.. _model-tnt-cates: =========================================================== TNT Cates (Living Polymers / Wormlike Micelles) — Handbook =========================================================== .. contents:: Table of Contents :local: :depth: 3 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:** .. list-table:: :header-rows: 1 :widths: 15 15 15 55 * - Symbol - Default - Units - Description * - :math:`G_0` - 100 - Pa - Plateau modulus * - :math:`\tau_\text{rep}` - 10.0 - s - Reptation time * - :math:`\tau_\text{break}` - 0.1 - s - Mean breaking time * - :math:`\eta_s` - 0.0 - Pa·s - Solvent viscosity **Key Equations:** Effective relaxation time (fast-breaking limit): .. math:: \tau_d = \sqrt{\tau_\text{rep} \cdot \tau_\text{break}} Breaking parameter: .. math:: \zeta = \frac{\tau_\text{break}}{\tau_\text{rep}} Zero-shear viscosity: .. math:: \eta_0 = G_0 \tau_d **Test Modes:** All six protocols supported: - OSCILLATION (SAOS): :math:`G'(\omega)`, :math:`G''(\omega)` - FLOW_CURVE: :math:`\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 (:math:`\zeta \ll 1`) - Perfect semicircular Cole-Cole plot (:math:`G''` vs :math:`G'`) - Monoexponential stress relaxation - Non-monotonic flow curve (constitutive instability) - Shear banding for :math:`\text{Wi}_d > 1` - Crossover frequency :math:`\omega_c = 1/\tau_d` Notation Guide ============== .. list-table:: :header-rows: 1 :widths: 15 15 70 * - Symbol - Units - Description * - :math:`G_0` - Pa - Plateau modulus (related to mesh size) * - :math:`\tau_\text{rep}` - s - Reptation time (curvilinear diffusion along tube) * - :math:`\tau_\text{break}` - s - Mean breaking time (Poisson scission) * - :math:`\tau_d` - s - Effective relaxation time = :math:`\sqrt{\tau_\text{rep} \tau_\text{break}}` * - :math:`\zeta` - -- - Breaking parameter = :math:`\tau_\text{break}/\tau_\text{rep}` * - :math:`\eta_s` - Pa·s - Solvent viscosity * - :math:`\eta_0` - Pa·s - Zero-shear viscosity = :math:`G_0 \tau_d` * - :math:`S` - -- - Conformation tensor (end-to-end vector average) * - :math:`\boldsymbol{\kappa}` - :math:`s^{-1}` - Velocity gradient tensor * - :math:`D` - :math:`s^{-1}` - Rate of deformation tensor = :math:`(\boldsymbol{\kappa} + \boldsymbol{\kappa}^T)/2` * - :math:`\boldsymbol{\sigma}` - Pa - Stress tensor * - :math:`\text{Wi}_d` - -- - Weissenberg number = :math:`\tau_d \dot{\gamma}` * - :math:`L` - nm - Mean micelle contour length * - :math:`\xi` - nm - Mesh size (entanglement length scale) * - :math:`\omega_c` - rad/s - Crossover frequency = :math:`1/\tau_d` * - :math:`k_B T` - J - Thermal energy * - :math:`E_\text{scission}` - J/mol - Activation energy for scission Overview ======== Physical Background ------------------- 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: 1. **Scission**: Random breaking at any point along the contour 2. **Recombination**: End-to-end fusion when micelle tips meet 3. **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 ----------------------- 1. **Explains Maxwell behavior in surfactants**: Conventional polymers show broad spectra (many modes); wormlike micelles show single-mode behavior 2. **Predictive power**: Quantitatively explains linear and nonlinear rheology with just 3 parameters 3. **Shear banding mechanism**: First model to predict flow curve instability from microscopic dynamics 4. **Industrial relevance**: Wormlike micelles are used in consumer products (shampoos, detergents), enhanced oil recovery, drag reduction 5. **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: .. math:: \tau_\text{rep} \sim \frac{L^3}{\pi^2 D} where :math:`L` is the contour length and :math:`D` is the curvilinear diffusion coefficient. For permanent polymers, :math:`\tau_\text{rep}` is the dominant relaxation time. The stress relaxes via a spectrum of modes: .. math:: 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 :math:`L` is: .. math:: \tau_\text{break}(L) = \frac{\tau_\text{break}^0}{L/L_0} where :math:`\tau_\text{break}^0` is the breaking time of a reference length :math:`L_0`. **Key insight:** Breaking randomizes the tube position. If :math:`\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:** .. math:: \zeta = \frac{\tau_\text{break}}{\tau_\text{rep}} \ll 1 **Consequence:** The effective stress relaxation becomes **single-mode** with a geometric mean relaxation time: .. math:: \tau_d = \sqrt{\tau_\text{rep} \cdot \tau_\text{break}} **Physical picture:** - Reptation requires diffusion over length :math:`L` - Breaking cuts the micelle into pieces of size approximately :math:`L/2` every :math:`\tau_\text{break}` - The micelle escapes its tube when the diffusion length :math:`\sqrt{D t}` equals the breaking length :math:`\sim \sqrt{D \tau_\text{break}}` - Solving :math:`L \sim \sqrt{D \tau_\text{break}}` with :math:`\tau_\text{rep} \sim L^3/D` gives :math:`\tau_d \sim \sqrt{\tau_\text{rep} \tau_\text{break}}` **Scaling:** .. math:: \tau_d \sim L \quad \text{(linear in length)} compared to :math:`\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 :math:`\tau_b` to reptation time :math:`\tau_{\text{rep}}`: **Fast-breaking regime** (:math:`\tau_b \ll \tau_{\text{rep}}`): The effective relaxation time is the geometric mean: .. math:: \tau_d = \sqrt{\tau_{\text{rep}} \tau_b} The relaxation modulus follows a stretched exponential: .. math:: 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** (:math:`\tau_b \gg \tau_{\text{rep}}`): Standard reptation dominates: .. math:: 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: .. math:: k_\text{break} n_\text{micelles} = k_\text{recomb} n_\text{ends}^2 where: - :math:`k_\text{break}` is the scission rate constant - :math:`k_\text{recomb}` is the recombination rate constant - :math:`n_\text{micelles}` is the number of micelles - :math:`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 :math:`S` represents the average end-to-end vector orientation. In the tube model: .. math:: S = \langle \mathbf{u} \otimes \mathbf{u} \rangle where :math:`\mathbf{u}` is the unit tangent vector along the tube. The stress is: .. math:: \boldsymbol{\sigma} = G_0 (S - I) + 2 \eta_s D where :math:`G_0 \sim k_B T / \xi^3` is the plateau modulus (:math:`\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 :math:`\tau_d`: .. math:: \frac{DS}{Dt} - \boldsymbol{\kappa} \cdot S - S \cdot \boldsymbol{\kappa}^T = -\frac{1}{\tau_d}(S - I) where: - :math:`\frac{D}{Dt}` is the material derivative - :math:`\boldsymbol{\kappa} = \nabla \mathbf{v}` is the velocity gradient tensor - :math:`I` is the identity tensor **Expanded form:** .. math:: \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 (:math:`\nabla S = 0`): .. math:: \frac{dS}{dt} = \boldsymbol{\kappa} \cdot S + S \cdot \boldsymbol{\kappa}^T - \frac{1}{\tau_d}(S - I) Stress Tensor ------------- .. math:: \boldsymbol{\sigma} = G_0 (S - I) + 2 \eta_s D where: - :math:`G_0` is the plateau modulus - :math:`\eta_s` is the solvent viscosity - :math:`D = (\boldsymbol{\kappa} + \boldsymbol{\kappa}^T)/2` is the rate of deformation tensor **Total stress:** .. math:: \boldsymbol{\sigma}_\text{total} = -p I + \boldsymbol{\sigma} where :math:`p` is the pressure (isotropic part). Effective Relaxation Time -------------------------- **Computed internally:** .. math:: \tau_d = \sqrt{\tau_\text{rep} \cdot \tau_\text{break}} This is **not** a fitted parameter. The model fits :math:`\tau_\text{rep}` and :math:`\tau_\text{break}` separately, and :math:`\tau_d` is derived. **Physical interpretation:** - :math:`\tau_d` is the **observable** relaxation time in SAOS (crossover frequency :math:`\omega_c = 1/\tau_d`) - :math:`\tau_\text{rep}` and :math:`\tau_\text{break}` are **microscopic** timescales - Requires temperature-dependent or concentration-dependent data to separate :math:`\tau_\text{rep}` and :math:`\tau_\text{break}` Steady Shear Flow ----------------- **Velocity field:** .. math:: \mathbf{v} = (\dot{\gamma} y, 0, 0) **Velocity gradient:** .. math:: \boldsymbol{\kappa} = \begin{pmatrix} 0 & \dot{\gamma} & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix} **Steady-state solution (analytical):** Weissenberg number: .. math:: \text{Wi}_d = \tau_d \dot{\gamma} Shear stress: .. math:: \sigma_{xy} = \frac{G_0 \text{Wi}_d}{1 + \text{Wi}_d^2} + \eta_s \dot{\gamma} Normal stress differences: .. math:: N_1 = \sigma_{xx} - \sigma_{yy} = \frac{2 G_0 \text{Wi}_d^2}{1 + \text{Wi}_d^2} .. math:: N_2 = 0 \quad \text{(UCM model)} **Flow curve instability:** The shear stress is **non-monotonic**: it increases for :math:`\text{Wi}_d < 1`, reaches a maximum at :math:`\text{Wi}_d = 1`, then decreases for :math:`\text{Wi}_d > 1`. Maximum shear stress: .. math:: \sigma_{xy}^\text{max} = \frac{G_0}{2} + \eta_s \dot{\gamma}_\text{max} where :math:`\dot{\gamma}_\text{max} = 1/\tau_d`. **Constitutive instability:** For :math:`\text{Wi}_d > 1`, the flow curve has **negative slope** :math:`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**: .. math:: \frac{d\sigma}{d\dot{\gamma}} < 0 \quad \text{for} \quad \dot{\gamma} > \dot{\gamma}_c **Stress maximum:** .. math:: \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 :math:`\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:** .. math:: \gamma(t) = \gamma_0 \sin(\omega t) **Complex modulus:** .. math:: G^*(\omega) = G'(\omega) + i G''(\omega) **Storage modulus:** .. math:: G'(\omega) = \frac{G_0 (\omega \tau_d)^2}{1 + (\omega \tau_d)^2} **Loss modulus:** .. math:: G''(\omega) = \frac{G_0 (\omega \tau_d)}{1 + (\omega \tau_d)^2} + \omega \eta_s **Limiting behavior:** Low frequency (:math:`\omega \tau_d \ll 1`): .. math:: G' \sim G_0 \omega^2 \tau_d^2, \quad G'' \sim G_0 \omega \tau_d + \omega \eta_s High frequency (:math:`\omega \tau_d \gg 1`): .. math:: G' \to G_0, \quad G'' \sim \frac{G_0}{\omega \tau_d} **Crossover frequency:** .. math:: \omega_c = \frac{1}{\tau_d} \quad \text{where } G'(\omega_c) = G''(\omega_c) - \omega_c \eta_s **Loss tangent:** .. math:: \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 :math:`G''` vs :math:`G'` (parametric in :math:`\omega`). For a single Maxwell mode with :math:`\eta_s = 0`: .. math:: \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 :math:`(G_0/2, 0)` - Radius :math:`G_0/2` - Passes through origin :math:`(0, 0)` at :math:`\omega \to 0` - Passes through :math:`(G_0, 0)` at :math:`\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: - :math:`\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: .. math:: G^*(\omega) = \frac{G_0}{1 + \sqrt{\tau_b / (2i\omega)}} Plotting :math:`G''` vs :math:`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** :math:`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:** .. math:: \dot{\gamma}(t) = \begin{cases} 0 & t < 0 \\ \dot{\gamma}_0 & t \geq 0 \end{cases} **ODE system:** .. math:: \frac{dS_{xx}}{dt} = 2 \dot{\gamma}_0 S_{xy} - \frac{1}{\tau_d}(S_{xx} - 1) .. math:: \frac{dS_{xy}}{dt} = \dot{\gamma}_0 S_{yy} - \frac{1}{\tau_d} S_{xy} .. math:: \frac{dS_{yy}}{dt} = -\frac{1}{\tau_d}(S_{yy} - 1) **Initial condition:** .. math:: S(0) = I \quad \text{(isotropic state)} **Transient shear stress:** .. math:: \sigma_{xy}(t) = G_0 S_{xy}(t) + \eta_s \dot{\gamma}_0 **Analytical solution (for :math:`\eta_s = 0`):** .. math:: \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 :math:`\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:** 1. Apply steady shear :math:`\dot{\gamma}_0` until steady state 2. At :math:`t = 0`, set :math:`\dot{\gamma} = 0` and monitor stress decay **Relaxation:** .. math:: \sigma_{xy}(t) = \sigma_{xy}(0) e^{-t/\tau_d} **Monoexponential decay** with time constant :math:`\tau_d`. Creep ----- **Protocol:** Apply constant stress :math:`\sigma_0` at :math:`t = 0` and measure strain :math:`\gamma(t)`. **Compliance:** .. math:: J(t) = \frac{\gamma(t)}{\sigma_0} **Analytical solution:** .. math:: J(t) = \frac{1}{G_0} \left[ 1 - e^{-t/\tau_d} \right] + \frac{t}{\eta_0} where :math:`\eta_0 = G_0 \tau_d`. **Limits:** - Short time: :math:`J(t) \sim t/(G_0 \tau_d) = t/\eta_0` (viscous flow) - Long time: :math:`J(t) \to 1/G_0 + t/\eta_0` (steady-state flow) Large Amplitude Oscillatory Shear (LAOS) ----------------------------------------- **Input:** .. math:: \gamma(t) = \gamma_0 \sin(\omega t) **Nonlinearity parameter:** .. math:: \text{Wi}_\text{LAOS} = \gamma_0 \omega \tau_d **Fourier decomposition:** For :math:`\gamma_0 \omega \tau_d > 1`, the stress waveform contains odd harmonics: .. math:: \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 :math:`\gamma_0` — the stress saturates at :math:`\sigma_{\max} \approx G_0` regardless of further strain increase - **Secondary loops** in the viscous Lissajous (:math:`\sigma` vs :math:`\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 =============== .. list-table:: :header-rows: 1 :widths: 15 15 20 15 35 * - Parameter - Symbol - Default - Bounds - Physical Meaning * - Plateau modulus - :math:`G_0` - 100 Pa - (1, 1e6) Pa - Elastic modulus at high frequency * - Reptation time - :math:`\tau_\text{rep}` - 10.0 s - (1e-3, 1e6) s - Relaxation time for unbreakable chain to reptate out of tube * - Breaking time - :math:`\tau_\text{break}` - 0.1 s - (1e-6, 1e4) s - Mean time between scission events for a micelle * - Solvent viscosity - :math:`\eta_s` - 0.0 Pa·s - (0, 1e4) Pa·s - Viscosity of solvent (water, glycerol mixtures, etc.) Derived Quantities ------------------ .. list-table:: :header-rows: 1 :widths: 25 20 55 * - Quantity - Formula - Meaning * - Effective relaxation time - :math:`\tau_d = \sqrt{\tau_\text{rep} \tau_\text{break}}` - Observable relaxation time in SAOS * - Breaking parameter - :math:`\zeta = \tau_\text{break}/\tau_\text{rep}` - Fast-breaking if :math:`\zeta \ll 1` * - Zero-shear viscosity - :math:`\eta_0 = G_0 \tau_d + \eta_s` - Viscosity at :math:`\dot{\gamma} \to 0` * - Crossover frequency - :math:`\omega_c = 1/\tau_d` - Frequency where :math:`G' = G''` * - Critical shear rate - :math:`\dot{\gamma}_c = 1/\tau_d` - Onset of shear thinning Parameter Interpretation ======================== Plateau Modulus --------------- Related to the mesh size :math:`\xi`: .. math:: 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: .. math:: G_0 \sim c^{2.3} where :math:`c` is surfactant concentration. Reptation Time -------------- Time for a micelle to diffuse curvilinearly along its tube over a distance equal to its contour length :math:`L`. Scaling: .. math:: \tau_\text{rep} \sim \frac{L^3}{D} Length dependence: .. math:` \tau_\text{rep} \sim L^3 Concentration dependence: .. math:: \tau_\text{rep} \sim c^{1.5} Breaking Time ------------- Mean time between scission events. Related to scission energy barrier: .. math:: \tau_\text{break} \sim \exp\left(\frac{E_\text{scission}}{k_B T}\right) Temperature dependence (Arrhenius): .. math:: \tau_\text{break}(T) = \tau_\text{break}^0 \exp\left(\frac{E_\text{scission}}{k_B T}\right) Length dependence: .. math:: \tau_\text{break} \sim \frac{1}{L} Breaking Parameter ------------------ .. math:: \zeta = \frac{\tau_\text{break}}{\tau_\text{rep}} Regimes: - Fast-breaking: :math:`\zeta \ll 1` (single-mode Maxwell) - Intermediate: :math:`\zeta \sim 1` (multi-mode spectrum) - Unbreakable: :math:`\zeta \gg 1` (pure reptation) Critical value: :math:`\zeta \lesssim 0.1` for single-mode approximation. Effective Relaxation Time -------------------------- .. math:: \tau_d = \sqrt{\tau_\text{rep} \cdot \tau_\text{break}} Scaling with length: .. math:: \tau_d \sim L Observable in SAOS crossover frequency: :math:`\omega_c = 1/\tau_d`. Decomposition challenge: Measuring :math:`\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:** .. math:: \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 :math:`\zeta < 0.1` Indicators of validity: - Perfect semicircular Cole-Cole plot - Monoexponential stress relaxation - Single crossover in :math:`G'`, :math:`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: :math:`\gamma_0 \ll 1` or :math:`\text{Wi}_d \ll 1` Behavior: - Single Maxwell mode with :math:`\tau_d` - Perfect semicircular Cole-Cole plot - :math:`G'(\omega) \sim \omega^2` at low :math:`\omega` Nonlinear Regime ---------------- Condition: :math:`\text{Wi}_d = \tau_d \dot{\gamma} \sim 1` Shear thinning: .. math:: \eta(\dot{\gamma}) = \frac{G_0 \tau_d}{1 + (\tau_d \dot{\gamma})^2} + \eta_s Flow curve maximum at: .. math:: \dot{\gamma}_\text{max} = \frac{1}{\tau_d}, \quad \sigma_\text{max} = \frac{G_0}{2} Shear Banding Regime -------------------- For :math:`\text{Wi}_d > 1`, negative slope in flow curve leads to shear banding. What You Can Learn ================== From SAOS Data -------------- 1. Effective relaxation time: :math:`\tau_d = 1/\omega_c` 2. Plateau modulus: :math:`G_0 = \lim_{\omega \to \infty} G'(\omega)` 3. Zero-shear viscosity: :math:`\eta_0 = \lim_{\omega \to 0} G''(\omega)/\omega` 4. Breaking parameter estimate from Cole-Cole plot From Temperature Series ----------------------- Arrhenius plot of :math:`\ln \tau_d` vs :math:`1/T` yields scission energy. From Concentration Series -------------------------- Scaling: .. math:: 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:** 1. Frequency sweep: 0.01 to 100 rad/s 2. Strain amplitude: 0.01 to 0.1 (linear regime) 3. Cole-Cole plot validation Secondary: Flow Curves ---------------------- **Why:** Test nonlinear predictions, identify shear banding. **Protocol:** 1. Steady shear sweep: 0.001 to 1000 1/s 2. Wait > 10 :math:`\tau_d` for equilibration 3. Look for stress plateau Computational Implementation ============================ Numerical Integration --------------------- ODE solver for conformation tensor evolution using adaptive Runge-Kutta. Effective Relaxation Time -------------------------- Computed internally: .. code-block:: python tau_d = jnp.sqrt(tau_rep * tau_break) Analytical Solutions -------------------- **Steady shear:** .. code-block:: python Wi_d = tau_d * gamma_dot sigma_xy = G_0 * Wi_d / (1 + Wi_d**2) + eta_s * gamma_dot **SAOS:** .. code-block:: python 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 (:math:`G_0`, :math:`\tau_d`, :math:`\eta_s`). **Step 2:** Validate with Cole-Cole plot. **Step 3:** Decompose :math:`\tau_d` using temperature or concentration series. **Step 4:** Fit flow curve (optional validation). Usage Examples ============== Basic Fitting ------------- .. code-block:: python 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 :math:`\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. See Also -------- **TNT Shared Reference:** - :doc:`tnt_protocols` — Full protocol equations and numerical methods - :doc:`tnt_knowledge_extraction` — Model identification and fitting guidance **TNT Base Model:** - :ref:`model-tnt-tanaka-edwards` — Base model (constant breakage) **Related TNT Variants:** - :ref:`model-tnt-loop-bridge` — Alternative two-species model for micellar networks - :ref:`model-tnt-multi-species` — Multi-mode generalization for broad relaxation - :ref:`model-tnt-bell` — Force-dependent breakage (complementary thinning mechanism) API Reference ============= .. autoclass:: rheojax.models.tnt.TNTCates :members: :undoc-members: :show-inheritance: References ========== 1. Cates (1987) Macromolecules 20:2289-2296 https://doi.org/10.1021/ma00175a038 2. Cates (1990) J Phys Chem 94:371-375 https://doi.org/10.1021/j100364a063 3. Cates and Candau (1990) J Phys Condens Matter 2:6869-6892 https://doi.org/10.1088/0953-8984/2/33/001 4. Turner and Cates (1991) Langmuir 7:1590-1594 https://doi.org/10.1021/la00056a009 5. Rehage and Hoffmann (1991) Mol Phys 74:933-973 https://doi.org/10.1080/00268979100102721 6. Berret (2006) Molecular Gels, Springer https://doi.org/10.1007/1-4020-3689-2_20 7. Fielding (2007) Soft Matter 3:1262-1279 https://doi.org/10.1039/b707980j 8. Doi and Edwards (1986) Theory of Polymer Dynamics, Oxford ISBN: 978-0198519768