VLB Transient Network Models¶

This section documents the Vernerey-Long-Brighenti (VLB) family of models for polymers with dynamic (reversible) cross-links.

VLB Transient Network Family

The VLB framework serves as the foundation for a hierarchy of models:

VLB — Base transient network theory (this section)
HVM — Hybrid Vitrimer Model: extends VLB with evolving natural-state tensor and TST kinetics for vitrimers (HVM (Hybrid Vitrimer Model))
HVNM — Hybrid Vitrimer Nanocomposite Model: extends HVM with a 4th interphase subnetwork for NP-filled vitrimers (HVNM (Hybrid Vitrimer Nanocomposite Model))

Inheritance: BaseModel → VLBBase → HVMBase → HVNMBase

Overview¶

The VLB framework (Vernerey, Long & Brighenti, 2017) provides a statistically grounded continuum theory for transient polymer networks. Starting from the chain end-to-end vector distribution \(\varphi(\mathbf{r},t)\), one derives a second-moment distribution tensor \(\boldsymbol{\mu}\) whose evolution is governed by:

\[\dot{\boldsymbol{\mu}} = k_d(\mathbf{I} - \boldsymbol{\mu}) + \mathbf{L} \cdot \boldsymbol{\mu} + \boldsymbol{\mu} \cdot \mathbf{L}^T\]

where \(k_d\) is the bond dissociation rate and \(\mathbf{L}\) is the velocity gradient. The Cauchy stress is:

\[\boldsymbol{\sigma} = G_0 (\boldsymbol{\mu} - \mathbf{I}) + p\mathbf{I}\]

With constant \(k_d\) the single-network model is exactly Maxwell: relaxation time \(t_R = 1/k_d\), zero-shear viscosity \(\eta_0 = G_0 / k_d\). All six standard protocols admit closed-form solutions. The multi-network variant extends this to a generalized Maxwell spectrum with M transient networks, an optional permanent network, and solvent viscosity.

Model Variants¶

Model	Description
`VLBLocal`	Single transient network (2 params: \(G_0, k_d\)). All protocols analytical.
`VLBMultiNetwork`	M transient + optional permanent network + solvent viscosity (\(2M+1\) or \(2M+2\) params).
`VLBVariant`	Bell breakage + FENE-P + temperature flags (2-6 params). Shear thinning, bounded extension.
`VLBNonlocal`	Spatially-resolved PDE with tensor diffusion (4-6 params). Shear banding detection.

Model Hierarchy¶

VLB Family (4 Classes)
│
├── VLBLocal (Single transient network)
│   ├── Parameters: G₀, k_d
│   ├── Relaxation time: t_R = 1/k_d
│   ├── Viscosity: η₀ = G₀/k_d
│   └── All 6 protocols: analytical (LAOS via ODE)
│
├── VLBMultiNetwork (Generalized Maxwell)
│   ├── N transient networks: {G_I, k_d_I} for I = 0..N-1
│   ├── Optional permanent network: G_e (include_permanent=True)
│   ├── Solvent viscosity: η_s
│   ├── Relaxation spectrum: G(t) = G_e + Σ G_I e^{-k_d_I·t}
│   └── Creep: ODE-based (analytical for 1 transient + permanent)
│
├── VLBVariant (Bell + FENE-P + Temperature)
│   ├── Bell breakage: k_d(μ) = k_d₀·exp(ν·(λ_c - 1))
│   ├── FENE-P stress: σ = G₀·f(tr(μ))·(μ - I)
│   ├── Temperature: Arrhenius k_d(T), G_0(T) = G_0_ref·T/T_ref
│   └── All 6 protocols via ODE (SAOS analytical)
│
└── VLBNonlocal (Spatial PDE)
    ├── 1D gap-resolved PDE with tensor diffusion D_μ∇²μ
    ├── Shear banding detection and band width analysis
    ├── Cooperativity length: ξ = √(D_μ/k_d₀)
    └── Protocols: steady shear, startup, creep

When to Use Which Model¶

Supported Protocols¶

Protocol	Method	Notes
FLOW_CURVE	Analytical	Newtonian: \(\sigma = (G_0/k_d) \dot{\gamma}\)
OSCILLATION	Analytical	Maxwell SAOS: \(G'(\omega), G''(\omega)\)
STARTUP	Analytical	\(\sigma(t) = (G_0 \dot{\gamma}/k_d)(1 - e^{-k_d t})\)
RELAXATION	Analytical	\(G(t) = G_0 e^{-k_d t}\)
CREEP	Analytical / ODE	Single: \(J(t) = (1 + k_d t)/G_0\); multi: ODE
LAOS	ODE (diffrax)	Linear \(\sigma_{12}\), \(N_1\) has \(2\omega\) harmonics

Quick Start¶

Single network:

from rheojax.models import VLBLocal

model = VLBLocal()
model.fit(omega, G_star, test_mode='oscillation')

# Properties
print(f"G₀ = {model.G0:.1f} Pa")
print(f"k_d = {model.k_d:.3f} 1/s")
print(f"t_R = {model.relaxation_time:.3f} s")
print(f"η₀ = {model.viscosity:.1f} Pa·s")

Multi-network:

from rheojax.models import VLBMultiNetwork

model = VLBMultiNetwork(n_modes=3, include_permanent=True)
model.fit(omega, G_star, test_mode='oscillation')

# Relaxation spectrum
spectrum = model.get_relaxation_spectrum()
for G_i, t_R_i in spectrum:
    print(f"G = {G_i:.1f} Pa, t_R = {t_R_i:.3f} s")

Bayesian inference:

model = VLBLocal()
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,
)

Relation to TNT Models¶

VLB and TNT both describe transient polymer networks but differ in their tensorial variables and derivation:

Aspect	VLB	TNT
State variable	Distribution tensor \(\boldsymbol{\mu} = \langle \mathbf{r r} \rangle / \langle r_0^2 \rangle\)	Conformation tensor \(\mathbf{S} = \langle \mathbf{Q Q} \rangle\)
Derivation	Statistical distribution \(\varphi(\mathbf{r},t)\)	Network theory (Green-Tobolsky)
Equilibrium	\(\boldsymbol{\mu}_{eq} = \mathbf{I}\)	\(\mathbf{S}_{eq} = \mathbf{I}\)
Stress	\(\boldsymbol{\sigma} = G_0(\boldsymbol{\mu} - \mathbf{I})\)	\(\boldsymbol{\sigma} = G(\mathbf{S} - \mathbf{I})\)
With constant \(k_d\)	Maxwell (identical predictions)	Maxwell (identical predictions)
Extensions	Langevin chains, Bell \(k_d(\mu)\)	Bell, FENE-P, non-affine, loop-bridge

At the constant-\(k_d\) level the models are mathematically equivalent. The VLB formulation provides a clearer path to molecular extensions (Langevin chains, entropy-based \(k_d\)).

Model Documentation¶

References¶

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.
Green, M.S. & Tobolsky, A.V. (1946). “A New Approach to the Theory of Relaxing Polymeric Media.” J. Chem. Phys., 14(2), 80-92.
Tanaka, F. & Edwards, S.F. (1992). “Viscoelastic properties of physically crosslinked networks.” J. Non-Newtonian Fluid Mech., 43(2-3), 247-271.