Polymer solution

A chemical solution with a polymer as a solute. The polymer conformation and interactions depend on the solvent and concentration.

A solution above the polymer overlap concentration but below the concentrated regime, with interpenetrating chains. Local polymer structure is characterized by a correlation length or concentration-blob size. Scaling formulas describe a broad semidilute regime most reliably when the concentration greatly exceeds overlap while remaining small compared with one in volume fraction.

A region of size $\xi$ containing a typical correlated chain segment whose internal monomer concentration matches the bulk concentration. If it contains $g$ statistical units with isolated size law $\xi \sim b{g}^{\nu }$ and unit volume $v_0$, concentration matching gives $\varphi \sim g{v}_0/{\xi }^3$. Eliminating $g$ gives $\xi \sim b(\varphi b^3/v_0{)}^{-\nu /(3\nu -1)}$. Numerical prefactors and distinctions among scattering, osmotic and geometric correlation lengths require additional definitions.

For an ideal chain inside a blob, $\nu =1/2$ and $\xi \sim b\sqrt{g}$. Together with $\varphi \sim g{v}_0/{\xi }^3$, this gives the displayed concentration-blob size. Under the site-volume approximation $v_0\sim b^3$, $\xi \sim b/\varphi$ and $g\sim {\varphi }^{-2}$. The estimate describes local overlap rather than the full chain size and requires $1\ll g\ll N$ for a well-developed semidilute regime.

A solvent under conditions where the leading effective pair excluded volume of a long dilute polymer vanishes. In the simplest Flory–Huggins solution theory this means $\chi =1/2$. The chain is approximately an ideal chain, while three-segment interactions can still affect the osmotic pressure of a concentrated solution. A particular polymer-solvent pair often meets this condition only near its theta temperature.

The overlap concentration is the concentration at which distinct dilute polymer coils first substantially interpenetrate. It corresponds to roughly one chain per coil volume. If a chain contains $N$ repeat units, has molar mass $M$ and occupies a coil volume $V_{\mathrm{coil}}$, then its chain number density is approximately $1/V_{\mathrm{coil}}$, its repeat-unit number density is $N/V_{\mathrm{coil}}$, and its mass concentration is $M/(N_A{V}_{\mathrm{coil}})$. A scaling convention uses $V_{\mathrm{coil}}\sim R^3$; a commonly chosen geometric convention uses $V_{\mathrm{coil}}=4\pi R_g^3/3$. These give different order-one prefactors because a coil has no sharp surface. For an ideal chain, $R\sim b{N}^{1/2}$ and a site-volume estimate $v_0\sim b^3$ gives overlap volume fraction ${\varphi }^{\ast }\sim N^{-1/2}$. A good solvent chain with the Flory approximation $R\sim b{N}^{3/5}$ instead gives ${\varphi }^{\ast }\sim N^{-4/5}$.

A poor solvent favors effective attraction between polymer segments relative to solvent contacts. A dilute flexible chain tends to contract, and sufficiently unfavorable conditions produce a compact polymer globule. Solvent quality depends on the polymer, solvent and temperature. The simple Flory–Huggins solution theory identifies the poor-solvent side of the theta condition with $\chi >1/2$; this does not guarantee that every finite-density mixture is inside its spinodal.

A polymer globule is a compact conformation favored by effective segment attractions in a poor solvent. If its internal segment density is independent of chain length, its volume is proportional to $N$ and a size measure scales as $N^{1/3}$. The proportionality between physical volume and the cube of a radius of gyration depends on shape and density profile. Compactness alone supplies scaling, not a uniquely determined numerical monomer volume.

The theta temperature is the temperature at which the leading effective two-segment interaction of a dilute long polymer vanishes, so its large-scale size is approximately that of an ideal chain. In the simplest Flory–Huggins solution theory this is the condition $\chi =1/2$. It is the infinite-chain limit of the upper critical temperature in the usual energetic model, not the critical temperature of every finite chain.

Flory–Huggins solution theory is an incompressible lattice description of mixing a polymer with a solvent. A chain occupies $N$ connected sites, while a solvent molecule occupies one; the displayed mixing free energy is per site. The first two terms are translational entropy of mixing contributions and the last is a mean-field excess contact interaction. A positive interaction parameter $\chi$ penalizes polymer–solvent contacts. If the site volume is $v_0$, the physical free-energy density is $f/v_0$ and the polymer volume fraction gives monomer number density $\varphi /v_0$. Composition-dependent interaction parameters require extra derivative terms in stability calculations.

Let $f(\varphi )$ be the mixing free energy per lattice site in Flory–Huggins solution theory, $v_0$ the site volume, and $\varphi$ the polymer volume fraction. The solvent mixing chemical potential is $f-\varphi f^{\prime }$, so its balance against the pure-solvent pressure contribution gives $\Pi =(\varphi f^{\prime }-f)/v_0$. For composition-independent $\chi$,The first term is the ideal chain translational contribution; the second encodes the effective two-segment excluded volume. At the theta temperature condition it vanishes but the positive third-order term remains. Treating $f$ as a free-energy density without dividing by $v_0$ loses the physical dimensions of osmotic pressure.

The Flory interaction parameter measures the excess free-energy cost of unlike contacts in Flory–Huggins solution theory. For a simple lattice of coordination number $z_c$ and contact energies ${ϵ}_{ps}$, ${ϵ}_{pp}$ and ${ϵ}_{ss}$,Real effective parameters may also contain entropic contributions and composition dependence. With site volume $v_0$ and composition-independent $\chi$, the effective two-segment excluded volume is $v_{\mathrm{eff}}=v_0(1-2\chi )$. Positive, zero and negative $v_{\mathrm{eff}}$ correspond respectively to a good solvent, the theta temperature condition and a poor solvent. In particular, $\chi =1/2$ means cancellation of leading repulsive and attractive contributions; it does not mean that every molecular interaction vanishes.

For composition-independent $\chi$, the spinodal of Flory–Huggins solution theory is $2\chi =1/(N\varphi )+1/(1-\varphi )$. At its thermodynamic critical point, a second condition $-1/(N{\varphi }^2)+1/(1-\varphi {)}^2=0$ gives $1-{\varphi }_c=\sqrt{N}{\varphi }_c$, proving the displayed values. Hence ${\varphi }_c\sim N^{-1/2}$ and ${\chi }_c-1/2\sim N^{-1/2}$. Under $\chi =C/T$, the theta temperature is $\Theta =2C$ and $T_c=\Theta /(1+N^{-1/2}{)}^2$. Thus $T_c$ approaches a constant while $\Theta -T_c$ scales as $N^{-1/2}$. Converting ${\chi }_c$ to a temperature always requires a specified or inferred $\chi (T)$ law.

A good solvent favors contact with polymer segments strongly enough that the effective excluded volume is positive. A long dilute flexible chain swells relative to an ideal chain; the three-dimensional size exponent is approximately $0.588$, often replaced by $3/5$ in a scaling estimate. This designation depends on the polymer, solvent and temperature, rather than on the solvent alone.