TNT Bell (Force-Dependent Breakage) — Handbook¶

Quick Reference ¶

Use when:

Associating polymer networks where bond breakage accelerates with chain stretch or applied force
Systems exhibiting shear thinning due to force-sensitive crosslinks
Materials with slip-bond behavior (force accelerates dissociation)
Networks where chain tension controls dissociation kinetics
Biological gels showing force-dependent mechanics (fibrin, collagen)
Vitrimers or dynamic covalent networks with mechanically activated exchange

Parameters:

4 base parameters: \(G\) (modulus, Pa), \(\tau_b\) (bond lifetime, s), \(\nu\) (force sensitivity, dimensionless), \(\eta_s\) (solvent viscosity, Pa·s)
Typical ranges: \(G \in [1, 10^8]\) Pa, \(\tau_b \in [10^{-6}, 10^4]\) s, \(\nu \in [0.01, 20]\), \(\eta_s \in [0, 10^4]\) Pa·s

Key equation:

Bell breakage rate with exponential force dependence:

\[k_{\text{off}}(\mathbf{S}) = \frac{1}{\tau_b} \exp\left[\nu \left(\text{tr}(\mathbf{S}) - 3\right)\right]\]

where \(\text{tr}(\mathbf{S}) - 3\) measures mean chain stretch above equilibrium.

Test modes:

All 6 protocols supported: FLOW_CURVE, STARTUP, OSCILLATION, RELAXATION, CREEP, LAOS

Material examples:

Fibrin networks (blood clots with force-sensitive fibrinogen cross-linking)
Collagen gels (mechanosensitive extracellular matrix)
Vitrimers (exchangeable covalent networks with force-activated exchange)
Catch-bond protein assemblies (selectin-ligand, integrin-ECM)
Force-sensitive supramolecular hydrogels (hydrogen bonds, metal-ligand)
Telechelic associating polymers under strong flow

Key characteristics:

Shear thinning viscosity \(\eta \sim \dot{\gamma}^{-n}\) with \(n\) controlled by \(\nu\)
Strain-rate-dependent stress overshoot in startup (Bell signature)
Stretch-dependent relaxation (non-exponential decay)
Linear SAOS identical to Maxwell model (Bell effect is second-order in strain)
Flow curve requires numerical root-finding (no closed-form solution)

Notation Guide ¶

Mathematical Symbols¶
Symbol	Units	Description
\(\mathbf{S}\)	dimensionless	Conformation tensor (mean square end-to-end distance normalized by equilibrium)
\(G\)	Pa	Network elastic modulus (plateau modulus)
\(\tau_b\)	s	Equilibrium bond lifetime (zero-force reference)
\(\nu\)	dimensionless	Force sensitivity parameter (Bell exponent)
\(\eta_s\)	Pa·s	Solvent viscosity contribution
\(k_{\text{off}}(\mathbf{S})\)	\(s^{-1}\)	Force-dependent bond breakage rate
\(\text{tr}(\mathbf{S})\)	dimensionless	Trace of conformation tensor (3 times mean square stretch)
\(\boldsymbol{\kappa}\)	\(s^{-1}\)	Velocity gradient tensor
\(\mathbf{D}\)	\(s^{-1}\)	Rate of deformation tensor (symmetric part of \(\boldsymbol{\kappa}\))
\(\boldsymbol{\sigma}\)	Pa	Total stress tensor
\(\boldsymbol{\sigma}_p\)	Pa	Polymeric (network) stress: \(G(\mathbf{S} - \mathbf{I})\)
\(\mathrm{Wi}\)	dimensionless	Weissenberg number: \(\tau_b \dot{\gamma}\) (startup) or \(\tau_b \omega\) (SAOS)
\(F_{\text{chain}}\)	\(k_B T / a\)	Effective chain force (entropic spring tension)
\(N_1\)	Pa	First normal stress difference: \(\sigma_{xx} - \sigma_{yy}\)
\(d_b\)	nm	Barrier distance in bond potential (transition state location)
\(k_0\)	\(s^{-1}\)	Zero-force breakage rate: \(1/\tau_b\)
\(E_a\)	J	Activation energy for bond dissociation (without force)

The Bell model, introduced by G.I. Bell in 1978 for cell adhesion kinetics, describes how applied force accelerates bond dissociation by tilting the energy landscape. Originally developed for single-molecule force spectroscopy, it has been adapted to continuum polymer network models where chain tension (measured by the conformation tensor \(\mathbf{S}\)) controls the breakage rate.

The Tanaka-Edwards transient network theory (1992) provides the base framework with constant breakage rate \(1/\tau_b\). The Bell variant extends this by making the breakage rate exponentially dependent on chain stretch:

\[k_{\text{off}}(\mathbf{S}) = \frac{1}{\tau_b} \exp\left[\nu \left(\text{tr}(\mathbf{S}) - 3\right)\right]\]

This modification introduces shear thinning and stretch-dependent dynamics absent in the base model.

Physical Picture ¶

In a transient network:

Chains connect junction points (physical or chemical crosslinks)
Stretch accumulates as flow deforms the network affinely
Tension increases the probability of bond rupture exponentially
Breakage accelerates with chain force (slip-bond mechanism)
Reformation occurs into the equilibrium (unstretched) state \(\mathbf{I}\)

The Bell mechanism captures the intuition that pulling on a bond makes it break faster. The force sensitivity \(\nu\) quantifies how much the activation barrier is lowered by chain tension.

Exponential Sensitivity ¶

The Bell model assumes Kramers’ escape rate:

\[k_{\text{off}}(F) = k_0 \exp\left(\frac{F \cdot d_b}{k_B T}\right)\]

where:

\(F\) is the applied force on the bond
\(d_b\) is the distance to the transition state
\(k_B T\) is thermal energy

In the continuum coarse-graining:

Force proxy: \(F \propto (\text{tr}(\mathbf{S}) - 3)\) measures mean chain stretch
Bell exponent: \(\nu = d_b / a\) where \(a\) is the Kuhn length
Typical values: \(\nu \sim 0.5-5\) for physical bonds, \(\nu \sim 5-20\) for weak non-covalent interactions

Distinguished from Constant-Breakage ¶

Comparison with Tanaka-Edwards Base Model¶
Feature	Tanaka-Edwards (Constant)	Bell (Force-Dependent)
Breakage rate	\(k_{\text{off}} = 1/\tau_b\)	\(k_{\text{off}} = (1/\tau_b) \exp[\nu(\text{tr}\mathbf{S} - 3)]\)
Steady shear viscosity	\(\eta \sim \text{const}\) (Newtonian)	\(\eta \sim \dot{\gamma}^{-n}\) (shear thinning)
Startup overshoot	Strain independent: \(\gamma_{\text{peak}} \sim 1\)	Decreases with \(\nu\): \(\gamma_{\text{peak}} \sim 1/\sqrt{\nu}\)
Relaxation	Single exponential	Stretched exponential (faster initial decay)
SAOS (linear)	Maxwell	Maxwell (identical in linear regime)
Parameter coupling	Independent \(G, \tau_b\)	Strong \(\nu\)-\(\tau_b\) correlation

Physical Foundations ¶

Kramers’ Escape Problem ¶

A bond is modeled as a particle in a potential well \(U(x)\) with barrier at \(x = d_b\):

\[U(x) = U_0 - F \cdot x\]

Applied force \(F\) tilts the potential, lowering the effective barrier:

\[\Delta U_{\text{eff}} = \Delta U_0 - F \cdot d_b\]

The Arrhenius escape rate becomes:

\[k_{\text{off}}(F) = k_0 \exp\left(-\frac{\Delta U_{\text{eff}}}{k_B T}\right) = k_0 \exp\left(\frac{F \cdot d_b}{k_B T}\right)\]

This is the Bell model for single-molecule kinetics (dynamic force spectroscopy).

Coarse-Graining to Continuum ¶

For a network of \(N\) chains with mean conformation tensor \(\mathbf{S}\):

Entropic force: \(F_{\text{chain}} \sim k_B T (\langle R^2 \rangle / R_0^2 - 1) \sim \text{tr}(\mathbf{S}) - 3\)
Mean-field assumption: All chains experience average stretch
Effective force: Replace \(F\) with \((\text{tr}(\mathbf{S}) - 3)\)
Bell rate:

\[k_{\text{off}}(\mathbf{S}) = \frac{1}{\tau_b} \exp\left[\nu \left(\text{tr}(\mathbf{S}) - 3\right)\right]\]

where \(\nu = d_b / a\) is dimensionless (barrier distance over Kuhn length).

Mechanical Analog ¶

The Bell TNT model can be visualized as:

┌─────────┬────────────────────┐
│  Spring │  Force-Dependent   │
│   (G)   │  Dashpot (η_eff(S))│
└─────────┴────────────────────┘
     ∥
Solvent viscosity (η_s)

where the dashpot viscosity \(\eta_{\text{eff}} \sim G \tau_b \exp[-\nu(\text{tr}\mathbf{S} - 3)]\) decreases with chain stretch (shear thinning).

Material Examples ¶

Biological networks:

Fibrin (blood clots): \(\nu \sim 1-3\), force-sensitive cross-linking by Factor XIII
Collagen: Mechanosensitive cross-links in extracellular matrix
Actin networks: Some binding proteins exhibit force-dependent unbinding

Synthetic polymers:

Vitrimers: Exchangeable bonds with force-dependent exchange kinetics
Supramolecular hydrogels: Hydrogen bonds, \(\nu \sim 0.5-2\)
Metal-ligand coordination: \(\nu \sim 2-10\) depending on coordination number
Telechelic polymers: Associating end-groups with force-sensitive dissociation

Governing Equations ¶

Conformation Tensor Evolution ¶

The conformation tensor \(\mathbf{S}\) evolves under:

\[\frac{D\mathbf{S}}{Dt} = \boldsymbol{\kappa} \cdot \mathbf{S} + \mathbf{S} \cdot \boldsymbol{\kappa}^T - k_{\text{off}}(\mathbf{S}) \left(\mathbf{S} - \mathbf{I}\right)\]

where:

\(\frac{D}{Dt}\) is the material derivative
\(\boldsymbol{\kappa} = \nabla \mathbf{v}\) is the velocity gradient
\(k_{\text{off}}(\mathbf{S}) = \frac{1}{\tau_b} \exp\left[\nu \left(\text{tr}(\mathbf{S}) - 3\right)\right]\)

Physical interpretation:

First two terms: Affine chain stretch by flow
Last term: Relaxation by bond breakage (rate depends exponentially on stretch)

Total Stress ¶

\[\boldsymbol{\sigma} = G \left(\mathbf{S} - \mathbf{I}\right) + 2 \eta_s \mathbf{D}\]

where:

\(G(\mathbf{S} - \mathbf{I})\) is the polymeric (network) stress
\(2\eta_s \mathbf{D}\) is the Newtonian solvent contribution

Steady Shear Flow ¶

For simple shear \(\boldsymbol{\kappa} = \dot{\gamma} \mathbf{e}_x \otimes \mathbf{e}_y\), at steady state \(\frac{D\mathbf{S}}{Dt} = 0\):

\[\begin{split}\begin{aligned} 0 &= 2 \dot{\gamma} S_{xy} - k_{\text{off}}(\mathbf{S}) (S_{xx} - 1) \\ 0 &= \dot{\gamma} S_{yy} - k_{\text{off}}(\mathbf{S}) (S_{xy}) \\ 0 &= -2 \dot{\gamma} S_{xy} - k_{\text{off}}(\mathbf{S}) (S_{yy} - 1) \end{aligned}\end{split}\]

where \(k_{\text{off}}(\mathbf{S}) = \frac{1}{\tau_b} \exp[\nu(S_{xx} + S_{yy} + S_{zz} - 3)]\).

Critical difference from constant breakage: This is a nonlinear implicit system with no closed-form solution. Numerical root-finding is required.

Flow Curve ¶

The shear stress at steady state is:

\[\sigma_{xy} = G S_{xy} + \eta_s \dot{\gamma}\]

where \(S_{xy}\) is obtained by solving the coupled nonlinear system above.

Typical behavior:

Low rates (\(\mathrm{Wi} \ll 1\)): \(\sigma \sim G \tau_b \dot{\gamma}\) (Newtonian)
High rates (\(\mathrm{Wi} \gg 1\)): \(\sigma \sim G \tau_b^{1/(1+\nu)} \dot{\gamma}^{\nu/(1+\nu)}\) (shear thinning)

The viscosity follows approximate power-law:

\[\eta(\dot{\gamma}) \sim \eta_0 \left(1 + (\tau_b \dot{\gamma})^{1+\nu}\right)^{-\nu/(1+\nu)}\]

Small Amplitude Oscillatory Shear (SAOS)¶

Linearizing around \(\mathbf{S} = \mathbf{I} + \delta \mathbf{S}\) with \(\gamma(t) = \gamma_0 e^{i\omega t}\):

\[k_{\text{off}}(\mathbf{I} + \delta \mathbf{S}) \approx \frac{1}{\tau_b} \left[1 + \nu \text{tr}(\delta \mathbf{S}) + O(\delta \mathbf{S}^2)\right]\]

The linear response gives standard Maxwell behavior:

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

The Bell correction enters only at second order (\(\gamma_0^2\)), so SAOS cannot distinguish Bell from constant breakage in the linear regime.

Startup of Steady Shear ¶

Starting from \(\mathbf{S}(0) = \mathbf{I}\), impose \(\dot{\gamma}\) at \(t > 0\). The ODE system must be integrated numerically:

\[\frac{d\mathbf{S}}{dt} = \boldsymbol{\kappa} \cdot \mathbf{S} + \mathbf{S} \cdot \boldsymbol{\kappa}^T - \frac{1}{\tau_b} \exp[\nu(\text{tr}\mathbf{S} - 3)] (\mathbf{S} - \mathbf{I})\]

Key predictions:

Stress overshoot at strain \(\gamma_{\text{peak}} \sim 1 / \sqrt{\nu}\) (decreases with force sensitivity)
Peak stress scales as \(\sigma_{\text{peak}} \sim G \mathrm{Wi}^{1/(1+\nu)}\)
Approach to steady state faster than constant breakage (stretch accelerates relaxation)

Overshoot Strain Scaling ¶

The peak strain in startup scales inversely with force sensitivity:

\[\gamma_{\text{peak}} \approx \frac{1}{\sqrt{\nu}}\]

This scaling arises because higher \(\nu\) means bonds break earlier in the startup transient (less strain needed to trigger accelerated breakage). At the overshoot, the breakage rate first exceeds the affine stretching rate.

Effective Relaxation Time ¶

Under finite strain, the relaxation time becomes strain-dependent:

\[\tau_{\text{eff}}(\gamma_0) = \tau_b \exp\!\left(-\nu\sqrt{1 + \gamma_0^2}\right)\]

Large strains relax much faster than small strains (Type I damping behavior). The damping function for step-strain relaxation is:

\[h(\gamma_0) = \exp\!\left(-\frac{t}{\tau_b}\left[\exp\!\left(\nu\sqrt{1 + \gamma_0^2}\right) - 1\right]\right)\]

This strain-dependent relaxation is a key signature of Bell kinetics: plotting the relaxation modulus \(G(t, \gamma_0)\) at multiple step strains reveals time-strain separability failure at large \(\gamma_0\).

Stress Relaxation ¶

After step strain \(\gamma_0\), with \(\boldsymbol{\kappa} = 0\) for \(t > 0\):

\[\frac{d\mathbf{S}}{dt} = -\frac{1}{\tau_b} \exp[\nu(\text{tr}\mathbf{S} - 3)] (\mathbf{S} - \mathbf{I})\]

For small strains (\(\gamma_0 \ll 1\)): Single exponential \(\sigma(t) \sim G \gamma_0 e^{-t/\tau_b}\) (Bell term negligible)

For large strains (\(\gamma_0 > 1\)): Stretched exponential or faster-than-exponential decay due to initial high \(k_{\text{off}}\).

Creep ¶

Under constant stress \(\sigma_0\), solve the 5-state ODE system:

\[\begin{split}\begin{aligned} \frac{d\mathbf{S}}{dt} &= \dot{\gamma}(t) \left(\mathbf{e}_x \otimes \mathbf{e}_y + \mathbf{e}_y \otimes \mathbf{e}_x\right) \cdot \mathbf{S} + \ldots - k_{\text{off}}(\mathbf{S})(\mathbf{S} - \mathbf{I}) \\ \frac{d\gamma}{dt} &= \dot{\gamma}(t) \end{aligned}\end{split}\]

where \(\dot{\gamma}(t)\) is determined implicitly by \(\sigma_{xy} = \sigma_0\).

Large Amplitude Oscillatory Shear (LAOS)¶

Impose \(\gamma(t) = \gamma_0 \sin(\omega t)\), integrate the full nonlinear ODE:

\[\frac{d\mathbf{S}}{dt} = \gamma_0 \omega \cos(\omega t) (\mathbf{e}_x \otimes \mathbf{e}_y + \mathbf{e}_y \otimes \mathbf{e}_x) \cdot \mathbf{S} - k_{\text{off}}(\mathbf{S})(\mathbf{S} - \mathbf{I})\]

Extract stress \(\sigma(t) = G S_{xy}(t)\) and perform Fourier decomposition to get higher harmonics:

\[\sigma(t) = \sum_{n=1,3,5,\ldots} \left[G_n'(\omega, \gamma_0) \sin(n\omega t) + G_n''(\omega, \gamma_0) \cos(n\omega t)\right]\]

Bell effect: Nonlinearity from \(\exp[\nu(\text{tr}\mathbf{S} - 3)]\) generates odd harmonics (\(n = 3, 5, 7, \ldots\)).

LAOS Mechanism: Oscillating Breakage Rate ¶

In LAOS, the breakage rate oscillates with the imposed strain:

\[\beta(t) \approx \beta_0 \exp\!\left(\nu \gamma_0 |\sin \omega t|\right)\]

This produces strong odd harmonics due to “clipping” of the stress waveform — at peak strain, the breakage rate spikes and the stress drops relative to the Hookean prediction. The Lissajous curve develops a characteristic intra-cycle softening (stress drops faster at peak strain).

MAOS Third Harmonic Scaling ¶

In the medium-amplitude regime (MAOS, \(\gamma_0 \sim 0.1-1\)), perturbation analysis gives the third-harmonic intensity scaling:

\[I_{3/1} \approx C(\omega, \beta) \, \gamma_0^2\]

where \(C(\omega, \beta)\) is a function of frequency and equilibrium breakage rate. The \(\gamma_0^2\) scaling (intrinsic nonlinearity) is diagnostic: FENE models give a different scaling exponent at the same amplitude range.

Parameter Table ¶

TNT Bell Model Parameters¶
Parameter	Symbol	Default	Bounds	Units	Description
Network modulus	\(G\)	1000.0	\([1, 10^8]\)	Pa	Elastic modulus of the network (plateau modulus)
Bond lifetime	\(\tau_b\)	1.0	\([10^{-6}, 10^4]\)	s	Equilibrium bond lifetime at zero force
Force sensitivity	\(\nu\)	1.0	\([0.01, 20]\)	dimensionless	Bell exponent (barrier distance / Kuhn length)
Solvent viscosity	\(\eta_s\)	0.0	\([0, 10^4]\)	Pa·s	Newtonian solvent contribution (high-frequency viscosity)

Parameter dependencies:

\(\tau_b\) and \(\nu\) are strongly correlated (both affect thinning)
\(G\) and \(\tau_b\) set the zero-shear viscosity: \(\eta_0 = G \tau_b\)
\(\eta_s\) becomes important at \(\dot{\gamma} \gg 1/\tau_b\)

Parameter Interpretation ¶

Force Sensitivity \(\nu\)¶

Physical meaning: Dimensionless ratio \(\nu = d_b / a\) where:

\(d_b\): Distance to transition state in bond potential (nm)
\(a\): Kuhn length of the polymer chain (nm)

Typical values:

\(\nu \to 0\): No force sensitivity (recovers Tanaka-Edwards)
\(\nu \sim 0.1-1\): Weak sensitivity (mild shear thinning)
\(\nu \sim 1-5\): Moderate sensitivity (physical bonds, hydrogen bonding)
\(\nu \sim 5-20\): Strong sensitivity (weak non-covalent, metal-ligand)

Effect on rheology:

\[\eta(\dot{\gamma}) \sim \eta_0 (\tau_b \dot{\gamma})^{-\nu/(1+\nu)}\]

Shear thinning exponent: \(n = \nu/(1+\nu)\).

Network Modulus \(G\)¶

Determines the plateau modulus in linear viscoelasticity:

\[G = \frac{\rho k_B T}{M_c}\]

where:

\(\rho\): Polymer density
\(M_c\): Molecular weight between crosslinks

From SAOS: \(G = \lim_{\omega \to \infty} G'(\omega)\).

Bond Lifetime \(\tau_b\)¶

The zero-force relaxation time:

\[\tau_b = \frac{1}{k_0} = \frac{1}{k_{\text{off}}(\mathbf{S} = \mathbf{I})}\]

From SAOS: Crossover frequency \(\omega_c = 1/\tau_b\) where \(G' = G''\).

Relation to activation energy:

\[\tau_b = \tau_0 \exp\left(\frac{\Delta U_0}{k_B T}\right)\]

where \(\tau_0 \sim 10^{-12}\) s (attempt time) and \(\Delta U_0\) is the barrier height.

Solvent Viscosity \(\eta_s\)¶

Physical origin:

Newtonian contribution from unentangled solvent or free chains
Provides high-frequency damping

Identifiability:

Clear at \(\omega \gg 1/\tau_b\) in SAOS: \(G'' \to \eta_s \omega\)
At low rates, \(\eta_s\) is masked by network viscosity \(G \tau_b\)

Parameter Correlations ¶

Correlation Structure¶
Parameter Pair	Correlation	Mitigation Strategy
\(\tau_b\) vs \(\nu\)	Strong (both control thinning rate)	Fit SAOS first for \(\tau_b\), then flow curve for \(\nu\)
\(G\) vs \(\tau_b\)	Moderate (both set \(\eta_0\))	Use plateau modulus constraint from SAOS
\(G\) vs \(\nu\)	Weak	Independent fitting possible
\(\eta_s\) vs others	Weak (only at high rates)	Fix \(\eta_s = 0\) unless high-frequency data available

Validity and Assumptions ¶

Core Assumptions ¶

Affine deformation: Chains deform with the macroscopic flow (no slip, no reptation)
Gaussian statistics: Chains are entropic springs with \(F \sim (R - R_0)\)
Instantaneous reformation: Broken chains immediately rejoin into equilibrium state \(\mathbf{S} = \mathbf{I}\)
Mean-field: All chains experience the same average conformation
Force-stretch proportionality: \(F \propto \text{tr}(\mathbf{S}) - 3\)

Validity Ranges ¶

Material classes:

✓ Associating polymers with reversible bonds
✓ Physical gels (low crosslink density)
✗ Entangled melts (need reptation)
✗ Permanent networks (bonds don’t break)

Deformation regimes:

✓ Linear viscoelasticity (\(\gamma_0 \ll 1\))
✓ Weakly nonlinear (\(\gamma_0 \sim 1\))
✗ Large strains near chain extensibility limit (use FENE variant)

Time scales:

✓ Dynamics slower than bond vibration (\(\omega \ll 10^{12}\) rad/s)
✗ Sub-nanosecond dynamics (molecular detail needed)

Known Limitations ¶

No finite extensibility: Chains can stretch indefinitely. For \(\text{tr}(\mathbf{S}) \gg 10\), use FENE-Bell hybrid.
No non-affine motion: Shear banding or localized flow requires non-affine variant.
No stretch-dependent creation: Assumes reformation into \(\mathbf{S} = \mathbf{I}\). For oriented reformation, use stretch-creation variant.
Mean-field: Fluctuations neglected (important near gel point).

When to Use Alternatives ¶

Model Selection Guide¶
Observation	Recommended Variant
Stress saturates at high rates	FENE-Bell (finite extensibility)
Shear banding observed	Non-affine Bell
Second normal stress difference \(N_2 \neq 0\)	Non-affine or add Giesekus-like term
Thixotropy (time-dependent)	Add fluidity evolution (SGR-Bell hybrid)
Strong strain hardening	FENE-Bell or Pom-Pom

Regimes and Behavior ¶

Linear Regime (\(\mathrm{Wi} \ll 1\))¶

For \(\omega \tau_b \ll 1\) or \(\dot{\gamma} \tau_b \ll 1\):

Conformation near equilibrium: \(\mathbf{S} \approx \mathbf{I}\)
Bell correction negligible: \(\exp[\nu(\text{tr}\mathbf{S} - 3)] \approx 1\)
Identical to Maxwell model:

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

Plateau modulus: \(G_N^0 = G\)

Zero-shear viscosity: \(\eta_0 = G \tau_b\)

Weakly Nonlinear Regime (\(\mathrm{Wi} \sim 1\))¶

Onset of shear thinning:

\[\eta(\dot{\gamma}) \approx \eta_0 \left[1 - \frac{\nu}{2} (\tau_b \dot{\gamma})^2 + \ldots\right]\]

Startup overshoot: Peak stress at \(\gamma \sim 1\) (slightly reduced from constant breakage).

First normal stress difference:

\[N_1 \approx 2 G \tau_b^2 \dot{\gamma}^2\]

Strongly Nonlinear Regime (\(\mathrm{Wi} \gg 1\))¶

Shear thinning power-law:

\[\eta \sim G \tau_b (\tau_b \dot{\gamma})^{-\nu/(1+\nu)}\]

For \(\nu = 1\): \(\eta \sim \dot{\gamma}^{-0.5}\) (square-root thinning).

Conformation stretch:

\[\text{tr}(\mathbf{S}) \sim (\tau_b \dot{\gamma})^{1/(1+\nu)}\]

Effective relaxation time:

\[\tau_{\text{eff}} = \frac{\tau_b}{\exp[\nu(\text{tr}\mathbf{S} - 3)]} \ll \tau_b\]

Chains break much faster under high stretch.

What You Can Learn ¶

From SAOS (Linear Viscoelasticity)¶

Plateau modulus: \(G = G'_{\text{plateau}}\)
Relaxation time: \(\tau_b = 1/\omega_c\) where \(G' = G''\)
Zero-shear viscosity: \(\eta_0 = G \tau_b\)

Note: SAOS cannot distinguish Bell from constant breakage (need nonlinear tests).

From Flow Curve (Steady Shear)¶

Force sensitivity: \(\nu\) from shear thinning slope \(n = \nu/(1+\nu)\)

\[\log \eta \sim -n \log \dot{\gamma}\]
Verification: Check if \(\eta_0 = G \tau_b\) matches SAOS
Solvent viscosity: \(\eta_s\) from high-rate plateau (if observed)

From Startup Tests ¶

Overshoot strain: \(\gamma_{\text{peak}} \sim 1/\sqrt{\nu}\) gives independent \(\nu\) estimate
Rate dependence: Peak stress scaling confirms \(\sigma_{\text{peak}} \sim \dot{\gamma}^{1/(1+\nu)}\)
Cross-validation: Compare \(\nu\) from thinning vs overshoot

Experimental Design ¶

Protocol Selection ¶

Recommended test sequence:

SAOS (strain amplitude sweep + frequency sweep)
Flow curve (steady shear rate sweep)
Startup tests at 3-5 rates
Optional: Step strain relaxation

Best Practices for Measuring \(\nu\)¶

Method 1: Flow curve slope

\[\nu = \frac{n}{1 - n} \quad \text{where} \quad n = -\frac{d \log \eta}{d \log \dot{\gamma}}\]

Method 2: Startup overshoot strain

\[\nu \sim \frac{1}{\gamma_{\text{peak}}^2}\]

Method 3: Joint fitting

Simultaneous NLSQ fit to flow curve + startup curves.

Computational Implementation ¶

JAX Architecture ¶

Key function:

def breakage_bell(S: Array, tau_b: float, nu: float) -> float:
    trace_S = jnp.trace(S)
    return (1.0 / tau_b) * jnp.exp(nu * (trace_S - 3.0))

Fitting Guidance ¶

NLSQ Strategy ¶

Sequential fitting (recommended):

Fit SAOS first (fix \(G\), \(\tau_b\), \(\eta_s\))
Fit flow curve with \(G\), \(\tau_b\) fixed (optimize \(\nu\) only)
Refine with startup (joint optimization)

Usage Examples ¶

Example 1: Basic NLSQ Fitting ¶

from rheojax.models.tnt import TNTSingleMode
import numpy as np

model = TNTSingleMode(breakage="bell")
result = model.fit(gamma_dot, sigma_true, test_mode='flow_curve')

Example 2: Bayesian Inference ¶

result = model.fit_bayesian(
    gamma_dot, sigma_obs,
    test_mode='flow_curve',
    num_warmup=1000,
    num_samples=2000,
    num_chains=4,
    seed=42
)

Example 3: Startup Simulation ¶

t, S = model.simulate_startup(gamma_dot=10.0, t_end=10.0)
stress = model.G * S[:, 0, 1]

Composition with Other Variants ¶

Bell + FENE ¶

model = TNTSingleMode(breakage="bell", stress_type="fene")

Failure Mode: Runaway Breakage ¶

For sufficiently large \(\nu\), the Bell model predicts a non-monotonic flow curve where stress decreases with increasing shear rate. This occurs because the exponential acceleration of bond breakage outpaces the linear increase of chain stretch.

Shear banding criterion: The flow curve becomes non-monotonic when

\[\frac{d\sigma}{d\dot{\gamma}} < 0\]

This occurs above a critical \(\nu\) value (typically \(\nu \gtrsim 1-2\) depending on \(Wi\)). In this regime:

The material cannot sustain homogeneous flow at intermediate rates
Shear banding develops: coexistence of high and low shear-rate bands
The stress-controlled flow curve shows a plateau (stress selection)

Warning

Non-monotonic flow curves indicate the model predicts mechanical instability. Physically, this corresponds to runaway bond breakage where stretched chains break so fast that the network cannot sustain stress. If observed in fits, consider whether the material truly exhibits shear banding, or whether \(\nu\) is overestimated.

API Reference ¶

class rheojax.models.tnt.TNTSingleMode(breakage='constant', stress_type='linear', xi=0.0)[source]

Single-mode Transient Network Theory model.

Implements the Green-Tobolsky / Tanaka-Edwards transient network model with composable physics variants. The conformation tensor S tracks the average chain configuration between reversible crosslinks.

The constitutive equation is:

dS/dt = L·S + S·L^T + g₀·I - β(S)·S

Stress is computed from S via σ = G·f(S)·(S - I) + η_s·γ̇.

Parameters:

breakage (Literal['constant', 'bell', 'power_law', 'stretch_creation']) – Breakage rate function: - “constant”: β = 1/τ_b (Tanaka-Edwards, UCM-like) - “bell”: β = (1/τ_b)·exp(ν·(stretch-1)) (force-dependent) - “power_law”: β = (1/τ_b)·stretch^m - “stretch_creation”: β = (1/τ_b), g₀ = (1+κ·stretch)/τ_b
stress_type (Literal['linear', 'fene']) – Stress formula: - “linear”: σ = G·(S - I) (Gaussian chains) - “fene”: σ = G·f(tr(S))·(S - I) (finitely extensible)
xi (float) – Gordon-Schowalter slip parameter (0=upper-convected, 1=corotational)

parameters

Model parameters

Type:: ParameterSet

fitted_

Whether the model has been fitted

Type:: bool

Examples

Basic Tanaka-Edwards model:

>>> model = TNTSingleMode()
>>> gamma_dot = np.logspace(-2, 2, 50)
>>> sigma = model.predict(gamma_dot, test_mode='flow_curve')

Bell force-dependent breakage:

>>> model = TNTSingleMode(breakage="bell")
>>> # Now has additional parameter 'nu' (force sensitivity)

References ¶

Bell, G.I. (1978). Science 200, 618-627. https://doi.org/10.1126/science.347575
Tanaka, F., & Edwards, S.F. (1992). JNFM 43, 247-271. https://doi.org/10.1016/0377-0257(92)80027-U
Evans, E., & Ritchie, K. (1997). Biophys J 72, 1541-1555. https://doi.org/10.1016/S0006-3495(97)78802-7
Hänggi, P., et al. (1990). Rev Mod Phys 62, 251-341. https://doi.org/10.1103/RevModPhys.62.251
Vaccaro, A., & Marrucci, G. (2000). JNFM 92, 261-273. https://doi.org/10.1016/S0377-0257(00)00095-1
Tripathi, A., et al. (2006). Macromolecules 39, 1981-1999. https://doi.org/10.1021/ma051614x