Polymer Chemistry

Polymerization, molecular weight, chain conformations, and mechanical properties

Polymerization Mechanisms

Addition (Chain-Growth) Polymerization

Monomers with double bonds add one at a time to a growing chain via a reactive intermediate (free radical, cation, or anion). Steps: initiation, propagation, termination.

Examples: polyethylene, polystyrene, PVC, PMMA, Teflon (PTFE).

Condensation (Step-Growth) Polymerization

Bifunctional monomers react with each other, eliminating a small molecule (e.g., H₂O) at each step. Any two species can react, so molecular weight builds gradually.

Examples: nylon (polyamide), PET (polyester), Bakelite (phenol-formaldehyde).

Molecular Weight Averages

Polymers have a distribution of chain lengths. Two key averages characterize this distribution:

Number-Average Molecular Weight

$$M_n = \frac{\sum N_i M_i}{\sum N_i}$$

Weighted by the number of chains of each length. Measured by osmometry or end-group analysis.

Weight-Average Molecular Weight

$$M_w = \frac{\sum N_i M_i^2}{\sum N_i M_i}$$

Weighted by mass. Measured by light scattering. Always $M_w \geq M_n$.

Degree of Polymerization and PDI

$$\text{DP} = \frac{M_n}{M_0}$$

$$\text{PDI} = \frac{M_w}{M_n} \geq 1$$

where $M_0$ is the monomer molecular weight. A PDI of 1 means all chains are identical (monodisperse). Typical PDI values: condensation polymers $\sim 2$, free-radical addition $\sim 1.5\text{--}2$, living polymerization $\sim 1.01\text{--}1.1$.

Polymer Chain Conformations

A flexible polymer chain in solution or the melt adopts a random coil conformation, well-described by a random walk model. For $N$ bonds of length $l$:

$$\langle R^2 \rangle = Nl^2 \quad \Rightarrow \quad R_{\text{rms}} = l\sqrt{N}$$

$$P(R) = 4\pi R^2 \left(\frac{3}{2\pi N l^2}\right)^{3/2} \exp\!\left(-\frac{3R^2}{2Nl^2}\right)$$

The end-to-end distance scales as $\sqrt{N}$, much smaller than the contour length $L = Nl$. This is why polymer coils are compact. For real chains with excluded volume, $R \sim N^{0.588}$ (Flory exponent in 3D).

Thermal & Mechanical Properties

Glass Transition ($T_g$)

Below $T_g$, the amorphous regions become rigid and glassy. Above $T_g$, chain segments gain mobility and the material becomes rubbery. $T_g$ depends on chain stiffness, side groups, and intermolecular forces.

Melting ($T_m$)

Only semicrystalline polymers have a $T_m$. The crystalline regions melt at $T_m$, while amorphous regions transition at $T_g$. Typically $T_g \approx (2/3)T_m$ (in Kelvin) for symmetric polymers.

Polymer Classification by Thermal Behavior

Thermoplastics: Soften and flow above $T_g$ or $T_m$; recyclable by remelting (PE, PS, nylon)
Thermosets: Covalently crosslinked; do not melt or dissolve once cured (epoxy, Bakelite, vulcanized rubber)
Elastomers: Lightly crosslinked; exhibit large reversible deformation above $T_g$ (natural rubber, silicone)

Stress-Strain Behavior

The Young's modulus $E = \sigma/\varepsilon$ (in the linear elastic region) quantifies stiffness. Polymers span a huge range: elastomers ($E \sim 0.01$ GPa), glassy polymers ($E \sim 2\text{--}4$ GPa), and oriented fibers like Kevlar ($E \sim 130$ GPa).

Python: Random Walk Polymer Simulation

Simulates thousands of 3D random walk polymer chains and compares the end-to-end distance distribution with the Gaussian chain theory prediction.

Random Walk Polymer Chain

Python

3D random walk simulation of polymer chains with Gaussian theory comparison

script.py82 lines

import numpy as np
import matplotlib
matplotlib.use('Agg')
import matplotlib.pyplot as plt
# ─────────────────────────────────────────────────────
# Random Walk Polymer Chain Simulation
# End-to-end distance distribution vs Gaussian theory
# ─────────────────────────────────────────────────────
np.random.seed(42)

N = 100          # number of monomers (bonds)
l = 1.0          # bond length
n_chains = 5000  # number of chains to simulate

# Simulate 3D random walk chains
end_to_end = np.zeros(n_chains)
sample_chains = []

for c in range(n_chains):
    # Random unit vectors on the sphere (isotropic)
    theta = np.arccos(2 * np.random.rand(N) - 1)
    phi = 2 * np.pi * np.random.rand(N)
    dx = l * np.sin(theta) * np.cos(phi)
    dy = l * np.sin(theta) * np.sin(phi)
    dz = l * np.cos(theta)

x = np.cumsum(np.concatenate([[0], dx]))
    y = np.cumsum(np.concatenate([[0], dy]))
    z = np.cumsum(np.concatenate([[0], dz]))

R = np.sqrt(x[-1]**2 + y[-1]**2 + z[-1]**2)
    end_to_end[c] = R

if c < 5:
        sample_chains.append((x, y))

# Theoretical: Gaussian chain
# P(R) = 4*pi*R^2 * (3/(2*pi*N*l^2))^(3/2) * exp(-3*R^2/(2*N*l^2))
R_theory = np.linspace(0, 30, 300)
sigma2 = N * l**2 / 3.0
P_R = 4 * np.pi * R_theory**2 * (3.0 / (2 * np.pi * N * l**2))**1.5 * \
      np.exp(-3 * R_theory**2 / (2 * N * l**2))

R_rms_theory = l * np.sqrt(N)
R_rms_sim = np.sqrt(np.mean(end_to_end**2))

print(f"N = {N} bonds, l = {l}")
print(f"Theoretical <R^2>^(1/2) = l*sqrt(N) = {R_rms_theory:.3f}")
print(f"Simulated   <R^2>^(1/2) = {R_rms_sim:.3f}")
print(f"Mean R = {np.mean(end_to_end):.3f}")

fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(13, 5))

# Left: sample chains in 2D projection
colors = ['cyan', 'lime', 'magenta', 'yellow', 'orange']
for i, (x, y) in enumerate(sample_chains):
    ax1.plot(x, y, color=colors[i], alpha=0.7, linewidth=0.8)
    ax1.plot(x[-1], y[-1], 'o', color=colors[i], markersize=6)
ax1.plot(0, 0, 'w*', markersize=10, label='Origin')
ax1.set_xlabel('x', fontsize=12)
ax1.set_ylabel('y', fontsize=12)
ax1.set_title(f'Random Walk Chains (N={N})', fontsize=14)
ax1.set_aspect('equal')
ax1.legend(fontsize=10)
ax1.grid(True, alpha=0.2)

# Right: end-to-end distance distribution
ax2.hist(end_to_end, bins=50, density=True, alpha=0.7, color='cyan',
         edgecolor='white', linewidth=0.5, label=f'Simulation ({n_chains} chains)')
ax2.plot(R_theory, P_R, 'r-', linewidth=2.5, label='Gaussian theory')
ax2.axvline(R_rms_theory, color='yellow', linestyle='--', linewidth=1.5,
            label=f'$R_{{rms}}$ = {R_rms_theory:.1f}')
ax2.set_xlabel('End-to-end distance R', fontsize=12)
ax2.set_ylabel('P(R)', fontsize=12)
ax2.set_title('End-to-End Distance Distribution', fontsize=14)
ax2.legend(fontsize=10)
ax2.grid(True, alpha=0.3)

plt.tight_layout()
plt.savefig('output.png', dpi=150, bbox_inches='tight')
print("Plot saved.")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

Fortran: Molecular Weight Averages & PDI

Computes number-average, weight-average molecular weights and polydispersity index from a simulated Flory (most probable) distribution of chain lengths.

Molecular Weight Averages & PDI

Fortran

Computes Mn, Mw, and PDI from a Flory (most probable) molecular weight distribution

mw_averages.f9060 lines

program mw_averages
  implicit none
  integer, parameter :: dp = selected_real_kind(15)
  integer, parameter :: max_dp_val = 500   ! max degree of polymerization
  real(dp) :: M0                           ! monomer molar mass
  real(dp) :: p                            ! extent of reaction
  real(dp) :: N_i(max_dp_val)              ! number of chains of length i
  real(dp) :: M_i                          ! molar mass of chain i
  real(dp) :: sum_Ni, sum_NiMi, sum_NiMi2
  real(dp) :: Mn, Mw, PDI, DP_n
  integer :: i

M0 = 104.0_dp   ! styrene monomer: 104 g/mol
  p = 0.99_dp      ! 99% conversion

write(*,'(A)') '=================================================='
  write(*,'(A)') '  Molecular Weight Averages from Flory Distribution'
  write(*,'(A)') '=================================================='
  write(*,'(A,F8.2,A)') 'Monomer M0 = ', M0, ' g/mol'
  write(*,'(A,F8.4)') 'Extent of reaction p = ', p
  write(*,'(A)') ''

! Flory (most probable) distribution:
  ! N_i proportional to (1-p) * p^(i-1)
  sum_Ni = 0.0_dp
  sum_NiMi = 0.0_dp
  sum_NiMi2 = 0.0_dp

do i = 1, max_dp_val
    N_i(i) = (1.0_dp - p) * p ** (i - 1)
    M_i = i * M0
    sum_Ni = sum_Ni + N_i(i)
    sum_NiMi = sum_NiMi + N_i(i) * M_i
    sum_NiMi2 = sum_NiMi2 + N_i(i) * M_i**2
  end do

Mn = sum_NiMi / sum_Ni
  Mw = sum_NiMi2 / sum_NiMi
  PDI = Mw / Mn
  DP_n = Mn / M0

write(*,'(A,F12.1,A)') 'Number-average Mn = ', Mn, ' g/mol'
  write(*,'(A,F12.1,A)') 'Weight-average Mw = ', Mw, ' g/mol'
  write(*,'(A,F8.4)') 'Polydispersity PDI = Mw/Mn = ', PDI
  write(*,'(A,F8.1)') 'Degree of polymerization DP_n = Mn/M0 = ', DP_n
  write(*,'(A)') ''

! Analytic formulas for Flory distribution
  write(*,'(A)') '--- Analytic (Flory distribution) ---'
  write(*,'(A,F12.1)') 'Mn = M0/(1-p) = ', M0 / (1.0_dp - p)
  write(*,'(A,F12.1)') 'Mw = M0*(1+p)/(1-p) = ', M0*(1.0_dp+p)/(1.0_dp-p)
  write(*,'(A,F8.4)') 'PDI = 1 + p = ', 1.0_dp + p

write(*,'(A)') ''
  write(*,'(A)') '--- Distribution sample (first 10 chain lengths) ---'
  write(*,'(A6, A12, A15)') 'DP', 'N_i/N_total', 'M_i (g/mol)'
  do i = 1, 10
    write(*,'(I6, ES12.4, F15.1)') i, N_i(i)/sum_Ni, i*M0
  end do
end program mw_averages

Click Run to execute the Fortran code

Code will be compiled with gfortran and executed on the server

Video Lectures

Lecture 32: Polymers I

Lecture 33: Polymers II

Goodie Bag 9: Polymers

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