15.17.01 · orgchem / polymers-photochemistry

Polymer chemistry and photochemistry

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Anchor (Master): Odian, Principles of Polymerization, 4th ed. (Wiley, 2004); Turro, Modern Molecular Photochemistry (1991); Balzani & Scandola, Supramolecular Photochemistry (Ellis Horwood, 1991)

Intuition Beginner

Two ideas run through this unit: building very long molecules by joining many small ones, and using light to drive chemistry that heat cannot.

A polymer is a giant molecule built by linking hundreds or thousands of small monomer units into a chain. Polyethylene is thousands of ethylene ( $CH_{2} = CH_{2}$ ) units joined end to end. The way the chain is assembled decides its properties. In chain-growth polymerisation, a reactive centre at one end of a growing chain grabs the next monomer, one at a time, like adding cars to a train. In step-growth polymerisation, any two monomers can react, then those dimers react with others, the chain doubling in length at each stage.

Not all polymer chains are the same length. Some are short, some long. This spread is the molecular weight distribution, and two averages describe it. The number average $M_{n}$ treats every chain equally; the weight average $M_{w}$ gives long chains more weight. The ratio $M_{w} / M_{n}$ tells you how broad the spread is.

A polymer also has a glass transition temperature $T_{g}$ : below it the chain is rigid and glassy; above it the chains can slide and the material turns rubbery.

Photochemistry is the second idea. When a molecule absorbs a particle of light (a photon), one electron jumps to a higher energy level. That excited molecule can release the energy as light (fluorescence or phosphorescence), as heat, or by reacting. Some reactions only work with light: a thermal Diels-Alder reaction closes six-membered rings, but a $[2 + 2]$ cycloaddition to make a four-membered ring is forbidden by heat and allowed only by light.

Visual Beginner

Two panels sit side by side. The left panel shows a chain growing one monomer at a time from a reactive end (a star marks the active centre). The right panel shows the Jablonski diagram: a molecule absorbs a photon and jumps from the ground state $S_{0}$ up to an excited singlet $S_{1}$ , then drops back by fluorescence, or crosses to a triplet $T_{1}$ and returns slowly by phosphorescence.

Worked example Beginner

Compute $M_{n}$ , $M_{w}$ , and the breadth ratio for a small polyethylene sample.

A sample contains 100 chains in three length groups:

Chains	Mass of each chain (Da)
10	1 000
60	5 000
30	10 000

Step 1. Number-average $M_{n}$ . Add up the total mass, then divide by the number of chains.

Total mass $= 10 \times 1000 + 60 \times 5000 + 30 \times 10000 = 10000 + 300000 + 300000 = 610000$ Da.

$M_{n} = 610000/100 = 6100$ Da.

Step 2. Weight-average $M_{w}$ . Weight each chain by its own mass, so long chains count more.

Numerator $= 10 \times 100 0^{2} + 60 \times 500 0^{2} + 30 \times 1000 0^{2} = 1 0^{7} + 1.5 \times 1 0^{9} + 3.0 \times 1 0^{9} = 4.51 \times 1 0^{9}$ .

$M_{w} = (4.51 \times 1 0^{9}) /610000 \approx 7393$ Da.

Step 3. Breadth. The ratio $M_{w} / M_{n} = 7393/6100 \approx 1.21$ . A ratio of 1 means every chain is the same length; values near 2 are typical for step-growth polymers. This sample is fairly uniform.

Check your understanding Beginner

Exercise (easy, multiple choice).

Which statement best distinguishes chain-growth from step-growth polymerisation?

A. Chain-growth joins any two molecules; step-growth adds one monomer at a time to an active chain end. B. Chain-growth adds monomers one at a time to an active centre; step-growth joins any two molecules bearing complementary reactive groups. C. Both proceed identically; only the monomer differs. D. Chain-growth requires light; step-growth requires heat.

Hint

Think about where reaction happens in each. In one mode only a growing chain end is reactive; in the other every monomer carries two reactive ends.

Answer

B. Chain-growth uses an active centre (a radical, ion, or coordinated metal) that adds monomers one at a time. Step-growth uses monomers each bearing two functional groups, so any two molecules can condense, and chains double in length early on.

Formal definition Intermediate+

A polymer is a molecule of high relative molecular mass composed of repeating structural units (repeat units or mers) connected by covalent bonds, in which the number of repeat units (the degree of polymerisation, $D P$ or $X$ ) is large enough that adding or removing one unit does not change the bulk properties. The threshold conventionally sits near $D P \approx 50$ -- $100$ .

Polymerisation mechanisms

Chain-growth (addition) polymerisation. A monomer bearing a carbon-carbon double bond ( $CH_{2} = CHR$ ) is added to a reactive chain end. The chain grows only at the active centre. Four classes of active centre define four mechanisms:

Free-radical. An initiator (a peroxide or azo compound) fragments into radicals $R \cdot$ that open the vinyl double bond, leaving a carbon radical at the chain end. This radical adds the next monomer, regenerating a radical one unit further along. Termination occurs when two radicals meet (combination or disproportionation). The radical chemistry is exactly that of 15.08.01.
Cationic. A Lewis acid ( $BF_{3}$ , $AlCl_{3}$ ) protonates the monomer to give a carbocation at the chain end. Suitable only for electron-rich vinyl monomers (isobutylene, vinyl ethers) whose cation is stabilised.
Anionic. A strong base or organolithium opens the double bond to give a carbanion chain end. With careful purification, anionic polymerisation has no termination step (living polymerisation), giving narrow molecular weight distributions and enabling block copolymers.
Coordination (Ziegler-Natta and metallocene). A transition-metal catalyst (TiCl $_{3}$ + AlR $_{3}$ , or a metallocene such as $Cp_{2} ZrCl_{2}$ ) coordinates the monomer to a vacant site on the metal, then inserts it into the metal-carbon bond. This is the route to high-density polyethylene (HDPE) and to stereoregular polypropylene.

Step-growth (condensation) polymerisation. Each monomer carries two functional groups (a diol and a diacid, a diamine and a diacid, etc.). Any two molecules bearing complementary groups can react, releasing a small molecule (often water). Chains grow by pairwise coupling: monomers form dimers, dimers form tetramers, and so on. Nylon-6,6 (from hexamethylenediamine + adipic acid) and polyesters (polyethylene terephthalate, PET) are step-growth polymers.

Molecular weight averages

For a sample containing $N_{i}$ chains each of molar mass $M_{i}$ :

M_{n} = \frac{\sum _{i} N _{i} M _{i}}{\sum _{i} N _{i}}, M_{w} = \frac{\sum _{i} N _{i} M _{i}^{2}}{\sum _{i} N _{i} M _{i}}

The polydispersity index $PDI = M_{w} / M_{n} \geq 1$ measures the breadth of the distribution. Living polymerisation reaches $PDI \approx 1.01$ -- $1.10$ ; conventional radical polymerisation gives $PDI \approx 1.5$ -- $2.0$ ; step-growth approaches $PDI \approx 2$ .

The glass transition and the amorphous state

Below the glass transition temperature $T_{g}$ , amorphous polymer segments are frozen in place and the material is glassy and brittle. Above $T_{g}$ , segments acquire enough thermal energy to rotate and translate locally (not the whole chain), and the material becomes leathery, then rubbery. $T_{g}$ rises with chain stiffness, with the strength of interchain forces, and with crosslink density. Crystalline polymers additionally show a sharp melting temperature $T_{m}$ from the crystalline domains.

Photochemistry: the Jablonski diagram and the laws

Absorption. A molecule absorbs a photon of energy $h ν$ matching the gap between the ground electronic state $S_{0}$ and an excited singlet state $S_{1}$ (or higher $S_{n}$ ). The transition is fast ( $\sim 1 0^{- 15}$ s) and, by the spin-conservation rule, preserves spin multiplicity: singlet to singlet.

Vibrational relaxation and internal conversion. Within $1 0^{- 12}$ s the excited molecule sheds its excess vibrational energy to the surroundings and settles at the lowest vibrational level of $S_{1}$ (Kasha's rule: emission usually occurs from the lowest excited state of a given multiplicity).

Fluorescence. The molecule returns from $S_{1}$ to $S_{0}$ by emitting a photon ( $1 0^{- 9}$ -- $1 0^{- 7}$ s). Because emission starts from the relaxed $S_{1}$ and the absorption ended at a higher $S_{n}$ , the fluorescence photon has lower energy (longer wavelength) than the absorbed photon — the Stokes shift.

Intersystem crossing and phosphorescence. A spin flip converts the singlet $S_{1}$ to the triplet $T_{1}$ (intersystem crossing, ISC). The $T_{1} \to S_{0}$ transition is spin-forbidden and slow ( $1 0^{- 3}$ to $1 0^{2}$ s); its emission is phosphorescence, red-shifted relative to fluorescence because $T_{1}$ lies below $S_{1}$ .

The Stark-Einstein photochemical equivalence law. Each photon absorbed activates exactly one molecule: one primary absorption event per quantum ^{[StarkEinstein1908]}. The quantum yield $Φ$ of a process is the ratio of molecules undergoing that process to photons absorbed:

Φ = \frac{number of molecules reacting}{number of photons absorbed}

For fluorescence, $Φ_{F} \leq 1$ . For a chain reaction (radical halogenation, photopolymerisation), the overall quantum yield can reach $1 0^{4}$ -- $1 0^{6}$ , because one photon initiates a chain of thousands of propagation steps — but the primary yield remains one, in keeping with the equivalence law.

Photochemical pericyclic reactions. The Woodward-Hoffmann selection rules derived in 15.08.01 predict that some pericyclic reactions allowed thermally are forbidden photochemically and vice versa. The thermal $[4 + 2]$ Diels-Alder cycloaddition is allowed; the thermal $[2 + 2]$ is forbidden. Under irradiation, one electron is promoted from HOMO to LUMO, inverting the orbital symmetry: the photochemical $[2 + 2]$ becomes allowed, and the photochemical $[4 + 2]$ becomes forbidden. Electrocyclic ring closures likewise invert: a $4 n$ system closes conrotatorily under heat but disrotatorily under light ^{[WoodwardHoffmann1969]}.

Counterexamples to common slips

"Step-growth polymers must release water." The defining feature is reaction between bifunctional monomers, not the leaving group. Polyurethanes form by step-growth addition of a diol to a diisocyanate with no small-molecule byproduct; they are step-growth but not condensation.
"A quantum yield above 1 violates the Stark-Einstein law." It does not. The law governs the primary absorption step (one molecule excited per photon). Chain reactions amplify the overall yield through subsequent dark steps that consume no new photons.
"Phosphorescence is just slow fluorescence." Phosphorescence is emission from a triplet state of different spin multiplicity from the singlet ground state. The long lifetime reflects the spin-forbidden $T_{1} \to S_{0}$ transition, not merely a slow radiative rate within the same manifold.
"All polymers are plastics." Proteins, DNA, cellulose, and natural rubber are polymers; many are water-soluble or elastomeric. "Plastic" describes a processing behaviour, not a chemical class.

Key derivation Intermediate+

Proposition (Carothers equation). Consider an equimolar, bifunctional step-growth polymerisation of monomers $A - A$ and $B - B$ (for example a diol and a diacid), in which every functional group has equal reactivity and the only bond-forming event is the condensation of an $A$ group with a $B$ group. Let $N_{0}$ be the initial number of monomer molecules and let $p \in [0, 1)$ be the extent of reaction, the fraction of $A$ (equivalently $B$ ) functional groups consumed. Then the number-average degree of polymerisation is

X_{n} = \frac{1}{1 - p} .

Proof. Each condensation event consumes exactly two functional groups (one $A$ , one $B$ ) and joins two previously separate molecules into one. The number of molecules therefore drops by one per condensation. If $p$ is the fraction of functional groups consumed, the number of condensation events that have occurred is $p N_{0}$ (there are $N_{0}$ $A$ groups initially, one per monomer molecule, and $p N_{0}$ have reacted). Hence the number of molecules remaining is

N = N_{0} - p N_{0} = N_{0} (1 - p) .

The number-average degree of polymerisation is the total number of monomer units incorporated divided by the number of molecules that contain them. Every initial monomer molecule is incorporated into some chain, so the total number of repeat units equals $N_{0}$ . Therefore

X_{n} = \frac{N _{0}}{N} = \frac{N _{0}}{N _{0} ( 1 - p )} = \frac{1}{1 - p} . □

The consequence is severe. To reach $X_{n} = 10$ (an oligomer), the reaction need only reach $p = 0.90$ . To reach $X_{n} = 100$ requires $p = 0.99$ , and a useful textile-grade polyester near $X_{n} = 200$ requires $p = 0.995$ . This is why step-growth polymerisation demands scrupulous stoichiometric balance, high monomer purity, and efficient removal of the condensation byproduct: a single percent of unreactive impurity caps the attainable chain length. With a small stoichiometric imbalance $r < 1$ between the two monomer types, the Carothers equation generalises to $X_{n} = (1 + r) / (1 + r - 2 r p)$ , and complete conversion still yields only a finite chain length set by the deficient monomer.

Bridge. The Carothers equation builds toward the Flory-Schulz distribution derived in the Full proof set below, where the same extent-of-reaction parameter $p$ generates the entire chain-length distribution and the polydispersity index $PDI = 1 + p \to 2$ . This is exactly the content of the most-probable distribution: the foundational reason step-growth polymers cluster near $PDI \approx 2$ is that each chain-terminating event is a rare independent draw with probability $p$ at each link. The central insight — that a single kinetic parameter controls both the average and the spread — appears again in the radical-polymerisation rate law of the Advanced results, where the initiator concentration sets the kinetic chain length, and the bridge is that both chain-growth and step-growth reduce to a competition between a propagation rate and a termination probability. The Carothers relation generalises to branched systems (the Flory-Stockmayer gel point) and is dual to the living-polymerisation limit, where $X_{n}$ is set not by conversion but by the monomer-to-initiator ratio.

Exercises Intermediate+

Exercise 4 (medium, short-answer).

Explain the difference between fluorescence and phosphorescence in terms of (i) the spin multiplicity of the emitting state, (ii) the typical lifetime, and (iii) the wavelength of the emitted light relative to the absorbed light.

Hint

Use the Jablonski diagram. Which transition is spin-allowed and which is spin-forbidden? Where does the triplet sit relative to the lowest singlet?

Answer

(i) Fluorescence is emission from the excited singlet $S_{1}$ to the singlet ground state $S_{0}$ (spin-allowed); phosphorescence is emission from the triplet $T_{1}$ to $S_{0}$ (spin-forbidden). (ii) Fluorescence lifetimes are $1 0^{- 9}$ -- $1 0^{- 7}$ s; phosphorescence lifetimes are $1 0^{- 3}$ -- $1 0^{2}$ s because the spin-forbidden transition is slow. (iii) Both are Stokes-shifted to longer wavelength than the absorbed light, but phosphorescence is further red-shifted because $T_{1}$ lies below $S_{1}$ (the intersystem crossing $S_{1} \to T_{1}$ is exergonic).

Exercise 6 (medium, short-answer).

Explain why the thermal $[2 + 2]$ cycloaddition of two ethylenes is forbidden, while the photochemical $[2 + 2]$ is allowed, using the frontier-molecular-orbital picture of 15.08.01.

Hint

In the thermal reaction, match the HOMO of one ethylene with the LUMO of the other and inspect the sign of overlap at both forming bonds. In the photochemical reaction, one ethylene has an electron promoted from HOMO to LUMO; how does the relevant frontier orbital change?

Answer

Thermally, the HOMO of one ethylene ( $π$ ) and the LUMO of the other ( $π^{*}$ ) overlap constructively at one forming bond but destructively at the other: one bonding plus one antibonding interaction makes the suprafacial-suprafacial $[2 + 2]$ symmetry-forbidden. Photochemically, one ethylene has an electron promoted to $π^{*}$ , so its frontier orbital that meets the partner's HOMO is now the LUMO with inverted phase at one terminus. The phase inversion repairs the destructive overlap, both forming bonds become bonding, and the photochemical $[2 + 2]$ is symmetry-allowed. This is precisely the Woodward-Hoffmann selection rule: promoting one electron inverts the orbital symmetry and swaps which cycloadditions are allowed.

Exercise 7 (medium, short-answer).

A sample of polypropylene produced by a Ziegler-Natta catalyst is rigid and crystalline, whereas a sample made by radical polymerisation is amorphous and oily. Explain the structural origin of this difference and why crystallinity raises $T_{g}$ and $T_{m}$ .

Hint

Consider the stereochemistry of successive methyl-bearing stereocentres along the chain. What does "isotactic" mean, and why does it help chains pack?

Answer

Ziegler-Natta catalysis enforces stereoregular insertion, producing isotactic polypropylene (all methyl groups on the same side of the backbone). The regular chain folds and packs into crystalline lamellae, giving a sharp $T_{m}$ near $165^{\circ} C$ and a high modulus. Radical polymerisation gives atactic polypropylene (random methyl stereochemistry), which cannot pack regularly and remains amorphous, rubbery, and low-melting. Crystallinity raises the effective $T_{g}$ and $T_{m}$ because the crystalline domains act as physical crosslinks that lock amorphous segments in place and resist chain slippage.

Exercise 8 (hard, symbolic).

The Stern-Volmer equation for dynamic quenching of fluorescence is $F_{0} / F = 1 + K_{S V} [Q]$ , where $K_{S V} = k_{q} τ_{0}$ . A fluorophore has unquenched lifetime $τ_{0} = 12 ns$ . Adding quencher at $[Q] = 0.010 mol L^{- 1}$ reduces the fluorescence intensity to 40% of its original value ( $F / F_{0} = 0.40$ ). Compute $K_{S V}$ and the bimolecular quenching rate constant $k_{q}$ in $L mol^{- 1} s^{- 1}$ , and comment on whether $k_{q}$ is close to the diffusion limit ( $\sim 1 0^{10} L mol^{- 1} s^{- 1}$ in water).

Hint

From $F_{0} / F = 1/0.40 = 2.5$ , subtract 1 to get $K_{S V} [Q]$ , divide by $[Q]$ to isolate $K_{S V}$ , then $k_{q} = K_{S V} / τ_{0}$ .

Answer

$F_{0} / F = 1/0.40 = 2.5$ , so $K_{S V} [Q] = 2.5 - 1 = 1.5$ . Thus $K_{S V} = 1.5/0.010 = 150 L mol^{- 1}$ . With $τ_{0} = 12 ns = 1.2 \times 1 0^{- 8} s$ , $k_{q} = K_{S V} / τ_{0} = 150/ (1.2 \times 1 0^{- 8}) = 1.25 \times 1 0^{10} L mol^{- 1} s^{- 1}$ . This is at the diffusion-controlled limit in water, indicating that essentially every encounter between an excited fluorophore and a quencher molecule results in quenching.

Exercise 9 (hard, symbolic).

The free-radical polymerisation of styrene follows the rate law $R_{p} = k_{p} (f k_{d} [I] / k_{t})^{1/2} [M]$ , with propagation rate constant $k_{p} = 50 L mol^{- 1} s^{- 1}$ , termination $k_{t} = 1 0^{8} L mol^{- 1} s^{- 1}$ , initiator efficiency $f = 0.6$ , initiator decomposition $k_{d} = 1 0^{- 4} s^{- 1}$ , monomer $[M] = 8.0 mol L^{- 1}$ , initiator $[I] = 0.010 mol L^{- 1}$ . Compute the rate of polymerisation $R_{p}$ in $mol L^{- 1} s^{- 1}$ , and briefly explain why the rate depends on the square root of initiator concentration.

Hint

Compute the steady-state radical concentration $[M \cdot] = (f k_{d} [I] / k_{t})^{1/2}$ first, then $R_{p} = k_{p} [M \cdot] [M]$ . The square root comes from the steady-state balance between bimolecular termination and unimolecular initiation.

Answer

$[M \cdot] = (0.6 \times 1 0^{- 4} \times 0.010/1 0^{8})^{1/2} = (6.0 \times 1 0^{- 8} /1 0^{8})^{1/2} = (6.0 \times 1 0^{- 16})^{1/2} = 2.45 \times 1 0^{- 8} mol L^{- 1}$ . Then $R_{p} = k_{p} [M \cdot] [M] = 50 \times 2.45 \times 1 0^{- 8} \times 8.0 = 9.8 \times 1 0^{- 6} mol L^{- 1} s^{- 1}$ .

The square-root dependence arises because initiation is unimolecular in initiator (rate $f k_{d} [I]$ ) but termination is bimolecular in radicals (rate $k_{t} [M \cdot]^{2}$ ). Setting the two rates equal at steady state gives $[M \cdot] \propto [I]^{1/2}$ , and since propagation is first-order in radical, $R_{p} \propto [I]^{1/2}$ . Doubling the initiator raises the rate by only $2$ .

Advanced results Master

Theorem 1 (free-radical polymerisation rate law). For a free-radical polymerisation with initiator decomposition rate constant $k_{d}$ and efficiency $f$ , propagation rate constant $k_{p}$ , and bimolecular termination rate constant $k_{t}$ , the steady-state rate of monomer consumption is

R_{p} = k_{p} (\frac{f k _{d} [ I ]}{k _{t}})^{1/2} [M] .

The kinetic chain length $ν$ — the average number of monomers added per radical generated — is

ν = \frac{R _{p}}{R _{i}} = \frac{k _{p} [ M ]}{( 2 f k _{d} k _{t} [ I ] ) ^{1/2}},

and, in the absence of chain transfer, $X_{n} = 2 ν$ for termination by combination and $X_{n} = ν$ for disproportionation.

The square-root initiator dependence is the diagnostic signature of any radical chain process and reappears in the radical halogenation kinetics of 15.08.01. The Mayo equation extends $X_{n}$ to include chain transfer to monomer, solvent, or polymer: $1/ X_{n} = 1/ X_{n, 0} + C_{M} + C_{S} [S] / [M]$ , where $C_{M} = k_{t r, M} / k_{p}$ is the chain-transfer constant.

Theorem 2 (Stark-Einstein equivalence law and quantum yield). In the primary photochemical act, one photon activates one molecule. The quantum yield of a process $j$ is

Φ_{j} = \frac{number of molecules undergoing process j}{number of photons absorbed} = \frac{rate of process j}{rate of photon absorption} .

For a molecule excited to $S_{1}$ that may fluoresce ( $k_{r}$ ), undergo internal conversion ( $k_{i c}$ ), or cross to $T_{1}$ by intersystem crossing ( $k_{i sc}$ ), the fluorescence quantum yield is $Φ_{F} = k_{r} / (k_{r} + k_{i c} + k_{i sc})$ and the triplet yield is $Φ_{T} = k_{i sc} / (k_{r} + k_{i c} + k_{i sc})$ , with $Φ_{F} + Φ_{i c} + Φ_{T} = 1$ .

Theorem 3 (Stern-Volmer dynamic quenching). If a quencher $Q$ deactivates the excited state $S_{1}$ with bimolecular rate constant $k_{q}$ , the ratio of unquenched to quenched fluorescence intensity is

\frac{F _{0}}{F} = 1 + K_{S V} [Q], K_{S V} = k_{q} τ_{0},

where $τ_{0}$ is the unquenched excited-state lifetime. A linear Stern-Volmer plot with matching lifetime ratio $τ_{0} / τ$ indicates purely dynamic (collisional) quenching; an upward curvature signals combined dynamic and static quenching through a ground-state complex.

Theorem 4 (photochemical Woodward-Hoffmann selection rules). For an electrocyclic reaction involving $N$ $π$ -electrons:

$N$	Thermal mode	Photochemical mode
$4 n$	conrotatory	disrotatory
$4 n + 2$	disrotatory	conrotatory

The photochemical rule inverts the thermal rule because the promoted electron swaps the symmetry of the highest occupied frontier orbital that controls the ring closure. Cyclobutene opens conrotatorily under heat (as in 15.08.01) and disrotatorily under irradiation; 1,3,5-hexatriene (6 $π$ electrons) closes disrotatorily under heat and conrotatorily under light. The photochemical electrocyclic ring opening of 7-dehydrocholesterol in skin to previtamin D $_{3}$ is the photochemical step of vitamin D biosynthesis ^{[WoodwardHoffmann1969]}.

Theorem 5 (Cossee-Arlman mechanism of Ziegler-Natta polymerisation). In a Ti(III) chloride catalyst with alkylaluminium cocatalyst, the growing polymer chain is bound to a titanium centre bearing one vacant coordination site. An olefin coordinates to the vacant site, migratory insertion into the Ti-C bond lengthens the chain by one unit and regenerates the vacancy on the opposite face. The stereochemistry of successive insertions is enforced by the chiral, ligand-defined geometry of the metal site, producing isotactic polypropylene. Metallocene catalysts ( $Cp_{2} ZrCl_{2}$ with methylaluminoxane) replace the heterogeneous Ti surface with a well-defined single-site catalyst whose ligand symmetry ( $C_{2}$ , $C_{s}$ , or $C_{2 v}$ ) prescribes isotactic, syndiotactic, or atactic stereochemistry on demand.

Applications: photolithography, photoredox catalysis, vision

Photolithography. A photoresist is a polymer film bearing a photoacid generator or photoinitiator. Irradiation through a mask generates acid (chemically amplified resists) or radicals locally, altering solubility so that the exposed (positive-tone) or unexposed (negative-tone) regions dissolve in developer. The entire integrated-circuit manufacturing sequence relies on progressively shorter wavelengths: 436 nm (g-line), 365 nm (i-line), 248 nm (KrF excimer), 193 nm (ArF), and 13.5 nm (extreme ultraviolet, EUV). The $[2 + 2]$ photocycloaddition of cinnamate side chains is a negative-tone crosslinking mechanism; the acid-catalysed deprotection of a $t$ -butyl ester on a poly(hydroxystyrene) is a positive-tone mechanism.

Photoredox catalysis. A visible-light-absorbing photocatalyst — typically a ruthenium or iridium polypyridyl complex ( $[Ru(bpy)_{3}]^{2 +}$ , $[Ir(ppy)_{2} (dtbbpy)]^{+}$ ) or an organic dye — reaches a long-lived triplet excited state on irradiation. This excited state is both a stronger reductant and a stronger oxidant than the ground state, enabling single-electron transfer to or from an organic substrate that would be inaccessible by thermal chemistry. Coupled with a co-catalyst or sacrificial reductant/oxidant, photoredox catalysis drives $α$ -aminoalkylation, trifluoromethylation, and dual nickel-photoredox cross-coupling — reactions that have reshaped pharmaceutical process chemistry since 2008.

Vision. The 11-cis-retinal chromophore is covalently bound as a Schiff base to the protein opsin in rhodopsin. Absorption of a single photon isomerises 11-cis-retinal to all-trans-retinal by a photochemical $[1, 5]$ -type torsion about the C11=C12 bond in roughly 200 femtoseconds. The resulting conformational change triggers the G-protein cascade of phototransduction: one photon activates one rhodopsin, which activates hundreds of transducins, amplifying the signal enough that a human eye can detect a single photon. This is the Stark-Einstein law exploited at its physiological limit.

Synthesis. The Carothers equation, the radical rate law, the Stark-Einstein equivalence law, and the Stern-Volmer relation are four instantiations of a single abstract pattern: a macroscopic observable (chain length, reaction rate, quantum yield, fluorescence intensity) set by the ratio of a productive rate to a competing loss rate, and the foundational reason each form looks the way it does is that every polymerisation and every photochemical act reduces to a kinetic competition. This is exactly the unity that appears again in 15.08.01, where the radical chain length is the same ratio of propagation to termination, and in 15.11.01, where NMR observables encode structural parameters through analogous rate and population ratios. Putting these together with the photochemical Woodward-Hoffmann rules, the central insight is that light and heat are not different reagents but different orbital populations, so a photochemical reaction is a thermal reaction run on an electronically reconfigured frontier manifold. The bridge is that polymer chemistry and photochemistry meet in photolithography, photopolymerisation, and photoredox catalysis, where a photon both initiates a chain and sets its stereochemistry, and the field generalises toward single-site catalysis, EUV resist design, and photopharmacology.

Full proof set Master

Proposition 1 (Flory-Schulz most-probable distribution and step-growth polydispersity). For an equimolar bifunctional step-growth polymerisation at extent of reaction $p$ , the mole fraction of chains with degree of polymerisation $x$ is

N_{x} = (1 - p) p^{x - 1},

the number-average and weight-average degrees of polymerisation are $X_{n} = 1/ (1 - p)$ and $X_{w} = (1 + p) / (1 - p)$ , and the polydispersity index is $PDI = X_{w} / X_{n} = 1 + p$ , approaching 2 as $p \to 1$ .

Proof. A chain of length $x$ contains $x$ repeat units connected by $x - 1$ formed bonds. Each bond is present with probability $p$ (the probability that a given pair of functional groups has reacted), and the single unreacted terminal group that caps the chain is present with probability $1 - p$ . The $x - 1$ bonds and the terminal cap are independent, so

N_{x} = p^{x - 1} (1 - p), x = 1, 2, 3, \dots

This is a geometric distribution, and the normalisation follows from the geometric series:

x = 1 \sum \infty N_{x} = (1 - p) x = 1 \sum \infty p^{x - 1} = (1 - p) \cdot \frac{1}{1 - p} = 1.

The number-average degree of polymerisation is

X_{n} = x = 1 \sum \infty x N_{x} = (1 - p) x = 1 \sum \infty x p^{x - 1} .

Using the standard series $\sum_{x = 1}^{\infty} x p^{x - 1} = 1/ (1 - p)^{2}$ for $∣ p ∣ < 1$ ,

X_{n} = (1 - p) \cdot \frac{1}{( 1 - p ) ^{2}} = \frac{1}{1 - p} .

The second moment is

x = 1 \sum \infty x^{2} N_{x} = (1 - p) x = 1 \sum \infty x^{2} p^{x - 1} = (1 - p) \cdot \frac{1 + p}{( 1 - p ) ^{3}} = \frac{1 + p}{( 1 - p ) ^{2}},

using $\sum x^{2} p^{x - 1} = (1 + p) / (1 - p)^{3}$ . The weight-average degree of polymerisation is

X_{w} = \frac{\sum x ^{2} N _{x}}{\sum x N _{x}} = \frac{( 1 + p ) / ( 1 - p ) ^{2}}{1/ ( 1 - p )} = \frac{1 + p}{1 - p} .

Therefore

PDI = \frac{X _{w}}{X _{n}} = \frac{( 1 + p ) / ( 1 - p )}{1/ ( 1 - p )} = 1 + p p \to 1 2. □

The Flory-Schulz distribution explains why every high-conversion step-growth polymer — nylon, polyester, polyurethane — clusters near $PDI = 2$ : the most-probable distribution is the universal attractor of pairwise, statistically independent chain growth. Narrower distributions ( $PDI$ near 1) require suppressing the random termination that broadens the distribution, which is precisely what living anionic or controlled-radical polymerisation achieves by maintaining a constant, slowly growing number of chains.

Proposition 2 (Stern-Volmer quenching from the Jablonski rate equations). Consider a fluorophore excited to $S_{1}$ with total unquenched decay rate $Γ_{0} = k_{r} + k_{i c} + k_{i sc}$ , so the unquenched lifetime is $τ_{0} = 1/ Γ_{0}$ and the steady-state concentration of excited molecules under a photon absorption rate $I_{a}$ is $[S_{1}]_{0} = I_{a} / Γ_{0}$ . Adding a quencher at concentration $[Q]$ introduces an additional decay pathway with rate $k_{q} [Q]$ . Then the ratio of unquenched to quenched fluorescence intensities satisfies $F_{0} / F = 1 + k_{q} τ_{0} [Q]$ .

Proof. Fluorescence intensity is proportional to the rate of radiative decay, $F \propto k_{r} [S_{1}]$ . Under steady-state excitation, the rate of formation of $S_{1}$ equals the rate of its decay. Without quencher, $I_{a} = Γ_{0} [S_{1}]_{0}$ , so $[S_{1}]_{0} = I_{a} / Γ_{0}$ and $F_{0} \propto k_{r} I_{a} / Γ_{0}$ . With quencher, the total decay rate becomes $Γ = Γ_{0} + k_{q} [Q]$ , so $[S_{1}] = I_{a} / (Γ_{0} + k_{q} [Q])$ and $F \propto k_{r} I_{a} / (Γ_{0} + k_{q} [Q])$ . Taking the ratio,

\frac{F _{0}}{F} = \frac{k _{r} I _{a} / Γ _{0}}{k _{r} I _{a} / ( Γ _{0} + k _{q} [ Q ])} = \frac{Γ _{0} + k _{q} [ Q ]}{Γ _{0}} = 1 + \frac{k _{q} [ Q ]}{Γ _{0}} = 1 + k_{q} τ_{0} [Q],

since $τ_{0} = 1/ Γ_{0}$ . Writing $K_{S V} = k_{q} τ_{0}$ gives the stated Stern-Volmer form $F_{0} / F = 1 + K_{S V} [Q]$ . The same algebra applied to lifetimes gives $τ_{0} / τ = 1 + K_{S V} [Q]$ , so dynamic quenching quenches intensity and lifetime by the same factor — the diagnostic that distinguishes it from static quenching, which quenches intensity only. $□$

Connections Master

Radical and pericyclic reactions 15.08.01. Free-radical polymerisation is a direct application of the radical chain chemistry developed there: the same steady-state radical balance, the same propagation-versus-termination competition, and the same persistent-radical logic reappear in controlled-radical methods (ATRP, RAFT, NMP). The photochemical Woodward-Hoffmann rules of this unit invert the thermal selection rules derived in 15.08.01, because promoting one electron reconfigures the controlling frontier orbital — so the two units together give the complete (thermal + photochemical) pericyclic selection table.
NMR spectroscopy of organic molecules 15.11.01. Molecular weight and microstructure of synthetic polymers are measured by NMR: end-group analysis gives absolute $M_{n}$ , and the tacticity sequence (meso/racemic dyads and triads) is read directly from $^{1}$ H and $^{13}$ C chemical shifts. The same spin-coupling and integration logic of 15.11.01 becomes a stereochemical probe of how a Ziegler-Natta or metallocene catalyst inserts each monomer.
Carbonyl chemistry 15.07.01. Step-growth polyesterification and polyamidation are condensations of the carbonyl functional groups of 15.07.01 — the same nucleophilic acyl substitution and Claisen/Michael chemistry, iterated to macromolecular length. Photoredox catalysis, in turn, exploits the excited-state redox chemistry of carbonyls and polypyridyl complexes, drawing on the carbonyl electronic structure treated there.
Natural products and organocatalysis 15.16.01. Living polymerisation and organocatalysis share the design principle of a reversible, chiral or persistent active centre that controls the stereochemical outcome of each bond-forming event. The photoredox-nickel dual catalysis of this unit and the iminium/enamine manifolds of 15.16.01 together define modern mild C-C bond formation, and both draw on the same notion of catalyst turnover.
General-chemistry spectroscopy 14.12.01. The Jablonski diagram and the Stokes shift rest on the energy-level and absorption principles introduced at introductory breadth in 14.12.01. This unit deepens the photochemical half of that survey: quantum yield, fluorescence lifetime, and Stern-Volmer quenching are the detailed content that the introductory unit only names.

Historical & philosophical context Master

The existence of covalent macromolecules was contested for two decades. Hermann Staudinger proposed in 1920 that rubber, starch, and cellulose consist of long chains joined by ordinary covalent bonds rather than colloidal aggregates held together by weak forces ^{[Staudinger1920]}. The establishment resisted: a leading colloid chemist told Staudinger to "drop the idea of large molecules." Staudinger persisted, and by the 1930s the viscosity and osmometry measurements from his group and from Wallace Carothers at DuPont had settled the question. Staudinger received the Nobel Prize in Chemistry in 1953, the citation naming his work "for his discoveries in the field of macromolecular chemistry." Carothers' 1929 derivation of the extent-of-reaction relation ^{[Carothers1929]} turned polymerisation from empirical art into predictive kinetics; his synthesis of nylon in 1935 was the first fully synthetic fibre, decisive in the Second World War.

Stereospecific catalysis arrived in 1953-1954. Karl Ziegler found that triethylaluminium with titanium tetrachloride polymerises ethylene at low pressure to linear high-density polyethylene, and Giulio Natta extended the system to propylene, discovering that the catalyst controls the stereochemistry of successive insertions to give crystalline isotactic polypropylene ^{[ZieglerNatta1955]}. Natta coined the terms isotactic, syndiotactic, and atactic to describe the three tacticities. Ziegler and Natta shared the 1963 Nobel Prize in Chemistry. The single-site metallocene catalysts of the 1980s and 1990s completed the programme, making tacticity a dial rather than a discovery.

Photochemistry's foundational law emerged even earlier. Johannes Stark stated in 1908, and Albert Einstein derived from the quantum theory of radiation in 1912, that each photon absorbed activates exactly one molecule — the photochemical equivalence law ^{[StarkEinstein1908]}. The law anchored the young quantum theory to a measurable chemical prediction: one quantum in, one excited molecule out. The Jablonski diagram, introduced by Aleksander Jablonski in 1933, organised the singlet and triplet manifolds and the radiative and non-radiative transitions between them, explaining fluorescence, phosphorescence, and the Stokes shift in a single figure.

The unification of photochemistry with the orbital-symmetry theory of pericyclic reactions came in 1965. Woodward and Hoffmann's conservation of orbital symmetry ^{[WoodwardHoffmann1969]} predicted that a pericyclic reaction's stereochemical outcome depends on whether it runs thermally or photochemically, because promoting one electron swaps which frontier orbital controls the closure. The photochemical $[2 + 2]$ cycloaddition and the photochemical electrocyclic ring opening of 7-dehydrocholesterol to previtamin D $_{3}$ in skin are direct consequences. Hoffmann shared the 1981 Nobel Prize with Kenichi Fukui; Woodward had died in 1979.

The philosophical thread connecting these histories is the maturation of chemistry from cataloguing substances to predicting transformations from first principles: Staudinger's chains, Carothers' kinetics, the Stark-Einstein quantum, and the Woodward-Hoffmann rules each converted an empirical observation into a quantitative, predictive law. Polymer chemistry and photochemistry together epitomise that maturation — one shows how repetition of a single bond-forming event builds macroscopic materials, the other how a single photon rewrites the rules of an entire reaction class.

Bibliography Master

@article{Staudinger1920,
  author = {Staudinger, H.},
  title = {Uber Polymerisation},
  journal = {Ber. Dtsch. Chem. Ges.},
  volume = {53},
  pages = {1073--1085},
  year = {1920}
}

@article{Carothers1929,
  author = {Carothers, W. H.},
  title = {Studies on Polymerization and Ring Formation. {I.} Introduction to the General Theory of Condensation Polymers},
  journal = {J. Am. Chem. Soc.},
  volume = {51},
  pages = {2548--2559},
  year = {1929}
}

@article{Flory1936,
  author = {Flory, P. J.},
  title = {Molecular Size Distribution in Linear Condensation Polymers},
  journal = {J. Am. Chem. Soc.},
  volume = {58},
  pages = {1877--1885},
  year = {1936}
}

@article{StarkEinstein1908,
  author = {Stark, J.},
  title = {Weitere Bemerkungen uber die physikalische Entstehung der chemischen Valenzkr{\"a}fte},
  journal = {Phys. Z.},
  volume = {9},
  pages = {889--894},
  year = {1908}
}

@article{Einstein1912,
  author = {Einstein, A.},
  title = {Thermodynamische Begrundung des photochemischen \"{A}quivalentgesetzes},
  journal = {Ann. Phys.},
  volume = {338},
  pages = {819--838},
  year = {1912}
}

@article{ZieglerNatta1955,
  author = {Ziegler, K. and Holzkamp, E. and Breil, H. and Martin, H.},
  title = {Das Mulheimer Normaldruck-Polyathylen-Verfahren},
  journal = {Angew. Chem.},
  volume = {67},
  pages = {425},
  year = {1955}
}

@article{Natta1955,
  author = {Natta, G.},
  title = {Une nuova classe di polimeri di $\alpha$-olefine con regolarita di struttura},
  journal = {J. Polym. Sci.},
  volume = {16},
  pages = {143--154},
  year = {1955}
}

@article{Cossee1964,
  author = {Cossee, P.},
  title = {Ziegler-Natta Catalysis {I}. Mechanism of Polymerization of $\alpha$-Olefins with Ziegler-Natta Catalysts},
  journal = {J. Catal.},
  volume = {3},
  pages = {80--88},
  year = {1964}
}

@article{WoodwardHoffmann1969,
  author = {Woodward, R. B. and Hoffmann, R.},
  title = {The Conservation of Orbital Symmetry},
  journal = {Angew. Chem. Int. Ed.},
  volume = {8},
  pages = {781--853},
  year = {1969}
}

@article{SternVolmer1919,
  author = {Stern, O. and Volmer, M.},
  title = {Uber die Abklingungszeit der Fluoreszenz},
  journal = {Phys. Z.},
  volume = {20},
  pages = {183--188},
  year = {1919}
}

@book{Odian2004,
  author = {Odian, G.},
  title = {Principles of Polymerization},
  edition = {4th},
  publisher = {Wiley-Interscience},
  address = {Hoboken},
  year = {2004}
}

@book{Turro1991,
  author = {Turro, N. J.},
  title = {Modern Molecular Photochemistry},
  publisher = {University Science Books},
  address = {Sausalito},
  year = {1991}
}

@book{BalzaniScandola1991,
  author = {Balzani, V. and Scandola, F.},
  title = {Supramolecular Photochemistry},
  publisher = {Ellis Horwood},
  address = {Chichester},
  year = {1991}
}

@book{Clayden2012,
  author = {Clayden, J. and Greeves, N. and Warren, S.},
  title = {Organic Chemistry},
  edition = {2nd},
  publisher = {Oxford University Press},
  address = {Oxford},
  year = {2012}
}

Prerequisites

15.08.01

Tier anchors

beginner: Klein, Organic Chemistry as a Second Language — polymer and photochemistry chapters in plain English; Crash Course Organic Chemistry — polymers and photochemistry episodes
intermediate: Clayden, Greeves & Warren — Organic Chemistry 2nd ed., polymer chemistry chapter; Turro, Modern Molecular Photochemistry (University Science Books, 1991), Chs. 1–5
master: Odian, Principles of Polymerization, 4th ed. (Wiley, 2004); Turro, Modern Molecular Photochemistry (1991); Balzani & Scandola, Supramolecular Photochemistry (Ellis Horwood, 1991)

References

Clayden, Greeves & Warren — Organic Chemistry, 2nd ed. (Oxford UP, 2012) · Polymer chemistry chapter: chain-growth and step-growth polymerisation, molecular weight distributions, glass transition
Odian, G. — Principles of Polymerization, 4th ed. (Wiley, 2004) · Radical, ionic, and coordination (Ziegler-Natta) chain-growth kinetics; Carothers and Flory-Schulz distributions; copolymerisation
Turro, N. J. — Modern Molecular Photochemistry (University Science Books, 1991) · Jablonski diagram, Stark-Einstein law, quantum yield, Stern-Volmer quenching, pericyclic photochemistry
Staudinger, H. — Uber Polymerisation, Ber. Dtsch. Chem. Ges. 53, 1073-1085 (1920) · Primary source proposing covalent macromolecules; foundation of polymer science
Carothers, W. H. — Studies on Polymerization and Ring Formation. I. Introduction to the General Theory of Condensation Polymers, J. Am. Chem. Soc. 51, 2548-2559 (1929) · Primary source deriving the extent-of-reaction relation for step-growth polymers
Stark, J. & Einstein, A. — Physikalische Zeitschrift 9, 893 (1908); Einstein, A., Ann. Phys. 338, 819-838 (1912) · Primary sources stating the photochemical equivalence law (one photon per absorbed quantum)
Ziegler, K. et al. — Angew. Chem. 67, 425 (1955); Natta, G. — J. Polym. Sci. 16, 143-154 (1955) · Primary sources on stereospecific olefin polymerisation (Ziegler-Natta catalysts)
Woodward, R. B. & Hoffmann, R. — The Conservation of Orbital Symmetry, Angew. Chem. Int. Ed. 8, 781-853 (1969) · Primary source for thermal and photochemical pericyclic selection rules

Estimated time

beginner: 15m
intermediate: 40m
master: 95m