## Abstract

The conformational dynamics of individual, flexible polymers in steady shear flow were directly observed by the use of video fluorescence microscopy. The probability distribution for the molecular extension was determined as a function of shear rate, γ˙, for two different polymer relaxation times, τ. In contrast to the behavior in pure elongational flow, the average polymer extension in shear flow does not display a sharp coil-stretch transition. Large, aperiodic temporal fluctuations were observed, consistent with end-over-end tumbling of the molecule. The rate of these fluctuations (relative to the relaxation rate) increased as the Weissenberg number,γ˙τ, was increased.

The dynamics of flexible polymers in shear is of great practical interest because this type of flow occurs whenever a fluid flows past a surface. Macroscopic, non-Newtonian rheological properties of polymer solutions, such as flow-dependent viscosities and normal stresses, result from microscopic stresses that arise when polymeric molecules are stretched and affect the solvent motion. Thus, much effort has been directed at predicting the molecular dynamics of polymers in shear flows (1–5). However, it has been difficult to rigorously test these predictions because the dynamics of a polymer molecule in shear have not been observed directly. Experimental efforts have mainly focused on measuring bulk rheological properties or on measuring the scattering of light or neutrons by polymer solutions (6–10). Here we describe how single-molecule imaging techniques (11) can be used to study the configurations of polymers in shear flow so that the detailed molecular predictions of theories can be tested.

It has long been recognized that the amount of distortion of a molecule is strongly dependent on the nature of the flow (4,5). In general, any planar flow of the form *v⃗ = v _{x}x̂ + v_{y}ŷ *may be represented as a linear superposition of a rotational flow with a vorticity ω = [(∂

*v*

_{y}/∂

*x*) − (∂

*v*/∂

_{x}*y*)]/2 and an elongational flow with a strain rate ɛ˙ = [(∂

*v*

_{y}/∂

*x*) + (∂

*v*/∂

_{x}*y*)]/2. In a pure elongational flow (ω = 0) one expects large deformations of a polymer, whereas in a pure rotational flow (ɛ˙ = 0) one expects only rotation and not deformation (4, 5).

Most practical flows consist of a mixture of both rotational and elongational components, and the resulting polymer deformation depends on the relative magnitudes of ɛ˙ and ω. In general, the response is not necessarily a linear superposition of the responses to each component. In a simple shear flow (*v _{y}
* = 0 and

*v*= γ˙

_{x}*y*, where γ˙=

*dv*/

_{x}*dy*is the shear rate) the magnitude of the elongational and rotational components are equal (|ɛ˙| = |ω|) (Fig. 1A). In this case, it has been suggested that the polymer will not attain a stable, strongly stretched state (4, 5). In fact, large fluctuations in the extension due to an end-over-end tumbling of the molecule have been observed in some simulations (3). These fluctuations presumably occur because the stretched state is destabilized by the rotational component of the shear flow (Fig. 1B). By observing individual molecules, we have experimentally confirmed the existence of such fluctuations and studied their statistical and temporal properties.

In our experiment, a uniform shear flow was created in a ∼50-μm gap between two parallel glass plates (Fig. 1C) (12). Individual, isolated molecules of fluorescently labeled lambda bacteriophage DNA (λ-DNA) in viscous sugar solutions were imaged at a depth of 25 ± 15 μm (13–15). These DNA molecules of contour length *L* ≅ 22 μm may be considered “flexible” polymers because they contain roughly 440 persistence lengths (13). Images of the molecules were digitized, and a computer-generated cursor was used to measure (by inspection) the maximum extension, *x*, of the molecule along the shear direction.

The strength of an applied shear flow is specified by the dimensionless number *Wi* = γ˙τ (called the Weissenberg number), which is the ratio of the applied shear rate, γ˙, to the polymer's natural relaxation rate, 1/τ. In our experiment, data were recorded at shear rates ranging from γ˙ = 0 to 4.0 s^{−1} and with polymer relaxation times of τ = 6.3 and 19 s such that *Wi* ranged from 0 to 76. The relaxation time was adjusted by using two different solvent viscosities, η = 60 and 220 centipoise (cP) (16). Molecular extension data (Fig. 2) were recorded after the mean extension reached a steady-state value. The initial transient period, during which the chain began to stretch, lasted for fewer than 50 units of strain (γ = γ˙*t* < 50) for all of the recorded data sets. The raw data show that, even at relatively low *Wi*, there are large fluctuations in the extension of the molecule. As*Wi* is increased, the molecule fluctuates more rapidly (relative to its relaxation rate) and reaches larger extensions. These two basic trends are in qualitative agreement with computer simulations (3). The amount of extension increases as *Wi* is increased because the velocity gradient, and hence the net hydrodynamic forces acting across the molecule, increase.

The exact configuration of the polymer backbone cannot be extracted from the images because of the limits of optical resolution (∼0.3 μm). However, a reasonably detailed picture of the mass distribution (projected into a plane parallel to the shear plates) can be inferred from the relative brightness of various parts of the image (Fig. 3). After time-averaging many images, we find that, on average, the shape is approximately elliptical. However, the conformation changed continuously, and at various times dumbbell, half-dumbbell, kinked, and folded shapes, similar to those observed in elongational flow (11), were observed. Folded and kinked shapes were seen more often at higher*Wi*. Lumps of mass density that propagated up or down the length of partially stretched chains were also seen (Fig. 3B). Molecules that developed a hairpin fold (or U shape) while stretching or contracting tended to unfold slowly, like a rope traveling over a pulley (Fig. 3C). The polymer never made large excursions in a direction perpendicular to the shear plates. The depth of field of the microscope objective was about 1 μm, and we found that the entire chain contour could always be kept in focus as the polymer stretched.

To calculate the mean extension at each *Wi*, we time-averaged the data over all of the measurements at each shear rate and viscosity (Fig. 4). In steady shear, the mean fractional extension, 〈*x*〉/*L*, increases gradually with *Wi* and appears to approach an asymptotic value of ∼0.4 to 0.5. This behavior differs markedly from that which occurs in pure elongational flow, where the extension rises very rapidly to a value close to the full contour length of the chain at relatively low strain rates (11). These steady-state results, in shear (|ɛ˙| = |ω|) and in elongational flow (ω = 0), qualitatively agree with the basic coil-stretch “phase diagram” picture proposed by de Gennes (5). However, whether or not the elongational component will dominate the behavior as soon as |ɛ˙| is made greater than |ω| remains to be explored (10). The sharp transition seen in elongational flow was not observed in birefringence measurements on DNA (17) because in the birefringence measurements there was a wide distribution of different residence times of the molecules in the flow (18).

Previous experiments with light and neutron scattering indicated that polymers did not deform substantially in shear (7–9). However, this seems to be due to the small shear rates applied to the molecule in these experiments. Because of technical difficulties the scattering measurements were limited to*Wi *≲ 2.5, where there is very little deformation. In our experiment we were able to reach a value of *Wi *= 76, and we saw substantial (>100%) deformation for *Wi* ≳ 5. The scattering experiments measure a different quantity than our experiment—they determine the components of the radius of gyration tensor of a time-averaged mass distribution. It is difficult to compare our small-deformation data with the scattering data because blooming of the image due to the intensified camera made it difficult to accurately estimate the radius of gyration of the coiled polymer when it is only slightly deformed. This is unfortunate because the values obtained in the light- and the neutron-scattering experiments do not agree. We do observe less deformation as *Wi* is increased than that predicted by the Zimm model (1). However, good agreement is not really expected, because the Zimm model is only meant to describe small fluctuations of the polymer about its equilibrium shape and does not constrain the polymer to have a finite extensibility.

Because we recorded the dynamics of individual molecules, the probability distribution for molecular extension in shear flow can be determined directly. Histograms of the extension were calculated by binning the extension data obtained from many molecules (Fig. 5). The extension is always positive because when the polymer is coiled we cannot usually distinguish the ends of the chain and therefore do not track the orientation of the end-to-end vector. The minimum measured extension is ∼1 μm, which is close to the inferred ∼0.7-μm radius of gyration,*R*
_{G,0}, of the coiled polymer at equilibrium (19). As *Wi* is increased, the shape of the extension distribution changes dramatically. At low values of*Wi*, the distribution is strongly skewed to lower values, but as *Wi* is increased it becomes more symmetric: At higher shear rates, the molecules make more frequent and larger excursions in extension. The standard deviation, which indicates the size of the fluctuations, increases with *Wi* and reaches a value of ∼4 μm (∼20% of the full contour length) for *Wi* = 76.

To investigate the spectral properties of the fluctuations, we calculated the power spectral density per unit time (PSD) using a fast Fourier transform (Fig. 6A) (20). The rotational component of a simple shear flow causes solid objects of finite aspect ratio, such as ellipsoids (21), to tumble periodically in steady shear flow and might lead one to expect periodic tumbling for a polymer coil (2). However, these simplest models neglect Brownian motion. When Brownian motion is important we might expect a tumbling motion that is stochastic rather than periodic. Indeed, no distinct peaks were distinguishable in the power spectra, indicating that there was no periodicity detectable within the observed frequency range (22). The data suggest a random stretching and contraction process that occurs more frequently at higher shear rates. As γ˙is increased, the fluctuations are larger and more frequent because both the stretching and tumbling are driven by the velocity gradient as indicated in Fig. 1B. Alternatively, by adjusting the solvent viscosity, we can change the polymer relaxation time, τ (23). As τ is decreased, the PSD shifts to higher frequencies. An increase in the role of Brownian motion causes the polymer to diffuse more rapidly from the stretched state (left hand side of Fig. 1B) to the tumbling state (right hand side of Fig. 1B). Furthermore, if the polymer relaxes into a coiled shape, Brownian fluctuations may be needed to begin the elongation process again (top of Fig. 1B).

Commonly accepted polymer physics theory holds that the relevant dimensionless parameters describing the dynamics are*Wi* and the frequency made dimensionless by multiplying by the relaxation time. To investigate the validity of this assumption, we plotted PSD/τ*R*
_{G,0}
^{2} (dimensionless PSD) versus frequency × τ (dimensionless frequency) for two data sets that had the same value of *Wi* but were acquired at different viscosities and shear rates (Fig. 6B). In agreement with theoretical expectations, we found that the PSD for these data sets do collapse when plotted with these dimensionless parameters.

In zero flow (*Wi* = 0), there are still fluctuations in the extension that are entirely due to Brownian motion. We find that at low and intermediate frequencies the PSD of these small fluctuations from equilibrium is reasonably well fit by a Lorentzian function, PSD ∼1/(*a* + *f*
^{2}), where *a* is a constant. This function is the form of the PSD for a Brownian particle confined by a harmonic potential well (24), which indicates that in this frequency range the dynamics may be described by a “dumbbell” model in which the polymer is modeled as two Brownian particles connected by a harmonic spring. When a shear flow is applied, the shape of the PSD function changes. For the lowest *Wi* studied (*Wi* = 1.3, γ˙ = 0.2), the chain is only weakly deformed and the PSD deviates only slightly from the Lorentzian form. As *Wi* is increased, the deviation increases because the PSD falls off faster at intermediate frequencies. In all cases, however, the PSD declined much more slowly than the Lorentzian at the highest observed frequencies. This behavior is also seen in some computer simulations of polymers in shear (25).

Another way of characterizing the temporal properties of random fluctuations is to calculate the autocorrelation function 〈*x*(*t*)*x*(*t* +*T*)〉, where *t* is time and *T*is a fixed time delay interval. The autocorrelation decay shows that, although the fluctuations are not periodic, they occur on a characteristic time scale that depends on the shear rate. The autocorrelation function may also be plotted against a dimensionless time, namely the amount of strain γ˙*T *(Fig. 6C). When plotted this way, one sees that the deformation remains correlated for higher strains at higher *Wi*. Larger changes in conformation occur at higher *Wi*, and more strain is apparently required to produce these changes. Empirically, the data for all *Wi*may be approximately, but not exactly, collapsed by plotting the autocorrelation versusγ˙*T*/(*Wi*)^{2/3}
*.*

We have presented experimental data on the conformational dynamics of individual polymer molecules in steady shear flow. Possible improvements in the technique include better spatial and temporal resolution and specific labeling of portions of the polymer. The ability to clearly distinguish the ends of the chain should allow the tumbling dynamics to be studied in greater detail. In future studies, it is expected that these types of measurements may be extended to time-dependent (transient) flows and to the study of semi-dilute and entangled polymer solutions. Such experiments, in conjunction with the development of accurate molecular models, should reveal many details of the molecular processes that underlie non-Newtonian rheological effects. This approach should be useful for rigorously testing polymer physics models and lay the foundation for a complete microscopic understanding of the rheology of polymer solutions.

↵* To whom correspondence should be addressed. E-mail: schu{at}leland.stanford.edu