TNT Multi-Species (Multiple Bond Types) — Handbook¶

Quick Reference ¶

Use When ¶

The TNT Multi-Species model is appropriate for network materials containing multiple distinct bond types with different relaxation timescales:

Dual-crosslinked hydrogels (physical + chemical crosslinks)
Interpenetrating polymer networks (IPNs)
Heterogeneous biological networks (fibrin + collagen, actin + intermediate filaments)
Multi-strength supramolecular assemblies (hydrogen bonds + hydrophobic interactions)
Hierarchical networks with discrete energy barriers
Reversible networks with multiple association strengths

Key diagnostic: Multiple distinct relaxation peaks in \(G''(\omega)\) or multi-modal relaxation spectrum.

Parameters ¶

The model requires \(2N + 1\) parameters for \(N\) bond species:

Symbol	Typical Value	Units	Description
\(G_i\)	\(10^2\) to \(10^5\)	Pa	Modulus contribution from species \(i\)
\(\tau_{b,i}\)	\(10^{-3}\) to \(10^3\)	s	Bond lifetime of species \(i\)
\(\eta_s\)	0.0 to \(10^{-2}\)	Pa·s	Solvent viscosity (background)
\(N\)	2 to 5	dimensionless	Number of bond species (constructor argument)

Total parameters: \(2N + 1\) (e.g., 5 for \(N=2\), 11 for \(N=5\))

Key Equation ¶

Each bond species evolves independently:

\[\frac{d\mathbf{S}_i}{dt} = \boldsymbol{\kappa} \cdot \mathbf{S}_i + \mathbf{S}_i \cdot \boldsymbol{\kappa}^T - \frac{1}{\tau_{b,i}} (\mathbf{S}_i - \mathbf{I})\]

Total stress is the superposition of all species contributions:

\[\boldsymbol{\sigma} = \sum_{i=1}^{N} G_i (\mathbf{S}_i - \mathbf{I}) + 2\eta_s \mathbf{D}\]

where \(\mathbf{S}_i\) is the conformation tensor for species \(i\), \(\boldsymbol{\kappa}\) is the velocity gradient tensor, and \(\mathbf{D}\) is the rate-of-strain tensor.

Supported Test Modes ¶

All six rheological protocols are supported:

FLOW_CURVE: Steady shear viscosity \(\eta(\dot{\gamma})\)
OSCILLATION: SAOS moduli \(G'(\omega)\), \(G''(\omega)\)
STARTUP: Stress growth \(\sigma^+(t, \dot{\gamma})\)
CREEP: Strain response \(\gamma(t, \sigma_0)\)
RELAXATION: Stress relaxation \(G(t)\) (analytical)
LAOS: Large amplitude oscillatory shear

Material Examples ¶

Dual-Crosslinked Hydrogels:

Fast species: hydrogen bonds (\(\tau_{b,1} \sim 0.01\) s)
Slow species: covalent crosslinks (\(\tau_{b,2} \sim 100\) s)
Example: Alginate-polyacrylamide (Sun et al. 2012)

Biological Networks:

Species 1: Fibrin (\(\tau_{b,1} \sim 0.1\) s)
Species 2: Collagen (\(\tau_{b,2} \sim 10\) s)
Interpenetrating structure in blood clots

Supramolecular Networks:

Fast species: H-bonds (\(\tau_{b,1} \sim 0.001\) s)
Medium: Hydrophobic (\(\tau_{b,2} \sim 1\) s)
Slow: Ionic (\(\tau_{b,3} \sim 100\) s)

Key Characteristics ¶

Multi-modal relaxation: Discrete peaks at \(\omega_i = 1/\tau_{b,i}\)
Broad relaxation spectrum: From discrete bond types, not distribution
Superposition principle: Total response = sum of Maxwell modes
Physical interpretation: Each mode corresponds to a specific bond type
Network equivalence: Mathematically equivalent to Generalized Maxwell, but with network interpretation

Notation Guide ¶

Primary Variables ¶

Symbol	Definition
\(\mathbf{S}_i\)	Conformation tensor for species \(i\) (dimensionless, 3x3 symmetric)
\(S_{xx,i}, S_{yy,i}, S_{zz,i}\)	Normal components of conformation tensor for species \(i\)
\(S_{xy,i}\)	Shear component of conformation tensor for species \(i\)
\(\boldsymbol{\sigma}\)	Total Cauchy stress tensor (Pa, 3x3 symmetric)
\(\boldsymbol{\kappa}\)	Velocity gradient tensor (1/s, 3x3)

Parameters and Constants ¶

Symbol	Definition
\(N\)	Number of distinct bond species (integer \(\geq 2\))
\(G_i\)	Elastic modulus contribution from species \(i\) (Pa)
\(\tau_{b,i}\)	Bond lifetime for species \(i\) (s)
\(\eta_s\)	Solvent viscosity (Pa·s)
\(\mathbf{I}\)	Identity tensor (3x3)
\(\mathbf{D}\)	Rate-of-strain tensor: \(\mathbf{D} = (\boldsymbol{\kappa} + \boldsymbol{\kappa}^T)/2\)

Derived Quantities ¶

Symbol	Definition
\(G_{\text{total}}\)	Total plateau modulus: \(G_{\text{total}} = \sum_{i=1}^{N} G_i\)
\(\lambda_i\)	Relaxation time: \(\lambda_i = \tau_{b,i}\) (equivalent for single-mode TNT)
\(\eta_i\)	Viscosity contribution: \(\eta_i = G_i \tau_{b,i}\)
\(\eta_0\)	Zero-shear viscosity: \(\eta_0 = \sum_{i=1}^{N} G_i \tau_{b,i} + \eta_s\)
\(\omega_i\)	Characteristic frequency: \(\omega_i = 1/\tau_{b,i}\)

Protocol-Specific Variables ¶

Symbol	Definition
\(\dot{\gamma}\)	Shear rate (1/s, for flow curve and startup)
\(\omega\)	Angular frequency (rad/s, for oscillation)
\(\gamma_0\)	Strain amplitude (dimensionless, for LAOS)
\(\sigma_0\)	Applied stress (Pa, for creep)
\(G'(\omega)\)	Storage modulus (Pa, oscillation)
\(G''(\omega)\)	Loss modulus (Pa, oscillation)
\(G(t)\)	Relaxation modulus (Pa, relaxation)

The TNT Multi-Species model extends the single-mode Tanaka-Edwards transient network theory to materials with multiple distinct bond types. Unlike the Generalized Maxwell model (which uses empirical mode decomposition), this model assigns a physical interpretation to each mode: a chemically distinct bond population with its own modulus \(G_i\) and lifetime \(\tau_{b,i}\).

Physical picture:

Each bond type (species) forms an independent transient network
Species do not interconvert or couple during deformation
Each species has its own energy barrier for bond breakage, yielding distinct \(\tau_{b,i}\)
Total stress is the additive superposition of species contributions
Mathematically equivalent to multi-mode Maxwell, but with network interpretation

Relationship to Other Models ¶

Generalized Maxwell:

Identical mathematics for linear response (SAOS, relaxation)
Different interpretation: TNT assigns physical meaning to each mode
Nonlinear regime: TNT includes conformation evolution; GMM is linear only

Single-Mode Tanaka-Edwards:

Special case: \(N=1\) reduces to base TNT
Extension: Multi-species adds independent bond populations

Sticky Rouse:

Different physics: Sticky Rouse has distributed lifetimes (Rouse modes)
Multi-species: Discrete lifetimes from distinct bond types

Loop-Bridge Model:

Coupling: Loop-Bridge has inter-species conversion (loop to bridge)
Multi-species: No coupling, independent evolution

Physical Foundations ¶

Multiple Bond Types ¶

Real network materials often contain chemically distinct bond types:

Hydrogen Bonds:

Typical \(E_a \sim 10\) to 30 kJ/mol
Lifetime \(\tau_b \sim 10^{-3}\) to \(10^{-1}\) s at room temperature
Common in protein gels, synthetic polymers with H-bonding groups

Hydrophobic Interactions:

Typical \(E_a \sim 20\) to 50 kJ/mol
Lifetime \(\tau_b \sim 0.1\) to 10 s
Entropy-driven, temperature-sensitive

Ionic Bonds:

Typical \(E_a \sim 30\) to 80 kJ/mol
Lifetime \(\tau_b \sim 1\) to \(10^3\) s
Strong dependence on ionic strength

Covalent Crosslinks:

Typical \(E_a > 150\) kJ/mol
Lifetime \(\tau_b \to \infty\) (permanent on experimental timescales)
Provides long-time elasticity

Each bond type has a different energy landscape, yielding different Arrhenius factor:

\[\tau_{b,i} = \tau_0 \exp\left(\frac{E_{a,i}}{k_B T}\right)\]

Independent Evolution ¶

The model assumes no coupling between species:

No conversion: Bonds of type 1 do not convert to type 2
No cooperative breaking: Breaking of one species does not affect others
Independent kinetics: Each species obeys its own relaxation equation
Additive stress: Total stress = sum of species contributions (linear superposition)

This is valid when:

Bond types are chemically distinct (different functional groups)
Timescale separation is sufficient (\(\tau_{b,i}/\tau_{b,j} > 10\))
No synergistic effects (e.g., one bond type does not stabilize another)
Mean-field applies per species (high bond density within each type)

Stress Additivity ¶

Total stress is the sum of contributions from each species:

\[\boldsymbol{\sigma}_{\text{total}} = \sum_{i=1}^{N} \boldsymbol{\sigma}_i + \boldsymbol{\sigma}_{\text{solvent}}\]

where:

\[ \begin{align}\begin{aligned}\boldsymbol{\sigma}_i = G_i (\mathbf{S}_i - \mathbf{I})\\\boldsymbol{\sigma}_{\text{solvent}} = 2\eta_s \mathbf{D}\end{aligned}\end{align} \]

Physical interpretation:

\(G_i\) is the modulus from species \(i\) alone (proportional to number density of type-\(i\) bonds)
\(\mathbf{S}_i - \mathbf{I}\) is the extra stress from deformation of species \(i\) network
No cross-terms between species (independence assumption)

Governing Equations ¶

Per-Species Evolution ¶

Each species evolves according to the Tanaka-Edwards equation independently:

\[\frac{d\mathbf{S}_i}{dt} = \boldsymbol{\kappa} \cdot \mathbf{S}_i + \mathbf{S}_i \cdot \boldsymbol{\kappa}^T - \frac{1}{\tau_{b,i}} (\mathbf{S}_i - \mathbf{I})\]

where:

\(\mathbf{S}_i\): Conformation tensor for species \(i\) (3x3 symmetric)
\(\boldsymbol{\kappa} = \nabla \mathbf{v}\): Velocity gradient tensor
\(\tau_{b,i}\): Bond lifetime for species \(i\)
\(\mathbf{I}\): Identity tensor (equilibrium conformation)

Component form (simple shear \(\dot{\gamma}\)):

\[ \begin{align}\begin{aligned}\frac{dS_{xx,i}}{dt} &= 2\dot{\gamma} S_{xy,i} - \frac{1}{\tau_{b,i}}(S_{xx,i} - 1)\\\frac{dS_{yy,i}}{dt} &= -\frac{1}{\tau_{b,i}}(S_{yy,i} - 1)\\\frac{dS_{zz,i}}{dt} &= -\frac{1}{\tau_{b,i}}(S_{zz,i} - 1)\\\frac{dS_{xy,i}}{dt} &= \dot{\gamma} S_{yy,i} - \frac{1}{\tau_{b,i}} S_{xy,i}\end{aligned}\end{align} \]

Total Stress ¶

The total stress tensor is:

\[\boldsymbol{\sigma} = \sum_{i=1}^{N} G_i (\mathbf{S}_i - \mathbf{I}) + 2\eta_s \mathbf{D}\]

For simple shear (measuring \(\sigma_{xy}\)):

\[\sigma(t) = \sum_{i=1}^{N} G_i S_{xy,i}(t) + \eta_s \dot{\gamma}(t)\]

State Vector Representation ¶

For numerical integration, the state is represented as a 4N-dimensional vector:

\[\mathbf{y}(t) = [S_{xx,1}, S_{yy,1}, S_{zz,1}, S_{xy,1}, \ldots, S_{xx,N}, S_{yy,N}, S_{zz,N}, S_{xy,N}]^T\]

This enables efficient vectorization via JAX vmap over species index.

Analytical Solutions: SAOS ¶

For small-amplitude oscillatory shear \(\gamma(t) = \gamma_0 \sin(\omega t)\), the complex modulus is:

\[G^*(\omega) = \sum_{i=1}^{N} \frac{G_i (\omega \tau_{b,i})^2 + i G_i \omega \tau_{b,i}}{1 + (\omega \tau_{b,i})^2} + i\omega\eta_s\]

Storage modulus:

\[G'(\omega) = \sum_{i=1}^{N} \frac{G_i (\omega \tau_{b,i})^2}{1 + (\omega \tau_{b,i})^2}\]

Loss modulus:

\[G''(\omega) = \sum_{i=1}^{N} \frac{G_i \omega \tau_{b,i}}{1 + (\omega \tau_{b,i})^2} + \omega\eta_s\]

Key features:

Each species contributes a Maxwell peak at \(\omega_i = 1/\tau_{b,i}\)
\(G'(\omega)\) transitions from \(G_{\text{total}}\) (high \(\omega\)) to 0 (low \(\omega\)) through plateaus
\(G''(\omega)\) shows \(N\) distinct peaks if \(\tau_{b,i}\) are well-separated

Analytical Solutions: Relaxation ¶

For step strain \(\gamma(t) = \gamma_0 H(t)\) (Heaviside function), the relaxation modulus is:

\[G(t) = \sum_{i=1}^{N} G_i \exp\left(-\frac{t}{\tau_{b,i}}\right)\]

Multi-exponential decay with \(N\) discrete relaxation times.

Analytical Solutions: Flow Curve ¶

For steady simple shear at rate \(\dot{\gamma}\), the steady-state shear stress is:

\[\sigma_{\infty}(\dot{\gamma}) = \sum_{i=1}^{N} \frac{G_i \tau_{b,i} \dot{\gamma}}{1 + (\tau_{b,i} \dot{\gamma})^2} + \eta_s \dot{\gamma}\]

Viscosity:

\[\eta(\dot{\gamma}) = \sum_{i=1}^{N} \frac{G_i \tau_{b,i}}{1 + (\tau_{b,i} \dot{\gamma})^2} + \eta_s\]

Limiting behavior:

Zero-shear viscosity: \(\eta_0 = \sum_{i=1}^{N} G_i \tau_{b,i} + \eta_s\)
High-shear viscosity: \(\eta_{\infty} = \eta_s\) (all networks relax)
Shear thinning: Each species contributes thinning at \(\dot{\gamma}_i \sim 1/\tau_{b,i}\)

Parameter Table ¶

Comprehensive Parameter List ¶

Parameter	Symbol	Typical Range	Units	Description
Species moduli	\(G_i\)	\(10^2\) to \(10^5\)	Pa	Modulus contribution from species \(i\) (\(N\) values)
Species lifetimes	\(\tau_{b,i}\)	\(10^{-3}\) to \(10^3\)	s	Bond lifetime for species \(i\) (\(N\) values)
Solvent viscosity	\(\eta_s\)	0.0 to \(10^{-2}\)	Pa·s	Background Newtonian viscosity (1 value)
Number of species	\(N\)	2 to 5	dimensionless	Number of bond types (constructor argument)

Total parameters: \(2N + 1\)

Parameter Interpretation ¶

Number of Species (N)¶

Physical meaning:

\(N\) = number of chemically distinct bond types
Each species represents a different molecular interaction or crosslink

Common cases:

N = 1:

Reduces to single-mode Tanaka-Edwards
Single bond type (homogeneous network)

N = 2:

Most common for dual-crosslinked materials
Example: Fast physical + slow chemical crosslinks

N = 3:

Triple network or three interaction types

N >= 4:

High parameter count (>= 9 parameters)
Risk of overfitting without sufficient data

Species Moduli (G_i)¶

Physical meaning:

\(G_i\) is proportional to the number density of type-\(i\) bonds
Reflects the strength of species \(i\) contribution to total elasticity

Relative contributions:

\[f_i = \frac{G_i}{\sum_{j=1}^{N} G_j}\]

where \(f_i\) is the fraction of total modulus from species \(i\).

Species Lifetimes (tau_b_i)¶

Physical meaning:

\(\tau_{b,i}\) is the average time before a type-\(i\) bond breaks
Inversely related to bond breakage rate: \(k_{\text{break},i} = 1/\tau_{b,i}\)

Timescale separation:

Distinct peaks in \(G''(\omega)\) require \(\tau_{b,i}/\tau_{b,j} > 10\)
If \(\tau_{b,i}/\tau_{b,j} < 3\), species \(i\) and \(j\) may be indistinguishable

Validity and Assumptions ¶

Core Assumptions ¶

Independent species: - No conversion between bond types - No cooperative breaking or stabilization - Each species evolves according to its own kinetics
Mean-field per species: - High bond density within each type - No spatial correlations between species - Uniform distribution of each bond type
Gaussian chains: - All species obey Gaussian network statistics - Valid for small to moderate deformations - Breaks down for chain stretching (strain > 100 percent)
Constant breakage rate: - \(1/\tau_{b,i}\) independent of stress (affine network) - No stress-assisted bond breaking (unlike Sticky Rouse)
Affine deformation: - Chains deform with the bulk medium - No chain slippage or retraction

Regimes ¶

Frequency Regimes (SAOS)¶

High frequency (\(\omega \gg 1/\tau_{b,\text{min}}\)):

All species are elastic (no relaxation)
\(G'(\omega) \approx \sum_{i=1}^{N} G_i = G_{\text{total}}\)
\(G''(\omega) \approx \omega \eta_s\) (if \(\eta_s > 0\))

Intermediate frequency (\(1/\tau_{b,i} < \omega < 1/\tau_{b,i-1}\)):

Slowest \(i\) species have relaxed, faster species still elastic
\(G'(\omega) \approx \sum_{j=i}^{N} G_j\) (partial sum)
\(G''(\omega)\) shows local minimum between peaks

Low frequency (\(\omega \ll 1/\tau_{b,\text{max}}\)):

All species have relaxed (if all bonds are transient)
\(G'(\omega) \sim \omega^2\), \(G''(\omega) \sim \omega\) (terminal regime)
\(\eta_0 = \sum_{i=1}^{N} G_i \tau_{b,i} + \eta_s\)

What You Can Learn ¶

Structural Information ¶

Number of bond types:

\(N\) determined from number of distinct peaks in \(G''(\omega)\)
Requires wide frequency range (3+ decades)

Bond population fractions:

\(f_i = G_i / G_{\text{total}}\) quantifies relative importance
Dominant species (\(f_i > 0.5\)) controls long-time behavior

Lifetime hierarchy:

Order of \(\tau_{b,i}\) reveals fastest to slowest bond types
Lifetime ratios \(\tau_{b,i}/\tau_{b,j}\) indicate timescale separation

Experimental Design ¶

Essential Measurements ¶

1. Wide-frequency SAOS (critical):

Range: At least 3 decades to resolve \(N=2\) modes, 4+ decades for \(N=3\)
Example: 0.01 to 100 rad/s for \(\tau_{b,1} = 0.1\) s, \(\tau_{b,2} = 10\) s
Why: Each mode needs approximately 1 decade around \(\omega_i = 1/\tau_{b,i}\) to be resolved

2. Stress relaxation (recommended):

Protocol: Step strain \(\gamma_0 = 1\) percent to 10 percent (linear regime)
Duration: \(t_{\max} > 10 \tau_{b,\text{max}}\) to capture slowest decay
Why: Direct measurement of \(G(t) = \sum_i G_i \exp(-t/\tau_{b,i})\)

3. Flow curve (optional but valuable):

Range: \(\dot{\gamma}\) from \(0.01/\tau_{b,\text{max}}\) to \(10/\tau_{b,\text{min}}\)
Why: Validates nonlinear predictions, reveals shear thinning transitions

Computational Implementation ¶

Efficient Multi-Mode Computation ¶

Key strategy: Use JAX vmap to vectorize over species dimension.

State vector structure:

State is a 4N-dimensional flattened array: [S_xx_0, S_yy_0, S_zz_0, S_xy_0, S_xx_1, …, S_xy_N-1]

Vectorized ODE:

Each species evolves independently via the TNT equation. JAX vmap applies the single-species evolution to all species in parallel.

Benefits:

Parallel computation of all species (GPU-friendly)
Code reuse from single-mode TNT
Memory efficient (no loops)

Fitting Guidance ¶

Model Selection (Determining N)¶

Strategy:

Start with :math:`N=2` (most common, 5 parameters)
Fit to wide-frequency SAOS data
Increase :math:`N` if: - Residuals show systematic structure (e.g., extra peaks) - BIC or AIC significantly decreases - Visual inspection suggests additional modes

Bayesian Information Criterion (BIC):

\[\text{BIC} = n \ln\left(\frac{\text{RSS}}{n}\right) + k \ln(n)\]

where \(n\) = number of data points, \(k = 2N+1\) = number of parameters, RSS = residual sum of squares.

Decision rule:

If delta BIC = BIC(N+1) - BIC(N) < -10: Strong evidence for \(N+1\)
If -10 < delta BIC < -2: Weak evidence for \(N+1\)
If delta BIC > -2: No evidence, keep \(N\)

Practical limit: \(N \leq 5\) (11 parameters) for most datasets.

Primary Data Source ¶

SAOS is best for parameter estimation:

Analytical solution (fast, no ODE integration)
Directly reveals \(N\) distinct peaks in \(G''(\omega)\)
Linear regime (simple interpretation)

Recommended workflow:

Fit SAOS first to get initial \(G_i\), \(\tau_{b,i}\), \(\eta_s\)
Validate with relaxation (if available) to confirm \(\tau_{b,i}\)
Test predictions on flow curve, startup (nonlinear validation)

Usage Examples ¶

Basic Fitting (N=2)¶

from rheojax.models.tnt import TNTMultiSpecies
from rheojax.core import RheoData
import numpy as np

# Experimental data
omega = np.logspace(-2, 2, 50)
G_star = measure_saos(omega)

# Create model with 2 species (dual-crosslinked)
model = TNTMultiSpecies(n_species=2)

# Fit to SAOS data
rheo_data = RheoData(x=omega, y=G_star, test_mode='oscillation')
result = model.fit(rheo_data)

# Inspect fitted parameters
params = model.get_parameters()
print(f"G_1 = {params['G_1'].value:.1f} Pa")
print(f"tau_b_1 = {params['tau_b_1'].value:.3f} s")

Model Selection (N=2 vs N=3)¶

# Fit with N=2
model_2 = TNTMultiSpecies(n_species=2)
result_2 = model_2.fit(rheo_data)
rss_2 = result_2.residual_sum_of_squares
k_2 = 5
bic_2 = len(omega) * np.log(rss_2 / len(omega)) + k_2 * np.log(len(omega))

# Fit with N=3
model_3 = TNTMultiSpecies(n_species=3)
result_3 = model_3.fit(rheo_data)
rss_3 = result_3.residual_sum_of_squares
k_3 = 7
bic_3 = len(omega) * np.log(rss_3 / len(omega)) + k_3 * np.log(len(omega))

# Compare
if bic_3 - bic_2 < -10:
    print("Strong evidence for N=3")
else:
    print("N=2 is sufficient")

Permanent + Transient Dual Network ¶

The multi-species model naturally accommodates networks with both permanent (chemical) and transient (physical) crosslinks. This is the most common application of the multi-species framework, representing materials such as dual-crosslinked hydrogels, hybrid organogels, and biological tissues with both covalent and non-covalent bonds.

Total Stress Decomposition ¶

The total stress separates into a non-relaxing permanent contribution and transient species contributions:

\[\boldsymbol{\sigma}(t) = G_{\text{chem}} \left(\mathbf{B} - \mathbf{I}\right) + \sum_{i=1}^{N_{\text{phys}}} G_i \left(\mathbf{S}_i(t) - \mathbf{I}\right) + 2\eta_s \mathbf{D}\]

where \(G_{\text{chem}}\) is the modulus of the permanent (chemical) network and \(\mathbf{B}\) is the left Cauchy-Green deformation tensor. The permanent species has \(\tau_{b} \to \infty\) (no breakage), contributing a non-relaxing elastic plateau.

Stress Relaxation With Permanent Bonds ¶

Under step strain, the relaxation modulus approaches a non-zero equilibrium:

\[G(t) = G_{\text{chem}} + \sum_{i=1}^{N_{\text{phys}}} G_i \exp\left(-\frac{t}{\tau_{b,i}}\right) \quad \xrightarrow{t \to \infty} \quad G_{\text{chem}}\]

The stress never fully relaxes to zero. This is the fundamental difference from purely physical (transient) networks: the permanent bonds provide a long-time elastic restoring force, so that:

\[\sigma(t) = G_{\text{chem}} \cdot \gamma_0 + \sum_{i=1}^{N_{\text{phys}}} G_i \exp\left(-\frac{t}{\tau_{b,i}}\right) \cdot \gamma_0 \quad \xrightarrow{t \to \infty} \quad G_{\text{chem}} \cdot \gamma_0\]

Mathematically, a “permanent” species simply means \(\tau_{b,i} \gg t_{\text{exp}}\), where \(t_{\text{exp}}\) is the experimental timescale. In practice, any species with \(\tau_{b,i}\) exceeding the longest measurement time by a factor of \(\sim 100\) is effectively permanent.

Creep Saturation ¶

Unlike purely transient networks which flow indefinitely under constant stress, dual networks with permanent bonds exhibit creep saturation: the strain reaches a finite asymptote rather than growing without bound.

Strain Saturation ¶

Under constant applied stress \(\sigma_0\), the strain saturates at:

\[\gamma_{\infty} = \frac{\sigma_0}{G_{\text{chem}}}\]

No steady-state flow occurs because the permanent network acts as a spring in parallel with the transient species, providing an equilibrium restoring force that eventually balances the applied stress.

Creep Compliance ¶

The creep compliance for a dual network takes the form:

\[J(t) = \frac{1}{G_{\text{chem}}} - \sum_{i=1}^{N_{\text{phys}}} \frac{G_i}{G_{\text{chem}}\left(G_{\text{chem}} + \sum_j G_j\right)} \exp\left(-\frac{t}{\tau_i^*}\right)\]

where \(\tau_i^*\) are the retardation times (related to but distinct from the relaxation times \(\tau_{b,i}\)).

Physical insight: The strain asymptote directly gives the permanent network modulus:

\[G_{\text{chem}} = \frac{\sigma_0}{\gamma_{\infty}}\]

This provides a model-independent route to measuring the permanent network contribution from a single creep experiment.

“Double-Yielding” in LAOS ¶

When bond species have well-separated lifetimes, large amplitude oscillatory shear (LAOS) reveals sequential yielding — a phenomenon where different bond populations break at different strain amplitudes.

Sequential Yielding Mechanism ¶

The progression with increasing strain amplitude \(\gamma_0\) is:

Small \(\gamma_0\) (linear regime):: All species remain elastic. The Lissajous figure is a linear ellipse with stiffness reflecting \(G_{\text{total}} = \sum_i G_i\).
Intermediate \(\gamma_0\) (first yielding):: Weak bonds (short \(\tau_{b}\)) break and reform within a cycle. The first yielding event occurs when \(\gamma_0 \dot{\gamma}_{\max} \sim 1/\tau_{b,\text{weak}}\). The Lissajous figure begins to distort, showing the onset of nonlinearity.
Large \(\gamma_0\) (second yielding):: Strong bonds (long \(\tau_{b}\)) also break, leading to a second yielding event. The material becomes fully fluidized, and the Lissajous figure shows a characteristic butterfly or rectangular shape.

LAOS Signatures ¶

The Lissajous figure shows two distinct stiffness changes as the strain amplitude increases, corresponding to the progressive loss of each bond population.

This “double-yielding” signature is also visible in the intensity ratio \(I_{3/1}\) (third harmonic normalized by fundamental) plotted against \(\gamma_0\): for \(N=2\) species, \(I_{3/1}(\gamma_0)\) shows two peaks, one per bond type. Each peak corresponds to the onset of nonlinear response from a specific bond population. More generally, \(N\) well-separated species produce \(N\) peaks in \(I_{3/1}(\gamma_0)\).

Progressive Loss Under Flow ¶

The flow curve of a multi-species network shows staged shear thinning: each bond species yields at a different critical shear rate, producing a characteristic stepped viscosity profile.

Staged Shear Thinning ¶

The critical shear rate for each species is:

\[\dot{\gamma}_{c,i} \approx \frac{1}{\tau_{b,i}}\]

The flow curve exhibits distinct regimes as the shear rate increases:

Low rates (\(\dot{\gamma} \ll 1/\tau_{b,\max}\)):: All bond species are intact. The full network modulus \(G_{\text{total}} = \sum_i G_i\) is active, and the material exhibits Newtonian behavior with zero-shear viscosity \(\eta_0 = \sum_i G_i \tau_{b,i} + \eta_s\).
Intermediate rates (\(1/\tau_{b,i+1} < \dot{\gamma} < 1/\tau_{b,i}\)):: Species \(i\) (and all weaker species) have yielded. The effective modulus is reduced to \(G_{\text{eff}} = \sum_{j>i} G_j\), and the viscosity contribution from species \(i\) is lost.
High rates (\(\dot{\gamma} \gg 1/\tau_{b,\min}\)):: All transient species have yielded. Only the solvent viscosity remains: \(\sigma \to \eta_s \cdot \dot{\gamma}\).

Staircase Flow Curve ¶

On a log-log plot, the viscosity \(\eta(\dot{\gamma})\) shows a staircase pattern with \(N\) steps, each corresponding to the loss of one bond type. The height of each step is proportional to \(G_i \tau_{b,i}\), and the transition between steps occurs at \(\dot{\gamma} \approx 1/\tau_{b,i}\).

This staircase structure is a direct fingerprint of the discrete multi-species nature and distinguishes it from continuous spectrum models (e.g., KWW stretched exponential) which produce smooth power-law thinning.

Failure Mode: Bond Hierarchy ¶

Failure Mode

Mechanism: Under increasing deformation, bond types fail sequentially from weakest (shortest \(\tau_{b}\)) to strongest (longest \(\tau_{b}\)). This progressive failure cascade determines the material’s nonlinear response.

Signature: Multi-step yielding in flow curves and LAOS, visible as multiple inflection points in \(\eta(\dot{\gamma})\) and multiple peaks in \(I_{3/1}(\gamma_0)\).

Physical origin: Each bond species has a characteristic force scale; exceeding it causes that species to dynamically break faster than it can reform. The critical rate for species \(i\) is \(\dot{\gamma}_{c,i} \approx 1/\tau_{b,i}\).

Implication: The material’s response depends on which bonds have “survived” at a given rate — the effective constitutive law changes character as bonds are progressively lost. At low rates the material behaves as a full \(N\)-species network; at high rates only the strongest (or permanent) species remain.

Design insight: By tuning the ratio \(G_{\text{strong}}/G_{\text{weak}}\) and \(\tau_{\text{strong}}/\tau_{\text{weak}}\), one can engineer materials with prescribed yielding sequences. This is important for self-healing materials (where weak bonds sacrifice first to protect the network), drug delivery gels (where sequential release requires staged degradation), and impact-absorbing materials (where energy dissipation is maximized by progressive bond failure).

API Reference ¶

class rheojax.models.tnt.TNTMultiSpecies(n_species=2)[source]¶

Bases: TNTBase

Multi-species Transient Network Theory model.

Implements a network with N independent bond populations, each with its own modulus G_i and lifetime τ_b_i. The total stress is a superposition of N Maxwell modes.

This is equivalent to a generalized Maxwell model where each mode represents a distinct physical crosslink species rather than a mathematical decomposition.

Parameters:: n_species (int) – Number of distinct bond species (N ≥ 1)

parameters¶

Model parameters: [G_0, tau_b_0, G_1, tau_b_1, …, G_{N-1}, tau_b_{N-1}, eta_s]

Type:: ParameterSet

fitted_¶

Whether the model has been fitted

Type:: bool

_n_species¶

Number of species

Type:: int

Notes

The state vector has 4*N components:: [S_xx_0, S_yy_0, S_zz_0, S_xy_0, …, S_xy_{N-1}]

Each species evolves independently via the upper-convected derivative with constant breakage/creation rates.

References ¶

Foundational Theory ¶

Tanaka, F., & Edwards, S. F. (1992). Viscoelastic properties of physically crosslinked networks: Transient network theory. Macromolecules, 25(5), 1516-1523. DOI: 10.1021/ma00031a024
Rubinstein, M., & Semenov, A. N. (2001). Dynamics of entangled solutions of associating polymers. Macromolecules, 34(4), 1058-1068. DOI: 10.1021/ma0013049

Dual-Crosslinked Networks ¶

Narita, T., Mayumi, K., Ducouret, G., & Hébraud, P. (2013). Viscoelastic properties of poly(vinyl alcohol) hydrogels having permanent and transient cross-links studied by microrheology, classical rheometry, and dynamic light scattering. Macromolecules, 46(10), 4174-4183. DOI: 10.1021/ma400600f
Sun, J.-Y., Zhao, X., Illeperuma, W. R. K., et al. (2012). Highly stretchable and tough hydrogels. Nature, 489(7414), 133-136. DOI: 10.1038/nature11409

Reversible Network Rheology ¶

Skrzeszewska, P. J., Sprakel, J., de Wolf, F. A., et al. (2010). Fracture and self-healing in a well-defined self-assembled polymer network. Macromolecules, 43(7), 3542-3548. DOI: 10.1021/ma1000173
Long, R., Mayumi, K., Creton, C., Narita, T., & Hui, C.-Y. (2014). Time dependent behavior of a dual cross-link self-healing gel: Theory and experiments. Macromolecules, 47(20), 7243-7250. DOI: 10.1021/ma501290h