Sources and References

Primary texts — Young, R.J. and Lovell, P.A., Introduction to Polymers, 3rd ed., CRC Press, 2011; Painter, P.C. and Coleman, M.M., Fundamentals of Polymer Science, 2nd ed., CRC Press, 1997.

Supplementary texts — Rubinstein, M. and Colby, R.H., Polymer Physics, Oxford, 2003; Odian, G., Principles of Polymerization, 4th ed., Wiley, 2004.

Online resources — MIT OCW 3.064 “Polymer Engineering” and 3.063 “Polymer Physics”; University of Bayreuth polymer science open lectures; IUPAC recommendations on nomenclature; ISO 472 polymer terminology (public summaries).

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Chapter 1: Polymer Molecules

1.1 What Makes a Polymer

A polymer is a molecule of many (\( 10^2 \)–\( 10^6 \)) repeating units. The degree of polymerization \( DP \) counts these units; molecular weight \( M = DP \cdot M_0 \) where \( M_0 \) is repeat-unit mass. Above a critical molecular weight (typically 20–100 kg/mol), entanglements dominate mechanical behavior, distinguishing polymer melts from small molecules.

1.2 Molecular Weight Distributions

Synthesis is statistical; samples contain chains of many lengths. Two averages dominate:

\[ M_n = \frac{\sum N_i M_i}{\sum N_i}, \quad M_w = \frac{\sum N_i M_i^2}{\sum N_i M_i}. \]

The polydispersity \( PDI = M_w/M_n \geq 1 \). Step-growth polymerization gives PDI ≈ 2; free-radical chain polymerization typically 1.5–3; living/controlled polymerization < 1.1.

Characterization: size-exclusion chromatography (SEC, aka GPC), MALDI-TOF, light scattering, osmometry (absolute \( M_n \)), viscometry.

1.3 Architecture

Linear, branched (short- and long-chain), star, comb, dendritic, hyperbranched, cyclic, network. Each architecture correlates with different processing and property profiles. Cross-linked networks (thermosets) do not melt or dissolve; linear/branched (thermoplastics) do.

1.4 Nomenclature and Classification

IUPAC source-based (poly(ethylene), poly(vinyl chloride)) and structure-based names coexist. Commodity thermoplastics (PE, PP, PVC, PS, PET) dominate by volume; engineering plastics (PA, PC, PBT, POM) offer higher performance; specialty polymers (PEEK, PTFE, PI) serve demanding niches. Thermosets (epoxy, phenolics, unsaturated polyesters, rubbers) add covalent network stability.

Chapter 2: Step-Growth Polymerization

2.1 Mechanism

Bi-functional monomers (AA + BB or AB) react by any free pair of functional groups. Unlike chain polymerization, reaction proceeds at all sizes simultaneously; high polymer forms only near complete conversion.

2.2 Carothers Equation

For equal stoichiometry and extent of reaction \( p \):

\[ \bar{X}_n = \frac{1}{1 - p}. \]

For high molecular weight (\( \bar X_n > 100 \)), \( p > 0.99 \). Stoichiometric imbalance \( r = N_A/N_B < 1 \) limits achievable \( X_n \):

\[ \bar X_n = \frac{1 + r}{1 + r - 2 r p}. \]

Monofunctional impurities act as chain stoppers, with similar catastrophic effects on achievable molecular weight.

2.3 Molecular Weight Distribution

Most probable distribution (Flory):

\[ N_x = N_0 (1-p) p^{x-1}, \quad PDI = 1 + p \to 2 \text{ as } p \to 1. \]

2.4 Industrial Examples

Nylon-6,6 from hexamethylenediamine and adipic acid; PET from ethylene glycol and terephthalic acid; polycarbonate from bisphenol A and phosgene (or diphenyl carbonate). Melt polymerization at high temperature drives equilibrium by removing byproduct (water, methanol) under vacuum.

Chapter 3: Chain-Growth Polymerization

3.1 Free-Radical Polymerization

Initiation: \( I_2 \to 2 I\cdot \), rate \( R_i = 2 f k_d [I_2] \) with efficiency \( f \). Propagation: \( P\cdot + M \to P\cdot \), rate \( R_p = k_p [M][P\cdot] \). Termination (combination or disproportionation): \( R_t = 2 k_t [P\cdot]^2 \).

Steady-state \( [P\cdot] \): \( [P\cdot] = (R_i/2k_t)^{1/2} \). Rate of polymerization:

\[ R_p = k_p [M]\left(\frac{f k_d [I_2]}{k_t}\right)^{1/2}. \]

Kinetic chain length \( \nu = R_p/R_i = k_p[M]/(2(fk_d k_t [I_2])^{1/2}) \). \( \bar X_n = \nu \) for disproportionation; \( 2\nu \) for combination.

3.2 Chain Transfer

Transfer to monomer, solvent, polymer, or chain transfer agent (CTA) reduces \( M_n \) without affecting \( R_p \). Mayo equation:

\[ \frac{1}{\bar X_n} = \frac{1}{\bar X_{n,0}} + \sum_i C_i\frac{[S_i]}{[M]}, \]

with \( C_i = k_{tr,i}/k_p \). Chain transfer to polymer produces long-chain branches—deliberate in LDPE manufacture.

3.3 Controlled/Living Radical Polymerization

ATRP, RAFT, NMP suppress termination by reversible capping of the growing radical. Result: narrow dispersity (PDI ~1.1), predictable \( M_n \propto \) conversion, access to block copolymers and complex architectures.

3.4 Ionic and Coordination Polymerization

Anionic (butyllithium initiators): truly living for nonpolar monomers in dry aprotic solvents, narrow distributions, block copolymer synthesis (styrene-butadiene-styrene). Cationic (for vinyl ethers, isobutylene). Coordination (Ziegler-Natta, metallocene, post-metallocene): stereocontrol for polyolefins; isotactic polypropylene is a signature industrial product.

3.5 Copolymerization

For A + B monomer mixture with reactivity ratios \( r_A, r_B \), instantaneous copolymer composition (Mayo-Lewis):

\[ \frac{d[A]}{d[B]} = \frac{[A](r_A[A] + [B])}{[B]([A] + r_B[B])}. \]

\( r_A r_B \approx 1 \): random (statistical); \( r_A r_B \approx 0 \): alternating; \( r_A, r_B \gg 1 \): blocky. Composition drift with conversion must be managed for uniform product.

Chapter 4: Solid-State Structure

4.1 Amorphous State and Glass Transition

Below \( T_g \), chain segmental motion freezes; polymer is a brittle glass. Above \( T_g \), it is rubbery or fluid. Factors raising \( T_g \): chain stiffness, bulky side groups, hydrogen bonding, crosslinking. Free-volume theory (WLF equation) relates temperature-dependent relaxation.

4.2 Crystallinity

Semi-crystalline polymers (PE, PP, PET, nylon) contain folded-chain lamellae (10 nm thick) organized into spherulites. Crystallization requires regular chain structure: linear PE (HDPE) crystallizes readily; heavily branched LDPE less; atactic PS not at all. Percent crystallinity (by DSC, X-ray, density) controls modulus, strength, and clarity.

4.3 Thermal Transitions

DSC traces show \( T_g \) as a step in heat capacity, \( T_m \) as an endotherm, and crystallization as an exotherm (on cooling). For semi-crystalline polymers both transitions exist; for amorphous only \( T_g \).

4.4 Structure-Property Correlation

Property windows: service above \( T_g \) (rubber), between \( T_g \) and \( T_m \) (leathery to rubbery), above \( T_m \) (melt). Processing windows: melt-processable above \( T_m \) (crystalline) or above \( T_g \) (amorphous).

Chapter 5: Mechanical Properties

5.1 Viscoelasticity

Polymers exhibit both elastic (solid-like) and viscous (fluid-like) response depending on timescale and temperature. Time-temperature superposition: curves at different temperatures overlay when plotted against reduced time \( t/a_T \). The WLF equation,

\[ \log a_T = \frac{-C_1(T - T_{ref})}{C_2 + T - T_{ref}}, \]

typically has \( C_1 \approx 17 \), \( C_2 \approx 52 \) K when \( T_{ref} = T_g \).

5.2 Rubber Elasticity

Statistical mechanics of a network of flexible chains gives stress-strain for uniaxial deformation \( \lambda \):

\[ \sigma = \frac{\rho R T}{M_c}\left(\lambda - \frac{1}{\lambda^2}\right), \]

with \( M_c \) the molecular weight between crosslinks. The modulus scales with \( T \), not \( 1/T \) as for crystals—a counterintuitive fingerprint of entropic elasticity.

5.3 Yield, Failure, and Toughness

Ductile (PC, PA) vs. brittle (PS, PMMA) response depends on temperature, strain rate, and notch geometry. Rubber toughening (HIPS, ABS) disperses rubber particles in a brittle matrix to absorb crack energy. Craze formation precedes brittle failure in glassy polymers.

5.4 Polymer Melts

Melt viscosity of entangled polymer:

\[ \eta_0 \propto M^{3.4}, \quad M > M_c, \]

where \( M_c \) is the critical entanglement molecular weight. Shear thinning at processing rates reflects chain disentanglement under flow. Rheology directly dictates extrudability, moldability, and fiber-spinnability.

Chapter 6: Characterization and Properties Mapping

6.1 Size and Shape

Radius of gyration \( R_g \sim M^\nu \) with \( \nu = 0.5 \) in \( \theta \)-solvent, 0.588 in good solvent (Flory exponent), 1/3 in melt (ideal chain). SANS, SAXS, light scattering, and intrinsic viscosity measurements probe chain dimensions.

6.2 Spectroscopic Identification

FTIR identifies functional groups; NMR determines tacticity and sequence distribution; mass spectrometry verifies repeat unit and end groups; XPS probes surfaces.

6.3 Thermal and Mechanical Testing

DSC, DMA, TGA, tensile/compressive/flexural, impact (Izod, Charpy), fatigue, creep. ASTM and ISO standards prescribe specimen geometry and conditions for comparable results.

6.4 Applications Mapping

Engineering selection matches property demands (modulus, toughness, chemical resistance, temperature range, optical, electrical, biocompatibility) to a candidate polymer class. Processing constraints (injection, extrusion, blow molding, thermoforming, casting) further narrow choice. The Ashby-style property map is the systematic tool; experience guides the finer judgment.