# Rolie-Double-Poly equations¶

## Summary¶

Rolie-Double-Poly equations for the nonlinear predictions of polydisperse melts of entangled linear polymers

• Function
$\boldsymbol \sigma = G_N^0 \sum_i g(Z_{\text{eff},i}) \text{fene}(\lambda_i) \phi_i \boldsymbol A_i$
where
$\begin{split}\boldsymbol A_i &= \sum_j \phi_j \boldsymbol A_{ij}\\ \lambda_i &= \left( \dfrac{\text{Tr} \boldsymbol A_i}{3} \right)^{1/2}\\ \stackrel{\nabla}{\boldsymbol A_{ij}} &= -\dfrac{1}{\tau_{\mathrm d,i}} (\boldsymbol A_{ij} - \boldsymbol I) -\dfrac{2}{\tau_{\mathrm s,i}} \dfrac{\lambda_i - 1}{\lambda_i} \boldsymbol A_{ij} -\left( \dfrac{\beta_\text{th}}{\tau_{\mathrm d,j}} + \beta_\text{CCR}\dfrac{2}{\tau_{\mathrm s,j}} \dfrac{\lambda_j - 1}{\lambda_j}\lambda_i^{2\delta} \right) (\boldsymbol A_{ij} - \boldsymbol I)\\ \text{fene}(\lambda) &= \dfrac{1-1/\lambda_\text{max}^2}{1-\lambda^2/\lambda_\text{max}^2}\end{split}$

with $$\beta_\text{th}$$ the thermal constrain release parameter, set to 1. If the “modulus correction” button is pressed, $$g(z) = 1- \dfrac{c_1}{z^{1/2}} + \dfrac{c_2}{z} + \dfrac{c_3}{z^{3/2}}$$ is the Likhtman-McLeish CLF correction function to the modulus ($$c_1=1.69$$, $$c_2=2$$, $$c_3=-1.24$$), $$g(z) = 1$$ otherwise; $$Z_{\text{eff},i}=Z_i\phi_{\text{dil},i}$$ is the effective entanglement number of the molecular weight component $$i$$, and $$\phi_{\text{dil},i}$$ the dilution factor ($$\phi_{\text{dil},i}\leq \phi_i$$).

• Parameters
• GN0 $$\equiv G_N^0$$: Plateau modulus

• beta $$\equiv\beta_\text{CCR}$$: Rolie-Poly CCR parameter

• delta $$\equiv\delta$$: Rolie-Poly CCR exponent

• phi_i $$\equiv\phi_i$$: Volume fraction of species $$i$$

• tauD_i $$\equiv\tau_{\mathrm d,i}$$: Reptation time of species $$i$$ (including CLF)

• tauR_i $$\equiv\tau_{\mathrm s,i}$$: Stretch relaxation time of species $$i$$

• lmax $$\equiv\lambda_\text{max}$$: Maximum stretch ratio (active only when the “fene button” is pressed)