HVNMLocal — Full Model Reference ================================= .. note:: HVNM extends HVM (:doc:`/models/hvm/hvm`), which builds on VLB (:doc:`/models/vlb/vlb`). The E-network and D-network equations are identical to HVM. This page focuses on the **I-network** (interphase) additions and HVNM-specific behavior. Quick Reference --------------- .. list-table:: :widths: 25 75 :header-rows: 0 * - Class - ``HVNMLocal`` * - Registry names - ``"hvnm_local"``, ``"hvnm"`` * - Parameters - 15 (base) to 25 (all flags on) * - Parent class - ``HVNMBase(HVMBase(VLBBase(BaseModel)))`` * - Feature flags - ``include_dissociative``, ``include_damage``, ``include_interfacial_damage``, ``include_diffusion`` * - ODE state - 17 or 18 components (simple shear) Notation Guide -------------- .. list-table:: :widths: 15 20 10 55 :header-rows: 1 * - Symbol - Parameter - Units - Description * - :math:`G_P` - ``G_P`` - Pa - Permanent (covalent) network modulus * - :math:`G_E` - ``G_E`` - Pa - Exchangeable (vitrimer) network modulus * - :math:`G_D` - ``G_D`` - Pa - Dissociative (physical) network modulus * - :math:`\nu_0` - ``nu_0`` - 1/s - Matrix TST attempt frequency * - :math:`E_a` - ``E_a`` - J/mol - Matrix activation energy * - :math:`V_{act}` - ``V_act`` - m³/mol - Matrix activation volume * - :math:`T` - ``T`` - K - Temperature * - :math:`k_{d,D}` - ``k_d_D`` - 1/s - Dissociative rate constant * - :math:`\beta_I` - ``beta_I`` - — - Interphase reinforcement ratio :math:`G_I / G_E` * - :math:`\nu_0^{int}` - ``nu_0_int`` - 1/s - Interfacial TST attempt frequency * - :math:`E_a^{int}` - ``E_a_int`` - J/mol - Interfacial activation energy * - :math:`V_{act}^{int}` - ``V_act_int`` - m³/mol - Interfacial activation volume * - :math:`\phi` - ``phi`` - — - NP volume fraction * - :math:`R_{NP}` - ``R_NP`` - m - NP radius * - :math:`\delta_m` - ``delta_m`` - m - Mobile interphase thickness * - :math:`D` - ``D`` - — - Permanent-network damage variable :math:`\in [0,1]` * - :math:`D_{int}` - ``D_int`` - — - Interfacial damage variable :math:`\in [0,1]` * - :math:`\phi_I` - (derived) - — - Interphase volume fraction from NP geometry * - :math:`X(\phi)` - (derived) - — - Guth-Gold strain amplification factor * - :math:`G_{I,eff}` - (derived) - Pa - Effective interphase modulus :math:`\beta_I G_E \phi_I` * - :math:`k_{diff}` - ``k_diff`` - 1/s - Diffusion-limited slow mode rate (``include_diffusion=True``) * - :math:`h_{int}` - (derived) - 1/s - Interfacial self-healing rate (Arrhenius) * - :math:`\Gamma_0` - ``Gamma_0`` - 1/s - Damage rate coefficient (``include_damage=True``) Physical Foundations -------------------- **4-Subnetwork Architecture** The HVNM Cauchy stress in simple shear is: .. math:: \sigma_{tot} = \underbrace{(1-D) G_P X(\phi) \gamma}_{\text{permanent}} + \underbrace{G_E (\mu^E_{xy} - \mu^{E,nat}_{xy})}_{\text{exchangeable}} + \underbrace{G_D (\mu^D_{xy} - \delta_{xy})}_{\text{dissociative}} + \underbrace{(1-D_{int}) G_{I,eff} X_I (\mu^I_{xy} - \mu^{I,nat}_{xy})}_{\text{interphase}} where: - :math:`X(\phi) = 1 + 2.5\phi + 14.1\phi^2` is the Guth-Gold strain amplification - :math:`G_{I,eff} = \beta_I G_E \phi_I` is the effective interphase modulus - :math:`\phi_I` is the interphase volume fraction from NP geometry - :math:`X_I = X(\phi_I)` is the interphase amplification factor **Dual TST Kinetics** Matrix and interfacial BER rates are independent: .. math:: k_{BER}^{mat} &= \nu_0 \exp\!\left(-\frac{E_a}{RT}\right) \cosh\!\left(\frac{V_{act} \sigma_{VM}^E}{RT}\right) \\ k_{BER}^{int} &= \nu_0^{int} \exp\!\left(-\frac{E_a^{int}}{RT}\right) \cosh\!\left(\frac{V_{act}^{int} \sigma_{VM}^I}{RT}\right) **I-Network Evolution** The interphase distribution tensor evolves with amplified affine deformation: .. math:: \dot{\mu}^I_{xx} &= 2 X_I \dot{\gamma} \, \mu^I_{xy} + k_{BER}^{int}(\mu^{I,nat}_{xx} - \mu^I_{xx}) \\ \dot{\mu}^I_{yy} &= k_{BER}^{int}(\mu^{I,nat}_{yy} - \mu^I_{yy}) \\ \dot{\mu}^I_{xy} &= X_I \dot{\gamma} \, \mu^I_{yy} + k_{BER}^{int}(\mu^{I,nat}_{xy} - \mu^I_{xy}) The I-network natural-state tensor evolves symmetrically with the E-network: .. math:: \dot{\mu}^{I,nat}_{ij} = k_{BER}^{int}(\mu^I_{ij} - \mu^{I,nat}_{ij}) .. _hvnm-dual-factor-of-2: Dual Factor-of-2 ^^^^^^^^^^^^^^^^^ This coupled evolution gives the same **factor-of-2** as the E-network (:ref:`hvm-factor-of-2`): the I-network stress relaxes with :math:`\tau_I = 1/(2k_{BER,0}^{int})`. **How HVNM Differs from HVM:** - **P-network**: modulus amplified by :math:`X(\phi)` — rigid inclusions increase effective strain - **I-network**: entirely new fourth subnetwork with independent TST kinetics - **Steady state**: both :math:`\sigma_E = 0` and :math:`\sigma_I = 0` (all natural states track deformation) - **SAOS**: three Maxwell modes instead of two (E, D, I) plus amplified plateau - **Parameter count**: 15-25 vs HVM's 6-10 - **Damage**: optional interfacial damage :math:`D_{int}` with self-healing (see :ref:`hvnm-damage-mechanics`) - **Diffusion**: optional slow mode :math:`k_{diff}` for long-time relaxation tail (see :ref:`hvnm-diffusion-mode`) Interphase Volume Fraction -------------------------- The interphase volume fraction is computed from NP geometry: .. math:: \phi_I = \phi \left[\left(\frac{R_{NP} + \delta_m}{R_{NP}}\right)^3 - 1\right] For dilute suspensions (:math:`\phi < 0.2`), the interphase shells do not overlap. At higher :math:`\phi`, percolation occurs when :math:`\phi_I` exceeds a critical threshold. See :ref:`hvnm-interphase-model` for the full three-layer interphase model and percolation analysis. **Guth-Gold Strain Amplification:** .. math:: X(\phi) = 1 + 2.5\phi + 14.1\phi^2 This applies to the P-network modulus (:math:`G_P X(\phi)`) and to the interphase amplification (:math:`X_I = X(\phi_I)`). The quadratic term captures hydrodynamic interactions between NPs. Parameter Table --------------- .. list-table:: :widths: 12 12 15 10 51 :header-rows: 1 * - Parameter - Default - Bounds - Units - Description * - ``G_P`` - 1e4 - (0, 1e9) - Pa - Permanent network modulus (covalent crosslinks) * - ``G_E`` - 1e4 - (0, 1e9) - Pa - Exchangeable network modulus (matrix vitrimer bonds) * - ``nu_0`` - 1e10 - (1e6, 1e14) - 1/s - Matrix TST attempt frequency * - ``E_a`` - 80e3 - (20e3, 200e3) - J/mol - Matrix activation energy for BER * - ``V_act`` - 1e-5 - (1e-8, 1e-2) - m³/mol - Matrix activation volume * - ``T`` - 300 - (200, 500) - K - Temperature * - ``phi`` - 0.05 - (0.0, 0.5) - -- - NP volume fraction * - ``R_NP`` - 20e-9 - (1e-9, 1e-6) - m - NP radius * - ``delta_m`` - 10e-9 - (1e-9, 1e-7) - m - Mobile interphase thickness * - ``beta_I`` - 3.0 - (1.0, 10.0) - -- - Interphase reinforcement ratio :math:`G_I/G_E` * - ``nu_0_int`` - 1e10 - (1e6, 1e14) - 1/s - Interfacial TST attempt frequency * - ``E_a_int`` - 90e3 - (30e3, 250e3) - J/mol - Interfacial activation energy (typically > :math:`E_a`) * - ``V_act_int`` - 5e-6 - (1e-8, 1e-2) - m³/mol - Interfacial activation volume * - ``G_D`` - 1e3 - (0, 1e8) - Pa - Dissociative network modulus (``include_dissociative=True``) * - ``k_d_D`` - 1.0 - (1e-6, 1e6) - 1/s - Dissociative bond rate (``include_dissociative=True``) * - ``Gamma_0`` - 1e-4 - (0, 0.1) - 1/s - Damage rate coefficient (``include_damage=True``) * - ``lambda_crit`` - 2.0 - (1.001, 10) - -- - Critical stretch for damage onset (``include_damage=True``) * - ``Gamma_0_int`` - 1e-3 - (0, 1.0) - 1/s - Interfacial damage rate (``include_interfacial_damage=True``) * - ``lambda_crit_int`` - 1.5 - (1.001, 5.0) - -- - Interfacial critical stretch (``include_interfacial_damage=True``) * - ``h_0`` - 1e-4 - (0.0, 1.0) - 1/s - Interfacial healing prefactor (``include_interfacial_damage=True``) * - ``E_a_heal`` - 100e3 - (30e3, 300e3) - J/mol - Healing activation energy (``include_interfacial_damage=True``) * - ``k_diff_0_mat`` - 1e-4 - (0.0, 1.0) - 1/s - Matrix diffusion rate constant (``include_diffusion=True``) * - ``k_diff_0_int`` - 1e-6 - (0.0, 0.1) - 1/s - Interphase diffusion rate constant (``include_diffusion=True``) * - ``E_a_diff`` - 120e3 - (50e3, 400e3) - J/mol - Diffusion activation energy (``include_diffusion=True``) .. _hvnm-protocol-summary: Protocol Summary ---------------- For complete derivations and closed-form solutions, see :doc:`hvnm_protocols`. .. list-table:: :widths: 15 15 70 :header-rows: 1 * - Protocol - Method - Key Result * - :ref:`Flow Curve ` - Analytical - :math:`\sigma_E = \sigma_I = 0` at steady state; :math:`\sigma^{ss} = (1-D) G_P X \gamma + \eta_D \dot{\gamma}` * - :ref:`SAOS ` - Analytical - Three Maxwell modes + :math:`G_P X` plateau; dual factor-of-2 * - :ref:`Startup ` - ODE - Dual TST overshoot; amplified initial slope :math:`G_{tot}^{NC}` * - :ref:`Relaxation ` - ODE - Quad-exponential + :math:`G_P X` plateau; optional :math:`k_{diff}` tail * - :ref:`Creep ` - ODE - Three retardation modes; NP reduces compliance * - :ref:`LAOS ` - ODE - Payne onset at :math:`\gamma_c / X_I`; Lissajous + harmonic extraction Limiting Cases -------------- **Factory Methods:** .. list-table:: :widths: 25 35 25 15 :header-rows: 1 * - Limiting Case - Conditions - Factory Method - Behavior * - HVM (unfilled) - :math:`\phi = 0` - ``unfilled_vitrimer()`` - Exact HVM * - Filled elastomer - :math:`G_E = G_D = 0` - ``filled_elastomer()`` - Neo-Hookean + NP * - Partial vitrimer NC - :math:`G_D = 0` - ``partial_vitrimer_nc()`` - P + E + I * - Conventional filled rubber - :math:`G_E = 0`, frozen I - ``conventional_filled_rubber()`` - P + D + elastic I * - Matrix-only exchange - Frozen interphase - ``matrix_only_exchange()`` - P + E + D **Additional Limiting Regimes:** .. list-table:: :widths: 25 40 35 :header-rows: 1 * - Regime - Conditions - Physical Interpretation * - Low-:math:`T` (glassy) - :math:`T < T_v^{mat}` - All exchange frozen; elastic solid * - Intermediate-:math:`T` - :math:`T_v^{mat} < T < T_v^{int}` - Matrix relaxes; interphase frozen * - High-:math:`T` - :math:`T > T_v^{int}` - Both networks relax; :math:`G_P X` plateau only * - Dilute filler - :math:`\phi \ll 0.05` - :math:`X \approx 1`, negligible interphase * - Percolation - :math:`\phi_I > \phi_I^{perc}` - Interphase shells overlap, enhanced modulus * - Strong confinement - :math:`\beta_I \gg 1` - I-network dominates at high :math:`\phi` * - No damage - :math:`D = D_{int} = 0` - Conservative system (default) Advanced Theory --------------- For thermodynamic foundations (Helmholtz energy with 4 networks + 2 damage variables), the three-layer interphase model, enhanced damage mechanics with self-healing, diffusion-limited slow modes, and numerical implementation details, see :doc:`hvnm_advanced`. For troubleshooting, cross-protocol validation, knowledge extraction workflows, and Payne effect interpretation, see :doc:`hvnm_knowledge`.