Hybrid Vitrimer Model (HVM) – Literature Review & Equations

1. Key References

Primary papers for the HVM theoretical framework:

  1. Vernerey, Long, & Brighenti (2017) “A statistically-based continuum theory for polymers with transient networks.” Journal of the Mechanics and Physics of Solids, 107, 1-20.

  2. Meng, Saed, & Terentjev (2019) “Elasticity and Relaxation in Full and Partial Vitrimer Networks.” Macromolecules, 52, 7423-7429.

    • Develops continuum model for dynamic-mechanical response of vitrimers.

    • Introduces “partial vitrimer” concept: permanent sub-network + exchangeable sub-network.

    • Partial vitrimer in linear regime behaves as a Zener (SLS) model.

    • Eq. (3): k_BER = nu_0 * exp(-E_a / RT) (Arrhenius bond exchange rate).

    • Eq. (5): tau_v = 1/(2*k_BER) (vitrimer relaxation time with factor-of-2).

    • https://pubs.acs.org/doi/10.1021/acs.macromol.9b01123

  3. Meng, Saed, & Terentjev (2022) “Rheology of vitrimers.” Nature Communications, 13, 5753.

    • Full rheological characterization spanning small to large deformation.

    • Analytical expressions for Master Curves across 22 decades of frequency.

    • Treats partial vitrimers (permanent + exchangeable sub-networks).

    • https://www.nature.com/articles/s41467-022-33321-w

  4. Stukalin, Cai, Kumar, Leibler, & Rubinstein (2013) “Self-Healing of Unentangled Polymer Networks with Reversible Bonds.” Macromolecules, 46(18), 7525-7541.

  5. Constitutive Modeling of Vitrimers Based on Transient Network Theory (2025) Macromolecules (recent).

2. Physical Model: Three-Subnetwork Architecture

The HVM decomposes the polymer into three parallel subnetworks:

Subnetwork

Symbol

Bond Type

Relaxation

Behavior

Permanent (P)

G_P

Covalent crosslinks

None (infinite)

Neo-Hookean elastic

Exchangeable (E)

G_E

Associative vitrimer bonds (BER)

tau_E_eff = 1/(2*k_BER)

Maxwell-like with evolving natural state

Dissociative (D)

G_D

Physical/reversible bonds

tau_D = 1/k_d^D

Standard Maxwell

Total stress:

sigma = sigma_P + sigma_E + sigma_D

3. Constitutive Equations

3.1 Stress Decomposition (General Tensorial Form)

Permanent network (neo-Hookean):

sigma_P = (1 - D) * G_P * (B - I)

where B is the left Cauchy-Green tensor, D is damage [0,1].

Exchangeable network (vitrimer hallmark):

sigma_E = G_E * (mu^E - mu^E_nat)

Stress depends on the difference between current distribution and its evolving natural state.

Dissociative network (standard Maxwell/VLB):

sigma_D = G_D * (mu^D - I)

3.2 Simple Shear Stress Components

sigma_P_xy = (1 - D) * G_P * gamma
sigma_E_xy = G_E * (mu^E_xy - mu^E_nat_xy)
sigma_D_xy = G_D * mu^D_xy
sigma_total = sigma_P_xy + sigma_E_xy + sigma_D_xy

3.3 Evolution ODEs (Simple Shear)

E-network distribution tensor (upper-convected + BER relaxation):

d(mu^E_xx)/dt = 2*gamma_dot*mu^E_xy + k_BER*(mu^E_nat_xx - mu^E_xx)
d(mu^E_yy)/dt = k_BER*(mu^E_nat_yy - mu^E_yy)
d(mu^E_xy)/dt = gamma_dot*mu^E_yy + k_BER*(mu^E_nat_xy - mu^E_xy)

E-network natural-state tensor (VITRIMER HALLMARK):

d(mu^E_nat_ij)/dt = k_BER * (mu^E_ij - mu^E_nat_ij)

The natural state continuously drifts toward the current state at rate k_BER. This is the key distinction from conventional transient networks (where natural state = I).

D-network distribution tensor (standard VLB):

d(mu^D_xx)/dt = 2*gamma_dot*mu^D_xy - k_d^D*(mu^D_xx - 1)
d(mu^D_yy)/dt = -k_d^D*(mu^D_yy - 1)
d(mu^D_xy)/dt = gamma_dot*mu^D_yy - k_d^D*mu^D_xy

3.4 Factor-of-2 in Relaxation

The stress difference Delta_mu_ij = mu^E_ij - mu^E_nat_ij satisfies:

d(Delta_mu_ij)/dt = -2 * k_BER * Delta_mu_ij

Both tensors relax toward each other at rate k_BER, so the difference (which determines stress) decays at 2*k_BER. Therefore:

tau_E_eff = 1 / (2 * k_BER_0)

4. Analytical Solutions (Linear Regime, Constant k_BER)

4.1 Storage and Loss Moduli (SAOS)

G'(omega) = G_P + G_E * (omega*tau_E)^2 / (1 + (omega*tau_E)^2)
                 + G_D * (omega*tau_D)^2 / (1 + (omega*tau_D)^2)

G''(omega) =      G_E * (omega*tau_E)   / (1 + (omega*tau_E)^2)
                 + G_D * (omega*tau_D)   / (1 + (omega*tau_D)^2)

where:

  • tau_E = tau_E_eff = 1/(2*k_BER_0) (effective E-network relaxation time)

  • tau_D = 1/k_d^D (D-network relaxation time)

Physical interpretation:

  • G_P provides a frequency-independent elastic plateau (permanent crosslinks)

  • G_E contributes a Maxwell mode with the vitrimer relaxation time

  • G_D contributes a second Maxwell mode with the dissociative time

Limiting behavior:

  • omega -> 0: G’(0) = G_P, G’’(0) = 0

  • omega -> inf: G’(inf) = G_P + G_E + G_D

  • Loss peaks at omega = 1/tau_E and omega = 1/tau_D

4.2 Relaxation Modulus G(t)

G(t) = (1-D)*G_P + G_E*exp(-2*k_BER_0*t) + G_D*exp(-k_d^D*t)
     = (1-D)*G_P + G_E*exp(-t/tau_E_eff) + G_D*exp(-t/tau_D)

Physical interpretation:

  • Permanent plateau (1-D)*G_P survives at all times

  • E-network decays exponentially with tau_E_eff = 1/(2*k_BER_0)

  • D-network decays exponentially with tau_D = 1/k_d^D

  • Initial value: G(0+) = G_P + G_E + G_D

  • Long-time value: G(inf) = G_P (or (1-D)*G_P with damage)

4.3 Startup Stress (Linear Regime, Constant Rate)

sigma(t) = G_P * gamma_dot * t
         + G_E * gamma_dot * tau_E * (1 - exp(-t/tau_E))
         + G_D * gamma_dot * tau_D * (1 - exp(-t/tau_D))

4.4 Steady-State Shear Stress (Flow Curve)

At steady state, the E-network fully relaxes (mu^E -> mu^E_nat, so sigma_E -> 0):

sigma_ss = eta_D * gamma_dot    (viscous contribution only)

where eta_D = G_D / k_d^D.

Note: sigma_P = G_P * gamma grows unbounded under continuous shear (elastic storage).

4.5 Creep Compliance J(t)

J(t) = 1/G_tot + (G_E / (G_P * G_tot)) * (1 - exp(-t/tau_ret_E))
               + (G_D / (G_P * G_tot)) * (1 - exp(-t/tau_ret_D))

where:

  • G_tot = G_P + G_E + G_D

  • tau_ret_E = G_tot / (G_P * 2*k_BER_0) (E-network retardation time)

  • tau_ret_D = G_tot / (G_P * k_d^D) (D-network retardation time)

Limiting behavior:

  • J(0+) = 1/G_tot (instantaneous elastic compliance)

  • J(inf) = 1/G_P (long-time compliance set by permanent network)

5. TST Kinetics (Bond Exchange Rate)

5.1 Thermal BER Rate (Arrhenius)

k_BER_0 = nu_0 * exp(-E_a / (R*T))

5.2 Stress-Coupled BER Rate (TST)

k_BER = k_BER_0 * cosh(V_act * sigma_VM^E / (R*T))

where sigma_VM^E is the von Mises equivalent stress on the E-network:

sigma_VM = sqrt(sigma_xx^2 + sigma_yy^2 - sigma_xx*sigma_yy + 3*sigma_xy^2)

5.3 Stretch-Coupled BER Rate

k_BER = k_BER_0 * cosh(V_act * G_E * delta_stretch / (R*T))

where delta_stretch = sqrt(tr(mu^E - mu^E_nat) / dim).

6. Parameter Definitions and Physical Meaning

Parameter

Symbol

Units

Default

Physical Meaning

G_P

G_P

Pa

1e4

Permanent network modulus. Proportional to covalent crosslink density: c_P = G_P/(k_B*T)

G_E

G_E

Pa

1e4

Exchangeable network modulus. Proportional to exchangeable crosslink density

G_D

G_D

Pa

1e3

Dissociative network modulus. Proportional to physical bond density

nu_0

nu_0

1/s

1e10

TST attempt frequency. ~10^8-10^12 for small-molecule bond rearrangements

E_a

E_a

J/mol

80e3

Activation energy for bond exchange reaction. Typical: 40-150 kJ/mol

V_act

V_act

m^3/mol

1e-5

Activation volume. Controls mechanochemical coupling (stress-accelerated exchange)

T

T

K

300

Absolute temperature

k_d^D

k_d_D

1/s

1.0

Dissociative bond breakage rate

Gamma_0

Gamma_0

1/s

1e-4

Damage rate coefficient (optional)

lambda_crit

lambda_crit

–

2.0

Critical stretch for damage onset (optional)

Derived quantities:

  • k_BER_0 = nu_0 * exp(-E_a/(R*T)) – thermal BER rate at zero stress

  • tau_E_eff = 1/(2*k_BER_0) – effective vitrimer relaxation time

  • tau_D = 1/k_d^D – dissociative relaxation time

  • eta_D = G_D/k_d^D – dissociative viscosity

  • f_E = G_E/(G_P + G_E + G_D) – exchange fraction

  • T_v = topology freezing temperature (where k_BER_0 * tau_obs ~ 1)

8. Comparison: Vitrimer vs Conventional Transient Network

Feature

VLB / TNT (Conventional)

HVM (Vitrimer)

Natural state

Fixed (I)

Evolving (mu^E_nat)

Steady-state stress

sigma = eta * gamma_dot

sigma_E = 0 (BER erases E-stress)

Permanent memory

None (fully relaxes)

G_P plateau preserved

Relaxation form

Single exponential

Bi-exponential + plateau

Bond exchange

Dissociative (network breaks)

Associative (topology changes, network intact)

Temperature dependence

k_d ~ T (simple)

Arrhenius k_BER ~ exp(-E_a/RT) (TST)