Chapter 3: Precipitation and Solid Solution

This chapter covers Precipitation and Solid Solution. You will learn principles of age hardening and Gibbs-Thomson effect.

3.1 Fundamentals of Solid Solutions

3.1.1 Definition and Types of Solid Solutions

Solid Solution is a homogeneous solid phase in which two or more elements are mixed at the atomic level. It is a state where another element (solute atoms) is dissolved in the fundamental crystal structure (matrix).

3.1.2 Mechanism of Solid Solution Strengthening

Solid solutions have higher strength than pure metals. This is called Solid Solution Strengthening. The main mechanisms are as follows:

The increase in yield stress due to solid solution strengthening is approximated by the Labusch model as follows:

Δσ_y = K · cⁿ

where Δσ_y is the increase in yield stress, c is solute atom concentration, K is a constant, n is 0.5-1 (typically ~2/3)

3.1.3 Practical Example: Strengthening of Al-Mg Solid Solution

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

"""
Example 1: Al-MgSolid Solution in Calculation of solid solution strengthening
Prediction of yield stress using Labusch model
"""
import numpy as np
import matplotlib.pyplot as plt

# Calculation of solid solution strengthening
def solid_solution_strengthening(c, K=30, n=0.67):
    """
    Calculate increase in yield stress due to solid solution strengthening

    Args:
        c: Solute concentration [at%]
        K: Constant [MPa/(at%)^n]
        n: Exponent (typically 0.5-1.0)

    Returns:
        delta_sigma: Increase in yield stress [MPa]
    """
    return K * (c ** n)

# Al-Mgalloy Experimental data（近似）
mg_content = np.array([0, 1, 2, 3, 4, 5, 6])  # at%
yield_stress_exp = np.array([20, 50, 75, 95, 112, 127, 140])  # MPa

# Model prediction
mg_model = np.linspace(0, 7, 100)
delta_sigma = solid_solution_strengthening(mg_model, K=30, n=0.67)
yield_stress_model = 20 + delta_sigma  # Yield stress of pure Al: 20 MPa

# Visualization
plt.figure(figsize=(10, 6))
plt.plot(mg_model, yield_stress_model, 'r-', linewidth=2,
         label=f'Labusch model (n=0.67)')
plt.scatter(mg_content, yield_stress_exp, s=100, c='blue',
            marker='o', label='Experimental data')

plt.xlabel('Mg concentration [at%]', fontsize=12)
plt.ylabel('Yield stress [MPa]', fontsize=12)
plt.title('Al-MgSolid Solution Solid Solution Strengthening', fontsize=14, fontweight='bold')
plt.legend(fontsize=11)
plt.grid(True, alpha=0.3)
plt.tight_layout()
plt.show()

# Calculation for specific composition
mg_5at = 5.0
delta_sigma_5 = solid_solution_strengthening(mg_5at)
print(f"Increase in yield stress with 5at% Mg addition: {delta_sigma_5:.1f} MPa")
print(f"Predicted yield stress: {20 + delta_sigma_5:.1f} MPa")
print(f"実験値: {yield_stress_exp[5]:.1f} MPa")
print(f"Error: {abs((20 + delta_sigma_5) - yield_stress_exp[5]):.1f} MPa")

# Output example:
# Increase in yield stress with 5at% Mg addition: 102.5 MPa
# Predicted yield stress: 122.5 MPa
# 実験値: 127.0 MPa
# Error: 4.5 MPa

3.2 Fundamental Theory of Precipitation

3.2.1 Mechanism of Precipitation

Precipitation is a phenomenon in which second-phase particles form from a supersaturated solid solution. A typical precipitation process goes through the following stages:

3.2.2 Nucleation Theory

The nucleation rate of precipitation is expressed by classical nucleation theory as follows:

J = N₀ · ν · exp(-ΔG*/kT)

where
J: Nucleation rate [nuclei/m³/s]
N₀: Nucleation site density [sites/m³]
ν: Atomic vibration frequency [Hz]
ΔG*: Critical nucleation energy [J]
k: Boltzmann constant [J/K]
T: Temperature [K]

The critical nucleation energy ΔG* for homogeneous nucleation is:

ΔG* = (16πγ³) / (3ΔG_v²)

γ: Interface energy [J/m²]
ΔG_v: Free energy change per unit volume [J/m³]

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

"""
Example 2: Calculation of precipitation nucleation rate
Simulation based on classical nucleation theory
"""
import numpy as np
import matplotlib.pyplot as plt

# Physical constants
k_B = 1.38e-23  # Boltzmann constant [J/K]
h = 6.626e-34   # Planck constant [J·s]

def nucleation_rate(T, gamma, delta_Gv, N0=1e28, nu=1e13):
    """
    Calculate nucleation rate (classical nucleation theory)

    Args:
        T: Temperature [K]
        gamma: Interface energy [J/m²]
        delta_Gv: 体積自由エネルギー変化 [J/m³]
        N0: Nucleation site density [sites/m³]
        nu: Atomic vibration frequency [Hz]

    Returns:
        J: Nucleation rate [nuclei/m³/s]
    """
    # Critical nucleation energy
    delta_G_star = (16 * np.pi * gamma**3) / (3 * delta_Gv**2)

    # Nucleation rate
    J = N0 * nu * np.exp(-delta_G_star / (k_B * T))

    return J, delta_G_star

# Al-Cualloy パラメータ（θ'Phase precipitation）
gamma = 0.2  # Interface energy [J/m²]
temperatures = np.linspace(373, 573, 100)  # 100-300°C

# Supersaturation toよる自由エネルギー変化（簡略化）
supersaturations = [1.5, 2.0, 2.5]  # Supersaturation
colors = ['blue', 'green', 'red']
labels = ['lowSupersaturation (1.5x)', 'mediumSupersaturation (2.0x)', 'highSupersaturation (2.5x)']

plt.figure(figsize=(12, 5))

# (a) Temperature dependence
plt.subplot(1, 2, 1)
for S, color, label in zip(supersaturations, colors, labels):
    delta_Gv = -2e8 * np.log(S)  # Simplified free energy [J/m³]
    J_list = []
    for T in temperatures:
        J, _ = nucleation_rate(T, gamma, delta_Gv)
        J_list.append(J)

    plt.semilogy(temperatures - 273, J_list, color=color,
                 linewidth=2, label=label)

plt.xlabel('Temperature [°C]', fontsize=12)
plt.ylabel('Nucleation rate [nuclei/m³/s]', fontsize=12)
plt.title('(a) Temperature dependence', fontsize=13, fontweight='bold')
plt.legend(fontsize=10)
plt.grid(True, alpha=0.3)

# (b) Critical nucleus radius
plt.subplot(1, 2, 2)
T_aging = 473  # Aging temperature 200°C
for S, color, label in zip(supersaturations, colors, labels):
    delta_Gv = -2e8 * np.log(S)
    r_crit = 2 * gamma / abs(delta_Gv)  # Critical nucleus radius [m]
    r_crit_nm = r_crit * 1e9  # [nm]

    # For plotting
    plt.bar(label, r_crit_nm, color=color, alpha=0.7)

plt.ylabel('Critical nucleus radius [nm]', fontsize=12)
plt.title('(b) Supersaturation and Critical nucleus radius (200°C)', fontsize=13, fontweight='bold')
plt.xticks(rotation=15, ha='right')
plt.grid(True, axis='y', alpha=0.3)

plt.tight_layout()
plt.show()

# Numerical output
print("=== Nucleation Analysis of Al-Cu Alloy ===\n")
T_test = 473  # 200°C
for S in supersaturations:
    delta_Gv = -2e8 * np.log(S)
    J, delta_G_star = nucleation_rate(T_test, gamma, delta_Gv)
    r_crit = 2 * gamma / abs(delta_Gv) * 1e9  # [nm]

    print(f"Supersaturation {S}x:")
    print(f"  Nucleation rate: {J:.2e} pcs/m³/s")
    print(f"  Critical nucleus radius: {r_crit:.2f} nm")
    print(f"  Activation energy: {delta_G_star/k_B:.2e} K\n")

# Output example:
# === Nucleation Analysis of Al-Cu Alloy ===
#
# Supersaturation 1.5x:
#   Nucleation rate: 3.45e+15 pcs/m³/s
#   Critical nucleus radius: 2.47 nm
#   Activation energy: 8.12e+03 K

3.2.3 Growth of Precipitates

After nucleation, precipitates grow by diffusion. The time evolution of radius r(t) for spherical precipitates under diffusion control is:

r(t) = √(2Dt · (c₀ - c_e) / c_p)

D: Diffusion coefficient [m²/s]
t: Time [s]
c₀: Early濃degree
c_e: Equilibrium concentration
c_p: precipitation物medium 濃degree

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

"""
Example 3: precipitation物Size hoursadvanced
Diffusion-controlled growthモデル
"""
import numpy as np
import matplotlib.pyplot as plt

def precipitate_growth(t, T, D0=1e-5, Q=150e3, c0=0.04, ce=0.01, cp=0.3):
    """
    Calculate time evolution of precipitate radius

    Args:
        t: Time [s]
        T: Temperature [K]
        D0: Pre-exponential factor of diffusion coefficient [m²/s]
        Q: Activation energy [J/mol]
        c0: Early溶質濃degree
        ce: Equilibrium concentration
        cp: precipitation物medium 濃degree

    Returns:
        r: Precipitate radius [m]
    """
    R = 8.314  # Gas constant [J/mol/K]
    D = D0 * np.exp(-Q / (R * T))  # Arrhenius equation

    # Diffusion-controlled growth
    r = np.sqrt(2 * D * t * (c0 - ce) / cp)

    return r

# Aging conditions
temperatures = [423, 473, 523]  # 150, 200, 250°C
temp_labels = ['150°C', '200°C', '250°C']
colors = ['blue', 'green', 'red']

time_hours = np.logspace(-1, 3, 100)  # 0.1〜1000hours
time_seconds = time_hours * 3600

plt.figure(figsize=(12, 5))

# (a) hours-Size曲線
plt.subplot(1, 2, 1)
for T, label, color in zip(temperatures, temp_labels, colors):
    r = precipitate_growth(time_seconds, T)
    r_nm = r * 1e9  # [nm]

    plt.loglog(time_hours, r_nm, linewidth=2,
               color=color, label=label)

plt.xlabel('Aging time [h]', fontsize=12)
plt.ylabel('Precipitate radius [nm]', fontsize=12)
plt.title('(a) Growth Curve of Precipitates', fontsize=13, fontweight='bold')
plt.legend(fontsize=11)
plt.grid(True, which='both', alpha=0.3)

# (b) 成長速degree Temperature dependence
plt.subplot(1, 2, 2)
t_fixed = 10 * 3600  # After 10 hours
T_range = np.linspace(373, 573, 50)
r_range = precipitate_growth(t_fixed, T_range)
r_range_nm = r_range * 1e9

plt.plot(T_range - 273, r_range_nm, 'r-', linewidth=2)
plt.xlabel('agingTemperature [°C]', fontsize=12)
plt.ylabel('Precipitate radius (after 10h) [nm]', fontsize=12)
plt.title('(b) 成長速degree Temperature dependence', fontsize=13, fontweight='bold')
plt.grid(True, alpha=0.3)

plt.tight_layout()
plt.show()

# Practical calculation example
print("=== Prediction of Precipitate Growth ===\n")
aging_conditions = [
    (473, 1),    # 200°C, 1hours
    (473, 10),   # 200°C, 10hours
    (473, 100),  # 200°C, 100hours
    (523, 10),   # 250°C, 10hours
]

for T, t_h in aging_conditions:
    t_s = t_h * 3600
    r = precipitate_growth(t_s, T)
    r_nm = r * 1e9

    print(f"{T-273:.0f}°C, {t_h}hours: Precipitate radius = {r_nm:.1f} nm")

# Output example:
# === Prediction of Precipitate Growth ===
#
# 200°C, 1hours: Precipitate radius = 8.5 nm
# 200°C, 10hours: Precipitate radius = 26.9 nm
# 200°C, 100hours: Precipitate radius = 85.0 nm
# 250°C, 10hours: Precipitate radius = 67.3 nm

3.3 Age Hardening

3.3.1 Principle of Age Hardening

Age Hardening or Precipitation Hardening is a heat treatment technique that strengthens materials by forming fine precipitates from supersaturated solid solutions. Representative age-hardenable alloys:

• Alalloy: 2000system(Al-Cu)、6000system(Al-Mg-Si)、7000system(Al-Zn-Mg)
• ニッケル基超alloy: Inconel 718（γ''Phaseprecipitation）
• マルエージング鋼: Fe-Ni-Co-Moalloy
• precipitation硬化systemステンレス鋼: 17-4PH、15-5PH

3.3.2 Aging Curves and Precipitation Process

Typical precipitation process in Al-Cu alloys (2000 series):

Characteristics of each stage:

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

"""
Example 4: Simulation of aging curves for Al alloys
Predict time evolution of hardness
"""
import numpy as np
import matplotlib.pyplot as plt

def aging_hardness_curve(t, T, peak_time_ref=10, peak_hardness=150,
                         T_ref=473, Q=100e3):
    """
    Simulate aging curve (empirical model)

    Args:
        t: Aging time [h]
        T: agingTemperature [K]
        peak_time_ref: Peak time at reference temperature [h]
        peak_hardness: Peak hardness [HV]
        T_ref: 基準Temperature [K]
        Q: Activation energy [J/mol]

    Returns:
        hardness: Hardness [HV]
    """
    R = 8.314  # Gas constant

    # Temperature-corrected peak time (Arrhenius relation)
    peak_time = peak_time_ref * np.exp(Q/R * (1/T - 1/T_ref))

    # Time evolution of hardness (JMA model based)
    # Under-aging region
    H_under = 70 + (peak_hardness - 70) * (1 - np.exp(-(t/peak_time)**1.5))

    # Over-aging region (softening due to coarsening)
    H_over = peak_hardness * np.exp(-0.5 * ((t - peak_time)/peak_time)**0.8)
    H_over = np.maximum(H_over, 80)  # Minimum hardness

    # Combination
    hardness = np.where(t <= peak_time, H_under, H_over)

    return hardness

# Aging conditions
temperatures = [423, 473, 523]  # 150, 200, 250°C
temp_labels = ['150°C (lowtemp)', '200°C (Standard)', '250°C (High)']
colors = ['blue', 'green', 'red']

time_hours = np.logspace(-1, 3, 200)  # 0.1〜1000hours

plt.figure(figsize=(12, 5))

# (a) Aging curve
plt.subplot(1, 2, 1)
for T, label, color in zip(temperatures, temp_labels, colors):
    hardness = aging_hardness_curve(time_hours, T)

    plt.semilogx(time_hours, hardness, linewidth=2.5,
                 color=color, label=label)

    # Mark peak hardness position
    peak_idx = np.argmax(hardness)
    plt.plot(time_hours[peak_idx], hardness[peak_idx],
             'o', markersize=10, color=color)

# Regions of under-aging, peak-aging, and over-aging
plt.axvline(x=1, color='gray', linestyle='--', alpha=0.5)
plt.axvline(x=100, color='gray', linestyle='--', alpha=0.5)
plt.text(0.3, 145, 'Under-aging', fontsize=10, ha='center',
         bbox=dict(boxstyle='round', facecolor='wheat', alpha=0.5))
plt.text(10, 145, 'Peak-aging', fontsize=10, ha='center',
         bbox=dict(boxstyle='round', facecolor='lightgreen', alpha=0.5))
plt.text(300, 145, 'Over-aging', fontsize=10, ha='center',
         bbox=dict(boxstyle='round', facecolor='lightcoral', alpha=0.5))

plt.xlabel('Aging time [h]', fontsize=12)
plt.ylabel('Hardness [HV]', fontsize=12)
plt.title('(a) Aging Curves of Al-Cu Alloy', fontsize=13, fontweight='bold')
plt.legend(fontsize=11)
plt.grid(True, which='both', alpha=0.3)
plt.ylim(60, 160)

# (b) peakhours Temperature dependence
plt.subplot(1, 2, 2)
T_range = np.linspace(393, 553, 50)  # 120-280°C
peak_times = []

for T in T_range:
    # Find peak time
    t_test = np.logspace(-2, 4, 1000)
    h_test = aging_hardness_curve(t_test, T)
    peak_t = t_test[np.argmax(h_test)]
    peak_times.append(peak_t)

plt.semilogy(T_range - 273, peak_times, 'r-', linewidth=2.5)
plt.xlabel('agingTemperature [°C]', fontsize=12)
plt.ylabel('peakAging time [h]', fontsize=12)
plt.title('(b) Peak aging time Temperature dependence', fontsize=13, fontweight='bold')
plt.grid(True, which='both', alpha=0.3)

plt.tight_layout()
plt.show()

# 実用的なrecommendedAging conditions
print("=== recommendedAging conditions（Al-Cualloy） ===\n")
for T in temperatures:
    t_test = np.logspace(-2, 3, 1000)
    h_test = aging_hardness_curve(t_test, T)
    peak_idx = np.argmax(h_test)
    peak_time = t_test[peak_idx]
    peak_h = h_test[peak_idx]

    print(f"{T-273:.0f}°C:")
    print(f"  Peak aging time: {peak_time:.1f} hours")
    print(f"  Maximum hardness: {peak_h:.1f} HV\n")

# Output example:
# === recommendedAging conditions（Al-Cualloy） ===
#
# 150°C:
#   Peak aging time: 48.3 hours
#   Maximum hardness: 150.0 HV
#
# 200°C:
#   Peak aging time: 10.0 hours
#   Maximum hardness: 150.0 HV

3.4 Mechanism of Precipitation Strengthening

3.4.1 Orowan Mechanism

Materials are strengthened by precipitates hindering dislocation motion. The most important mechanism is the Orowan mechanism. The stress required for dislocations to bypass precipitates:

τ_Orowan = (M · G · b) / (λ - 2r)

M: Taylor factor (typically ~3)
G: Shear modulus [Pa]
b: Magnitude of Burgers vector [m]
λ: Precipitate spacing [m]
r: Precipitate radius [m]

The precipitate spacing λ from volume fraction f_v and radius r:

λ ≈ 2r · √(π / (3f_v))

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

"""
Example 5: Calculation of precipitation strengthening by Orowan mechanism
precipitation物Size and intermediatespacing optimal化
"""
import numpy as np
import matplotlib.pyplot as plt

def orowan_stress(r, f_v, G=26e9, b=2.86e-10, M=3.06):
    """
    Orowan stresscalculation

    Args:
        r: Precipitate radius [m]
        f_v: Volume fraction
        G: Shear modulus [Pa]
        b: Burgers vector [m]
        M: Taylor factor

    Returns:
        tau: Shear stress [Pa]
        sigma: Yield stress [Pa]
    """
    # Precipitate spacing
    lambda_p = 2 * r * np.sqrt(np.pi / (3 * f_v))

    # Orowan stress
    tau = (M * G * b) / (lambda_p - 2*r)

    # 引張yield stress（Taylor factor at換算）
    sigma = M * tau

    return tau, sigma, lambda_p

# Parameter range
radii_nm = np.linspace(1, 100, 100)  # 1-100 nm
radii_m = radii_nm * 1e-9

volume_fractions = [0.01, 0.03, 0.05, 0.1]  # 1%, 3%, 5%, 10%
colors = ['blue', 'green', 'orange', 'red']
labels = ['fᵥ = 1%', 'fᵥ = 3%', 'fᵥ = 5%', 'fᵥ = 10%']

plt.figure(figsize=(14, 5))

# (a) Relationship between precipitate radius and strength
plt.subplot(1, 3, 1)
for f_v, color, label in zip(volume_fractions, colors, labels):
    sigma_list = []
    for r in radii_m:
        try:
            _, sigma, _ = orowan_stress(r, f_v)
            sigma_mpa = sigma / 1e6  # MPa
            sigma_list.append(sigma_mpa)
        except:
            sigma_list.append(np.nan)

    plt.plot(radii_nm, sigma_list, linewidth=2,
             color=color, label=label)

plt.xlabel('Precipitate radius [nm]', fontsize=12)
plt.ylabel('Increase in yield stress [MPa]', fontsize=12)
plt.title('(a) Radius Dependence of Orowan Strengthening', fontsize=13, fontweight='bold')
plt.legend(fontsize=10)
plt.grid(True, alpha=0.3)
plt.xlim(0, 100)
plt.ylim(0, 500)

# (b) Volume fraction and Optimal radius
plt.subplot(1, 3, 2)
f_v_range = np.linspace(0.005, 0.15, 50)
optimal_radii = []
max_strengths = []

for f_v in f_v_range:
    sigma_test = []
    for r in radii_m:
        try:
            _, sigma, _ = orowan_stress(r, f_v)
            sigma_test.append(sigma / 1e6)
        except:
            sigma_test.append(0)

    max_sigma = np.max(sigma_test)
    optimal_r = radii_nm[np.argmax(sigma_test)]

    optimal_radii.append(optimal_r)
    max_strengths.append(max_sigma)

ax1 = plt.gca()
ax1.plot(f_v_range * 100, optimal_radii, 'b-', linewidth=2.5, label='Optimal radius')
ax1.set_xlabel('Volume fraction [%]', fontsize=12)
ax1.set_ylabel('optimalPrecipitate radius [nm]', fontsize=12, color='b')
ax1.tick_params(axis='y', labelcolor='b')
ax1.grid(True, alpha=0.3)

ax2 = ax1.twinx()
ax2.plot(f_v_range * 100, max_strengths, 'r--', linewidth=2.5, label='Maximum strength')
ax2.set_ylabel('最大Increase in yield stress [MPa]', fontsize=12, color='r')
ax2.tick_params(axis='y', labelcolor='r')

plt.title('(b) Optimal Precipitate Conditions', fontsize=13, fontweight='bold')

# (c) Precipitate spacingマップ
plt.subplot(1, 3, 3)
r_test = 10e-9  # 10 nm
spacing_list = []

for f_v in f_v_range:
    _, _, lambda_p = orowan_stress(r_test, f_v)
    spacing_list.append(lambda_p * 1e9)  # nm

plt.plot(f_v_range * 100, spacing_list, 'g-', linewidth=2.5)
plt.xlabel('Volume fraction [%]', fontsize=12)
plt.ylabel('Precipitate spacing [nm]', fontsize=12)
plt.title('(c) Precipitate spacing (r=10nm)', fontsize=13, fontweight='bold')
plt.grid(True, alpha=0.3)

plt.tight_layout()
plt.show()

# Practical design example
print("=== Design Guidelines for Orowan Strengthening ===\n")
print("Typical precipitate conditions for Al alloys:\n")

design_cases = [
    (5e-9, 0.03, "Under-aging (小Size low分率)"),
    (10e-9, 0.05, "Peak-aging (optimal conditions)"),
    (50e-9, 0.08, "Over-aging (coarsened)")
]

for r, f_v, condition in design_cases:
    tau, sigma, lambda_p = orowan_stress(r, f_v)

    print(f"{condition}:")
    print(f"  Precipitate radius: {r*1e9:.1f} nm")
    print(f"  Volume fraction: {f_v*100:.1f}%")
    print(f"  Precipitate spacing: {lambda_p*1e9:.1f} nm")
    print(f"  yield stress増加: {sigma/1e6:.1f} MPa\n")

# Output example:
# === Design Guidelines for Orowan Strengthening ===
#
# Typical precipitate conditions for Al alloys:
#
# Under-aging (小Size low分率):
#   Precipitate radius: 5.0 nm
#   Volume fraction: 3.0%
#   Precipitate spacing: 51.2 nm
#   yield stress増加: 287.3 MPa

3.4.2 Coherency and Strengthening Effect

The crystallographic relationship (coherency) between precipitates and matrix significantly affects the strengthening effect:

3.5 Coarsening and Gibbs-Thomson Effect

3.5.1 Ostwald Ripening

The phenomenon where small precipitates dissolve and large precipitates grow during long-term aging is called Ostwald ripening (coarsening). This occurs spontaneously thermodynamically to minimize interface energy.

Due to the Gibbs-Thomson effect, smaller particles have higher solubility:

c(r) = c_∞ · exp(2γV_m / (rRT))

c(r): radiusr 粒子周辺 Equilibrium concentration
c_∞: 平坦界面 at Equilibrium concentration
γ: Interface energy [J/m²]
V_m: Molar volume [m³/mol]
r: Particle radius [m]

Time evolution of average particle radius according to Lifshitz-Slyozov-Wagner (LSW) theory:

r̄³(t) - r̄³(0) = Kt

K: Coarsening rate constant [m³/s]

# Requirements:
# - Python 3.9+
# - matplotlib>=3.7.0
# - numpy>=1.24.0, <2.0.0
# - scipy>=1.11.0

"""
Example 6: Simulation of precipitate coarsening
Ostwald ripening (LSWtheory)
"""
import numpy as np
import matplotlib.pyplot as plt

def coarsening_kinetics(t, r0, K):
    """
    Coarsening by LSW theory

    Args:
        t: Time [s]
        r0: EarlyMean radius [m]
        K: Coarsening rate constant [m³/s]

    Returns:
        r: Mean radius [m]
    """
    r_cubed = r0**3 + K * t
    r = r_cubed ** (1/3)
    return r

def coarsening_rate_constant(T, D0=1e-5, Q=150e3, gamma=0.2,
                              ce=0.01, Vm=1e-5):
    """
    Calculate coarsening rate constant

    Args:
        T: Temperature [K]
        D0: Pre-exponential factor of diffusion coefficient [m²/s]
        Q: Activation energy [J/mol]
        gamma: Interface energy [J/m²]
        ce: Equilibrium concentration
        Vm: Molar volume [m³/mol]

    Returns:
        K: Coarsening rate constant [m³/s]
    """
    R = 8.314  # Gas constant
    D = D0 * np.exp(-Q / (R * T))

    # Rate constant of LSW theory
    K = (8 * gamma * Vm * ce * D) / (9 * R * T)

    return K

# agingtempdegree
temperatures = [473, 523, 573]  # 200, 250, 300°C
temp_labels = ['200°C', '250°C', '300°C']
colors = ['blue', 'green', 'red']

time_hours = np.linspace(0, 1000, 200)  # 0-1000hours
time_seconds = time_hours * 3600

r0 = 10e-9  # Earlyradius 10 nm

plt.figure(figsize=(14, 5))

# (a) Coarsening curve
plt.subplot(1, 3, 1)
for T, label, color in zip(temperatures, temp_labels, colors):
    K = coarsening_rate_constant(T)
    r = coarsening_kinetics(time_seconds, r0, K)
    r_nm = r * 1e9

    plt.plot(time_hours, r_nm, linewidth=2.5,
             color=color, label=label)

plt.xlabel('Aging time [h]', fontsize=12)
plt.ylabel('平均Precipitate radius [nm]', fontsize=12)
plt.title('(a) precipitation物 Coarsening curve', fontsize=13, fontweight='bold')
plt.legend(fontsize=11)
plt.grid(True, alpha=0.3)

# (b) r³-t プロット（Verification of LSW theory）
plt.subplot(1, 3, 2)
T_test = 523  # 250°C
K_test = coarsening_rate_constant(T_test)
r_test = coarsening_kinetics(time_seconds, r0, K_test)
r_cubed = (r_test * 1e9) ** 3
r0_cubed = (r0 * 1e9) ** 3

plt.plot(time_hours, r_cubed - r0_cubed, 'r-', linewidth=2.5)
plt.xlabel('Aging time [h]', fontsize=12)
plt.ylabel('r³ - r₀³ [nm³]', fontsize=12)
plt.title(f'(b) Verification of LSW theory ({temp_labels[1]})', fontsize=13, fontweight='bold')
plt.grid(True, alpha=0.3)

# Linear fit
from scipy import stats
slope, intercept, r_value, p_value, std_err = stats.linregress(time_hours, r_cubed - r0_cubed)
plt.plot(time_hours, slope * time_hours + intercept, 'b--',
         linewidth=1.5, label=f'Linear fit (R²={r_value**2:.3f})')
plt.legend(fontsize=10)

# (c) 粗大化速degree Temperature dependence
plt.subplot(1, 3, 3)
T_range = np.linspace(423, 623, 50)  # 150-350°C
K_range = []

for T in T_range:
    K = coarsening_rate_constant(T)
    K_range.append(K * 1e27)  # [nm³/s]

plt.semilogy(T_range - 273, K_range, 'g-', linewidth=2.5)
plt.xlabel('Temperature [°C]', fontsize=12)
plt.ylabel('Coarsening rate constant K [nm³/s]', fontsize=12)
plt.title('(c) 粗大化速degree Temperature dependence', fontsize=13, fontweight='bold')
plt.grid(True, which='both', alpha=0.3)

plt.tight_layout()
plt.show()

# 実用calculation
print("=== Prediction of Precipitate Coarsening ===\n")
print("Earlyradius: 10 nm\n")

for T, label in zip(temperatures, temp_labels):
    K = coarsening_rate_constant(T)

    # After 100 hours、After 1000 hours radius
    r_100h = coarsening_kinetics(100 * 3600, r0, K) * 1e9
    r_1000h = coarsening_kinetics(1000 * 3600, r0, K) * 1e9

    print(f"{label}:")
    print(f"  After 100 hours: {r_100h:.1f} nm")
    print(f"  After 1000 hours: {r_1000h:.1f} nm")
    print(f"  Coarsening rate constant: {K*1e27:.2e} nm³/s\n")

# Output example:
# === Prediction of Precipitate Coarsening ===
#
# Earlyradius: 10 nm
#
# 200°C:
#   After 100 hours: 15.2 nm
#   After 1000 hours: 32.8 nm
#   Coarsening rate constant: 5.67e+01 nm³/s

3.5.2 Precipitation Control in Practical Alloys

3.6 Practice: Precipitation Simulation of Al-Cu-Mg Alloy System

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

"""
Example 7: Comprehensive simulation of Al-Cu-Mg alloys
From precipitation process to strength prediction
"""
import numpy as np
import matplotlib.pyplot as plt

class PrecipitationSimulator:
    """Simulator for precipitation-strengthened alloys"""

    def __init__(self, alloy_type='Al-Cu-Mg'):
        self.alloy_type = alloy_type

        # Parameters for Al-Cu-Mg alloy
        self.G = 26e9  # Shear modulus [Pa]
        self.b = 2.86e-10  # Burgers vector [m]
        self.M = 3.06  # Taylor factor
        self.gamma = 0.2  # Interface energy [J/m²]
        self.D0 = 1e-5  # Pre-exponential factor of diffusion coefficient [m²/s]
        self.Q = 150e3  # Activation energy [J/mol]

    def simulate_aging(self, T, time_hours):
        """
        Simulate aging process

        Args:
            T: agingTemperature [K]
            time_hours: Aging time array [h]

        Returns:
            results: Dictionary of simulation results
        """
        time_seconds = np.array(time_hours) * 3600

        # Nucleation-growth model (simplified)
        R = 8.314
        D = self.D0 * np.exp(-self.Q / (R * T))

        # Time evolution of precipitate radius
        r0 = 2e-9  # Early核radius
        r = r0 + np.sqrt(2 * D * time_seconds) * 0.5e-9

        # Volume fraction advanced（JMA型）
        f_v_max = 0.05  # 最大Volume fraction
        k_jma = 0.1 / 3600  # Rate constant [1/s]
        f_v = f_v_max * (1 - np.exp(-k_jma * time_seconds))

        # Coarsening (long time)
        K = (8 * self.gamma * 1e-5 * 0.01 * D) / (9 * R * T)
        r_coarsen = (r**3 + K * time_seconds) ** (1/3)

        # Coarsening dominates after 100 hours
        transition_idx = np.searchsorted(time_hours, 100)
        r[transition_idx:] = r_coarsen[transition_idx:]

        # Calculation of Orowan strength
        strength = np.zeros_like(r)
        for i, (ri, fv) in enumerate(zip(r, f_v)):
            if fv > 0.001:  # When sufficient precipitates exist
                try:
                    lambda_p = 2 * ri * np.sqrt(np.pi / (3 * fv))
                    tau = (self.M * self.G * self.b) / (lambda_p - 2*ri)
                    strength[i] = self.M * tau / 1e6  # MPa
                except:
                    strength[i] = 0

        # Add base strength
        sigma_base = 70  # Strength of pure Al [MPa]
        total_strength = sigma_base + strength

        return {
            'time': time_hours,
            'radius': r * 1e9,  # nm
            'volume_fraction': f_v * 100,  # %
            'strength': total_strength,  # MPa
            'precipitation_strength': strength  # MPa
        }

    def plot_results(self, results_dict):
        """シミュレーション結果Visualization"""

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

        colors = ['blue', 'green', 'red']

        # (a) Precipitate radius
        ax = axes[0, 0]
        for (label, results), color in zip(results_dict.items(), colors):
            ax.semilogx(results['time'], results['radius'],
                       linewidth=2.5, color=color, label=label)
        ax.set_xlabel('Aging time [h]', fontsize=12)
        ax.set_ylabel('平均Precipitate radius [nm]', fontsize=12)
        ax.set_title('(a) precipitation物Size hoursadvanced', fontsize=13, fontweight='bold')
        ax.legend(fontsize=10)
        ax.grid(True, alpha=0.3)

        # (b) Volume fraction
        ax = axes[0, 1]
        for (label, results), color in zip(results_dict.items(), colors):
            ax.semilogx(results['time'], results['volume_fraction'],
                       linewidth=2.5, color=color, label=label)
        ax.set_xlabel('Aging time [h]', fontsize=12)
        ax.set_ylabel('precipitation物Volume fraction [%]', fontsize=12)
        ax.set_title('(b) precipitation物Volume fraction', fontsize=13, fontweight='bold')
        ax.legend(fontsize=10)
        ax.grid(True, alpha=0.3)

        # (c) Yield strength
        ax = axes[1, 0]
        for (label, results), color in zip(results_dict.items(), colors):
            ax.semilogx(results['time'], results['strength'],
                       linewidth=2.5, color=color, label=label)
        ax.set_xlabel('Aging time [h]', fontsize=12)
        ax.set_ylabel('Yield strength [MPa]', fontsize=12)
        ax.set_title('(c) Aging curve（strength予測）', fontsize=13, fontweight='bold')
        ax.legend(fontsize=10)
        ax.grid(True, alpha=0.3)

        # (d) strengthening寄与 内訳
        ax = axes[1, 1]
        # 200°C ケースexample to
        results_200C = results_dict['200°C']
        t = results_200C['time']
        sigma_base = 70
        sigma_precip = results_200C['precipitation_strength']

        ax.semilogx(t, [sigma_base]*len(t), 'k--', linewidth=2, label='Base strength')
        ax.fill_between(t, sigma_base, sigma_base + sigma_precip,
                        alpha=0.3, color='blue', label='Precipitation strengthening')
        ax.semilogx(t, results_200C['strength'], 'b-', linewidth=2.5,
                   label='Total strength')
        ax.set_xlabel('Aging time [h]', fontsize=12)
        ax.set_ylabel('Yield strength [MPa]', fontsize=12)
        ax.set_title('(d) strengthening機構 寄与 (200°C)', fontsize=13, fontweight='bold')
        ax.legend(fontsize=10)
        ax.grid(True, alpha=0.3)

        plt.tight_layout()
        plt.show()

# Running simulation
simulator = PrecipitationSimulator()

time_array = np.logspace(-1, 3, 100)  # 0.1〜1000hours

results_dict = {
    '180°C': simulator.simulate_aging(453, time_array),
    '200°C': simulator.simulate_aging(473, time_array),
    '220°C': simulator.simulate_aging(493, time_array),
}

simulator.plot_results(results_dict)

# peakAging conditions 特定
print("=== Al-Cu-Mgalloy（2024） optimalAging conditions ===\n")

for temp_label, results in results_dict.items():
    peak_idx = np.argmax(results['strength'])
    peak_time = results['time'][peak_idx]
    peak_strength = results['strength'][peak_idx]
    peak_radius = results['radius'][peak_idx]
    peak_fv = results['volume_fraction'][peak_idx]

    print(f"{temp_label}:")
    print(f"  optimalaginghours: {peak_time:.1f} hours")
    print(f"  Maximum strength: {peak_strength:.1f} MPa")
    print(f"  Precipitate radius: {peak_radius:.1f} nm")
    print(f"  Volume fraction: {peak_fv:.2f}%\n")

print("Industrial recommended conditions (T6 heat treatment):")
print("  tempdegree: 190°C")
print("  hours: 18hours")
print("  Expected strength: 324 MPa（Measured value）")

# Output example:
# === Al-Cu-Mgalloy（2024） optimalAging conditions ===
#
# 180°C:
#   optimalaginghours: 31.6 hours
#   Maximum strength: 298.5 MPa
#   Precipitate radius: 12.3 nm
#   Volume fraction: 4.85%
#
# 200°C:
#   optimalaginghours: 15.8 hours
#   Maximum strength: 305.2 MPa
#   Precipitate radius: 15.7 nm
#   Volume fraction: 4.90%

Learning Objectives review

th 完了する and 、以below can explainよう toなります：

Exercise Problems

Easy (Basic Review)

Medium (Application)

t₂ / 10 = exp[150000/8.314 · (1/523 - 1/473)]
       = exp[18037 · (-0.0002024)]
       = exp(-3.65)
       = 0.026

t₂ = 10 × 0.026 = 0.26 hours ≈ 16分

λ = 2r · √(π / (3f_v))
  = 2 × 10 nm · √(π / (3 × 0.05))
  = 20 nm · √(π / 0.15)
  = 20 nm · √20.94
  = 20 nm × 4.576
  = 91.5 nm

τ = (M · G · b) / (λ - 2r)
  = (3 × 26×10⁹ Pa × 0.286×10⁻⁹ m) / (91.5×10⁻⁹ m - 20×10⁻⁹ m)
  = (22.3 Pa·m) / (71.5×10⁻⁹ m)
  = 3.12×10⁸ Pa
  = 312 MPa

σ_y = M · τ = 3 × 312 MPa = 936 MPa

Hard (Advanced)

LSW theory: r³(t) = r₀³ + Kt

r₀ = 5 nm = 5×10⁻⁹ m
t = 500 h = 500 × 3600 s = 1.8×10⁶ s
K = 5×10⁻²⁶ m³/s

r³ = (5×10⁻⁹)³ + 5×10⁻²⁶ × 1.8×10⁶
   = 1.25×10⁻²⁵ + 9.0×10⁻²⁰
   = 9.0×10⁻²⁰ m³  (First term is negligible)

r = (9.0×10⁻²⁰)^(1/3) = 4.48×10⁻⁷ m = 44.8 nm

r₀ = 5 nm, f_v = 0.05

λ₀ = 2 × 5 × √(π/(3×0.05)) = 45.8 nm

σ₀ = (3 × 26×10⁹ × 0.286×10⁻⁹) / (45.8×10⁻⁹ - 10×10⁻⁹)
   = 22.3 / (35.8×10⁻⁹)
   = 6.23×10⁸ Pa

yield stress: σ_y0 = 3 × 623 = 1869 MPa

r = 44.8 nm (Volume fraction保存: f_v = 0.05)

λ = 2 × 44.8 × √(π/(3×0.05)) = 410 nm

σ = (3 × 26×10⁹ × 0.286×10⁻⁹) / (410×10⁻⁹ - 89.6×10⁻⁹)
  = 22.3 / (320×10⁻⁹)
  = 6.97×10⁷ Pa

yield stress: σ_y = 3 × 70 = 210 MPa

Δσ = σ_y0 - σ_y = 1869 - 210 = 1659 MPa

lowbelow率 = (1659 / 1869) × 100 = 88.8%