Stress Relaxation: Theories

Maxwell modes

Summary

Fit a generalized Maxwell model to a time dependent relaxation function.

  • Function
    \[\begin{eqnarray} G(t) & = & \sum_{i=1}^{n_{modes}} G_i \exp (-t/\tau_i) \end{eqnarray}\]
  • Parameters
    • \(n_{modes}\): number of Maxwell modes equally distributed in logarithmic scale between \(\omega_{min}\) and \(\omega_{max}\).

    • logtmin = \(\log(t_{min})\): decimal logarithm of the minimum time.

    • logtmax = \(\log(t_{max})\): decimal logarithm of the maximum time.

    • logGi = \(\log(G_{i})\): decimal logarithm of the amplitude of Maxwell mode \(i\).

Description

Fit of step strain + relaxation data with a set of \(N\) discrete Maxwell modes. The number of modes can be selected by pressing the Up/Down arrows in the theory window. The relaxation times of the Maxwell modes are equally distributed in logarithmic scale between a minimum time, \(t_\text{min}\), and a maximum time, \(t_\text{max}\), which can be fixed or set free by ticking the corresponding checkboxes. The position of \(t_\text{min}\) and \(t_\text{max}\) 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 relaxation modulus through the following formulas:

\[G_i(t) = G_i \exp (-t/\tau_i)\]

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), \(t_\text{min}\), \(t_\text{max}\), 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.

Rouse Time

Summary

Fit Rouse modes to a time dependent relaxation function

  • Function

    Continuous Rouse model (valid for “large” \(N\)):

    \[G(t) = G_0 \dfrac 1 N \sum_{p=1}^N \exp\left(\dfrac{-2p^2t}{N^2\tau_0}\right)\]
  • 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

DTD Stars Time

Summary

Fit DTD Theory for stars

  • Function

    See Milner-McLeish (1997) and Larson et al. (2003) for details.

  • Parameters
    • G0 \(\equiv G_N^0\): Plateau modulus

    • tau_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 time

    • Me \(\equiv M_\mathrm e^\mathrm G = \dfrac 4 5 \dfrac{\rho R T} {G_N^0}\): Entanglement molecular weight

    • alpha: 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

ReSpect

Summary

Extract continuous and discrete relaxation spectra from relaxation modulus G(t)

  • 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

1

F. R. Schwarzl. Numerical calculation of storage and loss modulus from stress relaxation data for linear viscoelastic materials. Rheologica Acta, 10(2):165–173, jun 1971. doi:10.1007/bf02040437.

2

Manlio Tassieri, Marco Laurati, Dan J. Curtis, Dietmar W. Auhl, Salvatore Coppola, Andrea Scalfati, Karl Hawkins, Phylip Rhodri Williams, and Jonathan M. Cooper. I-rheo: measuring the materials\textquotesingle linear viscoelastic properties \textquotedblleft in a step\textquotedblright ! Journal of Rheology, 60(4):649–660, jul 2016. doi:10.1122/1.4953443.