Rheological Observation of Glassy Dynamics of Dilute Polymer Solutionsnear the Coil-Stretch Transition in Elongational Flows

T. Sridhar,1 D. A. Nguyen,1 R. Prabhakar,2 and J. Ravi Prakash1,*1Department of Chemical Engineering, Monash University, Melbourne, VIC-3800, Australia

2Research School of Chemistry, Australian National University, Canberra, ACT-0200, Australia(Received 23 November 2006; published 20 April 2007)

It has long been conjectured that the macroscopic dynamics of dilute polymer solutions may exhibit aglasslike slowdown caused by ergodicity breaking, in the vicinity of the coil-stretch transition inelongational flows. We report experimental observations using a filament stretching rheometer thatconfirm the existence of such glassy states. It is observed that different time-dependent elongationalstrain-rate profiles lead to a pronounced history dependence and aging effects within a narrow range ofstrain rates. The results have a direct bearing on the analysis and design of processes employing dilutepolymer solutions, such as ink-jet printing, surface coating, and turbulent-drag reduction.

DOI: 10.1103/PhysRevLett.98.167801 PACS numbers: 61.25.Hq, 83.10.Gr, 83.10.Mj, 83.50.Jf

In materials near a glass transition, ergodicity breaksdown as a result of the emergence of deep local minima inthe phase-space energy landscapes. In addition, the tem-perature is not high enough for systems trapped in theenergy basins to scale their walls and sample all of theaccessible phase-space ergodically within the time scale ofthe observer. In a landmark paper in 1974, de Gennes [1]suggested that in a certain range of elongational strainrates, the coupling of the flow field with the hydrodynamiccharacteristics of long, flexible polymer molecules in adilute solution could create and sustain kinetically frozenstates. That is, a polymer solution could behave as arheoglass. As was recognized at about the same time byHinch and Tanner [2,3], this phenomenon has profoundimplications, since dilute solutions of long flexible poly-mer molecules find wide application in situations whereextensional flows dominate.

In the years following the 1974 paper, there was consid-erable doubt and debate over de Gennes’ contention thatone could observe hysteretic effects in bulk properties ofdilute solutions of polymers. Fan et al. [4], for example,argued that the hysteretic effects predicted by de Gennesmust be artifacts of the approximations employed therein.These issues remained largely unresolved until Shaqfehand co-workers demonstrated recently in a series of papers[5–8] using single-molecule experiments and computersimulations that it is indeed possible to observe the kindphenomena that de Gennes had conjectured. In addition,theoretical work by these authors and others [9–12] hasalso helped place the original arguments of de Gennes,Hinch, and Tanner on a more rigorous footing. There has,however, been no systematic experimental exploration ofglassy dynamics of bulk solution properties in elongationalflows of dilute polymer solutions. We report in this Letterthe observation of an appreciable glassy slowdown andhistory dependence of bulk stresses in a dilute solution ofpolystyrene within a small window of elongational strainrates.

Arguments for the existence of hysteresis [1,5–7] can besummarized as follows. In a uniaxial elongational flow, theexact nonequilibrium steady-state probability distributionfunction for a polymer molecule’s end-to-end distance Qhas the Boltzmann form, �Q� � exp��E�Q; _"�=�kBT��,where the function E�Q; _"� is interpreted as an effectivenonequilibrium energy function parameterized by thestrain rate [5,6], kB is the Boltzmann constant, and T isthe absolute temperature. The minima (maxima) in theeffective energy E (probability ) occur at the values ofQ at which the total friction force on the polymer exertedby the flowing solvent is balanced by the entropic resist-ance of the molecule to stretching. For _" < _"min, E has asingle minimum at Q � 0, whereas for _" > _"max, thisminimum is shifted to a value of Q close to its maximumvalue L. However, for _"min < _" < _"max, the convex non-linearities in the drag coefficient and entropic resistanceconspire to lead to two minima in E: one at Q � 0, andanother closer to L. For molecules that are large enough,the intervening maximum between the two minima can bemuch larger than kBT, the mean energy of thermal fluctua-tions in the solvent. Therefore, if an ensemble of initiallyfully stretched chains is subjected to a strain rate _"min <_" < _"max, the molecules are kinetically ‘‘frozen’’ in the

stretched-state energy minimum. Similarly, in experimentswhich start with a solution initially at equilibrium, mole-cules are trapped in the coiled-state energy minimum. Inthis case, as de Gennes showed using a simple estimate,even for polymer molecules of moderately large molecularweights, the time scale for ergodicity to be establishedcould be far greater than typical experimental observationtimes. More recently, Beck and Shaqfeh [8] have shownthat the transition rate between the energy minima de-creases exponentially with molecular weight, rigorouslyproving that ergodicity does break down in the limit ofinfinite system size. As a result, there is a significantslowing of the dynamics of macroscopic observableswithin the ‘‘hysteresis window’’, that is, for strain rates

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that lie between the two critical values _"min and _"max. Forsolutions of long polymer molecules, therefore, observ-ables such as the stress measured in experiments may settleto long-lived quasisteady states, which strongly depend onthe initial conditions, and the experimental protocolsfollowed.

The filament stretching rheometer (FSR) currently pro-vides the most reliable measurement of the elongationalstress for polymer solutions [13,14]. We have used solu-tions of monodisperse polystyrene in an oligomeric styrenesolvent [15]. Figure 1(a) is a composite plot of data forquasisteady dimensionless stress �� � �=�npkBT� againstthe Weissenberg number Wi � _"�coil, for a solution ofpolystyrene of molecular weight 1:12 106. Here, npdenotes the number density of polymer chains in the solu-tion and �coil is the largest relaxation time in the character-istic relaxation spectrum of an isolated polymer moleculein solution at equilibrium. The data in Fig. 1(a) wereobtained using three different experimental protocols,each of which used a different time course for the imposedstrain rate.

The white circles in Fig. 1(a) represent data obtainedwith the FSR in its ‘‘rate-controlled’’ mode of operation[13,16]. In this mode, a small sample of polymer solution isplaced between two plates, which are vertically movedapart such that the midfilament diameter of liquid filamentformed decreases exponentially in time. Thus, the exten-sional strain rate _" in the midfilament region is constant.The normal-stress difference � between the axial andradial components of the stress tensor at the midfilamentplane is extracted from force measurements at the top endplate. It is usually observed that� attains a plateau which isinterpreted here as a quasisteady state.

In this usual mode of operation of the FSR, the solutionis at equilibrium initially, and polymer molecules aremostly coiled. In Fig. 1(a), the white circles appear toform a lower bound for the quasisteady stress measure-ments. This is consistent with the de Gennes’ picture, asinitially coiled molecules are likely to remain in the coiled-state energy basin and thus manifest lower values for thestress. With increasing values of _", the barrier height forthe coiled-to-stretched transition decreases, permittingmore molecules to transit to the stretched state within theexperimental observation time, leading to increasing stressvalues.

Though it has become routine since the development ofthe FSR to characterize the steady-state extensional stressin a polymer solution in the constant strain-rate mode, tostudy the glassy dynamics associated with the coil-stretchtransition, we have extended the operation of the FSR tothe following two additional modes: (i) a constant stressmode, where a time-dependent _" is imposed to achieve andmaintain � at a desired value, and (ii) a strain-rate quenchmode, where a downward step change is made in _" from aninitially high value.

Although constant stress experiments are regularly usedin the characterization of complex fluids in shear flows, thismeasurement protocol has not so far been implemented inelongational flows of dilute polymer solutions. The strain-rate imposed on the polymer solution is quickly decreasedfrom a high value until � attains a predetermined set value.The motion of the end plates is then adjusted to maintainthe total fluid stress constant. We find that the strain-raterequired to maintain the set value of stress also stabilizes toa nearly constant value (for about 1 Hencky strain unit).This is interpreted as a quasisteady state. The actual im-posed strain-rate time course required to achieve the con-stant stress is determined by trial and error. Initially, theviscous stress due to the solvent dominates. As the polymermolecules unravel and stretch, their contribution to thestress grows, and in order to keep the stress constant, thestrain rate needs to be decreased. This not only decreasesthe solvent stress, but also decreases the frictional dragforce stretching the polymers. As the transitions of mole-

FIG. 1. (a) Quasi-steady-state stress measurements demon-strate glassy dynamics associated with the coil-stretch transition.The gray lines in (a) outline the coil-stretch hysteresis window.The inset in (a) shows data at high strain rates. The data shownin (a) have been obtained with three different protocols: constantimposed strain-rate (white circles), imposed total stress (blacksquares), and strain-rate quench (gray triangles). The numbersalongside the gray triangles report the Hencky strains at quench.(b) Transient dimensionless stress in strain-rate quench experi-ments. (c) Slowing down of postquench dynamics with decreasein quenched strain rate.

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cules between coiled and stretched states slow down due tothe energy barrier within the coil-stretch hysteresis win-dow, the rate at which the imposed strain rate has to bemodified in order to keep the stress constant also decreases.

The black squares in Fig. 1(a) represent data at thestress- or strain-rate plateaus obtained with this protocol.At high strain rates [Fig. 1(a), inset], the data from theconstant stress experiments are close to those obtained inthe constant strain-rate experiments. In sharp contrast,however, at the lower strain rates, we see that the datafrom the two protocols do not collapse onto a single curve.This suggests that, at these strain rates, true steady statesare not attained within the experimental observation time.Furthermore, the plateaus in stress and strain-rate signal aslowdown in microstructural dynamics.

Perhaps the most compelling evidence for glassy dy-namics in our experiments comes from data obtained withthe strain-rate quench protocol. In all the quench experi-ments, the polymer solution is initially subjected to a highstrain rate ( _" � 3 s�1; Wi� 10). Then, at some intermedi-ate value of the strain (1 & " & 5), the strain rate is sud-denly quenched to a lower value (0:5 & Wi & 1:5) and,subsequently, held constant. Following the quench in strainrate, it is observed that the stress drops rapidly initially, butthen levels off towards a plateau value [e.g., triangle sym-bols in Fig. 1(b)]. The triangle symbols in Fig. 1(a) denotethe post-quench plateau data obtained in these experi-ments. The numbers alongside the triangle symbols inFig. 1 indicate the Hencky strains at quench.

We observe that quenching at different strains lead toclearly different plateau levels of � for the same value ofthe final strain rate, with higher values of the stress forlarger quench strains. On the other hand, when the finalstrain rate of quench is high [the two triangles near the topright-hand corner of Fig. 1(a)], the plateau level of thestress is much less sensitive to quench strain, and matchesthe stress data obtained in the constant stress and strain-rateexperiments. Interestingly, in some cases, the plateau in thestress is brief, and is followed by a slow increase in thestress [Fig. 1(b)]. This behavior is akin to aging phe-nomena observed in conventional glass-forming materials.Furthermore, for the same quench strain, we also find thatquenching to a lower strain rate typically leads to a slower‘‘aging’’ of the stress after the plateau [Fig. 1(c)].

We have also performed Brownian dynamics (BD)simulations of the strain-rate quench protocol to assist usin the interpretation of experimental observations. Thebead-spring chain model with finitely extensible springsand solvent-mediated hydrodynamic interactions has beenshown in the past few years to be excellent in predicting thebehavior of dilute polymer solutions under �-conditions[17–19]. Details of the numerical algorithm [20] and pa-rameters used [15] are given elsewhere.

Figure 2(a) shows results of the simulations for theevolution of the stress � with Hencky strain for different

values of the quench strain, for a final strain rate corre-sponding to Wi � 0:5 [21]. Also shown in that figure arethe predictions obtained in constant strain-rate simulations,starting with an ensemble of chains initially at equilibrium,and with chains stretched to 90% their full length andaligned in the axial flow direction. In the quench simula-tions, the stress variation is qualitatively similar to thatobserved in the experiments, and depending on the strain atquench, different levels of postquench stress plateaus areobtained. The quasisteady nature of the stress plateausobtained with the simulations are quite clear.

In our simulations, we have also examined the frequencydistributions of the stretch x—defined as the projectedextent of the molecule in the flow direction—in the simu-lation ensemble. The peaks in the quasisteady stretch dis-tributions for simulations starting with equilibrium coils[black squares in Fig. 2(b)], and stretched chains [whitesquares in Fig. 2(b)], give a good indication of the locationof the coiled and stretched basins of attraction, respec-tively. The stretch distributions in Figs. 2(c) and 2(d)demonstrate that when the strain rate is suddenly quenchedinto the hysteretic regime, the population of moleculesimmediately after the quench is first rapidly partitionedinto the two energy basins corresponding to the coiled andstretched states, followed by slower changes depending onthe strain rate after quenching. The quasisteady stress levelattained subsequent to the quench hence depends on thepartitioning of the chain population between the two en-ergy basins at high strains. This partitioning is effectivelydetermined by the configurational distribution at the instant

FIG. 2. Brownian dynamics simulations prediction of multiplequasisteady stress values in the strain-rate quench protocol.In (a), the dashed curves are the results at constant strain rate(Wi � 0:5) starting with chains initially coiled at equilibrium,and 90% fully stretched, as indicated. The continuous black linesin (a) are results for strain-rate quench simulations. Plots (b)–(d) show frequency distributions of the fraction in the simulationensemble at the strains indicated by symbols in plot (a). Thevertical dashed line running down plots (b)–(d) indicates thelocation of the stretched basin of attraction.

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prior to quenching. The initial application of a high strainrate causes the chains to rapidly unravel and stretch, and ifone waits longer before applying the strain-rate quench,more molecules are stretched out. As in the experiments,we observe in our simulations a slow postquench aging ofthe stress towards its true steady-state value [Fig. 2(a)].

The size of the coil-stretch hysteresis window and theprominence of glassy effects in this window, dependson the molecular weight of the dissolved polymer. Thevalue of the critical strain rate _"max is such that Wimax �_"max�coil � 0:5. The ratio _"max= _"min, on the other hand, is

expected to scale for very large molecules as M1=2= logM[7,8]. For smaller molecules, however, there are correc-tions to this scaling, and there is no coil-stretch hystere-sis for molecules below a certain critical size. It has beenrecently found through BD simulations that the mini-mum molecular weight of polystyrene required to observethe effects of ergodicity breaking is about 5 105 [9].Below this chain length, the barriers to transitions betweenthe basins of attraction are comparable to or less than kBTat all strain rates. In line with these simulation results, wefind in our experiments that for solutions of 4:26 105

molecular weight polystyrene, the steady-state data ob-tained with both constant stress, and strain-rate, protocolssuperimpose across the entire range of strain rates exam-ined [15], and no glassy dynamics are observed for thismolecular weight. Although, as recent simulations haveshown [7,9], the slowdown of dynamics should be moreprominent in solutions of larger molecular weights, theminimum operating Wi, for dilute polymer solutions inthe FSR grows linearly with the molecular weight [14].Below this limit, gravitational sagging of the liquid fila-ment makes it difficult to obtain reliable fluid stress mea-surements. Since the upper boundary of the coil-stretchhysteresis window is at Wimax � 0:5 for a dilute polymersolution, we are limited to molecular weights less thanabout 1:5 106 for polystyrene solutions, which is notconsiderably larger than that used for the results inFig. 1. Hence, with our present FSR setup, glassy dynamicscan only be observed within the narrow molecular weightrange of 0.5 to 1:5 106 for dilute polystyrene solutions.Such effects are also observable in other devices whereextensional flows are dominant, such as the capillarybreakup extensional rheometer (CABER), in which glassydynamics associated with the coil-stretch transition hasbeen recently implicated as one of the causes of anomalousbehavior [22].

This work has been supported by a grant from theAustralian Research Council. We thank the Australian

Partnership for Advanced Computing (APAC) for super-computing time.

*Corresponding author.Electronic address: ravi.jagadeeshan@eng.monash.edu

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[22] R. Prabhakar, J. R. Prakash, and T. Sridhar, J. Rheol.(N.Y.) 50, 925 (2006).

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