## Abstract

The stretching of individual polymers in a spatially homogeneous velocity gradient was observed through use of fluorescently labeled DNA molecules. The probability distribution of molecular extension was determined as a function of time and strain rate. Although some molecules reached steady state, the average extension did not, even after a ∼300-fold distortion of the underlying fluid element. At the highest strain rates, distinct conformational shapes with differing dynamics were observed. There was considerable variation in the onset of stretching, and chains with a dumbbell shape stretched more rapidly than folded ones. As the strain rate was increased, chains did not deform with the fluid element. The steady-state extension can be described by a model consisting of two beads connected by a spring representing the entropic elasticity of a worm-like chain, but the average dynamics cannot.

The behavior of dilute polymers in elongational flow has been an outstanding problem in polymer science for several decades (1, 2). In elongational flows, a velocity gradient along the direction of flow can stretch polymers far from equilibrium. Extended polymers exert a force back on the solvent that leads to the important, non-Newtonian properties of dilute polymer solutions, such as viscosity enhancement and turbulent drag reduction.

A homogeneous elongational flow is defined by a linear velocity gradient along the direction of flow such that ν_{y} =ɛ˙*y*, where ɛ˙ ≡ ∂ν_{y}/∂*y*, the strain rate, is constant. Theory suggests that the onset of polymer stretching occurs at a critical velocity gradient or strain rate ofɛ˙_{c} of
(1)where τ_{1} is the longest relaxation time of the polymer (3). For ɛ˙ <ɛ˙_{c}, the molecules are in a “coiled” state. But as ɛ˙ is increased aboveɛ˙_{c}, the hydrodynamic force exerted across the polymer just exceeds the linear portion of the polymer's entropic elasticity, and the polymer stretches until its nonlinear elasticity limits the further extension of this “stretched” state. De Gennes predicted that this “coil-stretch transition” would be sharpened by an increase in the hydrodynamic drag of the stretched state relative to the drag of the coiled state (1).

In many types of elongational flows, such as flow through a pipette tip, the residency time *t*
_{res} of the polymers in the velocity gradient is limited. To increase*t*
_{res}, flows in which there is a stagnation point are often used. As molecular trajectories approach the stagnation point, *t*
_{res}diverges. The classical techniques for inferring the degree of polymer deformation have been light scattering (4, 5) and birefringence (6-9). For example, Keller and Odell reported a rapid increase in the birefringence for ɛ˙ aboveɛ˙_{c} followed by a saturation (6). Such saturation was interpreted as an indication that the polymers had reached equilibrium in a highly extended state (10). Molecular weight analysis showed some chains are fractured in half, further supporting the hypothesis that the polymers reached full extension (8, 11). However, light-scattering experiments imply deformations of only two to four times the equilibrium size (4, 5). But, these “bulk” measurements average over a macroscopic number of molecules with a broad range of*t*
_{res}. Moreover, only recent experiments have been dilute enough to prevent the polymers from altering the flow field (9).

Many rheological effects also remain unexplained. James and Saringer measured a pressure drop in a converging flow that was significantly greater than that predicted by simple models (12). Recently, Tirtaatmadja and Sridhar measured extensional viscosities*η*
_{E} in filament stretching experiments that were several thousand times greater than the shear viscosities (13). At large deformations,*η*
_{E} saturated, suggesting again that the polymers were fully extended. However, the measured stress was significantly lower than expected for fully extended polymers, implying that full extension had not actually been achieved (14). Also, the stress relaxation in such experiments contained both a strain-rate independent “elastic” and a strain-rate dependent “dissipative” component. The molecular origin of the dissipative component is uncertain (15). Examples such as these indicate that, even after a tremendous amount of study, the deformation of polymers in elongational flows is still poorly understood (14,16).

We report the direct visualization of individual polymers in an elongational flow. The conformation and extension of each molecule was measured as a function of ɛ˙ and*t*
_{res}, thereby eliminating the ambiguities in conformation and *t*
_{res}. We further eliminated polymer-polymer interactions and polymer-induced alterations of the flow field by working with single isolated molecules. The inherent uniformity in size of lambda bacteriophage DNA [λ-DNA,*L*
_{strained} ≅ 22 μm (17)] also eliminated complications due to polydispersity and enabled accurate calculation of ensemble averages (18). Thus, we determined the probability distribution of molecular extension rather than just an average or a moment of that distribution. Another advantage is that the entropic elasticity (19, 20) and hydrodynamic drag (17, 21, 22) of single DNA molecules have been previously characterized.

Using a microfabricated flow cell, we generated a planar elongational flow with a cross-slot geometry adapted for fluorescence microscopy (23). The main design consideration of the flow cell was to ensure that we studied dynamics of polymers unwinding from equilibrium (24). Our imaging area was 100 μm by 94 μm with the stagnation point 15 μm from the center of one side. The onset of the elongational flow, where*t*
_{res}= 0, was 960 μm up the inlet channel from the stagnation point.

By tracking individual molecules, we measured the extension*x* and *t*
_{res} of each molecule in our imaging area. Some molecules deformed only slightly, whereas others rapidly reached a steady-state extension (Fig.1A). This large and previously unobservable heterogeneity was perhaps unexpected because these molecules were identical in size and had experienced the sameɛ˙ and *t*
_{res}. From an ensemble of individual measurements, we calculated the average extension*〈x(t _{res})〉* as well as the time evolution of the probability distribution for molecular extension (Fig.1B).

We characterized the conformation of each polymer in the ensemble. In general, the molecules were found in one of seven conformations which we refer to as dumbbell, half-dumbbell, folded, uniform, kinked, coiled, or extended. The first three types were dominant at ɛ˙ = 0.86 s^{−1}. As shown in Fig.2A, these are highly nonequilibrium conformations, and they occurred only at higher strain rates (ɛ˙ > 0.5 s^{–1}) (25). In this case, the molecules were subject to aɛ˙ significantly greater than the inverse relaxation time [τ_{relax}
^{−1} = 0.26; τ_{relax} = 3.89 s (26)]. For λ-DNA (∼400 persistence lengths), we saw only single folds at the highestɛ˙ investigated. However, for longer molecules, we observed multiple folds (27).

There were clear differences in dynamics for the three dominant conformations. To highlight these differences, we plotted data, using only those molecules that best typified each conformational class. Molecules in a dumbbell configuration stretched significantly faster than folded ones (Fig. 2B). In addition, the residency time*t*
_{onset}at which significant stretching begins for any particular molecule was highly variable (28).

An analysis of the rate of stretching *x˙*as function of *x* shows that once a molecule in a dumbbell configuration starts to stretch, its dynamics follows a specific time evolution (Fig. 2C, inset). This result indicates that the data would approximately collapse onto a single “master curve” by sliding the individual curves along the time axis. Up to*x/L* = 0.6, we observed a linear increase in 〈*x˙*(*x*)〉 with *x* up to*x* = 12 μm at ɛ˙ = 0.86 s^{–1}. When integrated, this yields an initial exponential growth of the master curve. We show three such master curves generated from the molecules that best typify each of the dominant conformations (Fig. 2C). For comparison, we show*〈x(t _{res})〉* for the full data set as well as for several of the different conformational classes arising from the first, general classification (Fig. 2D). Because of the large variation in

*t*

_{onset}, the master curve better represents the unwinding dynamics of individual molecules and is different in shape than

*〈x(t*.

_{res})〉We plotted the fractional average extension *〈x〉/L* as a function of the accumulated fluid strain or “Henky stain” (ɛ =ɛ˙*t _{res}
*) (Fig.3A). By analyzing the subset of molecules that reached steady state (Fig. 3B, inset), we determined the steady-state extension

*x*

_{steady}as a function of the dimensionless strain rate or “Deborah number”ɛ˙τ

_{relax}(Fig. 3B). Note that

*x*

_{steady}rises sharply at a critical strain rate ofɛ˙

_{c}τ

_{relax}≅ 0.4 and that for ɛ˙ ≅ 0.9ɛ˙

_{c}the fractional size of fluctuations is large (σ

_{x}

*/x*

_{steady}≅ 0.4). Similar behavior is often seen at phase transitions. In comparison with classical bulk measurements, we also plotted a spatio-temporal average

*x*

_{bulk}of all our data (Fig. 3B).

In a linear velocity-gradient flow (ɛ˙_{fluid} ≡ ɛ˙ = ∂ν_{y}/∂*y*), the distance between two fluid elements grows as *y* ∼ exp(ɛ˙_{fluid}
*t*
_{res}). There was a similar but slower exponential growth in the master curves of molecular extension for ɛ˙ > 0.21 s^{–1}. We defined a molecular strain rateɛ˙_{mol} from a fit of 〈*x˙*(*x*)〉 =ɛ˙_{mol} *x* + *b*over the region where 〈*x˙(x)〉* is a linear function of *x* (Fig. 4, inset) and compared ɛ˙_{mol} toɛ˙_{fluid} −ɛ˙_{c} (Fig. 4). This analysis averages over the conformation-dependent dynamics shown in Fig. 2. To single out the most rapid stretching conformation, we also plottedɛ˙_{mol} for the dumbbell configuration at ɛ˙ = 0.86 s^{–1}.

A polymer is said to “affinely” deform with the fluid if the molecular deformation equals the deformation of the surrounding fluid element. It has been suggested that when ɛ˙ >> 1/τ_{relax}, affine deformation becomes an increasingly valid approximation (29). In the simplest analysis, we note that *〈x(t _{res})〉*did not reach

*x*

_{steady}even after an accumulated fluid strain of ɛ = ɛ˙

*t*≅ 5.7, which corresponds to an e

_{res}^{5.7}or ∼300-fold distortion of the fluid element (Fig. 3). For comparison, the required molecular distortion to fully extend stained λ-DNA is

*L/R*

_{G}≅ 30 where

*R*

_{G}, the radius of gyration, is 0.73 μm (21).

In part, this lack of affine deformation in*〈x(t _{res})〉* arises from the large variation in

*t*

_{onset}. Notwithstanding this variation which is intrinsically nonaffine, we wanted to know if molecules deform affinely once they start to stretch. To do so, we analyzed the dynamics of the master curve because it suppresses the variation in

*t*

_{onset}by computing 〈

*x˙*(

*x*)〉 instead of 〈

*x˙*(

*t*)〉. At moderate strain rates, affine deformation is not expected, because there must be some slip between the polymer and the fluid to create the hydrodynamic force necessary to overcome the native elasticity of the polymer. Because there is no deformation for ɛ˙ <ɛ˙

_{res}_{c}, we plottedɛ˙

_{mol}versusɛ˙

_{fluid}−ɛ˙

_{c}, whereɛ˙

_{c}≅ 0.4/τ

_{relax}. At lower ɛ˙, the molecules are stretching near the theoretically expected limit (Fig. 4). At higher ɛ˙, the data shows a marked departure, and it is clear that the affine deformation approximation breaks down. Furthermore, when plotted asɛ˙

_{mol}/(ɛ˙

_{fluid}− ɛ˙

_{c}) versus (ɛ˙

_{fluid}−ɛ˙

_{c}), the data is decreasing at 0.86 s

^{–1}. Thus, the data shows neither an absolute nor a fractional approach toward affine deformation at higherɛ˙ even after eliminating the large variation in

*t*

_{onset}. This failure arises from the introduction of intramolecular constraints (folds) which dramatically slow down the average dynamics. On the other hand, the subset of molecules in a dumbbell configuration stretched almost as fast as can be theoretically expected.

Our steady-state results are approximately characterized by a simple “dumbbell” model consisting of two beads connected by a spring based on the Marko-Siggia force law (Fig. 3B, solid line) (20). Previously, the steady-state extension of a tethered polymer in a uniform flow was well described by this model (17), and we developed a molecular understanding of the origin of this agreement based on simulations (22). An extrapolation of the model to *x* = 0 gives a critical strain rate ofɛ˙_{c}τ_{relax}≅ 0.4, which is near the theoretical value of 0.5 calculated from the Zimm model and by the numerical calculation of Larson and Magda (3). This value ofɛ˙_{c}τ_{relax}≅ 0.4 is less than the values of 3 to 8 seen in recent birefringence measurements of polystyrene solutions by Nguyen *et al.*(9).

To see if this model could self-consistently describe the dynamics of the master curve, we calculated the expected dynamics, using parameters determined from the steady-state results. Whereas the predicted rate of extension is close to the measured dynamics for the dumbbell configuration, it overestimates the average measured dynamics (Fig. 4, inset). So, although this dumbbell model describes the steady-state extension, it fails to describe these simplified dynamics in which the large variation in *t*
_{onset} is suppressed. Therefore, we expect difficulty in trying to predict the transient stress in the fluid by constitutive equations based on a simple dumbbell model (2). Given the nonaffine deformation, a term proportional to −*x˙* which can describe an “internal viscosity” might be added (2). Such a term is suggested by the measurements of η_{E} because it leads a dissipative component of the stress relaxation (15). Although a term proportional to −*x˙ *can approximately compensate for the slower average dynamics, our data show that, in part, these slower dynamics arise from folded configurations which are meta-stable rather than arising from the monomer-monomer friction typically associated with internal viscosity. We note that there are additional terms besides −*x˙ *that can lead to dissipative stresses (30, 31).

Given our measurements of the dynamic, steady-state, and ensemble-averaged properties of polymers in an elongational flow, we now compare our data to previous experimental and theoretical results. Atkins and Taylor measured the birefringence of λ-DNA in a similar planar elongational flow (Fig. 3B) (8). Our ability to select only those molecules that have reached steady-state extensions reveals a much sharper transition occurring at a lowerɛ˙_{c}
*τ _{relax}
*. As discussed above, the higher value ofɛ˙

_{c}τ

_{relax}seen by birefringence occurred for synthetic polymers as well as for DNA (9). Evidently there is no direct correspondence between either

*x*

_{steady}or

*x*

_{bulk}and the birefringence at the stagnation point. Because birefringence measures orientation rather than extension, some disagreement would be expected based on the observed conformational features such as folds. However, folds would cause a premature saturation in birefringence with respect to

*x*

_{steady}. Our results highlight the difficulties in interpreting birefringence and other bulk measurements and suggest that this difficulty may be even greater for synthetic polymers, for which the larger ratio of

*L/R*

_{G}requires an even larger accumulated fluid strain than is needed to extend λ-DNA.

In contrast to previous light scattering results on synthetic polymers (4, 5), our data shows extensions significantly greater than ∼2 *R*
_{G}, though*R*
_{G}, by definition, is always less than*x*/2. In general, the large difference between*R*
_{G} and*x _{steady}/2* is caused by the broad distribution in

*t*

_{res}for the population of molecules measured by light scattering. Hence,

*x*, not

_{bulk}/2*x*, should be used for comparison. In addition, the highly asymmetric mass distribution of the most common conformation (half-dumbbell) would further reduce

_{steady}/2*R*

_{G}. In particular, we directly calculated

*R*from the image data forɛ˙τ

_{G}_{relax}= 1.2. A spatio-temporal average of this data yielded

*R*

_{G}

^{bulk}= 2.2 μm, which is three times the equilibrium coil size [

*R*

_{G}= 0.73 μm (21)] but is much smaller than steady-state extension (

*x*

_{steady}= 14.8 μm) at thisɛ˙. Thus, our results help explain the apparent discrepancy between light scattering and birefringence measurements.

Our results suggest that midpoint chain fracture in stagnation point flows does not imply that all chains are extended. The large variability in *x* (Fig. 1A) indicates that a number of molecules rapidly reach steady state. If we extrapolate our results to a ɛ˙ of 100 times higher, it is these highly extended, early-stretching molecules that will experience a force large enough to fracture at or near their center. Nonetheless, because of the limited*t*
_{res}, the number of such chains that are rapidly stretching and start stretching early is relatively small. Thus, only a fraction of the total number of chains fracture in agreement with the results of bulk experiments (8, 11), but this fracture of some chains does not imply that all chains are extended.

Rheologists often infer molecular deformation from bulk viscoelastic measurements (2). Given the data in Fig. 1, the known elasticity of DNA (20), and classical results in rheology (2), one can calculate the extensional stress σ_{E} = *n* 〈*x·F(x)*〉 and the extensional viscosity η_{E} = σ_{E}/ɛ˙ where *n* is density of molecules and *F(x)* is the steady-state elasticity. However, because these molecules are in highly nonequilibrium configurations (Fig. 2A), it is inaccurate to use the steady-state elasticity for molecules at ɛ˙ >> 1/τ_{relax}. From this and the lack of a physically significant mean as described below, our results suggest difficulties with inferring an average conformation from bulk rheological measurements. Additionally, our results reveal problems with the use of the Peterlin approximation (32), in which*x ^{2}(t_{res})* is replaced by

*〈x*, to derive constitutive equations that predict bulk rheological measurements from a micromechanical or kinetic theory (2). The heterogeneity in our data that leads to the breakdown of the Peterlin approximation is also seen in Keunings' stochastic simulations of the finitely extensible dumbbell model (33). Although this simplified model of polymer dynamics based on kinetic theory yields histograms that are in semiquantitative agreement with our data (Fig. 1B), simulations with the Peterlin approximation in conjunction with kinetic theory lead to qualitatively different results.

^{2}(t_{res})〉To account for the excess stress measured by James and Saringer (12), Ryskin, Larson, Hinch, and King and James have developed theories based on different hypothesized molecular configurations (16, 31). By direct observation of dumbbell, half-dumbbell, folded, and kinked conformations, we confirm the presence of conformations similar to those proposed. The presence of these conformations provides a qualitative explanation for the dissipative component of stress found in measurements of η_{E}. However, no one of the theories describes the complete range of observed conformations. Rather, the individual conformations assumed in these theories represent one of the several observed conformations.

From a theoretical point of view, the conformation-dependent dynamics implies that the commonly used approach of developing mean-field theories has an inherent disadvantage (34). The probability distribution is not a narrow distribution about a mean but rather a broad, oddly shaped distribution (Fig. 1B) because of several distinctly different dynamical processes (Fig. 2). Further, the differences in *x˙ *and*t*
_{onset} imply a sensitive dependence on the polymer's initial conformation when it enters the velocity gradient. Presumably, these variations arise directly from the multitude of accessible conformations at equilibrium where thermal fluctuations cause instantaneous deviations away from a spherically symmetric distribution. For instance, a polymer whose initial configuration has both ends on the same side of the center of mass and is subject to aɛ˙ >> 1/τ_{relax} would most likely become folded, because there is not enough time (τ_{relax}) for an end to move to the other side of the molecule. Variations similar to those in our experimental data have been observed in the simulations of Larson (31), Hinch (16), and Keunings (33).

Although we observe a sudden increase in the steady-state extension of polymers at a critical strain rate, our data indicates that the concept of a discrete and abrupt coil-stretch transition is limited to the steady state. Polymers do not undergo a simple, collective and simultaneous unwinding as soon as ɛ˙ >ɛ˙_{c}. The mismatch between 〈*x*(*t*
_{res})〉 and*x*
_{steady} implies that the non-Newtonian properties of dilute polymer solutions in most practical elongational flows (where ɛ˙*t _{res}
* < 5.5) are dominated by the dynamic and not the steady-state properties. Our data should serve as a guide in developing improved microscopic theories for polymer dynamics and the bulk rheological properties of such solutions.

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