# 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 **D**e Souza **M**endes-**T**hompson, 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 * (omega*theta_a)^2 / (1 + (omega*theta_a)^2) - G''(omega) = G_a * (omega*theta_a) / (1 + (omega*theta_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 a*lambda*|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, LAOS - `nonlocal_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: 1. Separate elastic/viscous stress tracking (sigma_e, sigma_v) 2. Aggregate strain variable gamma_a with its own evolution equation 3. Free exponents b and d in the structure kinetics 4. Separate k_B and k_D rate constants (not a single t_eq) 5. 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). --- ## Sources - [Dullaert & Mewis 2006 (Semantic Scholar)](https://www.semanticscholar.org/paper/A-structural-kinetics-model-for-thixotropy-Dullaert-Mewis/519930dde56571e90afd65f39184632eadb30cfc) - [Dullaert & Mewis 2006 (ScienceDirect)](https://www.sciencedirect.com/science/article/abs/pii/S0377025706001431) - [Dullaert & Mewis 2005 (J. Rheol.)](https://sor.scitation.org/doi/10.1122/1.2039868) - [Larson & Wei 2019 Review](https://pubs.aip.org/sor/jor/article/63/3/477/241126/A-review-of-thixotropy-and-its-rheological) - [Armstrong et al. 2016 LAOS study (J. Rheol.)](https://doi.org/10.1122/1.4943986) - [de Souza Mendes & Thompson 2013 (Rheol. Acta)](https://link.springer.com/article/10.1007/s00397-013-0699-1) - [Gas Bubbles in Dullaert-Mewis Fluid (ResearchGate)](https://www.researchgate.net/publication/270432311_Dynamic_of_Gas_Bubbles_Surrounded_by_a_Dullaert-Mewis_Thixotropic_Fluid) - [Turbulent Pipe Flow of Thixotropic Fluids (arXiv)](https://arxiv.org/abs/2501.01597)