Chapter 1: Fundamentals and Classification

1.1 Introduction: What are Nanomaterials?

Nanomaterials are materials with at least one dimension in the nanoscale range, typically defined as 1 to 100 nanometers (nm). At this scale, materials exhibit unique properties that differ dramatically from their bulk counterparts due to quantum mechanical effects and the dominance of surface phenomena.

To put the nanoscale in perspective: a nanometer is one-billionth of a meter (10^-9 m). A human hair is approximately 80,000-100,000 nm in diameter, a DNA helix is about 2 nm wide, and a typical atom is 0.1-0.3 nm in diameter.

1.1.1 Historical Development

While the term "nanotechnology" was coined by Norio Taniguchi in 1974 and popularized by K. Eric Drexler in the 1980s, humans have unknowingly used nanomaterials for centuries:

• 4th century CE: The Lycurgus Cup, a Roman artifact, contains gold and silver nanoparticles that cause it to change color when illuminated
• Medieval period: Stained glass windows contain metal nanoparticles for coloration
• Damascus steel: Carbon nanotube structures contributed to the legendary blade's strength
• 1857: Michael Faraday synthesized colloidal gold nanoparticles
• 1959: Richard Feynman's famous lecture "There's Plenty of Room at the Bottom"
• 1985: Discovery of fullerenes (C60) by Kroto, Curl, and Smalley
• 1991: Sumio Iijima's systematic study of carbon nanotubes
• 2004: Isolation of graphene by Geim and Novoselov

1.1.2 Why the Nanoscale is Special

Two fundamental phenomena make nanomaterials unique:

1.2 Classification by Dimensionality

Nanomaterials are classified based on the number of dimensions that are NOT in the nanoscale:

1.2.1 Zero-Dimensional (0D) Nanomaterials

0D nanomaterials are confined in all three spatial dimensions, with sizes typically ranging from 1 to 100 nm. The most important examples include:

• Metal nanoparticles: Au, Ag, Pt, Pd with applications in catalysis and sensing
• Oxide nanoparticles: TiO₂, ZnO, Fe₃O₄ for photocatalysis and medicine
• Fullerenes: C₆₀, C₇₀ cage-like carbon molecules
• Core-shell nanoparticles: Structures with different compositions at core and shell

1.2.2 One-Dimensional (1D) Nanomaterials

1D nanomaterials have two dimensions in the nanoscale while extending to microscale or macroscale in the third dimension:

• Carbon nanotubes (CNTs): Single-walled (SWCNT) and multi-walled (MWCNT) with exceptional mechanical and electrical properties
• Nanowires: Si, ZnO, Ag, Au for electronics and sensing
• Nanorods: CdSe, Au for optical applications
• Nanofibers: Polymeric and ceramic fibers for filtration and composites

1.2.3 Two-Dimensional (2D) Nanomaterials

2D nanomaterials are confined in one dimension (typically thickness of a few atoms to ~100 nm):

• Transition metal dichalcogenides (TMDCs): MoS₂, WS₂, MoSe₂ with tunable bandgaps
• Hexagonal boron nitride (h-BN): Insulating 2D material for heterostructures
• MXenes: Ti₃C₂, Ti₂C transition metal carbides/nitrides
• Phosphorene: 2D black phosphorus with anisotropic properties

1.2.4 Three-Dimensional (3D) Nanostructured Materials

3D nanostructured materials have bulk dimensions but contain nanoscale features:

• Aerogels: Ultra-low density porous materials with nanoscale pore structure
• Metal-organic frameworks (MOFs): Crystalline materials with nanoscale pores
• Zeolites: Microporous aluminosilicates
• Nanocomposites: Bulk materials reinforced with nanofillers

1.3 Surface-to-Volume Ratio

One of the most important characteristics of nanomaterials is their extraordinarily high surface-to-volume ratio. As particle size decreases, the fraction of atoms at the surface increases dramatically.

import numpy as np
import matplotlib.pyplot as plt

def surface_to_volume_ratio(diameter_nm):
    """Calculate surface-to-volume ratio for spherical particles."""
    return 6 / diameter_nm  # nm^-1

def surface_atom_fraction(diameter_nm, atom_diameter_nm=0.25):
    """
    Estimate fraction of atoms at the surface.
    Assumes atoms are arranged in a shell of thickness equal to atomic diameter.
    """
    r = diameter_nm / 2
    r_core = r - atom_diameter_nm
    if r_core < 0:
        return 1.0

    volume_total = (4/3) * np.pi * r**3
    volume_core = (4/3) * np.pi * r_core**3

    return (volume_total - volume_core) / volume_total

# Calculate for different particle sizes
diameters = np.logspace(0, 3, 100)  # 1 to 1000 nm
sv_ratios = [surface_to_volume_ratio(d) for d in diameters]
surface_fractions = [surface_atom_fraction(d) for d in diameters]

# Create visualization
fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(12, 5))

# Plot 1: Surface-to-volume ratio
ax1.loglog(diameters, sv_ratios, 'b-', linewidth=2)
ax1.set_xlabel('Particle Diameter (nm)')
ax1.set_ylabel('Surface-to-Volume Ratio (nm⁻¹)')
ax1.set_title('Surface-to-Volume Ratio vs. Particle Size')
ax1.grid(True, alpha=0.3)
ax1.axvline(x=10, color='r', linestyle='--', label='10 nm')
ax1.axvline(x=100, color='g', linestyle='--', label='100 nm')
ax1.legend()

# Plot 2: Surface atom fraction
ax2.semilogx(diameters, [f*100 for f in surface_fractions], 'r-', linewidth=2)
ax2.set_xlabel('Particle Diameter (nm)')
ax2.set_ylabel('Surface Atom Fraction (%)')
ax2.set_title('Fraction of Atoms at Surface')
ax2.grid(True, alpha=0.3)
ax2.axhline(y=50, color='gray', linestyle=':', label='50% surface atoms')
ax2.legend()

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

# Print example values
print("\nSurface Properties at Different Scales:")
print("-" * 50)
for d in [1, 5, 10, 50, 100, 1000]:
    sv = surface_to_volume_ratio(d)
    sf = surface_atom_fraction(d) * 100
    print(f"d = {d:4d} nm: S/V = {sv:.3f} nm⁻¹, Surface atoms = {sf:.1f}%")
        

1.4 Quantum Confinement

When a material's dimensions become comparable to the de Broglie wavelength of electrons (typically a few nanometers), quantum mechanical effects become significant. This phenomenon, called quantum confinement, leads to discrete energy levels rather than continuous energy bands.

1.4.1 The Brus Equation

For semiconductor nanocrystals (quantum dots), the size-dependent bandgap can be approximated by the Brus equation, which accounts for the kinetic energy of confined electrons and holes:

import numpy as np
import matplotlib.pyplot as plt

# Physical constants
hbar = 1.055e-34   # J·s
e = 1.602e-19      # C
m0 = 9.109e-31     # kg (electron mass)
eps0 = 8.854e-12   # F/m

def quantum_dot_bandgap(diameter_nm, material='CdSe'):
    """
    Calculate quantum dot bandgap using simplified Brus equation.

    Parameters:
    -----------
    diameter_nm : float
        Particle diameter in nanometers
    material : str
        Material type ('CdSe', 'CdTe', 'PbS', 'InP')

    Returns:
    --------
    float : Bandgap energy in eV
    """
    # Material parameters: (Eg_bulk eV, me*/m0, mh*/m0, epsilon)
    materials = {
        'CdSe': (1.74, 0.13, 0.45, 10.6),
        'CdTe': (1.50, 0.11, 0.35, 10.2),
        'PbS':  (0.41, 0.085, 0.085, 17.0),
        'InP':  (1.35, 0.077, 0.64, 12.4),
    }

    Eg_bulk, me_ratio, mh_ratio, epsilon = materials[material]

    d = diameter_nm * 1e-9  # Convert to meters
    me = me_ratio * m0
    mh = mh_ratio * m0

    # Confinement energy (kinetic term)
    E_conf = (hbar**2 * np.pi**2 / (2 * d**2)) * (1/me + 1/mh)

    # Coulomb correction
    E_coulomb = 1.8 * e**2 / (4 * np.pi * eps0 * epsilon * d)

    # Total bandgap in eV
    Eg = Eg_bulk + (E_conf - E_coulomb) / e

    return max(Eg, Eg_bulk)  # Bandgap cannot be less than bulk

# Calculate bandgap vs size for different materials
diameters = np.linspace(2, 15, 100)

plt.figure(figsize=(10, 6))
colors = {'CdSe': 'red', 'CdTe': 'blue', 'PbS': 'green', 'InP': 'purple'}

for material in ['CdSe', 'CdTe', 'PbS', 'InP']:
    bandgaps = [quantum_dot_bandgap(d, material) for d in diameters]
    plt.plot(diameters, bandgaps, color=colors[material],
             linewidth=2, label=material)

plt.xlabel('Quantum Dot Diameter (nm)')
plt.ylabel('Bandgap Energy (eV)')
plt.title('Size-Dependent Bandgap in Quantum Dots (Brus Equation)')
plt.legend()
plt.grid(True, alpha=0.3)
plt.xlim(2, 15)
plt.savefig('quantum_dot_bandgaps.png', dpi=150)
plt.show()

# Wavelength calculation
def bandgap_to_wavelength(Eg_eV):
    """Convert bandgap energy to emission wavelength."""
    h = 6.626e-34  # J·s
    c = 3e8        # m/s
    wavelength_m = h * c / (Eg_eV * e)
    return wavelength_m * 1e9  # nm

# Print example: CdSe quantum dots
print("\nCdSe Quantum Dot Size-Color Relationship:")
print("-" * 50)
for d in [2, 3, 4, 5, 6, 7]:
    Eg = quantum_dot_bandgap(d, 'CdSe')
    wavelength = bandgap_to_wavelength(Eg)
    color = "UV" if wavelength < 400 else "Violet" if wavelength < 450 else \
            "Blue" if wavelength < 495 else "Green" if wavelength < 570 else \
            "Yellow" if wavelength < 590 else "Orange" if wavelength < 620 else "Red"
    print(f"d = {d} nm: Eg = {Eg:.2f} eV, λ = {wavelength:.0f} nm ({color})")
        

1.5 Melting Point Depression

Nanomaterials exhibit lower melting points than their bulk counterparts. This phenomenon, known as melting point depression, arises from the high surface energy contribution and can be described by the Gibbs-Thomson equation.

import numpy as np
import matplotlib.pyplot as plt

def melting_point_depression(diameter_nm, material='Au'):
    """
    Calculate size-dependent melting point using Gibbs-Thomson equation.

    Parameters:
    -----------
    diameter_nm : float
        Particle diameter in nanometers
    material : str
        Material type ('Au', 'Ag', 'Pt', 'Al')

    Returns:
    --------
    float : Melting temperature in K
    """
    # Material parameters: (Tm_bulk K, gamma_sl J/m^2, Vm m^3/mol, dHm J/mol)
    materials = {
        'Au': (1337, 0.27, 1.02e-5, 12550),
        'Ag': (1235, 0.18, 1.03e-5, 11300),
        'Pt': (2041, 0.32, 9.09e-6, 22170),
        'Al': (933, 0.16, 1.00e-5, 10790),
        'Cu': (1358, 0.18, 7.11e-6, 13260),
    }

    Tm_bulk, gamma_sl, Vm, dHm = materials[material]
    d = diameter_nm * 1e-9  # Convert to meters

    # Gibbs-Thomson equation
    Tm = Tm_bulk * (1 - 4 * gamma_sl * Vm / (dHm * d))

    return max(Tm, 0)  # Temperature cannot be negative

# Calculate for different materials
diameters = np.linspace(2, 50, 100)

plt.figure(figsize=(10, 6))
colors = {'Au': 'gold', 'Ag': 'silver', 'Pt': 'gray', 'Al': 'skyblue', 'Cu': 'orange'}

for material in ['Au', 'Ag', 'Pt', 'Al']:
    Tm_bulk = {'Au': 1337, 'Ag': 1235, 'Pt': 2041, 'Al': 933}[material]
    temps = [melting_point_depression(d, material) for d in diameters]
    reduction = [(1 - T/Tm_bulk)*100 for T in temps]
    plt.plot(diameters, temps, color=colors[material],
             linewidth=2, label=f'{material} (bulk: {Tm_bulk} K)')

plt.xlabel('Particle Diameter (nm)')
plt.ylabel('Melting Temperature (K)')
plt.title('Melting Point Depression in Metal Nanoparticles')
plt.legend()
plt.grid(True, alpha=0.3)
plt.xlim(2, 50)
plt.savefig('melting_point_depression.png', dpi=150)
plt.show()

# Print specific values for gold nanoparticles
print("\nGold Nanoparticle Melting Points:")
print("-" * 50)
Tm_bulk_Au = 1337
for d in [2, 5, 10, 20, 50, 100]:
    Tm = melting_point_depression(d, 'Au')
    reduction = (1 - Tm/Tm_bulk_Au) * 100
    print(f"d = {d:3d} nm: Tm = {Tm:.0f} K ({Tm-273:.0f}°C), "
          f"ΔTm = -{reduction:.1f}%")
        

1.6 Scherrer Equation: Crystallite Size from XRD

The Scherrer equation relates the broadening of X-ray diffraction peaks to the crystallite size, providing a practical method to estimate nanomaterial dimensions:

import numpy as np

def scherrer_size(two_theta_deg, fwhm_deg, wavelength_nm=0.15406, K=0.9):
    """
    Calculate crystallite size using Scherrer equation.

    Parameters:
    -----------
    two_theta_deg : float
        Diffraction angle 2θ in degrees
    fwhm_deg : float
        Full width at half maximum in degrees
    wavelength_nm : float
        X-ray wavelength in nm (default: Cu Kα = 0.15406 nm)
    K : float
        Scherrer constant (0.89-0.94 for spheres)

    Returns:
    --------
    float : Crystallite size in nm
    """
    # Convert to radians
    theta_rad = np.radians(two_theta_deg / 2)
    beta_rad = np.radians(fwhm_deg)

    # Scherrer equation
    tau = (K * wavelength_nm) / (beta_rad * np.cos(theta_rad))

    return tau

def instrument_correction(fwhm_measured, fwhm_instrument):
    """
    Correct for instrumental broadening.

    Uses the approximation: β² = β_sample² + β_instrument²
    """
    return np.sqrt(fwhm_measured**2 - fwhm_instrument**2)

# Example: XRD analysis of TiO2 nanoparticles
# Anatase (101) peak at 2θ = 25.3°
print("XRD Crystallite Size Analysis")
print("=" * 50)

# Example data: different synthesis conditions
samples = [
    {"name": "Sample A (hydrothermal)", "two_theta": 25.3, "fwhm": 0.8},
    {"name": "Sample B (sol-gel)", "two_theta": 25.3, "fwhm": 1.2},
    {"name": "Sample C (calcined)", "two_theta": 25.3, "fwhm": 0.4},
]

fwhm_instrument = 0.1  # Instrumental broadening

for sample in samples:
    # Correct for instrumental broadening
    fwhm_corrected = instrument_correction(sample["fwhm"], fwhm_instrument)

    # Calculate crystallite size
    size = scherrer_size(sample["two_theta"], fwhm_corrected)

    print(f"\n{sample['name']}:")
    print(f"  2θ = {sample['two_theta']}°, FWHM = {sample['fwhm']}°")
    print(f"  FWHM (corrected) = {fwhm_corrected:.3f}°")
    print(f"  Crystallite size = {size:.1f} nm")

# Demonstrate effect of FWHM on size
print("\n\nRelationship between FWHM and Crystallite Size:")
print("-" * 50)
fwhm_values = [0.2, 0.5, 1.0, 2.0, 5.0]
for fwhm in fwhm_values:
    size = scherrer_size(25.3, fwhm)
    print(f"FWHM = {fwhm}° → τ = {size:.1f} nm")
        

1.7 Summary

Exercises