DMT Model: Literature Verification Report¶
CRITICAL FINDING: Naming Discrepancy¶
The RheoJAX “DMT” model is the de Souza Mendes-Thompson model, NOT the Dullaert-Mewis thixotropic model. These are two distinct models in the thixotropy literature:
Aspect |
Dullaert-Mewis (2006) |
de Souza Mendes-Thompson (2012/2013) |
|---|---|---|
Abbreviation in literature |
“DM” or “Dullaert-Mewis” |
“SMT” or “DSM-T” |
RheoJAX abbreviation |
– (not implemented) |
“DMT” |
Paper |
JNNFM 139, 21-30 (2006) |
JNNFM 187-188, 8-15 (2012); Rheol. Acta 52, 673-694 (2013) |
The RheoJAX _base.py correctly references de Souza Mendes & Thompson in its docstrings.
The “DMT” abbreviation stands for De Souza Mendes-Thompson, not Dullaert-Mewis-Thixotropic.
1. Dullaert-Mewis Model (2006) – Literature Equations¶
1.1 Primary References¶
Dullaert, K. & Mewis, J. (2006) “A structural kinetics model for thixotropy.” J. Non-Newtonian Fluid Mech. 139, 21-30. DOI: 10.1016/j.jnnfm.2006.06.002
Dullaert, K. & Mewis, J. (2005) “Thixotropy: Build-up and breakdown curves during flow.” J. Rheol. 49(6), 1213.
Dullaert, K. & Mewis, J. (2005) “A model system for thixotropy studies.” Rheol. Acta 45, 23-32.
Dullaert, K. (2005) PhD thesis, “Constitutive equations for thixotropic dispersions,” KU Leuven (supervisor: J. Mewis).
1.2 Stress Decomposition¶
The Dullaert-Mewis model decomposes total stress into elastic and viscous contributions:
sigma = sigma_e + sigma_v
where:
sigma_e: elastic stress from aggregate deformation (floc strain)
sigma_v: viscous stress from flow through/around the structure
The viscous contribution has three sub-terms:
sigma_v = sigma_y(lambda) + eta_str(lambda) * gamma_dot + eta_medium * gamma_dot
where:
sigma_y(lambda): structure-dependent yield stress
eta_str(lambda): structural viscosity (hydrodynamic interaction of aggregates)
eta_medium: viscosity of the fully de-structured suspending medium
1.3 Elastic Stress Component¶
The elastic stress is modeled via an aggregate strain variable gamma_a:
sigma_e = G_a(lambda) * gamma_a
where G_a(lambda) is the structure-dependent elastic (aggregate) modulus, and gamma_a evolves according to a relaxation/deformation equation:
d(gamma_a)/dt = gamma_dot - gamma_a / theta_a(lambda)
where theta_a(lambda) is a structure-dependent relaxation time for the aggregates.
This is formally a Maxwell-type evolution for the elastic stress:
d(sigma_e)/dt = G_a * gamma_dot - sigma_e / theta_a
1.4 Structure Kinetics (dλ/dt)¶
The structure parameter lambda in [0, 1] evolves via:
d(lambda)/dt = k_B * (1 - lambda)^b - k_D * lambda^d * |gamma_dot|^a
where:
k_B: Brownian buildup rate constant (thermal restructuring)
b: buildup exponent (Dullaert & Mewis used b ~ 0.5)
k_D: shear-induced breakdown rate constant
d: breakdown structure exponent (Dullaert & Mewis suggested d ~ 1-2)
a: shear rate exponent for breakdown (Dullaert & Mewis used a ~ 1)
Steady state (d(lambda)/dt = 0) gives:
k_B * (1 - lambda_eq)^b = k_D * lambda_eq^d * |gamma_dot|^a
1.5 Distribution of Time Constants¶
A key distinguishing feature of Dullaert-Mewis vs. simpler models: both the kinetic equation and the relaxation equation contain a distribution of time constants (weighted sum over multiple relaxation modes), which can be treated as a single effective time constant for simplified implementations.
1.6 Material Function Dependencies on lambda¶
Elastic modulus: G_a = G_a0 * lambda^alpha_G (power-law in lambda)
Structural viscosity: eta_str = eta_str0 * lambda^alpha_eta
Yield stress: sigma_y = sigma_y0 * lambda^alpha_y
Relaxation time: theta_a = eta_str / G_a (Maxwell relation)
1.7 Oscillatory/Viscoelastic Variant¶
The Dullaert-Mewis model inherently includes viscoelasticity through the elastic stress contribution sigma_e. For SAOS (small amplitude oscillatory shear):
The structure is assumed constant at equilibrium: lambda ~ lambda_0
The elastic modulus G_a(lambda_0) and relaxation time theta_a(lambda_0) are constant
Standard Maxwell SAOS expressions apply:
G’(omega) = G_a * (omegatheta_a)^2 / (1 + (omegatheta_a)^2)
G’’(omega) = G_a * (omegatheta_a) / (1 + (omegatheta_a)^2) + eta_medium * omega
For LAOS (large amplitude oscillatory shear), lambda varies within each cycle, and the full coupled ODE system must be integrated (as done by Armstrong et al., 2016).
2. de Souza Mendes-Thompson Model (2012/2013) – What RheoJAX Implements¶
2.1 Primary References¶
de Souza Mendes, P.R. & Thompson, R.L. (2012) “A critical overview of elasto-viscoplastic thixotropic modeling.” JNNFM 187-188, 8-15.
de Souza Mendes, P.R. & Thompson, R.L. (2013) “A unified approach to model elasto-viscoplastic thixotropic yield-stress materials and apparent yield-stress fluids.” Rheol. Acta 52, 673-694.
2.2 Stress Equation¶
Maxwell-type stress evolution:
d(sigma)/dt = G(lambda) * gamma_dot - sigma / theta_1(lambda)
where theta_1 = eta(lambda) / G(lambda) is the relaxation time.
2.3 Viscosity Closure¶
Two options (both in RheoJAX):
Exponential closure (original DSM-T 2013):
eta(lambda) = eta_inf * (eta_0 / eta_inf)^lambda
Herschel-Bulkley closure:
eta_eff = tau_y(lambda) * (1 - exp(-m*|gamma_dot|))/|gamma_dot|
+ K(lambda) * |gamma_dot|^(n-1) + eta_inf
with tau_y(lambda) = tau_y0 * lambda^m1, K(lambda) = K0 * lambda^m2
2.4 Structure Kinetics¶
d(lambda)/dt = (1 - lambda)/t_eq - a * lambda * |gamma_dot|^c / t_eq
This is a simplified form compared to Dullaert-Mewis (b=1, d=1 fixed; single timescale t_eq governs both buildup and breakdown).
Equilibrium structure:
lambda_eq = 1 / (1 + a * |gamma_dot|^c)
3. Key Differences Between the Two Models¶
Feature |
Dullaert-Mewis (2006) |
de Souza Mendes-Thompson (2013) |
|---|---|---|
Stress decomposition |
sigma_e + sigma_v (separate elastic/viscous) |
Single Maxwell equation |
Buildup exponent b |
Free parameter (~0.5) |
Fixed at 1 |
Breakdown exponent d |
Free parameter (~1-2) |
Fixed at 1 (linear in lambda) |
Kinetics timescale |
Separate k_B, k_D rate constants |
Single t_eq timescale |
Time constant distribution |
Multiple modes possible |
Single mode |
Viscous sub-contributions |
Yield + structural + medium |
Single viscosity function |
Parameter count |
Higher (more physics) |
Lower (more parsimonious) |
4. Comparison with RheoJAX Implementation¶
The RheoJAX DMT implementation (rheojax/models/dmt/) correctly implements the
de Souza Mendes-Thompson model:
_kernels.py: Implements exponential and HB closures, Maxwell stress evolution, structure kinetics with (1-lambda)/t_eq buildup and alambda|gamma_dot|^c/t_eq breakdown_base.py: Parameters match DSM-T formulation (eta_0, eta_inf, G0, m_G, t_eq, a, c)local.py: ODE integration for startup, creep, relaxation, SAOS, LAOSnonlocal_model.py: Adds diffusion term D_lambda * d^2(lambda)/dy^2 for shear banding
The implementation does NOT contain the Dullaert-Mewis model. If you need the Dullaert-Mewis model specifically, it would require a new model family with:
Separate elastic/viscous stress tracking (sigma_e, sigma_v)
Aggregate strain variable gamma_a with its own evolution equation
Free exponents b and d in the structure kinetics
Separate k_B and k_D rate constants (not a single t_eq)
Optional distribution of relaxation times
5. Review Papers for Further Reference¶
Larson, R.G. & Wei, Y. (2019) “A review of thixotropy and its rheological modeling.” J. Rheol. 63(3), 477-501.
Mewis, J. & Wagner, N.J. (2009) “Thixotropy.” Adv. Colloid Interface Sci. 147-148, 214-227.
Mewis, J. & Wagner, N.J. (2012) Colloidal Suspension Rheology, Chapter 7: Thixotropy. Cambridge University Press.
Armstrong, M.J. et al. (2016) “Dynamic Shear Rheology of Thixotropic Suspensions: Comparison of Structure-Based Models with Large Amplitude Oscillatory Shear Experiments.” J. Rheol. (applied Dullaert-Mewis framework to LAOS).