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The end-to-end vector joins the two ends of a polymer chain. In an isotropic unforced polymer solution, its vector expected value is zero; a useful measure of chain size is its root-mean-square length. For an ideal chain with $N$ independent isotropic steps of length $b$, the cross terms in the square of the step sum have zero expected value, so $R_{\mathrm{rms}}^2=N{b}^2$. For a long Gaussian chain, its radius of gyration obeys $R_g^2=R_{\mathrm{rms}}^2/6$. If $N$ counts chemical repeat units instead of independent Kuhn length segments, the statistical length used in $R_{\mathrm{rms}}^2=N{b}^2$ must be calibrated to that choice of $N$; it is not automatically the physical bond length or the Kuhn length itself.

Pierre-Gilles de Gennes's 1991 lecture describes Andrew Keller and collaborators using four rollers to deform a dilute polymer solution. A well-chosen exit trajectory kept coils under sustained extension. Above a threshold flow rate the solution became birefringent, revealing a change in chain orientation and conformation. The experiment connected an optical readout to a mechanically driven molecular response.^[ref]

This original functional diagram shows a central flow idealization, rather than historical dimensions or a photograph. A probe of birefringence detects different optical responses along different directions; it does not directly photograph a chain or establish its extension without an optical model. Comparing a flowing solution with its quiescent state and controlling temperature and concentration are useful proposed checks, not an assertion that this is the complete historical protocol.

For a homogeneous polymer solution, examine the curvature of the mixing Helmholtz free energy with respect to composition. A composition fluctuation into equal volumes at $\varphi \pm \delta \varphi$ changes the average free energy byThus $f^{\prime \prime }>0$ locally resists fluctuations and $f^{\prime \prime }<0$ amplifies them, leading to spinodal decomposition. The spinodal curve is the locus $f^{\prime \prime }=0$, separating locally stable or metastable homogeneous compositions from locally unstable ones. The coexistence binodal instead follows a common tangent to the free energy. Between the binodal and spinodal, separation needs nucleation.

The Flory interaction parameter $\chi$ measures the excess cost of unlike segment contacts. In a simple lattice with coordination number $z$, contact energies ${ϵ}_{AA},{ϵ}_{BB},{ϵ}_{AB}$ and temperature $T$,A positive Flory interaction parameter penalizes unlike contacts; a negative value favors them. Mixing still depends on the balance with entropy of mixing, rather than the sign of $\chi$ alone.

For two polymers of degrees of polymerization $N_A,N_B$, let $\varphi$ be the volume fraction of species $A$. The binary Flory-Huggins polymer blend has mixing free energy per lattice siteThe logarithmic terms favor mixing. Their factors $1/N_A$ and $1/N_B$ arise because an entire chain translates as one object; connecting many segments greatly reduces the mixing entropy per segment. Hence long polymers can demix for a much smaller positive Flory interaction parameter than small molecules.

For composition-independent $\chi$, the local stability criterion isPositive curvature means stability to small composition fluctuations, zero curvature locates the spinodal, and negative curvature means spontaneous growth of fluctuations. Actual two-phase coexistence is determined by the common-tangent construction, or equality of both species' chemical potentials; its boundary is the binodal. A locally stable but globally unfavourable homogeneous mixture can lie between the binodal and spinodal.

Setting the second and third derivatives of $f$ to zero givesFor the symmetric Flory-Huggins polymer blend, ${\varphi }_c=1/2$ and ${\chi }_c=2/N$. For a polymer solution with $N_B=1$ and long chains of species $A$, ${\chi }_c\to 1/2$, consistent with the theta solvent limit. An effective Flory interaction parameter can depend on temperature and composition; in the simple energetic case $\chi \propto 1/T$, heating encourages mixing, but that temperature trend is not universal.

The original illustration uses stipulated $N_A=N_B=100$. At $\chi =0.01$ the mixing free energy is convex, whereas at $\chi =0.03$ a central unstable region appears. The horizontal common-tangent construction in the latter case joins the symmetric coexisting compositions; its contacts differ from the spinodal boundaries. These are model curves, not supplied experimental data.

The Flory–Huggins solution theory treats a polymer solution as an incompressible lattice mixture, with one solvent molecule occupying one site and one chain occupying $N$ connected sites. Thus the supplied $f$ is naturally the mixing free energy per lattice site; if a site has volume $v_0$, the free-energy density is $f/v_0$.

The term $k_{B}T(\varphi /N)\ln \varphi$ is the polymer translational entropy of mixing contribution. There are only $1/N$ as many independently translating chains as monomers, which explains the factor $1/N$. The term $k_{B}T(1-\varphi )\ln (1-\varphi )$ is the solvent entropy of mixing contribution. Both lower the mixing free energy for $0<\varphi <1$.

The term $k_{B}T\chi \varphi (1-\varphi )$ is the mean-field excess interaction contribution from replacing like contacts with polymer–solvent contacts. In the simplest energetic lattice model,where $z$ counts neighboring contacts and the ${ϵ}_{ij}$ are pair contact energies. Positive $\chi$ penalizes unlike contacts and can favor demixing. Because chain connectivity suppresses the polymer's translational mixing entropy, relatively modest unfavorable interactions can cause separation of long chains. More general effective $\chi$ can also include entropic effects.

Paul Flory developed a lattice treatment of polymer solutions that accounts for the reduced mixing entropy of connected chain segments. A chain contributes many segments but remains one translating molecule.

This distinction helps explain why long-chain mixtures can separate under small unfavourable contact energies. Flory–Huggins solution theory connects molecular connectivity with the balance between mixing entropy and interaction energy.^[ref]