Polymer Chains Under Strong Flow: Stems and Flowers
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\ kT/
(v= 3/5, swollen).
(1) (2)
For DNA, which is a semirigid chain, ideal behavior has been observed for chain lengths up to 20 ^im w h e n / i s small. At h i g h e r / ( = 100 pN), Cluzel et al.3 found a structural phase transition, corresponding to a plateau in the curve / versus elongation. Going even further, one reaches the break point.4 Another method to stretch DNA is based on optical tweezers.5 Here one end of the DNA is attached to a bead of glass of high polarizability, which is attracted by a region of large (optical) electric field. This was used by the Stanford group:5 In one series of experiments, they observed the conformations of the chain, pulled at one end by a constant force/. The friction on monomers are cumulative, and the tension along the chain is not uniform— it increases from the free end to the teth48
ered end. (1) At low /, the shape is a "trumpet" (Figure lb): Near the bead, all the drag forces add up, giving a relatively large tension; in this region, the trumpet is thin. At the bottom end, the tension is lower and the chain is more contorted. (2) At higher/ the portion near the bead is under such high tension that it becomes completely aligned: We call this the "stem."6 Beyond the stem, the other end is still expanded: We call this the "flower" (Figure lc). The border between stem and flower is relatively sharp. (3) At large /, the flower shrinks to zero. In the next section, we discuss the unwinding of one chain under uniform flow for swollen chains in a good solvent and for ideal chains, which correspond to the case of DNA molecules. The Stanford group 5 and the Curie group3 have also monitored the relaxation of one chain when the flow is abruptly suppressed. In a later section on Relaxation Processes, we discuss the relaxation for chains in pure solvent. The Stanford group also looked at a labeled DNA chain not in water, but in a solution of other (unlabeled) chains. 5 This leads to entanglements: The test chain is trapped into a "tube" because it cannot intersect its neighbors. By pulling the test chain along a curved path, such as the letter C, they could show that the whole chain follows the trajectory of the "head" because it stays in the same tube. The idea of the tube was invented long ago by S.F. Edwards7 and exploited by P.G. de Gennes in the dynamics of entangled polymers (Reptation model).8 The Stanford experiment gave the first visual proof of the existence of these tubes. Keeping this situation of one test chain entangled with other chains, Wirtz9 measured both the force applied at one end (via a magnetic bead) and the
resulting steady-state velocity V. This is the first direct measurement of friction on one single chain. In a certain velocity range, they found that the friction coefficient t, =f/V is constant; also the conformation is progressively changing. At higher velocity, the chain undergoes a disentanglement transition and enters a "marginal state" where the friction force becomes constant. At high velocity, the force starts to increase again. We discuss in
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