Visualization of Molecular Fluctuations near the Critical Point of the Coil-Stretch Transition in Polymer Elongation Hazen P. Babcock, † Rodrigo E. Teixeira, †,‡ Joe S. Hur, ‡ Eric S. G. Shaqfeh, § and Steven Chu* ,† Departments of Physics and Applied Physics, Varian Building, Stanford University; Department of Chemical Engineering, Stauffer III, Stanford University; and Departments of Chemical and Mechanical Engineering, Stauffer III, Stanford University, Stanford, California 94305 Received January 21, 2003; Revised Manuscript Received April 17, 2003 ABSTRACT: The coil-stretch transition in the extension of polymers in two-dimensional flows was investigated near a critical boundary defined by simple shear flow. Visualization of individual molecules revealed a sharp coil-stretch transition in the steady-state length of the polymer with increasing Weissenberg number (Wi) in flows where the magnitude of the elongational component (||E||) slightly exceeded the rotational component (||Ω||). However, unlike in pure elongational flow, large fluctuations in the length of the polymer were observed near the critical point of the transition. These fluctuations result in a “softening” of the phase transition between coiled and extended states of the polymer. In flows where ||Ω|| is slightly greater than ||E||, significant transient polymer deformation was observed. However, the average length of the polymer as a function of Wi increased much more slowly than in simple shear flow. Introduction In a seminal paper in 1974, de Gennes proposed a phase diagram for arbitrary two-dimensional flows. 1 These flows can always be factored into an elongational component E, where neighboring fluid elements sepa- rate exponentially with time, and a rotational compo- nent, Ω, where fluid elements rotate with time. Poly- mers in flows where ||E|| > ||Ω|| would exhibit a sharp coil-to-stretched transition if Wi exceeds a critical value. In the limit of long polymers the transition would be discontinuous (first order). As ||E|| approached ||Ω|| from above, de Gennes asserted that this coil-stretch transi- tion would “soften”. In the special case of simple shear flow (||E|| ) ||Ω||), the sharp transition would vanish. In flows where ||E|| < ||Ω||, polymers would remain coiled regardless of Wi. De Gennes’ predictions triggered extensive investigations into the conformational changes of polymers induced by these types of flow. In particular, the limiting cases of purely elongation flow (||Ω|| ) 0) and shear flow were heavily studied. Experiments in purely elongational flow based on optical birefringence (reviewed in refs 2 and 3) sug- gested that the polymers reached equilibrium in a highly extended state. Measurements of extensional viscosities in filament stretching experiments 4,5 also showed that these flows were stretching the polymers. However, the measured stress due to filament exten- sion 6 indicated that the polymers were not reaching full extension. Light scattering 7 (and references contained therein) experiments showed that polymers were barely deformed in flow strengths high enough to saturate the birefringence signal. These conflicts were finally re- solved in favor of De Gennes’ prediction by imaging single polymers of DNA in extensional flows. 8-10 The ability to observe the real-time extension of each molecule showed that previous ensemble measurements blurred the inherently sharp phase transition by aver- aging over many molecules where a significant fraction of them had not yet reached their equilibrium extension. Light scattering and neutron experiments on poly- mers in simple shear flows suggested that these flows were not deforming the polymer significantly. 11-14 In agreement with de Gennes’ predictions, however, the direct measurement of individual polymer extension in simple shear flow by single molecule techniques re- vealed that the average extension of the polymers was increasing with Wi. Furthermore, it was found that the molecular extension did not show a sharp transition with increasing velocity gradient, continued to fluctuate in time, and never reached an equilibrium state. 15 These fluctuations, difficult to detect with observations of an ensemble of molecules, were easily observed with single molecule imaging methods. The existence of a coil-stretch transition in general two-dimensional flows where ||E|| > ||Ω|| has been treated theoretically 16-20 and probed experimentally with optical birefringence by Leal and collaborators. 21-24 The birefringence signal was observed to approach saturation with increasing Wi, indicating the polymers were nearing the fully stretched state. They found that the birefringence signal could be collapsed by rescaling the flow strength with the flow type parameter λ. However, when the birefringence was plotted in terms of this rescaled, dimensionless velocity gradient, the transition from the coiled to stretched state for all flows was fairly gradual, starting at ∼2 and still increasing at ∼38 (see for example Figure 15 of ref 22). One question that was not addressed by the earlier birefringence studies of mixed flows was how the transition from coiled to stretched state might “soften” as ||E|| approached ||Ω||, since those studies did not observe a sharp transition in either pure extensional or mixed flows. In this paper we report the first single molecule visualization of the polymer dynamics involved in the coil-stretch transition in the region of the phase diagram where ||E|| is either slightly greater or less than † Departments of Physics and Applied Physics. ‡ Department of Chemical Engineering. § Departments of Chemical and Mechanical Engineering. * Corresponding author. 4544 Macromolecules 2003, 36, 4544-4548 10.1021/ma034073p CCC: $25.00 © 2003 American Chemical Society Published on Web 05/16/2003