Metastability and T₀ Calculations in Pycalphad
===============================================

This example demonstrates how to use `pycalphad` to compute the **driving force** of a metastable phase and the **T₀ (t-zero) temperature**, where two phases have the same Gibbs energy.

Driving Force Calculation
-------------------------

The **driving force** represents the thermodynamic tendency for a phase transformation to occur. In this example, we compute the driving force for the **LIQUID** phase in the Al-Zn system while allowing only the metastable liquid phase.

### Code Example:

.. code-block:: python

   from pycalphad import Workspace, variables as v
   from pycalphad.property_framework.metaproperties import DormantPhase
   import matplotlib.pyplot as plt

   # Define the workspace with thermodynamic data
   wks3 = Workspace('alzn_mey.tdb', ['AL', 'ZN'],
                   ['FCC_A1', 'HCP_A3', 'LIQUID'],
                   {v.X('ZN'): (0,1,0.02), v.T: 600, v.P:101325, v.N: 1})

   # Create a metastable workspace with only the liquid phase
   metastable_liq_wks = wks3.copy()
   metastable_liq_wks.phases = ['LIQUID']

   # Compute the driving force for the liquid phase
   liq_driving_force = DormantPhase('LIQUID', metastable_liq_wks).driving_force
   liq_driving_force.display_name = 'Liquid Driving Force'

   # Plot the driving force as a function of composition
   fig = plt.figure()
   ax = fig.add_subplot()
   ax.plot(wks3.get(v.X('ZN')), wks3.get(liq_driving_force))
   ax.set_xlabel(f"{v.X('ZN').display_name} [{v.X('ZN').display_units}]")
   ax.set_ylabel(f"{liq_driving_force.display_name} [{liq_driving_force.display_units}]")
   plt.show()

**Expected Output:**  
A plot showing the **Liquid Driving Force [J/mol]** as a function of Zn composition.

.. image:: https://pycalphad.org/docs/latest/_images/examples_Metastability_6_1.png
   :alt: Liquid Driving Force Plot

T₀ (t-zero) Temperature Calculation
------------------------------------

The **T₀ temperature** is the temperature at which two phases have the same Gibbs energy, meaning that a phase transformation can occur without diffusion barriers. Below **T₀**, a **massive transformation** is thermodynamically favored.

### Code Example:

.. code-block:: python

   from pycalphad.property_framework.tzero import T0

   # Define a workspace for step calculation (1D conditions required for T₀)
   wks4 = Workspace('alzn_mey.tdb', ['AL', 'ZN'],
                   ['FCC_A1', 'HCP_A3', 'LIQUID'],
                   {v.X('ZN'): (0,1,0.02), v.T: 300, v.P:101325, v.N: 1})

   # Compute T₀ for FCC_A1 and HCP_A3 phases
   tzero = T0('FCC_A1', 'HCP_A3', wks4)
   tzero.maximum_value = 1700  # Set temperature limit

   # Plot T₀ as a function of composition
   fig = plt.figure()
   ax = fig.add_subplot()
   ax.plot(wks4.get(v.X('ZN')), wks4.get(tzero))
   ax.set_xlabel(f"{v.X('ZN').display_name} [{v.X('ZN').display_units}]")
   ax.set_ylabel(f"{tzero} [{tzero.display_units}]")
   plt.show()

**Expected Output:**  
A plot showing **T₀(FCC_A1, HCP_A3) [kelvin]** as a function of Zn composition.

.. image:: https://pycalphad.org/docs/latest/_images/examples_Metastability_8_1.png
   :alt: T0 Temperature Plot

Summary
-------

- The **driving force** calculation helps determine the thermodynamic potential for a phase transformation.
- The **T₀ temperature** indicates when two phases have the same Gibbs energy, allowing diffusionless transformations.

This example demonstrates how `pycalphad` can be used to study metastability and phase transformation conditions in a thermodynamic system.