LVE Theories

Fit of linear viscoelastic data with a set of \(N\) discrete equidistant Maxwell modes. The number of modes can be selected by pressing the Up/Down arrows in the theory window. The frequencies of the Maxwell modes are equally distributed in logarithmic scale between a minimum frequency, \(\omega_\text{min}\), and a maximum frequency, \(\omega_\text{max}\), which can be fixed or set free by ticking the corresponding checkboxes. The position of \(\omega_\text{min}\) and \(\omega_\text{max}\) can be changed by dragging the leftmost and rightmost modes with the mouse. The vertical position of the modes can be changed by dragging the yellow points.

Each mode contributes to the linear viscoelastic spectrum through the following formulas:

\[\begin{split}G_i'(\omega) &= G_i \frac{(\omega \tau_i)^2}{1+ (\omega \tau_i)^2}\\ G_i''(\omega) &= G_i \frac{\omega \tau_i}{1+ (\omega \tau_i)^2}\end{split}\]

with \(G_i\) the modulus and \(\tau_i\) the characteristic relaxation time of the mode \(i\) (inverse of the frequency).

The parameters of the theory are the number of modes (which is fixed by the user and is not minimized), \(\omega_\text{min}\), \(\omega_\text{max}\), and a value of \(G_i\) for each mode. \(G_i\) is calculated in logarithmic scale.

Warning

The theory can only be applied to one file per data set. If more than one file is active in the current data set, the theory will be applied to the first one in the list of active files.

Note

Roughly one mode per decade should be enough for the purpose of using the modes in flow simulations.
Also for flow simulations, do not use modes much faster than the fastest flow rate in the simulation.
For linear polymers, one mode close to the crossing point of \(G'\) and \(G''\) should be enough to fit the terminal time.

Likhtman-McLeish theory 

Summary 

Description 

Full quantitative theory for the linear dynamics of entangled linear polymers, based on the tube model and considering contour length fluctuations, constraint release and longitudinal stress relaxation along the tube.

The parameter \(c_\nu\) is related to the number of chains that are needed to create one entanglement. \(c_\nu=0\) means there is no constraint release. The recommended value is \(c_\nu=0.1\). This value gives slightly worse fitting than \(c_\nu=1\), but is more consistent with start-up shear experiments.

For full details about how the predictions are calculated, refer to [1]. A set of predictions has been precalculated and stored in binary form in the RepTate distribution. There are predictions for different values of \(Z=M_w/M_e\) = 2, 3, 4 … 299, 300, 305, 310, … 1000, and \(c_\nu\) = 0, 0.01, 0.03, 0.1, 0.3, 1, 3, 10. The molecular weight of the sample is read from the data file. For values of \(Z\) and \(c_\nu\) in the precalculated range, RepTate reads the prediction directly from the precalculated data. For other values of \(Z\) and \(c_\nu\), the prediction is interpolated.

After the calculation is done, the values of the number of entanglements \(Z\), the Rouse time \(\tau_R\) and the reptation time \(\tau_D\) of the polymer are printed in the Log Window. These values are calculated according to the following formulas:

\[\tau_R = \tau_e Z^2\]

\[\tau_D = 3\tau_e Z^3 \left( 1 - \frac{2C_1}{\sqrt{Z}} + \frac{C_2}{Z} + \frac{C_3}{Z^{3/2}} \right)\]

with the constants \(C_1=1.69\), \(C_2=4.17\) and \(C_3=-1.55\).

Recommendations 

The theory is able to fit several molecular weight samples of the same polymer at the same time. That is the recommended procedure to get the material parameters \(\tau_e\), \(G_e\) and \(M_e\).
The theory allows the user to link the values of \(G_e\) and \(M_e\) through the density \(\rho\) (in g/cm3) by means of the standard expression:

\[G_e = \frac{1000\rho R T}{M_e}\]

The user must input the correct density. If available, \(\rho\) is read from the Materials database. The theory fails to fit correctly the data when the values of \(G_e\) and \(M_e\) are linked in this way.

Carreau-Yasuda equation 

Summary 

Description 

Fit of the complex viscosity to the phenomenological Carreau-Yasuda equation. The equation used in the fit is:

\[\eta^*(\omega) = \eta_\infty + (\eta_0-\eta_\infty)\left( 1 + (\lambda\omega)^a \right)^{(n-1)/a}\]

In the equation, \(a\) is a dimensionless parameter that describes the transition from the zero-shear rate region to the power law region. It frequently takes the value 2. In some experimental data, it is safe to assume that \(\eta_\infty\) is equal to zero.

Warning

The fit only makes sense when applied using the logstar view (logarithm of the complex viscosity). The logarithm is needed, due to the wide range of viscosity values.

Warning

The theory can only be applied to one file per data set. If more than one file is active in the current data set, the theory will be applied to the first one in the list of active files.

CFSM 

Summary 

Parameters and units 

CFSM+Rouse can initialise MK and rho0 from the Materials Database when the first file in the dataset contains a recognised chem value.

The parameter table is unit-aware:

MK is stored internally as a molar mass but displayed by default in Da.
rho0 is displayed in g/cc.
Mc is displayed by default in Da.
tau_c is displayed in s.

The underlying DSM formulas use the traditional molar-mass convention in Da. RepTate therefore converts the unit-aware MK and Mc parameters to Da inside the calculation. This preserves the legacy numerical behavior while still allowing the parameter table and the Materials Database to use explicit unit metadata.

For example, a Materials Database value MK = 140.5 Da is stored internally as 0.1405 kg/mol but displayed as 140.5 Da in this theory. If the initial crossover estimate computes Mc = 2146 Da, the stored internal value is 2.146 kg/mol and the displayed value remains 2146 Da.

Rouse Frequency 

Summary 

Dynamic dilution equation for stars 

Summary 

BoB LVE 

Summary 

Publications related to BoB: [2] [3] [4].

Rolie-Double-Poly LVE 

Summary 

LP2R 

Summary 

Polymer components 

The LP2R LVE theory represents the polymer input as a list of components rather than as a single visible input-mode parameter. Open the LP2R components dialog from the theory toolbar to add, edit, remove, import, and normalise the components used in the next calculation.

Each component has a weight fraction and one of two forms:

lognormal: described by npoly, Mw and PDI.
mwd: described by discrete molar masses and weights.

Discrete MWD components can be imported from a RepTate Discretize MWD theory or from a .gpc file. RepTate expects .gpc files to provide columns M and W(logM). The unit declared in the M header is read when available; masses are converted to RepTate’s internal molar-mass unit kg/mol before being passed to LP2R.

If no component has been defined, LP2R creates a default lognormal component from the first file when possible. Mw and PDI are taken from the file parameters if they exist and are positive; otherwise reasonable defaults are used. The default number of lognormal bins is npoly = 8.

Material and numerical parameters 

The normal parameter table contains the material and numerical controls: MK, Me, G0, tau_e, glass parameters, freq_ratio and the advanced numerical controls. The hidden default-component parameters Mw, PDI and n are not part of the ordinary fitting workflow.

When the first file contains a recognised chem value, LP2R imports common material parameters from the Materials Database. MK, Me and tau_e are imported directly when available. G0 is initialised from the shifted Materials Database value of Ge using G0 = 0.8 Ge.

Calculation output 

When calculation starts, LP2R prints a table of the current polymer components in the theory output area. The calculation progress is shown with dashed markers from 0% to 100%, following the convention used by GLaMM/SCCR.

If the component list is empty or invalid, LP2R stops before starting the solver and reports the problem in the theory output area.

Numerical robustness 

The LP2R implementation uses an MSVC-safe KWW implementation for the sake of compatibility with Windows computers. If a numeric integration failure occurs, the error is caught and reported in the theory output area instead of terminating RepTate.

Advanced LP2R Controls 

The Advanced LP2R controls dialog exposes numerical controls that are not normally varied during routine fitting. These parameters are kept separate from the polymer component list and the main material parameters.

Sticky Reptation 

Summary 

Description 

Theory for the linear rheology of linear entangled polymers with a number of stickers that can form reversible intramolecular crosslinks. The relaxation modulus, \(G(t)\), is modeled as the sum of a Sticky-Rouse, \(G_\mathrm{SR}(t)\) [5] and a Double-Reptation, \(G_\mathrm{rep}(t)\) [6] contribution,

\[G(t) = G_\mathrm{SR}(t) + G_\mathrm{rep}(t)\]

with

\[G_\mathrm{SR}(t) = \dfrac{G_\mathrm{e}}{Z_\mathrm{e}} \sum_{q=1}^{Z_\mathrm{s}} \kappa \exp\left(\dfrac{q^2 t}{\tau_\mathrm{s} (Z_\mathrm{s})^2}\right).\]

This equation assumes that most stickers are bound, and after sticker dissociation a strand of length \(N/Z_\mathrm{s}\) can relax, with \(N\) the number of monomers per chain and \(Z_\mathrm{s}\) the number of stickers per chain. \(\tau_\mathrm{s}\) is the dissociation time of a sticker. (Note that this is approximate: after sticker dissociation a chain with a length twice \(N/Z_\mathrm{s}\) or more relaxes, see [5] and [7] ) The truncation of the sum at \(Z_\mathrm{s}\) implies that we ignore high-frequency (non-sticky) Rouse relaxation of the subchains between stickers. This is only valid if the sticker dissociation time is much larger than the Rouse time of those substrands.

Warning

The high-frequency Rouse modes with time scales shorter than the sticker dissociation time are not included. This is only valid if the sticker dissociation time is much larger than the Rouse time of those substrands.

Finally, the factor \(\kappa\) is 0.2 for long wavelengths (i.e., for \(q < Z_\mathrm{e}\) and unity for short wavelengths (i.e., for \(q \geq Z_\mathrm{e}\)) (see Likhtman-McLeish, 2002). The factor \(Z_\mathrm{e}\) is the number of entanglements per chain.

The other contribution to the relaxation modulus is the double-reptation model,

\[G_\mathrm{rep}(t) = G_\mathrm{e} \left( \frac{8}{\pi^2}\sum_{\mathrm{odd}\, q} \frac{1}{q^2}\exp\left(-q^2 U(t)\right) \right)^2,\]

with

\[U(t) = \frac{t}{\tau_\mathrm{rep}} + \frac{\alpha}{Z_\mathrm{e}}g\left(\frac{Z_\mathrm{e}}{\alpha} \frac{t}{\tau_\mathrm{rep}}\right)\]

and with \(g(x)=\sum_{m=1}^{\infty}m^{-2}\left[1-\exp(-m^2 x)\right]\). The dimensionless constant \(\alpha\): is in principle universal, but in practice varies between different polymers (see e.g. Ref. [8]). Note that \(Z_\mathrm{e}/\alpha\) is equivalent to the parameter \(H\) in Ref. [6].

The sticky reptation time is \(\tau_\mathrm{rep} = \tau_\mathrm{s}Z_\mathrm{s}^2Z_\mathrm{e}\).

Recommendations 

To verify the theory, the calculated viscosity \(\eta_0 = G_\mathrm{e}\times \tau_\mathrm{s}Z_\mathrm{s}^2 Z_\mathrm{e}/\alpha\): should be close to the experimental value. Further, the fitted value of the elastic plateau modulus should be close to

\[G_\mathrm{e} = \dfrac{4}{5}\dfrac{\phi Z_\mathrm{e}}{\upsilon N} k_\mathrm{B}T\]

with \(\phi\) the volume fraction occupied by the polymer, \(\upsilon\) the volume of a monomer and \(N\) the number of monomers per chain. \(k_\mathrm{B}\) is Boltzmann’s constant and \(T\) is the absolute temperature in Kelvin.

ReSpect 

Summary 

For details about how to use this theory, check [9].

References

[1]

Alexei E. Likhtman and Tom C. B. McLeish. Quantitative Theory for Linear Dynamics of Linear Entangled Polymers. Macromolecules, 35(16):6332–6343, jul 2002. doi:10.1021/ma0200219.

[2]

Chinmay Das, Nathanael J. Inkson, Daniel J. Read, Mark a. Kelmanson, and Tom C. B. McLeish. Computational linear rheology of general branch-on-branch polymers. J. Rheol., 50(2):207–234, mar 2006. URL: http://sor.scitation.org/doi/10.1122/1.2167487, doi:10.1122/1.2167487.

[3]

Chinmay Das, Daniel J. Read, Mark A. Kelmanson, and Tom C. B. McLeish. Dynamic scaling in entangled mean-field gelation polymers. Phys. Rev. E, 74(1):011404, jul 2006. URL: https://link.aps.org/doi/10.1103/PhysRevE.74.011404, doi:10.1103/PhysRevE.74.011404.

[4]

Pierre Chambon, Christine M. Fernyhough, Kyuhyun Im, Taihyun Chang, Chinmay Das, John Embery, Tom C. B. McLeish, and Daniel J. Read. Synthesis, Temperature Gradient Interaction Chromatography, and Rheology of Entangled Styrene Comb Polymers. Macromolecules, 41(15):5869–5875, aug 2008. URL: http://pubs.acs.org/doi/abs/10.1021/ma800599m, doi:10.1021/ma800599m.

[5] (1,2)

Ludwik Leibler, Michael Rubinstein, and Ralph H. Colby. Dynamics of Reversible Networks. Macromolecules, 24(16):4701–4707, aug 1991. doi:10.1021/ma00016a034.

[6] (1,2)

J. des Cloizeaux. Relaxationand Viscosity Anomaly of Melts Made of Long Entangled Polymers Time-Dependent Reptation. Macromolecules, 23(21):4678–4687, oct 1990. doi:10.1021/ma00223a028.

[7]

Michael Rubinstein and Alexander N. Semenov. Dynamics of Entangled Solutions of Associating Polymers. Macromolecules, 34(4):1058–1068, jan 2001. doi:10.1021/ma0013049.

[8]

E. van Ruymbeke, R. Keunings, V. Stéphenne, A. Hagenaars, and C. Bailly. Evaluation of Reptation Models for Predicting the Linear Viscoelastic Properties of Entangled Linear Polymers. Macromolecules, 35(7):2689–2699, feb 2002. doi:10.1021/ma011271c.

[9]

Arsia Takeh and Sachin Shanbhag. A Computer Program to Extract the Continuous and Discrete Relaxation Spectra from Dynamic Viscoelastic Measurements. Applied Rheology, April 2013. doi:10.3933/applrheol-23-24628.