LVE Theories¶
Maxwell modes¶
Summary¶
Fit a generalized Maxwell model to a frequency dependent relaxation function.
- Function
- \[\begin{split}\begin{eqnarray} G'(\omega) & = & \sum_{1}^{n_{modes}} G_i \frac{(\omega\tau_i)^2}{1+(\omega\tau_i)^2} \\ G''(\omega) & = & \sum_{1}^{n_{modes}} G_i \frac{\omega\tau_i}{1+(\omega\tau_i)^2} \end{eqnarray}\end{split}\]
- Parameters
\(n_{modes}\): number of Maxwell modes equally distributed in logarithmic scale between \(\omega_{min}\) and \(\omega_{max}\).
logwmin = \(\log(\omega_{min})\): decimal logarithm of the minimum frequency.
logwmax = \(\log(\omega_{max})\): decimal logarithm of the maximum frequency.
logGi = \(\log(G_{i})\): decimal logarithm of the amplitude of Maxwell mode \(i\).
Description¶
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:
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:
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:
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¶
Fit the complex viscosity with the Carreau-Yasuda equation.
- Function
- \[\eta^*(\omega) = \eta_\infty + (\eta_0-\eta_\infty)\left( 1 + (\lambda\omega)^a \right)^{(n-1)/a}\]
- Parameters
\(\eta_0\): Viscosity at zero shear rate.
\(\eta_\infty\): Viscosity at infinite shear rate.
\(\lambda\): Relaxation time.
\(n\): Power law index.
\(a\): Dimensionless parameter (2 in most cases)
Description¶
Fit of the complex viscosity to the phenomenological Carreau-Yasuda equation. The equation used in the fit is:
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¶
Calculate the Discrete Slip Link theory for the linear rheology of linear entangled polymers.
Parameters
Todo
Complete the docs of this theory
Rouse Frequency¶
Summary¶
Fit Rouse modes to a frequency dependent relaxation function
- Function
Continuous Rouse model (valid for “large” \(N\)):
\[\begin{split}G'(\omega) &= G_0 \dfrac 1 N \sum_{p=1}^N \dfrac{(\omega\tau_p)^2} {1 + (\omega\tau_p)^2}\\ G''(\omega) &= G_0 \dfrac 1 N \sum_{p=1}^N \dfrac{\omega\tau_p} {1 + (\omega\tau_p)^2}\\ \tau_p &= \dfrac{N^2 \tau_0 }{ 2 p^2}\end{split}\]
- Parameters
\(G_0 = ck_\mathrm B T\): “modulus”
\(\tau_0\): relaxation time of an elementary segment
\(M_0\): molar mass of an elementary segment
- where
\(c\): number of segments per unit volume
\(k_\mathrm B\): Boltzmann constant
\(T\): temperature
\(N=M_w/M_0\): number of segments par chain
\(M_w\): weight-average molecular mass
Dynamic dilution equation for stars¶
Summary¶
Fit DTD Theory for stars. Theory of stress relaxation in star polymer melts with no adjustable parameters beyond those measurable in linear melts
- Function
See Milner-McLeish (1997) and Larson et al. (2003) for details.
- Parameters
G0\(\equiv G_N^0\): Plateau modulustau_e\(\equiv \tau_\mathrm e = \left(\dfrac{M_\mathrm e^\mathrm G}{M_0}\right)^2 \dfrac{\zeta b^2}{3\pi^2k_\mathrm B T}\): Entanglement equilibration timeMe\(\equiv M_\mathrm e^\mathrm G = \dfrac 4 5 \dfrac{\rho R T} {G_N^0}\): Entanglement molecular weightalpha: Dilution exponent
- where:
\(\rho\): polymer density
\(\zeta\): monomeric friction coefficient
\(b\): monomer-based segment length
\(k_\mathrm B T\): thermal energy
\(M_0\): molar mass of an elementary segment
BoB LVE¶
Summary¶
Analyse the relaxation of polymers read from a polymer configuration file using BoB v2.5 (Chinmay Das and Daniel Read). These 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.
Rolie-Double-Poly LVE¶
Summary¶
Rolie-Double-Poly equation for the linear predictions of polydispere entangled linear polymers
- Function
- \[\begin{split}\begin{eqnarray} G'(\omega) & = & \sum_{i=1}^{n_{modes}}\sum_{j=1}^{n_{modes}} G \phi_i \phi_j \frac{(\omega\tau)^2}{1+(\omega\tau)^2} \\ G''(\omega) & = & \sum_{i=1}^{n_{modes}}\sum_{j=1}^{n_{modes}} G \phi_i \phi_j \frac{\omega\tau}{1+(\omega\tau)^2} \end{eqnarray}\end{split}\]
where, \(\tau = (\tau_{\text D,i}^{-1} + \tau_{\text D, j}^{-1})^{-1}\), and, if the “modulus correction” button is clicked, \(G=G_N^0 \times g(Z_\text{eff})\), with \(g\) the Likthman-McLeish CLF correction function, otherwise \(G=G_N^0\)
- Parameters
nmodes: number of molecular mass components.G_N^0: Plateau modulusphi0i: Volume fraction of component \(i\)tauD0i: Reptation time of component \(i\)
Sticky Reptation¶
Summary¶
Fit the Sticky Reptation theory for the linear rheology of linear entangled polymers with a number of stickers that can form reversible intramolecular crosslinks.
- Parameters
Ge: elastic plateau modulus.Ze: number of entanglements per chain.Zs: number of stickers per chain.tau_s: sticker dissociation time.alpha: dimensionless constant.
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,
with
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,
with
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
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¶
Extract continuous and discrete relaxation spectra from complex modulus G*(w)
- Parameters
plateau : is there a residual plateau in the data (default False).
ns : Number of grid points to represent the continuous spectrum (typical 50-100)
lamC : Specify lambda_C instead of using the one inferred from the L-curve (default 0, use the L-curve).
SmFacLam = Smoothing Factor.
MaxNumModes = Max Number of Modes (default 0, automatically determine the optimal number of modes).
lam_min = lower limit of lambda for lcurve calculation (default 1e-10).
lam_max = higher limit of lambda for lcurve calculation (default 1e3).
lamDensity = lambda density per decade (default 3, use 2 or more).
rho_cutoff = Threshold to avoid picking too small lambda for L-curve without (default 0).
deltaBaseWeightDist = how finely to sample BaseWeightDist (default 0.2).
minTauSpacing = how close do successive modes (tau2/tau1) have to be before we try to mege them (default 1.25).
Todo
Complete docs for this theory
References
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.
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.
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.
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.
Ludwik Leibler, Michael Rubinstein, and Ralph H. Colby. Dynamics of Reversible Networks. Macromolecules, 24(16):4701–4707, aug 1991. doi:10.1021/ma00016a034.
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.
Michael Rubinstein and Alexander N. Semenov. Dynamics of Entangled Solutions of Associating Polymers. Macromolecules, 34(4):1058–1068, jan 2001. doi:10.1021/ma0013049.
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.
