.. _model-itt-mct-isotropic: ITT-MCT Isotropic (ISM) ======================= Quick Reference --------------- - **Use when:** Quantitative predictions needed, S(k) available, wave-vector-dependent dynamics important - **Parameters:** 5 (:math:`\phi`, :math:`\sigma_d`, :math:`D_0`, :math:`k_BT`, :math:`\gamma_c`) + :math:`S(k)` input - **Key equation:** :math:`k`-resolved correlator :math:`\Phi(k,t)` with MCT vertex from :math:`S(k)` - **Test modes:** Flow curve, oscillation, startup, creep, relaxation, LAOS - **Material examples:** Dense colloidal suspensions, hard-sphere glasses, silica particles, PMMA colloids, concentrated emulsions **Data required:** Structure factor :math:`S(k)` from experiment or Percus-Yevick Notation Guide -------------- .. list-table:: :widths: 15 85 :header-rows: 1 * - Symbol - Meaning * - :math:`\Phi(k,t)` - k-resolved density correlator (one function per wave vector) * - :math:`S(k)` - Static structure factor (equilibrium pair correlation) * - :math:`c(k)` - Direct correlation function, :math:`c(k) = 1 - 1/S(k)` * - :math:`V(k,q,p)` - MCT vertex function (mode-coupling kernel) * - :math:`\phi` - Volume fraction (control parameter for glass transition) * - :math:`\phi_g` - Glass transition volume fraction (:math:`\approx 0.516` for hard spheres) * - :math:`\sigma_d` - Particle diameter (m) * - :math:`D_0` - Bare short-time diffusion coefficient (:math:`\text{m}^2/\text{s}`) * - :math:`k_B T` - Thermal energy (J) * - :math:`\Gamma(k)` - k-dependent bare relaxation rate, :math:`\Gamma(k) = k^2 D_0 / S(k)` * - :math:`\gamma_c` - Critical strain for cage breaking (dimensionless) * - :math:`n` - Number density (particles/:math:`\text{m}^3`) Overview -------- The Isotropically Sheared Model (ISM) is the full k-resolved MCT for nonlinear rheology. Unlike the :math:`F_{12}` schematic model, ISM tracks correlators at each wave vector :math:`k`, using the static structure factor :math:`S(k)` to compute the memory kernel. **Key differences from** :math:`F_{12}`: - :math:`k`-resolved correlators :math:`\Phi(k,t)` - Memory kernel from :math:`S(k)` via MCT vertex :math:`V(k,q,|k-q|)` - Quantitative predictions without empirical parameters - Higher computational cost **When to use ISM:** - :math:`S(k)` is known (from scattering experiments or simulation) - Wave-vector-dependent relaxation is important - Quantitative comparison with microscopic measurements - Systems where :math:`F_{12}` simplifications are too severe Physical Foundations -------------------- The ISM model extends the schematic :math:`F_{12}` theory (see :doc:`itt_mct_schematic`) to include full :math:`k`-dependence. All physical concepts from the schematic model apply: cage effect, :math:`\beta`-relaxation, :math:`\alpha`-relaxation, glass transition. The key addition is **wave-vector resolution** of the dynamics. **Why** :math:`k`-**dependence matters:** 1. **Length-scale-dependent relaxation**: Small :math:`k` (long wavelengths) relax slower than large :math:`k` (short wavelengths) 2. **Structure factor weighting**: Peaks in :math:`S(k)` indicate preferred length scales that dominate dynamics 3. **Quantitative stress predictions**: Integration over :math:`k`-space with :math:`S(k)` weighting gives absolute stress values without empirical modulus The ISM model is the most faithful representation of MCT for colloidal glasses, but requires :math:`S(k)` as input and is computationally more expensive than :math:`F_{12}`. Structure Factor Input ---------------------- Percus-Yevick (Default) ~~~~~~~~~~~~~~~~~~~~~~~ For hard spheres, the analytic Percus-Yevick solution provides :math:`S(k)`: .. code-block:: python model = ITTMCTIsotropic(phi=0.55) # Uses Percus-Yevick automatically The glass transition occurs at :math:`\phi_{MCT} \approx 0.516` for hard spheres. User-Provided :math:`S(k)` ~~~~~~~~~~~~~~~~~~~~~~~~~ For real experimental data: .. code-block:: python # From light scattering or X-ray experiments k_data = np.array([...]) # Wave vectors sk_data = np.array([...]) # Structure factor model = ITTMCTIsotropic( sk_source="user_provided", k_data=k_data, sk_data=sk_data ) Parameters ---------- .. list-table:: :widths: 15 15 15 15 40 :header-rows: 1 * - Name - Default - Bounds - Units - Physical Meaning * - :math:`\phi` - 0.55 - (0.1, 0.64) - — - Volume fraction (glass at :math:`\phi \approx 0.516`) * - :math:`\sigma_d` - :math:`10^{-6}` - (:math:`10^{-9}`, :math:`10^{-3}`) - m - Particle diameter * - :math:`D_0` - :math:`10^{-12}` - (:math:`10^{-18}`, :math:`10^{-6}`) - m\ :sup:`2`/s - Bare short-time diffusion coefficient * - :math:`k_BT` - :math:`4.1 \times 10^{-21}` - (:math:`10^{-24}`, :math:`10^{-18}`) - J - Thermal energy * - :math:`\gamma_c` - 0.1 - (0.01, 0.5) - — - Critical strain for cage breaking Isotropic Shear Approximation (ISM) ----------------------------------- The ISM simplifies the full anisotropic MCT equations by assuming that the correlator depends only on the **magnitude** of the advected wavevector. Wavevector Advection Derivation ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ For simple shear with rate :math:`\dot{\gamma}`, the advected wavevector is: .. math:: \mathbf{k}(t,t') = \big(k_x,\; k_y - \gamma(t,t')k_x,\; k_z\big) The advected magnitude squared is: .. math:: k(t,t')^2 = k_x^2 + (k_y - \gamma k_x)^2 + k_z^2 = k^2 - 2\gamma k_x k_y + \gamma^2 k_x^2 **Orientational averaging**: Averaging over all initial orientations of :math:`\mathbf{k}` on a sphere: .. math:: \langle k_x^2 \rangle = \langle k_y^2 \rangle = \langle k_z^2 \rangle = k^2/3 \langle k_x k_y \rangle = 0 This gives the **isotropically sheared** wavevector magnitude: .. math:: \boxed{ k(t,t') \approx k\sqrt{1 + \frac{1}{3}\gamma(t,t')^2} } **Physical interpretation**: - At :math:`\gamma = 0`: :math:`k(t,t') = k` (no advection) - At :math:`\gamma \sim 1`: :math:`k(t,t') \approx 1.15k` (moderate stretch) - At :math:`\gamma \gg 1`: :math:`k(t,t') \propto k\gamma/\sqrt{3}` (strong stretch) The increased wavevector magnitude accelerates relaxation via the bare decay rate :math:`\Gamma(k) = k^2 D_0/S(k)`. Governing Equations ------------------- :math:`k`-Resolved Correlator Dynamics ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Each wave vector :math:`k` has its own correlator equation: .. math:: \partial_t \Phi(k,t) + \Gamma(k) \left[ \Phi(k,t) + \int_0^t m(k,t-s) \partial_s \Phi(k,s) ds \right] = 0 with the :math:`k`-dependent bare relaxation rate: .. math:: \Gamma(k) = \frac{k^2 D_0}{S(k)} This shows that: - Modes with large :math:`S(k)` (strong correlations) relax slower - Short-wavelength modes (large :math:`k`) have faster bare rates - The memory kernel :math:`m(k,t)` couples all :math:`k`-modes together MCT Vertex Function ~~~~~~~~~~~~~~~~~~~ The memory kernel at wave vector :math:`k` involves coupling to all other wave vectors: .. math:: m(k,t) = \sum_q V(k,q,|\mathbf{k}-\mathbf{q}|) \Phi(q,t) \Phi(|\mathbf{k}-\mathbf{q}|,t) The vertex :math:`V` depends on :math:`S(k)` and its derivatives: .. math:: V(k,q,p) \propto n S(k) S(q) S(p) \left[ \frac{\mathbf{k} \cdot \mathbf{q}}{k^2} c(q) + \frac{\mathbf{k} \cdot \mathbf{p}}{k^2} c(p) \right]^2 where c(k) = 1 - 1/S(k) is the direct correlation function. :math:`k`-Resolved Correlators ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Each wave vector has its own relaxation dynamics: .. math:: \partial_t \Phi(k,t) + \Gamma(k) \left[ \Phi(k,t) + \int_0^t m(k,t-s) \partial_s \Phi(k,s) ds \right] = 0 with :math:`k`-dependent relaxation rate: .. math:: \Gamma(k) = \frac{k^2 D_0}{S(k)} Stress from :math:`k`-Space Integration ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ The stress tensor involves integration over all wave vectors: .. math:: \sigma = \frac{k_B T}{6\pi^2} \int_0^\infty dk \, k^4 S(k)^2 \left[\frac{\partial \ln S}{\partial \ln k}\right]^2 \int_0^\infty d\tau \, \Phi(k,\tau)^2 h(\dot{\gamma}\tau) Microscopic Stress Formula Detail ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ The full generalized Green-Kubo expression for the shear modulus is: .. math:: 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 of the weighting factors**: .. list-table:: :widths: 25 75 :header-rows: 1 * - Factor - Physical Meaning * - :math:`k^4` - Short wavelengths contribute more to stress (local rearrangements) * - :math:`[S'(k)]^2` - Modes where S(k) varies rapidly (near the peak) dominate * - :math:`[S(k)]^{-4}` - Modes with strong correlations contribute less (collective, slow) * - :math:`\Phi_k^2` - Only correlated (unrelaxed) modes carry stress **Quantitative predictions without adjustable modulus**: Unlike the schematic model where :math:`G_\infty` is fitted, ISM computes the stress magnitude directly from :math:`k_B T`, :math:`S(k)`, and :math:`\Phi_k`. This provides a first-principles prediction of yield stress and flow curves. Equilibrium vs Driven Correlators ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ The correlator dynamics differ between quiescent and driven states: **Quiescent MCT** (no shear, for SAOS): .. math:: \partial_t \Phi_k^{\text{eq}}(t) + \Gamma_k \left[\Phi_k^{\text{eq}}(t) + \int_0^t ds\; m_k(t-s)\;\partial_s \Phi_k^{\text{eq}}(s)\right] = 0 where :math:`\Gamma_k = k^2 D_0 / S(k)` is constant. **Driven ITT-MCT** (with shear): .. math:: \partial_t \Phi_k(t,t') + \Gamma_k(t,t') \left[\Phi_k(t,t') + \int_{t'}^t ds\; m_k(t,s,t')\;\partial_s \Phi_k(s,t')\right] = 0 where :math:`\Gamma_k(t,t') = D_0 k(t,t')^2 / S(k(t,t'))` depends on the advected wavevector. The key difference is that shear: 1. **Accelerates initial decay** via increased :math:`\Gamma_k(t,t')` 2. **Decorrelates the memory kernel** via :math:`h[\gamma(t,s)]` 3. **Creates two-time dependence** in the correlator This microscopic stress formula requires: - :math:`S(k)` and its derivative (from structure factor) - :math:`\Phi(k,\tau)` for all :math:`k` (from solving the :math:`k`-resolved MCT equations) - :math:`h(\dot{\gamma}\tau)` (strain decorrelation function) The integral weights contributions by :math:`k^4 S(k)^2 [S'(k)]^2`, meaning: - Modes near the :math:`S(k)` peak contribute most - Both large :math:`S(k)` and large :math:`S'(k)` enhance stress contribution - Short-wavelength modes (large :math:`k`) have higher weight due to :math:`k^4` Validity and Assumptions ------------------------- **When ISM works well:** - Dense colloidal suspensions (:math:`\phi > 0.4` for hard spheres) - Monodisperse or narrow size distribution - No attractive interactions (or weak compared to entropic caging) - Brownian dynamics (not granular or inertial) - :math:`S(k)` accurately known (from scattering or theory) **Limitations:** - Computationally expensive (:math:`O(n_k^2 \times N)` vs :math:`O(N)` for :math:`F_{12}`) - Requires accurate :math:`S(k)` input - Assumes isotropic structure under shear (no shear-induced ordering) - No hopping or activated processes (important deep in glass) - Underestimates relaxation times far from transition **When to simplify to** :math:`F_{12}`: If you don't have :math:`S(k)` data or if qualitative trends are sufficient, use the :math:`F_{12}` schematic model instead. ISM is for quantitative comparison with experiments where :math:`S(k)` is measured via light scattering, X-rays, or neutron scattering. What You Can Learn ------------------ The ISM model extends the :math:`F_{12}` schematic with full :math:`k`-resolution and quantitative predictions from the structure factor :math:`S(k)`. All parameters now have microscopic interpretation tied to colloidal physics. Parameter Interpretation ~~~~~~~~~~~~~~~~~~~~~~~~ :math:`\phi` **(Volume Fraction)**: The packing fraction of particles, controlling the glass transition. *For graduate students*: :math:`\phi` is the order parameter for the jamming/glass transition in hard spheres. The MCT glass transition occurs at :math:`\phi_{MCT} \approx 0.516`, slightly below the random close packing :math:`\phi_{RCP} \approx 0.64`. The Percus-Yevick structure factor :math:`S(k; \phi)` becomes singular at :math:`\phi_{MCT}`, where the self-consistent MCT equation develops a non-zero long-time limit :math:`f(k) > 0`. The separation from the transition scales as :math:`\varepsilon \sim (\phi - \phi_g)/\phi_g`. *For practitioners*: :math:`\phi < 0.4` is dilute (fluid), :math:`0.4 < \phi < 0.516` is dense fluid (slow but ergodic), :math:`\phi > 0.516` is glass (yield stress). Fitting :math:`\phi` from rheology requires knowing the particle size :math:`\sigma_d` to convert number density to volume fraction. Typical calibration: measure :math:`\phi` gravimetrically or via osmotic pressure. :math:`\sigma_d` **(Particle Diameter)**: The hard-sphere diameter used to compute S(k) and set the k-grid resolution. *For graduate students*: :math:`\sigma_d` sets the characteristic length scale for structural correlations. The :math:`S(k)` peak occurs at :math:`k^* \approx 2\pi/\sigma_d` (nearest-neighbor spacing). In the microscopic stress formula, :math:`\sigma_d` appears implicitly through the :math:`k`-grid: stress is dominated by modes near :math:`k^*` where :math:`S(k)` is maximal and :math:`S'(k)` is large. *For practitioners*: Use :math:`\sigma_d` from microscopy (e.g., dynamic light scattering radius), not the hydrodynamic radius. For polydisperse systems, use the number-average diameter. Typical values: 10 nm -- 10 :math:`\mu\text{m}` for colloids. :math:`D_0` **(Bare Diffusion Coefficient)**: The short-time (non-interacting) diffusion coefficient, :math:`D_0 = k_B T/(6\pi \eta_s a)` for Stokes-Einstein. *For graduate students*: :math:`D_0` sets the bare relaxation rate :math:`\Gamma(k) = k^2 D_0/S(k)`. At high :math:`k` (short wavelengths), :math:`S(k) \to 1` and :math:`\Gamma(k) \approx k^2 D_0` (free diffusion). At the :math:`S(k)` peak, :math:`\Gamma(k)` is strongly suppressed by the large :math:`S(k)`, leading to slow collective relaxation. The long-time diffusion coefficient :math:`D_L = D_0/S(0)` accounts for thermodynamic slowing. *For practitioners*: Measure :math:`D_0` from dilute suspension DLS (:math:`\phi \to 0` limit) or calculate from Stokes-Einstein using solvent viscosity :math:`\eta_s`. Typical values: :math:`10^{-12}` -- :math:`10^{-9}` m\ :sup:`2`/s for colloids in water. :math:`k_B T` **(Thermal Energy)**: The thermal energy scale, :math:`k_B \times` temperature in Kelvin. *For graduate students*: :math:`k_B T` sets the absolute stress scale in the microscopic formula :math:`\sigma \sim (k_B T / 60\pi^2) \int dk\, k^4 [S'(k)]^2 \Phi^2`. For hard spheres, the stress is purely entropic (no potential energy), so :math:`k_B T` is the only energy scale. At room temperature, :math:`k_B T \approx 4.11 \times 10^{-21}` J. *For practitioners*: Use :math:`k_B T = 4.11 \times 10^{-21}` J at 25 degrees C. For temperature-dependent studies, scale :math:`k_B T` linearly with :math:`T`. If fitted stress is off by a factor of 2, check if the effective temperature differs from solvent temperature (non-equilibrium heating). :math:`\gamma_c` **(Critical Strain)**: The cage-breaking strain scale (same as :math:`F_{12}` schematic). *For graduate students*: :math:`\gamma_c` appears in the strain decorrelation :math:`h(\gamma) = \exp[-(\gamma/\gamma_c)^2]`. For hard spheres, :math:`\gamma_c \approx 0.05\text{--}0.1` corresponds to the Lindemann parameter: the ratio of thermal vibration amplitude to particle spacing. Unlike the schematic model, :math:`\gamma_c` in ISM is the only remaining fit parameter --- all other quantities are determined by :math:`\phi`, :math:`\sigma_d`, :math:`D_0`, :math:`k_B T`, and :math:`S(k)`. *For practitioners*: Fit :math:`\gamma_c` from the shear-thinning onset in flow curves. Smaller :math:`\gamma_c` means easier cage breaking. Typical values: 0.05 (rigid hard spheres), 0.15 (soft microgels), 0.3 (polymeric cages). Material Classification ~~~~~~~~~~~~~~~~~~~~~~~ .. list-table:: Material Classification from ITT-MCT ISM Parameters :header-rows: 1 :widths: 20 20 30 30 * - :math:`\phi` Range - Glass State - Typical Materials - :math:`S(k)` Characteristics * - :math:`\phi` **< 0.45** - Dilute fluid - Low-concentration PMMA colloids, silica sols - :math:`S(k)` peak :math:`< 2`, weak correlations, fast relaxation at all :math:`k` * - **0.45 <** :math:`\phi` **< 0.516** - Dense fluid - Pre-jammed colloids, moderate emulsions - :math:`S(k)` peak :math:`= 2\text{--}3`, strong correlations, slow but ergodic, critical slowing as :math:`\phi \to \phi_g` * - **0.516 <** :math:`\phi` **< 0.55** - Marginal glass - Weakly jammed colloids, soft microgel pastes - :math:`S(k)` peak :math:`> 3`, non-ergodic :math:`\Phi(k, t \to \infty) > 0`, small yield stress * - **0.55 <** :math:`\phi` **< 0.58** - Moderate glass - Hard-sphere colloids, carbopol microgels - :math:`S(k)` peak :math:`> 4`, large :math:`f(k)`, clear yield stress, pronounced plateau * - :math:`\phi` **> 0.58** - Deep glass/jammed - Highly concentrated colloids, dense emulsions - :math:`S(k)` peak :math:`> 5`, near-complete arrest, large yield stress, approaching RCP Wave-Vector-Dependent Relaxation ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ By inspecting :math:`\Phi(k,t)` at different k: - **Small** :math:`k` (long wavelengths, :math:`k\sigma_d < 1`): Collective density fluctuations, slow relaxation, sensitive to hydrodynamic interactions - **Peak** :math:`k` (:math:`k \approx 2\pi/\sigma_d`): Nearest-neighbor cage length scale, dominates stress response - **Large** :math:`k` (:math:`k\sigma_d > 5`): Single-particle rattling, fast relaxation, nearly free diffusion **Diagnostic use**: If experimental dynamic light scattering provides :math:`\Phi(k,t)` at multiple :math:`k`, fit the ISM model to all :math:`k` simultaneously to validate MCT predictions. Quantitative Stress Predictions ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Unlike :math:`F_{12}` (which has a fitted modulus :math:`G_\infty`), ISM predicts stress from first principles given: - Volume fraction :math:`\phi` - Particle size :math:`\sigma_d` - Thermal energy :math:`k_B T` - :math:`S(k)` (from Percus-Yevick or experiment) **No adjustable stress scale**: The only rheological fit parameter is :math:`\gamma_c` (critical strain). The absolute stress magnitude is predicted from :math:`S(k)`. **Validation test**: Compare ISM predictions to experimental flow curves. If the magnitude is wrong by a factor >2, check: 1. Is :math:`S(k)` correct? (Use experimental scattering if available) 2. Are particles truly hard spheres? (Softness changes :math:`S(k)`) 3. Is temperature correct? (:math:`k_B T` enters the prefactor) Structure Factor Evolution ~~~~~~~~~~~~~~~~~~~~~~~~~~~ While the current ISM implementation uses static :math:`S(k)`, inspecting :math:`S(k)` features reveals: - :math:`S(k)` **peak position**: Nearest-neighbor distance :math:`2\pi / k_{\text{peak}}` - :math:`S(k)` **peak height**: Strength of structural correlations (higher = stronger caging) - :math:`S(0)`: Compressibility (diverges at jamming in hard spheres) **Connection to** :math:`\phi_g`: The glass transition volume fraction :math:`\phi_g \approx 0.516` is where :math:`S(k)` becomes so large that :math:`\Phi(k, t \to \infty) > 0` for some :math:`k`. Fitting Guidance ---------------- Parameter Initialization ~~~~~~~~~~~~~~~~~~~~~~~~ **Method 1: From known colloid properties** If you have a well-characterized colloidal suspension: .. code-block:: python phi = 0.55 # Volume fraction (measured) sigma_d = 1e-6 # Particle diameter (1 μm) T = 298 # Temperature (K) k_BT = 1.38e-23 * T # Thermal energy eta_s = 1e-3 # Solvent viscosity (water, Pa·s) D0 = k_BT / (3 * np.pi * eta_s * sigma_d) # Stokes-Einstein model = ITTMCTIsotropic(phi=phi, sigma_d=sigma_d, D0=D0, k_BT=k_BT) **Method 2: Fit to rheological data** If material properties are unknown, fit :math:`\phi` and :math:`\gamma_c` to flow curve data, keeping :math:`\sigma_d`, :math:`D_0`, :math:`k_B T` as physically reasonable estimates. Troubleshooting ~~~~~~~~~~~~~~~ **Problem: S(k) peak too sharp/broad** - Solution: Check if Percus-Yevick is appropriate for your system. For soft particles, provide user S(k) from scattering. **Problem: Predicted stress too high/low** - Solution: Adjust :math:`k_B T` (effective thermal energy may differ from room temperature in driven systems) or check if :math:`D_0` is correct (hydrodynamic interactions). **Problem: Slow computation** - Solution: Reduce :math:`k`-grid resolution (``n_k_points`` parameter) or use :math:`F_{12}` schematic for initial exploration. Usage ----- Basic Prediction ~~~~~~~~~~~~~~~~ .. code-block:: python from rheojax.models.itt_mct import ITTMCTIsotropic import numpy as np # Hard-sphere glass model = ITTMCTIsotropic(phi=0.55) # Check glass state info = model.get_glass_transition_info() print(f"Glass: {info['is_glass']}") # True for φ > 0.516 # Flow curve gamma_dot = np.logspace(-2, 2, 30) sigma = model.predict(gamma_dot, test_mode='flow_curve') Inspect :math:`S(k)` ~~~~~~~~~~~~~~~~~~~ .. code-block:: python # Get S(k) information sk_info = model.get_sk_info() print(f"S(k) peak at k = {sk_info['S_max_position']:.2f}") print(f"S(k) max = {sk_info['S_max']:.2f}") # Access k-grid and S(k) directly import matplotlib.pyplot as plt plt.loglog(model.k_grid, model.S_k) plt.xlabel('k') plt.ylabel('S(k)') Update Parameters ~~~~~~~~~~~~~~~~~ .. code-block:: python # Change volume fraction and recalculate S(k) model.update_structure_factor(phi=0.52) # Or provide new experimental S(k) model.update_structure_factor(k_data=k_new, sk_data=sk_new) Model Comparison ---------------- ISM vs :math:`F_{12}` ~~~~~~~~~~~~~~~~~~~~ .. list-table:: :widths: 25 35 40 :header-rows: 1 * - Aspect - :math:`F_{12}` Schematic - ISM * - Correlators - Single scalar :math:`\Phi(t)` - Array :math:`\Phi(k,t)`, n_k points * - :math:`S(k)` input - Not needed - Required * - Parameters - :math:`\varepsilon`, :math:`\Gamma`, :math:`\gamma_c`, :math:`G_\infty` - :math:`\phi`, :math:`D_0`, :math:`\sigma_d`, :math:`k_B T`, :math:`\gamma_c` * - Glass transition - At :math:`v_2 = 4` - At :math:`\phi \approx 0.516` * - Computation - :math:`O(N)` per step - :math:`O(n_k^2 \times N)` * - Best for - Fitting, exploration - Quantitative predictions See Also -------- - :doc:`itt_mct_schematic` --- Simplified :math:`F_{12}` schematic model (faster, no :math:`S(k)` required) - :doc:`../sgr/sgr_conventional` --- Alternative glass transition framework (trap model) - :doc:`../stz/stz_conventional` --- Shear transformation zone theory (effective temperature) **When to use ISM vs** :math:`F_{12}`: - **Use ISM** if: :math:`S(k)` is known, quantitative predictions needed, validating MCT theory - **Use** :math:`F_{12}` if: Fitting rheological data, qualitative trends, faster computation API Reference ------------- .. autoclass:: rheojax.models.itt_mct.ITTMCTIsotropic :members: :undoc-members: :show-inheritance: :no-index: References ---------- .. 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[7] Fuchs, M. & Cates, M. E. "Theory of nonlinear rheology and yielding of dense colloidal suspensions." *Physical Review Letters*, **89**, 248304 (2002). https://doi.org/10.1103/PhysRevLett.89.248304 .. [8] Fuchs, M. & Ballauff, M. "Nonlinear rheology of dense colloidal dispersions: A phenomenological model and its connection to mode coupling theory." *Colloids Surf. A*, 270-271, 232-238 (2005). https://doi.org/10.1016/j.colsurfa.2005.06.017 .. [9] Brader, J. M. "Nonlinear rheology of colloidal dispersions." *Journal of Physics: Condensed Matter*, **22**, 363101 (2010). https://doi.org/10.1088/0953-8984/22/36/363101 .. [10] Zausch, J., Horbach, J., Laurati, M., Egelhaaf, S. U., Brader, J. M., Voigtmann, T., & Fuchs, M. "From equilibrium to steady state: The transient dynamics of colloidal liquids under shear." *Journal of Physics: Condensed Matter*, **20**, 404210 (2008). https://doi.org/10.1088/0953-8984/20/40/404210