Chemistry · Organic chemistry · Year 3

Physics of macromolecules

How is the chain in a cotton fibre coiled, and why is a polymer solution thick?

The situation

Cotton fibre is natural cellulose chains; acrylic fibre, a rubber hose and a PVC pipe that has cracked in winter are also built of very long molecules. A piece of polystyrene is as hard as glass at room temperature, but at 130 °C it softens and stretches like rubber; if a little polymer is added to water, the solution suddenly thickens. In the laboratory a polymer solution flows through an Ubbelohde viscometer, crosslinked rubber swells in a solvent, and a thermomechanical curve is recorded. How is a long chain coiled in solution, how do we measure its mass, and why does it change its state when heated?

Topic

Physics of macromolecules and polymer solutions. Cotton fibre — natural cellulose chains; nitron fibre, rubber and a PVC pipe that turns brittle in winter. In the laboratory, a polymer solution flows through an Ubbelohde viscometer,…

Key ideas

Random walk, ⟨h²⟩
Viscometry and M
Glass transition temperature

Formulas

⟨ h^{2} ⟩ = C_{\infty} N b^{2}, A = \frac{⟨ h^{2} ⟩}{L}

The mean-square end-to-end distance (for a freely jointed chain $C_{\infty}$ = 1, that is, $⟨ h^{2} ⟩ = N b^{2}$ ) and the Kuhn segment $A$ ; $L$ is the length of the fully extended chain.

{\overline{M}}_{n} = \frac{\sum n_{i} M_{i}}{\sum n_{i}}, {\overline{M}}_{w} = \frac{\sum w_{i} M_{i}}{\sum w_{i}}, Đ = \frac{{\overline{M}}_{w}}{{\overline{M}}_{n}}

Number-average and weight-average molar masses, and the degree of polydispersity.

\frac{η_{s p}}{c} = [η] + k_{H} {[η]}^{2} c, [η] = K M^{a}

The Huggins line (intrinsic viscosity as $c \to 0$ ) and the Mark–Kuhn–Houwink equation: $a$ = 0.5 in a θ-solvent and 0.6–0.8 in a good solvent.

\frac{π}{c} = R T (\frac{1}{{\overline{M}}_{n}} + A_{2} c)

Osmometry: ${\overline{M}}_{n}$ from the intercept and the second virial coefficient $A_{2}$ from the slope ( $A_{2}$ = 0 in a θ-solvent).

Δ G_{a r} = R T [φ_{1} \ln φ_{1} + \frac{φ_{2}}{x} \ln φ_{2} + χ φ_{1} φ_{2}]

Flory–Huggins: the energy of mixing (per mole of lattice sites); in the θ-state $χ$ = 0.5.

Theory

The macromolecule and its shape

A macromolecule is a long chain of thousands of covalently bonded atoms; it is built from repeating monomer units. The number of units is the degree of polymerisation $P$ ; the molar mass is $M = P M_{0}$ , and the contour length $L$ of the fully extended chain is also proportional to $P$ . Chains of a few dozen units are oligomers; $M$ ≳ 10⁴ means polymers. Living nature is built from polymers: proteins, nucleic acids and polysaccharides — cellulose (β-1,4-glucan, an extended semi-rigid chain; the basis of cotton fibre, degree of polymerisation about 10 000–15 000), starch (helical amylose and α-1,6-branched amylopectin), the still more branched glycogen, chitin (N-acetylglucosamine), inulin (a polymer of fructose), pectin (galacturonic acid; forms a gel with Ca²⁺), hyaluronic acid in connective tissue and the anticoagulant drug heparin; dextran solutions are used as blood plasma substitutes. In Uzbekistan the school of polymer chemistry began with the study of cotton cellulose (Academician Kh. U. Usmanov); the Institute of Polymer Chemistry and Physics of the Academy of Sciences of Uzbekistan studies cellulose, chitosan and composites. Configurational isomerism does not change unless a bond is broken: head-to-tail linking, cis/trans (natural rubber is cis-, gutta-percha is trans-1,4-polyisoprene), and tacticity (side groups all on one side — isotactic, alternating — syndiotactic, random — atactic). Conformational isomerism covers the shapes that arise from internal rotation about single bonds (trans and gauche). The chain is a 3D random walk: if the bond directions are independent, $⟨ h^{2} ⟩ = N b^{2}$ and the size is proportional to $\sqrt{N}$ (like the displacement in diffusion — see “Brownian motion and diffusion”). For a bond angle of 109.5° and free rotation, $C_{\infty} = (1 + \cos α) / (1 - \cos α)$ = 2 ( $α$ = 180° − 109.5° = 70.5° is the angle between neighbouring bond vectors); with hindered rotation trans is more frequent and the chain is more extended. The statistical (Kuhn) segment is $A = ⟨ h^{2} ⟩ / L$ : the chain can be replaced by freely jointed pieces of this length. Polyethylene ≈ 1.4 nm (flexible), cellulose esters ≈ 10–25 nm (semi-rigid). Thermodynamic flexibility depends on the energy difference $Δ E$ between trans and gauche and on $T$ (on heating gauche increases and $C_{\infty}$ decreases); kinetic flexibility depends on the rotation barrier $U_{0}$ (C–C ≈ 12 kJ/mol) and the rate at which conformations interconvert. In a good solvent the coil swells ( $⟨ h^{2} ⟩ \propto N^{1.2}$ , Flory), in a θ-solvent it is ideal ( $\propto N$ ).

Molar mass, the MMD and how to determine them

A synthetic polymer is a mixture of chains of different lengths: the molar mass distribution (MMD). The number-average ${\overline{M}}_{n}$ (each chain counts equally), the weight-average ${\overline{M}}_{w}$ (heavier chains contribute more) and ${\overline{M}}_{z}$ ; the polydispersity Đ = ${\overline{M}}_{w} / {\overline{M}}_{n}$ ≥ 1 (≈ 1.05 in anionic “living” polymerisation, ≈ 1.5–2 in radical polymerisation). End groups (L49, L53, L54) and cryoscopy (L51) give $M_{n}$ , but only for relatively small $M$ (end groups — $M$ ≲ 2·10⁴, cryoscopy — $M$ < 10⁴): in large chains the groups are too few and ΔT is too small. Osmometry (L52): $π / c = R T (1 / M_{n} + A_{2} c)$ — $M_{n}$ from the intercept and the second virial coefficient $A_{2}$ from the slope (> 0 in a good solvent, 0 at θ). Light scattering: $K c / R_{θ} = 1 / M_{w} + 2 A_{2} c$ (Debye) — $M_{w}$ and the coil size. Sedimentation and diffusion (ultracentrifuge, Svedberg: $M = s R T / [D (1 - \overline{v} ρ)]$ ) — mass, size and shape. Viscometry (L50): in an Ubbelohde viscometer the relative viscosity $η_{r e l} = t / t_{0}$ , specific $η_{s p} = η_{r e l} - 1$ , reduced $η_{s p} / c$ and intrinsic $[η] = \lim_{c \to 0} η_{s p} / c$ (the Huggins line; the Kraemer line $\ln η_{r e l} / c$ also converges on the same point). Mark–Kuhn–Houwink: $[η] = K M^{a}$ , with $a$ = 0.5 in a θ-solvent, 0.6–0.8 in a good solvent and greater than 1 for rigid chains; the method gives $M_{η}$ : $M_{n} < M_{η} < M_{w}$ . Fractionation: in fractional precipitation (L55) the longest chains precipitate first as precipitant is added, and in fractional dissolution (L56) the shortest dissolve first; from the masses of the fractions and $M_{i}$ the integral $W (M)$ and differential MMD curves are built (the modern method is gel permeation chromatography).

Polymer solutions, swelling and gels

A polymer swells before it dissolves: small solvent molecules enter between the chains, solvate them and increase the volume. A linear polymer swells without limit and passes into solution, whereas a crosslinked (network) polymer swells to a limited extent. To prepare a solution the crushed polymer is first allowed to swell, then stirred until dissolved and filtered (dust particles spoil light scattering and viscometry); the polymer is purified by reprecipitation — its solution is poured into a precipitant. In concentrated solutions the chains associate and form temporary clusters. The degree of swelling is $Q = (m - m_{0}) / m_{0}$ , with kinetics $Q_{t} = Q_{\infty} (1 - e^{- k t})$ — it depends on the nature of the solvent (L45); from equilibrium swelling the Flory–Rehner equation gives the chain mass between crosslinks $M_{c}$ (L46). Thermodynamics of dissolution: $Δ G_{m i x} = Δ H - T Δ S$ < 0; the entropy of mixing of a long chain is small, so the polymer–solvent interaction — the Flory–Huggins parameter $χ$ — decides. $χ$ < 0.5 means a good solvent ( $A_{2}$ > 0, the coil is swollen); in the θ-state $χ$ = 0.5, $A_{2}$ = 0 and the chain is ideal; for polystyrene–cyclohexane θ ≈ 34.5 °C (the θ-solvent). Below θ the solution separates into two phases — in the phase diagram the upper critical solution temperature is $T_{c}$ < θ, and it approaches θ as $M$ increases: $1 / T_{c} = (1 / θ) [1 + (1 / ψ) (1 / \sqrt{x} + 1 / 2 x)]$ (Schulz–Flory, L47). This is why a polymer can be fractionated by precipitation and dissolution. Concentrated solutions are non-Newtonian: as the shear rate increases the chains align, entanglements are released and the viscosity falls ( $τ = K {\dot{γ}}^{n}$ , $n$ < 1, L48). Gels: a hot gelatin solution sets when cooled (gelation); starch paste and bentonite are thixotropic — they become fluid when stirred and set again at rest (a hysteresis loop on the flow curve); with time a gel contracts and releases liquid — syneresis (the whey of yoghurt). In gels there is no convection, and diffusion and reactions are slow: the periodic precipitate rings of silver chromate are the Liesegang phenomenon. Lyophilic colloids are covered in the “Colloids and adsorption” experiment and viscosity in “Viscosity and flow”.

Polyelectrolytes and polyampholytes

Polyelectrolytes are polymers whose units carry ionisable groups: polyacids (polyacrylic acid, pectin, nucleic acids), polybases (polyvinylpyridine, polyamines) and their salts; they may be strong or weak. Polyampholytes have both acidic and basic groups on the chain — proteins and gelatin are amphoteric polyelectrolytes: at the isoelectric point the net charge is zero, the chain coils most tightly, and viscosity, swelling and solubility are at a minimum (L58; amino acids and proteins are covered in the “Amino acids and proteins” experiment). When a polyelectrolyte solution in salt-free water is diluted, the counter-ions move away, the like charges repel and the coil unfolds, so $η_{s p} / c$ increases (the polyelectrolyte effect, Fuoss); salt screens the charges and the line resembles that of an ordinary polymer (L59). Potentiometric titration (L60): in a weak polyacid each dissociated proton makes it harder to ionise the neighbouring groups (the electrostatic field of the chain), so the Henderson–Hasselbalch equation changes: $pH = p K + n \lg [α / (1 - α)]$ , with $n$ > 1 ( $n$ = 1 for acetic acid); adding salt reduces $n$ . Donnan equilibrium: the polyion inside the membrane cannot get out, and the small ions distribute so that ${[N a^{+}]}_{i} {[C l^{-}]}_{i} = {[N a^{+}]}_{t} {[C l^{-}]}_{t}$ : there is less salt inside, and a Donnan potential $E = (R T / F) \ln ({[N a^{+}]}_{t} / {[N a^{+}]}_{i})$ and an extra osmotic pressure appear on the membrane (cell membranes, dialysis, ion exchangers). Interpolymer complexes (L61): polyacrylic acid and polyethylene glycol bind cooperatively through many hydrogen bonds at low pH, forming a 1 : 1 complex that turns cloudy; above a critical pH (–COOH ionises) the complex breaks up. A polycation + a polyanion give polyelectrolyte complexes (membranes, flocculants). Polyelectrolyte and protein gels absorb a great deal of water. Ion exchangers are crosslinked polyelectrolytes; the static exchange capacity of a cation exchanger (L57) is used in water softening (see “Water purification and softening”).

Physical states, crystallisation and strength

When an amorphous linear polymer is heated it passes through three physical states (the thermomechanical curve is the deformation under a constant small load as a function of temperature): glassy (the segments are “frozen”, deformation ≈ 1 %), high-elastic (the segments move but whole chains do not slip — a large reversible deformation) and viscous-flow (the chains slide past one another). The boundaries are the glass transition temperature $T_{g}$ (polystyrene ≈ 100 °C, PMMA ≈ 105 °C, PVC ≈ 80 °C, natural rubber ≈ −70 °C) and the flow temperature $T_{f}$ . $T_{g}$ rises slightly with molar mass and levels off (Fox–Flory: $T_{g} = T_{g}^{\infty} - K / M_{n}$ ), while $T_{f}$ keeps rising as $M$ increases — the high-elastic interval widens; oligomers do not have this state. A crosslinked polymer (rubber) does not flow. In crystalline polymers (polyethylene, isotactic polypropylene) the chains fold to form crystallites and spherulites — a supramolecular structure; they keep their shape up to $T_{m}$ . Crystallisation is affected by chain regularity (tacticity, branching), cooling rate, stretching (orientation), crystallisation nuclei and plasticisers. The transitions are relaxational: if the load is applied quickly or the sample cooled quickly, $T_{g}$ appears higher. Mechanical properties: glassy polystyrene breaks brittlely; rigid PVC near $T_{g}$ shows forced elasticity (a large deformation after the yield point); crystalline polyethylene stretches cold, forming a “neck”; rubber stretches by hundreds of per cent. Under load the time to failure is $τ = τ_{0} e^{(U_{0} - γ σ) / R T}$ — Zhurkov's kinetic theory. Plasticisation: small molecules (dibutyl phthalate, dioctyl phthalate) enter between the chains and weaken the interaction between them — $T_{g}$ falls, and rigid PVC becomes a soft hose or linoleum. In polar polymers the fall in $T_{g}$ is proportional to the number of moles of plasticiser (Zhurkov's rule), and in non-polar ones to its volume fraction (the Kargin–Malinsky rule).

Lab work, safety and the model

17 lab works (HMC L45–L61) are in the “Lab work” list: the apparatus, the procedure, the formula and what is measured; “Set up on the bench” opens the apparatus needed. In 3D: an Ubbelohde viscometer, a membrane osmometer, a titration setup, a swelling beaker, θ-ampoules, a rotational viscometer and a thermomechanical analyser. Safety: toluene, benzene (carcinogenic), cyclohexane, hexane and acetone are volatile and flammable: work only in a fume cupboard, away from flames, with gloves and safety goggles; phenol and benzyl alcohol are harmful to the skin; if acid or alkali solutions get into the eyes, rinse immediately with plenty of water; draw liquid into a viscometer with a rubber bulb, never by mouth; do not taste anything. Model: the chain is a model of rotational isomeric states with independent rotors (in real polyethylene $C_{\infty}$ ≈ 6.7, in the model ≈ 3.1); swelling in a good solvent is a uniform enlargement with the Flory coefficient; cellulose is a worm-like chain ( $P$ = 6 nm, model). The MKH constants for polystyrene–toluene, $A_{2}$ , $χ$ , Fuoss, gelatin, PAA–PEG, thixotropy, plasticiser effect and thermomechanical curves are teaching (model) values within the range found in the literature; the precipitation threshold is taken as sharp. The processes are sped up (flow 10 times, osmosis and swelling minutes in seconds, heating and cooling fast) — the displays show the measurement time and temperature. The solutions are ideal (activity = concentration), and the temperature is held constant by a thermostat.

Did you know?

Paul Flory received the Nobel Prize in Chemistry in 1974 for the statistics of polymer chains and the theory of polymer solutions; the θ-state and the Flory–Huggins parameter come from his work.
The cellulose chain in cotton fibre has about 10 000–15 000 glucose units; each unit is ≈ 0.5 nm along the chain, so a single chain, fully extended, would be about 5–8 µm long.

Inside the experiment

The experiment starts from a real-life situation: you predict the result, change the controls, run the experiment and draw a conclusion. Inside are missions, theory, a quiz and a guided tour in which the camera moves and explains the key points.

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