Statistical mechanical constitutive theory of polymer networks: The inextricable links between distribution, behavior, and ensemble

A fundamental theory is presented for the mechanical response of polymer networks undergoing large deformation which seamlessly integrates statistical mechanical principles with macroscopic thermodynamic constitutive theory. Our formulation permits the consideration of arbitrary polymer chain behaviors when interactions among chains may be neglected. This careful treatment highlights the naturally occurring correspondence between single-chain mechanical behavior and the equilibrium distribution of chains in the network, as well as the correspondences between different single-chain thermodynamic ensembles. We demonstrate these important distinctions with the extensible freely jointed chain model. This statistical mechanical theory is then extended to the continuum scale, where we utilize traditional macroscopic constitutive theory to ultimately retrieve the Cauchy stress in terms of the deformation and polymer network statistics. Once again using the extensible freely jointed chain model, we illustrate the importance of the naturally occurring statistical correspondences through their effects on the stress-stretch response of the network. We additionally show that these differences vanish when the number of links in the chain becomes sufficiently large enough, and discuss why certain methods perform better than others before this limit is reached.

Figure 1

Diagram describing the exact (Helmholtz) and approximate (Gibbs-Legendre) methods of arriving at the single-chain Helmholtz free energy ${\psi }^{*}\left(\mathit{\xi }\right)$ of a chain with end-to-end vector $\mathit{\xi }$ and equilibrium probability distribution $P^{\mathrm{eq}}\left(\mathit{\xi }\right)$ of chains with that end-to-end vector. $†$ The Gibbs-Legendre method is approximate since the necessary Legendre transformation is only valid in the thermodynamic limit of long chains.

Figure 2

Nondimensional free energy per link versus end-to-end stretch for $N_b=3$, 5, 10, and 25, for (a) $\kappa =50$ and (b) $\kappa =5$. The free energy is plotted using both the Helmholtz and Gibbs-Legendre methods, as well as using the ideal model valid for $\lambda \ll 1$; the Gibbs-Legendre and ideal results are independent of $N_b$.

Figure 3

Nondimensional equilibrium radial distribution function versus end-to-end stretch for $N_b=3$, 5, 10, and 25, for (a) $\kappa =50$ and (b) $\kappa =5$. The distribution is plotted using the Helmholtz, Gibbs-Legendre, and Gaussian methods.

Figure 4

Nondimensional uniaxial stress-stretch results for the EFJC network with (a) $\kappa =50$ and (b) $\kappa =5$, for $N_b=5$, 10, and 25. This mechanical response is plotted using the true method (Helmholtz) and two approximation methods (Gibbs-Legendre, Gibbs-Legendre-Gaussian). Shading indicates equal $N_b$ value. The neo-Hookean response is included as reference.

Figure 5

Nondimensional (a) equibiaxial and (b) simple shear stress-stretch results for the EFJC network with $\kappa =50$ for $N_b=5$, 10, and 25. This mechanical response is plotted using the true method (Helmholtz) and two approximation methods (Gibbs-Legendre, Gibbs-Legendre-Gaussian). Shading indicates equal $N_b$ value. The neo-Hookean response is included as reference.