Thermodynamics in Materials Discovery

Thermodynamics in Materials Discovery#

Learning Objectives#

After completing this section, you will be able to:

Understand basic thermodynamic principles in materials science
Apply thermodynamic concepts to predict material stability
Use computational tools to calculate thermodynamic properties
Interpret phase diagrams and stability plots

Why Thermodynamics Matters#

When we design new materials computationally, we’re essentially asking: “If we could make this material, would it actually exist?” Thermodynamics gives us the tools to answer this question by telling us whether a material is stable, metastable, or will spontaneously decompose into other phases.

Fundamental Thermodynamic Concepts#

Gibbs Free Energy: The Stability Scorecard#

The Gibbs free energy (G) is arguably the most important quantity in materials thermodynamics. It combines the internal energy of a system with the effects of temperature and pressure:

\[G = H - TS\]

where:

$H$ is the enthalpy (total heat content)
$T$ is the temperature
$S$ is the entropy (disorder)

Why it matters: Materials with lower Gibbs free energy are more stable. When comparing different possible structures or compositions, nature favours the configuration with the lowest G.

Enthalpy: The Energy Budget#

Enthalpy represents the total energy content of a material, including both internal energy and the work needed to make room for it:

\[H = U + PV\]

In computational materials science, we often work at constant pressure where volume changes are small, so enthalpy differences are approximately equal to energy differences calculated by DFT.

Practical insight: When DFT calculations give you the “formation energy” of a material, they’re essentially giving you the enthalpy of formation at 0K.

Entropy: The Disorder Factor#

Entropy quantifies disorder in a system. While often overlooked in basic computational screening, entropy becomes crucial at high temperatures. There are several types of entropy in materials:

Configurational entropy: Disorder from mixing different atoms on lattice sites
Vibrational entropy: From thermal vibrations of atoms
Electronic entropy: From thermal excitation of electrons

Rule of thumb: At room temperature, entropy effects typically contribute ~0.025 eV per atom to the free energy.

Chemical Potential: The Driving Force#

The chemical potential ($\mu$) represents the energy cost of adding one more atom of a particular element to the system:

\[\mu_i = \frac{\partial G}{\partial n_i}\]

This concept is crucial for understanding:

Why materials decompose
Which phases form under different conditions
How materials exchange atoms with their environment

Phase Stability: Will Your Material Survive?#

The Convex Hull: A Stability Map#

The convex hull (often called the “ehull” or energy hull) is a geometric construction that identifies the most stable phases at each composition. Here’s how it works:

Plot formation energies of all known phases vs. composition
Draw the lowest energy envelope connecting stable phases
Materials on the hull are stable; those above are unstable

Energy above hull (Ehull): The vertical distance from a material to the hull indicates its instability:

Ehull = 0: Thermodynamically stable
0 < Ehull < 25 meV/atom: Potentially synthesisable
Ehull > 50 meV/atom: Likely unstable

Reading Phase Diagrams#

Phase diagrams map out which phases are stable under different conditions. For a binary system (A-B), you might see:

Temperature
    ^
    |  Liquid
    |--------
    |  A+B  |  B
    | solid | solid
    |_______|_______> Composition
    A              B

Key features to identify:

Single phase regions: Only one structure is stable
Two-phase regions: Material separates into two phases
Phase boundaries: Where transitions occur

Metastability: Kinetically Trapped States#

Not all useful materials are thermodynamically stable! Many technologically important materials are metastable:

Diamond: Metastable form of carbon (graphite is stable)
Many battery cathodes: Delithiated states are often metastable
Amorphous materials: Higher energy than crystals but kinetically trapped

Practical consideration: If Ehull < 50 meV/atom and there are high kinetic barriers to decomposition, the material might be synthesisable.

Computational Methods for Thermodynamics#

DFT Energy Calculations#

Density Functional Theory provides the foundation for computational thermodynamics:

Total energy: Direct output from DFT calculations
Formation energy: Energy relative to standard reference states $$E_f = E_{compound} - \sum n_i E_i^{ref}$$
Mixing energy: Energy change from mixing pure components

The Materials Project Thermodynamics#

The Materials Project provides thermodynamic data for ~150,000 materials:

# Example: Getting thermodynamic data
from mp_api.client import MPRester

with MPRester("YOUR_API_KEY") as mpr:
    # Get thermodynamic data for Li-Fe-O system
    entries = mpr.get_entries_in_chemsys(["Li", "Fe", "O"])
    
    # Each entry contains:
    # - formation energy
    # - energy above hull
    # - equilibrium reaction energy

Temperature Effects: Beyond 0K#

Most DFT calculations are performed at 0K, but real applications need finite temperature properties:

Vibrational contributions: Use phonon calculations $$G_{vib} = \sum_{\mathbf{q},j} \left[\frac{\hbar\omega_{\mathbf{q}j}}{2} + k_BT\ln(1-e^{-\hbar\omega_{\mathbf{q}j}/k_BT})\right]$$
Configurational entropy: For disordered systems $$S_{config} = -k_B \sum_i x_i \ln(x_i)$$
Electronic entropy: Important for metals and small-gap semiconductors

Pressure Effects#

While often neglected in initial screening, pressure can dramatically affect stability:

Phase transitions: Many materials transform under pressure
Synthesis conditions: High-pressure synthesis can access metastable phases
Operational conditions: Battery materials experience pressure during cycling

Quick estimate: Pressure contributes $PV$ to Gibbs energy. For typical solid volumes (~10 Å³/atom) and moderate pressures (1 GPa), this adds ~0.01 eV/atom.

Practical Thermodynamic Screening#

Workflow for Stability Assessment#

Calculate formation energy using DFT or retrieve from databases
Construct convex hull for the relevant chemical system
Determine Ehull for your target composition
Check decomposition products to understand failure modes
Estimate temperature effects if near stability boundary
Consider synthesis conditions that might stabilise the phase

Rules of Thumb#

Stable materials: Ehull = 0 meV/atom
Synthesisable materials: Ehull < 25-50 meV/atom
Temperature stabilisation: ~25 meV/atom per 300K from entropy
Polymorphs: Often within 10-50 meV/atom of each other
Hydration/oxidation: Check stability against H₂O and O₂

Common Pitfalls#

Ignoring competing phases: Always check the full phase diagram
0K approximation: Temperature can stabilise high-entropy phases
Missing metastable phases: Databases might not include all polymorphs
Environmental stability: Consider reaction with air/moisture

Connecting to Other Course Concepts#

Thermodynamics provides the foundation for understanding:

Why certain compositions form (Chemical Filters section)
Which structures are preferred (Structure Prediction section)
How to design stable materials (Compositional Screening section)

In the next sections on DFT and MLFF, we’ll see how these thermodynamic properties are actually calculated from first principles.