Keller four-roller polymer experiment

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.

An original local flow model has velocity $\mathbf{v}=(\dot{ϵ}x,-\dot{ϵ}y,0)$ for constant extensional strain rate $\dot{ϵ}>0$. Its divergence is zero, and its trajectories satisfy the displayed equations. Material approaches along $y$ and leaves along $x$. If the idealized observation region ends at $|x|=L$, a trajectory starting at $0<|x_0|<L$ has residence timeFor chosen $\dot{ϵ}=1{\mathrm{s}}^{-1}$, $L=1\mathrm{mm}$ and $|x_0|=0.01\mathrm{mm}$, this is $\log 100\mathrm{s}\simeq 4.61\mathrm{s}$. Starting ten times farther from the symmetry line halves this particular residence time. The mathematical trajectory $x_0=0$ never exits in this model; diffusion, imperfect flow and finite apparatus prevent treating that ideal trajectory as a guaranteed infinite exposure in an experiment.