Chapter 4: Size-Dependent Properties - Introduction to Nanomaterials

Learning Objectives

Understand localized surface plasmon resonance (LSPR) in metal nanoparticles
Explain quantum dot optical properties and size-tunable emission
Analyze superparamagnetism in magnetic nanoparticles
Evaluate size effects on catalytic activity and selectivity
Apply knowledge to predict nanomaterial behavior for specific applications

4.1 Optical Properties

4.1.1 Localized Surface Plasmon Resonance (LSPR)

When light interacts with metal nanoparticles smaller than the wavelength of light, it induces collective oscillation of conduction electrons. This phenomenon, called localized surface plasmon resonance (LSPR), results in strong light absorption and scattering at specific wavelengths.

Surface Plasmon

A coherent oscillation of free electrons at a metal-dielectric interface. For nanoparticles, the resonance condition depends on particle size, shape, composition, and the surrounding medium's dielectric constant.

Mie Theory (Dipole Approximation)

For spherical particles much smaller than the wavelength (d << λ), the extinction cross-section is:

\[ C_{ext} = \frac{24\pi^2 R^3 \varepsilon_m^{3/2}}{\lambda} \cdot \frac{\varepsilon_2}{(\varepsilon_1 + 2\varepsilon_m)^2 + \varepsilon_2^2} \]

where \(\varepsilon_1, \varepsilon_2\) are real and imaginary parts of the metal dielectric function, and \(\varepsilon_m\) is the medium dielectric constant.

Python Example: Gold Nanoparticle LSPR Simulation


import numpy as np
import matplotlib.pyplot as plt

def gold_dielectric(wavelength_nm):
    """
    Drude-Lorentz model for gold dielectric function.

    Returns complex dielectric function ε = ε1 + i*ε2
    """
    # Gold parameters
    eps_inf = 9.5      # High-frequency dielectric constant
    omega_p = 9.0      # Plasma frequency (eV)
    gamma = 0.07       # Damping constant (eV)

    # Convert wavelength to energy
    hc = 1239.8  # eV·nm
    omega = hc / wavelength_nm

    # Drude model
    eps = eps_inf - omega_p**2 / (omega**2 + 1j * gamma * omega)

    return eps

def mie_extinction(wavelength_nm, radius_nm, n_medium=1.33):
    """
    Calculate extinction cross-section using Mie theory (dipole approximation).

    Parameters:
    -----------
    wavelength_nm : array
        Wavelengths in nm
    radius_nm : float
        Particle radius in nm
    n_medium : float
        Refractive index of surrounding medium

    Returns:
    --------
    array : Extinction cross-section (normalized)
    """
    eps_m = n_medium**2
    eps_metal = gold_dielectric(wavelength_nm)
    eps_1 = np.real(eps_metal)
    eps_2 = np.imag(eps_metal)

    # Extinction cross-section
    numerator = eps_2
    denominator = (eps_1 + 2*eps_m)**2 + eps_2**2

    C_ext = (24 * np.pi**2 * radius_nm**3 * eps_m**1.5 / wavelength_nm) * \
            (numerator / denominator)

    return C_ext / np.max(C_ext)  # Normalize

# Calculate LSPR for different sizes
wavelengths = np.linspace(400, 800, 500)

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

# Plot 1: Size dependence
plt.subplot(1, 2, 1)
radii = [5, 10, 20, 40]
colors = plt.cm.plasma(np.linspace(0.2, 0.8, len(radii)))

for r, c in zip(radii, colors):
    ext = mie_extinction(wavelengths, r)
    peak_idx = np.argmax(ext)
    peak_wl = wavelengths[peak_idx]
    plt.plot(wavelengths, ext, color=c, linewidth=2,
             label=f'r = {r} nm (λ_max = {peak_wl:.0f} nm)')

plt.xlabel('Wavelength (nm)')
plt.ylabel('Normalized Extinction')
plt.title('LSPR of Gold Nanoparticles: Size Effect')
plt.legend()
plt.grid(True, alpha=0.3)

# Plot 2: Medium dependence
plt.subplot(1, 2, 2)
media = [(1.0, 'Air'), (1.33, 'Water'), (1.5, 'Glass'), (1.7, 'High-n polymer')]
colors = plt.cm.viridis(np.linspace(0.2, 0.8, len(media)))

for (n, name), c in zip(media, colors):
    ext = mie_extinction(wavelengths, 20, n_medium=n)
    peak_idx = np.argmax(ext)
    peak_wl = wavelengths[peak_idx]
    plt.plot(wavelengths, ext, color=c, linewidth=2,
             label=f'{name} (n={n}, λ={peak_wl:.0f} nm)')

plt.xlabel('Wavelength (nm)')
plt.ylabel('Normalized Extinction')
plt.title('LSPR of 20 nm Au NPs: Medium Effect')
plt.legend()
plt.grid(True, alpha=0.3)

plt.tight_layout()
plt.savefig('lspr_simulation.png', dpi=150)
plt.show()

# LSPR peak position vs size
print("\nLSPR Peak Wavelength vs Particle Size:")
print("-" * 40)
for r in [5, 10, 15, 20, 30, 40, 50]:
    ext = mie_extinction(wavelengths, r)
    peak_wl = wavelengths[np.argmax(ext)]
    print(f"Radius = {r:2d} nm: λ_LSPR = {peak_wl:.1f} nm")

4.1.2 Quantum Dot Photoluminescence

Quantum dots (QDs) exhibit size-tunable photoluminescence due to quantum confinement. Smaller dots emit higher energy (blue) light, while larger dots emit lower energy (red) light.

Quantum Dot Properties

High quantum yield: Up to 90% for core-shell structures
Narrow emission: FWHM of 20-40 nm vs 50-100 nm for organic dyes
Photostability: Resistant to photobleaching
Broad absorption: Can be excited by multiple wavelengths
Large Stokes shift: Separation between absorption and emission

Python Example: Quantum Dot Emission Spectra


import numpy as np
import matplotlib.pyplot as plt

def quantum_dot_emission(wavelength_nm, peak_nm, fwhm_nm=30, intensity=1.0):
    """
    Gaussian emission spectrum for quantum dot.
    """
    sigma = fwhm_nm / (2 * np.sqrt(2 * np.log(2)))
    return intensity * np.exp(-(wavelength_nm - peak_nm)**2 / (2 * sigma**2))

def size_to_emission_wavelength(diameter_nm, material='CdSe'):
    """
    Empirical relationship between QD size and emission wavelength.
    Based on experimental calibration curves.
    """
    if material == 'CdSe':
        # Empirical formula for CdSe QDs
        wavelength = 350 + 50 * diameter_nm - 1.5 * diameter_nm**2
        wavelength = max(400, min(700, wavelength))
    elif material == 'InP':
        wavelength = 450 + 40 * diameter_nm
        wavelength = max(500, min(750, wavelength))
    return wavelength

def wavelength_to_rgb(wavelength):
    """Convert wavelength to approximate RGB color."""
    if wavelength < 440:
        r, g, b = (440-wavelength)/(440-380), 0, 1
    elif wavelength < 490:
        r, g, b = 0, (wavelength-440)/(490-440), 1
    elif wavelength < 510:
        r, g, b = 0, 1, (510-wavelength)/(510-490)
    elif wavelength < 580:
        r, g, b = (wavelength-510)/(580-510), 1, 0
    elif wavelength < 645:
        r, g, b = 1, (645-wavelength)/(645-580), 0
    else:
        r, g, b = 1, 0, 0

    return (min(1, max(0, r)), min(1, max(0, g)), min(1, max(0, b)))

# Generate emission spectra for different sizes
wavelengths = np.linspace(400, 750, 500)
sizes = [2.5, 3.5, 4.5, 5.5, 6.5]

fig, axes = plt.subplots(1, 2, figsize=(14, 5))

# Plot 1: Emission spectra
ax1 = axes[0]
for size in sizes:
    peak = size_to_emission_wavelength(size)
    emission = quantum_dot_emission(wavelengths, peak, fwhm_nm=30)
    color = wavelength_to_rgb(peak)
    ax1.fill_between(wavelengths, emission, alpha=0.6, color=color,
                     label=f'd = {size} nm (λ = {peak:.0f} nm)')
    ax1.plot(wavelengths, emission, color=color, linewidth=1)

ax1.set_xlabel('Wavelength (nm)')
ax1.set_ylabel('Emission Intensity (a.u.)')
ax1.set_title('CdSe Quantum Dot Size-Tunable Emission')
ax1.legend()
ax1.grid(True, alpha=0.3)
ax1.set_xlim(400, 750)

# Plot 2: Size-wavelength relationship
ax2 = axes[1]
diameters = np.linspace(2, 8, 50)
wavelengths_peak = [size_to_emission_wavelength(d) for d in diameters]

# Color each point by its emission color
for i in range(len(diameters)-1):
    color = wavelength_to_rgb(wavelengths_peak[i])
    ax2.plot(diameters[i:i+2], wavelengths_peak[i:i+2],
             color=color, linewidth=4)

ax2.set_xlabel('Quantum Dot Diameter (nm)')
ax2.set_ylabel('Emission Wavelength (nm)')
ax2.set_title('Size-Wavelength Calibration Curve')
ax2.grid(True, alpha=0.3)

# Add color bar representation
for wl in [450, 500, 550, 600, 650, 700]:
    ax2.axhline(wl, color=wavelength_to_rgb(wl), alpha=0.3, linestyle='--')

plt.tight_layout()
plt.savefig('quantum_dot_emission.png', dpi=150)
plt.show()

4.2 Magnetic Properties

4.2.1 Superparamagnetism

Below a critical size, magnetic nanoparticles become superparamagnetic: they exhibit strong magnetization in an applied field but have zero remanence when the field is removed. This occurs when thermal energy can overcome the magnetic anisotropy energy barrier.

Superparamagnetic Blocking Temperature

The blocking temperature \(T_B\) is the temperature below which a particle appears ferromagnetic on the measurement timescale:

\[ T_B = \frac{K_V V}{k_B \ln(\tau_m / \tau_0)} \]

where \(K_V\) is the magnetic anisotropy constant, \(V\) is particle volume, \(\tau_m\) is measurement time (~100 s for DC magnetization), and \(\tau_0 \approx 10^{-9}\) s is the attempt time.

Python Example: Superparamagnetic Nanoparticles


import numpy as np
import matplotlib.pyplot as plt

def blocking_temperature(diameter_nm, K_V, tau_m=100):
    """
    Calculate superparamagnetic blocking temperature.

    Parameters:
    -----------
    diameter_nm : float
        Particle diameter in nm
    K_V : float
        Anisotropy constant (J/m³)
    tau_m : float
        Measurement time (s)

    Returns:
    --------
    float : Blocking temperature (K)
    """
    kB = 1.381e-23  # J/K
    tau_0 = 1e-9    # Attempt time (s)

    r = diameter_nm * 1e-9 / 2
    V = (4/3) * np.pi * r**3

    TB = K_V * V / (kB * np.log(tau_m / tau_0))
    return TB

def langevin_function(x):
    """Langevin function L(x) = coth(x) - 1/x"""
    # Handle x = 0 case
    result = np.zeros_like(x)
    mask = np.abs(x) > 1e-6
    result[mask] = 1/np.tanh(x[mask]) - 1/x[mask]
    return result

def superparamagnetic_magnetization(H, T, diameter_nm, Ms, K_V):
    """
    Calculate magnetization of superparamagnetic nanoparticles.

    M = Ms * L(μH/kT) where μ = Ms * V
    """
    kB = 1.381e-23
    mu0 = 4 * np.pi * 1e-7

    r = diameter_nm * 1e-9 / 2
    V = (4/3) * np.pi * r**3
    mu = Ms * V  # Magnetic moment

    x = mu0 * mu * H / (kB * T)
    M = Ms * langevin_function(x)

    return M

# Material parameters
materials = {
    'Fe3O4': {'Ms': 4.8e5, 'K_V': 1.35e4, 'name': 'Magnetite'},
    'gamma-Fe2O3': {'Ms': 3.9e5, 'K_V': 4.7e3, 'name': 'Maghemite'},
    'CoFe2O4': {'Ms': 4.2e5, 'K_V': 2.0e5, 'name': 'Cobalt Ferrite'},
}

# Plot 1: Blocking temperature vs size
plt.figure(figsize=(14, 5))

plt.subplot(1, 2, 1)
diameters = np.linspace(3, 30, 100)

for mat_id, props in materials.items():
    TB = [blocking_temperature(d, props['K_V']) for d in diameters]
    plt.plot(diameters, TB, linewidth=2, label=props['name'])

plt.axhline(300, color='gray', linestyle='--', label='Room temperature')
plt.xlabel('Particle Diameter (nm)')
plt.ylabel('Blocking Temperature (K)')
plt.title('Superparamagnetic Blocking Temperature')
plt.legend()
plt.grid(True, alpha=0.3)
plt.ylim(0, 500)

# Plot 2: M-H curves at different temperatures
plt.subplot(1, 2, 2)
H = np.linspace(-1e6, 1e6, 500)  # A/m
temperatures = [10, 100, 300]
diameter = 10  # nm
mat = materials['Fe3O4']

for T in temperatures:
    M = superparamagnetic_magnetization(H, T, diameter, mat['Ms'], mat['K_V'])
    plt.plot(H/1e6, M/mat['Ms'], linewidth=2, label=f'T = {T} K')

plt.xlabel('Magnetic Field (MA/m)')
plt.ylabel('M/Ms')
plt.title(f'M-H Curves: {diameter} nm Fe₃O₄ Nanoparticles')
plt.legend()
plt.grid(True, alpha=0.3)
plt.xlim(-1, 1)

plt.tight_layout()
plt.savefig('superparamagnetism.png', dpi=150)
plt.show()

# Print critical sizes
print("\nCritical Superparamagnetic Size at 300 K:")
print("-" * 50)
for mat_id, props in materials.items():
    # Find diameter where TB = 300 K
    for d in np.linspace(1, 50, 1000):
        if blocking_temperature(d, props['K_V']) >= 300:
            print(f"{props['name']}: d_critical = {d:.1f} nm")
            break

4.3 Catalytic Properties

Nanoparticles exhibit enhanced catalytic activity due to their high surface-to-volume ratio and the presence of under-coordinated surface atoms with higher reactivity.

4.3.1 Size Effects in Catalysis

Size Effect	Mechanism	Examples
Geometric	Fraction of edge, corner, terrace atoms	CO oxidation on Au, structure-sensitive reactions
Electronic	Quantum size effects, d-band shifts	ORR activity on Pt, bandgap-dependent photocatalysis
Support interaction	Metal-support electronic coupling	Au/TiO2, Pt/CeO2 for water-gas shift

Python Example: Surface Atom Fraction and Catalytic Activity


import numpy as np
import matplotlib.pyplot as plt

def cuboctahedral_atom_counts(n_shells):
    """
    Calculate number of atoms in different positions for cuboctahedral cluster.

    Parameters:
    -----------
    n_shells : int
        Number of complete shells (1 = 13 atoms, 2 = 55 atoms, etc.)

    Returns:
    --------
    dict : Counts of corner, edge, face, and core atoms
    """
    # Magic numbers for cuboctahedra
    total = 10 * n_shells**3 // 3 + 5 * n_shells**2 + 11 * n_shells // 3 + 1

    if n_shells == 0:
        return {'total': 1, 'corner': 1, 'edge': 0, 'face': 0, 'core': 0}

    # Surface atoms
    corner = 12  # Always 12 for cuboctahedron
    edge = 24 * (n_shells - 1)
    face_square = 6 * (n_shells - 1)**2
    face_triangle = 8 * ((n_shells - 1) * (n_shells - 2) // 2) if n_shells > 1 else 0

    surface = corner + edge + face_square + face_triangle
    core = total - surface

    return {
        'total': total,
        'corner': corner,
        'edge': edge,
        'face': face_square + face_triangle,
        'core': max(0, core),
        'surface': surface
    }

def diameter_from_shells(n_shells, atom_radius_nm=0.144):
    """Estimate cluster diameter from number of shells."""
    return 2 * atom_radius_nm * (2 * n_shells + 1)

def turnover_frequency(diameter_nm, TOF_terrace=1, TOF_edge=5, TOF_corner=10):
    """
    Calculate effective TOF based on surface site distribution.

    Assumes different intrinsic activities for different sites.
    """
    # Estimate number of shells from diameter
    n_shells = max(1, int(diameter_nm / 0.5))

    counts = cuboctahedral_atom_counts(n_shells)
    if counts['surface'] == 0:
        return TOF_terrace

    # Weighted average TOF
    f_corner = counts['corner'] / counts['surface']
    f_edge = counts['edge'] / counts['surface']
    f_face = counts['face'] / counts['surface']

    TOF_eff = f_corner * TOF_corner + f_edge * TOF_edge + f_face * TOF_terrace

    return TOF_eff

# Calculate for different cluster sizes
shells = list(range(1, 15))
diameters = []
surface_fractions = []
corner_fractions = []
edge_fractions = []
tof_values = []

for n in shells:
    counts = cuboctahedral_atom_counts(n)
    d = diameter_from_shells(n)
    diameters.append(d)
    surface_fractions.append(counts['surface'] / counts['total'])
    corner_fractions.append(counts['corner'] / counts['surface'])
    edge_fractions.append(counts['edge'] / counts['surface'])
    tof_values.append(turnover_frequency(d))

# Visualization
fig, axes = plt.subplots(2, 2, figsize=(12, 10))

# Plot 1: Surface fraction vs size
ax1 = axes[0, 0]
ax1.plot(diameters, [f*100 for f in surface_fractions], 'bo-', linewidth=2)
ax1.set_xlabel('Cluster Diameter (nm)')
ax1.set_ylabel('Surface Atoms (%)')
ax1.set_title('Surface Atom Fraction vs Size')
ax1.grid(True, alpha=0.3)

# Plot 2: Site distribution
ax2 = axes[0, 1]
ax2.stackplot(diameters,
              [f*100 for f in corner_fractions],
              [f*100 for f in edge_fractions],
              [100 - 100*c - 100*e for c, e in zip(corner_fractions, edge_fractions)],
              labels=['Corner', 'Edge', 'Face'], alpha=0.7)
ax2.set_xlabel('Cluster Diameter (nm)')
ax2.set_ylabel('Site Fraction (%)')
ax2.set_title('Surface Site Distribution')
ax2.legend(loc='upper right')
ax2.set_xlim(min(diameters), max(diameters))

# Plot 3: Effective TOF vs size
ax3 = axes[1, 0]
ax3.plot(diameters, tof_values, 'ro-', linewidth=2)
ax3.axhline(1, color='gray', linestyle='--', label='Terrace TOF')
ax3.set_xlabel('Cluster Diameter (nm)')
ax3.set_ylabel('Effective TOF (relative)')
ax3.set_title('Catalytic Activity vs Size\n(Structure-Sensitive Reaction)')
ax3.legend()
ax3.grid(True, alpha=0.3)

# Plot 4: Mass activity
ax4 = axes[1, 1]
mass_activity = [tof * sf for tof, sf in zip(tof_values, surface_fractions)]
ax4.plot(diameters, mass_activity, 'g^-', linewidth=2)
ax4.set_xlabel('Cluster Diameter (nm)')
ax4.set_ylabel('Mass Activity (a.u.)')
ax4.set_title('Mass-Normalized Catalytic Activity')
ax4.grid(True, alpha=0.3)

plt.tight_layout()
plt.savefig('catalytic_size_effects.png', dpi=150)
plt.show()

# Print data table
print("\nCuboctahedral Cluster Properties:")
print("-" * 70)
print(f"{'Shells':<8} {'Atoms':<8} {'Diameter':<10} {'Surface %':<12} {'Corner %':<10}")
print("-" * 70)
for n in [1, 2, 3, 5, 7, 10]:
    counts = cuboctahedral_atom_counts(n)
    d = diameter_from_shells(n)
    sf = counts['surface'] / counts['total'] * 100
    cf = counts['corner'] / counts['surface'] * 100 if counts['surface'] > 0 else 0
    print(f"{n:<8} {counts['total']:<8} {d:.2f} nm{'':<4} {sf:.1f}%{'':<6} {cf:.1f}%")

4.4 Thermal Properties

Nanomaterials exhibit modified thermal properties due to surface effects and phonon confinement. Key phenomena include:

Melting point depression: Surface atoms have lower binding energy (Gibbs-Thomson effect)
Reduced thermal conductivity: Phonon scattering at boundaries
Enhanced specific heat: Surface phonon contributions
Modified phase transitions: Size-dependent transition temperatures

Python Example: Phonon Mean Free Path and Thermal Conductivity


import numpy as np
import matplotlib.pyplot as plt

def thermal_conductivity_nanostructure(d_nm, kappa_bulk, mfp_bulk):
    """
    Calculate thermal conductivity reduction in nanostructures.

    Uses simple boundary scattering model:
    κ/κ_bulk = d / (d + 3*mfp_bulk)

    Parameters:
    -----------
    d_nm : float
        Characteristic dimension (nm)
    kappa_bulk : float
        Bulk thermal conductivity (W/m·K)
    mfp_bulk : float
        Bulk phonon mean free path (nm)

    Returns:
    --------
    float : Nanostructure thermal conductivity (W/m·K)
    """
    ratio = d_nm / (d_nm + 3 * mfp_bulk)
    return kappa_bulk * ratio

# Material properties
materials = {
    'Si': {'kappa_bulk': 148, 'mfp': 300, 'color': 'blue'},
    'Ge': {'kappa_bulk': 60, 'mfp': 200, 'color': 'red'},
    'GaAs': {'kappa_bulk': 55, 'mfp': 100, 'color': 'green'},
}

dimensions = np.logspace(0, 4, 100)  # 1 to 10000 nm

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

# Plot 1: Thermal conductivity vs size
plt.subplot(1, 2, 1)
for mat, props in materials.items():
    kappa = [thermal_conductivity_nanostructure(d, props['kappa_bulk'], props['mfp'])
             for d in dimensions]
    plt.semilogx(dimensions, kappa, color=props['color'], linewidth=2,
                 label=f'{mat} (bulk: {props["kappa_bulk"]} W/m·K)')

plt.xlabel('Characteristic Dimension (nm)')
plt.ylabel('Thermal Conductivity (W/m·K)')
plt.title('Size-Dependent Thermal Conductivity')
plt.legend()
plt.grid(True, alpha=0.3)

# Plot 2: Reduction ratio
plt.subplot(1, 2, 2)
for mat, props in materials.items():
    ratio = [thermal_conductivity_nanostructure(d, props['kappa_bulk'], props['mfp']) /
             props['kappa_bulk'] for d in dimensions]
    plt.semilogx(dimensions, [r*100 for r in ratio], color=props['color'],
                 linewidth=2, label=mat)

plt.axhline(50, color='gray', linestyle='--', alpha=0.5)
plt.xlabel('Characteristic Dimension (nm)')
plt.ylabel('κ / κ_bulk (%)')
plt.title('Thermal Conductivity Reduction')
plt.legend()
plt.grid(True, alpha=0.3)

plt.tight_layout()
plt.savefig('thermal_conductivity.png', dpi=150)
plt.show()

4.5 Summary

Key Takeaways

LSPR: Metal nanoparticles exhibit size- and shape-dependent plasmon resonance; used in sensing and photothermal applications
Quantum dots: Size-tunable emission from quantum confinement; applications in displays, imaging, and solar cells
Superparamagnetism: Zero remanence below critical size; crucial for biomedical applications and data storage
Catalysis: High surface area and under-coordinated sites enhance activity; size affects selectivity
Thermal properties: Reduced thermal conductivity from boundary scattering; useful for thermoelectrics

Exercises

Exercise 1: Biosensor Design

Design a gold nanoparticle-based colorimetric biosensor. What particle size would you choose for maximum sensitivity to refractive index changes? How would the LSPR shift upon analyte binding?

Exercise 2: Magnetic Hyperthermia

For magnetic hyperthermia cancer treatment, you need Fe₃O₄ nanoparticles that are superparamagnetic at body temperature (37°C). What is the maximum allowable particle size? How would you optimize heating efficiency?

Exercise 3: Catalyst Optimization

A structure-sensitive reaction has TOF_corner = 100, TOF_edge = 20, and TOF_terrace = 1. At what particle size is mass activity maximized? Consider both surface fraction and site distribution.