HVNMLocal — Full Model Reference¶

Note

HVNM extends HVM (HVM Model Reference), which builds on VLB (VLB Transient Network Models). 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¶

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¶

Symbol	Parameter	Units	Description
\(G_P\)	`G_P`	Pa	Permanent (covalent) network modulus
\(G_E\)	`G_E`	Pa	Exchangeable (vitrimer) network modulus
\(G_D\)	`G_D`	Pa	Dissociative (physical) network modulus
\(\nu_0\)	`nu_0`	1/s	Matrix TST attempt frequency
\(E_a\)	`E_a`	J/mol	Matrix activation energy
\(V_{act}\)	`V_act`	m³/mol	Matrix activation volume
\(T\)	`T`	K	Temperature
\(k_{d,D}\)	`k_d_D`	1/s	Dissociative rate constant
\(\beta_I\)	`beta_I`	—	Interphase reinforcement ratio \(G_I / G_E\)
\(\nu_0^{int}\)	`nu_0_int`	1/s	Interfacial TST attempt frequency
\(E_a^{int}\)	`E_a_int`	J/mol	Interfacial activation energy
\(V_{act}^{int}\)	`V_act_int`	m³/mol	Interfacial activation volume
\(\phi\)	`phi`	—	NP volume fraction
\(R_{NP}\)	`R_NP`	m	NP radius
\(\delta_m\)	`delta_m`	m	Mobile interphase thickness
\(D\)	`D`	—	Permanent-network damage variable \(\in [0,1]\)
\(D_{int}\)	`D_int`	—	Interfacial damage variable \(\in [0,1]\)
\(\phi_I\)	(derived)	—	Interphase volume fraction from NP geometry
\(X(\phi)\)	(derived)	—	Guth-Gold strain amplification factor
\(G_{I,eff}\)	(derived)	Pa	Effective interphase modulus \(\beta_I G_E \phi_I\)
\(k_{diff}\)	`k_diff`	1/s	Diffusion-limited slow mode rate (`include_diffusion=True`)
\(h_{int}\)	(derived)	1/s	Interfacial self-healing rate (Arrhenius)
\(\Gamma_0\)	`Gamma_0`	1/s	Damage rate coefficient (`include_damage=True`)

Physical Foundations¶

4-Subnetwork Architecture

The HVNM Cauchy stress in simple shear is:

\[\sigma_{tot} = \underbrace{(1-D) G_P X(\phi) \gamma}_{\text{permanent}} + \underbrace{G_E (\mu^E_{xy} - \mu^{E,nat}_{xy})}_{\text{exchangeable}} + \underbrace{G_D (\mu^D_{xy} - \delta_{xy})}_{\text{dissociative}} + \underbrace{(1-D_{int}) G_{I,eff} X_I (\mu^I_{xy} - \mu^{I,nat}_{xy})}_{\text{interphase}}\]

where:

\(X(\phi) = 1 + 2.5\phi + 14.1\phi^2\) is the Guth-Gold strain amplification
\(G_{I,eff} = \beta_I G_E \phi_I\) is the effective interphase modulus
\(\phi_I\) is the interphase volume fraction from NP geometry
\(X_I = X(\phi_I)\) is the interphase amplification factor

Dual TST Kinetics

Matrix and interfacial BER rates are independent:

\[\begin{split}k_{BER}^{mat} &= \nu_0 \exp\!\left(-\frac{E_a}{RT}\right) \cosh\!\left(\frac{V_{act} \sigma_{VM}^E}{RT}\right) \\ k_{BER}^{int} &= \nu_0^{int} \exp\!\left(-\frac{E_a^{int}}{RT}\right) \cosh\!\left(\frac{V_{act}^{int} \sigma_{VM}^I}{RT}\right)\end{split}\]

I-Network Evolution

The interphase distribution tensor evolves with amplified affine deformation:

\[\begin{split}\dot{\mu}^I_{xx} &= 2 X_I \dot{\gamma} \, \mu^I_{xy} + k_{BER}^{int}(\mu^{I,nat}_{xx} - \mu^I_{xx}) \\ \dot{\mu}^I_{yy} &= k_{BER}^{int}(\mu^{I,nat}_{yy} - \mu^I_{yy}) \\ \dot{\mu}^I_{xy} &= X_I \dot{\gamma} \, \mu^I_{yy} + k_{BER}^{int}(\mu^{I,nat}_{xy} - \mu^I_{xy})\end{split}\]

The I-network natural-state tensor evolves symmetrically with the E-network:

\[\dot{\mu}^{I,nat}_{ij} = k_{BER}^{int}(\mu^I_{ij} - \mu^{I,nat}_{ij})\]

Dual Factor-of-2¶

This coupled evolution gives the same factor-of-2 as the E-network (Factor-of-2 in Relaxation): the I-network stress relaxes with \(\tau_I = 1/(2k_{BER,0}^{int})\).

How HVNM Differs from HVM:

P-network: modulus amplified by \(X(\phi)\) — rigid inclusions increase effective strain
I-network: entirely new fourth subnetwork with independent TST kinetics
Steady state: both \(\sigma_E = 0\) and \(\sigma_I = 0\) (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 \(D_{int}\) with self-healing (see Enhanced Damage Mechanics)
Diffusion: optional slow mode \(k_{diff}\) for long-time relaxation tail (see Diffusion-Limited Slow Mode)

Interphase Volume Fraction¶

The interphase volume fraction is computed from NP geometry:

\[\phi_I = \phi \left[\left(\frac{R_{NP} + \delta_m}{R_{NP}}\right)^3 - 1\right]\]

For dilute suspensions (\(\phi < 0.2\)), the interphase shells do not overlap. At higher \(\phi\), percolation occurs when \(\phi_I\) exceeds a critical threshold. See Nanoparticle Interphase Model for the full three-layer interphase model and percolation analysis.

Guth-Gold Strain Amplification:

\[X(\phi) = 1 + 2.5\phi + 14.1\phi^2\]

This applies to the P-network modulus (\(G_P X(\phi)\)) and to the interphase amplification (\(X_I = X(\phi_I)\)). The quadratic term captures hydrodynamic interactions between NPs.

Parameter Table¶

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 \(G_I/G_E\)
`nu_0_int`	1e10	(1e6, 1e14)	1/s	Interfacial TST attempt frequency
`E_a_int`	90e3	(30e3, 250e3)	J/mol	Interfacial activation energy (typically > \(E_a\))
`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`)

Protocol Summary¶

For complete derivations and closed-form solutions, see HVNM Protocol Equations & Derivations.

Protocol	Method	Key Result
Flow Curve	Analytical	\(\sigma_E = \sigma_I = 0\) at steady state; \(\sigma^{ss} = (1-D) G_P X \gamma + \eta_D \dot{\gamma}\)
SAOS	Analytical	Three Maxwell modes + \(G_P X\) plateau; dual factor-of-2
Startup	ODE	Dual TST overshoot; amplified initial slope \(G_{tot}^{NC}\)
Relaxation	ODE	Quad-exponential + \(G_P X\) plateau; optional \(k_{diff}\) tail
Creep	ODE	Three retardation modes; NP reduces compliance
LAOS	ODE	Payne onset at \(\gamma_c / X_I\); Lissajous + harmonic extraction

Limiting Cases¶

Factory Methods:

Limiting Case	Conditions	Factory Method	Behavior
HVM (unfilled)	\(\phi = 0\)	`unfilled_vitrimer()`	Exact HVM
Filled elastomer	\(G_E = G_D = 0\)	`filled_elastomer()`	Neo-Hookean + NP
Partial vitrimer NC	\(G_D = 0\)	`partial_vitrimer_nc()`	P + E + I
Conventional filled rubber	\(G_E = 0\), frozen I	`conventional_filled_rubber()`	P + D + elastic I
Matrix-only exchange	Frozen interphase	`matrix_only_exchange()`	P + E + D

Additional Limiting Regimes:

Regime	Conditions	Physical Interpretation
Low-\(T\) (glassy)	\(T < T_v^{mat}\)	All exchange frozen; elastic solid
Intermediate-\(T\)	\(T_v^{mat} < T < T_v^{int}\)	Matrix relaxes; interphase frozen
High-\(T\)	\(T > T_v^{int}\)	Both networks relax; \(G_P X\) plateau only
Dilute filler	\(\phi \ll 0.05\)	\(X \approx 1\), negligible interphase
Percolation	\(\phi_I > \phi_I^{perc}\)	Interphase shells overlap, enhanced modulus
Strong confinement	\(\beta_I \gg 1\)	I-network dominates at high \(\phi\)
No damage	\(D = D_{int} = 0\)	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 HVNM Advanced Theory & Numerical Methods.

For troubleshooting, cross-protocol validation, knowledge extraction workflows, and Payne effect interpretation, see HVNM Knowledge Extraction Guide.