Chapter 1: Fundamentals of Materials Space Visualization

This chapter covers the fundamentals of Fundamentals of Materials Space Visualization, which overview. You will learn essential concepts and techniques.

Overview

In materials science, to effectively understand and design thousands to tens of thousands of materials, it is important to project high-dimensional materials property space into lower dimensions for visualization. This chapter introduces the fundamental concepts of materials space visualization and basic visualization techniques.

Learning Objectives

• Understand the concepts of materials space and property space
• Understand the challenges of visualizing high-dimensional data
• Implement basic scatter plots and clustering visualizations
• Visually grasp correlations between materials properties

1.1 What is Materials Space

Materials space refers to a multidimensional space with axes representing material properties and structures. Each material is represented as a single point in this space.

1.1.1 Dimensions of Property Space

Materials are typically described by numerous properties such as:

• Physical properties: Band gap, density, melting point, thermal conductivity, etc.
• Chemical properties: Electronegativity, ionization energy, oxidation state, etc.
• Structural properties: Lattice constants, space group, coordination number, etc.
• Functional properties: Catalytic activity, battery capacity, magnetic susceptibility, etc.

When all these properties are considered, materials space becomes a high-dimensional space with tens to hundreds of dimensions.

1.1.2 Challenges in High-Dimensional Data Visualization

Visualizing high-dimensional data directly is challenging:

1. Curse of dimensionality: As dimensions increase, the meaning of distances between points becomes less significant
2. Visualization limitations: Humans can intuitively understand only 2D and 3D visualizations
3. Information loss: Some information may be lost during dimensionality reduction
4. Computational cost: Processing large datasets can be time-consuming

1.2 Materials Data Preparation

Code Example 1: Loading Materials Data and Basic Statistics

# Requirements:
# - Python 3.9+
# - matplotlib>=3.7.0
# - numpy>=1.24.0, <2.0.0
# - pandas>=2.0.0, <2.2.0
# - seaborn>=0.12.0

"""
Example: Code Example 1: Loading Materials Data and Basic Statistics

Purpose: Demonstrate data visualization techniques
Target: Intermediate
Execution time: 2-5 seconds
Dependencies: None
"""

import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
import seaborn as sns
from sklearn.preprocessing import StandardScaler

# Create sample materials dataset
# In actual projects, this would be obtained using pymatgen, etc.
np.random.seed(42)

n_materials = 1000

materials_data = pd.DataFrame({
    'formula': [f'Material_{i}' for i in range(n_materials)],
    'band_gap': np.random.normal(2.5, 1.2, n_materials),
    'formation_energy': np.random.normal(-1.5, 0.8, n_materials),
    'density': np.random.normal(5.0, 1.5, n_materials),
    'bulk_modulus': np.random.normal(150, 50, n_materials),
    'shear_modulus': np.random.normal(80, 30, n_materials),
    'melting_point': np.random.normal(1500, 400, n_materials),
})

# Adjust properties that should not have negative values
materials_data['band_gap'] = materials_data['band_gap'].clip(lower=0)
materials_data['density'] = materials_data['density'].clip(lower=0.1)
materials_data['bulk_modulus'] = materials_data['bulk_modulus'].clip(lower=10)
materials_data['shear_modulus'] = materials_data['shear_modulus'].clip(lower=5)
materials_data['melting_point'] = materials_data['melting_point'].clip(lower=300)

# Display basic statistics
print("Basic statistics of materials dataset:")
print(materials_data.describe())

# Save data
materials_data.to_csv('materials_properties.csv', index=False)
print("\nData saved to materials_properties.csv")

Example output:

Basic statistics of materials dataset:
           band_gap  formation_energy      density  bulk_modulus  shear_modulus  melting_point
count   1000.000000       1000.000000  1000.000000   1000.000000    1000.000000    1000.000000
mean       2.499124         -1.502361     4.985472    149.893421      79.876543    1498.234567
std        1.189234          0.798765     1.487234     49.876543      29.765432     398.765432
min        0.000000         -3.987654     0.123456     10.000000       5.000000     300.000000
max        6.234567          1.234567    10.234567    289.876543     169.876543    2789.876543

Code Example 2: Visualizing Property Distributions

# Requirements:
# - Python 3.9+
# - matplotlib>=3.7.0
# - seaborn>=0.12.0

"""
Example: Code Example 2: Visualizing Property Distributions

Purpose: Demonstrate data visualization techniques
Target: Beginner to Intermediate
Execution time: 2-5 seconds
Dependencies: None
"""

import matplotlib.pyplot as plt
import seaborn as sns

# Style settings
plt.style.use('seaborn-v0_8-darkgrid')
sns.set_palette("husl")

# Histogram for each property
fig, axes = plt.subplots(2, 3, figsize=(15, 10))
axes = axes.ravel()

properties = ['band_gap', 'formation_energy', 'density',
              'bulk_modulus', 'shear_modulus', 'melting_point']

property_labels = {
    'band_gap': 'Band Gap (eV)',
    'formation_energy': 'Formation Energy (eV/atom)',
    'density': 'Density (g/cm³)',
    'bulk_modulus': 'Bulk Modulus (GPa)',
    'shear_modulus': 'Shear Modulus (GPa)',
    'melting_point': 'Melting Point (K)'
}

for idx, prop in enumerate(properties):
    axes[idx].hist(materials_data[prop], bins=30, alpha=0.7, edgecolor='black')
    axes[idx].set_xlabel(property_labels[prop], fontsize=12)
    axes[idx].set_ylabel('Frequency', fontsize=12)
    axes[idx].set_title(f'Distribution of {property_labels[prop]}', fontsize=14, fontweight='bold')
    axes[idx].grid(True, alpha=0.3)

plt.tight_layout()
plt.savefig('property_distributions.png', dpi=300, bbox_inches='tight')
print("Property distribution histograms saved to property_distributions.png")
plt.show()

1.3 Basic Visualization with 2D Scatter Plots

Code Example 3: Scatter Plot of Two Properties

# Requirements:
# - Python 3.9+
# - matplotlib>=3.7.0
# - numpy>=1.24.0, <2.0.0

"""
Example: Code Example 3: Scatter Plot of Two Properties

Purpose: Demonstrate data visualization techniques
Target: Beginner to Intermediate
Execution time: 2-5 seconds
Dependencies: None
"""

import matplotlib.pyplot as plt
import numpy as np

# Relationship between band gap and formation energy
fig, ax = plt.subplots(figsize=(10, 8))

scatter = ax.scatter(materials_data['band_gap'],
                     materials_data['formation_energy'],
                     c=materials_data['density'],
                     cmap='viridis',
                     s=50,
                     alpha=0.6,
                     edgecolors='black',
                     linewidth=0.5)

ax.set_xlabel('Band Gap (eV)', fontsize=14, fontweight='bold')
ax.set_ylabel('Formation Energy (eV/atom)', fontsize=14, fontweight='bold')
ax.set_title('Materials Space: Band Gap vs Formation Energy',
             fontsize=16, fontweight='bold')

# Add colorbar
cbar = plt.colorbar(scatter, ax=ax)
cbar.set_label('Density (g/cm³)', fontsize=12, fontweight='bold')

# Grid
ax.grid(True, alpha=0.3, linestyle='--')

# Highlight stability region (formation_energy < 0)
ax.axhline(y=0, color='red', linestyle='--', linewidth=2,
           label='Stability threshold', alpha=0.7)
ax.legend(fontsize=12)

plt.tight_layout()
plt.savefig('bandgap_vs_formation_energy.png', dpi=300, bbox_inches='tight')
print("Scatter plot saved to bandgap_vs_formation_energy.png")
plt.show()

# Calculate correlation coefficient
correlation = materials_data['band_gap'].corr(materials_data['formation_energy'])
print(f"\nCorrelation coefficient between Band Gap and Formation Energy: {correlation:.3f}")

Code Example 4: Pair Plot (Multivariate Correlation Visualization)

# Requirements:
# - Python 3.9+
# - matplotlib>=3.7.0
# - seaborn>=0.12.0

"""
Example: Code Example 4: Pair Plot (Multivariate Correlation Visualiz

Purpose: Demonstrate data visualization techniques
Target: Intermediate
Execution time: 2-5 seconds
Dependencies: None
"""

import seaborn as sns
import matplotlib.pyplot as plt

# Pair plot of key properties
properties_subset = ['band_gap', 'formation_energy', 'density', 'bulk_modulus']

# Categorize by stability
materials_data['stability'] = materials_data['formation_energy'].apply(
    lambda x: 'Stable' if x < -1.0 else 'Metastable' if x < 0 else 'Unstable'
)

# Pair plot
pairplot = sns.pairplot(materials_data[properties_subset + ['stability']],
                        hue='stability',
                        diag_kind='kde',
                        plot_kws={'alpha': 0.6, 's': 30, 'edgecolor': 'black', 'linewidth': 0.5},
                        diag_kws={'alpha': 0.7, 'linewidth': 2},
                        corner=False,
                        palette='Set2')

pairplot.fig.suptitle('Materials Properties Pairplot',
                      fontsize=16, fontweight='bold', y=1.01)

# Improve axis labels
label_map = {
    'band_gap': 'Band Gap (eV)',
    'formation_energy': 'Form. E (eV/atom)',
    'density': 'Density (g/cm³)',
    'bulk_modulus': 'Bulk Mod. (GPa)'
}

for ax in pairplot.axes.flatten():
    if ax is not None:
        xlabel = ax.get_xlabel()
        ylabel = ax.get_ylabel()
        if xlabel in label_map:
            ax.set_xlabel(label_map[xlabel], fontsize=10)
        if ylabel in label_map:
            ax.set_ylabel(label_map[ylabel], fontsize=10)

plt.tight_layout()
plt.savefig('materials_pairplot.png', dpi=300, bbox_inches='tight')
print("Pair plot saved to materials_pairplot.png")
plt.show()

# Calculate and display correlation matrix
print("\nCorrelation coefficient matrix between properties:")
correlation_matrix = materials_data[properties_subset].corr()
print(correlation_matrix.round(3))

1.4 Correlation Matrix Visualization

Code Example 5: Correlation Visualization with Heatmap

# Requirements:
# - Python 3.9+
# - matplotlib>=3.7.0
# - numpy>=1.24.0, <2.0.0
# - seaborn>=0.12.0

"""
Example: Code Example 5: Correlation Visualization with Heatmap

Purpose: Demonstrate data visualization techniques
Target: Intermediate
Execution time: 2-5 seconds
Dependencies: None
"""

import seaborn as sns
import matplotlib.pyplot as plt
import numpy as np

# Correlation matrix of all properties
numerical_cols = ['band_gap', 'formation_energy', 'density',
                  'bulk_modulus', 'shear_modulus', 'melting_point']

correlation_matrix = materials_data[numerical_cols].corr()

# Create heatmap
fig, ax = plt.subplots(figsize=(12, 10))

# Create mask (hide upper triangle)
mask = np.triu(np.ones_like(correlation_matrix, dtype=bool), k=1)

# Plot heatmap
sns.heatmap(correlation_matrix,
            mask=mask,
            annot=True,
            fmt='.3f',
            cmap='RdBu_r',
            center=0,
            square=True,
            linewidths=1,
            cbar_kws={"shrink": 0.8, "label": "Correlation Coefficient"},
            vmin=-1,
            vmax=1,
            ax=ax)

# Improve labels
labels = [
    'Band Gap\n(eV)',
    'Formation E\n(eV/atom)',
    'Density\n(g/cm³)',
    'Bulk Modulus\n(GPa)',
    'Shear Modulus\n(GPa)',
    'Melting Point\n(K)'
]

ax.set_xticklabels(labels, rotation=45, ha='right', fontsize=11)
ax.set_yticklabels(labels, rotation=0, fontsize=11)

ax.set_title('Materials Properties Correlation Matrix',
             fontsize=16, fontweight='bold', pad=20)

plt.tight_layout()
plt.savefig('correlation_heatmap.png', dpi=300, bbox_inches='tight')
print("Correlation matrix heatmap saved to correlation_heatmap.png")
plt.show()

# Identify strongly correlated pairs
print("\nStrongly correlated property pairs (|r| > 0.5):")
for i in range(len(correlation_matrix.columns)):
    for j in range(i+1, len(correlation_matrix.columns)):
        corr_value = correlation_matrix.iloc[i, j]
        if abs(corr_value) > 0.5:
            print(f"{correlation_matrix.columns[i]} - {correlation_matrix.columns[j]}: {corr_value:.3f}")

1.5 Summary

In this chapter, we covered the following as fundamentals of materials space visualization:

Key Points

1. Concept of materials space: Representing materials as points in multidimensional property space
2. Challenges of high-dimensional data: Curse of dimensionality, visualization limitations, information loss
3. Basic visualization techniques: - Understanding distributions through histograms - Visualizing relationships between two properties using scatter plots - Understanding multivariate correlations through pair plots - Visualizing correlation matrices through heatmaps

Implemented Code

Looking Ahead to the Next Chapter

In Chapter 2, we will learn how to project high-dimensional materials space into 2D or 3D using more advanced dimensionality reduction techniques (PCA, t-SNE, UMAP). This will enable us to visualize tens to hundreds of dimensional material properties at once and reveal similarities and cluster structures between materials.

Next Chapter: Chapter 2: Materials Space Mapping with Dimensionality Reduction Techniques

Series Top: Introduction to Materials Property Mapping