ITT-MCT Protocol Equations

This document provides a comprehensive reference for the protocol-specific equations used in ITT-MCT (Integration Through Transients Mode-Coupling Theory) rheological modeling. Each protocol has distinct kinematics and stress formulas.

Quick Reference

Protocol	Input	Key Output
Flow Curve	Constant \(\dot{\gamma}\)	Steady stress \(\sigma(\dot{\gamma})\), yield stress \(\sigma_y\)
Startup	Step from rest to \(\dot{\gamma}\)	\(\sigma(t)\) with overshoot
Cessation	Stop shear at \(t=0\)	Relaxing \(\sigma(t)\), residual stress
Creep	Constant \(\sigma_0\)	\(\gamma(t)\), \(J(t)\), viscosity bifurcation
SAOS	Small \(\gamma_0 \sin(\omega t)\)	\(G'(\omega)\), \(G''(\omega)\)
LAOS	Finite \(\gamma_0 \sin(\omega t)\)	Harmonics \(\sigma'_n\), \(\sigma''_n\)

Notation Guide

Symbol	Definition
\(\sigma(t)\)	Shear stress at time \(t\)
\(\dot{\gamma}(t)\)	Shear rate at time \(t\)
\(\gamma(t,t')\)	Accumulated strain from \(t'\) to \(t\): \(\int_{t'}^{t}\dot{\gamma}(s)\,ds\)
\(G(t,t')\)	Generalized shear modulus (history-dependent)
\(G_{\text{eq}}(t)\)	Equilibrium (quiescent) modulus
\(\Phi_k(t,t')\)	Transient density correlator at wavevector \(k\)
\(\Phi(t,t')\)	Schematic (scalar) correlator
\(h(\gamma)\)	Strain decorrelation function: \(\exp[-(\gamma/\gamma_c)^2]\)
\(S(k)\)	Static structure factor
\(G_\infty\)	High-frequency elastic modulus

Overview: The ITT Stress Functional

ITT-MCT is not a closed-form constitutive equation. It is a procedure that expresses stress as a history integral over past deformations, weighted by a generalized modulus built from transient density correlators.

The General History Integral

The shear stress at time \(t\) is given by the generalized Green-Kubo relation:

(1)\[\boxed{ \sigma_{xy}(t) = \int_{-\infty}^{t} dt'\; \dot{\gamma}(t')\,G(t,t') }\]

where \(G(t,t')\) is a history-dependent shear modulus functional of transient correlators under the full strain history between times \(t'\) and \(t\).

The Microscopic Modulus

For Brownian colloids using the isotropized MCT approximation:

(2)\[G(t,t') = \frac{k_B T}{60\pi^2} \int_0^{\infty} dk\; k^4 \left[\frac{S'(k)}{S(k)^2}\right]^2\,\Phi_k(t,t')^2\]

Physical interpretation:

• Stress arises from distorted microstructure: The integral over \(k\) weights contributions from different length scales

• Relaxation is controlled by cage breaking: As density correlators \(\Phi_k\) decay (cages break), the modulus decreases

• \(S(k)\) weighting: Modes near the \(S(k)\) peak contribute most to stress

Schematic Approximation

For the \(F_{12}\) schematic model:

(3)\[G(t,t') = G_\infty \Phi(t,t')^2\]

where \(\Phi(t,t')\) is a single scalar correlator and \(G_\infty\) is a fitted high-frequency modulus.

Protocol 1: Flow Curve (Steady Shear)

Protocol definition: Constant shear rate applied indefinitely.

\[\dot{\gamma}(t) = \dot{\gamma} = \text{constant}\]

Kinematics

• Accumulated strain: \(\gamma(t,t') = \dot{\gamma}(t - t')\)

• Advected wavevector (ISM): \(k(\tau) = k\sqrt{1 + (\dot{\gamma}\tau)^2/3}\)

Steady-State Stress

At steady state, the modulus becomes time-translation invariant: \(G(t,t') \to G_{\dot{\gamma}}(t-t') = G_{\dot{\gamma}}(\tau)\).

(4)\[\boxed{ \sigma_{xy}(\dot{\gamma}) = \dot{\gamma} \int_0^{\infty} d\tau\; G_{\dot{\gamma}}(\tau) }\]

where the steady-state modulus:

\[G_{\dot{\gamma}}(\tau) = \frac{k_B T}{60\pi^2} \int_0^{\infty} dk\; k^4 \left[\frac{S'(k)}{S(k)^2}\right]^2\,\Phi_k(\tau;\dot{\gamma})^2\]

Dynamic Yield Stress

In the glass state (\(\varepsilon > 0\)):

(5)\[\boxed{ \sigma_y = \lim_{\dot{\gamma} \to 0} \sigma_{xy}(\dot{\gamma}) }\]

Physical behavior:

• Low rates: Stress approaches yield stress \(\sigma_y\)

• Intermediate rates: Power-law shear thinning \(\sigma \sim \dot{\gamma}^n\)

• High rates: Linear viscous regime \(\sigma \sim \eta_\infty \dot{\gamma}\)

Protocol 2: Start-up of Steady Shear

Protocol definition: Heaviside switch-on of shear rate at \(t=0\).

\[\begin{split}\dot{\gamma}(t) = \begin{cases} 0, & t < 0 \\ \dot{\gamma}_0, & t \geq 0 \end{cases}\end{split}\]

Stress Evolution

For \(t \geq 0\):

(6)\[\boxed{ \sigma_{xy}(t) = \dot{\gamma}_0 \int_0^{t} d\tau\; G_{\dot{\gamma}_0}(\tau) }\]

Under constant rate (homogeneous flow):

\[\sigma_{xy}(t) = \dot{\gamma}_0 \int_0^{t} d\tau\; G_\infty \Phi(\tau;\dot{\gamma}_0)^2\]

Stress Overshoot Physics

The stress overshoot is a signature of cage breaking:

Strain Regime	Behavior
\(\gamma \ll \gamma_c\)	Linear elastic: \(\sigma \approx G_\infty \gamma\)
\(\gamma \sim \gamma_c\)	Stress overshoot (cages begin to break)
\(\gamma \gg \gamma_c\)	Approach to steady state

Overshoot strain: \(\gamma_{\text{peak}} \sim 0.05-0.3\) depending on \(\varepsilon\) and \(\dot{\gamma}\).

Rate dependence: Higher \(\dot{\gamma}\) leads to larger overshoot amplitude and earlier peak in time (but similar peak strain).

Protocol 3: Cessation (Stress Relaxation)

Protocol definition: Shear at constant rate until \(t=0\), then stop.

\[\begin{split}\dot{\gamma}(t) = \begin{cases} \dot{\gamma}_{\text{pre}}, & t < 0 \\ 0, & t \geq 0 \end{cases}\end{split}\]

Stress Relaxation

For \(t \geq 0\):

(7)\[\boxed{ \sigma_{xy}(t \geq 0) = \int_{-\infty}^{0} dt'\; \dot{\gamma}_{\text{pre}}\; G(t,t') }\]

Or, rewriting with \(\tau = -t'\):

\[\sigma_{xy}(t) = \dot{\gamma}_{\text{pre}} \int_0^{\infty} d\tau\; G_{\text{stop}}(t; \tau, \dot{\gamma}_{\text{pre}})\]

Mixed History

The correlators involve mixed history:

• Pre-shear phase (\(t' < 0\)): Accumulated strain \(\gamma(0,t') = \dot{\gamma}_{\text{pre}}|t'|\)

• Relaxation phase (\(t > 0\)): No further strain, but correlators continue relaxing

Key Predictions

State	Relaxation Behavior
Fluid (\(\varepsilon < 0\))	Complete decay to zero (exponential or stretched exponential)
Glass (\(\varepsilon > 0\))	Residual stress \(\sigma_{\text{res}} > 0\) (frozen cages)

The residual stress magnitude depends on the pre-shear rate and distance from the glass transition.

Protocol 4: Creep (Step Stress)

Protocol definition: Constant stress applied at \(t=0\).

\[\sigma_{xy}(t) = \sigma_0 H(t)\]

The Volterra Equation

ITT is naturally strain/rate-controlled. For stress control, we must solve an inverse problem (Volterra integral equation) for \(\dot{\gamma}(t)\):

(8)\[\boxed{ \sigma_0 = \int_0^{t} dt'\; \dot{\gamma}(t')\; G(t,t') \quad (t > 0) }\]

while simultaneously evolving \(\Phi_k(t,t')\) under the resulting \(\dot{\gamma}(t)\) history.

Creep Compliance

The creep strain and compliance are:

\[\gamma(t) = \int_0^{t} \dot{\gamma}(s)\,ds, \qquad J(t) = \frac{\gamma(t)}{\sigma_0}\]

Viscosity Bifurcation

ITT-MCT predicts a sharp viscosity bifurcation at the yield stress:

Stress Regime	Response
\(\sigma_0 < \sigma_y\) (glass)	\(\dot{\gamma}(t) \to 0\), \(J(t)\) saturates (solid-like)
\(\sigma_0 > \sigma_y\) (glass)	Delayed yielding: \(\dot{\gamma}(t)\) grows, then steady flow
Fluid state	\(J(t) \sim t\) at long times (viscous flow)

The transition between creeping and flowing states is discontinuous - a hallmark of the MCT glass transition.

Protocol 5: SAOS (Small Amplitude Oscillatory Shear)

Protocol definition: Small amplitude oscillatory strain.

\[\gamma(t) = \gamma_0 \sin(\omega t), \qquad \gamma_0 \ll 1\]

Linear Response Regime

For \(\gamma_0 \ll \gamma_c\), advection is negligible. The modulus reduces to its quiescent (equilibrium) form:

(9)\[\boxed{ \sigma_{xy}(t) = \int_{-\infty}^{t} dt'\; \dot{\gamma}(t')\; G_{\text{eq}}(t-t') }\]

Equilibrium Modulus

\[G_{\text{eq}}(t) = \frac{k_B T}{60\pi^2} \int_0^{\infty} dk\; k^4 \left[\frac{S'(k)}{S(k)^2}\right]^2\,\Phi_k^{\text{eq}}(t)^2\]

where \(\Phi_k^{\text{eq}}(t)\) satisfies the quiescent MCT equation (no advection).

Complex Modulus

The complex modulus is obtained via Fourier transform:

(10)\[\boxed{ G^*(\omega) = i\omega \int_0^{\infty} dt\; e^{-i\omega t}\; G_{\text{eq}}(t) }\]

with storage and loss moduli:

\[\begin{split}G'(\omega) &= \omega \int_0^{\infty} G_{\text{eq}}(t) \sin(\omega t)\, dt \\ G''(\omega) &= \omega \int_0^{\infty} G_{\text{eq}}(t) \cos(\omega t)\, dt\end{split}\]

MCT Predictions

State	\(G^*(\omega)\) Behavior
Fluid (\(\varepsilon < 0\))	\(G' \sim \omega^2\) at low \(\omega\), crossover to plateau at high \(\omega\)
Glass (\(\varepsilon > 0\))	\(G'(\omega \to 0) \to G_\infty f\) (non-zero plateau)
Critical (\(\varepsilon = 0\))	Power-law behavior \(G' \sim G'' \sim \omega^a\)

Protocol 6: LAOS (Large Amplitude Oscillatory Shear)

Protocol definition: Finite amplitude oscillatory strain.

\[\gamma(t) = \gamma_0 \sin(\omega t), \qquad \gamma_0 \sim O(\gamma_c)\]

Accumulated Strain

The strain between times \(t'\) and \(t\):

(11)\[\boxed{ \gamma(t,t') = \gamma_0 \left[\sin(\omega t) - \sin(\omega t')\right] }\]

Full ITT Stress

The stress involves the full oscillatory history:

(12)\[\boxed{ \sigma_{xy}(t) = \int_{-\infty}^{t} dt'\; \dot{\gamma}(t')\; G(t,t') }\]

where \(G(t,t')\) depends on the time-dependent accumulated strain through advected wavevectors and the strain decorrelation function.

Harmonic Decomposition

By symmetry, only odd harmonics appear:

(13)\[\boxed{ \sigma_{xy}(t) = \sum_{n=1,3,5,...} \left[\sigma'_n \sin(n\omega t) + \sigma''_n \cos(n\omega t)\right] }\]

The nonlinear moduli are:

\[G'_n(\omega, \gamma_0) = \frac{\sigma'_n}{\gamma_0}, \qquad G''_n(\omega, \gamma_0) = \frac{\sigma''_n}{\gamma_0}\]

Third Harmonic Ratio

A key nonlinearity measure is the intrinsic nonlinearity \(I_3/I_1\):

\[\frac{I_3}{I_1} = \frac{|\sigma_3^*|}{|\sigma_1^*|}\]

ITT-MCT predictions:

• Higher harmonics emerge when \(\gamma_0\) is large enough to break cages each cycle

• \(I_3/I_1\) increases with \(\gamma_0/\gamma_c\)

• Intra-cycle yielding: stress peak occurs before strain peak

• Strain softening: \(G'_1\) decreases with increasing \(\gamma_0\)

Schematic \(F_{12}\) Protocol Implementations

For the schematic model, the protocol equations simplify considerably.

Scalar Correlator Equation

\[\partial_t \Phi(t,t_0) + \Gamma \left[\Phi(t,t_0) + \int_{t_0}^{t} ds\; m(t,s,t_0)\;\partial_s\Phi(s,t_0)\right] = 0\]

\(F_{12}\) Memory with Strain Cutoff

\[m(t,s,t_0) = h[\gamma(t,t_0)] \cdot h[\gamma(t,s)] \cdot \left(v_1\Phi(t,s) + v_2\Phi(t,s)^2\right)\]

with the strain decorrelation function:

\[h[\gamma] = \exp\left[-(\gamma/\gamma_c)^2\right]\]

Schematic Stress

\[\sigma(t) = \int_{-\infty}^{t} dt'\; \dot{\gamma}(t')\; G_\infty\; \Phi(t,t')^2\]

This schematic model is widely used for:

• Creep and stress-controlled simulations (with feedback)

• LAOS and Fourier-Transform rheology

• Qualitative flow curves and yielding studies

See Also

• ITT-MCT Schematic (F_1_2) — \(F_{12}\) schematic model theory and implementation

• ITT-MCT Isotropic (ISM) — Full \(k\)-resolved ISM model with \(S(k)\) input

• Models Handbook — ITT-MCT models overview

References

[1]

Fuchs, M. & Cates, M. E. “Theory of nonlinear rheology and yielding of dense colloidal suspensions.” Phys. Rev. Lett. 89, 248304 (2002). https://doi.org/10.1103/PhysRevLett.89.248304

[2]

Brader, J. M., Cates, M. E. & Fuchs, M. “First-Principles Constitutive Equation for Suspension Rheology.” Phys. Rev. Lett. 101, 138301 (2008). https://doi.org/10.1103/PhysRevLett.101.138301

[3]

Voigtmann, T. “Nonlinear glassy rheology.” Curr. Opin. Colloid Interface Sci. 19, 549-560 (2014). https://doi.org/10.1016/j.cocis.2014.11.001