REFERENCES AND NOTES___________________________

1. R. W. Simpson, S. S. Schulz, L. D. Dietz, R. O.Burford, Pure Appl. Geophys. 126, 665 (1988).

2. Working Group on California Earthquake Probabili-ties, U.S. Geol. Surv. Circ. 1053, 51 (1990).

3. J. J. Lienkaemper, G. Borchardt, M. Lisowski [J.Geophys. Res. 96, 18261 (1991)], being revised as J.J. Lienkaemper and J. S. Galehouse, in preparation;J. J. Lienkaemper, J. S. Galehouse, R. W. Simpson,Eos 73, 119 (1992);iiii, K. S. Breckenridge, ibid.74, 192 (1993).

4. P. A. Reasenberg and R. W. Simpson, Science 255,1687 (1992).

5. R. W. Simpson and P. A. Reasenberg, U.S. Geol.Surv. Prof. Pap. 1550-F, 55 (1994).

6. J. S. Galehouse, Geophys. Res. Lett. 17, 1219(1990); U.S. Geol. Surv. Prof. Pap., in press.

7. iiii, B. D. Brown, B. Pierce, J. J. Thordsen, Calif.Div. Mines Geol. Spec. Publ. 62, 239 (1982).

8. J. Langbein and H. Johnson, J. Geophys. Res. 102,591 (1997).

9. J. C. Savage and M. Lisowski, ibid. 98, 787 (1993).10. The previous largest event on the fault was 10 mm in

1983 during a 124-day interval at km 62. The date ofthe 18-mm event may be 9 February 1996, when awater main broke at km 65.9 on the fault. This 1996event extended more than 1.7 km (from km 65.3 to67.0), as shown on four arrays but less than 3.7 km,as limited by other data. We observed no triggeredslip on the Hayward fault in 1989 except for 3 mmthat showed on fresh cracks at km 66.3 (3).

11. Suddenness of resumption of fast creep from km65 to 67 may be caused by two extremely rainyseasons following a drought. North of km 63, allsites had resumed pre–Loma Prieta rates by 1994.Two sites, km 63 and 64, resumed nearly normalrates during 1994. A creepmeter at km 63.6 re-sumed a 9 mm/year rate by late 1995 with no largeevent in early 1996. We note that a borehole strain-meter located 5 km northeast of the ;18-mmcreep event recorded a strain event starting on 7February 1996. These data are consistent with slipof 18 mm on a patch ;1 km in extent (M. Johnston,personal communication). Possibly, creep begansteadily at depth months earlier, not slipping at thesurface until induced by a water table raised byheavy rains.

12. M. Lisowski, W. H. Prescott, J. C. Savage, M. J.Johnston, Geophys. Res. Lett. 17, 1437 (1990). Dis-location model parameters: 37 km in length; 316°strike, 70°SW dip; depth of 5 to 17.5 km; 1.66-mdextral slip; and 1.19-m reverse. Other models test-ed yield similar results for the Hayward fault (4, 5).

13. Funded by the U.S. Geological Survey National Earth-quake Hazards Reduction Program (NEHRP) and SanFrancisco State University (NEHRP contract 1434-94-G2420). We thank many student assistants, espe-cially C. Garrison, O. Graves, T. Hoyt, and F. McFar-land; the governments of Fremont, Hayward, San Le-andro, and Oakland; W. Hazelwood of the Bay AreaRapid Transit District; D. J. Russell; and two anony-mous reviewers, W. Prescott and J. Langbein.

11 March 1997; accepted 5 May 1997

Single Polymer Dynamicsin an Elongational Flow

Thomas T. Perkins, Douglas E. Smith, Steven Chu*

The stretching of individual polymers in a spatially homogeneous velocity gradient wasobserved through use of fluorescently labeled DNA molecules. The probability distri-bution 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 aftera ;300-fold distortion of the underlying fluid element. At the highest strain rates, distinctconformational shapes with differing dynamics were observed. There was considerablevariation in the onset of stretching, and chains with a dumbbell shape stretched morerapidly than folded ones. As the strain rate was increased, chains did not deform withthe fluid element. The steady-state extension can be described by a model consistingof two beads connected by a spring representing the entropic elasticity of a worm-likechain, but the average dynamics cannot.

The behavior of dilute polymers in elon-gational flow has been an outstanding prob-lem in polymer science for several decades(1, 2). In elongational flows, a velocitygradient along the direction of flow canstretch polymers far from equilibrium. Ex-tended polymers exert a force back on thesolvent that leads to the important, non-Newtonian properties of dilute polymer so-lutions, such as viscosity enhancement andturbulent drag reduction.

A homogeneous elongational flow is de-fined by a linear velocity gradient along the

direction of flow such that ny 5 εy, where ε§ ]ny/]y, the strain rate, is constant. Theorysuggests that the onset of polymer stretch-ing occurs at a critical velocity gradient orstrain rate of εc of

εc '0.5t1

(1)

where t1 is the longest relaxation time ofthe polymer (3). For ε , εc, the moleculesare in a “coiled” state. But as ε is increasedabove εc, the hydrodynamic force exertedacross the polymer just exceeds the linearportion of the polymer’s entropic elasticity,and the polymer stretches until its nonlin-ear elasticity limits the further extension ofthis “stretched” state. De Gennes predictedthat this “coil-stretch transition” would be

sharpened by an increase in the hydrody-namic drag of the stretched state relative tothe drag of the coiled state (1).

In many types of elongational flows,such as flow through a pipette tip, the res-idency time tres of the polymers in the ve-locity gradient is limited. To increase tres,flows in which there is a stagnation pointare often used. As molecular trajectoriesapproach the stagnation point, tres diverges.The classical techniques for inferring thedegree of polymer deformation have beenlight scattering (4, 5) and birefringence(6–9). For example, Keller and Odell re-ported a rapid increase in the birefringencefor ε above εc followed by a saturation (6).Such saturation was interpreted as an indi-cation that the polymers had reached equi-librium in a highly extended state (10).Molecular weight analysis showed somechains are fractured in half, further support-ing the hypothesis that the polymersreached full extension (8, 11). However,light-scattering experiments imply deforma-tions of only two to four times the equilib-rium size (4, 5). But, these “bulk” measure-ments average over a macroscopic numberof molecules with a broad range of tres.Moreover, only recent experiments havebeen dilute enough to prevent the polymersfrom altering the flow field (9).

Many rheological effects also remain un-explained. James and Saringer measured apressure drop in a converging flow that wassignificantly greater than that predicted bysimple models (12). Recently, Tirtaatmadjaand Sridhar measured extensional viscosi-ties hE in filament stretching experimentsthat were several thousand times greaterthan the shear viscosities (13). At largedeformations, hE saturated, suggesting againthat the polymers were fully extended.However, the measured stress was signifi-cantly lower than expected for fully extend-ed polymers, implying that full extensionhad not actually been achieved (14). Also,the stress relaxation in such experimentscontained both a strain-rate independent“elastic” and a strain-rate dependent “dissi-pative” component. The molecular origin ofthe dissipative component is uncertain(15). Examples such as these indicate that,even after a tremendous amount of study,the deformation of polymers in elongationalflows is still poorly understood (14, 16).

We report the direct visualization ofindividual polymers in an elongationalflow. The conformation and extension ofeach molecule was measured as a functionof ε and tres, thereby eliminating the am-biguities in conformation and tres. We fur-ther eliminated polymer-polymer interac-tions and polymer-induced alterations ofthe flow field by working with single iso-lated molecules. The inherent uniformity

Department of Physics, Stanford University, Stanford, CA94305, USA.

*To whom correspondence should be addressed. E-mail:schu@leland.stanford.edu

SCIENCE z VOL. 276 z 27 JUNE 1997 z www.sciencemag.org2016

in size of lambda bacteriophage DNA[l-DNA, Lstrained > 22 mm (17)] also elim-inated complications due to polydispersityand enabled accurate calculation of en-semble averages (18). Thus, we deter-mined the probability distribution of mo-lecular extension rather than just an av-erage or a moment of that distribution.Another advantage is that the entropicelasticity (19, 20) and hydrodynamic drag(17, 21, 22) of single DNA moleculeshave been previously characterized.

Using a microfabricated flow cell, wegenerated a planar elongational flow with across-slot geometry adapted for fluorescencemicroscopy (23). The main design consider-ation of the flow cell was to ensure that westudied dynamics of polymers unwindingfrom equilibrium (24). Our imaging area was100 mm by 94 mm with the stagnation point15 mm from the center of one side. Theonset of the elongational flow, where tres 50, was 960 mm up the inlet channel from thestagnation point.

By tracking individual molecules, wemeasured the extension x and tres of eachmolecule in our imaging area. Some mole-cules deformed only slightly, whereas othersrapidly reached a steady-state extension(Fig. 1A). This large and previously unob-servable heterogeneity was perhaps unex-pected because these molecules were iden-tical in size and had experienced the same εand tres. From an ensemble of individualmeasurements, we calculated the averageextension ^x(tres)& as well as the time evo-lution of the probability distribution formolecular extension (Fig. 1B).

We characterized the conformation ofeach polymer in the ensemble. In general,the molecules were found in one of sevenconformations which we refer to as dumb-bell, half-dumbbell, folded, uniform, kinked,coiled, or extended. The first three typeswere dominant at ε 5 0.86 s21. As shown inFig. 2A, these are highly nonequilibriumconformations, and they occurred only athigher strain rates (ε . 0.5 s–1) (25). In thiscase, the molecules were subject to a ε sig-nificantly greater than the inverse relaxationtime [trelax

21 5 0.26; trelax 5 3.89 s (26)]. Forl-DNA (;400 persistence lengths), we sawonly single folds at the highest ε investigat-ed. However, for longer molecules, we ob-served multiple folds (27).

There were clear differences in dynam-ics for the three dominant conformations.To highlight these differences, we plotteddata, using only those molecules that besttypified each conformational class. Mole-cules in a dumbbell configurationstretched significantly faster than foldedones (Fig. 2B). In addition, the residencytime tonset at which significant stretchingbegins for any particular molecule was

highly variable (28).An analysis of the rate of stretching x

as function of x shows that once a mole-cule in a dumbbell configuration starts tostretch, its dynamics follows a specifictime evolution (Fig. 2C, inset). This resultindicates that the data would approxi-mately collapse onto a single “mastercurve” by sliding the individual curvesalong the time axis. Up to x/L 5 0.6, weobserved a linear increase in ^x(x)& with xup to x 5 12 mm at ε 5 0.86 s–1. Whenintegrated, this yields an initial exponen-tial growth of the master curve. We showthree such master curves generated fromthe molecules that best typify each of thedominant conformations (Fig. 2C). Forcomparison, we show ^x(tres)& for the fulldata set as well as for several of the differ-ent conformational classes arising fromthe first, general classification (Fig. 2D).Because of the large variation in tonset, themaster curve better represents the un-winding dynamics of individual moleculesand is different in shape than ^x(tres)&.

We plotted the fractional average exten-sion ^x&/L as a function of the accumulatedfluid strain or “Henky stain” (ε 5 εtres) (Fig.

3A). By analyzing the subset of moleculesthat reached steady state (Fig. 3B, inset), wedetermined the steady-state extension xsteadyas a function of the dimensionless strainrate or “Deborah number” εtrelax (Fig. 3B).Note that xsteady rises sharply at a criticalstrain rate of εctrelax > 0.4 and that for ε >0.9 εc the fractional size of fluctuations islarge (sx /xsteady > 0.4). Similar behavior isoften seen at phase transitions. In compar-ison with classical bulk measurements, wealso plotted a spatio-temporal average xbulkof all our data (Fig. 3B).

In a linear velocity-gradient flow (εfluid[ ε 5 ]ny/]y), the distance between twofluid elements grows as y ; exp(εfluidtres).There was a similar but slower exponentialgrowth in the master curves of molecularextension for ε . 0.21 s–1. We defined amolecular strain rate εmol from a fit of ^x(x)&5 εmol x 1 b over the region where ^x(x)& isa linear function of x (Fig. 4, inset) andcompared εmol to εfluid 2 εc (Fig. 4). Thisanalysis averages over the conformation-dependent dynamics shown in Fig. 2. Tosingle out the most rapid stretching confor-mation, we also plotted εmol for the dumb-bell configuration at ε 5 0.86 s–1.

Fig. 1. (A) The extension x as a function of the polymer’sinteraction or residency time tres in the elongational flow at« 5 0.86 s21 for 992 molecules. Several individual traces are highlighted. We also plotted the averageextension ^x(tres)& as open circles. Notice in particular, the large heterogeneity in the dynamics of thesemolecules. To slow down the dynamics, we used an aqueous sugar solution (h 5 41 cP). We imagedthose molecules whose center of mass started within 22 mm of the center line of the inlet channel. Withinthis region, the measured velocity gradient along the full length of the inlet (]vx/]x 5 2«) was linear andwithin 2% of the measured velocity gradient along the outgoing axis (]ny/]y 5 «). Because the fluid isincompressible (¹zWn 5 0) and it is a planar flow (]vz /]z 5 0), the molecules experienced a constant strainrate independent of position. Furthermore, the calibrated strain rate measured by tracking fluorescentbeads agreed within 2% with the strain rate calculated from the motion of the center of intensity ofindividual DNA molecules. The data starts at tres . 0, because the onset of elongational flow xonset is960 mm upstream from the stagnation point, whereas the edge of the imaging area xscreen is 86 mmupstream. The interaction time of the polymer with the applied velocity gradient before imaging is givenby t 5 ln(xonset/xscreen)/« . The raw extension data was smoothed by weighted averages with its nearestneighbor of xi 5 0.21 xi–1 1 0.58 xi 1 0.21 xi11. (B) Time evolution of the probability distribution ofmolecular extension calculated from an ensemble of at least 40 individual molecules at « 5 0.86 s–1. Thesecondary peak arises from molecules in a folded configuration (Fig. 2). Keuning’s simulations of abead-spring-bead model generated similar broadly shaped histograms (33), though such simulationsare incapable of producing the secondary peak associated with folded configuration seen in theexperimental data.

REPORTS

www.sciencemag.org z SCIENCE z VOL. 276 z 27 JUNE 1997 2017

A polymer is said to “affinely” deformwith the fluid if the molecular deformationequals the deformation of the surroundingfluid element. It has been suggested thatwhen ε .. 1/trelax, affine deformation be-comes an increasingly valid approximation(29). In the simplest analysis, we note that^x(tres)& did not reach xsteady even after anaccumulated fluid strain of ε 5 εtres > 5.7,which corresponds to an e5.7 or ;300-folddistortion of the fluid element (Fig. 3). Forcomparison, the required molecular distor-tion to fully extend stained l-DNA is L/RG> 30 where RG, the radius of gyration, is0.73 mm (21).

In part, this lack of affine deformation in^x(tres)& arises from the large variation intonset. Notwithstanding this variation whichis intrinsically nonaffine, we wanted toknow if molecules deform affinely oncethey start to stretch. To do so, we analyzedthe dynamics of the master curve because itsuppresses the variation in tonset by comput-ing ^x(x)& instead of ^x(tres)&. At moderate

strain rates, affine deformation is not ex-pected, because there must be some slipbetween the polymer and the fluid to createthe hydrodynamic force necessary to over-come the native elasticity of the polymer.Because there is no deformation for ε , εc,we plotted εmol versus εfluid 2 εc, where εc >0.4/trelax. At lower ε, the molecules arestretching near the theoretically expectedlimit (Fig. 4). At higher ε, the data shows amarked departure, and it is clear that theaffine deformation approximation breaksdown. Furthermore, when plotted as εmol/(εfluid 2 εc) versus (εfluid 2 εc), the data isdecreasing at 0.86 s–1. Thus, the data showsneither an absolute nor a fractional ap-proach toward affine deformation at higherε even after eliminating the large variationin tonset. This failure arises from the intro-duction of intramolecular constraints(folds) which dramatically slow down theaverage dynamics. On the other hand, thesubset of molecules in a dumbbell configu-ration stretched almost as fast as can be

theoretically expected.Our steady-state results are approximate-

ly characterized by a simple “dumbbell”model consisting of two beads connected bya spring based on the Marko-Siggia forcelaw (Fig. 3B, solid line) (20). Previously,the steady-state extension of a tetheredpolymer in a uniform flow was well de-scribed by this model (17), and we devel-oped a molecular understanding of the ori-gin of this agreement based on simulations(22). An extrapolation of the model to x 50 gives a critical strain rate of εctrelax > 0.4,which is near the theoretical value of 0.5calculated from the Zimm model and by thenumerical calculation of Larson and Magda(3). This value of εctrelax > 0.4 is less thanthe values of 3 to 8 seen in recent birefrin-gence measurements of polystyrene solu-tions by Nguyen et al. (9).

To see if this model could self-consistent-ly describe the dynamics of the master curve,we calculated the expected dynamics, usingparameters determined from the steady-state

A B

C D

4 6 80

5

10

15

20

10

15

20

DumbbellFolded

Residency time (s)

Ext

ensi

on

(µm

)

00

0 10 20

2 4 6 8 2 4 6 8

5

15

20

10

0

5

10

0

5

Master curve Ensemble averagex (µm)

x (µ

m s

-1)

Time (s) Residency time (s)

Ext

ensi

on

(µm

)

< x (t

res)

> (

µm)

Dumbbell Half-dumbbell Folded

All Dumbbell Half-dumbbell Folded Coil

•

Fig. 2. Conformation-de-pendent rate of stretching:(A) Images of molecularconfigurations spaced ev-ery 0.13 s at the higheststrain rate investigated («5 0.86 s–1). We classifiedthese configurations as(from top to bottom)dumbbell, kinked, half-dumbbell, and folded. Themolecular extension of thelast image in the first row is13.9 mm. Sketches ofpossible molecular config-urations are shown on theleft. (Inset) A schematic il-lustration of the flow pat-tern where the red rectan-gle indicates the observation region. (B) Individualtraces of x versus tres for those molecules thatbest typify dumbbell or folded conformations. Forclassification purposes, the molecules that besttypified a dumbbell configuration had approxi-mately symmetric “coils” at each end. For classi-fication as folded, we required the initial percent-age of the folded section to be .75%. This clas-sification yielded 30 dumbbells, 34 folded, and 43half-dumbbells out of 992 molecules, and thesemolecules were reanalyzed. A general classifica-tion of all molecules at « 5 0.86 s21 yielded 5.4%kinked, 24% folded, 20% dumbbell, 35% half-dumbbell, 8.3% uniform, 5% extended, and 3%coils where precedence for classification is in thelisted order (for example, the molecule in the sec-ond row of Fig. 2A was classified as kinked). At «5 0.51s21, this classification yielded 3.7% kinked,10% folded, 31% dumbbells, 33% half-dumbbells, 9.8% uniform, 3.7% ex-tended, and 9.1% coils. A coil is a ball-shaped configuration that did notdeform during the observation time. (C) Comparison of the “master curves” fordominant conformations. To obtain these curves, we integrated ^x(x)& versus xfor those molecules that best typified each conformation where ^x (x)& wascalculated by binning the individual x(x) measurements every 0.5 mm. (Inset)

The rate of extension x for each dumbbell molecule in Fig. 2B was calculated bya linear fit over five successive data points. (D) The average extension ^x& as afunction tres for all of the data (black) and the different conformations (colors) for« 5 0.86 s21. Averages were calculated from at least 40 molecules except forthe coiled configuration, which because of the limited number of coils and theirlack of variability, was calculated from an ensemble of at least 10 molecules.

SCIENCE z VOL. 276 z 27 JUNE 1997 z www.sciencemag.org2018

results. Whereas the predicted rate of exten-sion is close to the measured dynamics forthe dumbbell configuration, it overestimatesthe average measured dynamics (Fig. 4, in-set). So, although this dumbbell model de-scribes the steady-state extension, it fails todescribe these simplified dynamics in whichthe large variation in tonset is suppressed.Therefore, we expect difficulty in trying topredict the transient stress in the fluid byconstitutive equations based on a simpledumbbell model (2). Given the nonaffinedeformation, a term proportional to 2xwhich can describe an “internal viscosity”might be added (2). Such a term is suggestedby the measurements of hE because it leads adissipative component of the stress relax-ation (15). Although a term proportional to2x can approximately compensate for theslower average dynamics, our data show that,in part, these slower dynamics arise fromfolded configurations which are meta-stablerather than arising from the monomer-monomer friction typically associated withinternal viscosity. We note that there areadditional terms besides 2x that can lead todissipative stresses (30, 31).

Given our measurements of the dynamic,

steady-state, and ensemble-averaged proper-ties of polymers in an elongational flow, wenow compare our data to previous experimen-tal and theoretical results. Atkins and Taylormeasured the birefringence of l-DNA in asimilar planar elongational flow (Fig. 3B) (8).Our ability to select only those molecules thathave reached steady-state extensions reveals amuch sharper transition occurring at a lowerεctrelax. As discussed above, the higher valueof εctrelax seen by birefringence occurred forsynthetic polymers as well as for DNA (9).Evidently there is no direct correspondencebetween either xsteady or xbulk and the birefrin-gence at the stagnation point. Because bire-fringence measures orientation rather thanextension, some disagreement would be ex-pected based on the observed conformationalfeatures such as folds. However, folds wouldcause a premature saturation in birefringencewith respect to xsteady. Our results highlightthe difficulties in interpreting birefringenceand other bulk measurements and suggest thatthis difficulty may be even greater for synthet-ic polymers, for which the larger ratio of L/RGrequires an even larger accumulated fluidstrain than is needed to extend l-DNA.

In contrast to previous light scattering

results on synthetic polymers (4, 5), our datashows extensions significantly greater than;2 RG, though RG, by definition, is alwaysless than x /2. In general, the large differencebetween RG and xsteady/2 is caused by thebroad distribution in tres for the population ofmolecules measured by light scattering.Hence, xbulk/2, not xsteady/2, should be usedfor comparison. In addition, the highlyasymmetric mass distribution of the mostcommon conformation (half-dumbbell)would further reduce RG. In particular, wedirectly calculated RG from the image datafor εtrelax 5 1.2. A spatio-temporal averageof this data yielded RG

bulk 5 2.2 mm, which isthree times the equilibrium coil size [RG 50.73 mm (21)] but is much smaller thansteady-state extension (xsteady 5 14.8 mm) atthis ε. Thus, our results help explain theapparent discrepancy between light scatter-ing and birefringence measurements.

Our results suggest that midpoint chainfracture in stagnation point flows does notimply that all chains are extended. Thelarge variability in x (Fig. 1A) indicates thata number of molecules rapidly reach steadystate. If we extrapolate our results to a ε of100 times higher, it is these highly extend-ed, early-stretching molecules that will ex-perience a force large enough to fracture ator near their center. Nonetheless, becauseof the limited tres, the number of suchchains that are rapidly stretching and startstretching early is relatively small. Thus,only a fraction of the total number of chainsfracture in agreement with the results ofbulk experiments (8, 11), but this fractureof some chains does not imply that allchains are extended.

Rheologists often infer molecular deforma-tion from bulk viscoelastic measurements (2).Given the data in Fig. 1, the known elasticityof DNA (20), and classical results in rheology(2), one can calculate the extensional stresssE 5 n ^xzF(x)& and the extensional viscosityhE 5 sE/ε where n is density of molecules andF(x) is the steady-state elasticity. However,because these molecules are in highly non-equilibrium configurations (Fig. 2A), it is in-accurate to use the steady-state elasticity formolecules at ε .. 1/trelax. From this and thelack of a physically significant mean as de-scribed below, our results suggest difficultieswith inferring an average conformation frombulk rheological measurements. Additionally,our results reveal problems with the use of thePeterlin approximation (32), in which x2(tres)is replaced by ^x2(tres)&, to derive constitutiveequations that predict bulk rheological mea-surements from a micromechanical or kinetictheory (2). The heterogeneity in our data thatleads to the breakdown of the Peterlin ap-proximation is also seen in Keunings’ stochas-tic simulations of the finitely extensibledumbbell model (33). Although this simpli-

Fig. 3. (A) Fractional average extension ^x&/L calculated from an ensemble of individual measurementsas a function of the accumulated fluid strain « 5 «tres. The deformation of the surrounding fluid elementis given by exp(«). Averages were calculated from up to ;1000 individual molecules for the fivehighest strain rates and up to ;400 molecules at the lower strain rates depending on tres. Averagesfor ,40 molecules were not plotted. For « 5 0.51 s21, the stage was moved 100 mm up an inlet toobserve the earlier time evolution. Note the similarity in the slope between the two highest « impliesthe difference in magnitude of ^x&/L probably arises more from a reduction in ^«tonset& than from anincrease in x for the highest «. This supposition is verified by the analysis presented in Fig. 4. (B) Thesteady-state extension xsteady (open symbols) and “bulk” averaged extension xbulk (closed symbols)as a function of the dimensionless strain rate «trelax, where trelax is the longest relaxation time (34) or,more precisely, the slowest measured relaxation time from an extended state (8, 26). For the fivehighest «, xsteady was fit to a dumbbell model (solid line) with a worm-like spring (20) and twoparameters: the contour length (L5 21.1 mm) of the stained DNA and the effective bead radius (Rbead

5 0.16 mm). This value of L for stained l-DNA is close to our previous measurement of L 5 22 mm(17 ). Because of the large fluctuations at lower «, xsteady was calculated as average over all the dataafter the molecule reached the lower bound of the fluctuations about xsteady. xbulk was determined bya spatio-temporal average over all measurements. The dashed line is the normalized birefringencedata of dilute l-DNA by Atkins and Taylor (8). The dashed arrows indicate the relationship between^x(tres )& and xsteady for the same «. Note, ^x(tres)& does not reach xsteady within «tres # 5.7 even thoughsome individual molecules do (see inset). The symbol shape is preserved between plots and indicates«. (Inset) An individual molecule stretching to a steady-state value (open circles).

REPORTS

www.sciencemag.org z SCIENCE z VOL. 276 z 27 JUNE 1997 2019

fied model of polymer dynamics based onkinetic theory yields histograms that are insemiquantitative agreement with our data(Fig. 1B), simulations with the Peterlin ap-proximation in conjunction with kinetic the-ory lead to qualitatively different results.

To account for the excess stress measuredby James and Saringer (12), Ryskin, Larson,Hinch, and King and James have developedtheories based on different hypothesized mo-lecular configurations (16, 31). By directobservation of dumbbell, half-dumbbell,folded, and kinked conformations, we con-firm the presence of conformations similar tothose proposed. The presence of these con-formations provides a qualitative explana-tion for the dissipative component of stressfound in measurements of hE. However, noone of the theories describes the completerange of observed conformations. Rather,the individual conformations assumed inthese theories represent one of the severalobserved conformations.

From a theoretical point of view, theconformation-dependent dynamics impliesthat the commonly used approach of devel-oping mean-field theories has an inherentdisadvantage (34). The probability distribu-tion is not a narrow distribution about amean but rather a broad, oddly shaped dis-tribution (Fig. 1B) because of several dis-

tinctly different dynamical processes (Fig.2). Further, the differences in x and tonsetimply a sensitive dependence on the poly-mer’s initial conformation when it entersthe velocity gradient. Presumably, thesevariations arise directly from the multitudeof accessible conformations at equilibriumwhere thermal fluctuations cause instanta-neous deviations away from a sphericallysymmetric distribution. For instance, apolymer whose initial configuration hasboth ends on the same side of the center ofmass and is subject to a ε .. 1/trelax wouldmost likely become folded, because there isnot enough time (trelax) for an end to moveto the other side of the molecule. Variationssimilar to those in our experimental datahave been observed in the simulations ofLarson (31), Hinch (16), and Keunings(33).

Although we observe a sudden increasein the steady-state extension of polymers ata critical strain rate, our data indicates thatthe concept of a discrete and abrupt coil-stretch transition is limited to the steadystate. Polymers do not undergo a simple,collective and simultaneous unwinding assoon as ε . εc. The mismatch between^x(tres)& and xsteady implies that the non-Newtonian properties of dilute polymer so-lutions in most practical elongational flows

(where εtres , 5.5) are dominated by thedynamic and not the steady-state proper-ties. Our data should serve as a guide indeveloping improved microscopic theoriesfor polymer dynamics and the bulk rheo-logical properties of such solutions.

REFERENCES AND NOTES___________________________

1. P. G. de Gennes, J. Chem. Phys. 60, 5030 (1974).2. R. G. Larson, Constitutive Equations for Polymer

Melts and Solution (Buttersworths, New York, 1988);R. B. Bird, C. F. Curtiss, R. C. Armstrong, O. Has-sager, Dynamics of Polymeric Liquids ( Wiley, NewYork, ed. 2, 1987), vol. 2.

3. R. G. Larson and J. J. Magda, Macromolecules 22,3004 (1989).

4. M. J. Menasveta and D. A. Hoagland, ibid. 24, 3427(1991).

5. K. A. Smith, E. W. Merrill, L. H. Peebles, S. H. Bani-jamali, Colloq. Int. CNRS 233, 341 (1975); J. L. Lum-ley, Phys. Fluids 20, s64 (1977).

6. A. Keller and J. A. Odell, Colloid Polym. Sci. 263, 181(1985).

7. G. G. Fuller and L. G. Leal, Rheol. Acta 19, 580(1980); C. A. Cathey and G. G. Fuller, J. Non-New-tonian Fluid Mech. 34, 63 (1990); P. N. Dunlap and L.G. Leal, ibid. 23, 5 (1987).

8. E. D. T. Atkins and M. A. Taylor, Biopolymers 32, 911(1992). To plot the data in Fig. 3B, we normalizedbirefringence by the maximum birefringence and mul-tiplied their « by t1 5 0.067 s, where we determined t1for l-DNA by scaling t1 5 0.046 s reported by L. C.Klotz and B. H. Zimm [J. Mol. Biol. 72, 779 (1972)] forT7-DNA in a creep recovery experiment and scaledfor the slight difference in length between T7 andl-DNA with a scaling exponent of 1.66 (35 ). This valueof t1 5 0.067 s is in agreement, after scaling forlength, with t1 5 0.058 to 0.068 s from intrinsic vis-cosity [ L. C. Klotz and B. H. Zimm, J. Mol. Biol. 72,779 (1972)], light scattering [K. S. Schmitz and R.Percora, Biopolymers 14, 521 (1975)], birefringence[D. S. Thompson and S. J. Gill, J. Chem. Phys. 47,5008 (1967)], and flow dichroisim experiments [P. R.Callis and N. Davidson, Biopolymers 8, 379 (1969)].For a comparison of our trelax to t1, see (26).

9. T. Q. Nguyen, G. Yu, H.-H. Kausch, Macromolecules28, 4851 (1995).

10. Recent experiments by Nguyen et al (9) exhibited thesame saturation, but when the probe region ismoved toward the outlet, the birefringence in-creased, which suggests that polymers had notreached equilibrium. However, S. P. Carrington andJ. A. Odell [ J. Non-Newtonian Fluid Mech. 67, 269(1996)] argue that this is a concentration-dependenteffect.

11. J. A. Odell, A. Keller, Y. Rabin, J. Chem. Phys. 88, 4022(1988); J. A. Odell, A. Keller, A. J. Muller, Colloid Polym.Sci. 270, 307 (1992); J. A. Odell and M. A. Taylor,Biopolymers 34, 1483 (1994). See also H. R. Reese andB. H. Zimm [J. Chem. Phys. 92, 2650 (1990)], whichfocuses on DNA fracture through an orifice.

12. D. F. James and J. H. Saringer, J. Fluid Mech. 97,655 (1980).

13. V. Tirtaatmadja and T. Sridhar, J. Rheol. 37, 1081(1993).

14. D. F. James and T. Sridhar, ibid. 39, 713 (1995).15. S. H. Spiegelberg and G. H. McKinely, J. Non-New-

tonian Fluid Mech. 67, 49 (1996); N. V. Orr and T.Sridhar, ibid., p. 77.

16. E. J. Hinch, ibid. 54, 209 (1994).17. T. T. Perkins, D. E. Smith, R. G. Larson, S. Chu,

Science 268, 83 (1995).18. Some molecules (,5%) are fractured, possibly during

the mixing of the DNA into the high-viscosity buffer,but a visual inspection of the light intensity and sizecan generally rule out these smaller molecules.

19. S. B. Smith, L. Finzi, C. Bustamante, Science 258,1122 (1992); A. Vologodskii, Macromolecules 27,5623 (1994).

20. J. F. Marko and E. D. Siggia, Macromolecules 28,8759 (1995); C. Bustamante, J. F. Marko, E. D. Sig-

Fig. 4. A comparison between the rates ofdeformation of the fluid element («fluid § «)and the polymer as characterized by themaster curve. We subtract «c, the experi-mentally determined critical strain rate,from the «fluid to account for the necessarydifference between and «fluid and «mol. Thisdifference leads to the hydrodynamic forcethat stretches the polymer against its na-tive elasticity. The master curve is usedinstead of ^x(tres )&, as it better representsthe unwinding dynamics by suppressingthe variation in tonset. To define a moleculestrain rate from the master curve, «mol wascalculated from fitting ^x(x)& 5 «mol x 1 bover the region where ^x(x)& is a linear func-tion of x. As shown in the inset, this approx-imately linear behavior is seen for x ,11.8 mm at « 5 0.86 s21. For the lower «,^x(x)& was linear in x up to 5.2, 7.8, 8.8, and10.2 mm in increasing order of «. If affinedeformation becomes an increasingly valid approximation at higher «, then the data would line along theline described by «mol 5 «fluid 2 «c (dotted line). «mol determined for those molecules that best typify adumbbell configuration is plotted as an open symbol (see Fig. 2C, inset). (Inset) ^x(x)& versus x for allmolecules at « 5 0.86 s21 (filled circles) and molecules in the dumbbell configuration (open circles). Thesolid line represent the dynamics predicted by the dumbbell model using the values (Rbead, L) deter-mined by the fit of xsteady to « shown in Fig. 3B. More explicitly, we sum the hydrodynamic and entropicforces and set them to zero (Ftotal 5 Fhydro 2 Fspring 5 0), where Fhydro 5 6phRbead (nfluid 2 xpredicted (x)/2),nfluid 5 0.5«x, and Fspring 5 f(x/L) as given in (20). Because xpredicted is calculated at a given x, ratherthan a tres, Ftotal is a function on only x and « given Rbead and L from the fit to the steady-state data.We can then calculate the predicted rate of stretching of the total chain at each x to be xpredicted 52(6pnRbeadnfluid 2 Fspring) /6phRbead (dashed line). Thus, this simple model overestimates the aver-age dynamics although it agrees closely with the dynamics of molecules in the dumbbell configura-tion. The introduction of a term (1 1 a) multiplying xpredicted in Fhydro, where a 5 0.55 approximatelycompensates for this overestimation of the dynamics at « 5 0.86 s21. Note that the variation in Fig.4 away from «mol 5 «fluid 2 «c shows that this coefficient a is dependent on «.

SCIENCE z VOL. 276 z 27 JUNE 1997 z www.sciencemag.org2020

gia, S. Smith, Science 265, 1599 (1994). Their ana-lytic approximation for the elasticity is given by Fp/kbT 5 0.25*(12 x/L)22 2 0.25 1 x/L, where F is theforce applied across the ends, p is the persistencelength, and kBT is the thermal energy.

21. D. E. Smith, T. T. Perkins, S. Chu, Macromolecules29, 1372 (1996).

22. R. G. Larson, T. T. Perkins, D. E. Smith, S. Chu,Phys. Rev. E 55, 1794 (1997).

23. Following the example of W. D. Volkmuth and R. H.Austin [Nature 358, 600 (1992)], we made our flowcells by etching silicon wafers with KOH and anion-ically bonding Pyrex coverslips to the silicon to sealthe top surface of the channels. Vertical side wallsalong the inlet and outlet channels were achievedby rotating the pattern by 45° to the crystal axis [C.Hu and S. Kim, Appl. Phys. Lett. 29, 582 (1976)].To generate the flow, we used a syringe pump (Iscomodel 100 D) which was temperature-stabilized at22.7°C. A ;100 :1 fluid shunt bypassing the flowcell was used to operate the pump at flow rates.40 ml/min. The velocity field was calibrated bytracking fluorescent beads near the stagnationpoint and showed ny 5 « y, confirming that a uni-form elongational flow had been achieved. Our im-aging and digitization system was the same aspreviously described (17 ), except we used a waterimmersion objective ( Zeiss 403 C-APO NA 1.2)and stroboscopically illuminated the DNA to elimi-nate the blurring of the image due to camera lag.Our measurements of the equilibrium coil size gen-erally did not yield 2 RG but something slightly larg-er due to blooming in the camera. Once the chain isextended about two times the equilibrium size, theblooming is reduced and the measurements corre-spond closely to the actual extension. We stainedthe l-DNA (New England Biolabs) with YOYO-1(Molecular Probes) at a dye:base-pair ratio of 1:4for .1 hour. The experiment was performed in ahigh viscosity (h 5 41 centipoise) buffer consistingof 10 mM tris-HCl, 2 mM EDTA, 10 mM NaCl, 4%b-mercaptoethanol, ;50 mg/ml glucose oxidase(Boehringer Mannheim), ;10 mg/ml catalase(Boehringer Mannheim), ;18% (w/w) glucose and;40% (w/w) sucrose. The viscosity of each solu-tion was measured in a temperature-stabilized vis-cometer and adjusted as needed. The flow cell wasmounted on a copper block and stabilized to 22.7°6 0.2°C.

24. To prevent any predeformation of the polymer beforeentering the elongational flow, we used a cross-slotflow cell with channels 650 mm wide and 220 mmdeep and imaged the polymers at the center of depthof the channel (zcenter 5 110 mm) where the appliedshear (g § ]v,x/]z) was negligible. Because of themismatch in the index of refraction of the immersionfluid (water) and the high viscosity buffer, we deter-mined zcenter by measuring the velocity of fluores-cent beads in the inlet as a function of depth andfitting this velocity to a parabolic, Poiseuille velocityfield. Because of their finite size (RG > 0.7 mm),polymers at zcenter still experienced some shear.For our highest strain rate (« 5 0.86 s21), the cor-responding reduced shear rate in the inlet wasgtrelax 5 0.5, which is below the expected onset ofdeformations (2). We observed no deformation ofthe polymers in the inlet at « 5 0.86 s21, but adoubling of the flow rate led to modest (;4 mm)predeformation of some molecules. We used 25-mm-long inlet channels, which allowed any mole-cule deformed upon entering the flow cell sufficienttime (.8 trelax) to relax back to equilibrium beforeentering the elongational flow.

25. At the lower strain rates, there were not distinct, wellpreserved conformations, but the large variation tonsetstill occurred. The polymers had the general appear-ance of a dumbbell configuration subject to a largefluctuations in extension and shape.

26. We determined trelax 5 3.89 s by averaging the re-laxation of 14 individual, highly extended molecules.The relaxation time we report is from a fit over theregion where x/L , 0.3 to ^x(t)x(t)& 5 c exp (2t/trelax)2 2 RG, where trelax, c, and RG were free parame-ters. This is directly related to the relaxation of thestress in the fluid via sE 5 n ^xzF(x)&, because in the

limit of small deformation, F(x) is linear in x and thisyields sE(t) ; ^x(t)x(t)&. Thus, our characterization oftrelax via stress relaxation is made in the region (x/L ,0.3) where trelax > t1 because the contribution ofhigher order relaxation modes should be negligible(2). When scaled to h 5 1 centipoise and for thechange in length caused by staining, our trelax forunstained, l-DNA in water is 0.061 s (8). This anal-ysis does not take into account any possible chang-es in persistence length. No change in solvent qualitywas measured between sucrose and glycerol vis-cosity-enhanced aqueous buffered solution used forthe measurements of trelax and t1, respectively (35).Our trelax reported here is different than relaxationtime determined from ^x(t)& 5 c exp(t/t) 2 RG, whichyields t 5 6.2 s, and is the relaxation time we report-ed in an earlier experiment where the DNA was teth-ered to a bead (35).

27. The longer DNA molecules were concatemers ofl-DNA (up to 250 mm). We were unable to system-atically investigate the dynamics of longer molecules,because we did not have an adequately monodis-perse sample and could not independently measurethe length of individual molecules.

28. The results presented here should not be general-ized to polymers in a mixed elongational and shear-ing flow or to polymers in an elongational flow thatwere presheared. Our data indicates that the pro-cesses involved in the diverse dynamics arise fromthe variation in tonset and from internal configurations(that is, folds). In mixed flows, a large fraction of themolecules are partially extended because of shear-ing, and this effect may eliminate some of the internalconstraints that led to the observed dynamics.

29. S. Daoudi and F. Brochard, Macromolecules 11,

751 (1978); P. G. de Gennes, Scaling Concepts inPolymer Physics (Cornell Univ. Press, Ithaca, NY,1979); E. J. Hinch, Phys. Fluids 20, 522 (1977).

30. J. M. Rallison, J. Non-Newtonian Fluid Mech. 68, 61(1997). For simulations of longer chains, see P. S.Doyle, E. S. G. Shaqfeh, and A. P. Gast [J. FluidMech. 334, 251 (1997)].

31. G. Ryskin, Phys. Rev. Lett. 59, 2059 (1987); J. FluidMech. 178, 423 (1987); R. G. Larson, Rheol. Acta29, 371 (1990); D. H. King and D. F. James, J. Chem.Phys. 78, 4749 (1983).

32. A. Peterlin, Makromol. Chem. 44, 338 (1961).33. R. Keunings, J. Non-Newtonian Fluid Mech. 68, 85

(1997).34. B. H. Zimm, J. Chem. Phys. 24, 269 (1956); P. E.

Rouse, ibid. 21, 1272 (1953).35. T. T. Perkins, S. R. Quake, D. E. Smith, S. Chu,

Science 264, 822 (1994).36. We thank G. Fuller, D. Hoagland, R. Larson, R.

Pecora, and B. Zimm for helpful discussions and J.Shott for generous aid in lithographic design. Sup-ported in part by grants from the U.S. Air Force Officeof Scientific Research, NSF, and the Human FrontiersFoundation and by an endowment established byTheodore and Frances Geballe. We acknowledge thegenerous assistance of J. Spudich, including supportthrough NIH grant GM33289 to J. Spudich. D.E.S.was supported by a fellowship from the Program inMathematics and Molecular Biology at the Universityof California, Berkeley, through NSF grant DMS9406348. S.C. was supported in part by a Guggen-heim fellowship during the course of this work.

18 February 1997; accepted 8 May 1997

Surface Stress in the Self-Assembly ofAlkanethiols on Gold

Rudiger Berger,* Emmanuel Delamarche, Hans Peter Lang,Christoph Gerber, James K. Gimzewski, Ernst Meyer,

Hans-Joachim Guntherodt

Surface stress changes and kinetics were measured in situ during the self-assembly ofalkanethiols on gold by means of a micromechanical sensor. Self-assembly causedcompressive surface stress that closely followed Langmuir-type adsorption kinetics upto monolayer coverage. The surface stress at monolayer coverage increased linearly withthe length of the alkyl chain of the molecule. These observations were interpreted in termsof differences in surface potential. This highly sensitive sensor technique has a broadrange of applicability to specific chemical and biological interactions.

Molecular and biomolecular layers are scien-tifically appealing for a wide range of poten-tial applications (1, 2). Alkanethiols, whichare known to self-organize into well-ordered,densely packed films, represent a model mo-lecular system for controlling surface proper-ties (3, 4). These self-assembled monolayers(SAMs) are used in applications such as mi-

crocontact printing (5) and voltametric mi-crosensors (6), and they have recently beenapplied to molecular host-guest recognition(7).

There is little information available on themechanical properties of SAMs, particularlyconcerning the nature of surface stress in filmsduring the formation process, because it isdifficult to follow the structural evolution ofmonolayer self-assembly. One recent ap-proach (8) used scanning tunneling microsco-py to infer the growth kinetics of alkanethiolSAMs indirectly from snapshot images ob-tained at various coverages. Here, we usedmicromechanical sensors to gather quantita-tive data on surface stress changes that devel-op during the self-assembly process of HS-

R. Berger and H. P. Lang, IBM Research Division, ZurichResearch Laboratory, CH-8803 Ruschlikon, Switzerland,and Institute of Physics, University of Basel, CH-4056Basel, Switzerland.E. Delamarche, Ch. Gerber, J. K. Gimzewski, IBM Re-search Division, Zurich Research Laboratory, CH-8803Ruschlikon, Switzerland.E. Meyer and H.-J. Guntherodt, Institute of Physics, Uni-versity of Basel, CH-4056 Basel, Switzerland.

*To whom correspondence should be addressed.

REPORTS

www.sciencemag.org z SCIENCE z VOL. 276 z 27 JUNE 1997 2021