HVNM (Hybrid Vitrimer Nanocomposite Model)

This section documents the Hybrid Vitrimer Nanocomposite Model (HVNM) for nanoparticle-filled vitrimers — polymer networks containing rigid NP fillers that create an interphase subnetwork with distinct kinetics.

Part of VLB Transient Network Family

HVNM extends HVM (HVM (Hybrid Vitrimer Model)), which itself builds on VLB (VLB Transient Network Models). Full lineage: VLB → HVM → HVNM.

Inheritance: BaseModel → VLBBase → HVMBase → HVNMBase → HVNMLocal

Overview

The HVNM extends the Hybrid Vitrimer Model (HVM) with a fourth interphase (I) subnetwork that forms around nanoparticle surfaces. The key new physics are:

  • Guth-Gold strain amplification: \(X(\phi) = 1 + 2.5\phi + 14.1\phi^2\)

  • Dual TST kinetics: independent matrix and interfacial bond exchange rates

  • Interphase volume fraction: computed from NP geometry (\(\phi\), \(R_{NP}\), \(\delta_m\))

  • Optional interfacial damage with self-healing: reversible above \(T_v^{int}\)

The model employs four subnetworks:

  1. Permanent (P): Covalent crosslinks with amplified modulus \(G_P X(\phi)\)

  2. Exchangeable (E): Matrix vitrimer bonds with BER/TST kinetics (\(G_E\))

  3. Dissociative (D): Physical reversible bonds, standard Maxwell (\(G_D\))

  4. Interphase (I): NP-bound confined polymer with amplified affine deformation (\(G_{I,eff} X_I\))

These models are particularly well-suited for:

  • Nanoparticle-filled vitrimers and covalent adaptable networks

  • Silica/carbon-black reinforced polymer networks

  • Materials exhibiting Payne effect (strain-dependent modulus)

  • Systems with dual topological freezing temperatures

  • Multi-timescale relaxation from matrix vs interfacial exchange

Model Hierarchy

HVNM Family (extends HVM)
|
+-- HVNMLocal (Homogeneous, simple shear)
|   |
|   +-- Full HVNM: G_P + G_E + G_D + G_I (4-network)
|   |   +-- Dual TST kinetics: matrix + interphase
|   |   +-- Guth-Gold strain amplification
|   |   +-- Optional interfacial damage with self-healing
|   |   +-- Optional diffusion modes
|   |
|   +-- Limiting Cases (via factory methods):
|       +-- unfilled_vitrimer(...)         -> phi=0 (recovers HVM)
|       +-- filled_elastomer(G_P, phi)     -> G_E=0, G_D=0
|       +-- partial_vitrimer_nc(...)       -> G_D=0
|       +-- conventional_filled_rubber(...) -> G_E=0, frozen I
|       +-- matrix_only_exchange(...)      -> frozen interphase

Quick Reference

Class

HVNMLocal

Registry

"hvnm_local", "hvnm"

Parameters

13-25 (depending on feature flags)

Protocols

Flow curve, SAOS, Startup, Relaxation, Creep, LAOS

Inheritance

BaseModel -> VLBBase -> HVMBase -> HVNMBase -> HVNMLocal

Solver

Analytical (SAOS, flow curve) + diffrax ODE (startup, relaxation, creep, LAOS)

When to Use This Model

Behavior

HVNM Appropriate?

Alternative

NP-filled vitrimer

Yes (primary use case)

N/A

Unfilled vitrimer

Use phi=0 factory

HVMLocal (simpler)

Payne effect observed

Yes

N/A

Multi-timescale relaxation with phi dependence

Yes

N/A

Filled elastomer (no exchange)

Use limiting case

VLBMultiNetwork

Single relaxation mode

Overkill

VLBLocal or Maxwell

Supported Protocols

Protocol

Method

Notes

FLOW_CURVE

Analytical

\(\sigma_E = \sigma_I = 0\) at steady state; \(\sigma = \eta_D \dot{\gamma}\)

OSCILLATION

Analytical

Three Maxwell modes + \(G_P X\) plateau; dual factor-of-2

STARTUP

ODE (diffrax)

Dual TST overshoot; amplified initial slope

RELAXATION

ODE (diffrax)

Quad-exponential + \(G_P X\) plateau

CREEP

ODE (diffrax)

Three retardation modes; NP reduces compliance

LAOS

ODE (diffrax)

Payne onset at lower \(\gamma_0\); Lissajous + harmonic extraction

Quick Start

Full HVNM (4 subnetworks):

from rheojax.models import HVNMLocal

model = HVNMLocal(kinetics="stress", include_dissociative=True)
model.parameters.set_value("G_P", 5000.0)
model.parameters.set_value("G_E", 3000.0)
model.parameters.set_value("G_D", 1000.0)
model.parameters.set_value("phi", 0.1)
model.parameters.set_value("beta_I", 3.0)

# SAOS: three Maxwell modes + amplified plateau
omega = np.logspace(-3, 3, 100)
G_prime, G_double_prime = model.predict_saos(omega)

# Startup with dual TST feedback
t = np.linspace(0.01, 50, 200)
result = model.simulate_startup(t, gamma_dot=1.0, return_full=True)

Unfilled vitrimer (recovers HVM):

model = HVNMLocal.unfilled_vitrimer(G_P=5000, G_E=3000, G_D=1000)

Bayesian inference:

model = HVNMLocal()
model.fit(omega, G_star, test_mode='oscillation')
result = model.fit_bayesian(
    omega, G_star, test_mode='oscillation',
    num_warmup=1000, num_samples=2000,
)

Key Physics

Dual Factor-of-2: Both matrix and interphase relax with \(\hat{\tau}_E = 1/(2k_{BER,0}^{mat})\) and \(\hat{\tau}_I = 1/(2k_{BER,0}^{int})\). See Dual Factor-of-2 in the model reference.

Guth-Gold Strain Amplification: Rigid NPs amplify strain: \(X(\phi) = 1 + 2.5\phi + 14.1\phi^2\). See HVNMLocal — Full Model Reference for the full derivation.

Model Documentation

References

  1. Vernerey, F.J., Long, R. & Brighenti, R. (2017). “A statistically-based continuum theory for polymers with transient networks.” J. Mech. Phys. Solids, 107, 1-20. https://doi.org/10.1016/j.jmps.2017.05.016

  2. Karim, M.R., Vernerey, F. & Sain, T. (2025). “Constitutive Modeling of Vitrimers and Their Nanocomposites Based on Transient Network Theory.” Macromolecules, 58(10), 4899-4912. DOI: 10.1021/acs.macromol.4c02872 PDF

  3. Li, Z., Zhao, H., Duan, P., Zhang, L. & Liu, J. (2024). “Manipulating the Properties of Polymer Vitrimer Nanocomposites by Designing Dual Dynamic Covalent Bonds.” Langmuir, 40(14), 7769-7780. https://doi.org/10.1021/acs.langmuir.4c00699

See HVNM Advanced Theory & Numerical Methods for the full reference list (18 citations).