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 Giesekus model 

Summary 

Multi-mode Pom-Pom Model 

Summary 

Rolie-Double-Poly equations 

Summary 

BOB NLVE 

Summary 

PETS Model 

Summary 

For full details about how the predictions are calculated, refer to [4].

GLaMM Model 

Summary 

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]

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.