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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.

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.