pubs.acs.org/Macromolecules Published on Web 03/17/2010 r 2010 American Chemical Society

3556 Macromolecules 2010, 43, 3556–3565

DOI: 10.1021/ma100093b

Mechanism of Shear Thickening in Reversibly Cross-LinkedSupramolecular Polymer Networks

Donghua Xu, Jennifer L. Hawk, David M. Loveless, Sung Lan Jeon, and Stephen L. Craig*

Department of Chemistry and Center for Biologically Inspired Materials and Material Systems,Duke University, Durham, North Carolina 27708-0346

Received January 14, 2010; Revised Manuscript Received March 5, 2010

ABSTRACT: We report here the nonlinear rheological properties of metallo-supramolecular networksformed by the reversible cross-linking of semidilute unentangled solutions of poly(4-vinylpyridine) (PVP) indimethyl sulfoxide (DMSO). The reversible cross-linkers are bis-Pd(II) or bis-Pt(II) complexes thatcoordinate to the pyridine functional groups on the PVP. Under steady shear, shear thickening is observedabove a critical shear rate, and critical shear rate is experimentally correlated with the lifetime of themetal-ligand bond. The onset and magnitude of the shear thickening depend on the amount of cross-linkersadded. In contrast to the behavior observed inmost transient networks, the time scale of network relaxation isfound to increase during shear thickening. The primary mechanism of shear thickening is ascribed to theshear-induced transformation of intrachain cross-linking to interchain cross-linking, rather than nonlinearhigh tension along polymer chains that are stretched beyond the Gaussian range.

Introduction

The sophistication with which intermolecular interactions cannow be rationally engineered has had a major impact on a myriadof fields, including molecular recognition, self-assembly, nano-technology, and polymer science.1,2 It has long been appreciatedthat intermolecular forces between polymer chains contributeto their macroscopic mechanical properties and rheologicalbehavior.3,4 Recently, the incorporation of highly specific, stoichio-metric, and well-characterized associating units as defining interac-tions in polymers (often referred to as “supramolecular polymers”)has attracted much interest.5 The potential advantages of supramo-lecular polymers relative to their covalent counterparts includeopportunities for new, noncovalent approaches to polymer syn-thesis,6,7 the rational control of polymer properties, including stimuliresponsiveness,8,9 and new strategies by which to construct self-repairing materials.10

In many cases, the fundamental difference between supramo-lecular and covalent polymers is the reversibility (kinetic lability)of the supramolecular bond. Yount et al.11,12 recently demon-strated a conceptually simple, but practically useful, methodfor probing the contributions of molecular reversibility to therheological properties of polymers. The methodology takesadvantage of steric effects at theN-alkyl positions of vanKoten’sN,C,N-pincer Pd(II) andPt(II) complexes13 to change the rates ofligand exchange independently of the association constant. Thismethodology has been applied in our lab14 to explore themolecular mechanisms underlying the basic rheological proper-ties of linear supramolecular polymer solutions,11 supramole-cular polymers networks,12,15 and tribological properties of cross-linked polymer brushes.16 While the viscosities of solutions oflinear supramolecular polymers formed from thesemetal-ligandinteractions depend on the equilibrium constant (Keq) of themetal-ligand interaction,11 the low-strain, frequency-dependentdynamicmoduli ofmacroscopic polymer networks that are cross-linked via these same interactions (e.g., Figure 1) are related

quantitatively to the pyridine exchange rates (kd) measured onmodel Pd(II) and Pt(II) complexes.12,15 These studies are just apart of increasing amount of research regarding the rheologicalproperties of supramolecular polymers17-24 and their relation-ship to the nature of the defining supramolecular interactions.

In contrast to their linear rheological properties, the mecha-nisms underlying the nonlinear rheological properties of supra-molecular polymer networks (e.g., strain hardening and shearthickening) are often unclear and have received less attention.(A more involved discussion of shear thickening mechanismsin associative polymer networks is provided in the Discussionsection.) In this paper, we extend the methodology of Yountet al.11,12 to explore the shear thickening properties of metallo-supramolecular networks formed from the addition of bis-Pd(II)or bis-Pt(II) cross-linkers to semidilute unentangled solutions ofPVP in DMSO.

Experimental Section

Materials. Dimethyl sulfoxide (DMSO) and poly(4-vinyl-pyridine) (PVP),Mw=60000,were used as received fromAldrich.Four cross-linkers, [2,3,5,6-tetrakis{(dimethylamino)methy}-phenylene-1,4-bis(palladium trifluoromethanesulfonate)] (1),[2,3,5,6-tetrakis{(diethylamino)methy}phenylene-1,4-bis(palla-diumtrifluoromethanesulfonate)] (2), [2,3,5,6-tetrakis{(dimethyl-amino)methy}phenylene-1,4-bis(platinumtrifluoromethane-sulfonate)] (3), and [2,3,5,6-tetrakis{(diethylamino)methy}-phenylene-1,4-bis(platinumtrifluoromethanesulfonate)] (4),were synthesized as reported elsewhere.25 The reversiblecoordinative cross-linkers formed between PVP and bis-functional metal compounds are shown in Figure 2.

Sample Preparation. Samples were prepared as follows. PVPwas added toDMSO, stirred for 4 h, and then transferred to vials.Separate solutions of cross-linkers were formed by dissolvingthem in DMSO. These solutions were then added to the vialscontaining the PVP/DMSO solution. Samples with differentmolar ratios of cross-linkers to pyridine units in PVP wereprepared, and the molar ratios reported here are those betweenpalladium or platinum atoms (two per cross-linker) and pyridine*Corresponding author. E-mail: stephen.craig@duke.edu.

Article Macromolecules, Vol. 43, No. 7, 2010 3557

nitrogens in PVP.20 For 60 kDa, there are about 570 pyridineunits in each PVP chain (for purposes of this analysis, we assumeMn is equal to theMw provided by the supplier; the polydispersityof the PVP is not known). At different molar ratios betweenpalladium (or platinum atoms) and pyridine nitrogens (0.5%,1%, 2%, 3%, 5%, and 7% in this paper), there are about 3, 6, 11,17, 28, and 40 palladium (or platinum) atoms per PVP chain andhalf that number of bivalent cross-linkers. Additional DMSOwas added to the above vials to adjust the mass concentration ofPVP. The prepared samples were then stirred for 12 h at ∼90 �Cand subsequently allowed to cool to room temperature. Twodifferent series of samples were prepared, in which the massconcentration of PVP was held roughly constant at ∼0.08and ∼0.1 g/mL. These concentrations of PVP fall within thesemidilute unentangled regime (Figure S1 in the SupportingInformation). DMSO is close to a θ solvent for PVP at 25 �C(Supporting Information).

The micromorphology of the samples was checked for phaseseparation using a phase contrast microscope (Zeiss Axio Ob-server, Germany). Over sufficiently long times (∼2 h), phaseseparation was observed in some samples. Phase separation, asobserved by microscopy, was faster for samples that were notcovered with a cover glass. In addition, we observe no phaseseparation over a period ofmore than 6months (and counting atthe time this article was submitted) for samples left in sealedvials. We therefore infer that the phase separation is due toexposure to moist air and that it does not reflect an inherentinstability in the samples. All rheological data reported here

were obtained under conditions (experimental time scales of 1 hor less) in which no evidence for phase separation was observed.

Rheological Measurements. All rheological data were obtainedby using an AR G2 rheometer (TA Instruments) with cone-plategeometry (diameterof 20mm, coneangleof 2�, truncationheight of49 μm).A solvent trapwas usedwith this geometry tominimize theevaporation of solvent from the samples.

The samples were loaded into the plate at 25 �C and heated to80 �C for 2min to erase the thermal andmechanical history. Thesamples were then compressed to a height of 55 μm, excessmaterial was scraped from the edge of the geometry, and thesamples were cooled to 25 �C. Next, the samples were com-pressed to 49μm(the truncation height of the geometry), and theexperiment was started. Experiments were carried out at 25 �C(unless otherwise noted). The stiffness of samples with highconcentrations of bis-Pt(II) cross-linkers precluded proper load-ing, and as a result rheological experiments for those sampleswere not performed.

Strain sweep experiments were performed at a frequency of10 rad/s (unless otherwise noted) to determine the regionof linear response. Oscillatory frequency sweeps from 0.1 to500 rad/s were carried out with an appropriate strain within thelinear region. Steady shear measurements typically were per-formed over a range of shear rates between 10-3 and 103 s-1,although higher shear rates were occasionally employed forsamples with low viscosity.

Parallel superposition of oscillation onto steady-shear flowwas carried out at various applied stresses to determine thechange of the relaxation time of the samples under differentstress. A detailed introduction to the superposition techniquecan be found in an earlier report by Tam et al.26

Results

Three States of Cross-Linkers in the Samples. The bindingthermodynamics and exchange kinetics of the metal-pyridineinteractions have been characterized previously.12,15 The pre-viously reported equilibrium association constants (Keq) anddissociation rate constants (kd) are summarized for conveniencein Table 1.

In our samples, the cross-linkers might take any of threedifferent states, which we denote here as free, dangling, andbound states. The bound state is the state inwhich each of thetwo metal centers in the bis-Pd(II) (or bis-Pt(II)) cross-linkers is coordinated to a pyridine unit along the PVP. Ofcritical importance is the fact that the bound state may existas either an interchain or an intrachain cross-linker. Whileintrachain bonds neither contribute to the formation of thenetwork nor act as “elastically active chains”,27 interchainbonds are critical for gelation and contribute to the networkmodulus. By the dangling state, we refer to a bis-Pd(II) (orbis-Pt(II)) cross-linkers that is bound to a PVP pyridine unitonly on one end. The third state, that of the free bis-Pd(II) (orbis-Pt(II)) cross-linkers, is that in which neither metal centeris coordinated to a pyridine unit.

Because the Pd-pyridine (or Pt-pyridine) coordinationbond is labile, each bis-Pd(II) (or bis-Pt(II)) complex con-stantly changes between the three states. The distribution ofstates among all bis-Pd(II) (or bis-Pt(II)) complexes, how-ever, achieves (pseudo)equilibrium or steady-state values.

Figure 1. (a) Schematic of a supramolecular polymer network in whichpolymer side chains are cross-linked by bis-functional recognition units.(b) Thermodynamic and kinetic parameters of the supramolecularinteraction underlying network formation.

Figure 2. Schematic picture of networks formed from PVP chains andcross-linkingbimetallic compounds (1-4).Twocounterions 2[CF3SO3]

1-

are not shown in the schematic picture.

Table 1. Equilibrium Constants and Dissociation Rate Constants forPincer Pd and Pt Complexes (1-4) with Pyridine in DMSO at 25 �C

[Uncertainties: Keq (20%), kd (15%)]12

complex Keq (M-1) kd (s

-1)

1 3 pyridine 29 14502 3 pyridine 33 173 3 pyridine 8000 0.0264 3 pyridine 4000 0.0006

3558 Macromolecules, Vol. 43, No. 7, 2010 Xu et al.

Using the equation as shown inFigure 1b and accounting forthe presence of two binding sites on our cross-linkers, thefractions of free ( pf), dangling ( pd), and bound ( pb) cross-linkers can be derived:

pf ¼ 1

Keq2½A�eq2 þ 2Keq½A�eq þ 1

ð1Þ

pd ¼ 2Keq½A�eqKeq

2½A�eq2 þ 2Keq½A�eq þ 1ð2Þ

pb ¼ Keq2½A�eq2

Keq2½A�eq2 þ 2Keq½A�eq þ 1

ð3Þ

where [A]eq is the concentration of pyridine unit at equili-brium state. The fraction of different states of cross-linkers inan equilibrium state depends onKeq, the initial concentrationof PVP, and the initial concentration of cross-linkers. In ourexperiments, pb for bis-Pd(II) is about 0.91-0.94 and pb forbis-Pt(II) is about ∼0.999; pd for bis-Pd(II) is about0.056-0.086 and pd for bis-Pt(II) is about ∼10-4; andpf for bis-Pd(II) is about∼10-3 and pf for bis-Pt(II) is about∼10-7. (The calculations responsible for these values aredescribed in greater detail in the Supporting Information.)The overwhelming majority of the bis-Pd(II) (or bis-Pt(II))cross-linkers, therefore, are expected to be in the bound statein our samples, and the average ratio of bound state cross-linkers to PVP chain ranges from ∼1:1 to 20:1 across theconcentration ranges studied here.

Linear Rheological Properties. The linear rheology ofsimilar cross-linked PVP networks has been explored pre-viously in some detail by our group.12,15 Nevertheless, thespecific concentrations and conditions employed here areslightly different from those in previous studies, and so webegan by confirming the linear rheology of these samples.The storage (G0) and loss (G00) moduli for ∼0.1 g/mL PVPwith different concentrations of 2 are shown as a function offrequency in Figure 3. The modulus of∼0.1 g/mL PVP with0.5% 2 is small and outside of the measurement range of therheometer. As the concentration of 2 increases from 1% to7%, both the storage and loss moduli increase more than 3orders of magnitude. At these higher concentrations of 2, thebulk relaxation rate (β) of the networks can be obtained fromthe crossover points between G0 and G00. The plateau moduli(G0) are obtained from the high-frequency measurements.

As expected based on previous reports on the samesystems at different concentrations,12,15 the storage and lossmoduli curves are shifted to higher frequencies when thefaster bis-Pd(II) cross-linker 1 is used in place of 2. Also aspreviously reported, the dynamic mechanical properties ofthe PVP 3 1 and PVP 3 2 networks are superposed onto a singlemaster curve by scaling the frequency of the oscillatoryexperiment by kd measured for model Pd-pyridine com-plexes, consistent with the expectation that the Pd-pyridineinteractions define the elastically active segments in thenetwork. While the crossover points of G0 and G00 for thePVP 3 1 networks are not observed within the experimentalrange accessible on our rheometer, they can be inferredthrough the scaling relationship. In Figure 4, superpositionofG0 andG00 versus scaled frequency for∼0.1 g/mLPVPwith1 or 2 (above 1%) is shown. We note that the data for thenetwork formed from ∼0.1 g/mL PVP with 1% 1 cannot besuperposed with that formed from 1% of 2, consistent with

previous observations regarding cross-linker concentrationsthat are near the gel point.19

The slopes of G0 and G00 in the terminal zone for 0.1 g/mLPVP with 2 showed unusual terminal behavior with power-law dependencies for G0 and G00 versus frequency (ω) havingmuch smaller slopes than those expected of a single-elementMaxwell model:G0 ∼ω2 andG00 ∼ω1, respectively. With theincreasing concentration of 2 from 1% to 7%, the power-lawdependencies for G0 versus ω range from 1.86 to 1.33; thepower-law dependencies for G00 versus ω range from 0.98 to

Figure 4. Storage modulus (G0) and loss modulus (G0 0) versus scaledfrequency for ∼0.1 g/mL PVP with different concentrations of 1 or 2.

Figure 3. Storagemodulus (G0) and lossmodulus (G0 0) versus frequencyfor∼0.1 g/mLPVPwith different concentrations of 2. Concentration ofcross-linkers means molar ratio between Pd:N here and below unlessotherwise noted.

Article Macromolecules, Vol. 43, No. 7, 2010 3559

0.85. This behavior has also been noted previously,12,28 and itis apparent that the relaxation modes in these samples aremore complex than can be described by the single apparentrelaxation time obtained by the crossover of G0 and G00.

Shear Thickening. Steady shear viscosity versus shear ratefor ∼0.1 g/mL PVP with 1 or 2 is shown in Figure 5. Zero-shear viscosity of ∼0.1 g/mL PVP with 0.5% 1 or 2 is just alittle higher than the pure ∼0.1 g/mL PVP solution (0.021Pa 3 s). This indicates that the concentration of 0.5% cross-linker is below the gel concentration for the formation ofnetwork. With the increasing of concentration of cross-linker above 1%, the viscosities of the two mixtures aresignificantly different (and scale with the difference betweenthe dissociation rates of 1 and 2). This is consistent with theprevious results in our group that the gel concentration ofcross-linkers for forming network is ∼0.8%.12,19 For sam-ples with concentration of cross-linker above the gel con-centration, the steady shear viscosity exhibits three flowregimes with the increasing of shear rate, which are New-tonian, shear thickening, and shear thinning. The steepdecrease in viscosity at high shear rates (Figure 5) is attri-buted to network fracture29 and/or the subsequent ejection

of sample from the rheometer geometry (one video as anexample in the Supporting Information). Strain hardening isalso observed for those samples which show shear thicken-ing. The strain hardening behavior of these networks iscurrently under investigation and it is not explored furtherhere.

Table 2 shows a comparison of compiled parameters (fromFigure 3-5) for the ∼0.1 g/mL PVP solution with 1 or 2.From Table 2, we can get the Weissenberg number (shearrate divided by relaxation rate, _γ/β) at the lowest shear rate atwhich thickening is observed ( _γstart) and the shear rate atwhich the viscosity reaches its maximum point ( _γmax).In polymer solutions, a Weissenberg number that is largerthan 1 corresponds to orientation of polymer chains.3 For∼0.1 g/mL PVP solution with 1% 2 at _γstart, theWeissenbergnumber is about 1.4. With the increasing of concentration of1 or 2 (above 1%), the Weissenberg number ranges from0.3 to 0.76 at the shear thickening start point. The onset ofshear thickening at Weissenberg numbers that are less than1 might be due to the wide distribution of relaxation timesof our samples (see above), but the correlation supportsthe assertion that the shear thickening in our samples isconnected with the orientation of polymer chains.

As seen in Table 2, the ratio between the maximumviscosity at the shear thickening peak and the zero shearviscosity during shear thickening (ηmax/η0) ranges from 1.3to 5.9. The degree of shear thickening here is in the samerange as reported for hydrophobically modified urethane-ethoxylate (HEUR) or hydrophobically modified alkali-soluble emulsion (HASE) associative polymer solutionsunder different conditions.30-34 These and other parametersin Table 2 are discussed in more detail below.

Discussion

Shear Thickening of Associative Polymers. The molecularorigin of shear thickening has been debated over the pasttwo decades on the basis of experimental26,35,36 and theore-tical36-39 evidence. To date, the mechanisms for shear thicken-ing of associative polymers have been classified into two maincategories. The first ascribes thickening to the nonlinear hightension along stretched polymer chains beyond the Gaussianrange,30,31,38 while the second attributes thickening to anincrease in the number of elastically active chains.26,40-42

An example of the latter, a “structure-forming” mecha-nism is found in the work of Witten et al., who sketched apicture of shear thickening in which the flow producesincreased association between chains at the expense ofassociations within a chain.42 Witten’s theory was doubtedby Wang, who argued that when the lifetime of the associa-tion is much longer than the relaxation time of free chains,

Figure 5. Steady shear viscosity versus shear rate for ∼0.1 g/mL PVPwith different concentrations of 1 (a) and 2 (b).

Table 2. Parameters for ∼0.1 g/mL PVP Solution with 1 and 2 (From Oscillatory Frequency Sweep and Steady Shear Experiment)a

samples β (s-1) _γstart (s-1) _γstart/kd _γmax (s

-1) _γmax/kd σmax (Pa) ηmax/η0

7% 1 243.1 71.97 0.04963 268.3 0.185 38330 1.55% 1 252.5 100 0.06897 372.8 0.2571 16740 1.33% 1 302.8 193.1 0.1332 1000 0.6897 15540 1.42% 1 355.7 268.3 0.185 1931 1.332 12460 1.71% 1 1585 1.093 7944 5.478 6937 2.67% 2 2.85 1 0.05882 2.686 0.158 27250 1.45% 2 2.96 1.389 0.08171 3.727 0.2192 16380 1.33% 2 3.55 2.512 0.1478 7.944 0.4673 12400 1.62% 2 4.17 3.162 0.186 15.85 0.9324 10200 2.11% 2 13.92 19.31 1.136 100 5.882 3344 5.9a β (s-1) is the relaxation rate of samples. _γstart (s

-1) is the shear rate where the steady-shear viscosity starts to increase at the beginning of shearthickening. _γmax (s

-1) is the shear rate where the steady-shear viscosity reaches maximum viscosity during shear thickening. _γstart/kd and _γmax/kd are thescaled value of _γstart and _γmax, respectively. σmax is the shear stress where the steady-shear viscosity reaches maximum viscosity during shear thickening .η0 is the zero shear viscosity. ηmax is the maximum viscosity during shear thickening. ηmax/η0 represents the degree of shear thickening.

3560 Macromolecules, Vol. 43, No. 7, 2010 Xu et al.

flow cannot break strong intrachain associations until mostinterchain associations are broken.41 Wang subsequentlyintroduced free chains into a transient network model andpredicted that shear thickening is the result of coagulation offree chains into the existing network.41

In contrast, Marrucci et al.38 attributed shear thickeningto a non-Gaussian chain stretching effect within a networkwhose number of elastically active chains does not changeduring shear. Within the basis of a transient network model,they replaced the linear force law describing chain extensionwith an inverse Langevin function and found shear thicken-ing as a result of non-Gaussian chain stretching, althoughthe magnitude of shear thickening predicted by this theorywas smaller than that observed experimentally. Marrucciet al. further assumed that a polymer chain can only par-tially relax its extended conformation when the chain enddissociates from a network junction (the free path model).In the free path model, the critical shear rate is estimatedby assuming that the elastically active chains reach fullextension at the onset of shear thickening.38 Under theseassumptions, the shear rate at maximum viscosity ( _γmax) isapproximately

_γmax � N1=2=τ ð4Þwhere N is the number of Kuhn segments in the chain andτ is the network relaxation time. The critical shear stressat the shear thickening maximum point (σmax) is estimatedto be

σmax � νkBTN1=2 ð5Þ

where ν is the number of elastically active chains, kB isBoltzmann’s constant, and T is the temperature. Accordingto eq 5, the σmax is found to be independent of τ and dependsonly on chain molecular weight and concentration. The freepath model predicts that the viscosity attains a maximumvalue given by

ηmax � νEaτ ¼ ðEa=kBTÞη0 ð6Þ

whereEa is the activation energy of the association. Thus, thevalue of the viscosity maximum depends directly on theenergy barrier Ea/kBT. Subsequent work showed goodagreement with the prediction of the free path model forcritical shear rate, critical shear stress, and the magnitude ofshear thickening.30

Recently, Tripathi et al. proposed a modified nonlinearconstitutive model for telechelic associative polymers.33 Themodel incorporates contributions to the total stress tensorfrom both the “elastically active” bridging chains betweenmicelles and the dangling chains that continuously exit andreenter themicellar junctions.Nonlinear chain extension, theshear-induced enhancement of associations, and the stretch-induced dissociation of hydrophobic chains are essentialfeatures of the model. Tripathi et al. then compared themodel predictions with a systematic experimental study ofthe linear viscoelastic, steady shear, and transient exten-sional properties of a series of well-characterized modelhydrophobically modified ethoxylate-urethane (HEUR)polymers possessing varying degrees of hydrophobicity. Allexperimental results such as shear thickening at intermediateshear rate followed by shear thinning at higher shear rate (oronly shear thinning for some higher concentration samples)can be fit very well by the above model by changing a singledimensionless constitutive parameter Gm, which describes

the orientational and deformation-rate-dependent creationrate of the active chains. Gm is given by

Gm ¼ τs=τE ð7Þwhere τs is the characteristic interaction time of hydrophobicends with the surrounding fluid medium comprising a latticeof attractive micelles and τE is exit time of the hydrophobicends. For samples showing shear thinning,Gm is expected tobe less than 1. For samples show shear thickening, Gm isexpected to be larger than 1.33While τE has the clearmeaningas lifetime of reversible bond (1/kd in our experiment), theanalogous definition for τs (or a similar parameter) is notclear in our experiments.

Inherent Ambiguity between the Non-Gaussian Stretchingand Structural Reorganization Mechanisms. The initial ques-tion regarding the mechanism of shear thickening observed inour samples, therefore, is whether it involves non-Gaussianstretching of active chain segments within an effectively intactnetwork structure or whether the cross-linkers within thetransient network actually reorganize into a new structure inresponse to the applied shear. Such distinctions are typicallychallenging to realize, for reasons that we now discuss.Consider the behavior of the coordinative PVP networks(Figures 4 and 5) and the fact that the onset of shear thicken-ing during steady shear is well-correlated with the relaxationrateof thenetwork (Table 2),which in turn reflects the kineticsof metal-ligand dissociation in the defining cross-linkingcoordination bonds. This observation strongly implies arequirement for chain orientation within active chain seg-ments, since the opportunity for relaxation of the network(through dissociation of the metal-ligand bond) inhibits theshear thickening. The presence of chain orientation is furthersupported by the relationship between the onset of shearthickening and the Weissenberg number, as described pre-viously. The fundamental difficulty lies in the fact that bothstructural reorganization and non-Gaussian overstretchinginvolve chain orientation under applied shear, and so bothmechanisms by necessity will have many dynamic experi-mental signatures in common.

For example, the free path model described above, which isone example of non-Gaussian stretching within an effectivelystatic network, can be applied to our observations with somequalitative success: as noted above, the onset of shear thicken-ing as a function of shear rate scales with the metal-liganddissociation rate, and this is predicted by eq 4 . In addition, thecritical shear stress at the shear thickening maximum point(σmax in Table 2) is almost the same and independent of thedifferences in the kinetics of cross-link dissociation rate for 1and 2, as expected from eq 5. But similar scaling behaviorsshould, in general, be observed as a function of networkreorganization as well, under which circumstances the ulti-mate network responses for 1 and 2 are expected to beidentical except for the time scale on which they occur. Thus,consistency with the free path model is not direct proof thatshear thickening is caused by non-Gaussian stretching.

Evidence of a Shear-Induced Increase in the Number ofElastically Active Chains. One distinguishing feature bet-ween the reorganization and overstretching models involvesthe expected influence of each on network relaxation time.33

In Marrucci et al.’s work,38 the relaxation time of the net-work is calculated to be

τ � Nb2

a2β0

-1 ð8Þ

where a is the average spatial distance among neighboringaggregates (cross-linkers), N is the number of Kuhn chain

Article Macromolecules, Vol. 43, No. 7, 2010 3561

segments in the polymer chain, b is the Kuhn length, and β0is the chain detachment rate under equilibrium conditions(kd in our formalism). In Marrucci et al.’s work, a is relatedto the number of elastically active chains (ν) as38

νa3 ¼ 1 ð9ÞTripathi’s work results in a similar equation:33

τ∼Nb2=3

Æaæ2τE ð10Þ

In the case of an intact network in which shear thickeningoccurs as a result of non-Gaussian stretching, τ generallyremains constant in the linear regime and decrease during theshear thickening phenomenon.31 Under this mechanism, thedecrease of τmay result in the increase of a during stretchingof polymer chain or the decrease of νwhen the elastic energyof polymer chain compares with the activation barrier forcross-linkers.31 τmight also decrease because dissociation rateis accelerated by a coupled force.43 On the other hand, if shearthickening results froman increase in the number of elasticallyactive chains, a decrease in a is expected, which should in turnlead to an increase in the measured relaxation time of thenetwork. Characterizing the change (or lack thereof) of τduring shear thickening, therefore, provides strong evidencefor the mechanism underlying shear thickening.

We determine the relaxation time of the samples underdifferent stress through parallel superposition of oscillationon steady-shear flows (strain rate tensor applied by oscilla-tion in the direction of velocity).26 It is important to note thelimitations of the parallel superposition technique, in parti-cular when used to characterize the absolute value ofmoduli.For example, the moduli obtained by parallel superpositioncannot be treated as linear viscoelastic moduli, and in fact,apparently negative moduli have been obtained at lowfrequency; orthogonal superposition (strain rate tensorapplied by oscillation in the direction of vorticity) is thoughtto be a betterway to characterize themodulus.32On the otherhand, parallel superposition is still a sensitive method bywhich to obtain trends in the change of relaxation timewithin the sheared network. For example, Mewis et al. havereported that both parallel and orthogonal superpositiongive the same trends for network relaxation time as a functionof applied shear stress for a poly(isobutene)/decalin solutionand a hydrophobic alkali-swellable emulsion (HASE), eventhough the values of the relaxation times obtained by the twomethods differ, especially in the nonlinear regime.32,44 At thesame time, Tam et al.45 and Mewis et al.32 both reported thatthe relaxation time of a shear thinning HASE associativepolymer decreases with increasing shear stress.31,45 Munozet al. researched shear thickening in glyerine solutions of poly-(2-hydroxyethyl methacrylamide) and found that the relaxa-tion time does not change when the applied shear stress liesinside the limits of the linear regime but increases with theapplied shear flow for valuesof the shear stress associatedwithshear thickening. This observation is attributed to the shear-induced formation of hydrogen bonds.46 This prior workjustifies parallel superposition as a reliable method for char-acterizing the dependence of relaxation time on an appliedshear stress.

Results are shown in Figure 6 for four networks of PVPwith concentrations of cross-linker 2 ranging from 2 to 7%(average ratio of bound cross-linkers to number of PVPchains ranges from ∼5 to 20). In all cases, the maximumapplied shear stress is less than the critical shear stress defined

by the shear thickening maximum point, ensuring that thenetwork is not broken during the parallel superpositionexperiment. In Figure 6, shear stresses less than ∼103 Pacorrespond to the linear viscoelastic regime observed in thesteady shear experiments, and the relaxation times τ derivedfrom the parallel superposition in this regimeare close to thoseobtained from the network relaxation time under dynamicoscillatory stress without superposition (Table 2). In addition,τ is effectively independent of shear stress for shear stresses lessthan ∼103 Pa.

The onset of shear thickening occurs at shear stresses of∼103 Pa, and above this threshold value we observe anincrease in τ with increasing shear stress. The increasedrelaxation time is not typical for transient networks, and itis observed here at all concentrations of cross-linker. Asdiscussed above, we regard only the direction of the changein τ with increasing shear stress as reliable, rather than theabsolute value of τ in the nonlinear regime. In 0.1 g/mL PVPwith 2% 2, for example, τ changes from ∼0.23 to ∼0.55 s asthe applied shear increases. From eqs 8 and 9, τ scales with νas τ ∼ ν2/3. If the increase in τ derives entirely from anincreasing number of elastically active chains, ν wouldincrease by a factor of 2.8 with the increased shear. Asdiscussed below, this increase is larger than can reasonablybe expected (see, e.g., Figure 9 and related discussion). Weinfer that the exact value of τ in the shear thickening regimeshould be lower than the apparent value shown in Figure 6.

As described above and seen empirically in the low-stressregime of Figure 6, the increased relaxation time is consistentwith an increased density of active cross-linkers and contra-dicts the behavior expected from non-Gaussian overstretch-ing in an effectively intact network. While shear thickeningof the bis-Pd(II)-PVP networks is not dominated by non-Gaussian overstretching, however, the presence of structuralreorganization need not preclude contributions from non-Gaussian effects.

Evidence of Non-Gaussian Stretching of Polymer Chainsduring Steady Shear. As a polymer chain is stretched undershear, the stress along the polymer chain will increaseexponentially with the strain.47 When the elastic energyof the polymer chain is comparable to the dissociation energyof the cross-linker, the supramolecular polymer network willbreak. The break of such a supramolecular polymer networkwas experimentally observed by S�er�ero et al., who researchedthe rheology and microstructure of aqueous solutions oftelechelic polymers with a poly(ethylene oxide) middle blockand semiperfluorinated end-caps. Those authors referred tothe process as fluid fracture.29 A sharp decrease in viscosityafter shear thickening (called fluid fracture in their paper) wasobserved, similar to what we report here.29,31 S�er�ero et al.

Figure 6. Plot of relaxation time (τ) of ∼0.1 g/mL PVP solution withdifferent concentration of 2 versus applied shear stress imposed duringparallel superposition of oscillation on steady-shear flow measure-ments.

3562 Macromolecules, Vol. 43, No. 7, 2010 Xu et al.

conclude that above the linear regime the active chains arestretched in the non-Gaussian range of extensions, and con-sequently, the network hardens due to the increase of theentropic restoring forces. For rates and stresses large enough,there is a catastrophic breakdown or rupture of the networkleading to a low-viscosity flow. Similar behavior is observedhere, and sowe infer that non-Gaussian stretching of polymerchains in our samples does take place, especially during therupture of network (shear thinning regime during steadyshear). While network fracture clearly occurs, however, theshear thinning behavior cannot be attributed to networkfracture alone, as the shear thinning behavior here correlateswith the ejection of sample from the rheometer geometry.

We note that while forced dissociation of the cross-linksand fluid fracture could, in theory, occur at forces belowthose associated with non-Gaussian stretching, it is unlikely.Vaccaro et al. have estimated the elastic energy of a polymerchain needed to pull off cross-linkers.48 As long as the chainremains Gaussian, the elastic tension F in the chain growsproportionally with the end-to-end distanceR according to48

FðRÞ ¼ 3kBT

Nb2R ð11Þ

where b andN are the length and number of the chain Kuhnsegments, respectively. In the linear region, R remains muchsmaller than the fully extended chain length Nb, and thetensionF remainswell under kBT/b, or on the order of severalpiconewtons. Previous force spectroscopywork in our grouphas shown that more than 50 pN of force is required toaccelerate the rate of cross-link dissociation by a factor of 10,and so the forces associated with stretching of the polymerchains in the linear regime are not sufficient to explain thecatastrophic fracture observed at high shear rates.43 Thesearguments strongly suggest that high shear rates do even-tually lead to non-Gaussian stretching of active polymerchain segments within the network.

MolecularMechanismUnderlying Shear Thickening inOurSamples.Onemethod to determine the relative contributionsof these two shear thickening mechanisms is to compare theextent of shear thickening with theoretical expectations.According to the free path model, shear thickening fromnon-Gaussian overstretching leads to amaximum increase inviscosity ηmax which is proportional to the zero-shear visco-sity η0 through the activation energy for bond dissociation(eq 6).38 In our experiments, the free energies of activation(ΔG) of the four cross-linkers are calculated by the Eyringequation:49

kd ¼ 2:083� 1010Te-ΔG=RT ð12Þwhere R is the gas constant. In our experiments, ΔG of thefour cross-linkers at 25 �C are 55, 66, 82, and 91 kJ/mol for 1through 4, respectively. The internal energies of activation(Ea) of the four cross-linkers are also calculated to be 42, 53,64, and 74 kJ/mol for 1 through 4, respectively (see Support-ing Information). As seen in Figure 7, however, it is apparentthat ηmax/η0 does not increase with the increasingΔG (or Ea)of the four cross-linkers, in contrast to Marrucci’s free pathmodel. This is consistent with a picture in which non-Gaussian overstretching of polymer chains has a weakinfluence on shear thickening.

In Figure 7, we notice that ηmax/η0 of 0.08 g/mL PVPsolution with 1% of cross-linkers varies substantially fromone cross-linker to another. Because a cross-linker concen-tration of 1% is near the gel point (∼0.8%),19 ηmax/η0 here isexpected to be quite sensitive to the precise concentration

of cross-linkers. We therefore attribute the variations inηmax/η0 of 0.08 g/mL PVP solution with 1% cross-linkersto small differences in the concentration of cross-linkersduring preparation of samples. On the other hand, ηmax/η0of 0.08 g/mL PVP solution with higher concentrations ofcross-linkers remains in the range of 1.5-2.5 (same trends asshown in Table 2 for 0.1 g/mL PVP with 1 or 2). These dataare presented for the sake of completeness only and do notimpact the conclusions of this work.

Another approach to elucidating the shear thickeningmechanism is through kinetic studies of the shear thickeningprocess. While non-Gaussian stretching of polymer chainsoccurs on the time scale of chain elongation, the mechanismof shear-induced reorganization requires that cross-linkersmust dissociate in order to rearrange to form new networkstructure. In the latter case, the structural transformation isnot instantaneous upon application of sufficient shear stressbut requires cross-linker reorganization that occurs on thetime scale of 1/kd or longer. The kinetics of shear thickeningwere investigated using peak hold shear experiments follow-ing preshear. A faster sampling rate (1 point per second) wasemployed during these peak hold shear experiments than inthe steady shear experiments (∼1 point per minute). Thepeak hold shear rate was changed to a value within theobserved shear-thickening regime. In Figure 8, peak holdshears of three samples are shown as examples. For 0.1 g/mLPVPwith 7%of 1 (Figure 8a), the sampling rate of peak holdshear (1 point per second) is too slow to observe the transi-tion, consistentwith changes in the network that occur on thetime scale of network relaxation through cross-link dissocia-tion (∼0.004 s). For 0.1 g/mL PVP with 7% of 2 (Figure 8b),an initial decrease of viscosity at the beginning of peak holdshear is observed, followed by subsequent shear thickening.The kinetics are obviously not those of a smooth, single-steptransformation. Again, the time scale of this process is on theorder of the relaxation time of this sample (∼0.35 s). For 0.08g/mL PVP with 1% of the much slower cross-linker 3(relaxation time ∼21 s), the steady state under peak holdshear is reached around ∼100 s (Figure 8c). Berret et al.performed similar start-up shear experiments on telechelicpolymer networks (Figure 4 in ref 35).35 The steady state isachieved around ∼100 s, which is also comparable to therelaxation time of their network (∼26 s). In their work, theyalso show that the rupture of the network caused by non-Gaussian stretching of polymer chain takes place at fixedstrain during start-up shear (Figure 8 in ref 35). As the shearrate increases above the critical value so that the stresscannot relax via dissociation of cross-linkers, the ruptureof the network accelerates under the increased stress. Thisstrongly suggests that non-Gaussian stretching of elasticallyactive polymer chain segments occurs on shorter time scales

Figure 7. Degree of shear thickening (ηmax/η0) of ∼0.08 g/mL PVPsolutionwith four different cross-linkers (1-4) versus the concentrationof cross-linkers is shown.

Article Macromolecules, Vol. 43, No. 7, 2010 3563

than does the shear induced increase in the number ofelastically active chains. We therefore infer that shearthickening in our samples is a two (or more) state processthat comprises first partial breakage of the original networkstructure, followed by orientation of the polymer chainsand the subsequent re-formation of a network structurewith a higher density of elastically active interchain cross-linking.

Origin of the Shear-Induced Elastically Active Chain Seg-ments. We have concluded that a structural rearrangementwithin the network underlies the shear thickening behavior,but two possibilities for the molecular origins of this struc-tural transition exist: (i) the increased number of elasticallyactive chains may result from the shear-induced transforma-tion of intramolecular associations to intermolecular asso-ciations40,42 (here, conversion of bound cross-linkers froman intrachain to interchain state), or (ii) the newly activechains may result from shear-induced incorporation offloating chains into the network backbone41 (conversionfrom dangling to interchain bound state). To distinguishwhich effect dominates the shear thickening in our system,we compare the extent of shear thickening and apparentnumber of shear-induced elastically active chains with esti-mates of the number of cross-linkers in the intrachain boundand dangling states.

The frequency sweep data provide the plateau modulusG0, fromwhich the density of elastically active chains (ν) canbe calculated47

G0 ¼ νkBT ð13ÞIn the present work, ν corresponds to the number of inter-chain bound cross-linkers per unit volume in the 0.1 g/mLPVP solution. The total number of bis-Pd(II) complexes perunit volume (ν0) is calculated from the concentration ofcross-linkers. The ratio ν/ν0 in the 0.1 g/mL PVP solutionis then taken to be the fraction of interchain bound statecross-linkers ( pinter), and the fraction of intrachain boundstate cross-linkers ( pintra) can be calculated from the differ-ence between bound state cross-linkers (pb) and interchainbound state cross-linkers ( pinter)

pintra ¼ pb - pinter ð14ÞThe values of pb, pinter, and pintra change with concentration,and Figure 9 shows calculated equilibrium values for each at0.1 g/mL PVP and varying concentrations of 2, using eqs 3,13, and 14. As the density of cross-linkers (ν0) increases, sotoo does the fraction of those cross-linkers that form elasti-cally active chains (ν/ν0 or pinter), which increases smoothlyfrom ∼0.35 to ∼0.89 as the concentration of cross-linkersincreases from 2% to 7%. According to eqs 6, 8, and 9, thescaling law between zero shear viscosity and number ofinterchain bound cross-linkers is η0 ∼ ν5/3.38 By assumingthat the increased viscosity during shear-thickening is attri-buted entirely to the addition of new cross-linkers and thatthe same scaling law applies, we therefore infer that pinterincreases at the maximum viscosity during shear thickeningby a factor of (ηmax/η0)

3/5 from its zero-shear equilibriumvalue. From the ηmax/η0 data in Table 2, interchain boundcross-linkers at shear thickening maximum point (pinter*) isfrom 1.2 to 1.6 times larger than pinter at equilibrium state,with apparentmaximal values of from0.55 to 1.09 dependingon cross-linker concentration.

These values are clearly only approximate;it is impos-sible, for example, for p to exceed 1;but they provide areasonable framework through which to assess the extent ofstructural reorganization. Table 3 shows the calculatedfraction of bis-Pd(II) complexes that become active duringshear (from Figure 9) in comparison to the fraction of bis-Pd(II) complexes that are calculated to exist in the free,dangling, and intrachain bound states (from Figure S2 inSupporting Information). The comparisons in Table 3 showthat the fraction of free and dangling state cross-linkers is toosmall for their conversion to active cross-linkers to account

Figure 9. Fraction of interchain bound cross-linkers ( pinter) and intra-chain bound cross-linkers ( pintra) for ∼0.1 g/mL PVP solution withdifferent concentrations of cross-linkers 2: fraction of interchain boundcross-linkers ( pinter) at equilibrium state (0); fractionof intrachain boundcross-linkers ( pintra) at equilibrium state (O); estimated interchainbound cross-linkers at shear thickening maximum point ( pinter*) (1).

Figure 8. (a) Peak hold shear at different shear rates after preshearfrom 10-3 to 51.79 s-1 for ∼0.1 g/mL PVP with 7% 1. (b) Peak holdshear at different shear rates after preshear from 10-3 to 0.5179 s-1 for∼0.1 g/mL PVP with 7% 2. (c) Peak hold shear at different shear ratesafter preshear from10-3 to 0.037 28 s-1 for∼0.08 g/mLPVPwith 1% 3.

3564 Macromolecules, Vol. 43, No. 7, 2010 Xu et al.

for the full extent of shear thickening. For the data of0.1 g/mL PVP with 7% of 2, pd is larger than pintra. It seemsthat both dangling state cross-linkers and intrachain boundcross-linkers may transfer to interchain bound cross-linkers.But other examples show that the transfer of pd to pinter is notnecessarily needed to account for the magnitude of shearthickening. For example, ∼0.08 g/mL PVP with 3 or 4 haspd less than∼0.1%while pb is larger than∼99.9%. From thevalue of ηmax/η0 of them (Figure 9), we can see that thefraction of free and dangling state cross-linkers is far toosmall for their conversion to active cross-linkers to accountfor the extent of shear thickening. By comparison, the cross-linkers that are originally in an inactive intrachain boundstate are present in sufficient numbers to roughly accountfor the enhanced viscosity due to shear thickening. We con-clude, therefore, that the shear thickening is primarily dueto a shear-induced transformation of elastically inactiveintrachain cross-linkers to elastically active interchaincross-linkers.

Conclusions

Networks formed from semidilute unentangled solutions ofpoly(4-vinylpyridine) in DMSO and bis-Pd(II) (or bis-Pt(II))organometallic cross-linkers are observed to undergo shearthickening above critical rates of steady shear. The shear ratesinvolved in the nonlinear rheological properties scalewith the rateof dissociation of the defining metal-ligand bond. Parallelsuperposition of oscillation onto steady-shear flows facilitates acharacterization of the change in network relaxation time fromthe linear to the shear thickening regime. An increase in therelaxation time is observed, supporting a mechanism in whichshear thickening is caused by a shear-induced increase in thenumber of elastically active chains. The kinetics andmagnitudeofthe shear thickening support this conclusion. An analysis of thedifferent state of cross-linkers further supports that the increasedelastically active chains result from the conversion of intrachaincross-linkers to interchain cross-linkers, rather than from dang-ling state cross-linkers to interchain bound state cross-linkers inour experiments.

Acknowledgment. This work was supported by NSF (CHE-0646670) and NIH (EB-001037). We thank M. Rubinstein forhelpful comments.

Supporting Information Available: Characterization of thePVP concentration regime, calculation of the fraction of cross-linker states and activation energies of cross-linker dissociation,and video of the steady shear experiment. This material is avai-lable free of charge via the Internet at http://pubs.acs.org.

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Table 3. Fraction of Cross-Linkers of ∼0.1 g/mL PVP with 2a

samples pf (%) pd (%) pb (%) pinter (%) pintra (%) pinter* (%) Δpinter (%)

7% 2 0.1 6.1 93.8 89.0 4.8 108.9 19.95% 2 0.1 5.9 94.0 71.9 22.1 84.1 12.23% 2 0.1 5.8 94.2 45.4 48.8 60.2 14.82% 2 0.1 5.7 94.2 35.4 58.8 55.3 19.9a pf is the fraction of free state cross-linkers. pd is the fraction of dangling state cross-linkers. pb is the fraction of bound state cross-linkers. pinter is the

fraction of interchain bound cross-linkers at equilibrium state. pinter* is the fraction of interchain bound cross-linkers at shear thickening maximumpoint. pintra is the fraction of intrachain bound cross-linkers at equilibrium state. Δpinter is the difference between pinter and pinter*.

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