ITT-MCT Literature Review: Key Equations¶

References¶

Fuchs & Cates (2002) “Theory of nonlinear rheology and yielding of dense colloidal suspensions” Phys. Rev. Lett. 89, 248304
Fuchs & Cates (2003) “Schematic models for dynamic yielding of sheared colloidal glasses” Faraday Discuss. 123, 267-286
Fuchs & Cates (2009) “A mode coupling theory for Brownian particles in homogeneous steady shear flow” J. Rheol. 53(4), 957-1000
Brader, Voigtmann, Fuchs, Larson, Cates (2009) “Glass rheology: From mode-coupling theory to a dynamical yield criterion” PNAS 106(36), 15186-15191
Voigtmann, Brader, Fuchs, Cates (2012) “Schematic mode coupling theory of glass rheology: single and double step strains” Soft Matter 8, 4244-4253

1. Equilibrium MCT (No Flow)¶

1.1 Density correlator¶

\[\Phi(k,t) = \frac{\langle \rho_k(t) \rho_{-k}(0) \rangle}{\langle |\rho_k|^2 \rangle}\]

1.2 Full MCT equation of motion (underdamped)¶

\[\ddot{\Phi}(t) + \Omega^2 \left[ \Phi(t) + \int_0^t m(t-s) \dot{\Phi}(s)\, ds \right] = 0\]

where \(\Omega(k) = k^2 k_B T / (m S(k))\).

1.3 Full MCT memory kernel (microscopic)¶

\[m(k,t) = \sum_{\mathbf{q}} V(k, q, |\mathbf{k}-\mathbf{q}|)\; \Phi(q,t)\; \Phi(|\mathbf{k}-\mathbf{q}|, t)\]

with vertex:

\[V(\mathbf{k}, \mathbf{q}) = \frac{(\mathbf{k}\cdot\mathbf{q})\, c(q)}{k} + \frac{\mathbf{k}\cdot(\mathbf{k}-\mathbf{q})\, c(|\mathbf{k}-\mathbf{q}|)}{k}\]

where \(c(k)\) is the direct correlation function related to \(S(k)\) via \(S(k) = 1/(1 - \rho c(k))\).

1.4 Overdamped (Brownian/colloidal) limit¶

\[\partial_t \Phi(t) + \Gamma \left[ \Phi(t) + \int_0^t m(t-s)\, \partial_s \Phi(s)\, ds \right] = 0\]

where \(\Gamma = D_0 k^2 / S(k)\) is the initial decay rate.

2. F12 Schematic Model (Equilibrium)¶

2.1 Schematic equation of motion¶

\[\partial_t \Phi(t) + \Gamma \left[ \Phi(t) + \int_0^t m(\Phi(s))\, \partial_s \Phi(s)\, ds \right] = 0\]

with initial condition \(\Phi(0) = 1\).

2.2 Schematic memory kernel¶

\[m(\Phi) = v_1 \Phi + v_2 \Phi^2\]

\(v_1\): linear vertex (coupling constant)
\(v_2\): quadratic vertex (coupling constant)
The quadratic term creates the feedback mechanism for glass transition

2.3 Glass transition criterion¶

The non-ergodicity parameter \(f = \lim_{t\to\infty} \Phi(t)\) satisfies:

\[\frac{f}{1-f} = m(f) = v_1 f + v_2 f^2\]

Glass transition occurs when this has a non-zero solution, i.e. when:

\[v_2 > v_{2,c} = \frac{4}{(1 - v_1)^2}\]

For \(v_1 = 0\): \(v_{2,c} = 4\).

2.4 Separation parameter¶

\[\varepsilon = \frac{v_2 - v_{2,c}}{v_{2,c}}\]

\(\varepsilon < 0\): Ergodic fluid (\(\Phi \to 0\) at long times)
\(\varepsilon = 0\): Critical point (power-law decay)
\(\varepsilon > 0\): Glass state (\(\Phi \to f > 0\))

2.5 Non-ergodicity parameter (for \(v_1 = 0\))¶

At the critical point (\(v_2 = 4\)):

\[f_c = 1/2\]

Above the transition (\(\varepsilon > 0\)):

\[f = f_c + \sqrt{\varepsilon/(1-f_c)} + O(\varepsilon)\]

2.6 MCT exponents and two-step relaxation¶

Near the glass transition:

\(\beta\)-relaxation: \(\Phi(t) \approx f_c + h \cdot (t/t_0)^{-a}\)
\(\alpha\)-relaxation: \(\Phi(t) \approx f \cdot \exp[-(t/\tau_\alpha)^b]\)

The exponents \(a\) and \(b\) satisfy:

\[\frac{\Gamma(1-a)^2}{\Gamma(1-2a)} = \frac{\Gamma(1+b)^2}{\Gamma(1+2b)} = \lambda\]

For \(F_{12}\) with \(v_1 = 0\): \(\lambda = 1\).

3. Integration Through Transients (ITT) – Extension to Flow¶

3.1 Deformation gradient¶

For homogeneous flow with velocity gradient \(\boldsymbol{\kappa}(t) = \nabla\mathbf{v}(t)\):

\[\mathbf{E}(t,t') = \mathcal{T}\exp\left(\int_{t'}^{t} \boldsymbol{\kappa}(s)\, ds\right)\]

3.2 Wavevector advection¶

Wavevectors are back-advected:

\[\mathbf{q}(t,t') = \mathbf{q} \cdot \mathbf{E}^{-1}(t,t')\]

For simple shear (flow in x, gradient in y):

\[k_x(t,t') = k_x - k_y \gamma(t,t')\]

where \(\gamma(t,t') = \int_{t'}^t \dot{\gamma}(s)\, ds\).

3.3 Transient density correlator (under shear)¶

\[\Phi_{\mathbf{q}}(t,t') = \frac{\langle \rho_{\mathbf{q}(t,t')}(t)\, \rho_{-\mathbf{q}}(t') \rangle}{N S(q)}\]

3.4 Generalized Green-Kubo relation (ITT stress functional)¶

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

This is the central result of ITT: stress as a history integral over a generalized modulus.

3.5 Microscopic modulus (isotropized MCT)¶

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

3.6 Schematic modulus¶

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

where \(G_\infty\) is the high-frequency (instantaneous) modulus.

3.7 Full correlator equation under shear¶

\[\partial_t \Phi_{\mathbf{q}}(t,t') + \Gamma_{\mathbf{q}}(t,t') \left[ \Phi_{\mathbf{q}}(t,t') + \int_{t'}^{t} ds\; m_{\mathbf{q}}(t,s,t')\; \partial_s \Phi_{\mathbf{q}}(s,t') \right] = 0\]

with advected decay rate:

\[\Gamma_{\mathbf{q}}(t,t') = D_0\, \frac{q(t,t')^2}{S(q(t,t'))}\]

3.8 Memory kernel under shear (microscopic, bilinear)¶

\[m_{\mathbf{q}}(t,s,t') = \int \frac{d^3k}{(2\pi)^3}\; V_{\mathbf{q},\mathbf{k},\mathbf{p}}(t,s,t')\; \Phi_{\mathbf{k}}(t,s)\, \Phi_{\mathbf{p}}(t,s)\]

4. Schematic ITT-MCT Under Shear (F12-dot-gamma model)¶

4.1 Strain decorrelation function¶

The advected correlator factorizes as:

\[\Phi(t,t') = \Phi_{\text{eq}}(t-t') \cdot h(\gamma(t,t'))\]

Gaussian form (Fuchs & Cates 2002, most common):

\[h(\gamma) = \exp\left[-\left(\frac{\gamma}{\gamma_c}\right)^2\right]\]

Lorentzian form (Brader et al. 2008):

\[h(\gamma) = \frac{1}{1 + (\gamma/\gamma_c)^2}\]

where \(\gamma_c \approx 0.1\) is the critical cage strain.

4.2 Memory kernel – simplified form¶

\[m(\Phi) = h[\gamma_{\text{acc}}] \times (v_1 \Phi + v_2 \Phi^2)\]

Single decorrelation factor depending on total accumulated strain.

4.3 Memory kernel – full two-time form (Fuchs & Cates 2002)¶

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

Two decorrelation factors:

\(h[\gamma(t,t_0)]\): cage breaking since flow started
\(h[\gamma(t,s)]\): cage breaking during the memory integral window

5. Protocol-Specific Equations¶

5.1 Steady-state flow curve¶

At constant \(\dot{\gamma}\), \(\gamma(t,t') = \dot{\gamma}(t-t')\):

\[\sigma_{ss} = \dot{\gamma} \int_0^\infty G(s) \cdot h(\dot{\gamma}\, s)\, ds\]

with \(G(s) = G_\infty \Phi_{\text{eq}}(s)^2\) (schematic, Fuchs & Cates 2002).

Yield stress (glass, \(\varepsilon > 0\)):

\[\sigma_y = \lim_{\dot{\gamma}\to 0} \sigma_{ss} = G_\infty\, \gamma_c\, f^2\]

Effective viscosity:

\[\eta_{\text{eff}} = \sigma / \dot{\gamma}\]

5.2 Linear viscoelasticity (SAOS)¶

For \(\gamma_0 \ll \gamma_c\), linearize around equilibrium:

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

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

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

Glass plateau: For \(\varepsilon > 0\), \(G'(\omega \to 0) \to G_\infty f\) (non-zero plateau).

5.3 Startup flow¶

Starting from rest with constant \(\dot{\gamma}\):

\[\sigma(t) = \dot{\gamma} \int_0^t G(t-s) \cdot h(\dot{\gamma}(t-s))\, ds\]

Shows stress overshoot when \(\dot{\gamma}\, \tau_\alpha > 1\).

5.4 Creep¶

At constant applied stress \(\sigma_0\):

\[\sigma_0 = \int_0^t \dot{\gamma}(t')\, G(t,t')\, dt'\]

\(\sigma_0 < \sigma_y\): bounded deformation (solid-like)
\(\sigma_0 > \sigma_y\): continuous flow (fluidization, viscosity bifurcation)

5.5 Stress relaxation (cessation of flow)¶

After cessation at \(t=0\):

\[\sigma(t) = \sigma(0) \cdot \Phi_{\text{relax}}(t)\]

In the glass state: \(\lim_{t\to\infty} \sigma(t) = \sigma_{\text{res}} > 0\) (residual stress).

5.6 LAOS¶

For \(\gamma(t) = \gamma_0 \sin(\omega t)\), stress decomposes into odd harmonics:

\[\sigma(t) = \sum_{n=1,3,5,...} [\sigma'_n \sin(n\omega t) + \sigma''_n \cos(n\omega t)]\]

6. Key Physical Parameters¶

Parameter	Symbol	Typical Values	Meaning
Linear vertex	\(v_1\)	0	Linear coupling (often set to 0)
Quadratic vertex	\(v_2\)	2-6	Controls distance from glass transition
Critical vertex	\(v_{2,c}\)	4 (for \(v_1=0\))	Glass transition point
Separation parameter	\(\varepsilon\)	-0.1 to 0.5	Distance from glass transition
Bare relaxation rate	\(\Gamma\)	1-1000 s\(^{-1}\)	Short-time Brownian rate
Critical strain	\(\gamma_c\)	0.05-0.2	Cage-breaking strain
High-freq modulus	\(G_\infty\)	\(10^3\)-\(10^7\) Pa	Instantaneous elastic modulus
Non-ergodicity param	\(f\)	0-1	Glass plateau height
Dynamic yield stress	\(\sigma_y\)	1-1000 Pa	\(G_\infty \gamma_c f\)

7. Key Relations Between Parameters¶

v2 from epsilon: \(v_2 = v_{2,c}(1 + \varepsilon) = 4(1+\varepsilon)\) (for \(v_1 = 0\))
f from v2: Solve \(f/(1-f) = v_2 f^2\) giving \(f = 1 - 1/\sqrt{v_2}\) (for \(v_1 = 0\), \(v_2 > 4\))
Yield stress: \(\sigma_y = G_\infty \gamma_c f\)
Alpha relaxation time divergence: \(\tau_\alpha \sim |\varepsilon|^{-\gamma}\) with \(\gamma = 1/(2a) + 1/(2b)\)

8. Comparison: Full MCT vs Schematic F12¶

Aspect	Full MCT	Schematic F12
Correlator	\(\Phi(k,t)\) for all \(k\)	Single \(\Phi(t)\)
Memory kernel	\(k\)-space integral with \(V(\mathbf{k},\mathbf{q})\)	\(v_1\Phi + v_2\Phi^2\)
Stress	\(k\)-weighted integral with \(S(k)\)	\(G_\infty \Phi^2\)
Parameters	\(S(k)\) (from liquid state theory)	\(v_1, v_2, \Gamma, \gamma_c, G_\infty\)
Glass transition	Volume fraction \(\phi_g \approx 0.516\)	\(v_{2,c} = 4\)
Computational cost	Hours-days	Seconds-minutes
Quantitative	Yes (with good \(S(k)\))	Qualitative/semi-quantitative