HVM Knowledge Extraction Guide

This guide explains how to extract physical insights about vitrimer materials from HVM model parameters and predictions.

What Knowledge Can Be Extracted

Structural parameters:

  • Crosslink densities from moduli: \(c_i = G_i / (k_B T)\)

  • Exchange fraction: \(f_E = G_E / (G_P + G_E)\)

  • Network architecture: permanent vs exchangeable vs physical

Kinetic parameters:

  • Activation energy \(E_a\) from multi-temperature fits (Arrhenius plot)

  • TST attempt frequency \(\nu_0\) from rate prefactor

  • Mechanochemical coupling \(V_{act}\) from nonlinear startup

Material classification:

  • Thermoset (\(G_P \gg G_E\)): dominated by permanent crosslinks

  • Partial vitrimer (\(G_E \sim G_P\)): mixed permanent + exchangeable

  • Vitrimer liquid (\(G_P \approx 0\)): fully exchangeable network

  • Full HVM (\(G_D > 0\)): additional physical crosslinks

Predictive capabilities:

  • Temperature-rate superposition for processing windows

  • Topology freezing transition temperature \(T_v\)

  • Stress relaxation vs permanent elastic memory

Parameter-to-Physics Map

Parameter

Physical Meaning

How to Determine

\(G_P\)

Permanent crosslink density

Low-frequency SAOS plateau: \(G'(\omega \to 0) = G_P\)

\(G_E\)

Exchangeable crosslink density

Difference: \(G'(\omega \to \infty) - G_P - G_D = G_E\)

\(G_D\)

Physical bond density

Second loss peak position and height in \(G''(\omega)\)

\(E_a\)

BER activation barrier

Arrhenius fit of \(k_{BER,0}\) vs \(1/T\) from multi-T relaxation

\(V_{act}\)

Mechanochemical coupling

Stress overshoot magnitude in startup shear

\(\nu_0\)

Bond exchange attempt rate

Arrhenius intercept (hard to determine independently)

\(k_d^D\)

Physical bond lifetime

Second relaxation time in bi-exponential fit

\(\Gamma_0\)

Damage sensitivity

Strain softening rate under large deformation

\(\lambda_{crit}\)

Damage onset threshold

Strain at which softening begins

Diagnostic Decision Tree

Does SAOS show a low-frequency plateau?
|
+-- Yes: G_P > 0 (permanent crosslinks present)
|   |
|   +-- Single relaxation peak in G''?
|   |   +-- Yes: Partial vitrimer (G_D = 0)
|   |   +-- No (two peaks): Full HVM (G_D > 0)
|   |
|   +-- Is plateau modulus T-dependent?
|       +-- No: Covalent permanent network
|       +-- Yes: May have T-dependent damage
|
+-- No: G_P ~ 0 (vitrimer liquid or pure physical)
    |
    +-- Relaxation fully exponential?
    |   +-- Yes: Maxwell-like, use VLBLocal
    |   +-- No (stretched): TST kinetics active
    |
    +-- Does stress relax to zero?
        +-- Yes: No permanent network
        +-- No: Hidden G_P, re-fit with G_P > 0

Multi-Protocol Fitting Strategy

The recommended fitting workflow exploits information content of each protocol:

  1. SAOS first (linear regime, analytical):

    • Identify \(G_P\) from low-\(\omega\) plateau

    • Identify \(G_P + G_E + G_D\) from high-\(\omega\) plateau

    • Locate loss peaks for \(\tau_{E,eff}\) and \(\tau_D\)

    • Fix \(T\) at experimental value

  2. Relaxation (confirm time constants):

    • Verify bi-exponential + plateau structure

    • Confirm \(G(0^+) \approx G_P + G_E + G_D\)

    • Confirm \(G(\infty) \approx G_P\)

  3. Multi-temperature SAOS (activation energy):

    • Fit \(k_{BER,0}(T)\) at 3+ temperatures

    • Extract \(E_a\) from Arrhenius plot slope: \(E_a = -R \cdot d(\ln k_{BER,0}) / d(1/T)\)

    • Extract \(\nu_0\) from Arrhenius intercept

  4. Startup (TST parameters):

    • Fit \(V_{act}\) from stress overshoot magnitude and position

    • High \(V_{act}\) = strong mechanochemical coupling = prominent overshoot

    • Validate against SAOS parameters

  5. Creep (long-time behavior):

    • Verify elastic jump: \(J(0^+) = 1/G_{tot}\)

    • Check long-time compliance: \(J(\infty) \to 1/G_P\) (with permanent network)

    • Identify vitrimer plastic creep at intermediate times

Common Pitfalls

Factor-of-2 confusion:

A standard Maxwell fit to E-network relaxation data yields \(\tau_{fit} = 1/(2 k_{BER,0})\), not \(1/k_{BER,0}\). Always account for this factor when converting fitted time constants to BER rates. Use model.get_vitrimer_relaxation_time() to get the correct \(\tau_{E,eff}\).

Unbounded permanent stress:

The P-network stress \(\sigma_P = G_P \gamma\) grows without bound in steady shear. This is physically correct (permanent crosslinks store elastic energy) but means flow curve predictions diverge unless you examine the viscous contribution \(\sigma_D\) separately. Use return_components=True in flow curve predictions.

Parameter identifiability:

With single-protocol data at one temperature, several parameters may be correlated:

  • \(\nu_0\) and \(E_a\) are coupled: both affect \(k_{BER,0}\). Resolve with multi-T data.

  • \(G_E\) and \(\tau_{E,eff}\) can trade off in SAOS. Fix one using relaxation data.

  • \(V_{act}\) is only identifiable from nonlinear data (startup, LAOS).

Temperature vs vitrimer regime:

At low T (classify_vitrimer_regime() == "glassy"), exchange is frozen and the model behaves as a neo-Hookean + Maxwell solid. All vitrimer-specific behavior vanishes. Use model.compute_ber_rate_at_equilibrium() to check whether BER is active at your experimental temperature.

Cross-Protocol Validation

Use multiple protocols to validate the HVM fit:

Check

Criterion

Failing Suggests

\(G_P\) from SAOS = \(G(\infty)\) from relaxation

\(\lim_{\omega \to 0} G' \approx G(t \to \infty)\)

Incorrect \(G_P\) or hidden slow mode

\(\tau_{E,eff}\) from SAOS = \(\tau_{E,eff}\) from relaxation

Loss peak frequency \(\approx 1/\tau_{E,eff}\)

TST feedback distorting linear regime

\(\sigma_E \to 0\) at steady state

E-network stress vanishes in long startup

BER rate too slow; increase \(\nu_0\) or reduce \(E_a\)

Arrhenius \(\ln k_{BER,0}\) vs \(1/T\) is linear

\(R^2 > 0.99\) for 3+ temperatures

Non-Arrhenius exchange; consider WLF kinetics

This is analogous to the VLB cross-protocol validation workflow (Cross-Protocol Validation Workflow).

When to Upgrade to HVNM

Consider upgrading from HVM to HVNM (HVNM (Hybrid Vitrimer Nanocomposite Model)) when:

  • NP fillers present: material contains silica, carbon black, clay, or other nanoparticles at \(\phi > 0.01\)

  • Phi-dependent modulus: \(G'\) increases with filler loading beyond what \(G_P\) alone can explain

  • Payne effect: strain-amplitude-dependent modulus (nonlinear LAOS shows \(G'_1\) decrease at moderate \(\gamma_0\))

  • Dual relaxation separation: two well-separated loss peaks in \(G''\) that respond differently to temperature (dual \(E_a\))

  • Interfacial signature: slow relaxation mode that depends on NP surface treatment

If none of these apply, HVM is simpler and preferred.

Vitrimer vs Conventional Transient Network

Feature

VLB / TNT (Conventional)

HVM (Vitrimer)

Natural state

Fixed (\(\mathbf{I}\))

Evolving (\(\boldsymbol{\mu}^E_{nat}\))

Steady-state stress

\(\sigma = \eta \dot{\gamma}\)

\(\sigma_E = 0\) (BER erases all E-stress)

Permanent memory

None (fully relaxes)

\(G_P\) plateau preserved

Relaxation

Single exponential

Bi-exponential + plateau

Bond exchange

Dissociative (network breaks)

Associative (topology changes, network intact)

Temperature

\(k_d \sim T\) (simple)

Arrhenius \(k_{BER} \sim e^{-E_a/RT}\) (TST)

Troubleshooting

SAOS fit gives wrong relaxation time: Check for the factor-of-2: the fitted time constant from a Maxwell fit is \(\tau_{E,eff} = 1/(2k_{BER,0})\), not the bond exchange time \(\tau_E = 1/k_{BER,0}\).

Steady-state stress grows without bound: The permanent network stress \(\sigma_P = G_P \gamma\) grows linearly with strain. This is physical for bounded strain protocols (relaxation, LAOS) but produces unbounded stress in flow curve mode. Use return_components=True to isolate contributions.

ODE diverges at high shear rates: TST kinetics can create very stiff ODEs at high stress. Reduce gamma_dot or switch to kinetics="stretch" (less stiff coupling). See Numerical Implementation for solver details.

Damage produces unphysical behavior: Ensure \(\lambda_{crit} > 1\) (damage only activates under stretch beyond equilibrium). Set Gamma_0 small initially and increase gradually.

Application Examples

Processing window estimation: Use the Arrhenius BER rate (Temperature & Topological Freezing) to compute the temperature range where \(\tau_{BER}\) falls within the processing window (\(1\)–\(10^3\) s). Below \(T_v\), the material is a thermoset; above \(T_v\), it flows.

Shape-memory programming: Program a temporary shape by deforming at \(T > T_v\) (BER active) and cooling below \(T_v\) (BER frozen). The natural state \(\boldsymbol{\mu}^E_{nat}\) records the deformation, and the permanent network \(G_P\) provides the recovery driving force.

Vitrimer reprocessing: Estimate reprocessing time from \(t_{reprocess} \sim 5/k_{BER,0}(T)\). At high \(T\), BER rapidly equilibrates stress, enabling remolding. The creep compliance \(J(\infty) = 1/G_P\) (Creep Derivation) sets the long-time deformation limit.