Transient Network Theory (TNT)¶

Transient Network Theory Foundations

Transient Network Theory (TNT) describes materials with reversible crosslinks that continuously break and reform on characteristic timescales, creating dynamic networks with time-dependent mechanical properties.

Physical Basis:

Network strands: Polymer chains or other elements spanning crosslink junctions
Crosslink lifetime (\(\tau_b\)): Mean bond survival time before detachment
Creation rate: New bonds form to maintain equilibrium network density
Conformation tensor (\(\mathbf{S}\)): Tracks average chain stretch and orientation

Characteristic Experimental Signatures:

Single relaxation mode: Maxwell-like behavior with relaxation time \(\tau = \tau_b\)
Shear thinning: Viscosity decreases as flow disrupts network structure
Strain softening: Modulus reduction under large deformations (chain stretch)
Transient overshoot: Stress peaks during startup as network orientation saturates
Normal stress differences: \(N_1 > 0\) from chain anisotropy (extensional resistance)

Fundamental Constitutive Equation:

\[\frac{D\mathbf{S}}{Dt} = \mathbf{\kappa} \cdot \mathbf{S} + \mathbf{S} \cdot \mathbf{\kappa}^T - \frac{1}{\tau_b(\mathbf{S})}(\mathbf{S} - \mathbf{I})\]

where \(\mathbf{\kappa}\) is the velocity gradient and \(\mathbf{I}\) is the identity. The stress is given by \(\boldsymbol{\sigma} = G \cdot f(\mathbf{S})\) where \(f(\mathbf{S})\) depends on the chain model:

Hookean: \(f(\mathbf{S}) = \mathbf{S} - \mathbf{I}\) (linear)
FENE-P: \(f(\mathbf{S}) = \frac{L^2_{max}}{L^2_{max} - \text{tr}(\mathbf{S}) + 3}(\mathbf{S} - \mathbf{I})\) (finite extensibility)

Bond Kinetics Models:

Kinetics	Breakage Rate \(1/\tau_b\)	Use Case
Constant (Tanaka-Edwards)	\(1/\tau_0\)	Baseline Maxwell-like
Bell Model	\((1/\tau_0)\exp(\nu F/k_BT)\)	Force-activated unbinding
Power-law	\((1/\tau_0)(F/F_0)^m\)	Empirical force-weakening
Stretch-enhanced creation	Creation \(\propto \text{tr}(\mathbf{S})\)	Strain-induced crosslinking

Advanced Extensions:

Loop-bridge equilibrium: Two-species kinetics (\(f_B\) equilibrium bridge fraction)
Sticky Rouse: Multi-mode relaxation with sticker-limited dynamics
Cates model: Living polymers with scission/recombination
Non-affine slip: Gordon-Schowalter parameter \(\xi\) for partial coupling
Multi-species networks: Multiple bond types with different lifetimes and moduli

Model Selection Guide:

Model	Best For
TNTSingleMode (constant)	Simple physical gels, baseline characterization
TNTSingleMode (Bell)	Bio-networks with force-sensitive bonds (fibrin, collagen)
TNTSingleMode (FENE-P)	Polymeric gels near maximum extensibility
TNTLoopBridge	Telechelic polymers with two junction types
TNTStickyRouse	Multi-sticker associating polymers (broad relaxation)
TNTCates	Wormlike micelles, living polymers
TNTSingleMode (Non-Affine)	Networks with imperfect chain-flow coupling (\(N_2 \neq 0\))
TNTSingleMode (Stretch-Creation)	Strain-crystallizing or mechanophore-activated networks
TNTMultiSpecies	Dual-crosslinked hydrogels, multi-strength assemblies

Dual Formulation:

TNT models admit two mathematically equivalent formulations:

Differential (conformation tensor ODE): Evolve \(\mathbf{S}(t)\) via the constitutive ODE above — efficient for steady-state and simple histories
Integral (cohort/history): Track chain cohorts born at each time \(t'\) and integrate their stress contributions — natural for complex deformation histories

Both yield identical predictions; the choice is computational convenience. See TNT Protocol Equations — Shared Reference for details.

Typical Parameter Ranges:

Network modulus \(G\): 1–\(10^6\) Pa (depends on crosslink density)
Bond lifetime \(\tau_b\): \(10^{-6}\)–\(10^4\) s (wide range across materials)
Bell parameter \(\nu\): 0.01–20 (bond sensitivity to force)
FENE extensibility \(L_{\max}\): 2–100 (chain contour length ratio)
Slip parameter \(\xi\): 0 (affine) to 1 (full slip)

Overview¶

The TNT family in RheoJAX provides 5 model classes spanning 9 distinct physical variants, from simple single-mode Maxwell analogs to complex multi-species living polymer systems. All models support the full suite of rheological protocols (flow curves, SAOS, LAOS, startup, creep, relaxation) with validated predictions against experimental data. The mathematical framework is implemented in JAX with full automatic differentiation support, enabling GPU acceleration and Bayesian inference via NumPyro NUTS sampling. See the foundation box above for the physical basis, key signatures, and constitutive equations shared across all TNT variants.

Dual Formulation — Integral vs Differential

TNT admits two mathematically equivalent perspectives:

Differential (conformation tensor ODE) — the primary RheoJAX implementation:

\[\frac{D\mathbf{S}}{Dt} = \boldsymbol{\kappa} \cdot \mathbf{S} + \mathbf{S} \cdot \boldsymbol{\kappa}^T - \frac{1}{\tau_b(\mathbf{S})}(\mathbf{S} - \mathbf{I})\]

Integral (history / cohort) formulation — useful for step-strain analysis and multi-protocol understanding:

\[\boldsymbol{\tau}(t) = \int_{-\infty}^{t} \beta(t') \, S(t,t') \, G \bigl[\mathbf{B}(t,t') - \mathbf{I}\bigr] \, dt'\]

where \(\beta(t')\) is the birth rate of chains at time \(t'\), \(S(t,t') = \exp\!\bigl[-\int_{t'}^{t} k_d(s)\,ds\bigr]\) is the survival probability, and \(\mathbf{B}(t,t')\) is the Finger deformation tensor.

The integral form tracks cohorts of chains born at time \(t'\), each carrying its deformation history. The differential form evolves the ensemble average conformation \(\mathbf{S}(t)\). Both yield identical stress predictions. See TNT Protocol Equations — Shared Reference for the full derivation and numerical methods for each approach.

Model Hierarchy¶

TNT Family (5 Classes, 9 Variants)
│
├── TNTSingleMode (Composable, 5 variants)
│   │   Base: Constant breakage (Tanaka-Edwards)
│   │         Maxwell-like with tensorial stress
│   │         Parameters: G, τ_b, η_s
│   │
│   ├── breakage="constant" (default)
│   │   └── Tanaka-Edwards: 1/τ_b = const
│   │       Simplest TNT, baseline for comparison
│   │
│   ├── breakage="bell"
│   │   └── Force-dependent detachment
│   │       1/τ_b = (1/τ_0) exp(ν·F/k_B·T)
│   │       For bio-networks with catch/slip bonds
│   │       Additional parameter: ν (force sensitivity)
│   │
│   ├── breakage="power_law"
│   │   └── Power-law force weakening
│   │       1/τ_b = (1/τ_0)(F/F_0)^m
│   │       Empirical extension, m ~ 1-5
│   │
│   ├── stress_type="fene" (FENE-P finite extensibility)
│   │   └── Chain force: F = 3k_B·T·L²_max/(L²_max - tr(S) + 3)
│   │       Polymer gels with limited chain stretch
│   │       Additional parameter: L_max (extensibility)
│   │       Nonlinear softening at large strains
│   │
│   └── xi > 0 (Non-affine slip, Gordon-Schowalter)
│       └── Partial coupling: S evolves with ξ ∈ [0, 1]
│           ξ = 0: Affine (full coupling)
│           ξ = 1: Full slip (isotropic stress)
│           Reduces N₁ predictions, empirical correction
│
├── TNTLoopBridge (Two-species kinetics)
│   │   Telechelic polymers with loops + bridges
│   │   Equilibrium: f_B(eq) = bridge fraction
│   │   Kinetics: df_B/dt = k_loop→bridge - k_bridge→loop
│   │
│   └── Parameters: G, τ_b, τ_a, ν, f_B_eq, η_s
│       Only bridges contribute to stress
│       Loops act as dangling ends (viscous)
│       6 parameters (richer dynamics than SingleMode)
│
├── TNTStickyRouse (Multi-mode sticker dynamics)
│   │   Rouse modes limited by sticker lifetime
│   │   τ_p = τ_s + (N/p²)·τ_Rouse (hybrid timescale)
│   │
│   └── Parameters: G_k (N moduli), τ_R_k (N Rouse times), τ_s, η_s (2N+2 parameters)
│       Broad relaxation spectrum
│       Terminal time: τ_terminal ≈ τ_s + N·τ_Rouse
│       For multi-sticker associating polymers
│
├── TNTCates (Living polymers / wormlike micelles)
│   │   Scission/recombination kinetics
│   │   Effective relaxation: τ_d = √(τ_rep · τ_break)
│   │
│   └── Parameters: G_0, τ_rep, τ_break, η_s
│       Single effective mode (faster of rep/break)
│       Shear-thinning from reduced effective length
│       For micellar solutions (CTAB, CPCl, etc.)
│
└── TNTMultiSpecies (N independent bond types)
    │   Distinct G_i, τ_i for each species
    │   Additive stress: σ_total = Σ_i σ_i
    │
    └── Parameters: {G_i, τ_b_i} for i=0..N-1, η_s
        Discrete relaxation spectrum
        For heterogeneous crosslink populations
        N typically 2-5 (more = fit flexibility vs physics)

When to Use Which Model¶

Feature	SingleMode	LoopBridge	StickyRouse	Cates	MultiSpecies
Material type	Physical gels	Telechelics	Multi-sticker	Micelles	Mixed networks
Number of params	3-5	7-8	4-6	4-5	2N+1
Relaxation spectrum	Single mode	Two modes	Broad (N modes)	Effective single	Discrete (N)
Key physics	Bond lifetime	Loop↔bridge	Sticker-limited	Scission/recomb	Heterogeneity
Recommended for	Baseline, simple gels	Telechelic ionomers	Associating polymers	Wormlike micelles	Complex networks
Force-dependence	✓ (Bell/power)	~	~	~	✓ (per species)
Finite extensibility	✓ (FENE-P)	~	~	~	~
Computational cost	1× (fastest)	1.5×	2-3× (modes)	1×	1.5-2.5×

Decision Tree:

Is there a single dominant bond type?
- Yes → TNTSingleMode (start here for most gels)
- No, two distinct types → TNTLoopBridge or TNTMultiSpecies
Are bonds sensitive to force/stress?
- Yes, exponential → TNTSingleMode(breakage=”bell”)
- Yes, power-law → TNTSingleMode(breakage=”power_law”)
- No → TNTSingleMode(breakage=”constant”)
Is chain extensibility important?
- Yes (large strains) → TNTSingleMode(stress_type=”fene”)
- No (linear/moderate) → stress_type=”hookean”
Is the material a living polymer system?
- Yes, wormlike micelles → TNTCates
- Yes, but multi-sticker → TNTStickyRouse
- No → TNTSingleMode or TNTLoopBridge
Do you observe broad relaxation spectrum?
- Yes, continuous → TNTStickyRouse
- Yes, discrete peaks → TNTMultiSpecies
- No, single mode → TNTSingleMode

Failure Modes¶

Each TNT variant has a characteristic failure mode — the dominant nonlinear phenomenon that limits the range of validity or produces extreme behavior:

Variant	Primary Phenomenon	Key Parameter	Failure Mode	Physical Mechanism
Bell	Shear thinning / banding	\(\nu\)	Runaway breakage	Exponential bond weakening under stretch
FENE-P	Strain stiffening	\(L_{\max}\)	Chain snap	Stress divergence as chains approach maximum extension
Loop-Bridge	Concentration-dependent viscosity	\(k_{LB}/k_{BL}\)	Loop saturation	All chains convert to loops under extreme flow
Cates	Single-mode Maxwellian	\(\tau_{\text{break}}\)	Shear banding	Non-monotonic flow curve from scission kinetics
Sticky Rouse	Self-similar relaxation	\(N_{\text{stickers}}\)	Terminal flow	All stickers eventually release at long times
Multi-Species	Residual elasticity	\(G_{\text{chem}}/G_{\text{phys}}\)	Bond hierarchy	Sequential failure from weakest to strongest bonds
Non-Affine	\(N_2 \neq 0\)	\(\xi\)	Wall slip	Extreme non-affinity decouples chains from flow
Stretch-Creation	Shear thickening	\(\alpha\)	Gelation	Runaway network formation under sustained deformation

Feature Comparison Matrix¶

Predicted rheological features across all TNT variants (base Tanaka-Edwards plus 8 extensions). This matrix summarizes which nonlinear phenomena each variant can capture:

Feature	Base TE	Bell	FENE	NonAffine	StretchCreate	LoopBridge	StickyRouse	Cates	MultiSpecies
Shear thinning	-	Yes	Yes	Yes	-	Yes	Yes	Yes	Yes
Stress overshoot	-	Yes	Yes	Yes	Yes	Yes	Yes	Yes	Yes
\(N_2 \neq 0\)	-	-	-	Yes	-	-	-	-	-
Strain hardening	-	-	Yes	-	Yes	-	-	-	-
Higher harmonics	-	Yes	Yes	Yes	Yes	Yes	Yes	Yes	Yes
Shear thickening	-	-	-	-	Yes	-	-	-	-
Non-monotonic flow	-	(high \(\nu\))	-	(high \(\xi\))	-	-	-	Yes	-
Multi-mode spectrum	-	-	-	-	-	2 modes	N modes	-	N modes
Residual stress	-	-	-	-	-	-	-	-	Yes

Decision Framework¶

Three complementary decision trees help identify the best TNT variant. Use whichever matches your starting point:

Property-based (above, `Decision Tree`_): Start from known material class (e.g., “telechelic polymer” → LoopBridge). Best when the material type is known.
Observation-based (TNT Knowledge Extraction Guide, “Master Decision Tree”): Start from raw data features (e.g., “Cole-Cole plot is semicircular” → Cates). Best when you have data but are unsure of the material class.
Residual-based (TNT Knowledge Extraction Guide, “Iterative Refinement”): Start from a base fit and systematically add physics to reduce residuals (e.g., “startup overshoot too sharp” → add Bell breakage). Best when iterating on model fits.

Key Parameters¶

Parameter	Symbol	Typical Range	Physical Meaning
Network modulus	G	1-10⁶ Pa	Elastic modulus at short times (\(G \sim n k_B T\), n = crosslink density)
Bond lifetime	τ_b	10⁻⁶-10⁴ s	Mean survival time before bond detachment (sets relaxation time)
Solvent viscosity	η_s	0-10⁴ Pa·s	Background viscosity (can be zero for ideal network)
Bell force sensitivity	ν	0.01-20	Dimensionless activation barrier reduction (ΔE = ν·k_B·T per unit force)
FENE extensibility	L_max	2-100	Maximum chain stretch ratio (\(L_{\max}^2 \sim N_{\text{Kuhn}}\), chain stiffness)
Slip parameter	ξ	0-1	Gordon-Schowalter: ξ=0 (affine), ξ=1 (full slip), affects N₁
Bridge fraction (eq)	f_B_eq	0-1	Equilibrium fraction of bridge junctions (LoopBridge model)
Exchange rate	k_ex	10⁻⁴-10² s⁻¹	Loop↔bridge interconversion rate (LoopBridge)
Sticker lifetime	τ_s	10⁻⁶-10⁴ s	Mean sticker attachment time (StickyRouse, limits Rouse modes)
Chain length	N	10-1000	Number of Kuhn segments (StickyRouse, sets \(\tau_{\text{Rouse}} \sim N^2\))
Reptation time	τ_rep	10⁻³-10³ s	Tube escape time for entangled chains (Cates model)
Breakage time	τ_break	10⁻³-10³ s	Mean time for chain scission (Cates model, \(\tau_d \sim \sqrt{\tau_{\text{rep}} \cdot \tau_{\text{break}}}\))

Quick Start¶

Basic Tanaka-Edwards model (constant breakage):

from rheojax.models import TNTSingleMode

# Create model with default constant breakage
model = TNTSingleMode()

# Fit to oscillatory data (SAOS)
model.fit(omega, G_star, test_mode='oscillation')

# Check parameters
G = model.parameters.get_value('G')
tau_b = model.parameters.get_value('tau_b')
print(f"Network modulus: {G:.1f} Pa, Bond lifetime: {tau_b:.3f} s")

# Predict flow curve
gamma_dot = jnp.logspace(-2, 2, 50)
sigma = model.predict(gamma_dot, test_mode='flow_curve')

Force-dependent Bell model for bio-networks:

from rheojax.models import TNTSingleMode

# Bell model with force-activated unbinding
model = TNTSingleMode(breakage="bell")

# Set initial guesses for sensitive bonds
model.parameters.set_value('nu', 5.0)  # Moderate force sensitivity

# Fit to startup shear data (shows force-induced softening)
model.fit(t, sigma_startup, test_mode='startup', gamma_dot=1.0)

FENE-P model for finite extensibility:

from rheojax.models import TNTSingleMode

# FENE-P with finite chain length
model = TNTSingleMode(stress_type="fene")

# Set extensibility limit
model.parameters.set_value('L_max', 10.0)  # 10x equilibrium length

# Fit to large amplitude data (will show strain softening)
model.fit(gamma, sigma, test_mode='startup', gamma_dot=10.0)

Wormlike micelles (Cates model):

from rheojax.models import TNTCates

# Living polymer system
model = TNTCates()

# Fit to oscillatory data
model.fit(omega, G_star, test_mode='oscillation')

# Extract timescales
tau_rep = model.parameters.get_value('tau_rep')
tau_break = model.parameters.get_value('tau_break')
tau_d = jnp.sqrt(tau_rep * tau_break)
print(f"Effective relaxation: {tau_d:.3e} s")

Multi-sticker polymer (StickyRouse):

from rheojax.models import TNTStickyRouse

# Create model with 5 Rouse modes
model = TNTStickyRouse(n_modes=5)

# Fit to frequency sweep (broad spectrum)
model.fit(omega, G_star, test_mode='oscillation')

# Predict storage/loss moduli
G_prime, G_double_prime = model.predict(omega, test_mode='oscillation',
                                        return_components=True)

Bayesian Inference¶

All TNT models support full Bayesian inference via NumPyro with automatic warm-starting from NLSQ point estimates. The recommended workflow uses 4 chains for robust diagnostics:

from rheojax.models import TNTSingleMode
import jax.numpy as jnp

# Step 1: NLSQ point estimate (fast, ~seconds)
model = TNTSingleMode(breakage="bell", stress_type="fene")
model.fit(omega, G_star, test_mode='oscillation')

# Step 2: Bayesian inference with warm-start (num_chains=4 default)
result = model.fit_bayesian(
    omega, G_star,
    test_mode='oscillation',
    num_warmup=1000,
    num_samples=2000,
    num_chains=4,  # Parallel chains for diagnostics
    seed=42        # Reproducibility
)

# Step 3: Diagnostics (automatic R-hat, ESS checks)
intervals = model.get_credible_intervals(result.posterior_samples,
                                         credibility=0.95)

for param_name, (lower, upper) in intervals.items():
    point_est = model.parameters.get_value(param_name)
    print(f"{param_name}: {point_est:.3e} [{lower:.3e}, {upper:.3e}]")

# Step 4: Posterior predictive checks
G_pred = model.predict(omega, test_mode='oscillation')
# Compare G_pred to G_star to validate model

ArviZ diagnostics for complex models:

from rheojax.pipeline.bayesian import BayesianPipeline

# Full pipeline with automated diagnostics
pipeline = BayesianPipeline()
(pipeline
    .load('gel_data.csv', x_col='omega', y_col='G_star')
    .fit_nlsq('tnt_single_mode', breakage='bell')
    .fit_bayesian(num_warmup=1000, num_samples=2000, num_chains=4)
    .plot_trace()       # MCMC convergence
    .plot_pair(divergences=True)  # Parameter correlations
    .plot_forest(hdi_prob=0.95)   # Credible intervals
    .save('results.hdf5'))

# Check specific diagnostics
print(f"R-hat: {pipeline.get_diagnostic('r_hat')}")
print(f"ESS: {pipeline.get_diagnostic('ess')}")

Supported Protocols¶

All TNT models support the full suite of rheological test protocols:

Protocol	test_mode	Notes
Flow curve	‘flow_curve’	Steady shear stress σ(γ̇), shear thinning from network disruption
SAOS (oscillatory)	‘oscillation’	\(G'(\omega)\), \(G''(\omega)\), single-mode shows \(G' \sim G'' \sim \omega^2\) at low \(\omega\)
Startup shear	‘startup’	Transient σ(t, γ̇), stress overshoot from chain orientation
Stress relaxation	‘relaxation’	G(t) after step strain, exponential decay with τ_b
Creep	‘creep’	γ(t, σ₀), delayed compliance from bond reformation
LAOS	‘laos’	Large amplitude σ(t), Fourier/Chebyshev harmonics

Protocol-specific features:

Flow curve: Shear thinning η(γ̇) from reduced effective τ_b at high rates
SAOS: \(G'(\omega)\) crossover at \(\omega \approx 1/\tau_b\), \(\tan(\delta) = G''/G'\) diagnostic
Startup: Overshoot at γ ≈ 1-2 (network orientation saturation)
Relaxation: Single exponential \(G(t) \sim \exp(-t/\tau_b)\) for constant breakage
Creep: Power-law at short times, viscous flow at long times
LAOS: Strain softening (I₃/I₁ ratio) from FENE-P or Bell kinetics

Example multi-protocol characterization:

from rheojax.models import TNTSingleMode
import jax.numpy as jnp

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

# 1. Fit to SAOS for linear parameters
model.fit(omega, G_star, test_mode='oscillation')

# 2. Predict startup for validation
t = jnp.linspace(0, 10, 200)
sigma_startup = model.predict(t, test_mode='startup', gamma_dot=1.0)

# 3. Flow curve for nonlinear regime
gamma_dot = jnp.logspace(-3, 2, 50)
sigma_flow = model.predict(gamma_dot, test_mode='flow_curve')

# 4. Relaxation modulus
G_t = model.predict(t, test_mode='relaxation', gamma_0=0.1)

Model Documentation¶

References¶

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