ITT-MCT Models

Integration Through Transients Mode-Coupling Theory (ITT-MCT) models describe the nonlinear rheology of dense colloidal suspensions and glassy materials through microscopic physics: the cage effect.

Overview

Mode-Coupling Theory (MCT) provides a first-principles approach to understanding the dynamics of dense particulate systems. The theory predicts:

  • Glass transition at a critical volume fraction (\(\phi \approx 0.516\) for hard spheres)

  • Two-step relaxation with \(\beta\) (in-cage) and \(\alpha\) (cage-breaking) processes

  • Yield stress in the glass state from arrested structure

  • Shear thinning from flow-induced cage breaking

ITT-MCT extends MCT to nonlinear deformations by tracking how flow “advects” density fluctuations, destroying the cage structure above a critical strain.

Available Models

Model

Description

ITTMCTSchematic

\(F_{12}\) schematic model with scalar correlator. Fast computation, captures essential physics with ~6 parameters. Best for qualitative understanding and fitting experimental data.

ITTMCTIsotropic

Full isotropically sheared model with k-resolved correlators \(\Phi(k,t)\). Uses structure factor \(S(k)\) input. More quantitative but computationally expensive.

Model Selection Guide

Use ITTMCTSchematic when:

  • You need fast computations for fitting or exploration

  • Qualitative understanding of glass/yield phenomena is sufficient

  • You want to explore parameter space quickly

  • Working with systems where \(S(k)\) is unknown

Use ITTMCTIsotropic when:

  • Quantitative predictions are needed

  • \(S(k)\) is available (measured or from simulation)

  • Wave-vector-dependent relaxation is important

  • Comparing with microscopic measurements (DLS, X-ray scattering)

Supported Protocols

Both models support all six standard rheological protocols:

  1. Flow curve (steady shear): \(\sigma(\dot{\gamma})\) - shows yield stress and shear thinning

  2. SAOS (oscillation): \(G'(\omega)\), \(G''(\omega)\) - shows glass plateau and loss peak

  3. Startup: \(\sigma(t)\) at constant \(\dot{\gamma}\) - shows stress overshoot

  4. Creep: \(J(t)\) at constant \(\sigma\) - shows viscosity bifurcation

  5. Relaxation: \(\sigma(t)\) after cessation - shows residual stress in glass

  6. LAOS: \(\sigma(t)\) for \(\gamma = \gamma_0 \sin(\omega t)\) - shows nonlinear harmonics

For detailed mathematical formulation of each protocol including governing equations and physical interpretation, see ITT-MCT Protocol Equations.

Theoretical Framework

The ITT-MCT formalism consists of three key components:

  1. ITT Stress Functional: A history integral over past deformations weighted by a generalized shear modulus built from transient density correlators. This is the microscopic generalization of the Green-Kubo relation for driven systems.

  2. MCT Correlator Dynamics: The Zwanzig-Mori integro-differential equation with a mode-coupling memory kernel. This describes how density fluctuations decorrelate under the combined influence of Brownian motion and shear advection.

  3. Wavevector Advection: Flow “advects” density fluctuations, causing the wavevector \(\mathbf{k}\) to evolve as \(\mathbf{k}(t,t') = \mathbf{k} \cdot \mathbf{E}^{-1}(t,t')\) where \(\mathbf{E}\) is the deformation gradient. This advection destroys the cage structure above a critical accumulated strain.

Physical Context

MCT is most applicable to:

  • Hard-sphere colloids (PMMA, silica particles)

  • Dense emulsions (mayonnaise, cosmetics)

  • Concentrated polymer solutions near gelation

  • Soft glassy materials (pastes, gels)

The theory captures the universal features of the glass transition that emerge from the cage effect, independent of specific interparticle interactions.

Key Parameters

\(F_{12}\) Schematic Model:

  • \(\varepsilon\) (epsilon): Separation parameter controlling distance from glass transition

    • \(\varepsilon < 0\): Ergodic fluid

    • \(\varepsilon = 0\): Critical point

    • \(\varepsilon > 0\): Glass state

  • \(\gamma_c\): Critical strain for cage breaking (~0.05-0.2)

  • \(\Gamma\): Bare relaxation rate (microscopic timescale)

  • \(G_\infty\): High-frequency modulus

ISM Model:

  • \(\phi\) (phi): Volume fraction (glass at \(\phi \approx 0.516\))

  • \(S(k)\): Structure factor (from Percus-Yevick or experiment)

  • \(D_0\): Bare diffusion coefficient

References

[Gotze2009]

Götze W. (2009) “Complex Dynamics of Glass-Forming Liquids: A Mode-Coupling Theory”, Oxford University Press. https://doi.org/10.1093/acprof:oso/9780199235346.001.0001

[Fuchs2002]

Fuchs M. & Cates M.E. (2002) “Theory of Nonlinear Rheology and Yielding of Dense Colloidal Suspensions”, Phys. Rev. Lett. 89, 248304. https://doi.org/10.1103/PhysRevLett.89.248304

[Fuchs2009]

Fuchs M. & Cates M.E. (2009) “A mode coupling theory for Brownian particles in homogeneous steady shear flow”, J. Rheol. 53, 957. https://doi.org/10.1122/1.3119084

[Brader2008]

Brader J.M., Voigtmann T., Fuchs M., Larson R.G. & Cates M.E. (2009) “Glass rheology: From mode-coupling theory to a dynamical yield criterion”, Proc. Natl. Acad. Sci. USA 106, 15186-15191. https://doi.org/10.1073/pnas.0905330106

Models