TNT Knowledge Extraction Guide

This guide shows how to extract physical knowledge from fitted TNT (Transient Network Theory) model parameters. Learn how to map fitted parameters to physical properties, classify materials, apply scaling laws, and use diagnostic decision trees to select the right TNT variant.

Overview

What Knowledge Can Be Extracted

TNT models provide access to fundamental network physics through fitted parameters:

Structural Properties

  • Chain number density from plateau modulus

  • Entanglement density from network connectivity

  • Chain length from extensibility limits

  • Network topology from bridge/loop fractions

Kinetic Properties

  • Activation energies from temperature dependence

  • Force sensitivity from shear-rate dependence

  • Bond lifetime distributions from relaxation spectra

  • Mechanochemical coupling strengths

Material Classification

  • Network type: physical vs chemical crosslinks

  • Polymer architecture: linear, telechelic, miktoarm

  • Flow regime: dilute, semi-dilute, concentrated

  • Transition identification: gel point, glass transition

Predictive Capabilities

  • Nonlinear rheology from linear viscoelastic data

  • Temperature-rate superposition

  • Concentration scaling

  • Processing window optimization

How to Use This Guide

  1. Fit your TNT model using TNT Protocol Equations — Shared Reference

  2. Extract parameter values (point estimates or posteriors)

  3. Apply parameter-to-physics maps (Section 2) to get physical properties

  4. Classify your material using decision trees (Sections 3, 5)

  5. Validate using scaling laws (Section 4)

  6. Compare models if needed (Section 6)

  7. Report results following Section 11

Cross-Model Comparison Strategy

TNT family has 10 variants. Decision process:

  1. Start with TNTSingleMode (simplest)

  2. Check residuals and physical reasonableness

  3. If systematic deviations, add complexity:

  4. Compare via WAIC/BIC (Bayesian) or AIC (NLSQ)

  5. Use simplest model within 2 WAIC units of best

Parameter-to-Physics Map

This section provides explicit formulas to convert fitted TNT parameters into physical material properties.

Primary TNT Parameters

Core TNT Parameter Interpretations

Parameter

Symbol

Physical Meaning

Typical Range

Plateau Modulus

\(G\)

Network stiffness, entropic elasticity

10 Pa (dilute) to 100 kPa (gel)

Breakage Time

\(\tau_b\)

Bond lifetime at zero stress

\(10^{-6}\) s (micelles) to \(10^3\) s (reversible gels)

Solvent Viscosity

\(\eta_s\)

Matrix contribution to viscosity

0.001 Pa·s (water) to 10 Pa·s (polymer melt)

Derived Physical Properties

Chain Number Density

From rubber elasticity theory:

\[n_{\text{chains}} = \frac{G}{k_B T}\]

where \(k_B = 1.380649 \times 10^{-23}\) J/K is Boltzmann constant and \(T\) is absolute temperature (Kelvin).

Example: \(G = 1000\) Pa at \(T = 298\) K gives \(n_{\text{chains}} = 2.43 \times 10^{23}\) m-3.

Molar Concentration:

\[c_{\text{chains}} = \frac{n_{\text{chains}}}{N_A} = \frac{G}{k_B T N_A} = \frac{G}{RT}\]

where \(R = 8.314\) J/(mol·K) and \(c_{\text{chains}}\) is in mol/m3.

Activation Energy

If \(\tau_b\) measured at multiple temperatures, use Arrhenius:

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

Solve for activation energy:

\[E_a = k_B T \ln\left(\frac{\tau_b}{\tau_0}\right)\]

where \(\tau_0 \sim 10^{-9}\) s is molecular attempt time (phonon frequency).

Example: \(\tau_b = 0.1\) s at \(T = 298\) K gives \(E_a \approx 0.97\) eV.

Alternative: Plot \(\ln(\tau_b)\) vs \(1/T\) to get slope \(E_a/k_B\).

Bell Model Extensions

See also: TNT Bell (Force-Dependent Breakage) — Handbook for full variant handbook.

Force Sensitivity Parameter

Bell Parameter Interpretation

Parameter

Symbol

Physical Meaning

Typical Range

Bell Parameter

\(\nu\)

Force sensitivity of bond breakage

0.1 (weak) to 10 (strong)

Characteristic Stress

\(\sigma_c = G\)

Stress scale for shear thinning

Same as \(G\)

Barrier Distance:

\[d_b = \frac{\nu k_B T}{F_{\text{char}}}\]

where \(F_{\text{char}} = \sigma_c / n_{\text{chains}}^{1/3}\) is characteristic force per chain.

Physical Interpretation:

  • \(\nu < 1\): bonds weakly force-sensitive (small barrier distance)

  • \(\nu \sim 1\): typical hydrogen bond

  • \(\nu > 5\): strongly force-sensitive (large conformational change)

FENE Model Extensions

See also: TNT FENE-P (Finite Extensibility) — Handbook for full variant handbook.

Chain Extensibility

FENE Parameter Interpretation

Parameter

Symbol

Physical Meaning

Typical Range

Max Extension

\(L_{\text{max}}\)

Maximum chain stretch

1.5 (slight) to 20 (highly extensible)

Number of Kuhn Segments:

\[N_K = L_{\text{max}}^2\]

Contour Length: For known Kuhn length \(b_K\):

\[L_c = N_K b_K = L_{\text{max}}^2 b_K\]

Example: \(L_{\text{max}} = 10\) gives \(N_K = 100\) segments. With \(b_K = 1\) nm, \(L_c = 100\) nm.

Onset of Stiffening: Strain at which FENE effects appear:

\[\gamma_{\text{onset}} \approx \frac{1}{L_{\text{max}}}\]

Non-Affine Model Extensions

See also: TNT Non-Affine (Gordon-Schowalter) — Handbook for full variant handbook.

Entanglement Coupling

Non-Affine Parameter Interpretation

Parameter

Symbol

Physical Meaning

Typical Range

Non-Affine Parameter

\(\xi\)

Degree of non-affine deformation

-0.5 (slip) to 0.5 (affine)

Second Normal Stress Coefficient:

\[\Psi_2 = -\frac{G \tau_b^2 \xi}{2}\]

Interpretation:

  • \(\xi = 0\): affine (standard TNT, \(N_2 = 0\))

  • \(\xi > 0\): chain slip suppressed (\(N_2 < 0\))

  • \(\xi < 0\): enhanced slip (\(|N_2|\) larger)

Stretch-Creation Model Extensions

See also: TNT Stretch-Creation (Enhanced Reformation) — Handbook for full variant handbook.

Mechanochemical Coupling

Stretch-Creation Parameter Interpretation

Parameter

Symbol

Physical Meaning

Typical Range

Coupling Strength

\(\kappa\)

Stretch-induced bond creation rate

0 (no coupling) to 5 (strong)

Creation Enhancement:

\[k_c(\lambda) = k_c^0 \exp(\kappa \lambda)\]

where \(\lambda\) is chain stretch and \(k_c^0 = 1/\tau_c\) is baseline creation rate.

Physical Meaning:

  • \(\kappa = 0\): no mechanochemical coupling

  • \(\kappa > 0\): strain stiffening via bond creation

  • \(\kappa < 0\): strain softening (rare)

Loop-Bridge Model Extensions

See also: TNT Loop-Bridge (Two-Species Kinetics) — Handbook for full variant handbook.

Network Topology

Loop-Bridge Parameter Interpretation

Parameter

Symbol

Physical Meaning

Typical Range

Association Time

\(\tau_a\)

Time for chain end to find binding site

\(10^{-3}\) to \(10^2\) s

Equilibrium Bridge Fraction

\(f_{B,\text{eq}}\)

Fraction of chains forming bridges at rest

0.1 (mostly loops) to 0.9 (mostly bridges)

Network Connectivity:

\[\nu_{\text{eff}} = f_{B,\text{eq}} \nu_{\text{total}}\]

where \(\nu_{\text{total}} = G/(k_B T)\) is total chain density.

Shear Thickening Onset:

\[\dot{\gamma}_{\text{thick}} \sim \frac{1}{\tau_a}\]

Cates Model Extensions

See also: TNT Cates (Living Polymers / Wormlike Micelles) — Handbook for full variant handbook.

Living Polymer Parameters

Cates Parameter Interpretation

Parameter

Symbol

Physical Meaning

Typical Range

Reptation Time

\(\tau_{\text{rep}}\)

Time for chain to diffuse its length

\(10^{-2}\) to \(10^2\) s

Breakage Time

\(\tau_{\text{break}}\)

Time for chain scission

\(10^{-4}\) to 1 s

Effective Relaxation Time (geometric mean):

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

Average Micelle Length:

\[L_{\text{avg}} \sim \sqrt{\frac{\tau_{\text{rep}}}{\tau_{\text{break}}}}\]

Scission Energy: From temperature dependence of \(\tau_{\text{break}}\).

Sticky Rouse Extensions

See also: TNT Sticky Rouse (Multi-Mode Sticker Dynamics) — Handbook for full variant handbook.

Sticker Dynamics

Sticky Rouse Parameter Interpretation

Parameter

Symbol

Physical Meaning

Typical Range

Rouse Times

\(\tau_{R,p}\)

Rouse mode \(p\) relaxation time

\(10^{-6}\) to 0.1 s

Sticker Time

\(\tau_s\)

Sticker dissociation time

\(10^{-4}\) to 10 s

Effective Mode Relaxation:

\[\tau_{p,\text{eff}} = \tau_{R,p} + \tau_s\]

Number of Kuhn Segments from Rouse time:

\[N \sim \left(\frac{\tau_R \eta_s}{k_B T}\right)^{1/2}\]

Material Classification

This section provides decision trees to identify material type from fitted TNT parameters.

Network Density Classification

Based on plateau modulus \(G\):

Plateau Modulus Decision Tree
==============================

G < 100 Pa
├─> Dilute Network
│   ├─> Likely: Simple associating polymer, transient gel
│   ├─> Examples: Low-concentration HEUR, weakly crosslinked
│   └─> Check: c < c* (overlap concentration)

100 Pa ≤ G < 10,000 Pa
├─> Semi-Dilute to Concentrated
│   ├─> Likely: HEUR, telechelic polymers, colloidal gels
│   ├─> Examples: PEO-PPO-PEO, hydrophobically modified polymers
│   └─> Check: c* < c < c** (entanglement concentration)

G ≥ 10,000 Pa
├─> Dense Network or Gel
│   ├─> Likely: Reversible elastomer, dense colloidal gel
│   ├─> Examples: Vitrimers, dense microgels
│   └─> Check: Gel point passed, G' > G'' at all frequencies

Quantitative Criteria:

  • Dilute: \(c < c^* \sim 1/[N b^3]\) (overlap)

  • Semi-dilute: \(c^* < c < c^{**} \sim 1/N_e b^3\) (entanglement)

  • Concentrated: \(c > c^{**}\)

where \(N\) is degree of polymerization, \(b\) is segment length, \(N_e\) is entanglement length.

Relaxation Time Classification

Based on breakage time \(\tau_b\):

Breakage Time Decision Tree
============================

τ_b < 0.001 s
├─> Fast Dynamics
│   ├─> Likely: Wormlike micelles, weak hydrogen bonds
│   ├─> Examples: CTAB/NaSal, PEO/tannic acid
│   └─> Accessible frequency range: f > 1000 Hz (SAOS challenging)

0.001 s ≤ τ_b < 1 s
├─> Intermediate Dynamics
│   ├─> Likely: Typical telechelic, moderate H-bonds
│   ├─> Examples: HEUR in water, supramolecular polymers
│   └─> Accessible: Standard SAOS (0.01 - 100 Hz)

1 s ≤ τ_b < 1000 s
├─> Slow Dynamics
│   ├─> Likely: Reversible gels, strong physical crosslinks
│   ├─> Examples: Vitrimers, ionomers
│   └─> Accessible: Creep, stress relaxation preferred

τ_b ≥ 1000 s
├─> Very Slow or Permanent
│   ├─> Likely: Near-permanent network, glassy
│   └─> Check: Is network truly reversible? May need chemical gel model

Shear Response Classification

Based on flow curve shape (from Bell, FENE-P, or loop-bridge fits):

Flow Curve Decision Tree
=========================

Shear Thinning (η decreases with γ̇)
├─> Bell Parameter ν > 1
│   ├─> Likely: Force-sensitive bonds (H-bonds, metal-ligand)
│   ├─> Power-law region: η ~ γ̇^(-α), α = ν/(ν+1)
│   └─> Onset: γ̇ ~ 1/τ_b

Shear Thickening (η increases with γ̇)
├─> Loop-Bridge f_B,eq increases
│   ├─> Likely: Telechelic with free ends (loops → bridges)
│   ├─> Onset: γ̇ ~ 1/τ_a
│   └─> Check: Does thickening saturate or diverge?

Strain Hardening (σ increases with γ at fixed γ̇)
├─> FENE L_max finite
│   ├─> Likely: Extensible chains (not infinitely stretchable)
│   ├─> Onset: γ ~ 1/L_max
│   └─> Alternative: Stretch-creation κ > 0

Oscillatory Response Classification

Based on SAOS Cole-Cole plot (\(G''\) vs \(G'\)):

SAOS Decision Tree
==================

Perfect Semicircle
├─> Cates Living Polymer
│   ├─> Single effective relaxation time τ_d
│   ├─> τ_rep and τ_break coupled
│   └─> G_max'' = G/2 at ω = 1/τ_d

Partial Semicircle (truncated at low ω)
├─> Multi-Species or Sticky Rouse
│   ├─> Multiple relaxation times
│   ├─> Fit spectrum of {G_i, τ_i}
│   └─> Check: Is there a slow mode not fully relaxed?

Skewed Arc
├─> Non-Affine (ξ ≠ 0)
│   ├─> Entanglements affect loss modulus
│   └─> Or: Distributed relaxation times (polydispersity)

No Semicircle (G'' increases monotonically)
├─> Not Single-Mode TNT
│   ├─> Try: Fractional models (KWW, Cole-Cole)
│   └─> Or: Broad distribution (multi-mode)

Tip

Variant Handbooks: For detailed physics, equations, protocol predictions, and failure modes of each variant, see:

Polymer Architecture Inference

Architecture from TNT Parameters

Architecture

TNT Signature

Example Systems

Linear Associating

Single-mode, moderate \(\tau_b\)

PEO with end groups

Telechelic

Loop-bridge, \(f_{B,\text{eq}} < 1\), shear thickening

HEUR, PEO-PPO-PEO

Miktoarm Star

Multi-species (different arms)

ABC star copolymers

Wormlike Micelle

Cates, \(\tau_d \ll \tau_{\text{rep}}\)

CTAB/NaSal

Reversible Gel

High \(G\), slow \(\tau_b\), FENE or stretch-creation

Vitrimers, ionomers

Supramolecular

Sticky Rouse, multiple \(\tau_s\)

Multi-H-bond systems

Scaling Laws

This section lists physical scaling relationships to validate fitted parameters and make predictions.

Fundamental Scaling Relations

Rubber Elasticity

Plateau modulus scales with chain density:

\[G \sim n_{\text{chains}} k_B T \sim c_{\text{polymer}} T\]

Validation: \(G/T\) should be roughly constant across temperatures (for entropic networks).

Concentration Scaling:

  • Dilute: \(G \sim c^{2.3}\) (scaling theory)

  • Semi-dilute: \(G \sim c^{2.0}\) (mean-field)

  • Concentrated: \(G \sim c^{2.0 - 2.3}\)

Arrhenius Kinetics

Breakage time temperature dependence:

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

Validation: Plot \(\ln(\tau_b)\) vs \(1/T\) to verify linearity and measure \(E_a\).

WLF Alternative (near \(T_g\)):

\[\log\left(\frac{\tau_b(T)}{\tau_b(T_0)}\right) = \frac{-C_1 (T - T_0)}{C_2 + T - T_0}\]

Use for glass-forming systems.

Viscosity Relations

Zero-Shear Viscosity

\[\eta_0 = G \tau_b + \eta_s\]

Validation: Measure \(\eta_0\) from flow curve plateau. Should match \(G \tau_b + \eta_s\) from SAOS fit.

Concentration Scaling:

\[\eta_0 \sim c^{3.0 - 3.9}\]

in semi-dilute regime (varies by polymer type).

Relaxation Time Scaling

For sticky Rouse:

\[\tau_R \sim N^2 \frac{\eta_s b^2}{k_B T}\]

where \(N\) is number of segments, \(b\) is segment size.

Validation: If \(N\) known, check \(\tau_R\) vs \(N^2\) scaling.

Normal Stress Relations

First Normal Stress Difference

In weak shear (Weissenberg number \(\text{Wi} \ll 1\)):

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

Validation: Plot \(N_1 / \dot{\gamma}^2\) vs \(\dot{\gamma}\) at low rates. Should plateau at \(2 G \tau_b^2\).

Normal Stress Coefficient:

\[\Psi_1 = \frac{N_1}{\dot{\gamma}^2} = 2 G \tau_b^2\]

Second Normal Stress Difference

For non-affine model:

\[N_2 = -G \tau_b^2 \xi \dot{\gamma}^2\]

Ratio:

\[\frac{N_2}{N_1} = -\frac{\xi}{2}\]

Typical Values: \(|N_2/N_1| \sim 0.1 - 0.3\) for polymer melts.

Oscillatory Shear Relations

Storage and Loss Moduli

Single-mode TNT:

\[G'(\omega) = G \frac{(\omega \tau_b)^2}{1 + (\omega \tau_b)^2}, \quad G''(\omega) = G \frac{\omega \tau_b}{1 + (\omega \tau_b)^2}\]

Crossover Frequency: \(G' = G''\) at

\[\omega_c = \frac{1}{\tau_b}\]

Validation: Measure \(\omega_c\) from SAOS. Should match \(1/\tau_b\) from fit.

High-Frequency Limit:

\[\lim_{\omega \to \infty} G'(\omega) = G\]

Low-Frequency Limit:

\[\lim_{\omega \to 0} G''(\omega) / \omega = \eta_0 = G \tau_b + \eta_s\]

Complex Viscosity

\[|\eta^*(\omega)| = \frac{\sqrt{G'^2 + G''^2}}{\omega}\]

Cox-Merz Rule (empirical, often holds for TNT):

\[\eta(\dot{\gamma}) \approx |\eta^*(\omega)| \quad \text{at } \dot{\gamma} = \omega\]

Validation: Overlay flow curve \(\eta(\dot{\gamma})\) with \(|\eta^*(\omega)|\). Deviations indicate Cox-Merz breakdown.

Cates Model Scaling

See also: TNT Cates (Living Polymers / Wormlike Micelles) — Handbook for the full Cates variant handbook.

Living Polymer Relations

Effective Relaxation Time:

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

Plateau Modulus: Standard rubber elasticity:

\[G = \frac{k_B T}{a^3} = c_{\text{polymer}} \frac{RT}{M_e}\]

where \(M_e\) is entanglement molecular weight.

Crossover: When \(\tau_{\text{break}} \ll \tau_{\text{rep}}\):

\[\tau_d \ll \tau_{\text{rep}} \quad \Rightarrow \quad \text{Rouse-like (no entanglements)}\]

When \(\tau_{\text{break}} \gg \tau_{\text{rep}}\):

\[\tau_d \approx \tau_{\text{rep}} \quad \Rightarrow \quad \text{Reptation-like (long chains)}\]

Sticky Rouse Scaling

See also: TNT Sticky Rouse (Multi-Mode Sticker Dynamics) — Handbook for the full Sticky Rouse variant handbook.

Effective Mode Times

For mode \(p\):

\[\tau_{p,\text{eff}} = \tau_{R,p} + \tau_s\]

Rouse Time Scaling:

\[\tau_{R,p} = \frac{\tau_R}{p^2}, \quad \tau_R = \frac{N^2 b^2 \eta_s}{3 \pi^2 k_B T}\]

Slowest Mode (\(p = 1\)):

\[\tau_1 = \tau_R + \tau_s\]

Fastest Mode (\(p \gg 1\)):

\[\tau_{p,\text{eff}} \approx \tau_s \quad \text{(sticker-limited)}\]

Bell Model Scaling

See also: TNT Bell (Force-Dependent Breakage) — Handbook for the full Bell variant handbook.

Shear Thinning Power Law

At intermediate shear rates \(1/\tau_b \ll \dot{\gamma} \ll \exp(\nu)/\tau_b\):

\[\eta(\dot{\gamma}) \sim \eta_0 \left(\frac{\dot{\gamma}}{\dot{\gamma}_0}\right)^{-\alpha}\]

where

\[\alpha = \frac{\nu}{\nu + 1}, \quad \dot{\gamma}_0 = \frac{1}{\tau_b}\]

Validation: Fit flow curve to power law, extract \(\alpha\), solve for \(\nu = \alpha/(1 - \alpha)\).

Limiting Cases:

  • \(\nu \to 0\): \(\alpha \to 0\) (Newtonian)

  • \(\nu \to \infty\): \(\alpha \to 1\) (strong shear thinning)

FENE Model Scaling

See also: TNT FENE-P (Finite Extensibility) — Handbook for the full FENE-P variant handbook.

Strain Hardening Onset

Stress upturn at strain:

\[\gamma_{\text{onset}} \sim \frac{1}{L_{\text{max}}}\]

Validation: In startup shear, identify \(\gamma\) where stress deviates from linear (constant-stress) regime.

Stress Enhancement:

\[\frac{\sigma(\gamma)}{\sigma_{\text{linear}}} \sim 1 + \left(\frac{\gamma}{1/L_{\text{max}}}\right)^2\]

for \(\gamma \lesssim 1/L_{\text{max}}\).

Diagnostic Decision Tree

This section provides a step-by-step flowchart to select the appropriate TNT variant based on experimental observations.

Master Decision Tree

TNT Variant Selection Flowchart
================================

START: You have rheological data (SAOS, flow curve, startup, or creep)

┌───────────────────────────────────────────────────────────────────┐
│ Step 1: MEASURE SAOS (Small-Amplitude Oscillatory Shear)         │
│ ----------------------------------------------------------------- │
│ Plot Cole-Cole: G'' vs G'                                        │
└───────────────────────────────────────────────────────────────────┘
           ↓
┌─────────────────────────────────────────────┐
│ Q1: Is the Cole-Cole plot a semicircle?    │
└─────────────────────────────────────────────┘
           ↓
     ┌─────┴─────┐
     │           │
    YES          NO
     │           │
     ↓           ↓
┌─────────┐  ┌──────────────────────────┐
│ CATES   │  │ Continue to Step 2       │
│ Model   │  │ (Not living polymer)     │
└─────────┘  └──────────────────────────┘
(Living         ↓
polymers)
             ┌───────────────────────────────────────────────────────┐
             │ Step 2: MEASURE FLOW CURVE                            │
             │ ----------------------------------------------------- │
             │ η vs γ̇ (steady shear viscosity)                      │
             └───────────────────────────────────────────────────────┘
                ↓
             ┌──────────────────────────────────────────┐
             │ Q2: Is there shear thickening?           │
             │ (η increases with γ̇ at high rates)      │
             └──────────────────────────────────────────┘
                ↓
           ┌────┴────┐
           │         │
          YES        NO
           │         │
           ↓         ↓
      ┌─────────┐  ┌──────────────────────────┐
      │ LOOP-   │  │ Continue to Step 3       │
      │ BRIDGE  │  │ (No shear thickening)    │
      └─────────┘  └──────────────────────────┘
      (Telechelic)    ↓

             ┌───────────────────────────────────────────────────────┐
             │ Step 3: MEASURE STARTUP SHEAR (Multiple Rates)       │
             │ ----------------------------------------------------- │
             │ σ vs γ at different γ̇                                │
             └───────────────────────────────────────────────────────┘
                ↓
             ┌──────────────────────────────────────────┐
             │ Q3: Is there strain stiffening?          │
             │ (σ increases faster than linear)         │
             └──────────────────────────────────────────┘
                ↓
           ┌────┴─────┐
           │          │
          YES         NO
           │          │
           ↓          ↓
      ┌─────────┐  ┌──────────────────────────┐
      │ Q3a:    │  │ Continue to Step 4       │
      │ Rate-   │  │ (No strain stiffening)   │
      │ depend? │  └──────────────────────────┘
      └─────────┘     ↓
           ↓
     ┌─────┴─────┐
     │           │
    YES          NO
     │           │
     ↓           ↓
┌──────────┐ ┌────────┐
│ STRETCH- │ │ FENE   │
│ CREATION │ │ Model  │
└──────────┘ └────────┘
(Mechanochem) (Chain
              extension)

             ┌───────────────────────────────────────────────────────┐
             │ Step 4: FIT BASIC TNT                                 │
             │ ----------------------------------------------------- │
             │ TNTSingleMode with constant breakage                  │
             └───────────────────────────────────────────────────────┘
                ↓
             ┌──────────────────────────────────────────┐
             │ Q4: Does constant-breakage fit well?     │
             │ (Check R² > 0.95, random residuals)      │
             └──────────────────────────────────────────┘
                ↓
           ┌────┴────┐
           │         │
          YES        NO
           │         │
           ↓         ↓
      ┌─────────┐  ┌──────────────────────────┐
      │ TANAKA- │  │ Q4a: What fails?         │
      │ EDWARDS │  │                          │
      │ (Basic) │  └──────────────────────────┘
      └─────────┘     ↓
      (Simplest    ┌────────────┬─────────────┐
       model)      │            │             │
                SHEAR-THIN  MULTIPLE    SECOND-
                   ↓        RELAXATIONS NORMAL
                ┌──────┐      ↓           ↓
                │ BELL │  ┌────────┐  ┌─────────┐
                │Model │  │ MULTI- │  │NON-AFFINE│
                └──────┘  │SPECIES │  │ Model   │
                (Force-   └────────┘  └─────────┘
                sensitive) (Polydisperse) (Entangle)

Tip

Variant Handbooks: For detailed physics, equations, protocol predictions, and failure modes of each variant, see:

Protocol-Specific Diagnostic Signatures

This section shows how each experimental protocol discriminates between TNT variants. Use this table to plan which experiments will be most informative for identifying your material’s dominant physics.

Protocol-Specific Diagnostic Signatures

Protocol

Bell

FENE-P

Non-Affine

Stretch-Creation

Loop-Bridge

Sticky Rouse

Cates

Multi-Species

SAOS

Same Maxwell

Same Maxwell

Same Maxwell

Same Maxwell

Reduced \(G'\) plateau

Multi-mode spectrum

Cole-Cole semicircle

Multi-peak \(G''\)

Flow Curve

Power-law thinning

Thinning + saturation

\(N_2 \neq 0\), same \(\sigma(\dot{\gamma})\)

Shear thickening

\(f_B\)-dependent thinning

Cox-Merz failure

Non-monotonic \(\to\) banding

Staged thinning (staircase)

Startup

Overshoot at \(\gamma \approx 1/\sqrt{\nu}\)

Strain stiffening (super-linear)

Reduced \(N_1\), \(N_2 \neq 0\)

Super-linear stress rise

Two timescales in \(\sigma(t)\)

Multi-mode relaxation

Large overshoot \(\to\) plateau

Sequential yielding

Relaxation

Strain-dependent \(\tau_{\text{eff}}\)

\(f\)-dependent initial decay

Same as base TE

Slower decay (hardening)

Bridge recovery (stress \(\uparrow\))

Multi-exponential

Stretched exponential

Multi-exponential

Creep

Eventual rupture at \(\sigma > \sigma_c\)

Strain saturates at \(L_{\text{max}}\)

Faster creep (slip)

Creep ringing / arrest

\(f_B\) collapse \(\to\) rupture

Multi-stage compliance

Viscosity bifurcation

Staged compliance

LAOS

Strong odd harmonics

Box-like Lissajous

\(N_2\) oscillates at \(2\omega\)

Enhanced odd harmonics

Asymmetric Lissajous

Complex multi-harmonic

Stress plateau

Double-yielding

Note

Diagnostic power is highest when comparing multiple protocols. A single protocol rarely uniquely identifies a variant. Plan experiments to cover at least two rows of this table for unambiguous variant selection.

Master Experimental Fingerprints

The following table maps observable experimental signatures to their diagnostic TNT variant and the confirmatory test needed to validate the identification.

Master Experimental Fingerprints

Observable Signature

Diagnostic Variant

Confirmatory Test

Power-law shear thinning with rate-dependent relaxation

Bell

Step strain: \(\tau_{\text{eff}}(\gamma_0)\) decreases with \(\gamma_0\)

Strain stiffening at large extensions

FENE-P

Extensional viscosity bounded, Lissajous becomes box-like

Non-zero \(N_2\) (negative, proportional to \(\xi\))

Non-Affine

Lodge-Meissner violation in step strain

Shear thickening (viscosity increases with rate)

Stretch-Creation

Startup shows super-linear stress growth

Concentration-dependent viscosity with two timescales

Loop-Bridge

Bridge fraction recovery after flow cessation

\(G'(\omega) \sim \omega^{1/2}\) at intermediate frequencies

Sticky Rouse

Multiple plateau regions in modulus

Cole-Cole semicircle in \(G''\) vs \(G'\)

Cates

Non-monotonic flow curve \(\to\) shear banding

Multiple peaks in \(G''(\omega)\)

Multi-Species

Sequential yielding in LAOS, staged flow curve

Non-zero equilibrium stress after relaxation

Multi-Species (permanent + transient)

Creep saturation: \(\gamma \to \gamma_\infty\)

Cross-Model Comparison

This section relates TNT parameters to other rheological model families.

TNT vs Maxwell Models

Relationship

Single-mode TNT is identical to Maxwell in linear viscoelasticity:

\[G_{\text{TNT}} = G_{\text{Maxwell}}, \quad \tau_{\text{TNT}} = \tau_{\text{Maxwell}}\]

Difference: TNT provides physical interpretation (bond breakage) and extends to nonlinear (Bell, FENE-P).

When to Use:

  • Use Maxwell for pure phenomenology

  • Use TNT when network physics is relevant

TNT vs Giesekus Model

Shear Thinning Comparison

Both TNT-Bell and Giesekus produce shear thinning, but via different mechanisms:

Shear Thinning Mechanisms

Model

Mechanism

Parameter

TNT-Bell

Force-accelerated bond breakage

\(\nu\) (force sensitivity)

Giesekus

Anisotropic drag (mobility tensor)

\(\alpha\) (anisotropy)

Power-Law Exponent:

  • TNT-Bell: \(\eta \sim \dot{\gamma}^{-\nu/(\nu+1)}\)

  • Giesekus: \(\eta \sim \dot{\gamma}^{-1/2}\) (fixed exponent at high \(\alpha\))

TNT vs Thixotropic Models (DMT, Fluidity)

Structural vs Kinetic Approaches

TNT vs Structure-Kinetics Models

Aspect

TNT

DMT / Fluidity

State Variable

Bond density (implicit)

Structure parameter \(\lambda\) or fluidity \(f\)

Dynamics

Bond breakage rate \(k_b(\sigma)\)

Structure evolution \(d\lambda/dt\)

Yield Stress

Emergent from network (high \(G\), slow \(\tau_b\))

Explicit \(\tau_y(\lambda)\) closure

Thixotropy

Via multi-species (different bond types)

Native (aging vs rejuvenation)

Strength

Network physics, normal stresses

Explicit history-dependence

Cohort Method: Alternative Numerical Approach

The cohort (history integral) method provides an alternative to the standard ODE-based approach for computing TNT model predictions. Instead of evolving the conformation tensor \(\mathbf{S}(t)\) forward in time via a differential equation, the cohort method tracks individual cohorts of chains born at time \(t'\) and sums their contributions to the total stress.

Integral Formulation

Each cohort of chains created at time \(t'\) contributes to the stress at time \(t\) proportionally to their birth rate \(\beta(t')\), their survival probability \(\mathcal{S}(t,t')\), and the deformation they have accumulated since birth. The total stress is the integral over all cohorts:

\[\boldsymbol{\sigma}(t) = \int_{-\infty}^{t} \beta(t') \, \mathcal{S}(t,t') \, G\left[\mathbf{B}(t,t') - \mathbf{I}\right] \, dt' + 2\eta_s \mathbf{D}(t)\]

where \(\beta(t')\) is the birth rate, \(\mathcal{S}(t,t') = \exp\left(-\int_{t'}^{t} k_d(s) \, ds\right)\) is the survival probability, and \(\mathbf{B}(t,t')\) is the Finger strain tensor.

Each individual cohort contributes:

\[d\boldsymbol{\sigma}(t) = G \cdot \beta(t') \cdot \mathbf{S}(t,t') \cdot \exp\left(-\int_{t'}^{t} k_d(s) \, ds\right) \cdot dt'\]

where \(k_d(s)\) is the destruction rate at time \(s\), which may depend on the local stress or strain rate (as in the Bell or stretch-creation variants).

Mathematical Equivalence

The ODE (conformation tensor) and integral (cohort) approaches are mathematically equivalent — they give identical predictions. The choice between them is purely one of numerical convenience for a given problem. The ODE form evolves a single state variable forward in time, while the integral form explicitly sums over the deformation history.

Advantages of the Cohort Method

  • Complex deformation histories: Naturally handles step-and-hold sequences, multi-rate protocols, and arbitrary time-dependent flows without special treatment at discontinuities.

  • Embarrassingly parallel on GPU: Each cohort is independent — the survival probability and strain accumulation for cohort \(t'\) can be computed without reference to any other cohort. This maps directly to GPU parallelism.

  • Direct access to age distribution: The cohort formulation gives immediate access to the age distribution of surviving chains, \(P(\text{age}) = \mathcal{S}(t, t-\text{age})\), which is a useful diagnostic for non-equilibrium states.

  • Numerical stability for large strain steps: The integral form avoids the stiffness issues that can arise in the ODE form when the deformation gradient changes abruptly.

Disadvantages of the Cohort Method

  • Memory grows with time steps: All cohort weights must be stored, leading to \(O(N_t)\) memory where \(N_t\) is the number of time steps. For long simulations, this can become prohibitive.

  • Less efficient for steady state: At steady state, the ODE form reaches a fixed point directly, while the cohort form must still integrate over the full history (or truncate at a sufficiently large age).

  • Deformation gradient computation: Each cohort pair \((t, t')\) requires computing the Finger tensor \(\mathbf{B}(t,t')\), which involves the full deformation gradient \(\mathbf{F}(t,t')\) from \(t'\) to \(t\).

When to Use Each Approach

  • ODE (conformation tensor): Preferred for steady-state calculations, simple flow histories (constant rate startup, steady shear), and when memory is limited.

  • Cohort (history integral): Preferred for complex multi-step protocols (e.g., pre-shear followed by relaxation followed by startup), when age-distribution information is needed, or when GPU parallelism can be exploited to offset the memory cost.

Bayesian Knowledge Extraction

Using posterior distributions to extract physical insights beyond point estimates.

Parameter Correlations

Physical Coupling from Posteriors

Posterior correlations reveal which parameters are physically coupled:

Strong Correlation (\(|\rho| > 0.7\)):

  • \(G\) vs \(\tau_b\): High correlation → data only constrains \(\eta_0 = G\tau_b\)

  • Solution: Use prior knowledge or multi-protocol fitting (SAOS + flow curve)

Moderate Correlation (\(0.3 < |\rho| < 0.7\)):

  • \(\nu\) vs \(\tau_b\) (Bell): Force sensitivity affects apparent lifetime

  • \(L_{\text{max}}\) vs \(G\) (FENE-P): Stiffening strain related to modulus

No Correlation (\(|\rho| < 0.3\)):

  • Parameters independently constrained

  • High identifiability

Credible Intervals

Report 95% highest density intervals (HDI) for all parameters with uncertainties on derived quantities.

Temperature Dependence

How to extract activation energies and characterize temperature effects.

Arrhenius Analysis

Multi-Temperature Fitting

  1. Measure SAOS at temperatures \(T_1, T_2, \ldots, T_n\)

  2. Fit TNT model at each \(T_i\) to extract \(\tau_b(T_i)\)

  3. Plot \(\ln(\tau_b)\) vs \(1/T\)

  4. Fit linear regression: slope = \(E_a / k_B\)

Expected Ranges:

  • H-bonds: 0.1 - 0.5 eV

  • Metal-ligand: 0.5 - 1.5 eV

  • Covalent: > 2 eV

Concentration Dependence

How parameters scale with polymer concentration to validate network physics.

Modulus Scaling

Power-Law Regimes

Dilute (\(c < c^*\)):

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

Semi-Dilute (\(c^* < c < c^{**}\)):

\[G \sim c^{2.0 - 2.25}\]

Concentrated (\(c > c^{**}\)):

\[G \sim c^{2.0}\]

Practical Recipes

Step-by-step guides for common analysis tasks.

Recipe 1: Determine if Material is a Living Polymer

  1. Perform SAOS: Frequency sweep from 0.01 to 100 rad/s

  2. Plot Cole-Cole: \(G''\) vs \(G'\)

  3. Check semicircle: Is \(G''_{\text{max}} \approx G/2\)?

  4. Fit Cates model

  5. Validate: Calculate \(\tau_{\text{rep}}\) and \(\tau_{\text{break}}\)

Reporting Guidelines

What to include in publications and technical reports.

Essential Reporting Elements

Best-Fit Parameters: Report with uncertainties (standard errors or credible intervals)

Model Selection Criteria: WAIC, AIC, BIC, goodness-of-fit statistics

Physical Interpretation Table: Map parameters to physical quantities

Comparison with Literature: Similar systems, parameter ranges

Data Quality Assessment: Number of points, noise level, replicates

See Also

TNT Model Handbooks:

Protocols and Workflows:

References

  1. Tanaka, F., & Edwards, S. F. (1992). Viscoelastic properties of physically crosslinked networks. Macromolecules, 25, 1516-1523. DOI: 10.1021/ma00031a024

  2. Bell, G. I. (1978). Models for the specific adhesion of cells to cells. Science, 200(4342), 618-627. DOI: 10.1126/science.347575

  3. Warner, H. R. (1972). Kinetic theory and rheology of dilute suspensions of finitely extendible dumbbells. Industrial & Engineering Chemistry Fundamentals, 11(3), 379-387. DOI: 10.1021/i160043a017

  4. Cates, M. E. (1987). Reptation of living polymers. Macromolecules, 20(9), 2289-2296. DOI: 10.1021/ma00175a038

  5. Leibler, L., Rubinstein, M., & Colby, R. H. (1991). Dynamics of reversible networks. Macromolecules, 24(16), 4701-4707. DOI: 10.1021/ma00016a034

  6. Rubinstein, M., & Colby, R. H. (2003). Polymer Physics. Oxford University Press. ISBN: 978-0198520597

  7. Annable, T., et al. (1993). The rheology of solutions of associating polymers. Journal of Rheology, 37(4), 695-726. DOI: 10.1122/1.550391

  8. Evans, E., & Ritchie, K. (1997). Dynamic strength of molecular adhesion bonds. Biophysical Journal, 72(4), 1541-1555. DOI: 10.1016/S0006-3495(97)78802-7

  9. McLeish, T. C. B. (2002). Tube theory of entangled polymer dynamics. Advances in Physics, 51(6), 1379-1527. DOI: 10.1080/00018730210153216