NLVE: Theories

Rolie-Poly equation

“Classic” Rolie-Poly model

The Rolie-Poly (ROuse LInear Entangled POLYmers) [1] constitutive equation is a single mode formulation of the full microscopic model [2] for linear entangled polymer chains with chain stretch and convective constraint release.

The evolution equations for the conformation tensor, \(\boldsymbol{A}\), solved by RepTate for every stretching mode is:

\[\dfrac{\text D \boldsymbol{A}}{\text Dt} = \boldsymbol{\kappa}\cdot\boldsymbol{A }+ \boldsymbol{A}\cdot\boldsymbol{\kappa}^{T} - \dfrac{1}{\tau_\mathrm{d}} (\boldsymbol{A}- \boldsymbol{I}) - \dfrac{2(1 - \lambda^{-1})}{\tau_\mathrm{R}} \left( \boldsymbol{A}+ \beta \lambda^{2\delta}(\boldsymbol{A }- \boldsymbol{I}) \right),\]

where the stretch ratio, \(\lambda\), is

\[\lambda = \left( \frac 1 3 \text{tr}\, \boldsymbol{A} \right)^{1/2}\]

and, for non-stretching modes, the equation for each mode is:

(1)\[\dfrac{\text D \boldsymbol{A}}{\text Dt} = \boldsymbol{\kappa}\cdot\boldsymbol{A }+ \boldsymbol{A}\cdot\boldsymbol{\kappa}^{T} - \dfrac{1}{\tau_\mathrm{d}} (\boldsymbol{A}- \boldsymbol{I}) - \dfrac{2}{3} \mathrm{tr} (\boldsymbol{\kappa}\cdot\boldsymbol{A}) \big(\boldsymbol{A}+ \beta(\boldsymbol{A}- \boldsymbol{I}) \big).\]

Given \(n\) modes \((G_i, \tau_{\mathrm d, i}, \tau_{\mathrm R, i})\), the total stress is calculated as

\[\boldsymbol{\sigma}(t) = \sum_{i=1}^n G_i\, \boldsymbol{A}_i\]

Finite extensibility

A non-Gaussian version of the Rolie–Poly constitutive equation, which accounts for finite extensibility of polymer chains, can be written in the following form

\[\dfrac{\text D \boldsymbol{A}}{\text Dt} = \boldsymbol{\kappa}\cdot\boldsymbol{A }+ \boldsymbol{A}\cdot\boldsymbol{\kappa}^{T} - \dfrac{1}{\tau_\mathrm{d}} (\boldsymbol{A}- \boldsymbol{I}) - \dfrac{2(1 - \lambda^{-1})}{\tau_\mathrm{R}} \text{fene}(\lambda) \left( \boldsymbol{A}+ \beta \lambda^{2\delta}(\boldsymbol{A }- \boldsymbol{I}) \right),\]

The nonlinear spring coefficient, \(\text{fene}(\lambda)\), is approximated by the normalized Padé inverse Langevin function [3]

\[\text{fene}(\lambda) = \dfrac{3 - \lambda^2/\lambda_\text{max}^2}{1 - \lambda^2/\lambda_\text{max}^2} \dfrac{1 - 1/\lambda_\text{max}^2}{3 - 1/\lambda_\text{max}^2},\]

where \(\lambda_\text{max}\) is the fixed maximum stretch ratio.

Given \(n\) modes \((G_i, \tau_{\mathrm d, i}, \tau_{\mathrm R, i})\), the total stress is calculated as

\[\boldsymbol{\sigma}(t) = \sum_{i=1}^n G_i\, \text{fene}(\lambda_i)\, \boldsymbol{A}_i\]

with \(\lambda_i = \left( \frac 1 3 \text{tr}\,\boldsymbol{A}_i \right)^{1/2}\).

Multi-mode Upper Convected Maxwell Model

Summary

Multi-mode Upper Convected Maxwell model (see Chapter 1 of [4]):

\[\begin{split}\boldsymbol \sigma &= \sum_{i=1}^n G_i \boldsymbol A_i\\ \dfrac {\mathrm D \boldsymbol A_i} {\mathrm D t} &= \boldsymbol \kappa \cdot \boldsymbol A_i + \boldsymbol A_i\cdot \boldsymbol \kappa ^T - \dfrac 1 {\tau_i} (\boldsymbol A_i - \boldsymbol I)\end{split}\]
  • Functions

    • Analytical solution in shear

      \[\eta^+(t) = \sum_{i=1}^n G_i \tau_i (1 - \exp(-t/\tau_i))\]

    • Analytical solution in uniaxial extension

      \[\eta^+_\mathrm E (t) = \dfrac 1 {\dot\varepsilon} \sum_{i=1}^n G_i (A_{xx, i}(t) - A_{yy, i}(t))\]

      with

      \[\begin{split}A_{xx, i}(t) &= \dfrac{ 1 - 2 \dot\varepsilon\tau_i \exp(-(1 - 2 \dot\varepsilon\tau_i) t / \tau_i) } {1 - 2 \dot\varepsilon\tau_i }\\ A_{yy, i}(t) &= \dfrac{ 1 + \dot\varepsilon\tau_i \exp(-(1 + \dot\varepsilon\tau_i)t/\tau_i) } { 1+ \dot\varepsilon\tau_i}\end{split}\]
    where for each mode \(i\):

    • \(G_i\): weight of mode \(i\)

    • \(\tau_i\): relaxation time of mode \(i\)

  • Parameters

    [none]

Multi-mode Giesekus model

Summary

Multi-mode Giesekus Model (see Chapter 6 of [4]):

\[\begin{split}\boldsymbol \sigma &= \sum_{i=1}^n G_i \boldsymbol {A_i},\\ \dfrac {\mathrm D \boldsymbol A_i} {\mathrm D t} &= \boldsymbol \kappa \cdot \boldsymbol A_i + \boldsymbol A_i\cdot \boldsymbol \kappa ^T - \dfrac {1} {\tau_i} (\boldsymbol A_i - \boldsymbol I) - \dfrac {\alpha_i} {\tau_i} (\boldsymbol A_i - \boldsymbol I)^2,\end{split}\]
where for each mode \(i\):

  • \(G_i\): weight of mode \(i\)

  • \(\tau_i\): relaxation time of mode \(i\)

  • \(\alpha_i\): constant of proportionality mode \(i\)

  • Parameters

    • alpha_i \(\equiv \alpha_i\)

Multi-mode Pom-Pom Model

Summary

Multi-mode PomPom Model based on [5]:

\[\begin{split}\boldsymbol \sigma &= 3 \sum_{i=1}^n G_i \lambda_i^2(t) \boldsymbol S_i (t),\\ \boldsymbol S_i &= \dfrac{\boldsymbol A_i } {\mathrm{Tr} \boldsymbol A_i}\\ \dfrac {\mathrm D \boldsymbol A_i} {\mathrm D t} &= \boldsymbol \kappa \cdot \boldsymbol A_i + \boldsymbol A_i\cdot \boldsymbol \kappa ^T - \dfrac {1} {\tau_{\mathrm b, i}} (\boldsymbol A_i - \boldsymbol I), \\ \dfrac {\mathrm D \lambda_i} {\mathrm D t} &= \lambda_i (\boldsymbol \kappa : \boldsymbol S_i) - \dfrac {1} {\tau_{\mathrm s, i}} (\lambda_i - 1) \exp\left( \nu^* (\lambda_i - 1) \right),\end{split}\]

where, for each mode \(i\):

  • \(G_i\): weight of mode \(i\)

  • \(\tau_{\mathrm b, i}\): backbone orientation relaxation time of mode \(i\)

  • \(\tau_{\mathrm s, i}\): backbone stretch relaxation time of mode \(i\)

  • \(\nu_i^* = \dfrac{2}{q_i - 1}\)

  • \(q_i\): the number of dangling arms of each mode

  • Parameters

    • q_i \(\equiv q_i\): the number of dangling arms of each mode

    • ratio_i \(\equiv \dfrac{\tau_{\mathrm b, i}}{\tau_{\mathrm s, i}}\):

    the ratio of orientation to stretch relaxation times of each mode

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)

BOB NLVE

Summary

Predict the nonlinear rheology of “branch-on-branch” polymers, read from a polymer configuration file, using BoB v2.5 (Chinmay Das and Daniel Read). Polymer configuration files can be generated from the React application in RepTate.

The original documentation of BoB can be found here: https://sourceforge.net/projects/bob-rheology/files/bob-rheology/bob2.3/bob2.3.pdf/download.

PETS Model

Summary

Preaveraged model for Entangled Telechelic Star polymers: This theory is intended for the prediction of non-linear transient flows of entangled telechelic (with sticky functional groups at the chain-ends) star polymers.

  • Parameters

    • G : Plateau Modulus

    • tauD : Orientation relaxation time

    • tauS : Stretch Relxation time

    • tau_as : Typical time the sticker spends associated

    • tau_free : Typical time the sticker spends free

    • lmax : Maximum extensibility

    • beta : CCR coefficient

    • delta : CCR exponent

    • Z : Entanglement number

    • r_a : Ratio of sticker size to tube diameter

For full details about how the predictions are calculated, refer to [6].

GLaMM Model

Summary

Full SCCR theory for the Non-linear transient flow of linear entangled polymers.

  • Parameters

    • tau_e : Rouse time of one entanglement segment (of length \(M_e\).

    • Ge : Entanglement modulus.

    • Me : Entanglement molecular weight.

    • c_nu : Constraint release parameter.

    • R_S : Retraction rate parameter

This theory is intended for the prediction of non-linear transient flows of linear entangled polymers. The model includes the contribution from contour length fluctuations, reptation, retraction, constraint release and variable number of entanglements. For full details about how the predictions are calculated, refer to [2].

References

[1]

Alexei E. Likhtman and Richard S. Graham. Simple constitutive equation for linear polymer melts derived from molecular theory: Rolie–Poly equation. J. Nonnewton. Fluid Mech., 114(1):1–12, 2003. URL: http://www.personal.reading.ac.uk/~sms06al2/papers/rolie-poly.pdf, doi:10.1016/S0377-0257(03)00114-9.

[2] (1,2)

Richard S. Graham, Alexei E. Likhtman, Tom C. B. McLeish, and Scott T. Milner. Microscopic theory of linear, entangled polymer chains under rapid deformation including chain stretch and convective constraint release. J. Rheol., 47(5):1171–1200, 2003. URL: http://www.che.psu.edu/faculty/milner/group/eprints/JournalofRheology2003Graham.pdf, doi:10.1122/1.1595099.

[3]

A Cohen. A Padé approximant to the inverse Langevin function. Rheol. Acta, 30(3):270–273, 1991. doi:10.1007/BF00366640.

[4] (1,2)

Ronald G. Larson. Constitutive equations for polymer melts and solutions. Butterworths, Stoneham, 1988. URL: https://www.elsevier.com/books/constitutive-equations-for-polymer-melts-and-solutions/larson/978-0-409-90119-1.

[5]

R. J. Blackwell, T. C. B. McLeish, and O. G. Harlen. Molecular drag–strain coupling in branched polymer melts. J. Rheol., 44(1):121–136, 2000. doi:10.1122/1.551081.

[6]

Victor AH Boudara and Daniel J Read. Stochastic and preaveraged nonlinear rheology models for entangled telechelic star polymers. Journal of Rheology, 61(2):339–362, 2017. URL: https://doi.org/10.1122/1.4974908, doi:10.1122/1.4974908.