Homogeneous rheological behavior of nanoparticle-based melt

Rheol Acta (2010) 49:971–977DOI 10.1007/s00397-010-0474-5

ORIGINAL CONTRIBUTION

Homogeneous rheological behaviorof nanoparticle-based melt

Xin Li · Shi-Qing Wang · Xiaorong Wang

Received: 4 August 2009 / Revised: 15 May 2010 / Accepted: 20 July 2010 / Published online: 5 August 2010© Springer-Verlag 2010

Abstract The present work explores nonlinear rheo-logical behavior of a strongly viscoelastic paste madeof nano-sized polybutadiene particles. Apart from con-ventional rheometric measurements, particle-trackingvelocimetric observations are carried out to determinethe macroscopic state of deformation during startupshear and after step strain. Despite its highly nonlinearrheological characteristics, the system shows no sign ofinhomogeneous response to large shear deformationsin sharp contrast to well-entangled polymeric liquidsmade of linear chains. Apparently strongly nonlinearrheological behavior can occur in absence of inhomo-geneous macroscopic deformation.

Keywords Gels · Nonlinear rheology · Yielding ·Particle tracking velocimetry · Granular materials

Introduction

Polymeric nanoparticles attracted significant attentiondue to their unique properties and a variety of appli-cations, ranging from drug delivery device (Soppimath

X. Li · S.-Q. Wang (B)Department of Polymer Science, University of Akron,Akron, OH 44325, USAe-mail: [email protected]

X. WangCenter for Research and Technology, BridgestoneAmericas, Akron, OH 44317, USA

et al. 2001) to templating agents for nanoporous micro-electronic materials (Hedrick et al. 1998). Different sizeand structure of nanoparticles can be made by control-ling synthesis conditions (Harth et al. 2002; Wang et al.2007). Micro gels (Goh et al. 2008; Seth et al. 2008;Caggioni et al. 2007) and pastes (Purnomo et al. 2006;Meeker et al. 2004; Wong and Kwan 2008; Saffour et al.2006) belong to this class of granular-like or gel-likematerials made of soft or hard particles. Dense colloids(Dawson 2002; Kobelev and Schweizer 2005; Phamet al. 2004; Russell and Israeloff 2000; Pham et al. 2002;Fortini et al. 2008; Jabbari-Farouji et al. 2007) haveoften been taken as model systems for glasses.

All viscoelastic and glassy materials must yield uponfast deformation. Under quick external deformationwhere it takes a time much shorter than the terminalrelaxation time of a well-entangled polymer liquid toproduce 100% shear deformation, it has to undergosignificant elastic deformation before reaching an even-tual state of flow. This transformation is most appro-priately captured by the phrase “yield”. Recent studiesshow that well-entangled polymeric liquids madeof flexible linear chains cannot undergo flow upona startup shear without yielding inhomogeneously(Tapadia and Wang 2006; Ravindranath and Wang2008). Particle-tracking velocimetric (PTV) observa-tions also reveal non-quiescent relaxation from a largestep strain (Wang et al. 2006; Ravindranath and Wang2007; Boukany et al. 2009). Unlike yielding in glassypolymers, entangled polymers undergo a great deal ofdeformation before the yield point is reached. The en-tanglement network breaks down only when the elasticretraction force within each load-bearing chain growsto exceed the intermolecular gripping force arisingfrom the inter-chain coupling (Wang and Wang 2009).

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In the present work, we are interested in exploringhow yielding takes place in a nanoparticle paste. Thispolybutadiene (PB) melt made of PB stars with ca. 27arms is highly viscoelastic. As a nano-granular material,various levels of jamming occur to produce a broadrelaxation spectrum. Since the constituent particles arenanometer in size, no yield stress exists in the exploredstress range for this paste of densely packed nanoparti-cles. In other words, the paste would eventually flowat any applied shear stress. However, this does notmean that the state of flow (unjammed) can be reachedinstantly. Actually, in the initial response to suddenstartup shear, the jammed paste undergoes elastic de-formation. We carry out PTV and rheometric measure-ments under both startup continuous shear and stepshear to explore how this polymeric paste would evolvefrom the initial jammed state to a final state of flow andwhether this transformation can occur homogeneouslyor not.

Experimental

Materials

The melt under study is made of a nanoparticle witha crosslinked polystyrene core and a shell of polybu-tadiene (PB) brushes. The vinyl content of PB shell is8.4%, and the 1,4-content consists of cis = 45.6% andtrans = 46.0%. The average size of the nanoparticle isaround 9 nm, and the core/shell ratio is around 1/4. Thepolydispersity of these nanoparticles is around 1.15.The PBD brush molecular weight is around 15 kg/mol,and the average number of PBD chains per particle isaround 27. More details about the nanoparticles can befound in Wang et al. (2007). This literature disclosedseven different methods of making nanoparticles. Forspherical core/shell nanoparticles, it also listed twomethods: option A and option B. The nanoparticles

used in this report were made using a similar methodto option B. The core is highly crosslinked. For thepurpose of PTV observations, we have incorporatedabout 600 ppm silver-coated particles with an averagediameter of 10 μm (Dantec Dynamics HGS-10) into thesample.

Apparatus and particle-tracking velocimetry setup

Measurements in stress mode were carried out using aPhysica MCR-301 rotational rheometer (Anton Paar)in a cone-plate setup at room temperature. A cone platewith a cone angle of 4◦ and diameter of 12.6 mm wasemployed.

Measurements of startup shear and step strain weremade with an advanced expansion rheometric system(ARES). To eliminate the edge fracture and avoidoverloading ARES, a cone/partitioned-plate assemblywas used on ARES, as shown in Fig. 1. The inner diskhas a diameter of 15 mm and is linked to the torquetransducer of ARES, and the outer ring as well as thebottom rotating cone has a diameter of 30 mm. Thecone angle of the bottom plate is 4◦. The clearancebetween the outer ring and inner disk is about 0.2 mm.

It is important to check whether the flow field ishomogeneous or not during our rheological measure-ments by making particle-tracking velocimetric obser-vations. As illustrated in Fig. 1, the large dimension ofthe ring provides enough room for a PTV assembly tobe set up from a transparent window on the stationaryring. A laser sheet is passed across the gap at an angleto illuminate the particles embedded in the sample.Movement of the illuminated particles can be capturedby a CCD camera at a speed of 30 frames per secondso that the time-dependent deformation field can berecorded by the video microscopy. In this setup, allsurfaces are smooth surfaces made by steel except fora small PTV window made of glass.

Fig. 1 Side view and top viewof the cone-partitioned platedevice along with aparticle-tracking velocimetricsetup

CCD Laser

2R1

2R2

Side view2R2

2R1

Top view

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Results

Startup shear

Controlled rate mode

A small amplitude oscillatory shear measurement wasfirst carried out to characterize the linear viscoelasticproperty of this nanoparticle paste. After loading thesample, we typically wait for 3 h before measurementbecause the sample does age considerably over time.A 3-h waiting time is chosen because only a smallvariation exists after this amount of aging. Figure 2shows the storage and loss moduli G′ and G′′ vs. theangular frequency ω at a strain amplitude of γ 0 = 0.02,for three discrete sample loadings. All the loadingsproduced a similar gel-like character with G′ aboveand nearly parallel to G′′ throughout the measuredfrequency range. The magnitudes of the moduli arein a range expected of PB melts. Lack of an elasticplateau, characteristic of well-entangled monodisperselinear PB melts (Wang et al. 2003), shows that thereis a broad relaxation spectrum arising from dynamicson various length scales. Since no terminal flow regimeis visible in Fig. 2, nonlinear rheological response mustarise at all shear rates higher than 0.01 s−1. In particular,shear stress overshoot can be expected.

Figure 3a and b present the stress growth and declineduring startup shear at different shear rates. Each mea-surement is waited for 3 h after the preceding test.These data are reproducible within an error bar of10%. Weak stress overshoot shows up at every shearrate. More importantly, the shear stress universallyrises from zero monotonically with the same level of

104

105

106

10-2

10-1

100

101

102

ω (s -1)

G' - filled symbols

G" - open symbols

circles: load 1squares: load 2diamonds: load 3

Fig. 2 Dynamic and loss moduli G′ and G′′ from SAOS measure-ments at γ 0 = 0.02 for three separate loadings

104

105

0.1 1 10 100

σ (P

a)

γ

20.05

0.1

0.2

0.5

1.0

2.0

γ y

0.01

0.1

1

0.01 0.1 1 10

rate 0.05rate 0.1rate 0.2rate 0.5rate 1.0rate 2.0

σ/σ y

γ/γy

0 0.1 0.2 0.3 0.4

σ (P

a)

γ

100

1000

10 4

10 5

10 6

0.01 0.1 1 10 100 1000

rate 0.05 s-1

rate 0.1 s -1

rate 0.2 s -1

rate 0.5 s -1

rate 1 s-1

rate 2 s-1

σ (P

a)

t (s)

σ y

a

b

c

Fig. 3 a Shear stress as a function of time at various applied shearrates. b Shear stress as a function of strain at various appliedshear rates. c Normalized stress versus normalized strain wherethe inset shows an initial common slope, i.e., linear relationshipbetween σ and γ , indicating the initial elastic modulus is the sameindependent of the shear rate

initial stiffness as shown in the inset of Fig. 3c. On theother hand, the stress peak scales quadratically withthe strain as shown in Fig. 3b. Thus, the collapse inFig. 3c is imperfect. The initial monotonic stress growth

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confirms that there is only elastic deformation at thebeginning, as is the case for entangled linear polymermelts. In steady state, we observe severe shear thinningbehavior as shown in Fig. 4, where also plotted are |G∗|vs. ω and the peak stress σy as a function of the appliedrate. Not surprisingly, the Cox-Merz rule is strongly vi-olated as expected for the present system made of neatnanoparticles. In contrast to monodisperse entangledlinear polymers, both peak and steady-state stressesscale with the applied rate in the same way, imply-ing the underlying structure of the paste is perhapsnot fundamentally altered by shearing. Conversely, themuch different scaling of the peak stress with shearrate from the steady state stress vs. rate relationship forentangled polymers indicate that the network structureduring steady shear is significantly different from itsequilibrium structure.

It is well known that highly viscoelastic materi-als tend to suffer interfacial failure during fast shear.Entangled polymers encounter inevitable wall slip(Boukany and Wang 2009a). Pastes (Meeker et al. 2004;Seth et al. 2008) and colloidal dispersions (Boersmaet al. 1991) containing small molecular liquids as sol-vent also tend to show wall slip. Moreover, when wallslip is not the dominant process, entangled polymersmay still experience shear banding upon startup shear(Tapadia and Wang 2006; Ravindranath and Wang2008). Therefore, such conventional rheometric mea-surements as shown in Fig. 3a, b should be supple-mented with direct determination of the deformationfield during shear.

1000

10

10

106

5

4

0.001 0.01 0.1 1 10 100

σ − steady state

σ − stress peaky

G* - frequency sweep

σ, G

* (P

a)

rate (s-1), ω (rad/s)

0.38

Fig. 4 Stress vs. shear rate in steady state (circles). Also shownis the absolute complex modulus |G∗| versus frequency ω fromSAOS (diamonds) as well as the peak (yield) shear stress (asso-ciate with the stress overshoot) vs. shear rate (squares)

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

rate 0.05 s-1

rate 0.5 s-1

rate 2 s-1

Y/Y

max

V/Vmax

Direction of shear

Fig. 5 Particle-tracking velocimetric observations, confirminghomogeneous deformation at rate of 0.05, 0.5, and 2 s−1 in steadystate

We apply an effective particle-tracking velocimet-ric method to determine whether the stress overshootis followed by interfacial failure or shear banding.Figure 5 shows the velocity profiles at shear rates of0.05, 0.5, and 2 s−1, respectively in steady state. The uni-form straining during the startup shear, observed at alltimes for all rates, indicates that shear inhomogeneitydoes not necessarily accompany the nonlinear rheolog-ical responses. Absence of wall slip in this solvent-freepaste underscores a fundamental difference from thesolvent-containing paste (Meeker et al. 2004) where itis the inviscous solvent that makes it possible to observesizable wall slip.

10-4

10-3

10-2

10-1

100

101

102

103

104

0.01 1 100 104

106

t (s)

shea

r ra

te (

s-1)

5 (kPa)

10 (kPa)

20 (kPa)

50 (kPa)

100 (kPa)

3/4

10-4

10-3

10-2

10-1

100

101

102

0.001 0.01 0.1 1γ

γ y

5 (kPa)

10 (kPa)

20 (kPa)50 (kPa)

100 (kPa)

Fig. 6 Apparent shear rate as a function of time at differentapplied stresses. The inset shows the rate vs. strain at thesestresses, where the minimum may be identified as the onset ofyielding

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104

105

0.1 1

stress mode

rate modey

γ y

2

σ

Fig. 7 The applied stress vs. the strain, at which the shear rateshows a minimum in creep mode (circles), read from the insetof Fig. 6 as well as the maximum stress of the overshoot vs. thecorresponding strain at the various applied shear rates, read fromFig. 3b

Controlled stress mode

Since G′ and G′′ do not show any plateau in Fig. 2, thereis not a single critical shear stress level for yielding. Inother words, flow will always take place in time underany given applied shear stress. In constant stress mode,response of many solid-like materials is well known(Rigby 1960): the apparent strain rate γ̇ shows power-law decrease in time: γ̇ ∼ t−n, with n < 1. We expectedour high viscoelastic paste to behave similarly beforeflow takes place. In Fig. 6, we apply stresses rangingfrom 5 to 100 kPa discretely to study the response of thepaste in terms of the apparent shear rate as a functionof the elapsed time. We see the initial shear rate indeeddecreases like a power law with n ranging from 3/4

104

105

0. 01 0.1 1 10 100

γ = 0.1γ = 0.2γ = 0.4γ = 0.7γ = 1.0γ = 2.0

σ (P

a)

t (s)

Fig. 8 Stress growth and relaxation upon step strain produced ata shear rate of 1 s−1, where the strain amplitude varies from 0.1to 2.0

0.01

0.1

1

0.01 0.1 1 10 100

γ = 0.1γ = 0.2γ = 0.4γ = 0.7γ = 1.0γ = 2.0

G(t

)/G

(0)

t (s)

Fig. 9 Normalized stress relaxation curve in terms of G(t)/G(0)vs. time at different applied strains

at 5 to 2/5 at 100 kPa. In some sense, this power-lawdecrease confirms that the sample responds like a solidinitially. It is indeed reasonable for γ̇ to drop as thesystem becomes more jammed during creep over time.

However, beyond a certain strain, which depends onthe magnitude of applied stress, the shear rate rises up,as shown in the inset of Fig. 6. If we are to identify thispoint as a point of yielding, we find the relationshipbetween the applied shear stress and the strain at theturning point (σy, γy) to obey the same scaling as thecharacteristics of the stress overshoot in rate-controlledstartup shear as shown in Fig. 7.

Step strain

In this section, we study stress relaxation behaviorafter step strain. Because G′ and G′′ are approximatelyparallel to each other in a wide range of frequency, stepstrain experiment can be carried out in a broad range ofshear rates. Figure 8 shows step strain data produced ata shear rate of 1 s−1, where the strain amplitude variesfrom 0.1 to 2.0. Unlike well-entangled monodisperselinear polymers, there is no stress plateau, and thestress declines continuously even in the linear responseregime. The stress relaxation behavior can be bestdepicted by the normalized relaxation modulus asshown in Fig. 9. Up to γ around 0.2, the normalizedstress nicely overlaps. Above γ = 0.2, the stress relaxesfaster with increasing γ .

Discussion

We have taken the point of minimum rate in the inset ofFig. 6 as the beginning of yielding. This interpretation

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stems from our past work on creep behavior of well-entangled polymer solutions (Tapadia and Wang2004; Ravindranath and Wang 2008). When the ap-plied stress is low enough, we can access terminal flowregime in a monodisperse linear polymer, where themeasured apparent rate decreases towards a constant.Beyond a certain shear stress, the apparent would risedramatically after the initial decrease, leading to aneventual state of much lower viscosity. We have phe-nomenologically termed this transformation a yield-like entanglement–disentanglement transition. Sincethe present paste of nano-polybutadiene-particles doesnot have a terminal flow regime, the shear rate willalways rise after the initial drop (due to the solid-likeresponse) at all applied stresses. For the present pastemade of the nano-polybutadiene-particles, the transi-tion from elastic deformation to flow, i.e., yielding,occurs when the particles hop over one another. Thismerely involves a strain on the order of unity. Theyielding at a higher strain for a higher applied rateor stress appears to suggest some level of interpene-tration of brushes from the neighboring particles. Ifthe nanoparticles were non-interpenetrating and non-deformable, we would expect the yielding to occur at acommon strain value independent of the applied stressor rate.

In any event, the fact that both characteristics foryielding resulted in the same scaling law in Fig. 7 sug-gests that perhaps the underlying physics is commonlyshared. This quadratic dependence is rather differentfrom the linear relationship between σy and γy thathas been uncovered for entangled linear chain systems(Boukany and Wang 2009b). So the yielding mecha-nism here is clearly different, which involves particlesovercoming the caging effect to hop over one another.

The faster relaxation at higher strains in Fig. 9 maybe associated with the structural loosening since thesystem has experienced yielding. This structural re-arrangement reveals a softer system since the stressgrowth starts to deviate from the initial slope beyondγ = 0.2 as shown in Fig. 8. This level of strain ∼0.2is consistent with that identified in creep as shown inFig. 7.

It is important to indicate that the relaxation at allstrains occurs quiescent according to the in situ PTVobservations in contrast to entangled polymeric liquidsmade of linear chains (Wang et al. 2006; Ravindranathand Wang 2007; Boukany et al. 2009). This quiescentrelaxation is expected because the system does notpossess a measurable slip length. In other words, theabsence of wall slip and dominance of homogeneousshear are ensured by the dense packing of nanoparticlesthat gives rise to a sizable viscosity and consequently

negligible magnitude of slip length. Figure 4 shows thatin our range of observation there is barely a three-orders-of-magnitude of drop in viscosity, which is adecrease clearly insufficient to produce visible wall slipand shear banding.

Summary

We have carried out rheometric and particle-trackingvelocimetric measurements of a strongly viscoelasticmaterial made of nano-sized soft particles. This pasteof particles packs densely and jams up in absence ofsolvent to behave rheologically like a gel. Analogousto granular materials, it does not exhibit any termi-nal flow behavior. Yet, in contrast to entangled linearpolymers, the nanoparticle melt yields and flows ho-mogeneously during startup simple shear. Unlike linearchain systems, this melt, made of 27-arm star-like parti-cles, derives its elasticity from close packing instead ofnetworking through chain entanglement. As a conse-quence, it cannot sustain much strain without structuralrearrangement. Large step strain experiments showthat the yielded structure possesses faster relaxationdynamics and relaxation proceeds quiescently. As amolecular fluid or a gel, the present paste is incapableof displaying shear banding, wall slip, and elastic yield-ing (i.e., non-quiescent relaxation) that are perhapsunique characters of linear chain networks.

Acknowledgement This work is supported, in part, by a grantfrom the National Science Foundation (DMR-0821697).

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