Vitrimer Nanocomposite Constitutive Models: Literature Review

Date: 2026-03-30 Purpose: Reference for implementing vitrimer nanocomposite rheological models in RheoJAX

1. Key Paper: Karim, Vernerey & Sain (2025)

“Constitutive Modeling of Vitrimers and Their Nanocomposites Based on Transient Network Theory”

Core Framework

  • Transient Network Theory (TNT) applied to vitrimers and their nanocomposites

  • Statistical mechanics of evolving polymer networks

  • Deformation-dependent bond exchange reaction (BER) kinetics under finite deformation explains nonlinear viscoelasticity in vitrimer short-term mechanical behavior

  • Long-term diffusion-driven chain dynamics incorporated via constant-rate kinetic term for slower cross-linked network mobility

  • Numerically implemented as a UMAT subroutine for commercial FE packages

  • Validated against: stress relaxation, loading-unloading, creep at multiple temperatures

2. Terentjev et al. – Vitrimer Rheology and Elasticity

2a. Elasticity and Relaxation in Full and Partial Vitrimer Networks

2b. Rheology of Vitrimers

  • Journal: Nature Communications, 2022

  • DOI: 10.1038/s41467-022-33321-w

  • Continuum model where elastic energy accounts for conserved crosslink number

  • Full rheology profile: small deformation (linear) to large deformation (nonlinear) viscoelasticity

  • Analytical expressions for experimental data analysis

  • Two relaxation modes identified:

    1. Mode I: Damped elastic motion

    2. Mode II: Reshuffling of crosslinks within the network

  • Elastic-plastic transition at timescale comparable to exchangeable bond lifetime

  • Topology freezing temperature (Tv): transition between viscoelastic solid and malleable viscoelastic liquid

Key Equations (Terentjev Framework)

Stress relaxation in vitrimers follows:

sigma(t) = sigma_elastic * exp(-t / tau_BER)

where:

  • tau_BER = characteristic bond exchange reaction time

  • tau_BER(T) = tau_0 * exp(E_a / (R * T)) (Arrhenius temperature dependence)

  • E_a = activation energy for bond exchange (typically 80-150 kJ/mol)

  • tau_0 = pre-exponential factor

  • R = gas constant

  • T = absolute temperature

3. Other Vitrimer Constitutive Models

3b. Mechanics of Vitrimer with Hybrid Networks

  • DOI: ScienceDirect

  • Models vitrimers containing both permanent and exchangeable crosslinks

3c. Nonsteady Fracture of Transient Networks: The Case of Vitrimer

4. Filler Reinforcement Models

4a. Guth-Gold Equation (1945)

Effective modulus of a filled elastomer with spherical particles:

G_eff = G_matrix * (1 + 2.5*phi + 14.1*phi^2)

Parameters:

  • G_matrix = shear modulus of the unfilled matrix

  • phi = filler volume fraction

  • 2.5*phi = Einstein viscosity correction (first-order hydrodynamic interaction)

  • 14.1*phi^2 = Guth’s second-order term for particle-particle interactions

Modified Guth-Gold (with shape factor f):

G_eff = G_matrix * (1 + 0.67*f*phi + 1.62*f^2*phi^2)

where f = aspect ratio of non-spherical fillers (f=1 for spheres recovers original).

Fukahori modification adds a cubic term:

G_eff = G_matrix * (1 + 2.5*phi + 14.1*phi^2 + k*phi^3)

for better agreement at high filler loadings.

References:

4b. Krieger-Dougherty Equation

Effective modulus (or viscosity) near maximum packing:

G_eff = G_matrix * (1 - phi/phi_max)^(-[eta]*phi_max)

Parameters:

  • phi_max = maximum packing fraction (random close packing ~0.64 for monodisperse spheres)

  • [eta] = intrinsic viscosity (= 2.5 for hard spheres)

  • Product [eta]*phi_max is typically ~1.6 for spheres

Notes:

  • Diverges as phi -> phi_max (captures jamming/gelation)

  • Improved Brinkman power-law model using Mooney’s crowding factor concept

  • Empirical modifications by Blissett add a second term with two constants

  • More accurate than Guth-Gold at high volume fractions (phi > 0.3)

References:

  • Krieger, I.M.; Dougherty, T.J. (1959). Trans. Soc. Rheol., 3, 137-152.

4c. Percolation Model for Filler Networks

Above the percolation threshold, filler contribution to modulus:

G_filler ~ (phi - phi_c)^t    for phi > phi_c
G_filler = 0                   for phi < phi_c

Full composite modulus:

G_composite = G_matrix(phi) + G_0_filler * (phi - phi_c)^t * H(phi - phi_c)

where H is the Heaviside step function.

Parameters:

  • phi_c = percolation threshold (critical filler volume fraction)

  • t = critical exponent

    • Universal values: t ~ 1.33 (2D), t ~ 2.0 (3D)

    • Experimental: t varies widely (1.3 to ~10) depending on filler geometry, dispersion, orientation

  • G_0_filler = scaling prefactor depending on filler stiffness and contact mechanics

Typical percolation thresholds:

Filler Type

phi_c (vol%)

Notes

Carbon black

10-20%

Depends on structure/grade

Silica nanoparticles

5-15%

Surface treatment dependent

CNT

0.1-2%

Very low due to high aspect ratio

Graphene/GO

0.1-1%

2D geometry, depends on exfoliation

Clay (montmorillonite)

1-5%

Depends on intercalation/exfoliation

References:

5. Payne Effect Models

5a. Kraus Model (1984) – Standard Payne Effect

Storage modulus as a function of strain amplitude:

G'(gamma_0) = G'_inf + (G'_0 - G'_inf) / (1 + (gamma_0 / gamma_c)^(2m))

Loss modulus (Kraus):

G''(gamma_0) = G''_inf + 2*(G''_max - G''_inf) * (gamma_0/gamma_c)^m / (1 + (gamma_0/gamma_c)^(2m))

Parameters:

  • G'_0 = storage modulus at zero/small strain amplitude (plateau value)

  • G'_inf = storage modulus at large strain amplitude (asymptotic lower bound)

  • gamma_c = critical strain amplitude (where G’’ is maximum and G’ drops most steeply)

  • m = phenomenological exponent (~0.5, independent of frequency, temperature, and CB content)

  • G''_max = maximum loss modulus

  • G''_inf = loss modulus at large strain

Physical basis:

  • Dynamic equilibrium between breakage and recovery of weak filler-filler contacts

  • Rate of breakage R_b = k_b * N * f_b(gamma_0) (proportional to existing contacts N)

  • Rate of recovery R_r = k_r * (N_0 - N) * f_r(gamma_0) (proportional to broken contacts)

  • At equilibrium: R_b = R_r

References:

5b. Lion-Kardelky-Haupt Model (Fractional Derivative Approach)

  • Constitutive modeling of Payne effect using fractional derivatives and intrinsic time scales

  • Framework for finite viscoelasticity

  • DOI: ScienceDirect

5c. Multiple Natural Configurations Framework

5d. Dynamic Strain-Dependent Relaxation Time Spectrum

  • Payne effect modeled via strain-amplitude-dependent relaxation spectrum

  • DOI: ScienceDirect

6. Nanoparticle Effects on Vitrimer Bond Exchange Reactions (BER)

6a. Catalytic Acceleration

  • Catalysts lower the topology freezing temperature (Tv) by reducing activation energy

  • Increasing catalytic loading increases overall exchange rate but does not shift Tv

  • Arrhenius kinetics for BER relaxation time:

tau*(T) = tau_0 * exp(E_a / (R * T))

  • Typical activation energies: E_a ~ 80-150 kJ/mol

  • With catalysts: E_a can drop by 20-50 kJ/mol

6b. Nanoparticle Effects on BER

  • Silica nanoparticles: Slow down stress relaxation compared to unfilled analogues (geometric hindrance to chain mobility), but full relaxation still achieved

  • Surface-functionalized fillers: Silane-modified silica ensures strong interfacial connections while maintaining dynamic characteristics

  • Fe3O4 nanoparticles: Enable photothermal triggering of BER via NIR light

  • CNT: 0-0.5 wt% in epoxy vitrimer; NIR light activates BER via photothermal effect

  • Graphite fillers: Activation energy 90.0-97.7 kJ/mol measured via stress relaxation analysis

6c. Modified BER Rate for Filled Vitrimers

Proposed modification accounting for filler hindrance and catalysis:

k_BER(T, phi) = k_0 * exp(-E_a(phi) / (R*T)) * f_hindrance(phi)

where:

  • E_a(phi) = E_a0 + Delta_E * phi (filler may increase or decrease E_a depending on surface chemistry)

  • f_hindrance(phi) = (1 - phi/phi_max)^alpha (geometric hindrance factor)

  • For catalytic fillers: E_a(phi) = E_a0 - Delta_E_cat * phi (decreasing activation energy)

7. Silica-Epoxy Vitrimer Nanocomposites

Key Experimental Results

  • Legrand & Soulie-Ziakovic (2016): Pioneering silica-epoxy vitrimer nanocomposites

  • Unmodified silica NPs (10-15 nm) enhance mechanical properties without preventing topology rearrangements:

    • Tensile stress: 48 -> 60 MPa

    • Elastic modulus: 1.8 -> 2.6 GPa

    • PMC

  • Reinforcing effect mainly in glassy state; near Tg, enhanced chain mobility reduces polymer-particle adhesion

  • Interfacial covalent binding + microphase separation for simultaneous strengthening and toughening

Leibler’s Foundational Work

  • Ludwik Leibler coined the term “vitrimer” (2011)

  • Developed silica-like networks using transesterification of epoxy with fatty dicarboxylic/tricarboxylic acids

  • Demonstrated Arrhenius-type stress relaxation (not WLF), distinguishing vitrimers from other dynamic networks

8. Comprehensive Vitrimer Nanocomposite Model (Proposed)

A full constitutive model for vitrimer nanocomposites combines:

Total Stress

sigma_total = sigma_matrix(phi) + sigma_filler_network

Matrix Contribution (Vitrimer TNT)

sigma_matrix = G_eff(phi) * f(B_e) * Phi(t, T, phi)

where:

  • G_eff(phi) = Guth-Gold or Krieger-Dougherty reinforced modulus

  • f(B_e) = strain energy derivative (e.g., neo-Hookean, Mooney-Rivlin)

  • B_e = elastic left Cauchy-Green tensor (evolving via BER)

  • Phi(t, T, phi) = relaxation function from BER kinetics

BER Evolution (Karim-Vernerey-Sain)

dPhi/dt = -k_BER(T, deformation, phi) * Phi

with deformation-dependent BER rate capturing nonlinear viscoelasticity.

Filler Network Contribution (Percolation + Payne)

sigma_filler = G_filler(phi, gamma_0) * f_filler(B)

where:

G_filler(phi, gamma_0) = G_0_filler * (phi - phi_c)^t / (1 + (gamma_0/gamma_c)^(2m))    for phi > phi_c

This combines percolation scaling with Kraus-type strain softening.

Temperature Dependence

tau_BER(T) = tau_0 * exp(E_a / (R*T))          [Arrhenius, above Tv]
G_matrix(T) = G_eq + (G_g - G_eq) * g(T/Tg)   [glass transition]

9. Review Articles

  1. Kausar & Ahmad (2024): “State-of-the-art epoxy vitrimer nanocomposites with graphene, CNT and silica”

  2. “Impact of nanofillers on vitrimerization and recycling strategies: a review”

  3. “Functional epoxy vitrimers and composites”

  4. “Molecular Simulation of Covalent Adaptable Networks and Vitrimers: A Review”

  5. “Vitrimer as a Sustainable Alternative to Traditional Thermoset”

  6. “Modeling and Simulation of Vitrimers” (Book Chapter)

  7. “A Review of Computational Modeling of Polymer Composites and Nanocomposites”

10. Summary of All Equations

Model

Equation

Application

Guth-Gold

G_eff = G_m * (1 + 2.5*phi + 14.1*phi^2)

Spherical filler reinforcement

Modified Guth-Gold

G_eff = G_m * (1 + 0.67*f*phi + 1.62*f^2*phi^2)

Non-spherical fillers (aspect ratio f)

Krieger-Dougherty

G_eff = G_m * (1 - phi/phi_max)^(-[eta]*phi_max)

High volume fraction, near jamming

Percolation

G_filler ~ (phi - phi_c)^t

Filler network above threshold

Kraus (G’)

G' = G'_inf + (G'_0 - G'_inf)/(1 + (gamma_0/gamma_c)^(2m))

Payne effect, storage modulus

Kraus (G’’)

G'' = G''_inf + 2*(G''_m - G''_inf)*(gamma_0/gamma_c)^m/(1+(gamma_0/gamma_c)^(2m))

Payne effect, loss modulus

Arrhenius BER

tau = tau_0 * exp(E_a/(R*T))

Vitrimer stress relaxation

BER evolution

dPhi/dt = -k(T, F, phi) * Phi

TNT vitrimer constitutive law