# Smooth-polySTRAND model for flow-induced nucleation¶

## Summary¶

Smooth-polyStrand model for flow-induced crystallisation in polydisperse melts of entangled linear polymers

• Rheological model: The Rolie-Double-Poly model

Evolution of chain structure under flow is computed by the Rolie-Double-Poly model. Implementation and parameters are the same as in the NVLE application.
$\boldsymbol \sigma = G_N^0 \sum_i g(Z_{\text{eff},i}) \text{fene}(\lambda_i) \phi_i \boldsymbol A_i$
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 ($$\phi_{\text{dil},i}\leq \phi_i$$).

• Nucleation model: The smooth-polyStrand model

This model takes the stress output from the Rolie-Double-Poly model for each mode, computes the flow-induced nucleation rate, using the Kuhn segment nematic order as the order parameter.

-Neglect quiescent nucleation button: this subtracts the quiescent nucleation rate and assumes all quiescent nucleation occurs from hetrogeneous nuclei.

-Average to single species button: this preaverages the chain configuration over all species in the melt and computes the nucleation rate with a single species based on this average.

• Crystal evolution model: The Schneider rate equations

From the computed nucleation rate and the crystal growth rate, the model computes the evolution of total crystallinity using the Schneider rate equations [W. Schneider, A. Koppl, and J. Berger, Int. Polym. Proc.II 3, 151 (1988)]. This calculatiom uses the Avrami expression to account approximately for impingement.

• Parameters

Rheological

• 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)

• Ne $$\equiv N_e$$: Number of Kuhn steps between entanglements

Quiescient Crystallisation

• epsilonB $$\equiv \epsilon_B$$: Bulk free energy gain of crystallisation per Kuhn step [dimensionless]

• muS $$\equiv \mu_S$$: Nucleus surface area cost [dimensionless]

• tau0 $$\equiv \tau_0$$: Kuhn step nucleation timescale [sec]

• rhoK $$\equiv \rho_K$$: Kuhn step density [$$\mu\mathrm{m}^{-3}$$]

• G_C $$\equiv G_C$$: Crystal growth rate [$$\mu\mathrm{m/sec}$$]

• N_0 $$\equiv N_0$$: Heterogeneous nucleation density [$$\mu\mathrm{m}^{-3}$$]

Flow-induced crystallisation

• Gamma $$\equiv \Gamma$$: Prefactor connecting the Kuhn segment nematic order and the monomer entropy loss [dimensionless].

• Kappa0 $$\equiv \kappa _0$$: Free energy penalty for the nucleus surface roughness [dimensionless].

• Qs0 $$\equiv Q_{s0}$$: Parameter setting the volume of the search region for new stems joining the nucleus [dimensionless].

## Description¶

For full details about how the predictions are calculated, refer to [1].