.. _material_classification:

Material Classification
========================

.. admonition:: Learning Objectives
   :class: note

   After completing this section, you will be able to:

   1. Classify materials as liquid-like, solid-like, or gel-like from rheological data
   2. Identify key signatures in :math:`G'` and :math:`G''` frequency sweeps
   3. Distinguish between physical gels, chemical gels, and viscoelastic liquids
   4. Predict material behavior from modulus frequency dependence
   5. Select appropriate models based on material classification

.. admonition:: Prerequisites
   :class: important

   - :doc:`what_is_rheology` — Core rheology concepts
   - Understanding of :math:`G'` (storage modulus) and :math:`G''` (loss modulus)

The Classification Problem
---------------------------

When you measure a material's rheological properties, the first question is:

**"What type of material is this?"**

The answer determines:

- Which mathematical models are appropriate
- What test modes to use
- How to interpret parameters physically
- What processing or application behavior to expect

Classification is based on observing **how moduli change with frequency** (or equivalently, how stress relaxes with time).

Three Main Categories
---------------------

Materials fall into three broad rheological classes:

1. **Viscoelastic Liquids** — Flow at long times (zero equilibrium modulus)
2. **Viscoelastic Solids** — Finite equilibrium modulus (network structure)
3. **Gels** — Power-law relaxation (critical or near-critical structure)

Let's explore each in detail.

1. Viscoelastic Liquids
-----------------------

Definition
~~~~~~~~~~

**Liquids flow** — they have no equilibrium modulus. At long times (low frequencies), all stress relaxes to zero.

Rheological Signature
~~~~~~~~~~~~~~~~~~~~~~

In a **frequency sweep** (Small-Amplitude Oscillatory Shear, SAOS):

**Low frequencies** (:math:`\omega \to 0`):

- :math:`G'' > G'` (loss dominates)
- :math:`G'' \sim \omega^1` (linear with frequency)
- :math:`G' \sim \omega^2` (quadratic with frequency)

**High frequencies**:

- :math:`G' > G''` (storage dominates --- elastic-like on short timescales)
- Both :math:`G'` and :math:`G''` increase

**Crossover frequency** (:math:`\omega_c`):

- Point where :math:`G' = G''` (transition from liquid-like to solid-like)
- Related to characteristic relaxation time: :math:`\tau \sim 1/\omega_c`

**Physical interpretation**: At low frequencies (long observation times), the material has time to flow --- viscous dissipation (:math:`G''`) dominates. At high frequencies (short observation times), molecular rearrangements can't keep up --- elastic storage (:math:`G'`) dominates.

Mathematical Models
~~~~~~~~~~~~~~~~~~~

Appropriate models for viscoelastic liquids:

- **Maxwell model**: Single relaxation time (simplest)
- **Fractional Maxwell Liquid (FML)**: Distribution of relaxation times
- **Generalized Maxwell model**: Discrete spectrum of relaxation times

Examples
~~~~~~~~

- **Polymer melts**: Polystyrene, polyethylene above :math:`T_g`
- **Polymer solutions**: DNA solutions, xanthan gum solutions
- **Micellar solutions**: Surfactant solutions above CMC

Code Example: Identifying a Viscoelastic Liquid
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

.. code-block:: python

   import numpy as np
   import matplotlib.pyplot as plt

   # Synthetic frequency sweep data for a viscoelastic liquid
   omega = np.logspace(-2, 2, 50)  # Angular frequency (rad/s)

   # Maxwell-like behavior
   tau = 1.0  # Relaxation time (s)
   G0 = 1e5   # Plateau modulus (Pa)

   G_prime = G0 * (omega * tau)**2 / (1 + (omega * tau)**2)
   G_double_prime = G0 * (omega * tau) / (1 + (omega * tau)**2)

   # Plot
   plt.figure(figsize=(8, 5))
   plt.loglog(omega, G_prime, 'o-', label="G' (storage)")
   plt.loglog(omega, G_double_prime, 's-', label='G" (loss)')
   plt.xlabel('Angular Frequency (rad/s)')
   plt.ylabel('Modulus (Pa)')
   plt.legend()
   plt.title('Viscoelastic Liquid: G" > G\' at low ω')
   plt.grid(True, alpha=0.3)

   # Identify crossover
   crossover_idx = np.argmin(np.abs(G_prime - G_double_prime))
   omega_c = omega[crossover_idx]
   print(f"Crossover frequency: {omega_c:.3f} rad/s")
   print(f"Characteristic time: {1/omega_c:.3f} s")

**Classification criteria**:

.. code-block:: python

   # Check low-frequency behavior
   low_freq_idx = 0
   if G_double_prime[low_freq_idx] > G_prime[low_freq_idx]:
       print("Classification: VISCOELASTIC LIQUID")
       print("(G\" > G' at low frequencies)")

2. Viscoelastic Solids
----------------------

Definition
~~~~~~~~~~

**Solids have a finite equilibrium modulus** — they can support stress indefinitely without flowing.
The network structure (chemical crosslinks or strong physical interactions) prevents flow.

Rheological Signature
~~~~~~~~~~~~~~~~~~~~~~

In a **frequency sweep**:

**Low frequencies** (:math:`\omega \to 0`):

- :math:`G' > G''` (storage dominates)
- :math:`G' \to G_e` (plateau at equilibrium modulus)
- :math:`G'' \to 0` or small constant

**High frequencies**:

- Both :math:`G'` and :math:`G''` increase
- :math:`G'` remains larger than :math:`G''`

**No terminal flow**: Unlike liquids, there is NO crossover where :math:`G''` exceeds :math:`G'` at low frequencies.

**Physical interpretation**: The crosslinked network can store elastic energy indefinitely. Even at long timescales, the material does not flow—it remains solid-like.

Mathematical Models
~~~~~~~~~~~~~~~~~~~

Appropriate models for viscoelastic solids:

- **Zener model** (Standard Linear Solid): Two elastic elements + one viscous
- **Fractional Zener Solid-Solid (FZSS)**: Two elastic elements + fractional damping
- **Kelvin-Voigt model**: Elastic + viscous in parallel

Examples
~~~~~~~~

- **Crosslinked rubbers**: Vulcanized rubber, silicone elastomers
- **Chemical gels**: Covalently crosslinked hydrogels
- **Strong physical gels**: Agar, gelatin (below melting temperature)

Code Example: Identifying a Viscoelastic Solid
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

.. code-block:: python

   # Zener model (solid)
   Ge = 1e4   # Equilibrium modulus (Pa)
   Gm = 5e4   # Arm modulus (Pa)
   tau = 1.0  # Relaxation time (s)

   G_prime_solid = Ge + Gm * (omega * tau)**2 / (1 + (omega * tau)**2)
   G_double_prime_solid = Gm * (omega * tau) / (1 + (omega * tau)**2)

   plt.figure(figsize=(8, 5))
   plt.loglog(omega, G_prime_solid, 'o-', label="G' (storage)")
   plt.loglog(omega, G_double_prime_solid, 's-', label='G" (loss)')
   plt.axhline(Ge, color='gray', linestyle='--', label=f'G_e = {Ge} Pa')
   plt.xlabel('Angular Frequency (rad/s)')
   plt.ylabel('Modulus (Pa)')
   plt.legend()
   plt.title('Viscoelastic Solid: G\' plateaus at G_e, G\' > G" everywhere')
   plt.grid(True, alpha=0.3)

**Classification criteria**:

.. code-block:: python

   # Check low-frequency behavior
   low_freq_idx = 0
   if G_prime_solid[low_freq_idx] > G_double_prime_solid[low_freq_idx]:
       # Check if G' plateaus
       G_prime_slope = np.gradient(np.log(G_prime_solid), np.log(omega))
       if abs(G_prime_slope[low_freq_idx]) < 0.1:  # Nearly flat
           print("Classification: VISCOELASTIC SOLID")
           print(f"Equilibrium modulus G_e ≈ {G_prime_solid[low_freq_idx]:.1e} Pa")

3. Gels and Power-Law Materials
--------------------------------

Definition
~~~~~~~~~~

**Gels exhibit power-law relaxation** — neither fully solid nor fully liquid. They sit at or near a critical gelation point, with a broad distribution of relaxation times.

Rheological Signature
~~~~~~~~~~~~~~~~~~~~~~

In a **frequency sweep**:

**Across all frequencies**:

- :math:`G' \sim \omega^\alpha` (power-law scaling)
- :math:`G'' \sim \omega^\alpha` (parallel scaling)
- :math:`G' \approx G''` (often within same order of magnitude)
- Exponent :math:`\alpha` between 0 and 1 (typically 0.1--0.5)

**Log-log plot**: Both :math:`G'` and :math:`G''` are nearly parallel straight lines

**Physical interpretation**: Gels have a fractal or near-critical network structure with relaxation times spanning many decades. The power-law exponent :math:`\alpha` characterizes the breadth of the relaxation spectrum.

Types of Gels
~~~~~~~~~~~~~

**Physical gels**:

- Weak transient networks (thermoreversible)
- Examples: Gelatin (warm), wormlike micelles, clay suspensions

**Chemical gels** (at gelation point):

- Crosslinking in progress (critical gel)
- Power-law behavior only AT gelation transition

**Weak gels / soft solids**:

- :math:`G'` slightly :math:`> G''` across frequencies
- Small equilibrium modulus
- Examples: Yogurt, mayonnaise, soft pastes

Mathematical Models
~~~~~~~~~~~~~~~~~~~

Appropriate models for gels:

- **Fractional Maxwell Gel (FMG)**: Pure power-law (critical gel)
- **Fractional Burgers**: Gel-like with weak plateau
- **SpringPot**: Simplest fractional element

Examples
~~~~~~~~

- **Weak gels**: Yogurt, soft cheese, mayonnaise
- **Colloidal gels**: Fumed silica dispersions, carbon black gels
- **Biopolymer networks**: Fibrin, collagen (weak crosslinking)

Code Example: Identifying a Gel
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

.. code-block:: python

   # Fractional power-law gel
   alpha = 0.3  # Fractional exponent
   S = 1e4      # Quasi-modulus (Pa·s^α)

   G_prime_gel = S * omega**alpha * np.cos(np.pi * alpha / 2)
   G_double_prime_gel = S * omega**alpha * np.sin(np.pi * alpha / 2)

   plt.figure(figsize=(8, 5))
   plt.loglog(omega, G_prime_gel, 'o-', label="G' (storage)")
   plt.loglog(omega, G_double_prime_gel, 's-', label='G" (loss)')
   plt.xlabel('Angular Frequency (rad/s)')
   plt.ylabel('Modulus (Pa)')
   plt.legend()
   plt.title(f'Gel: Power-law scaling G\', G" ~ ω^{alpha}')
   plt.grid(True, alpha=0.3)

**Classification criteria**:

.. code-block:: python

   # Check for power-law scaling
   log_omega = np.log(omega)
   log_Gprime = np.log(G_prime_gel)

   # Linear fit in log-log space
   slope, _ = np.polyfit(log_omega, log_Gprime, 1)

   if 0.1 < slope < 0.9:  # Power-law exponent
       print("Classification: GEL / POWER-LAW MATERIAL")
       print(f"Power-law exponent α ≈ {slope:.2f}")

Visual Summary: Frequency Sweep Signatures
-------------------------------------------

.. code-block:: text

   VISCOELASTIC LIQUID
   ─────────────────────
   log(G', G")
        │     G'
        │    /
        │   /
        │  /  G' > G" (high ω)
        │ /
   ─────┼───────── ω_c (crossover)
        │\
        │ \  G" > G' (low ω)
        │  \
        │   \ G"
        └─────────── log(ω)

   Low ω: G" > G' (liquid-like)
   High ω: G' > G" (solid-like)
   Crossover: τ ~ 1/ω_c


   VISCOELASTIC SOLID
   ──────────────────
   log(G', G")
        │
        │    G'
        │   ╱
   G_e  ├──╱─────── G' plateau
        │ ╱
        │╱  G' > G" everywhere
        │  G"
        └─────────── log(ω)

   Low ω: G' → G_e (equilibrium modulus)
   High ω: Both increase
   No terminal flow


   GEL (POWER-LAW)
   ───────────────
   log(G', G")
        │
        │    ╱ G'
        │   ╱
        │  ╱  Parallel lines
        │ ╱
        │╱ G"
        │
        └─────────── log(ω)

   All ω: G', G" ~ ω^α (parallel)
   G' ≈ G" (similar magnitude)
   No single relaxation time

Classification Flowchart
------------------------

.. code-block:: text

   START: Frequency sweep data (G' vs G" vs ω)
      │
      ▼
   [Low frequency: G' or G" larger?]
      │
      ├─→ G" > G' ──→ VISCOELASTIC LIQUID
      │                   │
      │                   ├─→ Models: Maxwell, FML
      │                   └─→ Examples: Polymer melts
      │
      ├─→ G' > G" ──→ [Does G' plateau?]
      │                   │
      │                   ├─→ YES ──→ VISCOELASTIC SOLID
      │                   │              │
      │                   │              ├─→ Models: Zener, FZSS
      │                   │              └─→ Examples: Rubbers, chemical gels
      │                   │
      │                   └─→ NO ──→ Check power-law
      │
      └─→ G' ≈ G" ──→ [Parallel lines in log-log?]
                          │
                          ├─→ YES ──→ GEL / POWER-LAW
                          │              │
                          │              ├─→ Models: FMG, SpringPot
                          │              └─→ Examples: Yogurt, soft gels
                          │
                          └─→ NO ──→ Re-examine data quality

Practical Classification Algorithm
-----------------------------------

Use this code to automatically classify materials:

.. code-block:: python

   def classify_material(omega, G_prime, G_double_prime):
       """
       Classify material from frequency sweep data.

       Parameters
       ----------
       omega : array
           Angular frequency (rad/s)
       G_prime : array
           Storage modulus (Pa)
       G_double_prime : array
           Loss modulus (Pa)

       Returns
       -------
       classification : str
           'liquid', 'solid', or 'gel'
       details : dict
           Classification metrics
       """
       # Low-frequency behavior (first 10% of data)
       low_freq_idx = len(omega) // 10

       G_prime_low = G_prime[:low_freq_idx]
       G_double_prime_low = G_double_prime[:low_freq_idx]

       # 1. Check low-frequency dominance
       if np.mean(G_double_prime_low) > np.mean(G_prime_low):
           return 'liquid', {'note': 'G" > G\' at low frequencies'}

       # 2. Check for plateau (solid)
       log_omega = np.log(omega[:low_freq_idx])
       log_Gprime = np.log(G_prime_low)
       slope, _ = np.polyfit(log_omega, log_Gprime, 1)

       if abs(slope) < 0.15:  # Nearly flat
           Ge = np.mean(G_prime_low)
           return 'solid', {'Ge': Ge, 'note': 'G\' plateau detected'}

       # 3. Check for power-law (gel)
       log_omega_all = np.log(omega)
       log_Gprime_all = np.log(G_prime)
       slope_all, _ = np.polyfit(log_omega_all, log_Gprime_all, 1)

       if 0.1 < slope_all < 0.9:
           return 'gel', {'alpha': slope_all, 'note': 'Power-law scaling'}

       # Default
       return 'unknown', {'note': 'Re-examine data or try different tests'}

   # Example usage
   material_type, info = classify_material(omega, G_prime, G_double_prime)
   print(f"Material classification: {material_type}")
   print(f"Details: {info}")

Key Concepts
------------

.. admonition:: Main Takeaways
   :class: tip

   1. **Viscoelastic liquids**: :math:`G'' > G'` at low frequencies, flow at long times (Maxwell, FML)

   2. **Viscoelastic solids**: :math:`G' > G''` everywhere, :math:`G'` plateaus at :math:`G_e` (Zener, FZSS)

   3. **Gels**: :math:`G' \approx G''` with power-law scaling :math:`G' \sim \omega^\alpha` (FMG, SpringPot)

   4. **Classification determines model selection** and expected behavior

   5. **Frequency sweeps** (SAOS) are the primary tool for material classification

.. admonition:: Self-Check Questions
   :class: tip

   1. **A material has** :math:`G' = 100` **Pa and** :math:`G'' = 1000` **Pa at 0.01 rad/s. What type of material is it likely to be?**

      Hint: Compare :math:`G'` and :math:`G''` at low frequency

   2. **You observe** :math:`G' = 5000` **Pa at all frequencies from 0.01 to 100 rad/s, while** :math:`G''` **varies from 100 to 500 Pa. What classification?**

      Hint: Look for a plateau in :math:`G'`

   3. **Both** :math:`G'` **and** :math:`G''` **scale as** :math:`\omega^{0.4}` **over 3 decades of frequency. What material type?**

      Hint: Parallel power-law scaling

   4. **Why can't you use a single-frequency measurement to classify a material?**

      Hint: Need to see frequency dependence to distinguish types

   5. **A material shows** :math:`G'' > G'` **at low frequencies but also has a small plateau in** :math:`G'`. **What might this indicate?**

      Hint: Could be intermediate case or weak gel

Further Reading
---------------

**Within this documentation**:

- :doc:`test_modes` — Experimental methods for probing material behavior
- :doc:`../02_model_usage/model_families` — Mathematical models for each material type
- :doc:`../02_model_usage/model_selection` — Flowcharts for choosing models

**External resources**:

- Winter, H.H. & Chambon, F. "Analysis of Linear Viscoelasticity of a Crosslinking Polymer at the Gel Point" *J. Rheol.* 30, 367 (1986) — Critical gels
- Ferry, J.D. *Viscoelastic Properties of Polymers*, Chapter 3 — Material classification

Summary
-------

Materials are classified rheologically into **viscoelastic liquids** (:math:`G'' > G'` at low :math:`\omega`, terminal flow), **viscoelastic solids** (:math:`G'` plateaus at :math:`G_e`, no flow), and **gels** (power-law scaling :math:`G' \sim \omega^\alpha`). Classification is based on frequency sweep signatures and determines appropriate models and expected behavior.

Next Steps
----------

Proceed to: :doc:`test_modes`

Learn about the four major experimental techniques (SAOS, relaxation, creep, flow) used to characterize materials.