Thermodynamic Phase Diagrams ML

# Thermodynamic Phase Diagrams ML

## Introduction & Motivation

Predicting phase stability across composition and temperature enables rapid alloy and materials design. ML models learn from computational and experimental phase data, generating phase diagrams without expensive CALPHAD calculations.

Motivation: Predict phase diagrams from composition and temperature.

Applications: Phase mapping, stability prediction, microstructure control, materials design.

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## Core Concepts & Theory

### Binary Phase Diagrams

Two-component systems and eutectic points.

### Ternary Systems

Three-component phase diagram complexity.

### Thermodynamic Stability

Gibbs energy minimization and equilibrium.

### Phase Boundaries

Solid-liquid and solid-solid equilibria.

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## Mathematical Formulation

Phase Stability Criterion:
$$G_{ ext{phase}} = H - TS + \sum_i \mu_i c_i$$

Phase Boundary Equation:
$$G_1(c,T) = G_2(c,T)$$

Liquidus Curve:
$$T_L = f_L(c_1, c_2, \ldots)$$

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## Advanced Theory & Extensions

### Multicomponent Systems

n-ary phase diagram prediction.

### Kinetic Effects

Metastable phases and quenching.

### Pressure Dependence

High-pressure phase maps.

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## Computational Considerations

Composition Grid: O(N^n) for n elements.

Thermodynamics: O(P·D²) per point.

Phase Mapping: O(N·T) grid points.

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## Practical Implementation Strategies

### Phase Representation

Stable and metastable phases.

### Composition Encoding

Multi-element fraction normalization.

### Temperature Dependence

Non-linear thermal effects.

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## Benchmark Datasets & Evaluation

NIST Phase Diagrams: Experimental data.

ASM Handbook: Alloy systems.

Literature Data: Published phase diagrams.

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## Key Challenges & Limitations

### Data Sparsity

Limited experimental measurements.

### Extrapolation

New composition ranges.

### Pressure Effects

Non-linear dependencies.

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## Hyperparameter Tuning

Hidden units: 128-256 neurons.

Dropout: 0.2-0.3 regularization.

Learning rate: 1e-4 to 1e-2 schedule.

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## Real-World Applications & Case Studies

Fe-C System: Steel design and microstructure.

Al-Cu System: Aluminum aerospace alloys.

Ni-Al System: Nickel superalloy engineering.

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## Integration with Other Methods

Phase diagrams + CALPHAD; + experiments; + kinetics.

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## Summary & Key Takeaways

ML accelerates phase diagram generation and discovery.

Principles:
1. Composition: Multi-element encoding.
2. Temperature: Thermal dependence.
3. Thermodynamics: Energy minimization.
4. Stability: Phase prediction.
5. Mapping: Diagram generation.

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## Appendix: Practical Labs

### Lab 1: Binary Phase Diagram Data

import numpy as np

def generate_phase_diagram_data(c_range, T_range, n_samples=50):
 """Generate synthetic phase diagram data"""
 c_points = np.linspace(c_range[0], c_range[1], n_samples)
 T_points = np.linspace(T_range[0], T_range[1], n_samples)
 
 phase_map = np.zeros((n_samples, n_samples), dtype=int)
 
 for i, c in enumerate(c_points):
 for j, T in enumerate(T_points):
 if T > 1000 + c * 500:
 phase_map[j, i] = 0
 else:
 phase_map[j, i] = 1
 
 assert phase_map.shape == (n_samples, n_samples), "Shape mismatch"
 return c_points, T_points, phase_map

c, T, phase = generate_phase_diagram_data([0, 1], [500, 1500])

assert phase.shape == (50, 50), "Phase diagram generation failed"
print(f"✓ Phase diagram generated: {phase.shape}")

### Lab 2: Phase Prediction

import numpy as np

class PhaseDiagramPredictor:
 def __init__(self, n_compositions=50, n_temperatures=50):
 self.weights = np.random.randn(2, 50) * 0.1
 self.phases = 3
 
 def predict_phase(self, composition, temperature):
 """Predict stable phase from composition and temperature"""
 features = np.array([composition, temperature/1000])
 logits = features @ self.weights
 phase_probs = np.exp(logits) / np.sum(np.exp(logits))
 return np.argmax(phase_probs)

predictor = PhaseDiagramPredictor()
phase = predictor.predict_phase(0.5, 1200)

assert 0 <= phase < 3, "Phase prediction failed"
print(f"✓ Predicted phase: {phase}")

### Lab 3: Phase Boundary Detection

import numpy as np

def detect_phase_boundary(phase_map, axis=0):
 """Detect phase transition boundaries"""
 boundaries = []
 
 for i in range(phase_map.shape[1-axis]):
 if axis == 0:
 slice_data = phase_map[:, i]
 else:
 slice_data = phase_map[i, :]
 
 transitions = np.where(np.diff(slice_data) != 0)[0]
 boundaries.extend(transitions)
 
 return np.array(boundaries)

phase_map = np.random.randint(0, 3, (50, 50))
boundaries = detect_phase_boundary(phase_map)

assert isinstance(boundaries, np.ndarray), "Boundary detection failed"
print(f"✓ Detected {len(boundaries)} phase transitions")

### Lab 4: Phase Diagram Optimization

import numpy as np

class PhaseOptimizer:
 def __init__(self, target_phase=1, target_temp=900):
 self.target_phase = target_phase
 self.target_temp = target_temp
 
 def find_composition(self, n_iterations=30):
 """Find composition for target phase"""
 best_c = 0.5
 best_error = float('inf')
 
 for _ in range(n_iterations):
 c = best_c + np.random.randn() * 0.05
 c = np.clip(c, 0, 1)
 
 predicted_temp = 1000 + c * 500
 error = abs(predicted_temp - self.target_temp)
 
 if error < best_error:
 best_error = error
 best_c = c
 
 return best_c

opt = PhaseOptimizer(target_phase=1, target_temp=1100)
composition = opt.find_composition()

assert 0 <= composition <= 1, "Optimization failed"
print(f"✓ Target composition: {composition:.3f}")

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