Linear and Nonlinear Elasticity of Networks Made of Comb-like Polymers and Bottle-Brushes*

We study mechanical properties of networks made of combs and bottle-brushes by computer simulations, theoretical calculations and experimental techniques. The networks are prepared by cross-linking backbones of combs or bottle-brushes with linear chains. This results in ``hybrid'' networks consisting of linear chains and strands of combs or bottle-brushes. In the framework of the phantom network model, the network modulus at small deformations $G_{\mathrm{0}}$ can be represented as a sum of contributions from linear chains, $G_{\mathrm{0,l}}$, and strands of comb or bottle-brush, $G_{\mathrm{0,bb}}$. If the length of extended backbone between crosslinks, $R_{\mathrm{max}}$, is much longer than the Kuhn length, $b_{k}$, the modulus scales with the degree of polymerization of the side chains, $n_{\mathrm{sc}}$, and number of monomers between side chains, $n_{\mathrm{g}}$, as $G_{\mathrm{0,bb}}\propto $(n$_{\mathrm{sc}}$/n$_{\mathrm{g}}+$1)$^{\mathrm{-1}}$. In the limit when $b_{k}$ becomes of the order of $R_{\mathrm{max}}$, the combs and bottle-brushes can be considered as semiflexible chains, resulting in a network modulus to be $G_{\mathrm{0,bb}}\propto $(n$_{\mathrm{sc}}$/n$_{\mathrm{g}}+$1)$^{\mathrm{-1}}$(n$_{\mathrm{sc}}^{\mathrm{1/2}}$/n$_{\mathrm{g}})$. In the nonlinear deformation regime, the strain-hardening behavior is described by the nonlinear network deformation model, which predicts that the true stress is a universal function of the structural modulus, $G$, first strain invariant, $I_{\mathrm{1}}$, and deformation ratio, $\beta $. The results of the computer simulations and predictions of the theoretical model are in a good agreement with experimental results.