Complex Disordered Systems
Polymers and solvents
Energy of mixing
On-lattice model:
- each site has z nearest neighbours
- N_{s} solvent molecules
- N_{p} monomers
- \mathrm{N}_{\mathrm{sp}} solvent-monomer contacts.
Effective interactions:
- \varepsilon_{s s} for the solvent-solvent
- \varepsilon_{\mathrm{pp}} for monomer-monomer
- \varepsilon_{s p} for solvent-monomer interactions.
Then the energy of mixing \Delta U_{\operatorname{mix}} is obtained as the difference between the total energy U of the mixed system and the sum of the energies of the pure solvent and pure polymer, U_{S}+U_{p}.
\Delta U_{\operatorname{mix}}=U-\left(U_{S}+U_{p}\right)
Energy of mixing
Energy of pure solvent U_{s}=\frac{z N_{s} \varepsilon_{s s}}{2}
Energy of pure polymer U_{p}=\frac{z N_{p} \varepsilon_{p p}}{2}
Energy of solution U=N_{s p} \varepsilon_{s p}+\dfrac{\left(z N_{s}-N_{s p}\right) \varepsilon_{s s}}{2}+\dfrac{\left(z N_{p}-N_{s p}\right) \varepsilon_{p p}}{2}
We obtain that the energy of mixing \Delta U_{\mathrm{mix}}=N_{s p}\left[\varepsilon_{s p}-\dfrac{1}{2}\left(\varepsilon_{s s}+\varepsilon_{p p}\right)\right]\equiv N_{sp}\,\Delta\varepsilon
which can change sign depending on our choices for the effective interaction energies.
Good solvents, poor solvents and theta solvents
- Good solvent: \varepsilon_{\rm eff}>0 (T>\Theta)
Excluded volume wins and expands the chain. This includes \Delta U_{\rm mix}<0, but also weakly positive \Delta U_{\rm mix}.
- Poor solvent: \varepsilon_{\rm eff}<0 (T<\Theta)
Requires \Delta U_{\rm mix}>0, with an attraction strong enough to overcome the excluded volume: the polymer collapses.
Collapsed chains: hydrophobic polypeptides, polystyrene in water, chromatin in the cell nucleus.
- Tuning solvent properties (e.g. the temperature) leads to conformational changes
Temperature dependence and theta (\Theta) solvents
The various affinities have temperature dependencies. Typically:
At high T, the coil expand and the solvent is good
At low T the coil collapses and phase separation is observed (polymer-rich from polymer-poor phase)
There is an intermediate temperature where the excluded volume and attractive interaction compensate each other (\varepsilon_{\rm eff}=0); in d=3 the polymer then behaves like an ideal chain (freely jointed chain).
- The temperature is conventionally named \Theta temperature (or Flory temperature, after physicist Paul Flory)
- The \Theta temperature increases with the effective monomer-monomer attractions (we need higher temperature to reach ideal behaviour)
Swelling behaviour of polymers in solvent.
Examples
Example: cold cataract. The eye lens is a concentrated solution of proteins (crystallins) in water.
- At body temperature water is a good solvent: the lens is transparent.
- On cooling, water becomes a poorer solvent: the mixture separates into protein-rich and protein-poor regions, which scatter light, and the lens turns opaque.
- Warming back to 37\,^\circC clears the lens within minutes: the change is reversible.
Mouse lenses at 4\,^\circC, 25\,^\circC and 37\,^\circC (left to right). Images from Li et al., Front. Chem. 9, 767696 (2021), CC BY. Phase separation in cold cataract: Tanaka, Ishimoto & Chylack, Science 197, 1010 (1977).
Dilute regime
The polymer coils are well-separated on average.
The system is dilute if the polymer concentration c is such that
\left(\dfrac{c}{M}N_A\right) \left(\dfrac{4\pi}{3}R_g^3\right)\ll 1
where N_A is Avogadro’s number, M is the molar mass of a single chain and R_g is the radius of gyration of the chain.
Essentially, the polymers do not overlap.
Overlap concentration c*
Overlap occurs when the volume fraction of coils reaches unity and thus
\frac{c^{*}}{M} N_{A} \frac{4 \pi}{3} R_{g}^{3} \sim 1 \quad \therefore c^{*}=\frac{3 M}{4 \pi N_{A} R_{g}^{3}}
Phenomenologically one has R g=\langle R_g^2\rangle^{1 / 2}=B M^{\nu} with an empirical exponent called the Flory scaling exponent. Hence
c^{*}=\frac{3}{4 \pi N_{A} B^{3}} M^{1-3 v}
Semi-dilute
The concentration is larger than the overlap concentration \mathrm{c}^{*}, but still much smaller than the bulk density.
The coils interpenetrate and entanglement begins
Change in the dynamics (slowing down)
Yet, the solution is still mostly solvent.
Start of the formation of a polymer network with transient crowded regions within the solvent.
Polymer entanglement
Overlapping polymer chains become topologically intertwined
Creates physical knots that restrict chain motion
- changes flow behaviour (rheology)
- changes material properties (e.g. elasticity, viscosity )
Introduces topological constraints (transient/permanent)
Concentrated: bulk polymers
Polymer concentration approaches bulk density: highly interpenetrated coils, entanglements restrict segment motion, solid-like behaviour.
Cross-linking forms covalent bonds between separate polymer chains, creating a three-dimensional network. Three main classes of bulk polymers, characterised by their degree of cross-linking:
- Elastomers (rubbers): low cross-linking → flexible and elastic (natural rubber, silicone)
- Thermosets: high cross-linking → rigid, hard, brittle network (epoxy resins)
- Thermoplastics: no cross-linking → most everyday plastics (PE, PET, PS); behaviour upon cooling shows similarities to colloids
Thermoplastics behaviour upon cooling
- At high temperature the free energy is dominated by entropy.
- random assembly of intertwined flexible coils
- Upon cooling, potential energy dominates and chain rotation becomes restricted, favoring more extended configurations.
- Below melting temperature T_m, crystals form (lowest free energy state), but require slow cooling and significant molecular ordering.
- Rapid cooling below glass transition temperature T_g (< T_m) prevents crystallization, forming an amorphous glassy state instead—a metastable, long-lived solid.
- Solid thermoplastics typically contain a mixture of crystalline and amorphous regions.
