Elasticity theory and crosslinking of reinforced rubber

For filler fraction C , molecular weight M c between crosslinks, and equilibrium swelling v r , the Flory function M c = F ( v r ) was corrected for a hard fraction C h = (1 + β) C , β C being rubber occluded within the primary filler structure. However, a small effect of a Graphon filler was unjustifiably attributed to rubber stretched hard by swelling; later research wrongly attempted to estimate β from F ( v r )/ F ( v c ) where v c considers the hard filler as rubber. By avoiding these mistakes, and with 1/ F o ( v r ) as unfilled crosslinking, Blanchard's constraint equations for linkage reinforcement ϕ and reinforced crosslinking 1/ M c are now simplified to $\phi = {1-C_h \over 1+C_h} {F_o (v_r)\over F(v_c)} = {1-\beta C \over 1+ \beta C} {F_o(v_r)\over F(v_r)}$ ${1\over M_c}={1-C_h\over 1+C_h} {1\over F(v_c)} = {1-\beta C\over 1+\beta C} {1\over F(v_r)}$ Alternatively, 1/ M c can be obtained from the theoretical modulus G = F /(α − 1/α 2 ), by the stress F at extension ratio α = 2 following two very different prestretches, α b ≫ 2. The choice of 100% strain (α = 2) is to minimize low‐strain Mooney‐Rivlin deviation from simple rubber theory and to avoid particle contact effects. The choice α b ≫ α ≫ 2 is to avoid stress upturn as α → α b . Then, for two prestretches α b = 3 and α b = 4.5 or 5 (400%), corresponding prestresses S 1 and S 2 , moduli G 1 and G 2 , and force per linkage factors X 1 = α b S 1 / $G^{2/3}_{1}$ and X 2 = α b S 2 / $G^{2/3}_{2}$ , the primary modulus G * is $G* = G_1 - G_rF(X_1) = G_2 - G_rF(X_2)$ $F(X) = (1 + kX^{1/2} + (k^2/2)X){\rm exp}(-kX^{1/2})$ Because k = 0.276 for all rubbers and fillers, the secondary modulus G r is known from ( G 1 − G 2 )/( F ( X 1 ) − F ( X 2 )). Hence, $G* = G_2 - {(G_1 - G_2)F(X_2)\over F(X_1) - F(X_2)}$ ${1\over M_c} = (1 + V) {1 - \beta C\over 1+ \beta C} {G*\over \rho RT}$ Here (1 + V ) allows for network dilution by filler volume V per milliliter of rubber, C = V /(1 + V ), and β C is rubber occluded within the particle aggregate structure of carbon blacks. The measured (effective) crosslinking $1/M^{\vert }_c$ is obtained by omitting (1 − β C )/(1 + β C ) from the above equations. The structure parameter β might be determined from G r using the present test prescription and modern furnace blacks with negligible to high structure. © 1998 John Wiley & Sons, Inc. J Appl Polym Sci 67: 119–129 1998