# 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"** - **Journal:** Macromolecules, 2025, Vol. 58, Issue 10, pp. 4899-4912 - **DOI:** [10.1021/acs.macromol.4c02872](https://pubs.acs.org/doi/10.1021/acs.macromol.4c02872) - **Open access preprint:** [Michigan Tech Digital Commons](https://digitalcommons.mtu.edu/michigantech-p2/1640/) ### 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 - **Authors:** Smallenburg, Leibler, Sciortino (with Terentjev framework) - **Journal:** Macromolecules, 2019 - **DOI:** [10.1021/acs.macromol.9b01123](https://pubs.acs.org/doi/10.1021/acs.macromol.9b01123) ### 2b. Rheology of Vitrimers - **Journal:** Nature Communications, 2022 - **DOI:** [10.1038/s41467-022-33321-w](https://www.nature.com/articles/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 ### 3a. Meng et al. (2016) -- Dual-Crosslink Networks - Constitutive model for polymers with dynamic bonds - Combines physically-based insights with continuum-level constitutive laws - Applicable to dual-crosslink gels and networks with temperature-sensitive dynamic covalent bonds ### 3b. Mechanics of Vitrimer with Hybrid Networks - **DOI:** [ScienceDirect](https://www.sciencedirect.com/science/article/abs/pii/S0167663620307183) - Models vitrimers containing both permanent and exchangeable crosslinks ### 3c. Nonsteady Fracture of Transient Networks: The Case of Vitrimer - **Journal:** PNAS, 2021 - **DOI:** [10.1073/pnas.2105974118](https://www.pnas.org/doi/10.1073/pnas.2105974118) - Rate-dependent fracture mechanics in vitrimer networks --- ## 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:** - Guth, E. (1945). "Theory of Filler Reinforcement." J. Appl. Phys., 16, 20-25. - [Modified Guth-Gold for CB-filled rubbers (ResearchGate)](https://www.researchgate.net/publication/274002275_Modified_guth-gold_equation_for_carbon_black-filled_rubbers) ### 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:** - Stauffer, D.; Aharony, A. "Introduction to Percolation Theory" (1992) - [Percolation overview (ScienceDirect)](https://www.sciencedirect.com/topics/materials-science/percolation) - [Explosive percolation in nanocomposites (Nature Comm.)](https://www.nature.com/articles/s41467-022-34631-9) --- ## 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:** - Kraus, G. (1984). "Mechanical losses in carbon-black-filled rubbers." J. Appl. Polym. Sci.: Appl. Polym. Symp., 39, 75-92. - [Payne effect overview (ScienceDirect)](https://www.sciencedirect.com/topics/engineering/payne-effect) - [Kraus model parameters (MDPI Polymers)](https://www.mdpi.com/2073-4360/15/7/1675) ### 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](https://www.sciencedirect.com/science/article/abs/pii/S0749641903001360) ### 5c. Multiple Natural Configurations Framework - Models Payne effect through evolving natural configurations - [DOI: ScienceDirect](https://www.sciencedirect.com/science/article/abs/pii/S002072252030183X) ### 5d. Dynamic Strain-Dependent Relaxation Time Spectrum - Payne effect modeled via strain-amplitude-dependent relaxation spectrum - [DOI: ScienceDirect](https://www.sciencedirect.com/science/article/abs/pii/S0167663620305688) --- ## 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 - [ResearchGate](https://www.researchgate.net/publication/306379300_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](https://pmc.ncbi.nlm.nih.gov/articles/PMC8467415/) - 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 - [ScienceDirect](https://www.sciencedirect.com/science/article/abs/pii/S0266353824002549) ### 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" - [SAGE Journals](https://journals.sagepub.com/doi/10.1177/87560879241226504) 2. **"Impact of nanofillers on vitrimerization and recycling strategies: a review"** - [PMC](https://pmc.ncbi.nlm.nih.gov/articles/PMC12355401/) 3. **"Functional epoxy vitrimers and composites"** - [ScienceDirect](https://www.sciencedirect.com/science/article/abs/pii/S0079642520300748) 4. **"Molecular Simulation of Covalent Adaptable Networks and Vitrimers: A Review"** - [PMC](https://pmc.ncbi.nlm.nih.gov/articles/PMC11125108/) 5. **"Vitrimer as a Sustainable Alternative to Traditional Thermoset"** - [ACS Polymers Au](https://pubs.acs.org/doi/10.1021/acspolymersau.5c00081) 6. **"Modeling and Simulation of Vitrimers" (Book Chapter)** - [Springer](https://link.springer.com/chapter/10.1007/978-981-19-6038-3_8) 7. **"A Review of Computational Modeling of Polymer Composites and Nanocomposites"** - [MDPI Polymers](https://www.mdpi.com/2073-4360/18/4/443) --- ## 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 |