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:
where the stretch ratio, \(\lambda\), is
and, for non-stretching modes, the equation for each mode is:
Given \(n\) modes \((G_i, \tau_{\mathrm d, i}, \tau_{\mathrm R, i})\), the total stress is calculated as
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
The nonlinear spring coefficient, \(\text{fene}(\lambda)\), is approximated by the normalized Padé inverse Langevin function [3]
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
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]):
- 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 moderatio_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 modulusbeta
\(\equiv\beta_\text{CCR}\): Rolie-Poly CCR parameterdelta
\(\equiv\delta\): Rolie-Poly CCR exponentphi_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
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.
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.
A Cohen. A Padé approximant to the inverse Langevin function. Rheol. Acta, 30(3):270–273, 1991. doi:10.1007/BF00366640.
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.
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.
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.