HVM (Hybrid Vitrimer Model)

This section documents the Hybrid Vitrimer Model (HVM) for polymers with permanent covalent crosslinks, associative (vitrimer-type) exchangeable bonds, and optionally dissociative (physical) reversible bonds.

Part of VLB Transient Network Family

HVM extends the VLB framework (VLB Transient Network Models) with vitrimer-specific physics: an evolving natural-state tensor and TST kinetics for associative bond exchange. For NP-filled vitrimers, see HVNM (Hybrid Vitrimer Nanocomposite Model).

Inheritance: BaseModel → VLBBase → HVMBase → HVMLocal

Overview

The HVM extends the VLB transient network framework (Vernerey, Long & Brighenti, 2017) with an evolving natural-state tensor that captures plastic flow through topology rearrangement without network breakdown. The key physics is bond exchange reactions (BER) accelerated by Transition State Theory (TST) kinetics, where mechanical stress lowers the activation barrier for bond exchange.

The model employs three subnetworks:

  1. Permanent (P): Covalent crosslinks, neo-Hookean elastic (\(G_P\))

  2. Exchangeable (E): Associative vitrimer bonds with BER kinetics (\(G_E\))

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

These models are particularly well-suited for:

  • Covalent adaptable networks (CANs) and vitrimers

  • Self-healing polymers with dynamic bonds

  • Shape-memory polymers with topology rearrangement

  • Multi-mechanism polymer networks

  • Temperature-dependent viscoelastic materials

Model Hierarchy

HVM Family
|
+-- HVMLocal (Homogeneous, simple shear)
|   |
|   +-- Full HVM: G_P + G_E + G_D (3-network)
|   |   +-- TST kinetics: stress or stretch coupling
|   |   +-- Optional cooperative damage shielding
|   |
|   +-- Limiting Cases (via factory methods):
|       +-- neo_hookean(G_P)       -> G_E=0, G_D=0
|       +-- maxwell(G_D, k_d_D)   -> G_P=0, G_E=0
|       +-- zener(G_P, G_D, ...)  -> G_E=0 (SLS)
|       +-- pure_vitrimer(G_E, ...)    -> G_P=0, G_D=0
|       +-- partial_vitrimer(G_P, G_E, ...) -> G_D=0 (Meng 2019)

Quick Reference

Class

HVMLocal

Registry

"hvm_local", "hvm"

Parameters

6-10 (depending on options)

Protocols

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

Inheritance

BaseModel -> VLBBase -> HVMBase -> HVMLocal

Solver

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

When to Use This Model

Behavior

HVM Appropriate?

Alternative

Vitrimer with BER kinetics

Yes (primary use case)

N/A

Permanent + exchangeable network

Yes (partial vitrimer)

VLBMultiNetwork (if no BER)

Single Maxwell relaxation

Use limiting case

VLBLocal (simpler)

Associating polymer (no BER)

Use D-network only

TNT or VLB models

TST stress-enhanced exchange

Yes

N/A

Temperature-dependent relaxation

Yes (Arrhenius BER rate)

TTS transforms

Supported Protocols

Protocol

Method

Notes

FLOW_CURVE

Analytical

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

OSCILLATION

Analytical

Two Maxwell modes + \(G_P\) plateau; \(\tau_{E,eff} = 1/(2k_{BER,0})\)

STARTUP

ODE (diffrax)

TST creates stress overshoot; analytical for constant-rate

RELAXATION

ODE (diffrax)

Bi-exponential + \(G_P\) plateau; TST gives non-exponential decay

CREEP

ODE (diffrax)

Two retardation modes; vitrimer plastic creep at intermediate times

LAOS

ODE (diffrax)

TST generates odd harmonics; Lissajous curves + harmonic extraction

Quick Start

Full HVM (3 subnetworks):

from rheojax.models import HVMLocal

model = HVMLocal(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)

# SAOS
omega = np.logspace(-3, 3, 100)
G_prime, G_double_prime = model.predict_saos(omega, return_components=False)

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

Partial vitrimer (Meng 2019):

model = HVMLocal.partial_vitrimer(G_P=5000, G_E=3000, nu_0=1e10, E_a=80e3)

Bayesian inference:

model = HVMLocal()
model.fit(omega, G_star, test_mode='oscillation')     # NLSQ warm start
result = model.fit_bayesian(
    omega, G_star, test_mode='oscillation',
    num_warmup=1000, num_samples=2000,
)

Key Physics: Factor-of-2

The exchangeable network relaxes with effective time constant \(\tau_{E,eff} = 1/(2k_{BER,0})\) because both \(\boldsymbol{\mu}^E\) and \(\boldsymbol{\mu}^E_{nat}\) relax toward each other at rate \(k_{BER}\). See Factor-of-2 in Relaxation in the 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. Meng, F., Saed, M.O. & Terentjev, E.M. (2019). “Elasticity and Relaxation in Full and Partial Vitrimer Networks.” Macromolecules, 52(19), 7423-7429. https://doi.org/10.1021/acs.macromol.9b01123

  3. Montarnal, D., Capelot, M., Tournilhac, F. & Leibler, L. (2011). “Silica-like malleable materials from permanent organic networks.” Science, 334, 965-968. https://doi.org/10.1126/science.1212648

See HVM Advanced Theory & Numerical Methods for the full reference list (12 citations).