Rolie-Double-Poly equations


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

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

    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)