Solid-State Chemistry/Part 5/Amorphous Solids & Glasses

5.2 Amorphous Solids & Glasses

Amorphous solids lack the long-range periodic order of crystals. Glasses, the most important class of amorphous solids, are supercooled liquids frozen below their glass transition temperature. Their unique properties make them essential in optics, construction, electronics, and telecommunications.

Glass Transition Temperature

The glass transition temperature $T_g$ marks the temperature below which a supercooled liquid becomes a glass. At $T_g$:

- Viscosity reaches approximately $10^{12}$ Pa$\cdot$s
- The material transitions from a viscous liquid to a rigid solid
- Heat capacity, thermal expansion, and other properties change discontinuously
- $T_g$ is not a sharp thermodynamic transition but depends on cooling rate

Zachariasen's Rules for Glass Formation

W.H. Zachariasen (1932) proposed rules for oxide glass formation based on random network theory:

An oxygen atom is linked to no more than two glass-forming cations
The coordination number of the glass-forming cation is small (3 or 4)
Oxygen polyhedra share corners, not edges or faces
At least 3 corners of each polyhedron must be shared

Network Formers, Modifiers & Intermediates

Network Formers

Form the glass network backbone. SiO$_2$, B$_2$O$_3$, P$_2$O$_5$, GeO$_2$. Can form glass on their own.

Network Modifiers

Break up the network by creating non-bridging oxygens. Na$_2$O, K$_2$O, CaO, MgO. Lower $T_g$ and viscosity.

Intermediates

Can act as either former or modifier depending on composition. Al$_2$O$_3$, TiO$_2$, ZnO, PbO.

Viscosity Models

Viscosity is the most important property for glass processing. Two models are commonly used:

Arrhenius

$$\eta = \eta_0\exp\!\left(\frac{E_a}{RT}\right)$$

Linear on Arrhenius plot. Good for "strong" glasses (SiO$_2$) that show nearly Arrhenius behavior.

Vogel-Fulcher-Tammann (VFT)

$$\eta = A\exp\!\left(\frac{B}{T - T_0}\right)$$

Curved on Arrhenius plot. Better for "fragile" glasses that deviate strongly from Arrhenius behavior near $T_g$.

Key viscosity reference points: strain point ($10^{13.5}$ Pa$\cdot$s), annealing point ($10^{12}$ Pa$\cdot$s, $\approx T_g$), softening point ($10^{6.6}$ Pa$\cdot$s), working point ($10^{4}$ Pa$\cdot$s).

Optical Properties

Glasses are valued for their optical transparency. The refractive index $n$ determines how light bends at interfaces, governed by Snell's law:

$$n_1\sin\theta_1 = n_2\sin\theta_2$$

Typical refractive indices: soda-lime glass $n \approx 1.52$, borosilicate $n \approx 1.47$, lead glass $n \approx 1.65$, fused silica $n \approx 1.46$. Higher PbO content increases $n$, giving lead crystal its brilliance.

Tempering and Annealing

Tempering

Rapid cooling creates compressive stress on the surface and tensile stress in the interior. This makes the glass 4--5 times stronger than annealed glass. Tempered glass shatters into small, relatively harmless pieces.

Annealing

Slow, controlled cooling through $T_g$ relieves internal stresses. The glass is held at the annealing point, then slowly cooled to the strain point. Essential for precision optics and laboratory glassware.

Video Lectures

25. Introduction to Glassy Solids

26. Engineering Glass Properties

Python: Viscosity vs Temperature

Comparison of Arrhenius and VFT viscosity models for soda-lime and borosilicate glass compositions, with key processing reference points.

Glass Viscosity: Arrhenius vs VFT

Python

Viscosity-temperature curves for soda-lime and borosilicate glass

script.py104 lines

import numpy as np
import matplotlib
matplotlib.use('Agg')
import matplotlib.pyplot as plt
# ─────────────────────────────────────────────────
# Viscosity vs Temperature: Arrhenius and VFT Models
# Comparison of soda-lime and borosilicate glass
# ─────────────────────────────────────────────────

R = 8.314  # J/(mol*K)

# Glass parameters
# Arrhenius: log10(eta) = A + B/T  (eta in Pa*s)
# VFT: log10(eta) = A + B/(T - T0)
glasses = {
    'Soda-lime': {
        'A_vft': -2.0, 'B_vft': 4500.0, 'T0': 500.0,
        'Tg': 840, 'Tm_work': 1300,
        'color': '#22d3ee', 'Ea': 350e3,  # J/mol
    },
    'Borosilicate': {
        'A_vft': -1.5, 'B_vft': 6200.0, 'T0': 420.0,
        'Tg': 820, 'Tm_work': 1500,
        'color': '#a78bfa', 'Ea': 450e3,
    },
}

fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(14, 6))
fig.patch.set_facecolor('#0a0a1a')
for ax in (ax1, ax2):
    ax.set_facecolor('#0a0a1a')
    for spine in ax.spines.values():
        spine.set_edgecolor('#334155')

print("=" * 65)
print("Glass Viscosity: Arrhenius and VFT Models")
print("=" * 65)

for name, props in glasses.items():
    T = np.linspace(props['T0'] + 50, 2000, 500)

# VFT model
    log_eta_vft = props['A_vft'] + props['B_vft'] / (T - props['T0'])
    eta_vft = 10**log_eta_vft

# Arrhenius model (using Ea)
    # log10(eta) = log10(eta0) + Ea/(2.303*R*T)
    # Fit eta0 so it matches VFT at Tg
    log_eta_at_Tg = props['A_vft'] + props['B_vft'] / (props['Tg'] - props['T0'])
    log_eta0_arr = log_eta_at_Tg - props['Ea'] / (2.303 * R * props['Tg'])
    log_eta_arr = log_eta0_arr + props['Ea'] / (2.303 * R * T)
    eta_arr = 10**log_eta_arr

# Plot vs temperature
    ax1.semilogy(T - 273, eta_vft, color=props['color'], linewidth=2, label=f'{name} (VFT)')
    ax1.semilogy(T - 273, eta_arr, color=props['color'], linewidth=1.5, linestyle='--',
                label=f'{name} (Arrhenius)', alpha=0.6)

# Arrhenius plot: log(eta) vs 1/T
    ax2.plot(10000/T, log_eta_vft, color=props['color'], linewidth=2, label=f'{name} (VFT)')
    ax2.plot(10000/T, log_eta_arr, color=props['color'], linewidth=1.5, linestyle='--',
            label=f'{name} (Arrhenius)', alpha=0.6)

print(f"\n{name}:")
    print(f"  VFT params: A={props['A_vft']}, B={props['B_vft']}, T0={props['T0']} K")
    print(f"  Tg = {props['Tg']} K ({props['Tg']-273} C)")
    print(f"  log10(eta) at Tg: {log_eta_at_Tg:.1f}")
    print(f"  Working range: ~{props['Tm_work']-273} C")

# Reference viscosity lines
for ax in (ax1,):
    ax.axhline(y=1e12, color='#ef4444', linestyle=':', alpha=0.5, linewidth=1)
    ax.text(1600, 2e12, 'Tg (~10$^{12}$ Pa s)', color='#ef4444', fontsize=9)
    ax.axhline(y=1e4, color='#f59e0b', linestyle=':', alpha=0.5, linewidth=1)
    ax.text(1600, 2e4, 'Working point (~10$^4$ Pa s)', color='#f59e0b', fontsize=9)
    ax.axhline(y=10, color='#22c55e', linestyle=':', alpha=0.5, linewidth=1)
    ax.text(1600, 15, 'Melting point (~10 Pa s)', color='#22c55e', fontsize=9)

ax1.set_xlabel('Temperature (C)', color='#94a3b8', fontsize=12)
ax1.set_ylabel('Viscosity (Pa s)', color='#94a3b8', fontsize=12)
ax1.set_title('Viscosity vs Temperature', color='white', fontsize=14)
ax1.legend(fontsize=9, facecolor='#0a0a1a', edgecolor='#334155', labelcolor='white')
ax1.tick_params(colors='#94a3b8')
ax1.grid(True, alpha=0.2)
ax1.set_ylim(1e-1, 1e16)

ax2.set_xlabel('10000/T (K$^{-1}$)', color='#94a3b8', fontsize=12)
ax2.set_ylabel('log$_{10}$(eta / Pa s)', color='#94a3b8', fontsize=12)
ax2.set_title('Arrhenius Plot of Viscosity', color='white', fontsize=14)
ax2.legend(fontsize=9, facecolor='#0a0a1a', edgecolor='#334155', labelcolor='white')
ax2.tick_params(colors='#94a3b8')
ax2.grid(True, alpha=0.2)

print("\nViscosity reference points:")
print("  Strain point:   ~10^13.5 Pa s")
print("  Annealing point: ~10^12 Pa s (Tg)")
print("  Softening point: ~10^6.6 Pa s")
print("  Working point:   ~10^4 Pa s")
print("  Melting point:   ~10^1 Pa s")

plt.tight_layout()
plt.savefig('viscosity.png', dpi=150, bbox_inches='tight', facecolor='#0a0a1a')
print("\nPlot saved as 'viscosity.png'")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

Fortran: Glass Composition & T$_g$ Calculator

Predicts the glass transition temperature from oxide composition using an empirical additive model. Compares soda-lime, borosilicate, and fused silica glasses.

Glass Composition & Tg Predictor

Fortran

Predict glass transition temperature from oxide composition

glass_tg.f90110 lines

program glass_tg_predictor
  implicit none
  integer, parameter :: dp = selected_real_kind(15)

! Glass composition calculator
  ! Predict Tg from composition using empirical additive model
  ! Tg = sum(xi * Tg_i) where xi is mole fraction of oxide i

integer, parameter :: n_oxides = 7
  character(len=10) :: oxide_names(n_oxides)
  real(dp) :: Tg_contrib(n_oxides)  ! Contribution to Tg (K) per mole fraction
  real(dp) :: comp(n_oxides), Tg_pred
  real(dp) :: total
  integer :: i, j

! Oxide names and their Tg contributions (empirical, approximate)
  ! Based on Priven-Mazurin model (simplified)
  oxide_names = (/ 'SiO2      ', 'B2O3      ', 'Na2O      ', &
                   'K2O       ', 'CaO       ', 'MgO       ', 'Al2O3     ' /)
  ! Approximate Tg contribution per mole% (for additive model)
  Tg_contrib = (/ 1473.0_dp, 533.0_dp, 243.0_dp, &
                  213.0_dp, 1113.0_dp, 923.0_dp, 1273.0_dp /)

write(*,'(A)') '============================================================'
  write(*,'(A)') ' Glass Composition Calculator: Predict Tg'
  write(*,'(A)') '============================================================'
  write(*,'(A)') ' Using simplified additive model: Tg = sum(xi * Tg_i)'
  write(*,'(A)') ''

! Reference values for individual oxide Tg contributions
  write(*,'(A)') ' Oxide Tg contributions (K):'
  do i = 1, n_oxides
    write(*,'(4X,A10,F8.0,A)') oxide_names(i), Tg_contrib(i), ' K'
  end do

! --- Glass 1: Soda-lime silicate (window glass) ---
  write(*,'(A)') ''
  write(*,'(A)') ' === Soda-Lime Silicate (Window Glass) ==='
  ! Typical: 72% SiO2, 1% B2O3, 14% Na2O, 1% K2O, 8% CaO, 3% MgO, 1% Al2O3
  comp = (/ 0.72_dp, 0.01_dp, 0.14_dp, 0.01_dp, 0.08_dp, 0.03_dp, 0.01_dp /)

total = sum(comp)
  Tg_pred = 0.0_dp
  write(*,'(A)') ' Composition (mole fraction):'
  do i = 1, n_oxides
    if (comp(i) > 0.001_dp) then
      write(*,'(4X,A10,F6.2,A,F8.1,A)') oxide_names(i), comp(i)*100, &
        '%  -> contrib = ', comp(i) * Tg_contrib(i), ' K'
    end if
    Tg_pred = Tg_pred + comp(i) * Tg_contrib(i)
  end do
  write(*,'(A,F7.1,A,F7.1,A)') ' Predicted Tg = ', Tg_pred, ' K (', Tg_pred - 273.0_dp, ' C)'
  write(*,'(A)') ' Experimental Tg ~ 840 K (567 C)'

! --- Glass 2: Borosilicate (Pyrex-type) ---
  write(*,'(A)') ''
  write(*,'(A)') ' === Borosilicate Glass (Pyrex-type) ==='
  ! Typical: 80% SiO2, 13% B2O3, 4% Na2O, 0% K2O, 0% CaO, 0% MgO, 3% Al2O3
  comp = (/ 0.80_dp, 0.13_dp, 0.04_dp, 0.00_dp, 0.00_dp, 0.00_dp, 0.03_dp /)

Tg_pred = 0.0_dp
  write(*,'(A)') ' Composition (mole fraction):'
  do i = 1, n_oxides
    if (comp(i) > 0.001_dp) then
      write(*,'(4X,A10,F6.2,A,F8.1,A)') oxide_names(i), comp(i)*100, &
        '%  -> contrib = ', comp(i) * Tg_contrib(i), ' K'
    end if
    Tg_pred = Tg_pred + comp(i) * Tg_contrib(i)
  end do
  write(*,'(A,F7.1,A,F7.1,A)') ' Predicted Tg = ', Tg_pred, ' K (', Tg_pred - 273.0_dp, ' C)'
  write(*,'(A)') ' Experimental Tg ~ 820 K (547 C)'

! --- Glass 3: High-silica (fused silica) ---
  write(*,'(A)') ''
  write(*,'(A)') ' === Fused Silica ==='
  comp = (/ 1.00_dp, 0.00_dp, 0.00_dp, 0.00_dp, 0.00_dp, 0.00_dp, 0.00_dp /)

Tg_pred = 0.0_dp
  do i = 1, n_oxides
    Tg_pred = Tg_pred + comp(i) * Tg_contrib(i)
  end do
  write(*,'(A)') ' Composition: 100% SiO2'
  write(*,'(A,F7.1,A,F7.1,A)') ' Predicted Tg = ', Tg_pred, ' K (', Tg_pred - 273.0_dp, ' C)'
  write(*,'(A)') ' Experimental Tg ~ 1473 K (1200 C)'

! --- Glass 4: Lead glass (crystal) ---
  write(*,'(A)') ''
  write(*,'(A)') ' === Effect of Network Modifiers ==='
  write(*,'(A)') ' Adding Na2O to SiO2 (binary system):'
  write(*,'(A)') ' Na2O%    Predicted Tg(K)    Tg(C)'
  write(*,'(A)') ' -----    ---------------    -----'

do j = 0, 40, 5
    comp = 0.0_dp
    comp(1) = 1.0_dp - real(j, dp) / 100.0_dp  ! SiO2
    comp(3) = real(j, dp) / 100.0_dp             ! Na2O
    Tg_pred = 0.0_dp
    do i = 1, n_oxides
      Tg_pred = Tg_pred + comp(i) * Tg_contrib(i)
    end do
    write(*,'(I5,A,F12.1,8X,F7.1)') j, '%', Tg_pred, Tg_pred - 273.0_dp
  end do

write(*,'(A)') ''
  write(*,'(A)') 'Note: Additive model is approximate. Real behavior'
  write(*,'(A)') 'shows non-linear mixing effects, especially at high'
  write(*,'(A)') 'modifier concentrations.'
  write(*,'(A)') ''
  write(*,'(A)') 'Done.'
end program glass_tg_predictor

Click Run to execute the Fortran code

Code will be compiled with gfortran and executed on the server

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