TNT Cates (Living Polymers / Wormlike Micelles) — Handbook¶

Quick Reference ¶

Use When:

Wormlike micelles (e.g., CTAB/NaSal, CPyCl/NaSal, SDS/LAPB)
Living polymer systems with reversible scission
Surfactant solutions showing single-mode Maxwell behavior
Systems with perfect semicircular Cole-Cole plots
Materials exhibiting shear banding in flow curves

Parameters:

Symbol	Default	Units	Description
\(G_0\)	100	Pa	Plateau modulus
\(\tau_\text{rep}\)	10.0	s	Reptation time
\(\tau_\text{break}\)	0.1	s	Mean breaking time
\(\eta_s\)	0.0	Pa·s	Solvent viscosity

Key Equations:

Effective relaxation time (fast-breaking limit):

\[\tau_d = \sqrt{\tau_\text{rep} \cdot \tau_\text{break}}\]

Breaking parameter:

\[\zeta = \frac{\tau_\text{break}}{\tau_\text{rep}}\]

Zero-shear viscosity:

\[\eta_0 = G_0 \tau_d\]

Test Modes:

All six protocols supported:

OSCILLATION (SAOS): \(G'(\omega)\), \(G''(\omega)\)
FLOW_CURVE: \(\sigma(\dot{\gamma})\), shear banding prediction
STARTUP: Transient stress overshoot
RELAXATION: Monoexponential stress decay
CREEP: Single-mode compliance
LAOS: Nonlinear oscillatory response

Material Examples:

CTAB/NaSal wormlike micelles (cetyl trimethylammonium bromide / sodium salicylate)
CPyCl/NaSal (cetyl pyridinium chloride / sodium salicylate)
SDS/LAPB (sodium dodecyl sulfate / lauryl amido propyl betaine)
Ionic surfactant solutions above critical micelle concentration
Living polymer melts with reversible cross-linking
Telechelic polymers with sticky ends

Key Characteristics:

Single Maxwell-like relaxation in fast-breaking limit (\(\zeta \ll 1\))
Perfect semicircular Cole-Cole plot (\(G''\) vs \(G'\))
Monoexponential stress relaxation
Non-monotonic flow curve (constitutive instability)
Shear banding for \(\text{Wi}_d > 1\)
Crossover frequency \(\omega_c = 1/\tau_d\)

Notation Guide ¶

Symbol	Units	Description
\(G_0\)	Pa	Plateau modulus (related to mesh size)
\(\tau_\text{rep}\)	s	Reptation time (curvilinear diffusion along tube)
\(\tau_\text{break}\)	s	Mean breaking time (Poisson scission)
\(\tau_d\)	s	Effective relaxation time = \(\sqrt{\tau_\text{rep} \tau_\text{break}}\)
\(\zeta\)	–	Breaking parameter = \(\tau_\text{break}/\tau_\text{rep}\)
\(\eta_s\)	Pa·s	Solvent viscosity
\(\eta_0\)	Pa·s	Zero-shear viscosity = \(G_0 \tau_d\)
\(S\)	–	Conformation tensor (end-to-end vector average)
\(\boldsymbol{\kappa}\)	\(s^{-1}\)	Velocity gradient tensor
\(D\)	\(s^{-1}\)	Rate of deformation tensor = \((\boldsymbol{\kappa} + \boldsymbol{\kappa}^T)/2\)
\(\boldsymbol{\sigma}\)	Pa	Stress tensor
\(\text{Wi}_d\)	–	Weissenberg number = \(\tau_d \dot{\gamma}\)
\(L\)	nm	Mean micelle contour length
\(\xi\)	nm	Mesh size (entanglement length scale)
\(\omega_c\)	rad/s	Crossover frequency = \(1/\tau_d\)
\(k_B T\)	J	Thermal energy
\(E_\text{scission}\)	J/mol	Activation energy for scission

The TNT Cates model describes the rheology of living polymers, systems where polymeric chains can reversibly break and recombine on timescales comparable to their stress relaxation. The most prominent experimental realization is wormlike micelles: long, flexible, cylindrical surfactant aggregates that form in concentrated surfactant solutions.

Unlike conventional polymers with permanent covalent bonds, wormlike micelles continuously undergo:

Scission: Random breaking at any point along the contour
Recombination: End-to-end fusion when micelle tips meet
Reversibility: Breaking and recombination rates are balanced at equilibrium

The model was developed by M.E. Cates in 1987-1990 and represents one of the most successful theories in surfactant rheology.

Historical Development ¶

1987 - Cates (Macromolecules):

Extended reptation theory to living polymers
Showed that reversible scission fundamentally alters stress relaxation
Predicted single-mode Maxwell behavior in fast-breaking limit

1990 - Cates (J Phys Chem):

Nonlinear rheology and flow curve predictions
Constitutive instability leading to shear banding
Connection to experimental observations

1990 - Cates and Candau (J Phys Condens Matter):

Comprehensive review of statics and dynamics
Scaling laws for micelle length and relaxation times

1991 - Turner and Cates (Langmuir):

Linear viscoelasticity in detail
Cole-Cole plot predictions

1991 - Rehage and Hoffmann (Mol Phys):

Experimental verification with CTAB/NaSal
Perfect Maxwell behavior and shear banding

Why This Model Matters ¶

Explains Maxwell behavior in surfactants: Conventional polymers show broad spectra (many modes); wormlike micelles show single-mode behavior
Predictive power: Quantitatively explains linear and nonlinear rheology with just 3 parameters
Shear banding mechanism: First model to predict flow curve instability from microscopic dynamics
Industrial relevance: Wormlike micelles are used in consumer products (shampoos, detergents), enhanced oil recovery, drag reduction
Theoretical foundation: Connects reptation theory to reversible kinetics

Physical Foundations ¶

Reptation Theory ¶

De Gennes (1971), Doi-Edwards (1978):

Entangled polymers are confined to a “tube” formed by neighboring chains. Stress relaxation occurs via curvilinear diffusion (reptation) along the tube axis. The reptation time scales as:

\[\tau_\text{rep} \sim \frac{L^3}{\pi^2 D}\]

where \(L\) is the contour length and \(D\) is the curvilinear diffusion coefficient.

For permanent polymers, \(\tau_\text{rep}\) is the dominant relaxation time. The stress relaxes via a spectrum of modes:

\[G(t) = G_0 \sum_{p \text{ odd}} \frac{8}{\pi^2 p^2} \exp\left(-\frac{p^2 t}{\tau_\text{rep}}\right)\]

Reversible Scission ¶

Cates addition (1987):

Wormlike micelles break at random positions with Poisson statistics. The mean scission time for a micelle of length \(L\) is:

\[\tau_\text{break}(L) = \frac{\tau_\text{break}^0}{L/L_0}\]

where \(\tau_\text{break}^0\) is the breaking time of a reference length \(L_0\).

Key insight: Breaking randomizes the tube position. If \(\tau_\text{break} \ll \tau_\text{rep}\), the micelle breaks many times before reptating out of its original tube. This scrambles the memory of the initial conformation.

Fast-Breaking Limit ¶

Condition:

\[\zeta = \frac{\tau_\text{break}}{\tau_\text{rep}} \ll 1\]

Consequence:

The effective stress relaxation becomes single-mode with a geometric mean relaxation time:

\[\tau_d = \sqrt{\tau_\text{rep} \cdot \tau_\text{break}}\]

Physical picture:

Reptation requires diffusion over length \(L\)
Breaking cuts the micelle into pieces of size approximately \(L/2\) every \(\tau_\text{break}\)
The micelle escapes its tube when the diffusion length \(\sqrt{D t}\) equals the breaking length \(\sim \sqrt{D \tau_\text{break}}\)
Solving \(L \sim \sqrt{D \tau_\text{break}}\) with \(\tau_\text{rep} \sim L^3/D\) gives \(\tau_d \sim \sqrt{\tau_\text{rep} \tau_\text{break}}\)

Scaling:

\[\tau_d \sim L \quad \text{(linear in length)}\]

compared to \(\tau_\text{rep} \sim L^3\) for unbreakable chains.

Fast-Breaking vs Slow-Breaking Regimes ¶

The Cates model exhibits two limiting regimes depending on the ratio of breakage time \(\tau_b\) to reptation time \(\tau_{\text{rep}}\):

Fast-breaking regime (\(\tau_b \ll \tau_{\text{rep}}\)):

The effective relaxation time is the geometric mean:

\[\tau_d = \sqrt{\tau_{\text{rep}} \tau_b}\]

The relaxation modulus follows a stretched exponential:

\[G(t) = G_0 \exp\!\left(-\sqrt{2t/\tau_b}\right)\]

This regime produces near-single-mode Maxwell behavior — the defining signature of wormlike micelles in the fast-breaking limit.

Slow-breaking regime (\(\tau_b \gg \tau_{\text{rep}}\)):

Standard reptation dominates:

\[G(t) = G_0 \exp(-t/\tau_{\text{rep}})\]

Breakage has negligible effect; the system behaves like an entangled polymer melt with the standard reptation spectrum.

Recombination and Equilibrium ¶

At equilibrium, the scission rate equals the recombination rate:

\[k_\text{break} n_\text{micelles} = k_\text{recomb} n_\text{ends}^2\]

where:

\(k_\text{break}\) is the scission rate constant
\(k_\text{recomb}\) is the recombination rate constant
\(n_\text{micelles}\) is the number of micelles
\(n_\text{ends}\) is the number of free ends

This gives an equilibrium micelle length distribution. For simplicity, the TNT Cates model assumes a mean-field description with average properties.

Tube Model Mapping ¶

The conformation tensor \(S\) represents the average end-to-end vector orientation. In the tube model:

\[S = \langle \mathbf{u} \otimes \mathbf{u} \rangle\]

where \(\mathbf{u}\) is the unit tangent vector along the tube.

The stress is:

\[\boldsymbol{\sigma} = G_0 (S - I) + 2 \eta_s D\]

where \(G_0 \sim k_B T / \xi^3\) is the plateau modulus (\(\xi\) is the mesh size).

Governing Equations ¶

Conformation Tensor Evolution ¶

The fast-breaking Cates model reduces to a single-mode upper-convected Maxwell (UCM) constitutive equation with relaxation time \(\tau_d\):

\[\frac{DS}{Dt} - \boldsymbol{\kappa} \cdot S - S \cdot \boldsymbol{\kappa}^T = -\frac{1}{\tau_d}(S - I)\]

where:

\(\frac{D}{Dt}\) is the material derivative
\(\boldsymbol{\kappa} = \nabla \mathbf{v}\) is the velocity gradient tensor
\(I\) is the identity tensor

Expanded form:

\[\frac{\partial S}{\partial t} + \mathbf{v} \cdot \nabla S - \boldsymbol{\kappa} \cdot S - S \cdot \boldsymbol{\kappa}^T = -\frac{1}{\tau_d}(S - I)\]

For homogeneous flows (\(\nabla S = 0\)):

\[\frac{dS}{dt} = \boldsymbol{\kappa} \cdot S + S \cdot \boldsymbol{\kappa}^T - \frac{1}{\tau_d}(S - I)\]

Stress Tensor ¶

\[\boldsymbol{\sigma} = G_0 (S - I) + 2 \eta_s D\]

where:

\(G_0\) is the plateau modulus
\(\eta_s\) is the solvent viscosity
\(D = (\boldsymbol{\kappa} + \boldsymbol{\kappa}^T)/2\) is the rate of deformation tensor

Total stress:

\[\boldsymbol{\sigma}_\text{total} = -p I + \boldsymbol{\sigma}\]

where \(p\) is the pressure (isotropic part).

Effective Relaxation Time ¶

Computed internally:

\[\tau_d = \sqrt{\tau_\text{rep} \cdot \tau_\text{break}}\]

This is not a fitted parameter. The model fits \(\tau_\text{rep}\) and \(\tau_\text{break}\) separately, and \(\tau_d\) is derived.

Physical interpretation:

\(\tau_d\) is the observable relaxation time in SAOS (crossover frequency \(\omega_c = 1/\tau_d\))
\(\tau_\text{rep}\) and \(\tau_\text{break}\) are microscopic timescales
Requires temperature-dependent or concentration-dependent data to separate \(\tau_\text{rep}\) and \(\tau_\text{break}\)

Steady Shear Flow ¶

Velocity field:

\[\mathbf{v} = (\dot{\gamma} y, 0, 0)\]

Velocity gradient:

\[\begin{split}\boldsymbol{\kappa} = \begin{pmatrix} 0 & \dot{\gamma} & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}\end{split}\]

Steady-state solution (analytical):

Weissenberg number:

\[\text{Wi}_d = \tau_d \dot{\gamma}\]

Shear stress:

\[\sigma_{xy} = \frac{G_0 \text{Wi}_d}{1 + \text{Wi}_d^2} + \eta_s \dot{\gamma}\]

Normal stress differences:

\[N_1 = \sigma_{xx} - \sigma_{yy} = \frac{2 G_0 \text{Wi}_d^2}{1 + \text{Wi}_d^2}\]

\[N_2 = 0 \quad \text{(UCM model)}\]

Flow curve instability:

The shear stress is non-monotonic: it increases for \(\text{Wi}_d < 1\), reaches a maximum at \(\text{Wi}_d = 1\), then decreases for \(\text{Wi}_d > 1\).

Maximum shear stress:

\[\sigma_{xy}^\text{max} = \frac{G_0}{2} + \eta_s \dot{\gamma}_\text{max}\]

where \(\dot{\gamma}_\text{max} = 1/\tau_d\).

Constitutive instability:

For \(\text{Wi}_d > 1\), the flow curve has negative slope \(d\sigma/d\dot{\gamma} < 0\). This is mechanically unstable and leads to shear banding: coexistence of high and low shear rate bands.

Non-Monotonic Flow Curve and Shear Banding ¶

A hallmark prediction of the Cates model is a non-monotonic flow curve:

\[\frac{d\sigma}{d\dot{\gamma}} < 0 \quad \text{for} \quad \dot{\gamma} > \dot{\gamma}_c\]

Stress maximum:

\[\sigma_{\max} \approx G_0 \quad \text{at critical rate} \quad \dot{\gamma}_c \sim 1/\tau_d\]

Above the stress maximum, the system cannot sustain homogeneous flow. Instead, shear banding develops: the material separates into coexisting bands of high and low shear rate, with a stress plateau \(\sigma_{\text{plateau}} < \sigma_{\max}\).

Physical mechanism: Scission accelerates with chain stretch. At high rates, chains break faster than they can recombine into stress-bearing configurations, causing an effective viscosity collapse.

Small Amplitude Oscillatory Shear (SAOS)¶

Input:

\[\gamma(t) = \gamma_0 \sin(\omega t)\]

Complex modulus:

\[G^*(\omega) = G'(\omega) + i G''(\omega)\]

Storage modulus:

\[G'(\omega) = \frac{G_0 (\omega \tau_d)^2}{1 + (\omega \tau_d)^2}\]

Loss modulus:

\[G''(\omega) = \frac{G_0 (\omega \tau_d)}{1 + (\omega \tau_d)^2} + \omega \eta_s\]

Limiting behavior:

Low frequency (\(\omega \tau_d \ll 1\)):

\[G' \sim G_0 \omega^2 \tau_d^2, \quad G'' \sim G_0 \omega \tau_d + \omega \eta_s\]

High frequency (\(\omega \tau_d \gg 1\)):

\[G' \to G_0, \quad G'' \sim \frac{G_0}{\omega \tau_d}\]

Crossover frequency:

\[\omega_c = \frac{1}{\tau_d} \quad \text{where } G'(\omega_c) = G''(\omega_c) - \omega_c \eta_s\]

Loss tangent:

\[\tan \delta = \frac{G''}{G'} = \frac{1}{\omega \tau_d} + \frac{\eta_s}{G_0} \frac{1}{(\omega \tau_d)^2}\]

Cole-Cole Plot ¶

Signature of single-mode Maxwell:

Plot \(G''\) vs \(G'\) (parametric in \(\omega\)). For a single Maxwell mode with \(\eta_s = 0\):

\[\left(G' - \frac{G_0}{2}\right)^2 + (G'')^2 = \left(\frac{G_0}{2}\right)^2\]

This is a perfect semicircle with:

Center at \((G_0/2, 0)\)
Radius \(G_0/2\)
Passes through origin \((0, 0)\) at \(\omega \to 0\)
Passes through \((G_0, 0)\) at \(\omega \to \infty\)

Experimental test:

If wormlike micelles truly follow the Cates model (fast-breaking limit), the Cole-Cole plot should be a perfect semicircle. Deviations indicate:

\(\zeta\) not small enough (intermediate breaking)
Branching (Y-junctions)
Polydispersity in micelle length
Multiple relaxation modes

Cole-Cole Diagnostic: Semicircular Plot ¶

In the fast-breaking regime, the complex modulus takes the approximate form:

\[G^*(\omega) = \frac{G_0}{1 + \sqrt{\tau_b / (2i\omega)}}\]

Plotting \(G''\) vs \(G'\) (Cole-Cole plot) produces a nearly perfect semicircle — this is the diagnostic fingerprint of Cates-type living polymers.

Deviations from the semicircle indicate:

Flattening at high \(G'\): Additional fast modes (Rouse spectrum at high frequency)
Asymmetry: Breakage time distribution (polydisperse scission)
Upturn: Contribution from unentangled chains

The semicircular Cole-Cole plot distinguishes Cates systems from multi-mode Maxwell models, which produce distorted or multi-lobed Cole-Cole curves.

Startup Flow ¶

Step shear rate:

\[\begin{split}\dot{\gamma}(t) = \begin{cases} 0 & t < 0 \\ \dot{\gamma}_0 & t \geq 0 \end{cases}\end{split}\]

ODE system:

\[\frac{dS_{xx}}{dt} = 2 \dot{\gamma}_0 S_{xy} - \frac{1}{\tau_d}(S_{xx} - 1)\]

\[\frac{dS_{xy}}{dt} = \dot{\gamma}_0 S_{yy} - \frac{1}{\tau_d} S_{xy}\]

\[\frac{dS_{yy}}{dt} = -\frac{1}{\tau_d}(S_{yy} - 1)\]

Initial condition:

\[S(0) = I \quad \text{(isotropic state)}\]

Transient shear stress:

\[\sigma_{xy}(t) = G_0 S_{xy}(t) + \eta_s \dot{\gamma}_0\]

Analytical solution (for :math:`eta_s = 0`):

\[\sigma_{xy}(t) = \frac{G_0 \text{Wi}_d}{1 + \text{Wi}_d^2} \left[ 1 - e^{-t/\tau_d} (1 + \text{Wi}_d^2) + \text{Wi}_d^2 e^{-t/\tau_d} \cos\left(\frac{\text{Wi}_d t}{\tau_d}\right) \right]\]

For \(\text{Wi}_d > 1\), the stress exhibits damped oscillations before reaching steady state. There is no stress overshoot in the UCM model (unlike shear-thinning models).

Stress Relaxation ¶

Protocol:

Apply steady shear \(\dot{\gamma}_0\) until steady state
At \(t = 0\), set \(\dot{\gamma} = 0\) and monitor stress decay

Relaxation:

\[\sigma_{xy}(t) = \sigma_{xy}(0) e^{-t/\tau_d}\]

Monoexponential decay with time constant \(\tau_d\).

Creep ¶

Protocol:

Apply constant stress \(\sigma_0\) at \(t = 0\) and measure strain \(\gamma(t)\).

Compliance:

\[J(t) = \frac{\gamma(t)}{\sigma_0}\]

Analytical solution:

\[J(t) = \frac{1}{G_0} \left[ 1 - e^{-t/\tau_d} \right] + \frac{t}{\eta_0}\]

where \(\eta_0 = G_0 \tau_d\).

Limits:

Short time: \(J(t) \sim t/(G_0 \tau_d) = t/\eta_0\) (viscous flow)
Long time: \(J(t) \to 1/G_0 + t/\eta_0\) (steady-state flow)

Large Amplitude Oscillatory Shear (LAOS)¶

Input:

\[\gamma(t) = \gamma_0 \sin(\omega t)\]

Nonlinearity parameter:

\[\text{Wi}_\text{LAOS} = \gamma_0 \omega \tau_d\]

Fourier decomposition:

For \(\gamma_0 \omega \tau_d > 1\), the stress waveform contains odd harmonics:

\[\sigma(t) = \sum_{n \text{ odd}} \sigma_n \sin(n \omega t + \delta_n)\]

Lissajous curves:

Stress vs strain and stress vs strain rate curves become ellipses distorted by nonlinearity.

LAOS and Shear Banding ¶

At large amplitude, the Cates model in LAOS shows signatures of shear banding:

Stress plateau in the Lissajous curve at large \(\gamma_0\) — the stress saturates at \(\sigma_{\max} \approx G_0\) regardless of further strain increase
Secondary loops in the viscous Lissajous (\(\sigma\) vs \(\dot{\gamma}\)) curve indicate transient banding within the oscillation cycle
The elastic Lissajous becomes increasingly rectangular (box-like) as the system alternates between banded and unbanded states within each half-cycle

Parameter Table ¶

Parameter	Symbol	Default	Bounds	Physical Meaning
Plateau modulus	\(G_0\)	100 Pa	(1, 1e6) Pa	Elastic modulus at high frequency
Reptation time	\(\tau_\text{rep}\)	10.0 s	(1e-3, 1e6) s	Relaxation time for unbreakable chain to reptate out of tube
Breaking time	\(\tau_\text{break}\)	0.1 s	(1e-6, 1e4) s	Mean time between scission events for a micelle
Solvent viscosity	\(\eta_s\)	0.0 Pa·s	(0, 1e4) Pa·s	Viscosity of solvent (water, glycerol mixtures, etc.)

Derived Quantities ¶

Quantity	Formula	Meaning
Effective relaxation time	\(\tau_d = \sqrt{\tau_\text{rep} \tau_\text{break}}\)	Observable relaxation time in SAOS
Breaking parameter	\(\zeta = \tau_\text{break}/\tau_\text{rep}\)	Fast-breaking if \(\zeta \ll 1\)
Zero-shear viscosity	\(\eta_0 = G_0 \tau_d + \eta_s\)	Viscosity at \(\dot{\gamma} \to 0\)
Crossover frequency	\(\omega_c = 1/\tau_d\)	Frequency where \(G' = G''\)
Critical shear rate	\(\dot{\gamma}_c = 1/\tau_d\)	Onset of shear thinning

Parameter Interpretation ¶

Plateau Modulus ¶

Related to the mesh size \(\xi\):

\[G_0 \sim \frac{k_B T}{\xi^3}\]

Typical values:

Dilute micelles: 1-10 Pa
Semi-dilute: 10-100 Pa
Concentrated: 100-1000 Pa

Concentration dependence:

\[G_0 \sim c^{2.3}\]

where \(c\) is surfactant concentration.

Reptation Time ¶

Time for a micelle to diffuse curvilinearly along its tube over a distance equal to its contour length \(L\).

Scaling:

\[\tau_\text{rep} \sim \frac{L^3}{D}\]

Length dependence:

Concentration dependence:

\[\tau_\text{rep} \sim c^{1.5}\]

Breaking Time ¶

Mean time between scission events. Related to scission energy barrier:

\[\tau_\text{break} \sim \exp\left(\frac{E_\text{scission}}{k_B T}\right)\]

Temperature dependence (Arrhenius):

\[\tau_\text{break}(T) = \tau_\text{break}^0 \exp\left(\frac{E_\text{scission}}{k_B T}\right)\]

Length dependence:

\[\tau_\text{break} \sim \frac{1}{L}\]

Breaking Parameter ¶

\[\zeta = \frac{\tau_\text{break}}{\tau_\text{rep}}\]

Regimes:

Fast-breaking: \(\zeta \ll 1\) (single-mode Maxwell)
Intermediate: \(\zeta \sim 1\) (multi-mode spectrum)
Unbreakable: \(\zeta \gg 1\) (pure reptation)

Critical value: \(\zeta \lesssim 0.1\) for single-mode approximation.

Effective Relaxation Time ¶

\[\tau_d = \sqrt{\tau_\text{rep} \cdot \tau_\text{break}}\]

Scaling with length:

\[\tau_d \sim L\]

Observable in SAOS crossover frequency: \(\omega_c = 1/\tau_d\).

Decomposition challenge: Measuring \(\tau_d\) alone is not enough. Need additional information from temperature series, concentration series, or scattering.

Validity and Assumptions ¶

Core Assumptions ¶

1. Fast-breaking limit:

\[\zeta = \frac{\tau_\text{break}}{\tau_\text{rep}} \ll 1\]

2. Mean-field: Ignores spatial heterogeneity.

3. Linear chains: No branching (Y-junctions), no ring closure.

4. Reversible scission: Breaking and recombination are reversible.

5. Equilibrium structure: Micelle length distribution at thermodynamic equilibrium.

When the Model Applies ¶

Ideal systems:

CTAB/NaSal
CPyCl/NaSal
Solutions with \(\zeta < 0.1\)

Indicators of validity:

Perfect semicircular Cole-Cole plot
Monoexponential stress relaxation
Single crossover in \(G'\), \(G''\)

When the Model Breaks Down ¶

1. Slow breaking (:math:`zeta gtrsim 1`): Multi-mode spectrum.

2. Branching: Y-junctions change topology.

3. Very concentrated solutions: Gel-like structures.

4. Non-equilibrium: Transient networks.

Regimes and Behavior ¶

Linear Viscoelastic Regime ¶

Condition: \(\gamma_0 \ll 1\) or \(\text{Wi}_d \ll 1\)

Behavior:

Single Maxwell mode with \(\tau_d\)
Perfect semicircular Cole-Cole plot
\(G'(\omega) \sim \omega^2\) at low \(\omega\)

Nonlinear Regime ¶

Condition: \(\text{Wi}_d = \tau_d \dot{\gamma} \sim 1\)

Shear thinning:

\[\eta(\dot{\gamma}) = \frac{G_0 \tau_d}{1 + (\tau_d \dot{\gamma})^2} + \eta_s\]

Flow curve maximum at:

\[\dot{\gamma}_\text{max} = \frac{1}{\tau_d}, \quad \sigma_\text{max} = \frac{G_0}{2}\]

Shear Banding Regime ¶

For \(\text{Wi}_d > 1\), negative slope in flow curve leads to shear banding.

What You Can Learn ¶

From SAOS Data ¶

Effective relaxation time: \(\tau_d = 1/\omega_c\)
Plateau modulus: \(G_0 = \lim_{\omega \to \infty} G'(\omega)\)
Zero-shear viscosity: \(\eta_0 = \lim_{\omega \to 0} G''(\omega)/\omega\)
Breaking parameter estimate from Cole-Cole plot

From Temperature Series ¶

Arrhenius plot of \(\ln \tau_d\) vs \(1/T\) yields scission energy.

From Concentration Series ¶

Scaling:

\[G_0 \sim c^{2.3}, \quad \tau_d \sim c^{0.5}\]

Experimental Design ¶

Primary Technique: SAOS ¶

Why start here: Non-destructive, reveals full linear spectrum.

Protocol:

Frequency sweep: 0.01 to 100 rad/s
Strain amplitude: 0.01 to 0.1 (linear regime)
Cole-Cole plot validation

Secondary: Flow Curves ¶

Why: Test nonlinear predictions, identify shear banding.

Protocol:

Steady shear sweep: 0.001 to 1000 1/s
Wait > 10 \(\tau_d\) for equilibration
Look for stress plateau

Computational Implementation ¶

Numerical Integration ¶

ODE solver for conformation tensor evolution using adaptive Runge-Kutta.

Effective Relaxation Time ¶

Computed internally:

tau_d = jnp.sqrt(tau_rep * tau_break)

Analytical Solutions ¶

Steady shear:

Wi_d = tau_d * gamma_dot
sigma_xy = G_0 * Wi_d / (1 + Wi_d**2) + eta_s * gamma_dot

SAOS:

G_prime = G_0 * (omega * tau_d)**2 / (1 + (omega * tau_d)**2)
G_double_prime = G_0 * (omega * tau_d) / (1 + (omega * tau_d)**2)

Fitting Guidance ¶

Step-by-Step Protocol ¶

Step 1: Fit SAOS to single Maxwell (\(G_0\), \(\tau_d\), \(\eta_s\)).

Step 2: Validate with Cole-Cole plot.

Step 3: Decompose \(\tau_d\) using temperature or concentration series.

Step 4: Fit flow curve (optional validation).

Usage Examples ¶

Basic Fitting ¶

from rheojax.models.tnt import TNTCates
import jax.numpy as jnp

model = TNTCates()
omega = jnp.logspace(-2, 2, 50)
result = model.fit(omega, G_star, test_mode='oscillation')

Failure Mode: Shear Banding ¶

The non-monotonic flow curve in the Cates model leads to constitutive instability:

Homogeneous flow is unstable for \(\dot{\gamma}_c < \dot{\gamma} < \dot{\gamma}_2\)
The material separates into two coexisting shear rate bands
The stress is selected by a plateau criterion (equal areas or diffusive selection)
Flow becomes spatially inhomogeneous — violating the assumption of homogeneous deformation used in point-wise constitutive models

Experimental signatures:

Stress plateau in flow curve (controlled rate) or strain rate jump (controlled stress)
Birefringence banding (optically visible bands in Couette flow)
Velocity profiles from PIV or NMR showing discontinuous shear rate

Note

RheoJAX’s TNTCates model predicts the homogeneous (constitutive) flow curve. For the banded solution, a spatially-resolved (1D) calculation would be needed. The predicted non-monotonic curve should be interpreted as the constitutive relation, with the plateau stress estimated from the stress maximum.

API Reference ¶

class rheojax.models.tnt.TNTCates[source]¶

Bases: TNTBase

Cates living polymer (wormlike micelle) model.

Implements the Cates theory for living polymers with reversible scission. In the fast-breaking limit, the system behaves as a single Maxwell mode with effective relaxation time τ_d = √(τ_rep · τ_break).

The constitutive equation is identical to the basic TNT model (constant breakage, linear stress, upper-convected derivative), but with τ_d replacing the single bond lifetime τ_b:

dS/dt = L·S + S·L^T + (1/τ_d)·I - (1/τ_d)·S σ = G₀·S_xy + η_s·γ̇

Parameters:

G_0 (float, default 1e3) – Plateau modulus (Pa). Network elastic modulus.
tau_rep (float, default 10.0) – Reptation time (s). Curvilinear diffusion time along the tube.
tau_break (float, default 0.1) – Average breaking time (s). Mean time between scission events.
eta_s (float, default 0.0) – Solvent viscosity (Pa·s). Newtonian background contribution.
Derived
-------
tau_d (float) – Effective relaxation time τ_d = √(τ_rep · τ_break)
eta_0 (float) – Zero-shear viscosity η₀ = G₀·τ_d + η_s

parameters¶

Model parameters for fitting

Type:: ParameterSet

fitted_¶

Whether the model has been fitted

Type:: bool

Examples

Basic usage with default parameters:

>>> model = TNTCates()
>>> print(model.tau_d)  # Effective time
1.0

Fit to SAOS data:

>>> omega = np.logspace(-2, 2, 50)
>>> G_star = generate_synthetic_data(omega)
>>> model.fit(omega, G_star, test_mode='oscillation')

Predict flow curve:

>>> gamma_dot = np.logspace(-2, 2, 50)
>>> sigma = model.predict_flow_curve(gamma_dot)

References ¶

Cates (1987) Macromolecules 20:2289-2296 https://doi.org/10.1021/ma00175a038
Cates (1990) J Phys Chem 94:371-375 https://doi.org/10.1021/j100364a063
Cates and Candau (1990) J Phys Condens Matter 2:6869-6892 https://doi.org/10.1088/0953-8984/2/33/001
Turner and Cates (1991) Langmuir 7:1590-1594 https://doi.org/10.1021/la00056a009
Rehage and Hoffmann (1991) Mol Phys 74:933-973 https://doi.org/10.1080/00268979100102721
Berret (2006) Molecular Gels, Springer https://doi.org/10.1007/1-4020-3689-2_20
Fielding (2007) Soft Matter 3:1262-1279 https://doi.org/10.1039/b707980j
Doi and Edwards (1986) Theory of Polymer Dynamics, Oxford ISBN: 978-0198519768