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. .. admonition:: Part of VLB Transient Network Family HVNM extends HVM (:doc:`/models/hvm/index`), which itself builds on VLB (:doc:`/models/vlb/index`). 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**: :math:`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 (:math:`\phi`, :math:`R_{NP}`, :math:`\delta_m`) - **Optional interfacial damage with self-healing**: reversible above :math:`T_v^{int}` The model employs four subnetworks: 1. **Permanent (P)**: Covalent crosslinks with amplified modulus :math:`G_P X(\phi)` 2. **Exchangeable (E)**: Matrix vitrimer bonds with BER/TST kinetics (:math:`G_E`) 3. **Dissociative (D)**: Physical reversible bonds, standard Maxwell (:math:`G_D`) 4. **Interphase (I)**: NP-bound confined polymer with amplified affine deformation (:math:`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 --------------- .. list-table:: :widths: 20 80 :header-rows: 0 * - **Class** - :class:`~rheojax.models.hvnm.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 ---------------------- .. list-table:: :widths: 35 30 35 :header-rows: 1 * - 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 ------------------- .. list-table:: :widths: 20 20 60 :header-rows: 1 * - Protocol - Method - Notes * - FLOW_CURVE - Analytical - :math:`\sigma_E = \sigma_I = 0` at steady state; :math:`\sigma = \eta_D \dot{\gamma}` * - OSCILLATION - Analytical - Three Maxwell modes + :math:`G_P X` plateau; dual factor-of-2 * - STARTUP - ODE (diffrax) - Dual TST overshoot; amplified initial slope * - RELAXATION - ODE (diffrax) - Quad-exponential + :math:`G_P X` plateau * - CREEP - ODE (diffrax) - Three retardation modes; NP reduces compliance * - LAOS - ODE (diffrax) - Payne onset at lower :math:`\gamma_0`; Lissajous + harmonic extraction Quick Start ----------- **Full HVNM (4 subnetworks):** .. code-block:: python 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):** .. code-block:: python model = HVNMLocal.unfilled_vitrimer(G_P=5000, G_E=3000, G_D=1000) **Bayesian inference:** .. code-block:: python 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 :math:`\hat{\tau}_E = 1/(2k_{BER,0}^{mat})` and :math:`\hat{\tau}_I = 1/(2k_{BER,0}^{int})`. See :ref:`hvnm-dual-factor-of-2` in the model reference. **Guth-Gold Strain Amplification:** Rigid NPs amplify strain: :math:`X(\phi) = 1 + 2.5\phi + 14.1\phi^2`. See :doc:`hvnm` for the full derivation. Model Documentation ------------------- .. toctree:: :maxdepth: 1 hvnm hvnm_protocols hvnm_advanced hvnm_knowledge 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 `_ :download:`PDF <../../../reference/karim_2025_vitrimer_nanocomposites.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 :doc:`hvnm_advanced` for the full reference list (18 citations).