$, green: $23{}^{\circ }\mathrm{C}$, blue: $3{}^{\circ }\mathrm{C}$, and black: $-25{}^{\circ }\mathrm{C}$) taken from Ref. [15]. The data are the same as the date depicted in the mastered representation of the measurements in Fig. 2 using corresponding colors. The solid lines show the comparison of the experiment to Eq. (1) for the three parameter sets obtained via the data points in Fig. 3 (line types correspond).

Figure 5

(a) Tensile strength ${\sigma }_T$ vs particle size relative to N339. The curves correspond to the theoretical results for $T=60{}^{\circ }\mathrm{C}$ (green) and $23{}^{\circ }\mathrm{C}$ (red) at $\varphi =0.1$ (solid lines) and $\varphi =0.2$ (dashed lines) shown in the previous figure. (b) Measurements (red dots) taken from Fig. 10.15 in Ref. [5]. This panel depicts the tensile strength for a series of silicon rubber compounds filled with 38 phr of silica with different surface areas, which here are converted to particle size. (c) Tensile strength ${\sigma }_T$ vs particle size relative to N220 in an SBR vulcanizate. The data points are taken from Fig. 4 in Ref. [13] [40 (red spheres) and 30 (blue diamonds) phr].

Figure 6

Open circles: bulk crosslinks; solid circles: surface bonds. The ratio of displaced bulk crosslinks to surface bonds is 2/5 for the small circle but 27/19 for the large circle.

Figure 7

Tensile strength ${\sigma }_T$ vs crosslink density $n_c$. Fit of the present model (solid lines) to data from Figs. 1 and 3 in Ref. [38]. The systems are SBR 1006 (red circles) and SBR 1500 (blue diamonds) containing 30 phr of HAF black.

Figure 8

$s$ vs $q$ according to Eq. (4). The other parameter values are the same as for the solid line in Fig. 3. Here $t=1$.

Figure 9

(a) Frequency of chain break $h_{\mathrm{break}}\left(T\right)$ vs temperature $T$ for $N=10$ ($a=1, D=0.6, f_p=0$, and $\zeta =0.1$). (b) Chain rupture temperature $T_{\mathrm{break}}$ for $N$ between 2 and 100 for different parameter values. The letters g, m, b, r, and b stand for the colors of the dots (green, magenta, blue, red, and black) distinguishing the indicated parameter sets. Horizontal lines are a guide for the eye.