第4章：キャラクタリゼーションと分析

4.1 比表面積測定：BET法

BET式

ブルナウアー・エメット・テラー（BET）法は、比表面積測定の標準的な手法です。ラングミュアの単分子層モデルを多層吸着に拡張したものです：

ここで：

• $n$ = 圧力 $P$ における吸着量
• $P_0$ = 吸着質の飽和蒸気圧
• $n_m$ = 単分子層吸着容量
• $C$ = BET定数（吸着熱に関連）

BET比表面積の計算

ここで、$N_A$ はアボガドロ数、$A_m$ は吸着分子の断面積（N₂では0.162 nm²）です。

コード例1：BET解析

"""
BET surface area analysis from N2 adsorption data
Demonstrates linearization and surface area calculation
"""
import numpy as np
import matplotlib.pyplot as plt
from scipy import stats

# Simulated N2 adsorption data (P/P0, Volume adsorbed in cm³/g STP)
p_p0 = np.array([0.05, 0.10, 0.15, 0.20, 0.25, 0.30])
volume_ads = np.array([45.2, 52.1, 58.3, 64.8, 72.1, 80.5])

# BET linearization: y = (P/P0) / [V(1-P/P0)] vs x = P/P0
# Linear form: y = 1/(Vm*C) + [(C-1)/(Vm*C)] * x
y = p_p0 / (volume_ads * (1 - p_p0))
x = p_p0

# Linear regression
slope, intercept, r_value, p_value, std_err = stats.linregress(x, y)

# Calculate BET parameters
Vm = 1 / (slope + intercept)  # Monolayer volume (cm³/g STP)
C = 1 + slope / intercept      # BET constant

# Surface area calculation
# At STP: 1 mole of gas = 22,414 cm³
# N2 cross-sectional area: 0.162 nm² = 0.162e-18 m²
Na = 6.022e23
Am = 0.162e-18  # m²
V_molar = 22414  # cm³/mol at STP

S_BET = Vm * Na * Am / V_molar  # m²/g

# Create figure
fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(14, 5))

# Left: Raw isotherm
ax1.scatter(p_p0, volume_ads, s=100, c='blue', edgecolors='black', zorder=5)
ax1.plot(p_p0, volume_ads, 'b-', linewidth=1.5, alpha=0.7)
ax1.set_xlabel('Relative Pressure P/P₀', fontsize=12)
ax1.set_ylabel('Volume Adsorbed (cm³/g STP)', fontsize=12)
ax1.set_title('N₂ Adsorption Isotherm', fontsize=13, fontweight='bold')
ax1.grid(alpha=0.3)
ax1.set_xlim(0, 0.35)

# Right: BET plot
ax2.scatter(x, y, s=100, c='red', edgecolors='black', zorder=5)
x_line = np.linspace(0, 0.35, 100)
y_line = slope * x_line + intercept
ax2.plot(x_line, y_line, 'r--', linewidth=2, label=f'y = {slope:.4f}x + {intercept:.5f}')
ax2.set_xlabel('P/P₀', fontsize=12)
ax2.set_ylabel('P/P₀ / [V(1-P/P₀)]', fontsize=12)
ax2.set_title('BET Plot (Linearized)', fontsize=13, fontweight='bold')
ax2.legend(fontsize=10)
ax2.grid(alpha=0.3)
ax2.set_xlim(0, 0.35)

# Add equation and results
textstr = f'$R^2$ = {r_value**2:.4f}\n$V_m$ = {Vm:.1f} cm³/g\nC = {C:.1f}\n$S_{{BET}}$ = {S_BET:.1f} m²/g'
ax2.text(0.05, 0.95, textstr, transform=ax2.transAxes, fontsize=11,
         verticalalignment='top', bbox=dict(boxstyle='round', facecolor='wheat'))

plt.tight_layout()
plt.show()

print("\nBET Analysis Results:")
print(f"• Monolayer volume (Vm): {Vm:.2f} cm³/g STP")
print(f"• BET constant (C): {C:.1f}")
print(f"• BET Surface Area: {S_BET:.1f} m²/g")
print(f"• R² of linear fit: {r_value**2:.4f}")

4.2 活性サイトの同定：化学吸着

異なる金属のためのプローブ分子

金属分散度

金属分散度は、表面に露出している金属原子の割合です：

ここで、$S_f$ は化学量論因子（例：H:Pt = 1:1 の場合は1）です。

コード例2：分散度と粒子サイズ

"""
Calculate metal dispersion and particle size from chemisorption data
"""
import numpy as np
import matplotlib.pyplot as plt

def calculate_dispersion(V_chem_cm3_g, metal_loading_wt_pct, metal_mw,
                         stoichiometry=1, VM=22414):
    """
    Calculate metal dispersion from chemisorption volume

    Args:
        V_chem_cm3_g: Chemisorbed volume (cm³/g catalyst at STP)
        metal_loading_wt_pct: Metal loading (wt%)
        metal_mw: Metal molecular weight (g/mol)
        stoichiometry: Chemisorption stoichiometry (probe:metal)
        VM: Molar volume at STP (cm³/mol)
    """
    # Moles of probe chemisorbed per g catalyst
    n_probe = V_chem_cm3_g / VM

    # Moles of surface metal atoms
    n_surface_metal = n_probe / stoichiometry

    # Total moles of metal in catalyst
    n_total_metal = (metal_loading_wt_pct / 100) / metal_mw

    # Dispersion
    D = n_surface_metal / n_total_metal

    return min(D, 1.0)

def dispersion_to_particle_size(D, metal='Pt'):
    """
    Estimate particle size from dispersion (spherical particles)
    d = 6 * V_atomic / (a * D)
    """
    # Atomic parameters (approximate)
    params = {
        'Pt': {'V_atomic': 15.1e-24, 'a': 8.0e-20},  # cm³, cm²
        'Pd': {'V_atomic': 14.7e-24, 'a': 7.9e-20},
        'Rh': {'V_atomic': 13.8e-24, 'a': 7.6e-20},
    }
    p = params.get(metal, params['Pt'])

    if D > 0:
        d_cm = 6 * p['V_atomic'] / (p['a'] * D)
        return d_cm * 1e7  # Convert to nm
    return np.inf

# Example: Pt/Al2O3 catalyst with 1 wt% Pt
metal_loading = 1.0  # wt%
metal_mw = 195.08    # g/mol for Pt

# Different chemisorption volumes
V_chem_values = np.linspace(0.05, 0.5, 50)

dispersions = [calculate_dispersion(V, metal_loading, metal_mw) for V in V_chem_values]
particle_sizes = [dispersion_to_particle_size(D) for D in dispersions]

# Create figure
fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(14, 5))

# Left: Dispersion vs chemisorption volume
ax1.plot(V_chem_values, np.array(dispersions) * 100, 'b-', linewidth=2)
ax1.set_xlabel('H₂ Chemisorption (cm³/g STP)', fontsize=12)
ax1.set_ylabel('Dispersion (%)', fontsize=12)
ax1.set_title('Pt Dispersion vs Chemisorption', fontsize=13, fontweight='bold')
ax1.grid(alpha=0.3)
ax1.axhline(y=50, color='green', linestyle='--', alpha=0.5, label='50% dispersion')
ax1.legend()

# Right: Particle size vs dispersion
D_range = np.linspace(0.05, 1.0, 100)
sizes = [dispersion_to_particle_size(D) for D in D_range]

ax2.plot(np.array(D_range) * 100, sizes, 'r-', linewidth=2)
ax2.set_xlabel('Dispersion (%)', fontsize=12)
ax2.set_ylabel('Particle Size (nm)', fontsize=12)
ax2.set_title('Particle Size vs Dispersion', fontsize=13, fontweight='bold')
ax2.grid(alpha=0.3)
ax2.set_ylim(0, 20)

# Add size regions
ax2.axhspan(0, 2, alpha=0.2, color='green', label='Clusters (<2 nm)')
ax2.axhspan(2, 5, alpha=0.2, color='yellow', label='Small NPs (2-5 nm)')
ax2.axhspan(5, 20, alpha=0.2, color='orange', label='Large NPs (>5 nm)')
ax2.legend(loc='upper right', fontsize=9)

plt.tight_layout()
plt.show()

# Example calculation
V_example = 0.25
D_example = calculate_dispersion(V_example, metal_loading, metal_mw)
size_example = dispersion_to_particle_size(D_example)

print(f"\nExample: 1 wt% Pt/Al₂O₃ with V_chem = {V_example} cm³/g")
print(f"• Dispersion: {D_example*100:.1f}%")
print(f"• Estimated particle size: {size_example:.1f} nm")

4.3 分光法

X線光電子分光法（XPS）

XPSは、表面組成と酸化状態に関する情報を提供します：

• 原理：X線による内殻電子の光電子放出
• 情報：元素組成、酸化状態、化学結合
• 深度：約1-10 nm（表面敏感）

コード例3：XPSピーク解析

"""
Simulate and analyze XPS spectrum for Pt catalyst
Demonstrates peak fitting and oxidation state identification
"""
import numpy as np
import matplotlib.pyplot as plt
from scipy.optimize import curve_fit

def gaussian(x, amp, cen, wid):
    """Gaussian peak function"""
    return amp * np.exp(-(x - cen)**2 / (2 * wid**2))

def multi_gaussian(x, *params):
    """Multiple Gaussian peaks"""
    n_peaks = len(params) // 3
    y = np.zeros_like(x)
    for i in range(n_peaks):
        amp = params[3*i]
        cen = params[3*i + 1]
        wid = params[3*i + 2]
        y += gaussian(x, amp, cen, wid)
    return y

# Binding energy range for Pt 4f
BE = np.linspace(68, 82, 200)

# Simulate Pt 4f spectrum with Pt(0) and Pt(II) components
# Pt(0) 4f7/2: 71.0 eV, 4f5/2: 74.5 eV (spin-orbit splitting ~3.5 eV)
# Pt(II) 4f7/2: 72.5 eV, 4f5/2: 76.0 eV (shifted +1.5 eV)

# True parameters (what we're trying to recover)
true_params = [
    100, 71.0, 0.8,   # Pt(0) 4f7/2
    67, 74.5, 0.8,    # Pt(0) 4f5/2 (ratio 0.67 from degeneracy)
    40, 72.5, 1.0,    # Pt(II) 4f7/2
    27, 76.0, 1.0,    # Pt(II) 4f5/2
]

# Generate synthetic spectrum with noise
np.random.seed(42)
spectrum = multi_gaussian(BE, *true_params)
noise = np.random.normal(0, 3, len(BE))
spectrum_noisy = spectrum + noise
background = 5 + 0.5 * (BE - 68)  # Shirley-like background
spectrum_noisy += background

# Fit the spectrum (simplified - in practice, more sophisticated fitting is used)
initial_guess = [90, 71.2, 0.9, 60, 74.6, 0.9, 35, 72.8, 1.1, 23, 76.3, 1.1]
popt, pcov = curve_fit(lambda x, *p: multi_gaussian(x, *p) + 5 + 0.5*(x-68),
                       BE, spectrum_noisy, p0=initial_guess, maxfev=5000)

# Create figure
fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(14, 5))

# Left: Raw spectrum with fit
ax1.plot(BE, spectrum_noisy, 'k-', linewidth=1.5, label='Experimental')
ax1.plot(BE, multi_gaussian(BE, *popt) + 5 + 0.5*(BE-68), 'r-',
         linewidth=2, label='Fitted')

# Individual components
colors = ['blue', 'blue', 'green', 'green']
labels = ['Pt(0) 4f₇/₂', 'Pt(0) 4f₅/₂', 'Pt(II) 4f₇/₂', 'Pt(II) 4f₅/₂']
for i in range(4):
    peak = gaussian(BE, popt[3*i], popt[3*i+1], popt[3*i+2])
    ax1.fill_between(BE, background, peak + background, alpha=0.3,
                     color=colors[i], label=labels[i])

ax1.set_xlabel('Binding Energy (eV)', fontsize=12)
ax1.set_ylabel('Intensity (a.u.)', fontsize=12)
ax1.set_title('Pt 4f XPS Spectrum', fontsize=13, fontweight='bold')
ax1.legend(fontsize=9, loc='upper left')
ax1.invert_xaxis()  # XPS convention: higher BE on left
ax1.grid(alpha=0.3)

# Right: Oxidation state quantification
pt0_area = popt[0] * popt[2] + popt[3] * popt[5]  # Pt(0) 4f7/2 + 4f5/2
pt2_area = popt[6] * popt[8] + popt[9] * popt[11]  # Pt(II)
total_area = pt0_area + pt2_area

pt0_pct = pt0_area / total_area * 100
pt2_pct = pt2_area / total_area * 100

categories = ['Pt(0)\nMetallic', 'Pt(II)\nOxidized']
values = [pt0_pct, pt2_pct]
colors_bar = ['#2196f3', '#4caf50']

bars = ax2.bar(categories, values, color=colors_bar, edgecolor='black', width=0.5)
ax2.set_ylabel('Relative Amount (%)', fontsize=12)
ax2.set_title('Pt Oxidation State Distribution', fontsize=13, fontweight='bold')
ax2.set_ylim(0, 100)

for bar, val in zip(bars, values):
    ax2.text(bar.get_x() + bar.get_width()/2, bar.get_height() + 2,
             f'{val:.1f}%', ha='center', fontsize=12, fontweight='bold')

plt.tight_layout()
plt.show()

print("\nXPS Analysis Results:")
print(f"• Pt(0) metallic: {pt0_pct:.1f}%")
print(f"• Pt(II) oxidized: {pt2_pct:.1f}%")
print(f"• Pt(0) 4f₇/₂ binding energy: {popt[1]:.2f} eV")
print(f"• Pt(II) 4f₇/₂ binding energy: {popt[7]:.2f} eV")

4.4 顕微鏡法

TEMとSEM

コード例4：粒子サイズ分布解析

"""
Analyze particle size distribution from TEM data
Demonstrates histogram fitting and statistics
"""
import numpy as np
import matplotlib.pyplot as plt
from scipy import stats

# Simulated TEM particle size measurements (nm)
np.random.seed(123)
# Lognormal distribution (common for nanoparticles)
particle_sizes = np.random.lognormal(mean=1.0, sigma=0.4, size=200)
particle_sizes = particle_sizes[particle_sizes < 10]  # Remove outliers

# Calculate statistics
mean_size = np.mean(particle_sizes)
std_size = np.std(particle_sizes)
median_size = np.median(particle_sizes)
d10 = np.percentile(particle_sizes, 10)
d50 = np.percentile(particle_sizes, 50)
d90 = np.percentile(particle_sizes, 90)

# Create figure
fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(14, 5))

# Left: Histogram with fitted distribution
n, bins, patches = ax1.hist(particle_sizes, bins=25, density=True,
                             alpha=0.7, color='steelblue', edgecolor='black')

# Fit lognormal distribution
shape, loc, scale = stats.lognorm.fit(particle_sizes, floc=0)
x_fit = np.linspace(0, 10, 200)
pdf_fit = stats.lognorm.pdf(x_fit, shape, loc, scale)
ax1.plot(x_fit, pdf_fit, 'r-', linewidth=2, label='Lognormal fit')

ax1.axvline(x=mean_size, color='green', linestyle='--', linewidth=2,
            label=f'Mean = {mean_size:.2f} nm')
ax1.axvline(x=median_size, color='orange', linestyle=':', linewidth=2,
            label=f'Median = {median_size:.2f} nm')

ax1.set_xlabel('Particle Size (nm)', fontsize=12)
ax1.set_ylabel('Probability Density', fontsize=12)
ax1.set_title('TEM Particle Size Distribution', fontsize=13, fontweight='bold')
ax1.legend(fontsize=9)
ax1.set_xlim(0, 8)
ax1.grid(alpha=0.3)

# Right: Cumulative distribution
sorted_sizes = np.sort(particle_sizes)
cumulative = np.arange(1, len(sorted_sizes) + 1) / len(sorted_sizes) * 100

ax2.plot(sorted_sizes, cumulative, 'b-', linewidth=2)
ax2.axhline(y=10, color='gray', linestyle='--', alpha=0.5)
ax2.axhline(y=50, color='gray', linestyle='--', alpha=0.5)
ax2.axhline(y=90, color='gray', linestyle='--', alpha=0.5)

ax2.scatter([d10, d50, d90], [10, 50, 90], s=100, c='red', zorder=5)
ax2.text(d10, 15, f'd₁₀={d10:.2f}', fontsize=10)
ax2.text(d50, 55, f'd₅₀={d50:.2f}', fontsize=10)
ax2.text(d90, 85, f'd₉₀={d90:.2f}', fontsize=10)

ax2.set_xlabel('Particle Size (nm)', fontsize=12)
ax2.set_ylabel('Cumulative %', fontsize=12)
ax2.set_title('Cumulative Size Distribution', fontsize=13, fontweight='bold')
ax2.grid(alpha=0.3)
ax2.set_xlim(0, 8)

plt.tight_layout()
plt.show()

print("\nTEM Particle Size Analysis:")
print(f"• Number of particles measured: {len(particle_sizes)}")
print(f"• Mean diameter: {mean_size:.2f} ± {std_size:.2f} nm")
print(f"• d₁₀/d₅₀/d₉₀: {d10:.2f}/{d50:.2f}/{d90:.2f} nm")
print(f"• Polydispersity index (PDI): {(d90-d10)/d50:.2f}")

4.5 オペランド分光法

主要なオペランド手法

4.6 ML支援スペクトル解析

コード例5：MLベースのスペクトル分類

"""
Machine learning classification of catalyst states from spectra
Demonstrates training and prediction for catalyst health monitoring
"""
import numpy as np
import matplotlib.pyplot as plt
from sklearn.ensemble import RandomForestClassifier
from sklearn.model_selection import train_test_split
from sklearn.metrics import classification_report, confusion_matrix
import seaborn as sns

# Generate synthetic spectral data for different catalyst states
np.random.seed(42)
n_samples = 200
n_features = 100  # Spectral points

def generate_spectrum(state, noise_level=0.1):
    """Generate synthetic IR spectrum for different catalyst states"""
    x = np.linspace(1000, 2000, n_features)

    if state == 'fresh':
        # Strong metal carbonyl peak at 2050, weak hydroxyl
        spectrum = 0.8 * np.exp(-(x - 2050)**2 / 100) + \
                   0.2 * np.exp(-(x - 1600)**2 / 200)
    elif state == 'active':
        # Medium carbonyl, reaction intermediates
        spectrum = 0.5 * np.exp(-(x - 2050)**2 / 100) + \
                   0.4 * np.exp(-(x - 1800)**2 / 150) + \
                   0.3 * np.exp(-(x - 1500)**2 / 100)
    elif state == 'deactivated':
        # Weak carbonyl, strong coke peaks
        spectrum = 0.2 * np.exp(-(x - 2050)**2 / 100) + \
                   0.6 * np.exp(-(x - 1400)**2 / 200) + \
                   0.5 * np.exp(-(x - 1300)**2 / 150)

    spectrum += np.random.normal(0, noise_level, n_features)
    return spectrum

# Generate dataset
states = ['fresh', 'active', 'deactivated']
X = []
y = []

for state in states:
    for _ in range(n_samples // len(states)):
        X.append(generate_spectrum(state))
        y.append(state)

X = np.array(X)
y = np.array(y)

# Split data
X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.3,
                                                      random_state=42)

# Train classifier
clf = RandomForestClassifier(n_estimators=100, random_state=42)
clf.fit(X_train, y_train)

# Predict
y_pred = clf.predict(X_test)

# Create figure
fig, axes = plt.subplots(2, 2, figsize=(14, 10))

# Top-left: Example spectra
ax1 = axes[0, 0]
x = np.linspace(1000, 2000, n_features)
for state, color in zip(states, ['blue', 'green', 'red']):
    spectrum = generate_spectrum(state, noise_level=0.02)
    ax1.plot(x, spectrum, color=color, linewidth=2, label=state.capitalize())

ax1.set_xlabel('Wavenumber (cm⁻¹)', fontsize=11)
ax1.set_ylabel('Absorbance (a.u.)', fontsize=11)
ax1.set_title('Example FTIR Spectra', fontsize=12, fontweight='bold')
ax1.legend()
ax1.invert_xaxis()
ax1.grid(alpha=0.3)

# Top-right: Feature importance
ax2 = axes[0, 1]
importance = clf.feature_importances_
ax2.plot(x, importance, 'purple', linewidth=1.5)
ax2.fill_between(x, importance, alpha=0.3, color='purple')
ax2.set_xlabel('Wavenumber (cm⁻¹)', fontsize=11)
ax2.set_ylabel('Feature Importance', fontsize=11)
ax2.set_title('ML Feature Importance', fontsize=12, fontweight='bold')
ax2.invert_xaxis()
ax2.grid(alpha=0.3)

# Bottom-left: Confusion matrix
ax3 = axes[1, 0]
cm = confusion_matrix(y_test, y_pred, labels=states)
sns.heatmap(cm, annot=True, fmt='d', cmap='Blues', xticklabels=states,
            yticklabels=states, ax=ax3)
ax3.set_xlabel('Predicted', fontsize=11)
ax3.set_ylabel('True', fontsize=11)
ax3.set_title('Confusion Matrix', fontsize=12, fontweight='bold')

# Bottom-right: Accuracy by class
ax4 = axes[1, 1]
accuracy = cm.diagonal() / cm.sum(axis=1) * 100
bars = ax4.bar(states, accuracy, color=['blue', 'green', 'red'], edgecolor='black')
ax4.set_ylabel('Classification Accuracy (%)', fontsize=11)
ax4.set_title('Accuracy by Catalyst State', fontsize=12, fontweight='bold')
ax4.set_ylim(0, 105)
for bar, acc in zip(bars, accuracy):
    ax4.text(bar.get_x() + bar.get_width()/2, bar.get_height() + 2,
             f'{acc:.1f}%', ha='center', fontsize=11)

plt.tight_layout()
plt.show()

# Classification report
print("\nML Classification Results:")
print(classification_report(y_test, y_pred, target_names=states))

4.7 触媒の失活

失活メカニズム

4.8 章のまとめ

演習

演習1：BET計算

傾き = 0.015、切片 = 0.001のBETデータから、Vm、C、比表面積を計算しなさい。

演習2：分散度解析

2 wt% Pt/SiO₂触媒がH₂化学吸着量0.35 cm³/gを示しました。分散度を計算し、粒子サイズを推定しなさい。

演習3：XPS解釈

XPSスペクトルで、Pt 4f₇/₂ピークが71.2 eVと73.5 eVに見られます。どのような酸化状態が存在しますか？高い結合エネルギー成分の原因として考えられることは何ですか？

演習4：失活診断

水素化触媒が100時間後に50%の活性を失いました。BET比表面積は変化していませんが、CO化学吸着量は60%低下しました。最も可能性の高い失活メカニズムは何ですか？