TNT Sticky Rouse (Multi-Mode Sticker Dynamics) — Handbook¶

Quick Reference ¶

Use when:	Multi-sticker associating polymers with hierarchical relaxation (multiple stickers per chain, Rouse-like sub-chain dynamics between stickers)
Materials:	Multi-sticker ionomers (sulfonated polystyrene), HEUR with multiple hydrophobes, beta-cyclodextrin/adamantane complexes, supramolecular polymers with multiple binding sites
Parameters:	2N+2 parameters: \(G_k\) (N moduli), \(\tau_{R,k}\) (N Rouse times), \(\tau_s\) (sticker lifetime), \(\eta_s\) (solvent viscosity), where N = n_modes
Key equation:	Multi-mode ODE: \(\frac{dS_k}{dt} = \kappa \cdot S_k + S_k \cdot \kappa^T - \frac{1}{\tau_{eff,k}}(S_k - I)\) with \(\tau_{eff,k} = \tau_{R,k} + \tau_s\)
Test modes:	FLOW_CURVE, STARTUP, CREEP, RELAXATION, OSCILLATION, LAOS (all 6 protocols)
Signature:	Broad relaxation spectrum, power-law stress relaxation at intermediate times, Rouse-like \(G' \sim \omega^{1/2}\) scaling, sticky plateau at low frequencies
Typical range:	\(\tau_s\): 1e-6 to 1e6 s; \(G_k\): 1e-2 to 1e6 Pa; \(\tau_{R,k} = \tau_{R,1}/k^2\) (Rouse scaling)
Related models:	TNT Tanaka-Edwards (Basic Transient Network) — Handbook, TNT Multi-Species (Multiple Bond Types) — Handbook, Generalized Maxwell Model (Multi-Mode), Rouse model

Notation Guide ¶

Primary Symbols¶
Symbol	Definition	Units
\(G_k\)	Modulus of Rouse mode k (k = 1..N)	Pa
\(\tau_{R,k}\)	Rouse relaxation time for mode k	s
\(\tau_s\)	Sticker (association) lifetime	s
\(\eta_s\)	Solvent viscosity	Pa·s
\(\tau_{eff,k}\)	Effective relaxation time for mode k: \(\tau_{R,k} + \tau_s\)	s
\(S_k\)	Conformation tensor for mode k (3x3 symmetric)	dimensionless
\(N\)	Number of Rouse modes (n_modes)	dimensionless
\(N_s\)	Number of stickers per chain (physical parameter)	dimensionless
\(\kappa\)	Velocity gradient tensor \((\nabla v)^T\)	1/s
\(D\)	Rate of deformation tensor \((D = (\kappa + \kappa^T)/2)\)	1/s
\(\sigma\)	Extra stress tensor	Pa
\(I\)	Identity tensor	dimensionless
\(p\)	Rouse mode index (alternative to k)	dimensionless

Derived Quantities¶
Symbol	Definition	Units
\(\eta_0\)	Zero-shear viscosity: \(\sum_k G_k \tau_{eff,k} + \eta_s\)	Pa·s
\(G_N^{(0)}\)	Plateau modulus: \(\sum_k G_k\)	Pa
\(\tau_{max}\)	Longest relaxation time: \(\max(\tau_{eff,k})\)	s
\(\lambda\)	Terminal relaxation time (sticky-limited)	s
\(\omega_c\)	Characteristic (crossover) frequency	rad/s

The TNT Sticky Rouse model describes the viscoelastic response of unentangled polymer chains bearing multiple reversible association sites (“stickers”). It extends the classical Rouse model to incorporate hierarchical relaxation: fast Rouse dynamics of sub-chain segments between stickers, combined with slow sticker exchange kinetics.

Historical Context:

Leibler, Rubinstein, Colby (1991): Introduced sticky reptation model for entangled associating polymers
Rubinstein & Semenov (1998): Developed thermoreversible gelation theory for multi-sticker networks
Chen, Liang, Colby (2013): Experimental validation of sticky Rouse dynamics in ionomers

Multi-Sticker Architecture:

Consider a flexible polymer chain with \(N_s\) regularly-spaced stickers along its backbone. Each sticker can reversibly bind to complementary sites (from other chains or matrix). Between consecutive stickers, the chain segment behaves as a Rouse sub-chain with its own relaxation spectrum.

Hierarchical Relaxation:

Short times (\(t \ll \tau_s\)): Stickers remain associated; chain appears permanently crosslinked; Rouse modes relax subject to sticker constraints
Intermediate times (\(t \sim \tau_s\)): Sticker exchange begins; interplay between Rouse dynamics and sticker unbinding
Long times (\(t \gg \tau_s\)): All stickers have exchanged; terminal relaxation governed by longest effective time \(\tau_{eff,1} = \tau_{R,1} + \tau_s\)

Key Signature:

The superposition of multiple Rouse modes with sticker-renormalized relaxation times creates a broad relaxation spectrum that manifests as:

Power-law stress relaxation \(G(t) \sim t^{-1/2}\) at intermediate times
Characteristic \(G' \sim \omega^{1/2}\) scaling in SAOS (Rouse regime)
Sticky plateau at frequencies \(1/\tau_s < \omega < 1/\tau_{R,N}\)

Material Examples ¶

Sulfonated Polystyrene Ionomers:

Multiple ionic groups along backbone
Reversible ionic clusters act as transient crosslinks
Exhibits sticky Rouse behavior below entanglement threshold

HEUR (Hydrophobically-modified Ethoxylated Urethanes):

Multiple hydrophobic end-groups per chain
Hydrophobic association creates reversible network
Broad relaxation spectrum from hierarchical dynamics

Beta-Cyclodextrin/Adamantane Complexes:

Polymers with multiple adamantane guest groups
Beta-cyclodextrin hosts provide reversible binding
Tunable sticker lifetime via pH, temperature

Supramolecular Polymers:

Hydrogen-bonded assemblies with multiple binding sites
Metal-ligand coordination polymers
Pi-stacking based reversible networks

Relationship to Rouse Model ¶

The classical Rouse model describes unentangled polymer melts as bead-spring chains with Gaussian statistics. Each mode \(p\) has:

Modulus: \(G_p \approx G_N^{(0)}/N\) (equal mode strength)
Relaxation time: \(\tau_p = \tau_R/p^2\) (harmonic spacing)
SAOS prediction: \(G'(\omega) \sim G''(\omega) \sim \omega^{1/2}\) in Rouse regime

The Sticky Rouse model modifies this by adding a sticker contribution to each mode’s relaxation:

\[\tau_{eff,p} = \tau_{R,p} + \tau_s = \frac{\tau_{R,1}}{p^2} + \tau_s\]

This shifts the mode spectrum and creates new regimes depending on the ratio \(\tau_s/\tau_{R,1}\).

Physical Foundations ¶

Rouse Dynamics ¶

For a flexible polymer chain with \(N_s\) beads (no hydrodynamic interactions), the Rouse model predicts:

Normal Mode Decomposition:

The chain’s configuration is decomposed into \(N_s\) normal modes with eigenvalues:

\[\lambda_p = 4 \sin^2\left(\frac{\pi p}{2N_s}\right) \quad p = 1, 2, \ldots, N_s\]

Relaxation Times:

Each mode has characteristic time:

\[\tau_p = \frac{\zeta b^2}{3\pi^2 k_B T} \cdot \frac{N_s^2}{p^2} = \frac{\tau_{R,1}}{p^2}\]

where \(\zeta\) is bead friction, \(b\) is segment length.

Stress Contribution:

Mode \(p\) contributes:

\[\sigma_p(t) = G_p \langle S_p(t) - I \rangle\]

with \(G_p = \frac{\nu k_B T}{N_s}\) (polymer number density \(\nu\)).

Sticker Kinetics ¶

Association/Dissociation:

Each sticker undergoes reversible binding:

\[\text{Bound} \xrightleftharpoons[k_{on}]{k_{off}} \text{Free}\]

with sticker lifetime:

\[\tau_s = \frac{1}{k_{off}}\]

Renewal Assumption:

When a sticker detaches, the sub-chain segment immediately relaxes its orientation via Rouse dynamics. This “renewal” process couples sticker exchange to Rouse modes.

Effective Relaxation Time ¶

The key insight is that mode \(k\) can only fully relax after:

Stickers on both sides of the sub-chain have detached (time \(\sim \tau_s\))
Rouse relaxation of the freed segment (time \(\sim \tau_{R,k}\))

This gives:

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

Physical Interpretation:

If \(\tau_s \ll \tau_{R,k}\): Stickers exchange rapidly; mode relaxes at Rouse time
If \(\tau_s \gg \tau_{R,k}\): Sticker exchange rate-limiting; mode relaxes at \(\tau_s\)
If \(\tau_s \sim \tau_{R,k}\): Cooperative effect; effective time is sum

Multi-Sticker Coupling ¶

For \(N_s\) stickers dividing the chain into \(N_s+1\) segments:

Independent Modes: Each Rouse mode “sees” the sticker network as a collection of independent obstacles
Spectrum Broadening: The range of \(\tau_{eff,k}\) values spans from \(\tau_{R,N} + \tau_s\) to \(\tau_{R,1} + \tau_s\)
Sticky Plateau: At frequencies \(1/\tau_s < \omega < 1/\tau_{R,N}\), stickers are effectively permanent crosslinks; \(G' \approx G_N^{(0)}\)

Scaling Predictions ¶

High-Frequency Rouse Regime (\(\omega \tau_{R,1} \gg 1\)):

\[G'(\omega) \approx G''(\omega) \approx G_N^{(0)} \left(\frac{\omega \tau_{R,1}}{N}\right)^{1/2}\]

Intermediate Sticky Regime (\(1/\tau_s < \omega < 1/\tau_{R,N}\)):

\[G'(\omega) \approx G_N^{(0)} \quad \text{(plateau)}\]

Terminal Regime (\(\omega \tau_s \ll 1\)):

\[G'(\omega) \approx \left(\sum_k G_k \tau_{eff,k}\right) \omega^2, \quad G''(\omega) \approx \eta_0 \omega\]

Leibler-Rubinstein-Colby (LRC) Scaling ¶

The original LRC theory (Leibler, Rubinstein, Colby 1991) provides scaling relations for sticky Rouse dynamics:

Terminal relaxation time:

\[\tau_{\text{term}} = \tau_s \left(\frac{N}{N_s}\right)^2\]

where \(\tau_s\) is the sticker lifetime and \(N_s\) is the number of monomers between stickers.

Zero-shear viscosity:

\[\eta_0 \sim \tau_s \, N_s^2 \, N^3\]

The \(N^3\) scaling (rather than \(N^{3.4}\) for entangled melts) reflects the Rouse-like dynamics between sticker release events.

Sticky Reptation Crossover ¶

For entangled sticky polymers, there is a crossover between sticky Rouse and sticky reptation dynamics:

\[\tau_{\text{term}} = \max\!\left(\tau_{\text{rep}}, \, \tau_s \cdot n_e\right)\]

where \(n_e\) is the number of entanglements per sticker. When \(\tau_s \cdot n_e > \tau_{\text{rep}}\), sticker dynamics dominate over reptation — the sticky regime. When reptation is faster, the system crosses over to standard entangled dynamics.

Governing Equations ¶

Multi-Mode Conformation Tensor Evolution ¶

For each Rouse mode \(k = 1, 2, \ldots, N\), the conformation tensor \(S_k\) (3x3 symmetric) evolves according to:

\[\frac{dS_k}{dt} = \kappa \cdot S_k + S_k \cdot \kappa^T - \frac{1}{\tau_{eff,k}} (S_k - I)\]

where:

\(\kappa = (\nabla v)^T\) is the velocity gradient tensor
\(I\) is the identity tensor
\(\tau_{eff,k} = \tau_{R,k} + \tau_s\) is the effective relaxation time for mode \(k\)

Tensor Components:

In 2D shear flow (\(\kappa_{xy} = \dot{\gamma}\), others zero):

\[\begin{split}\frac{dS_{xx,k}}{dt} &= 2\dot{\gamma} S_{xy,k} - \frac{1}{\tau_{eff,k}}(S_{xx,k} - 1) \\ \frac{dS_{yy,k}}{dt} &= - \frac{1}{\tau_{eff,k}}(S_{yy,k} - 1) \\ \frac{dS_{zz,k}}{dt} &= - \frac{1}{\tau_{eff,k}}(S_{zz,k} - 1) \\ \frac{dS_{xy,k}}{dt} &= \dot{\gamma} S_{yy,k} - \frac{1}{\tau_{eff,k}} S_{xy,k}\end{split}\]

Total Stress ¶

The extra stress tensor is the sum over all modes plus solvent contribution:

\[\sigma = \sum_{k=1}^{N} G_k (S_k - I) + 2\eta_s D\]

where \(D = (\kappa + \kappa^T)/2\) is the rate of deformation tensor.

Shear Stress:

\[\sigma_{xy} = \sum_{k=1}^{N} G_k S_{xy,k} + \eta_s \dot{\gamma}\]

Normal Stress Differences:

\[N_1 = \sigma_{xx} - \sigma_{yy} = \sum_{k=1}^{N} G_k (S_{xx,k} - S_{yy,k})\]

State Vector ¶

The model tracks \(4N\) degrees of freedom (4 independent components per mode):

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

Equilibrium State:

\[S_k = I \quad \forall k \implies \mathbf{y}_{eq} = [1, 1, 1, 0, \ldots, 1, 1, 1, 0]^T\]

Analytical Solutions ¶

Small-Amplitude Oscillatory Shear (SAOS):

For \(\gamma(t) = \gamma_0 \sin(\omega t)\), linearization gives:

\[G'(\omega) = \sum_{k=1}^{N} G_k \frac{(\omega \tau_{eff,k})^2}{1 + (\omega \tau_{eff,k})^2}\]

\[G''(\omega) = \sum_{k=1}^{N} G_k \frac{\omega \tau_{eff,k}}{1 + (\omega \tau_{eff,k})^2} + \omega \eta_s\]

Stress Relaxation:

For step strain \(\gamma_0\), the relaxation modulus is:

\[G(t) = \sum_{k=1}^{N} G_k \exp\left(-\frac{t}{\tau_{eff,k}}\right)\]

Flow Curve (Approximate):

At steady shear rate \(\dot{\gamma}\), assuming mode decoupling:

\[\eta(\dot{\gamma}) \approx \sum_{k=1}^{N} \frac{G_k \tau_{eff,k}}{1 + (\tau_{eff,1} \dot{\gamma})^2} + \eta_s\]

(Note: This neglects nonlinear coupling; full solution requires ODE integration.)

Startup Shear:

Requires numerical integration of the multi-mode ODE system with initial condition \(\mathbf{y}(0) = \mathbf{y}_{eq}\).

Creep:

For constant stress \(\sigma_0\), strain evolution requires coupled ODE solution (no closed form).

LAOS:

For \(\gamma(t) = \gamma_0 \sin(\omega t)\) with large \(\gamma_0\), full nonlinear ODE integration is necessary; harmonics extracted via Fourier analysis.

Parameter Table ¶

TNT Sticky Rouse Parameters¶
Parameter	Symbol	Default	Bounds	Physical Meaning
Mode moduli	\(G_k\)	[varied]	(1e-2, 1e6) Pa	Contribution of Rouse mode k to total modulus
Rouse times	\(\tau_{R,k}\)	[varied]	(1e-6, 1e4) s	Relaxation time of mode k without stickers
Sticker lifetime	\(\tau_s\)	1.0	(1e-6, 1e6) s	Average duration of sticker association
Solvent viscosity	\(\eta_s\)	0.0	(0.0, 1e4) Pa·s	Background viscosity (monomeric friction)

Derived Parameters:

Symbol	Definition	Units
\(\tau_{eff,k}\)	\(\tau_{R,k} + \tau_s\)	s
\(G_N^{(0)}\)	\(\sum_{k=1}^{N} G_k\)	Pa
\(\eta_0\)	\(\sum_{k=1}^{N} G_k \tau_{eff,k} + \eta_s\)	Pa·s
\(\lambda\)	\(\max_k(\tau_{eff,k})\)	s

Typical Constraints:

For ideal Rouse behavior:

Equal mode strengths: \(G_k = G_N^{(0)}/N\)
Harmonic time spacing: \(\tau_{R,k} = \tau_{R,1}/k^2\)

These can be relaxed for real materials, but imposing them reduces parameter count from \(2N+2\) to \(4\) (\(G_N^{(0)}\), \(\tau_{R,1}\), \(\tau_s\), \(\eta_s\)).

Parameter Interpretation ¶

Sticker Lifetime (\(\tau_s\))¶

Physical Meaning:

Average time a sticker remains bound before dissociating. Controlled by:

Binding energy: \(\tau_s \sim \exp(\Delta E_{bind}/k_B T)\)
Sticker chemistry (H-bonds, ionic, hydrophobic)
Temperature, pH, ionic strength

Regimes:

Condition	Behavior
\(\tau_s \gg \tau_{R,1}\)	Sticker-dominated; all modes relax at \(\sim \tau_s\); narrow spectrum; single-time Maxwell-like
\(\tau_s \ll \tau_{R,N}\)	Rouse-dominated; stickers irrelevant; pure Rouse spectrum \(G'(\omega) \sim \omega^{1/2}\)
\(\tau_{R,N} \ll \tau_s \ll \tau_{R,1}\)	Crossover regime; broad spectrum; sticky plateau visible at \(\omega \sim 1/\tau_s\)

Experimental Determination:

Onset of sticky plateau in \(G'(\omega)\) occurs near \(\omega \approx 1/\tau_s\)
Terminal relaxation time \(\lambda \approx \tau_{R,1} + \tau_s\) (from \(G(t)\) or \(G''\) peak)

Rouse Times (\(\tau_{R,k}\))¶

Physical Meaning:

Relaxation time of mode \(k\) in the absence of stickers. Determined by:

Segment friction \(\zeta\)
Molecular weight distribution
Solvent quality (affects \(b\), \(\zeta\))

Ideal Scaling:

For monodisperse chains:

\[\tau_{R,k} = \frac{\tau_{R,1}}{k^2}, \quad \tau_{R,1} = \frac{\zeta N_s^2 b^2}{3\pi^2 k_B T}\]

Polydispersity Effects:

Real materials may deviate from \(1/k^2\) scaling due to:

Molecular weight distribution
Chain branching
Non-ideal solvent conditions

Constraints in Fitting:

To reduce parameter count, enforce \(\tau_{R,k} = \tau_{R,1}/k^2\) and fit only \(\tau_{R,1}\).

Mode Moduli (\(G_k\))¶

Physical Meaning:

Contribution of mode \(k\) to plateau modulus \(G_N^{(0)} = \sum_k G_k\). Related to chain density and mode entropy.

Ideal Rouse:

\[G_k = \frac{G_N^{(0)}}{N} \quad \forall k\]

Real Materials:

Mode strengths may vary (non-ideal spectrum)
Typically \(G_k\) decreases slightly with \(k\) due to friction distribution

Fitting Strategy:

Unconstrained: Fit all \(N\) values of \(G_k\) independently (2N+2 total parameters)
Constrained: Set \(G_k = G_N^{(0)}/N\) and fit only \(G_N^{(0)}\) (4 total parameters)

Solvent Viscosity (\(\eta_s\))¶

Physical Meaning:

Background viscosity from solvent or unassociated monomers. Provides high-frequency dissipation floor.

Impact on Rheology:

Adds constant contribution to \(G''(\omega)\): \(\omega \eta_s\)
Shifts \(\tan\delta = G''/G'\) at high \(\omega\)
Negligible for polymer melts; important for solutions

Typical Values:

Melt: \(\eta_s \approx 0\) Pa·s
Dilute solution: \(\eta_s \approx \eta_{solvent}\) (e.g., 0.001 Pa·s for water)
Semi-dilute: \(\eta_s = \phi \eta_{solvent}\) (volume fraction \(\phi\))

Validity and Assumptions ¶

Underlying Assumptions ¶

Unentangled Regime:
- Chain length \(N_s < N_e\) (entanglement threshold)
- Rouse dynamics (no tube constraints)
- Violated for high-MW associating polymers (use sticky reptation instead)
Gaussian Statistics:
- Chains obey Gaussian elasticity (small to moderate deformations)
- Breaks down for \(\gamma > 1\) (finite extensibility)
- FENE corrections needed for large \(\gamma_0\) in LAOS
Homogeneous Stickers:
- All stickers identical (same \(\tau_s\), binding energy)
- No sticker-sticker variation along chain
- Real systems may have binding site heterogeneity
Independent Sticker Renewal:
- Sticker dissociation events uncorrelated
- No cooperative unbinding (e.g., zipper-like dissociation)
- Valid for dilute sticker networks
Mean-Field Binding:
- Sticker rebinding is instantaneous to available sites
- Neglects spatial correlations in binding site distribution
- Assumes well-mixed environment
No Excluded Volume:
- Ideal chain statistics
- Violated in good solvent conditions (swollen coils)
Affine Deformation:
- Chain deforms with the flow (no slip)
- Valid for homogeneous shear; breaks down in extensional flows with chain tumbling

Material Applicability ¶

Material Class	When Appropriate	When Inappropriate
Ionomers	Low MW (unentangled), dilute ionic groups	High MW (entangled), dense ionic clusters
Supramolecular polymers	Weak H-bonds, multiple sites	Strong coordination bonds (lifetime distribution)
Hydrogels	Unentangled precursors, reversible crosslinks	Chemical crosslinks, entangled networks
Associating solutions	Dilute/semi-dilute, multiple hydrophobes	Concentrated (overlap), micellar aggregation

Comparison with Other Models ¶

Model	Key Difference	When to Use Sticky Rouse Instead
Generalized Maxwell	No physical mode structure	Need mechanistic interpretation, Rouse scaling validation
TNT Tanaka-Edwards	Single-mode, simpler	Multiple relaxation times observed, broader spectrum
Sticky Reptation	Entangled regime	Unentangled polymers (MW < entanglement threshold)
GENERIC Fluidity	Thixotropic structure parameter	Thixotropy negligible, sticker exchange dominant

Regimes and Behavior ¶

Frequency-Domain Map ¶

High-Frequency Rouse Regime (\(\omega \tau_{R,1} \gg 1\)):

\[G'(\omega) \approx G''(\omega) \approx G_N^{(0)} \sqrt{\frac{\omega \tau_{R,1}}{N}}\]

Characteristic \(\omega^{1/2}\) scaling
Moduli roughly equal (\(\tan\delta \approx 1\))
Polymer segments undergoing sub-Rouse relaxation

Sticky Plateau Regime (\(1/\tau_s < \omega < 1/\tau_{R,N}\)):

\[G'(\omega) \approx G_N^{(0)}, \quad G''(\omega) \ll G'\]

Stickers effectively permanent
Temporary network behavior
Width of plateau scales as \(\log(\tau_s/\tau_{R,N})\) in frequency space

Terminal Relaxation Regime (\(\omega \tau_s \ll 1\)):

\[G'(\omega) \approx \eta_0 \lambda \omega^2, \quad G''(\omega) \approx \eta_0 \omega\]

Liquid-like terminal flow
\(G'' > G'\) (viscous dissipation dominates)
Longest time \(\lambda = \tau_{R,1} + \tau_s\)

Intermediate Frequency Signature ¶

The sticky Rouse model predicts a characteristic half-power-law scaling at intermediate frequencies:

\[G'(\omega) \sim \omega^{1/2} \quad \text{for} \quad 1/\tau_{\text{term}} \ll \omega \ll 1/\tau_s\]

This \(\omega^{1/2}\) dependence is the Rouse scaling, arising from the self-similar relaxation of chain segments between stickers. It appears as a characteristic slope in the log-log plot of \(G'\) vs \(\omega\).

Diagnostic value: The \(\omega^{1/2}\) intermediate regime distinguishes sticky Rouse from:

Single Maxwell (TNTSingleMode): Sharp transition from \(\omega^2\) to plateau
Multi-species (TNTMultiSpecies): Discrete steps between modes
Cates: Near-perfect Maxwell with single crossover

Plateau Identification ¶

For entangled sticky polymers, two plateaus may be visible in \(G'(\omega)\):

High-frequency plateau (\(G_N^0\)): Entanglement plateau — reflects topological constraints between chain entanglements
Low-frequency plateau (\(G_e\)): Sticker network plateau — reflects the elastic contribution of sticker-sticker associations

The ratio \(G_e / G_N^0\) gives the fraction of stress carried by the sticker network relative to entanglements.

Time-Domain Signatures ¶

Stress Relaxation after Step Strain:

\[G(t) = \sum_{k=1}^{N} G_k \exp\left(-\frac{t}{\tau_{eff,k}}\right)\]

Multi-exponential decay
At \(t \ll \tau_s\): Rapid Rouse decay (\(\sim t^{-1/2}\) envelope)
At \(t \sim \tau_s\): Crossover to slower decay
At \(t \gg \tau_{R,1}\): Final exponential tail \(\sim \exp(-t/\lambda)\)

Startup Shear Flow:

For constant \(\dot{\gamma}\), stress grows as:

Initial elastic response (fast modes)
Stress overshoot if \(\dot{\gamma} \tau_s > 1\) (sticker network stretches before yielding)
Steady-state flow at \(\sigma_{ss} = \eta(\dot{\gamma}) \dot{\gamma}\)

Creep under Constant Stress:

\[\gamma(t) \sim t^{1/2} \quad \text{at } t \ll \tau_s \quad \text{(sub-diffusive Rouse creep)}\]

\[\gamma(t) \sim t \quad \text{at } t \gg \tau_s \quad \text{(viscous flow)}\]

Nonlinear Flow Regimes ¶

Shear Rate Parameter:

Define Weissenberg number for mode \(k\):

\[Wi_k = \dot{\gamma} \tau_{eff,k}\]

Regimes:

\(Wi_1\) Range	Behavior
\(Wi_1 \ll 1\)	Linear viscoelastic (Newtonian plateau); \(\eta \approx \eta_0\)
\(Wi_1 \sim 1\)	Longest mode becomes nonlinear; stress overshoot in startup
\(Wi_1 \gg 1\)	Shear thinning; \(\eta \sim \dot{\gamma}^{-1}\) (power-law from mode superposition)

LAOS Nonlinearity:

For oscillatory strain \(\gamma_0 \sin(\omega t)\):

Deborah number: \(De_k = \omega \tau_{eff,k}\)
Strain amplitude: \(\gamma_0\)

\(De_1\)	\(\gamma_0\)	Response
\(\ll 1\)	Small	Terminal regime; linear viscous dissipation
\(\sim 1\)	Small	Viscoelastic transition; \(G' \approx G''\)
\(\gg 1\)	Small	Elastic regime; \(G' \gg G''\)
Any	\(> 1\)	Nonlinear LAOS; higher harmonics appear; chain stretching

Sticky vs. Rouse Crossover ¶

The relative importance of stickers vs. Rouse dynamics is governed by the ratio:

\[R = \frac{\tau_s}{\tau_{R,1}}\]

Rouse-Dominated (\(R \ll 1\)):

Stickers exchange much faster than Rouse relaxation
Effectively no stickers; pure Rouse model applicable
\(\tau_{eff,k} \approx \tau_{R,k} \propto 1/k^2\)

Sticky-Dominated (\(R \gg 1\)):

Sticker exchange rate-limits all modes
\(\tau_{eff,k} \approx \tau_s\) for all \(k\)
Narrow spectrum; single-mode Maxwell-like behavior

Crossover Regime (\(R \sim 1\)):

Broad spectrum with \(\tau_{eff,k}\) ranging from \(\tau_s\) (slow modes) to \(\tau_{R,N} + \tau_s\) (fast modes)
Richest rheological behavior; power-law relaxation
Sticky plateau visible in \(G'(\omega)\)

What You Can Learn from This Model ¶

Extracting Sticker Lifetime ¶

Method 1: Sticky Plateau Onset

In SAOS data, identify frequency \(\omega_s\) where \(G'\) begins to plateau (transition from terminal to sticky regime):

\[\tau_s \approx \frac{1}{\omega_s}\]

Method 2: Terminal Relaxation Time

From stress relaxation \(G(t)\), fit the long-time tail:

\[G(t \to \infty) \sim \exp(-t/\lambda), \quad \lambda = \tau_{R,1} + \tau_s\]

If \(\tau_{R,1}\) known (from mode fitting), extract \(\tau_s = \lambda - \tau_{R,1}\).

Method 3: Peak in :math:`G’’(omega)`

The terminal peak in \(G''\) occurs near \(\omega \approx 1/\lambda\), providing another estimate of \(\tau_s\).

Determining Number of Modes ¶

Spectral Width:

The breadth of the relaxation spectrum correlates with the number of distinguishable Rouse modes. Compare:

Frequency span of \(G'\) plateau: \(\Delta \log\omega \sim \log(N)\)
Number of inflection points in \(G'(\omega)\) or \(G''(\omega)\)

Parsimonious Fitting:

Start with \(N = 3\), increase until fit quality plateaus (adjusted \(R^2\), AIC). Overfitting risk if \(N > 10\) for typical experimental noise.

Validating Rouse Scaling ¶

Test 1: Harmonic Time Spacing

Plot fitted \(\tau_{R,k}\) vs. \(k\) on log-log axes. Expect slope \(-2\) if ideal Rouse:

\[\log(\tau_{R,k}) = \log(\tau_{R,1}) - 2\log(k)\]

Test 2: High-Frequency Scaling

In Rouse regime (\(\omega \tau_{R,1} \gg 1\)), verify:

\[\log G'(\omega) \sim \frac{1}{2} \log\omega + \text{const}\]

Slope of 0.5 on log-log plot confirms Rouse dynamics.

Test 3: Equal Mode Strengths

Check if \(G_k \approx G_N^{(0)}/N\) for all modes. Deviation indicates non-ideal distribution (polydispersity, branching).

Molecular Weight Estimation ¶

From Rouse theory, the longest Rouse time scales as:

\[\tau_{R,1} \sim M_w^2\]

where \(M_w\) is weight-average molecular weight. If \(\tau_{R,1}\) known:

\[M_w \propto \sqrt{\tau_{R,1}}\]

(Requires calibration with known standards.)

Sticker Binding Energy ¶

If \(\tau_s\) measured at multiple temperatures \(T\):

\[\tau_s(T) = \tau_0 \exp\left(\frac{\Delta E_{bind}}{k_B T}\right)\]

Arrhenius plot of \(\log\tau_s\) vs. \(1/T\) yields binding energy \(\Delta E_{bind}\) from slope.

Discriminating Material Classes ¶

Observable	Sticky Rouse	Alternative Mechanism
\(G'(\omega) \sim \omega^{1/2}\) at high \(\omega\)	Rouse modes active	Glassy/entangled: \(G' \sim \omega^0\) (plateau)
\(G'\) plateau at intermediate \(\omega\)	Sticky network	Chemical gel: permanent plateau
Stress overshoot in startup	Sticker stretching (\(Wi_1 > 1\))	Shear banding, yield stress (no overshoot)
Multi-exponential \(G(t)\)	Multiple modes	Single Maxwell: mono-exponential

Experimental Design ¶

Optimal Test Protocols ¶

Primary: Small-Amplitude Oscillatory Shear (SAOS)

Cover frequency range \(10^{-3}\) to \(10^2\) rad/s (at least 5 decades):

Low \(\omega\): Terminal regime (\(G' \sim \omega^2\), \(G'' \sim \omega\))
Intermediate \(\omega\): Sticky plateau (\(G' \approx G_N^{(0)}\))
High \(\omega\): Rouse regime (\(G' \sim \omega^{1/2}\))

Strain Amplitude: \(\gamma_0 = 0.01-0.1\) (confirm linear regime via amplitude sweep).

Secondary: Stress Relaxation

Step strain \(\gamma_0 = 0.1-0.5\), measure \(G(t)\) from \(10^{-2}\) to \(10^4\) s:

Validates multi-exponential spectrum
Direct access to \(\tau_{eff,k}\) via exponential fitting
Complementary to SAOS (covers same time scales in different representation)

Tertiary: Steady Shear Flow Curve

Measure \(\eta(\dot{\gamma})\) from \(10^{-3}\) to \(10^2\) 1/s:

Probes nonlinear regime (\(Wi_1 > 1\))
Validates shear thinning predictions
Tests multi-mode consistency (must match SAOS via Cox-Merz rule at low \(\dot{\gamma}\))

Advanced: LAOS

Strain sweeps at fixed \(\omega\) (e.g., \(\omega = 1/\tau_s\)):

\(\gamma_0\) from 0.1 to 10
Extract \(G'_1, G''_1\) (fundamental), \(G'_3, G''_3\) (third harmonic)
Quantifies nonlinear elasticity (chain stretching effects)

Time-Temperature Superposition ¶

Applicability:

Sticky Rouse is thermorheologically simple if:

\(\tau_s(T)\) and \(\tau_{R,k}(T)\) have the same activation energy
\(G_k\) temperature-independent (or weakly dependent)

Procedure:

Measure \(G'(\omega, T)\), \(G''(\omega, T)\) at multiple \(T\) (e.g., 10-60°C in 10°C steps).

Shift horizontally to reference \(T_{ref}\) using shift factor \(a_T\):

\[G'(\omega, T) \to G'(a_T \omega, T_{ref})\]

If successful, reveals extended frequency range (e.g., 8 decades from 5 temperatures).

Extract Activation Energy:

\[\log a_T = \frac{\Delta E_{a}}{R} \left(\frac{1}{T} - \frac{1}{T_{ref}}\right)\]

Sample Requirements ¶

Geometry:

Cone-plate (preferred): Homogeneous shear, small sample volume, gap angle 0.04-0.1 rad
Parallel plates: Large normal forces, edge effects at high \(\gamma_0\)
Couette: High-viscosity samples, but difficult LAOS interpretation

Volume: 0.5-2 mL (cone-plate), 5-10 mL (parallel plates)

Loading: Avoid air bubbles, ensure complete wetting of geometry

Temperature Control: ±0.1°C stability for TTS measurements

Equilibration: 5-10 minutes at each temperature before measurement

Data Quality Checks ¶

Linearity Verification:

Perform strain amplitude sweep at fixed \(\omega\):

\(G', G''\) should be constant for \(\gamma_0 < \gamma_{LVE}\)
Typical \(\gamma_{LVE} \sim 0.1-1\) for sticky Rouse systems

Instrument Compliance:

At high \(\omega\), check for artifacts:

\(G''\) should not exceed \(\omega \eta_s + G''_{max}\) (solvent limit)
Spurious peaks in \(G''\) indicate inertia effects

Torque Range:

Ensure measured torque \(> 10\%\) of instrument minimum for accurate data.

Repeatability:

Replicate SAOS at reference condition; coefficient of variation should be \(< 5\%\).

Computational Implementation ¶

State Vector and ODE System ¶

For \(N\) modes, the state vector has dimension \(4N\):

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

ODE Right-Hand Side:

For mode \(k\), with velocity gradient \(\kappa\):

\[\frac{d\mathbf{y}_k}{dt} = \mathbf{f}_k(\mathbf{y}_k, \kappa, \tau_{eff,k})\]

where \(\mathbf{y}_k = [S_{xx,k}, S_{yy,k}, S_{zz,k}, S_{xy,k}]^T\) and:

\[\begin{split}\mathbf{f}_k = \begin{bmatrix} 2\kappa_{xy} S_{xy,k} - (S_{xx,k} - 1)/\tau_{eff,k} \\ - (S_{yy,k} - 1)/\tau_{eff,k} \\ - (S_{zz,k} - 1)/\tau_{eff,k} \\ \kappa_{xy} S_{yy,k} - S_{xy,k}/\tau_{eff,k} \end{bmatrix}\end{split}\]

Vectorization via vmap:

Use JAX vmap to parallelize over modes:

def ode_single_mode(y_k, kappa, tau_eff_k):
    S_xx, S_yy, S_zz, S_xy = y_k
    dS_xx = 2*kappa_xy*S_xy - (S_xx - 1)/tau_eff_k
    dS_yy = - (S_yy - 1)/tau_eff_k
    dS_zz = - (S_zz - 1)/tau_eff_k
    dS_xy = kappa_xy*S_yy - S_xy/tau_eff_k
    return jnp.array([dS_xx, dS_yy, dS_zz, dS_xy])

ode_all_modes = jax.vmap(ode_single_mode, in_axes=(0, None, 0))

Then call ode_all_modes(y, kappa, tau_eff) where y is (N, 4), tau_eff is (N,).

Stress Calculation ¶

Total Shear Stress:

def compute_stress(y, G, eta_s, gamma_dot):
    S_xy = y[:, 3]  # Shape (N,)
    sigma_xy = jnp.sum(G * S_xy) + eta_s * gamma_dot
    return sigma_xy

Normal Stress Differences:

def compute_N1(y, G):
    S_xx = y[:, 0]
    S_yy = y[:, 1]
    N1 = jnp.sum(G * (S_xx - S_yy))
    return N1

SAOS Implementation ¶

Use analytical expressions for efficiency:

def saos(omega, G, tau_eff, eta_s):
    # G, tau_eff are arrays of length N
    omega_tau = omega[:, None] * tau_eff[None, :]  # (len(omega), N)
    G_prime = jnp.sum(G * omega_tau**2 / (1 + omega_tau**2), axis=1)
    G_double_prime = jnp.sum(G * omega_tau / (1 + omega_tau**2), axis=1) + omega * eta_s
    return G_prime, G_double_prime

Relaxation Modulus ¶

def relaxation_modulus(t, G, tau_eff):
    exp_terms = jnp.exp(-t[:, None] / tau_eff[None, :])  # (len(t), N)
    G_t = jnp.sum(G * exp_terms, axis=1)
    return G_t

Startup Shear Simulation ¶

def simulate_startup(gamma_dot, t_end, G, tau_eff, eta_s):
    y0 = jnp.tile(jnp.array([1.0, 1.0, 1.0, 0.0]), N)  # Equilibrium
    kappa = jnp.array([[0, gamma_dot], [0, 0]])

    def rhs(t, y):
        y_reshaped = y.reshape(N, 4)
        dy = ode_all_modes(y_reshaped, kappa, tau_eff)
        return dy.ravel()

    t_eval = jnp.linspace(0, t_end, 1000)
    solution = odeint(rhs, y0, t_eval)

    sigma_xy = jax.vmap(lambda y: compute_stress(y.reshape(N, 4), G, eta_s, gamma_dot))(solution)
    return t_eval, sigma_xy

Performance Optimization ¶

JIT Compilation:

Decorate all functions with @jax.jit for 10-100x speedups:

@jax.jit
def ode_all_modes(y, kappa, tau_eff):
    ...

Avoid Python Loops:

Use vmap, lax.scan, or lax.fori_loop instead of explicit for-loops over modes.

Precompute Constants:

Calculate \(\tau_{eff,k} = \tau_{R,k} + \tau_s\) once at initialization, not during ODE integration.

Fitting Guidance ¶

Primary Data: SAOS ¶

Why SAOS is Ideal:

Analytical solution (no ODE integration)
Direct access to all modes via frequency sweep
High signal-to-noise ratio
Well-defined linear regime

Objective Function:

Minimize log-space error to balance \(G'\) and \(G''\):

\[\mathcal{L} = \sum_i \left[\left(\log G'_{pred}(\omega_i) - \log G'_{data}(\omega_i)\right)^2 + \left(\log G''_{pred}(\omega_i) - \log G''_{data}(\omega_i)\right)^2\right]\]

Parameter Bounds:

\(G_k \in (0.01 \cdot G''_{max}, 100 \cdot G''_{max})\)
\(\tau_{R,k} \in (0.01/\omega_{max}, 100/\omega_{min})\)
\(\tau_s \in (0.01/\omega_{max}, 100/\omega_{min})\)
\(\eta_s \in (0, 10 \cdot G''_{max}/\omega_{max})\)

Constrained vs. Unconstrained Fitting ¶

Unconstrained (2N+2 parameters):

Fit all \(G_k, \tau_{R,k}\) independently plus \(\tau_s, \eta_s\).

Pros: Maximum flexibility; captures non-ideal spectra
Cons: High parameter count; risk of overfitting; non-unique solutions

Constrained (4 parameters):

Impose Rouse scaling:

\[G_k = \frac{G_N^{(0)}}{N}, \quad \tau_{R,k} = \frac{\tau_{R,1}}{k^2}\]

Fit only \(G_N^{(0)}, \tau_{R,1}, \tau_s, \eta_s\).

Pros: Parsimonious; physically motivated; stable fits
Cons: May not capture polydispersity or non-ideal behavior

Recommended Strategy:

Start with constrained fit (N=3-5)
If fit poor (\(R^2 < 0.95\)), relax to unconstrained
Validate by checking if fitted \(\tau_{R,k}\) obeys \(1/k^2\) scaling

Initialization Strategy ¶

Step 1: Estimate Plateau Modulus

\[G_N^{(0)} \approx \min_{\omega} G'(\omega) \quad \text{(sticky plateau value)}\]

Step 2: Estimate Sticker Lifetime

From peak in \(G''(\omega)\):

\[\tau_s \approx \frac{1}{\omega_{G''_{peak}}}\]

Step 3: Estimate Longest Rouse Time

From terminal slope in \(G'\):

\[\tau_{R,1} \approx \frac{1}{\omega_{terminal}} - \tau_s\]

Step 4: Set Mode Strengths

\[G_k = \frac{G_N^{(0)}}{N}, \quad \tau_{R,k} = \frac{\tau_{R,1}}{k^2}\]

Step 5: Estimate Solvent Viscosity

\[\eta_s \approx \frac{G''(\omega_{max})}{\omega_{max}}\]

Regularization and Constraints ¶

Monotonicity:

Enforce \(\tau_{eff,1} > \tau_{eff,2} > \cdots > \tau_{eff,N}\) to prevent mode crossing.

Positivity:

All \(G_k, \tau_{R,k}, \tau_s, \eta_s > 0\) (built into bounds).

Smoothness Penalty:

For unconstrained fits, add regularization term:

\[\mathcal{L}_{reg} = \mathcal{L} + \alpha \sum_{k=1}^{N-1} (G_{k+1} - G_k)^2\]

to discourage erratic mode strength variations.

Multi-Start Optimization ¶

Due to multi-modal likelihood surface, use multiple initial guesses:

Random sampling within bounds (10-20 starts)
Latin hypercube sampling for parameter space coverage
Select solution with lowest \(\mathcal{L}\) and physical consistency

Secondary Data: Relaxation ¶

If \(G(t)\) available, fit directly:

\[\mathcal{L} = \sum_i \left(\log G(t_i) - \log G_{pred}(t_i)\right)^2\]

Advantages:

Analytical solution (fast)
Exponentials easier to resolve than SAOS peaks

Disadvantages:

Requires high dynamic range in \(G(t)\) (6+ decades)
Experimental drift at long times
Edge effects in step strain

Validation Tests ¶

After fitting, check:

R-squared: \(R^2 > 0.95\) (0.99 for good fit)
Residual Randomness: Plot residuals vs. \(\omega\); should show no trends
Rouse Scaling: Plot \(\tau_{R,k}\) vs. \(k\) on log-log; expect slope -2
Mode Strength Distribution: \(G_k\) should be similar order of magnitude
Physical Bounds: \(\eta_0 = \sum G_k \tau_{eff,k} + \eta_s\) should match steady-shear viscosity
Cross-Validation: Predict startup shear using fitted parameters; compare to experiment

Usage Examples ¶

Basic SAOS Fitting ¶

from rheojax.models.tnt import TNTStickyRouse
from rheojax.core import RheoData
import jax.numpy as jnp

# Experimental SAOS data
omega = jnp.logspace(-2, 2, 50)  # rad/s
G_prime_data = ...  # Pa
G_double_prime_data = ...  # Pa
G_star = G_prime_data + 1j * G_double_prime_data

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

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

print(f"R-squared: {result.r_squared:.4f}")
print(f"Fitted parameters: {result.parameters}")

# Extract sticker lifetime
tau_s = result.parameters['tau_s']
print(f"Sticker lifetime: {tau_s:.2e} s")

Constrained Rouse Scaling ¶

# Enforce ideal Rouse mode structure
model = TNTStickyRouse(n_modes=5, constrain_rouse_scaling=True)

# Now only 4 free parameters: G_N0, tau_R1, tau_s, eta_s
result = model.fit(rheo_data)

# Check if constraint was beneficial
print(f"Constrained R^2: {result.r_squared:.4f}")

Predicting Startup Shear ¶

# After fitting to SAOS, predict startup shear response
gamma_dot = 1.0  # 1/s
t_startup = jnp.linspace(0, 100, 500)  # s

# Simulate startup
sigma_xy = model.predict(
    t_startup,
    test_mode='startup',
    gamma_dot=gamma_dot
)

# Plot stress growth
import matplotlib.pyplot as plt
plt.plot(t_startup, sigma_xy)
plt.xlabel('Time (s)')
plt.ylabel('Shear Stress (Pa)')
plt.title(f'Startup Shear at gamma_dot = {gamma_dot} 1/s')
plt.show()

Stress Relaxation ¶

# Predict relaxation modulus after step strain
t_relax = jnp.logspace(-3, 3, 100)  # s
G_t = model.predict(t_relax, test_mode='relaxation')

# Plot on log-log scale
plt.loglog(t_relax, G_t)
plt.xlabel('Time (s)')
plt.ylabel('G(t) (Pa)')
plt.title('Stress Relaxation Modulus')
plt.grid(which='both', alpha=0.3)
plt.show()

LAOS Simulation ¶

# Large-amplitude oscillatory shear
gamma_0 = 1.0  # Strain amplitude
omega_laos = 1.0  # rad/s
n_cycles = 10

t_laos = jnp.linspace(0, 2*jnp.pi*n_cycles/omega_laos, 1000)

sigma_laos = model.predict(
    t_laos,
    test_mode='laos',
    gamma_0=gamma_0,
    omega=omega_laos
)

# Extract harmonics via FFT
from rheojax.utils import extract_harmonics
harmonics = extract_harmonics(t_laos, sigma_laos, omega_laos, n_harmonics=5)

print(f"G'_1: {harmonics['G1_prime']:.2f} Pa")
print(f"G'_3: {harmonics['G3_prime']:.2f} Pa")

Bayesian Inference ¶

# Propagate uncertainty in fitted parameters
result_bayesian = model.fit_bayesian(
    rheo_data,
    num_warmup=1000,
    num_samples=2000,
    num_chains=4
)

# Get credible intervals
intervals = model.get_credible_intervals(
    result_bayesian.posterior_samples,
    credibility=0.95
)

print("95% Credible Intervals:")
for param, (low, high) in intervals.items():
    print(f"  {param}: [{low:.2e}, {high:.2e}]")

# Plot posterior distributions
import arviz as az
az.plot_pair(result_bayesian.posterior_samples, divergences=True)

Multi-Temperature TTS ¶

# Fit at multiple temperatures, extract activation energy
from rheojax.transforms import Mastercurve

temps = [20, 30, 40, 50, 60]  # °C
datasets = [load_saos_data(T) for T in temps]

# Apply TTS
mc = Mastercurve(reference_temp=40, auto_shift=True)
master_data, shift_factors = mc.transform(datasets)

# Fit sticky Rouse to master curve
model = TNTStickyRouse(n_modes=5)
result = model.fit(master_data)

# Extract activation energy from shift factors
import numpy as np
T_kelvin = np.array(temps) + 273.15
log_aT = np.log(shift_factors)

# Arrhenius fit: log(a_T) = E_a/R * (1/T - 1/T_ref)
from scipy.stats import linregress
slope, intercept, r_value, p_value, std_err = linregress(1/T_kelvin, log_aT)
E_a = slope * 8.314  # J/mol (R = 8.314 J/(mol·K))

print(f"Activation energy: {E_a/1000:.1f} kJ/mol")

Failure Mode: Terminal Flow ¶

The sticky Rouse model always predicts eventual viscous flow at sufficiently long times or low frequencies. When all stickers have released at least once (time \(\gg \tau_{\text{term}}\)), the chain loses all memory of its initial configuration and flows as a viscous liquid.

Physical signatures:

\(G'(\omega) \sim \omega^2\) and \(G''(\omega) \sim \omega\) at \(\omega \ll 1/\tau_{\text{term}}\)
Steady-state creep rate \(\dot{\gamma}_{ss} = \sigma_0/\eta_0\)
No residual elasticity (unlike multi-species with permanent bonds)

Distinction from gel behavior: If the material shows a low-frequency elastic plateau (\(G'\) does not decrease to zero), the sticky Rouse model is inappropriate. Consider TNT Multi-Species (Multiple Bond Types) — Handbook with a permanent bond component, or a yield-stress model.

API Reference ¶

class rheojax.models.tnt.TNTStickyRouse(n_modes=3)[source]¶

Bases: TNTBase

Sticky Rouse model for associative polymers.

Multi-mode Maxwell model where sticker dynamics impose a relaxation time floor: τ_eff_k = max(τ_R_k, τ_s).

Creates a plateau in G(t) at intermediate times (sticker-dominated regime) before terminal relaxation (slowest Rouse mode).

Parameters:: n_modes (int) – Number of Rouse modes

parameters¶

Model parameters: - G_0, G_1, …, G_{N-1}: Mode moduli (Pa) - tau_R_0, tau_R_1, …, tau_R_{N-1}: Rouse relaxation times (s) - tau_s: Sticker lifetime (s) - eta_s: Solvent viscosity (Pa·s)

Type:: ParameterSet

Notes

The model reduces to standard multi-mode Maxwell when tau_s → 0. For tau_s → ∞, all modes relax at tau_s (single network behavior).

Examples

>>> # 3-mode sticky Rouse
>>> model = TNTStickyRouse(n_modes=3)
>>> model.fit(omega, G_star, test_mode='oscillation')
>>>
>>> # Predict plateau modulus
>>> G_plateau = model.predict_plateau_modulus()
>>>
>>> # Predict startup with stress overshoot
>>> t = np.linspace(0, 10, 200)
>>> sigma = model.predict(t, test_mode='startup', gamma_dot=1.0)
>>>
>>> # Extract effective relaxation times
>>> tau_eff = model.get_effective_times()

__init__(n_modes=3)[source]¶

Initialize Sticky Rouse model.

Parameters:: n_modes (int) – Number of Rouse modes (must be >= 1)

property n_modes: int¶: Number of Rouse modes.

property tau_s: float¶: Sticker lifetime (s).

property eta_s: float¶: Solvent viscosity (Pa·s).

get_effective_times()[source]¶

Get effective relaxation times for all modes.

Returns:: Effective times τ_eff_k = τ_R_k + τ_s, shape (N,)
Return type:: ndarray

model_function(X, params, test_mode=None, **kwargs)[source]¶

Compute model prediction for given parameters.

Parameters:

X (Array) – Independent variable (time, frequency, or shear rate)
params (Array) – Parameter array [G_0, tau_R_0, G_1, tau_R_1, …, tau_s, eta_s] Length: 2*N + 2
test_mode (str | None) – Protocol: ‘oscillation’, ‘relaxation’, ‘flow_curve’, ‘startup’, ‘creep’, or ‘laos’

Returns:

Predicted response (protocol-dependent)

Return type:

Array

predict_plateau_modulus()[source]¶

Compute plateau modulus G_N = Σ G_k for modes with τ_R_k < τ_s.

The plateau modulus is the sum of mode moduli for modes dominated by sticker lifetime (fast Rouse modes).

Returns:: Plateau modulus G_N (Pa)
Return type:: float

predict_zero_shear_viscosity()[source]¶

Compute zero-shear viscosity η₀ = Σ G_k·τ_eff_k + η_s.

Returns:: Zero-shear viscosity η₀ (Pa·s)
Return type:: float

predict_saos(omega, return_components=True)[source]¶

Predict SAOS storage and loss moduli.

Analytical superposition for multi-mode Maxwell:: G’(ω) = Σ G_k·(ωτ_eff_k)² / (1 + (ωτ_eff_k)²) G’’(ω) = Σ G_k·(ωτ_eff_k) / (1 + (ωτ_eff_k)²) + η_s·ω

Parameters:

omega (ndarray) – Angular frequency array (rad/s)
return_components (bool) – If True, return (G’, G’’)

Returns:

(G’, G’’) if return_components=True, else |G*|

Return type:

tuple[ndarray, ndarray] | ndarray

predict_terminal_time()[source]¶

Return longest effective relaxation time (terminal mode).

Returns:: Terminal time τ_terminal = max(τ_eff_k) (s)
Return type:: float

predict_normal_stress_difference(gamma_dot)[source]¶

Predict first normal stress difference N₁(γ̇).

N₁ = Σ 2·G_k·τ_eff_k²·γ̇² / (1 + (τ_eff_k·γ̇)²)

Parameters:: gamma_dot (float | ndarray) – Shear rate (1/s)
Returns:: N₁ (Pa)
Return type:: ndarray

get_trajectory()[source]¶

Get stored ODE trajectory from last prediction.

Returns:: Dictionary with keys like ‘time’, ‘stress’, ‘strain’, ‘S_xy’
Return type:: dict[str, ndarray] | None

initialize_from_saos(omega, G_prime, G_double_prime)[source]¶

Initialize parameters from SAOS data.

Uses crossover frequency to estimate sticker lifetime and plateau modulus to distribute mode strengths.

Parameters:

omega (ndarray) – Angular frequency array (rad/s)
G_prime (ndarray) – Storage modulus G’ (Pa)
G_double_prime (ndarray) – Loss modulus G’’ (Pa)

Return type:

None

__repr__()[source]¶

Return string representation.

Return type:: str

References ¶

Foundational Papers ¶

Leibler, L., Rubinstein, M., & Colby, R. H. (1991). “Dynamics of reversible networks.” Macromolecules, 24(16), 4701-4707. DOI: 10.1021/ma00016a034
- Original sticky reptation model
- Introduced concept of renormalized Rouse time
Rouse, P. E. (1953). “A theory of the linear viscoelastic properties of dilute solutions of coiling polymers.” Journal of Chemical Physics, 21(7), 1272-1280. DOI: 10.1063/1.1699180
- Classical Rouse model
- Harmonic mode spacing \(\tau_p \propto 1/p^2\)
Rubinstein, M., & Semenov, A. N. (1998). “Thermoreversible gelation in solutions of associating polymers. 2. Linear dynamics.” Macromolecules, 31(4), 1386-1397. DOI: 10.1021/ma970617+
- Multi-sticker network dynamics
- Hierarchical relaxation theory

Experimental Validation ¶

Chen, Q., Tudryn, G. J., & Colby, R. H. (2013). “Ionomer dynamics and the sticky Rouse model.” Journal of Rheology, 57(5), 1441-1462. DOI: 10.1122/1.4818868
- Sticky Rouse behavior in ionomer solutions
- Power-law relaxation validation
Baxandall, L. G. (1989). “Dynamics of reversibly crosslinked chains.” Macromolecules, 22(4), 1982-1988. DOI: 10.1021/ma00194a076
- Early theoretical treatment of transient networks
- Crosslink kinetics coupling to Rouse modes

Review Articles ¶

Rubinstein, M., & Colby, R. H. (2003). Polymer Physics. Oxford University Press. ISBN: 978-0198520597
- Chapter 9: Rouse model (pages 372-399)
- Chapter 10: Sticky reptation (pages 431-450)
Tanaka, F., & Edwards, S. F. (1992). “Viscoelastic properties of physically crosslinked networks.” Macromolecules, 25(5), 1516-1523. DOI: 10.1021/ma00031a024
- Green-Tobolsky network theory
- Transient crosslink statistics

Computational Methods ¶

Padding, J. T., & Briels, W. J. (2001). “Uncrossability constraints in mesoscopic polymer melt simulations.” Journal of Chemical Physics, 115(6), 2846-2859. DOI: 10.1063/1.1385162
- Numerical integration of multi-mode constitutive equations
- Stability analysis for stiff ODE systems
Morrison, F. A. (2001). Understanding Rheology. Oxford University Press. ISBN: 978-0195141665
- Chapter 8: Multi-mode models (pages 441-488)
- SAOS vs. transient flow predictions

Applications ¶

Annable, T., Buscall, R., Ettelaie, R., & Whittlestone, D. (1993). “The rheology of solutions of associating polymers.” Journal of Rheology, 37(4), 695-726. DOI: 10.1122/1.550391
- HEUR associating polymer rheology
- Multi-mode sticky Rouse fits to experimental data