# LECTURE 3: Step Polymerization
## Introduction
- Extension of simple organic chemistry linking reactions
- First type of polymerization to be understood fundamentally (Carothers, 1920s-30s)
- Most polymers produced: M = 10-100 kg mol⁻¹ (typically 15-30 kg mol⁻¹)
- Requires VERY HIGH conversions (>99.9%) for useful properties
--## Types of Linear Step Polymerization
### 1. Polycondensation (elimination of small molecules)
**Polyesters**:
```
n HOOC-R₁-COOH + n HO-R₂-OH → polymer + (2n-1) H₂O
```
- Example: Poly(ethylene terephthalate) from terephthalic acid + ethylene glycol
- Requires acid catalyst, high temperature (80-300°C)
**Polyamides (Nylons)**:
```
n HOOC-R₁-COOH + n H₂N-R₂-NH₂ → polymer + (2n-1) H₂O
```
- Nylon 6.6: R₁ = (CH₂)₄, R₂ = (CH₂)₆
- Nylon 6.10: R₁ = (CH₂)₈, R₂ = (CH₂)₆
**Polyethers**:
```
n HO-R-OH → polymer + (n-1) H₂O
```
- Limited use; ring-opening polymerization preferred
### 2. Polyaddition (no elimination)
**Polyurethanes**:
```
n OCN-R₁-NCO + n HO-R₂-OH → polyurethane
```
- Proceeds rapidly at room temperature
- Diols can be simple or polyether-based
**Polyureas**:
```
n OCN-R₁-NCO + n H₂N-R₂-NH₂ → polyurea
```
- Very fast reaction (requires high-speed mixing)
--## Carothers Theory
**Extent of reaction (p)**:
```
p = (number of functional groups reacted) / (initial number)
```
**Carothers equation** (exact stoichiometry):
```
x̅n = 1 / (1 - p)
```
**For stoichiometric imbalance**:
```
r = Nₐ / Nᴮ (≤ 1)
x̅n = (1 + r) / (1 + r - 2rp)
```
**Key Insight**: To achieve x̅n > 100, need p ≥ 0.99
| p | x̅n |
|---|-----|
| 0.90 | 10 |
| 0.95 | 20 |
| 0.99 | 100 |
| 0.999 | 1000 |
--## Statistical Theory (Flory)
**Mole fraction of x-mers**:
```
P(x) = (1 - p) p^(x-1)
```
**Weight fraction of x-mers**:
```
wₓ = x(1 - p)² p^(x-1)
```
**Molar mass averages**:
```
M̅n = M₀ / (1 - p)
M̅w = M₀ (1 + p) / (1 - p)
M̅w/M̅n = 1 + p ≈ 2
```
--## Non-Linear Step Polymerization
### Gelation (when f > 2)
**Carothers gelation theory**:
```
p_c = 2 / f_av
```
- Overestimates p_c (flawed: based on x̅n → ∞)
**Statistical gelation theory (Flory)**:
```
α_c = 1 / (f - 1)
```
- More accurate; assumes equal reactivity
### Network Polymers
**Formaldehyde-based resins**:
- Phenol-formaldehyde (Bakelite): f = 3 (phenol)
- Urea-formaldehyde: f = 4 (urea)
- Melamine-formaldehyde: f = 6 (melamine)
**Epoxy resins**:
- Diglycidyl ether of bisphenol-A + polyfunctional amines
- Low shrinkage, used as adhesives, composites
**Network polyurethanes**:
- Diisocyanate + polyols (f = 3-6)
- Used for elastomers, flexible/rigid foams
--## Dendrimers and Hyperbranched Polymers
**Dendrimers**:
- Perfectly branched, single molar mass
- Synthesized generation by generation
- Divergent (core-outward) or convergent (surface-inward)
- Internal voids, high surface functionality
**Hyperbranched**:
- Less perfect, easier to synthesize
- ARBₓ step polymerization (x > 2)
- Branching factor typically 0.5-0.8
--**Key Takeaway**: Step polymerization requires very high conversions. Linear step
polymerization gives M̅w/M̅n ≈ 2. Functionality > 2 leads to branching and gelation (network
formation). Dendrimers and hyperbranched polymers are advanced branched structures with
unique properties.