at SciVerse ScienceDirect

Polymer 54 (2013) 4762e4775

Contents lists available

Polymer

journal homepage: www.elsevier .com/locate/polymer

Feature article

Linear viscoelasticity and dynamics of suspensions and moltenpolymers filled with nanoparticles of different aspect ratios

Philippe CassagnauUniversité de Lyon, Univ Lyon 1, CNRS, Ingénierie des Matériaux Polymères (IMP UMR 5223), 15 Boulevard Latarjet, 69622 Villeurbanne Cedex, France

a r t i c l e i n f o

Article history:Received 15 February 2013Received in revised form1 May 2013Accepted 9 June 2013Available online 18 June 2013

Keywords:SuspensionNanofillersViscoelasticity

E-mail address: philippe.cassagnau@univ-lyon1.fr.

0032-3861 � 2013 Elsevier Ltd.http://dx.doi.org/10.1016/j.polymer.2013.06.012

Open access under CC B

a b s t r a c t

In the present review, we report the linear viscoelasticity of suspensions and polymers filled with nano-size particles of different aspect ratios and structuration. The viscoelastic behaviour of liquid suspensionfilled with well-dispersed and stabilised particles proves that the Brownian motion is the dominantmechanism of relaxation. Accordingly, dilute and semi-dilute suspensions of stabilised carbon nanotubes,cellulose whisker and PS nanofibres obey a universal diffusion process according to the DoieEdwardstheory. Regarding spherical particles, the KriegereDougherty equation is generally successfully used topredict the zero shear viscosity of these suspensions. Regarding fractal fillers, two categories canbe considered: nanofillers such as fumed silica and carbon black due to their native structure; andsecondly exfoliated fillers such as organoclays, carbon nanotubes, graphite oxide and graphene. Theparticular rheological behaviour of these suspensions arises from the presence of the network structure(interparticle interaction), which leads to a drastic decrease in the percolation threshold at which thezero shear viscosity diverges to infinity. Fractal exponents are then derived from scaling concepts andrelated to the structure of the aggregate clusters. In the case of melt-filled polymers, the viscous forcesare obviously the dominant ones and the nanofillers are submitted to strong orientation under flow. It isgenerally observed from linear viscoelastic measurements that the network structure is broken up underflow and rebuilt upon the cessation of flow under static conditions (annealing or rest time experiments).In the case of platelet nanocomposites (organoclays, graphite oxide), a two-step process of recovery isgenerally reported: disorientation of the fillers followed by re-aggregation. Disorientation can beassumed to be governed by the Brownian motion; however, other mechanisms are responsible for the re-aggregation process.� 2013 Elsevier Ltd. Open access under CC BY-NC-ND license.

1. Introduction

The rheology of the suspensions started with the famous workof Einstein in 1905 and 1911 on the prediction of the viscosity ofhard-sphere suspensions at low particle concentrations. In thisdilute regime, the hydrodynamic disturbance of the flow fieldinduced by the particles in the suspending liquid leads to an in-crease in the energy dissipation and an increase in the relativeviscosity, hr, according to the equation (1):

hr ¼ h

hs¼ 1þ 2:5f (1)

where h is the viscosity of the suspension, hs the viscosity of thesuspending liquid and f the volume fraction of particles. In fact, the

Y-NC-ND license.

constant 2.5 is the intrinsic viscosity ð½h� ¼ limf/0

ðh� hs=fhsÞÞ, andtheoretically for rigid spheres [h] ¼ 2.5.

In the dilute regime of hard particles, the interparticle forces arenegligible compared with hydrodynamic forces and Browniandiffusion. In other words, there are no attractions between theparticles but only an excluded volume effect. At higher concentra-tions the probability of particle collisions increases so that the hy-drodynamic interactions become the dominant ones. As a result theEinstein law fails since a significant positive deviation of the rela-tive viscosity is observed. Furthermore, a shear thinning behaviourof the viscosity is observed with increasing volume fraction ofparticles. There are numerous models available for the descriptionof the rheology of suspensions of spherical spheres, which is still anopen domain of theoretical investigations [1]. However, the semi-empirical equation of Krieger and Dougherty [2] for mono-dispersed suspensions is one of the most used and developed inthe literature. The concentration dependence of the zero shearviscosity is expressed as:

P. Cassagnau / Polymer 54 (2013) 4762e4775 4763

h0 ¼ hs 1� f

fm

�½h�fm

(2)

� �

Where fm is the maximum packing fraction of particles. At thisconcentration, the zero shear viscosity rises to infinity and thesuspension exhibits a yield stress behaviour. The KriegereDough-erty equation can be simplified to the Quemada equation [3] h0 ¼hsð1� ðf=fmÞÞ�2 since it generally observed that [h]fm ¼ 2 in mostusual suspensions.

In model systems of monodispersed hard spheres, the criticalvolume fraction (fm z 0.63) reported by Bicerano et al. [4] andSmith and Zukoski [5] for the approach of the zero shear viscosity toinfinity is near the random close packing value of 0.64. The rightparticle fraction atwhich the zero shear viscosity diverges to infinityis still an open debate as most of the experimental results are farbelow the theoretical packing value. The KriegereDoughertymodelis generally used as a mathematical equation, where fm and [h] aretwo fitting parameters, to numerically fit the concentrationdependenceof the zero shear viscosity. Togive aphysicalmeaning toboth parameters, a common way is to derive fm as an effectivemaximum packing fraction fm,eff and [h] is related to the deviationfrom ideal spheres in terms of aspect ratio of the particles. The valueof fm,eff is no more a universal value as it has to be determined foreach suspension. In fact, fm,eff includes deviation from ideal parti-cles, as deformable spheres for example. On the other hand, aggre-gate particles can be expressed from an effective volume fractionfeff. Regarding the particle aspect ratio, Barnes [6] proposed thefollowing formula for the intrinsic viscosity ½h� ¼ 0:07ðL=dÞ5=3 and½h� ¼ 0:3ðL=dÞ for rod-like and disc-like particles respectively (L:longest length and d ¼ diameter or thickness respectively). Evenwithout any physicalmeaning, the KriegereDougherty equation is auseful tool to visualise at a glance the rheological trend of suspen-sions. In some cases (latex for example), it is of importance for usefuldevelopment from the point view of process-engineering applica-tions to have the lowest viscosity for the highest solid content. Fromequation (2) and keeping constant the viscosity of the suspendingliquid, the only way is to increase the packing fraction fm. This isgenerally achieved by using a bi-modal distribution of particles [7].In contrast, in some applications like suspensions filled withconductive fillers, the lowest percolation threshold is required interms of formulation costs and weight saving. The KriegereDougherty equation shows that particles with high aspect ratio (anincrease of [h] and/or with aggregation (feff > f) including fractalparticles) are required for such applications. Consequently, thephysic of suspensions and its extension to molten filled polymers isextremely complicated as the simple case of inert and rigid spheresat low concentration is generally far from real life. The suspensionsand filled polymers are the world of colloidal particles as they offermany possibilities of material developments. In the colloidal sizedomain, the Brownian forces, direct interparticle forces and viscoushydrodynamic forces are all of comparable magnitude. In somecases, the particleeparticle interactions play a dominant role inthese systems and results in aggregation or flocculation withpossible fractal organisation of such clusters. In fact, the balance ofthe different interaction can be tuned from a judicious chemicalmodification of the filler surface and then open a marvellous worldfor the control of the target rheological behaviours. Therefore, therheology of colloidal dispersions exhibits a rare diversity and hasbeen the subject of several publications and reviews. The last one ofgreatest and practical interest was published by Genovese [8]. Thispaper reviews the shear rheology of suspensions and matrix poly-mers filled with microscopic and colloid particles. The shearrheology of the suspensions has been discussed from the KriegereDougherty equation as it is well known, with some simple modifi-cations taking into account the deviation from ideal cases, to

effectively predict the concentration dependence of many types ofsuspensions. In the case of aggregated suspensions, the authorsdemonstrated the analogy between several theoretical modelsdevoted to yield stress and elastic modulus of these gels.

If the shear rheology (steady shear flow experiments) is welldescribed in the literature for suspensions, the linear viscoelas-ticity under oscillatory measurements is more limited in terms ofpublications. However, the linear viscoelasticity has beenextremely used to characterise the viscoelastic properties ofpolymer melts filled with nano-sized fillers (nanocomposites). Adirect consequence of filler incorporation in molten polymers isthe significant change in viscoelastic behaviour as they are sen-sitive to the structure, concentration, particle size, shape (aspectratio) and surface modification of the fillers. As a result, rheolog-ical methods are useful and suitable to assess the quality of fillerdispersion [9]. In recent years, nearly all types of nano-fillers havebeen used for the preparation of nanocomposites: organoclays,carbon black, fumed and colloid silica, carbon nanotubes, cellulosewhiskers, metallic oxide, etc. and more recently graphite oxideand graphene. From a literature survey, thousands of papers andreviews have been published in this broad scientific area andconsequently we can only cite the most recent reviews [10e14].From a physical point of view, nanoparticles in suspending fluidare submitted to particleeparticle forces, particleefluid in-teractions, viscous forces under flow and finally Brownian forces.The Brownian motion arises from thermal randomising forces thatlead to the dispersion of the nanoparticles. Consequently, Brow-nian motion is ever present even in highly viscous systems(entangled polymer melts). The aim of the present paper is toreview some linear viscoelastic behaviours of suspensions andmolten polymer filled with nanoparticles of different aspect ratiossuch as spheres, platelets and nanotubes (or nanofibres). Actually,the viscoelastic and dynamic behaviours have been discussed foreach system taking into account the dispersion at different scalesof nanoparticles as it was reported in the corresponding papers.Finally, this review is addressed in terms of a comprehensive studyof the viscoelastic behaviour and modelling from the Browniandiffusion of nanoparticles.

2. Particle diffusivity

In the dilute regime of suspensions, the particles can rotateabout their centre of mass. The particles are then able to rotatefreely without any interference interactionwith neighbouring ones.The particle diffusivity in this dilute regime is controlled by theBrownian forces (wkBT, kB is the Boltzmann constant) in the sus-pending liquid (viscosity hs), which exerts the Stokes friction(w6phsR) on the particle (radius of the particle R) and is written as:

D0 ¼ kBT6phsR

�m2s�1

�(3)

This equation is also known as the StokeseEinstein law. Therotary particle diffusivity (Unity: s�1) has been derived for non-spherical particles as following [15]:

Particles of nearly spherical shape (diameter d):

Dr0 ¼ kBTphsd3

(4)

Spheroid particles of the longest length L:

Dr0 ¼3kBT

�ln�2 Ld

�� 0:5

�phsL3

(5)

Platelet or circular disc-like particle of diameter d:

P. Cassagnau / Polymer 54 (2013) 4762e47754764

Dr0 ¼ 3kBT4hsd3

(6)

Rods or nanofibres of length L and diameter d:

Dr0 ¼3kBT

�ln�Ld

�� 0:8

�phsL3

(7)

In the semi-dilute regime, the particles are no longer able todiffuse or rotate freely without any interaction from surroundingones. For rigid spheres or disc-like particles, this transition is notgenerally well defined. However, the onset of the semi-diluteregime could be the concentration at which the concentrationdependence of the viscosity deviates from the Einstein law (w3%).Consequently, the diffusivity is expected to be slowed down asproved by Shikata and Pearson [16].

For rods, this transition (dilute/semi-dilute) occurs when therod concentration, expressed as n the number of objects per unitvolume ðf ¼ ðp=4Þd2LnÞ, reaches a value proportional to 1/L3.However, scaling developments [17,18], showed that the rotationalinterference becomes significant when:

nL3zb (8)

where b is a dimensionless constant. Experimentally, this constantwas found [19] to be roughly close to b z 30.

The molecular dynamics of straight rigid rods in suspensionhave been theoretically investigated by Doi and Edwards [20].Regarding the rotary diffusivity Dr of a rod in an isotropic suspen-sion, the DoieEdwards theory for semi-dilute regime predicts:

Dr ¼ ADr0

�nL3

��2(9)

where A is a dimensionless constant whose values are generallylarge. From theoretical [21] and simulation [22] considerations, thevalues of this constant have been found to be roughly w1000 forperfectly rigid rods. Experimentally, lower values of A have beenobserved for carbon nanotubes [23], A w 400 and nanofibres [24],A w 200.

When the concentration of the rods is higher than n > 1/dL2

(concentrated regime), the concentration of the rods is such thatthe rod diameter d is large enough compared to the distance be-tween two consecutive neighbouring rods. Consequently, theexcluded volume of interaction can no longer be neglected. Justabove the onset of this concentrated regime, the solution remainsisotropic at equilibrium (concentrated isotropic regime). At higherconcentration (n > 4.2/dL2), the rods can spontaneously orient intoa nematic crystalline phase (nematic regime). The molecular dy-namic of the rods is no longer Brownian.

To conclude, the diffusivity of particles depends on their shapeand volume concentration in the suspension. In fact, the diffusivityis considerably slowed down in the semi-dilute regime of the sus-pensions. On the other hand, from the StokeseEinstein law (equa-tion (3)), the diffusivity is inversely proportional to the viscosity ofthe suspending liquid. Consequently, the particle diffusivity isstrongly dependent of the suspending liquid. From a magnitudepoint of view, the diffusivity can be one-million-times lower in amelt polymer (hs w 103 Pa s) than inwater (hs ¼ 10�3 Pa s). A priori,the Brownian motion of nanoparticles can be neglected in moltenpolymer. However, we will show that this Brownian motion can beat the origin of some aggregation mechanisms at long rest time inhighly viscous polymer melts. The following parts of the presentpaper are devoted to the linear viscoelasticity of suspensions in lowviscous fluids like water (latex for example) or non-entangled

polymers (typically hs < 1 Pa s) and in entangled molten polymers(typically hs > 100 Pa s).

3. Liquid suspensions

3.1. Hard-spheres

Thefirst experimentson the linear viscoelastic properties of hard-sphere suspensions were reported first by van der Werff et al. [25]and Shikata and Pearson [16]. These results have been reviewed byLarson inhis book [15]. A terminal relaxation zone (G0au2 andG00au1)is well depicted for the suspensions (Silica particles of 60, 125 and225 nm) in a mixture of ethylene glycol and glycerol investigated inthese works. Furthermore, a master curve on concentration depen-dence ofG0 andG00 � h0NuaT versus the reduced frequencyuaTswwassuccessfully plotted. h0NðfÞ ¼ lim

u/NG00

.u is derived from the high

frequency behaviour. sw is the longest relaxation time, sw z 0.5sp.From Shikata and Pearson [16], the particles cannot move withoutgenerating lubrication forces with other neighbouring particles. Therelaxation time, assimilated to the diffusion time, is expected to beinversely proportional to the particle diffusivity:

spðfÞ ¼ R2

6DsðfÞ (10)

Note that the diffusion time for a particle, i.e. the time to diffusea distance equal to its radius, is given by equation (3)tp ¼ ð6phsR=kBTÞ. In other words and as explained by Larson, theelasticity of these suspensions is produced by the Brownian motionof the particles, which tends to restore the isotropic particleconfiguration under flow. By extending the StokeseEinstein law tothe high frequency domain, Shikata and Pearson [16] postulated:

DsðfÞ ¼ kBT6pRh0NðfÞ : (11)

The dependence of Ds(f)/D0 on the particle concentration hasbeen theoretically predicted by Beenakker [26] and Beenaker andMazur [27]:

DsðfÞ=D0 ¼ hsh0N

(12)

with

h’Nhs

¼ 1þ 52fð1� fÞ�1

We reported in one of our previous works [28] the viscoelasticproperties of two concentrated polystyrene lattices with differentelectrostatic properties around the high critical concentration. TwoPS lattices have been then synthesised and named PS and PSSrespectively. The major difference between the two lattices is thepresence of strong acid groups on the surface of PSS. These acidgroups change the degree of the hydrophilicity of the particlesurface. The diameter of these particles is nearly dw 200 nm. Froman experimental point of view, the complex shear modulus cannotbe measured at particle concentrations lower than 35%. The vari-ation of the complex shear modulus (G*(u) ¼ G0(u)þjG00(u)) versusfrequency for PS lattices is shown in Fig. 1. At lower concentrations(f < 55.3%), the terminal zone of relaxation is clearly depicted, andG0 and G00 are proportional to u2 and u1 respectively. The crossoverfrequency uc where G0 ¼ G00 is representative of a characteristicrelaxation time, sp ¼ 1/uc, which corresponds to the time for aparticle to diffuse a distance equal to its radius. This mean

Fig. 2. Variation of the complex shear modulus versus frequency for three typical PSfractions.a) PS latex and b) PSS latex. Filled symbol: loss modulus and open symbol:storage modulus. From Pishvaei et al. [28], with permission from Elsevier.

Fig. 1. Viscoelastic behaviour of latex for different PS volume fractions (%). Variation ofthe complex shear modulus versus frequency. a) storage modulus and b) loss modulusversus frequency. From Pishvaei et al. [28], with permission from Elsevier.

P. Cassagnau / Polymer 54 (2013) 4762e4775 4765

relaxation time is thus generally observed to be inversely propor-tional to the particle diffusivity Ds(f) according to equation (10).

The values of diffusion time and particle diffusivity are reportedin Table 1. As expected, the particle diffusivity is considerablyslowed down with increasing particle concentration.

The maximum packing fraction can be defined as the solidcontent at which the system begins to behave like an elastic solid,i.e. with G0 > G00 over the entire experimentally accessible frequencyrange, and where bothmoduli are nearly constant (i.e. independentof frequency). At this critical concentration, one can say that theparticle diffusion time tends to infinity (at least in the experi-mentally accessible frequency domain). This critical concentrationcan be assumed to be the maximum packing fraction fm. In thepresent system, fm must therefore be near 55.3%.

Surprisingly, a quite different trend was observed for the PSSlatex and these results contrast with those reported for PS samples.Fig. 2a, b (PS and PSS respectively) describe the viscoelastic

Table 1Diffusion time and particle diffusivity versus the concentration of PS particles.

f (%) sp (s) Ds (m2 s�1) Ds/Do

48.6 0.008 8.3 � 10�13 0.7553.3 0.02 3.3 � 10�13 0.3054.7 0.5 1.3 � 10�14 0.0155. 1 1.0 6.7 � 10�15 0.006

behaviour of the lattices far below the critical concentration,around this critical concentration, where the viscoelastic behaviourdrastically changes, and far beyond the critical concentration. ForPSS suspensions, a liquidesolid transition is observed in Fig. 2a atthis critical concentration as the complex shear modulus behavesas G0(u)fG00(u)fun with n ¼ 0.5. There is no evidence for such atransition from liquid-like to solid-like behaviour for PS suspen-sions. This result means that the fractal criterion of self-similarrelaxation as defined by Winter and Mours [29] can be applied tothe PSS suspension. For the PS suspensions, the network elasticityresulting from the particle interactions appears to be temporarilyelastic-like in entangled polymers. Actually, there is no arrange-ment of the particles in terms of particle aggregation and clus-tering. These different viscoelastic behaviours observed for PSS andPS can be attributed in large part to the fact that PSS has moreelectrically charged groups on its surface, therefore in an aqueousmedium it has more long-range interactions leading to a fractalbehaviour.

3.2. Nanotubes or nanofibres

As pointed out by Hobbie [30] in his recent review on shearrheology of carbon nanotubes (CNT), the main feature of semi-dilute suspensions of CNT under weak shear is a flow-inducedaggregation and finally macroscopic agglomeration [31]. It can beassumed that the driving mechanism is due to the attractiveinterparticle interactions that favour aggregation. In fact, under

Fig. 3. Variation of the storage modulus (a) and variation of the electrical conductivity (b) under different shear deformationsg0 ¼ 5, 30, 60, 100, 200 and 600%. From Moreira et al.[32]), with permission from Macromolecules.

P. Cassagnau / Polymer 54 (2013) 4762e47754766

Brownian motion and collisions induced by a weak shear flow, thenanotubes collide and stick to form at the end of the processmacroscopic clusters with a size comparable to the confining cellgap. When the nanotubes have finally formed a physical network,the thermal Brownian forces are insufficient to overcome thesephysical contacts. Only shear deformation above a critical intensitycan break up the network and disperse the nanotubes in the sus-pending liquid. The final isotropic dispersion of nanotubes isreached by their Brownian motion. In other words, this strikingphenomenon of CNT is totally reversible and can be controlled bycontrolling the rest time and the shear intensity. It must be pointedout that this reversibility can only be possible in low-viscosityfluids for which the Brownian diffusion time is short enough, i.e.in the same order of magnitude as the experimental time. From anexperimental point of view, a reasonable experiment time,including rest time, is one or two days. This discussion will bedeveloped later in highly viscous polymers.

This aggregation/clustering phenomenon was clearly evidencedin our recent work [32]. Simultaneous time-resolved measure-ments of electrical conductivity and dynamic shear modulus underdifferent strains of untreated CNTs (L z 1.5 mm, d z 10 nm), in alow-viscosity PDMS (h ¼ 5 Pa s, not entangled chains) matrix wereinvestigated as shown in Fig. 3. The concentration of nanotubes(f ¼ 0.2%) was chosen in order to be in the semi-dilute regime ofthe suspensions.

First of all, it is assumed that the nanotube dispersion at t ¼ 0 ofthe experiment is isotropic. At low shear rate the CNT aggregationrate increases with the shear dynamic rate until a critical value(gu0 w 1). Beyond this critical value, the level of network

Fig. 4. Optical observation of the aggregation of CNTs. a) the CNTs are well dispersed in PDM(Dt ¼ 24 h). From Moreira et al. [32], with permission from Macromolecules.

entanglement and/or contacts decrease as the shear rate increases.This behaviour can be explained in terms of the changing structuresuch as clusterecluster collision and CNT network formation [33e38]. The network structure is given by the balance of agglomerategrowth and destruction under shear deformation as quantitativelydemonstrated by Ma et al. [39]. This phenomenon of CNT clusteraggregation under low shear was simply proved by optical micro-scopy (Fig. 4a, b). After a period at rest of a few hours, the aggre-gation (clustering mechanism) of the CNTs is clearly seen. It can bepointed out that the scale of this clustering is relatively macro-scopic since it can be shown by simple optical microscopy.

It must be pointed out that such phenomenon on the influenceof shear on network formation or break up has been recently re-ported in the case of nanocrystalline cellulose suspensions. Indeed,Derakhshandeh et al. [40] observed that the increase of the vis-cosity with time was promoted at lower shear rate or under linearoscillatory shear deformation. They attributed this behaviour to anincreasing effect of Brownian motion under small deformationwhich facilitates fibres rearrangements.

Finally, these studies make the statements of Huang et al. [41]about the questioning aspect arising from some literature studiesabout the state of nanotube dispersion of many rheological exper-iments even more relevant. Fig. 5 shows that the viscoelasticbehaviour strongly depends on nanotube dispersion. When thenanotubes are supposed to be well dispersed in an isotropicconfiguration, the variation of the complex shear modulus (under5% strain) shows a rheological behaviour close to the viscoelasticbehaviour of the PDMS matrix, i.e. a liquid viscoelastic behaviour.After the aggregation process, the viscoelastic behaviour of the

S matrix (only micron-impurities are seen), b) the CNTs aggregate to build a network

Fig. 5. Variation of the complex shear modulus of the PDMS/CNT suspension. Filledsymbol: viscoelastic behaviour measured just after the compressive deformation, theCNTs are supposed to be well dispersed in the PDMS matrix. Open symbols: Visco-elastic behaviour measured at the end the aggregation process under g0 ¼ 30%. FromMoreira et al. [32], with permission from Macromolecules.

P. Cassagnau / Polymer 54 (2013) 4762e4775 4767

composite shows a solid-like behaviour, at least in the frequencywindow used in the present study, due to the formation of theCNT network from dynamic CNT aggregation. In the same way,Hobbie and Fry [41] reported a universal phase diagram (meancluster size versus stress) for the evolution from a solid-likenetwork to flowing.

Finally, it can be concluded that it is quite difficult to measurethe linear viscoelasticity of finely dispersed carbon nanotubes asthey are submitted to aggregation and clustering during measure-ment even under linear deformation. For example, the deviation ofthe rotary diffusivity to the Doi-Edward theory in the semi-diluteregime is related to a clustering phenomenon of CNTs that pro-duces additional slowing of the rotational motion [23]. Obviously,introducing functional groups onto the CNT surface is one of themost effective methods to enhance dispersion of CNT and to pre-vent their aggregation in suspending liquid. Consequently, theDoieEdwards theory has been rarely applied to semi-dilute solu-tion of carbon nanotubes. This is mainly due to the difficulties toaccess the terminal relaxation zone and to proceed with finedispersion of CNT in solvent as previously explained. However, thepioneering works of Hobbie and Fry [42] and Fry et al. [43] on rheo-optical studies of carbon nanotube suspensions under shear flowshowed that the DoieEdwards theory can be successful at lowPeclet number (Pe < 200, Pe ¼ ð4=pÞ2ððL=dÞ4=ADroÞ _gf2). In fact,their results are in agreement with the theory prediction as theyobserved a leading order scaling relation for the optical anisotropy

0.01

0.1

1

10

100

1000

0.01 0.1 1 10 100

G' (P

a)

Frequency (rad.s-1

)

Suspension A

13.3%

9.7%

7.3%

6.2%

5.5%

3.7%

2.2%

0.4%

Fig. 6. Overlay of the variation of the storage modulus G0(u) (left) and loss modulus G00(u)from Macromolecules.

such as Pe0.25. However, Ma et al. [44] proved that amild elasticity isstill observed despite CNT functionalisation. This elasticity is due tothe presence of a weakly interconnected CNT network. In otherwords, it seems that CNT cannot be dispersed at the scale of theprimary entity in the semi-dilute regime. At least, the nanotubesbehave like rod polymers in solutions which have been observed([45]) to form small aggregates in dilute regime. Petekidis et al. [45]studied such the association dynamics of hairy-rod (poly(p-phe-nylene) with flexible dodecyl side chains) in toluene. They observedfrom static and dynamic light scattering that the stiff polymermolecules are assembled into small aggregates (three to fourmolecules in aggregate) in dilute suspension and form largerclusters in semi-dilute regime. Actually, the molecules associationinto aggregates or clusters is governed by the balance of polymerepolymer and polymeresolvent interactions. The conclusion isfinally that toluene is not a good solvent at room temperature ofthese hairy-rod polymers.

Fortunately, new nanoparticles of high aspect ratio such asnanorods or nanofibres based on polymeric materials have beensuccessfully developed [46e48]. For example, the synthesis ofpolystyrene (PS) nanofibre aqueous dispersions was achieved viapolymerisation-induced self-assembly. The method leads to self-assembled amphiphilic block copolymer nanofibres directly inwater at high concentrations (>20 wt%). On the other hand, self-stabilised polystyrene nanofibres were again obtained [49] viapolymerisation-induced self-assembly in water from a P(MAA-co-PEOMA) hydrophilic precursor and their hydrophobic core wascrosslinked by the addition of a difunctional comonomer, di(eth-ylene glycol) dimethacrylate (DEGDMA), at >85% of conversion ofstyrene. Typically, nanofibres of diameter and length nearlyd z 50 nm and L z 3.5 mm were synthesised. Due to their hydro-philic surface, these nanofibres are finely and well dispersed inwater. As a result, aggregation and/or clustering are prevented inthe semi-dilute regime. Actually, PS fibres behave as rigid nanorodssubmitted to their Brownian motions in the water suspendingliquid. More precisely and as shown in Fig. 6, the suspensionsexhibit a clear terminal zone (G0au2 and G00au1) of the relaxationtimes for f < 6.2%. These experimental results prove that a Brow-nian motion of nanofibres is the dominant mechanism of relaxa-tion. As expected, this terminal relaxation zone is shifted towardslower frequencies with the increase of the nanofibre concentra-tions. Indeed, the diffusion time of the nanofibres is slowed downdue to the neighbouring nanofibres.

At high concentration, typically f> 7.3%, the terminal relaxationcannot longer be measured from the investigated frequency range.The dominance of G0 over G00 indicates that the nanofibre relaxation

0.001

0.01

0.1

1

10

100

0.01 0.1 1 10 100

G" (P

a)

Frequency (rad.s-1

)

Suspension A

13.3%

9.7%

7.3%

6.2%

5.5%

3.7%

2.2%

0.4%

(right) versus frequency for the suspension A. From Zhang et al. [24], with permission

-4,5

-4

-3,5

-3

-2,5

-2

-1,5

-1

-0,5

0

0 1 2 3 4

-2

log(

Dr/

Dr0

)

log(νL3)

Fig. 8. Variation of the rotary diffusivity (equation (9)) versus the concentration ofnanorods. Same symbols as in Fig. 7. :: CNTs in PDMS, A: Cellulose Whisker, A:Polymer nanofibre (T ¼ 25 �C), ,: Crosslinked polymer nanofibre.

P. Cassagnau / Polymer 54 (2013) 4762e47754768

time is longer than experimentally accessible. However, it seemsthat the highly concentrated suspensions are in the isotropicconcentrated regime where the Brownian motions of the nano-fibres cannot exist anymore. The network elasticity resulting fromthe nanofibre interactions appears to be temporarily elastic-like inentangled polymers. Note that such behaviour was previouslyobserved for hard-sphere PS suspensions. Fig. 6 show clearly thisargument as there is no evidence for a transition from liquid-like tosolid-like behaviour. Finally, the temporary elasticity, associatedwith the Brownian motion of nanofibres, appears to be the relevantviscoelastic behaviour.

In terms of Brownian motion, the rods are able to rotate freely inthe dilute regime without any interference interaction withneighbouring rods. According to the DoieEdwards theory, the ro-tary diffusivity Dr0 behaves as Dr0ff0 so that the concentrationdependence of the zero shear viscosity for the dilute regime(f < f*) obeys the following scaling law:

h0 ¼ hs þ2nkBT15Dr0

h0 � hsff1 (13)

In the semi-dilute regime the free rotation is hampered byinteraction with the surrounding fibres and the rotational diffu-sivity Dr, has been predicted to follow the power law Dr0f(nL3)�2.Consequently, in the semi-dilute regime (f > f*), the zero shearviscosity scales according to equation (14):

h0 ¼ nkBT10Dr

h0ff3 h0=hsf�nL3

�3(14)

It must be noted that in semi-dilute regime, rod flexibility,Brownian motion and van der Waals attractions strongly influencethe zero shear viscosity of the suspensions [50]. However, the DoieEdwards theory can be applied to any nanorod or nanofibre sus-pension provided that these particles exhibit a nanometric scale inorder to behave as solid Brownian entities in the suspending liquid.Consequently and from the equation derived by Doi and Edwards,the variation with nL3 of the reduced viscosity (equation (14) andFig. 7) and of the rotary diffusivity (equation 9 and Fig. 8) shouldboth obey a master curve. In other words, the Brownian motion ofnanorods (or nanofibres) must be universal and hence independentof their nature. The dynamic behaviour of different nanorod dis-persions (carbon nanotubes [23], cellulose whiskers [51]), polymer

0

1

2

3

4

5

6

0 1 2 3 4

log(νL3)

log(

η 0/η

s) 3

Fig. 7. Variation of the reduced viscosity (equation (14)) versus the concentration ofnanorods. :: CNTs in PDMS, from Marceau et al. [23], A: Cellulose Whisker, fromBercea and Navard [51], >: Polymer nanofibre (T ¼ 25 �C), from Zhang et al. [24], ,:Crosslinked polymer nanofibre (T ¼ 25 �C), from Zhang et al. [49].

nanofibres [24,49], was recently discussed [52]. Interestingly, inFigs. 7 and 8, it can be observed that, indeed, CNTs, cellulosewhiskers and polymer nanofibres obey a master curve in agree-ment with the DoieEdwards theory. This universal result meansthat the dynamic of these different rods are dominated by theBrownian forces.

3.3. Fractal fillers

The native specific structure of fractal fillers (from nano- tomicro-scale) such as fumed silica and carbon black, gives specificviscoelastic properties to their derivative suspensions. Exfoliatedfillers such as organoclays, carbon nanotubes and more recentlygraphene can also be classified in fractal fillers due to the long-rangeconnectivity that arises from interparticle physical interactions. Forexample, the exfoliation of organoclay layers increases the numberof frictional interactions between the layers. The cluster and/oraggregate structures can be viewed as an assembly of primaryparticles in a structure having a fractal dimension [53]. Due to theirfractal structure and their high specific area, these fillers are sub-jected to form a network of connected or interacting particles in thesuspending liquid. Compared with colloidal fillers without aggre-gation effect (our previous discussion), a direct consequence of theincorporation of fractal fillers in suspensions is the significantchange in the rheological and linear viscoelastic properties [54e58].Theoretically, the percolation threshold, fc, for a spherical spherewithout any interaction should be obtained at the close packingvolume fraction of fm ¼ 0.64. From a practical point of view, thisparticular rheological behaviour arises from the presence of thenetwork structure (interparticle interaction), which leads to adrastic decrease in fc, typically fc < 0.05 compared with fm ¼ 0.64.Above fc the storage modulus exceeds the viscous modulus and isalmost frequency-independent G0 ¼ lim

u/0G0ðuÞ. However, it is well

known that this gel structure is significantly altered by the surfacemodification of silica particles as recently discussed [59,60].Generally speaking, the surface of native fumed silica consists in partof silanol groups that [55,61]. The balance of these interactions re-sults in the stability of the suspension and the gelation behaviour.Consequently, the viscoelastic properties can be tuned by control-ling the balance of the hydrogen interactions as the silanol surfacegroups can easily be modified with different functional groups ofdifferent molecular lengths from monomer to polymer [62].

From the theoretical viewpoint, the scaling concept, based ona fractal dimension, is generally used to study the effect of

Fig. 10. Schematic diagram of the restructuring of the fumed silica aggregates fromchain-like structure to closely packed structure. From Wu et al. [60], with permissionfrom Soft Matter.

P. Cassagnau / Polymer 54 (2013) 4762e4775 4769

interparticle forces on the elasticity of aggregated suspensions[63e66]. To model the size dependence, the aggregates ofnanoparticles are described as a fractal structure with a charac-teristic size x, which is the radius of the smallest particle con-taining N primary particles of radius a.

NðxÞz�x

a

�df

(15)

with df the fractal dimension of the aggregate. The volume fractionof the primary particle inside the aggregate is then:

fz

�x

a

�df�3

(16)

Finally, the variation of the equilibrium storage modulus G0versus the volume fraction of silica can be expressed according tothe characteristics of the non-fluctuating fractal structure [65] asfollows:

GofF5=3�df (17)

Where df is the fractal dimension of silica clusters. Wolthers et al.[65] found df ¼ 2.25 for stearyl-coated silica particles. Paquien et al.[67] observed a fractal dimension df equal to 2.3 for unmodifiedsilica/PDMS composites. Furthermore, they demonstrated that thefractal dimension is very sensitive to the surface silica modificationas df can decrease to 1.4. Note that Shih and co-workers [68]

derived a close power laws for tactoids fillers: G0f43þx3�df where

x is an exponent related to the filler volume fraction and theaggregate structure. The exponent x depends on the number oflayer particles per aggregate and is thus related to the degree ofexfoliation. Klüppel [69] proposed the scaling behaviour

G0f4

3þdBf

3�df where dBf is the fractal dimension of the cluster backbone

dBf z1:3. In fact, all of these fractal power laws are close to eachother.

On the other hand, Wu et al. [60] suggested that temperature isanother variable that can control the microstructure of fumed silicasuspensions. For that purpose, they used hydrophobic and hydro-philic fumed silica to prepare colloidal dispersions in dodecane.They observed for hydrophilic silica a consequent increase of the

Fig. 9. Variation of the storage modulus as a function of time for dispersions of hy-drophilic fumed silica (6 wt.%) in dodecane at different temperatures. g ¼ 0.2% andu ¼ 1 rad/s. From Wu et al. [60], with permission from Soft Matter.

storage modulus G0 with rising temperature (Fig. 9) while hydro-phobic silica showed only low variation.

This variation is believed to be associated to the internalrestructuring of aggregates. In terms of fractal dimension, thetemperature tends to increase the fractal dimension df resulting in amore compact cluster and a denser network as schematicallyshown in Fig. 10.

Interestingly, Trappe and Weitz [70] and more recently Seli-movic et al. [71] reported the transformation from viscoelastic gelsto viscoelastic liquid (Fig. 11) within an ageing time of 101e103 h insuspensions of precipitated silica in silicone oil.

The aging process is slowed down by increasing the silica par-ticle concentration or by decreasing the molecular weight of PDMS.A priori the PDMS chains are in the Rouse regime of the viscosityand are not entangled. The breakdown of the particle network canbe explained qualitatively by the PDMS chain adsorption on silicaparticle surfaces. As the PDMS chains are not entangled, the PDMSchains bound onto silica surface cannot create gel chains asgenerally observed in rubber (bound rubber) causing the inter-particle hydrogen bonding to decrease and then to transform theinitial gel to viscous fluid.

Fig. 11. Variation of the complex shear modulus (G0: full symbol and G00: open sym-bols) versus the frequency. Silica concentration: 5.3%v. The samples ages are 0.25 h(circles) and 6006.5 h (diamonds). From Selimovic et al. [71], with permission from J.Rheol.

Fig. 12. Absolute complex viscosity of PDMS suspensions at different weight fractions: (a) Graphite oxide decorated with 3-(acryloxypropyl) trimethoxysilane, (b) Graphite oxide.From Guimont et al. [72], with permission from Macromolecules.

Fig. 13. Visual observation of PDMS-based droplets (upper pictures: t ¼ 0 h) after a24 h rest period: (a) Graphite oxide, (b) Graphite oxide decorated with 3-(acrylox-ypropyl) trimethoxysilane. The scheme is a representation of the filler morphology.

P. Cassagnau / Polymer 54 (2013) 4762e47754770

Regarding layer fillers, we studied [72] the linear viscoelasticbehaviour of graphite oxide (GO) and graphite oxide functionalisedwith 3-(acryloxypropyl) trimethoxysilane (GO-M) dispersed inlow molar mass PDMS (no entangled chains). The transmissionelectron microscopy and X-ray diffraction showed an intercalatedmorphology of graphite oxide potentially exfoliated in PDMS sus-pension. The PDMS suspension filled with the graphite oxideshowed a drastic change in the viscoelastic properties for weightfractions up to 6.5 wt%. In fact, for GO concentrations higher than1.5 wt%, the viscoelastic behaviour does not showany terminal flowzone and the elastic character of this suspension becomes domi-nant at low frequencies with the appearance of a secondary plateauand a shear thinning behaviour (rh*(u)rfu�1) as shown in Fig. 12a.Furthermore, the low percolation threshold of GO sheets is attrib-uted to their macro-scale aggregation/agglomeration driven bytheir Brownian motion in these low viscosity PDMS suspensions. Itwas further concluded that the surface modification of graphiteoxide by 3-(acryloxypropyl) trimethoxysilane prevents the aggre-gation of GO-M sheets resulting in a Newtonian viscous behaviourof the suspensions as shown in Fig. 12b. The surface functionali-sation with 3-(acryloxypropyl) trimethoxysilane of the GO layersprevents their aggregation by screening the depletion forces. Thedominant force in the present case is the repulsion force betweenthese submicron particles.

On the other hand, the visual observation of droplets depositedon a horizontal glass slide leads to some relevant and additionaldiscussions about these suspensions. After a 24 h rest period, it canbe observed in Fig. 13b that the GO-M/PDMS droplet spreads overthe glass slide whereas the GO/PDMS one shows a yield stressliquid behaviour (Fig. 13a).

Finally, the low percolation threshold of GO sheets is due to theiraggregation at a macro-scale under the driving force of theirBrownian motion. Such a mechanism is the dominant one in lowviscosity suspensions in the dilute regime for which the relaxationBrownian time is in the same order of the characteristic time of theexperiment. In the present case, the rotational relaxation time ofGO sheets was estimated according to Eq (6) at srz15 s meaningthat GO particles are able to recover their random orientation in afew seconds after their anisotropic orientation under flow.

As previously discussed, the carbon nanotubes in suspension arenot fractal if they are well stabilised. However, these ideal cases arescarce as the nanotubes tend to aggregate under Brownian motioncombined with weak flow. Actually, many works report a fractalrearrangement of carbon nanotubes in suspensions. From a varietyof light-scattering experiments Chen et al. [73] and Saltiel et al. [74]showed that carbon nanotubes in suspension formed networkswith a fractal structure. Chen et al. [75] showed that this fractal

structure, quantified via the fractal dimension df, depends on CNT(single-wall carbon nanotube, SWCNT) surface treatments and resttime. At the initial time of measurement df as-received ¼ 2.27 and dftreated ¼ 2.5 for as received (no treatment or at least unknowntreatment) and treated (acid treatment based on a mixture of HNO3

and H2SO4) SWNT respectively. This means that the treated SWNTform a compact structure at small-length scale. On the other hand,as-received nanotubes showed a stable network structure over fewdays as the fractal dimension is quite constant while treatedSWCNT showed a significant variability over the days. The fractaldimension was also observed to be dependent on the surfactantused to stabilise the suspension. From electrical experiments,Lisunova et al. [76] showed that the fractal dimension continuouslyincreased with surfactant concentration from df ¼ 2.5 to df ¼ 3. Infact, a more homogenous network is formed with increasing theCNT dispersion. However, the number of physical contacts betweennanotubes for the electrical conductivity decreases. Finally,Khalhkal and Carreau [36] showed from the scaling behaviour of

P. Cassagnau / Polymer 54 (2013) 4762e4775 4771

the linear viscoelastic properties that the CNT suspensions have afractal dimension df ¼ 2.15 corresponding to a weakly flocculatingnetwork. Furthermore, this fractal structure was observed to beslightly dependent on the flow history due to an initial compactstructure of the CNT network.

To conclude, carbon nanotubes in suspension generally behaveas fractal nanofillers due to the long-range connectivity that arisesfrom interparticle physical interactions. It is then generally acceptedthat the association of CNT into a fractal structure seems morerelevant for low percolation threshold than the random distributionof isolated particles finely dispersed and stabilised in the liquidmedium. On the other hand, it is extremely difficult to compareliterature results as most of the time the CNT suspensions are quitedifferent in terms of CNT nature and suspending liquid. Further-more, the determination of the fractal dimension from the CNTconcentration dependence of the equilibrium modulus (and/or thetransition between linear and non-linear regime) is questionable.Precise criteria for the equilibrium modulus have been defined forcrosslinked systems [29]. A priori these criteria can be extended tophysical systems; however it seems that these criteria are not sys-tematically used in most of the studies.

To conclude, the Brownian forces are the dominant ones at leastin the dilute and the semi-dilute regime of the suspensions. How-ever, the interparticle interactions generally lead to the formationof fractal network. The anisotropy of the materials in coating orlatexes applications for example can be then favoured andcontrolled. Taking for example a painting process, the anisotropy ofthe film can be controlled by combining the paint flow process(shear or elongational) with the kinetics of painting cure orfilmification.

4. Highly viscous suspensions: molten filled polymers

The use of nanoparticles has been of considerable interest inrecent years to improve the mechanical, electrical and barrierproperties of polymers. The last ten years have seen the emergenceand the developments of nanocomposites. A lot of papers and re-views e too many to cite them all e have addressed the rheology ofnanocomposites. For example, reviews were addressed by Litch-field and Baird [76] on the rheology of high aspect ratio nanoparticlesfilled liquids and by Cassagnau [11] on melt rheology of organoclaysand fumed silica nanocomposites. Many authors have discussed theconnection that can be made between the filler morphology(structure, particle size) and the melt viscoelastic properties ofpolymeric materials. Many papers have been devoted to themechanisms of polymer reinforcement [62,77e83]. Even thepolymer reinforcement is still an open debate in the literature, thelast work of Jouault et al. [83] on model nanocomposites seems tohave proved the relevant mechanisms. First of all, two regimes of

10

103

104

105

106

107

10-2

10-1

100

101

Strain (%)

G’ an

d G

” (P

a)

PP

Fig. 14. Variation of the complex shear modulus versus strain amplitude (u ¼ 1 rad/s) andT ¼ 200 �C. From Cassagnau et al. [84], with permission from Elsevier.

filler concentration must be considered: dilute and concentratedregimes. In the dilute regime, the reinforcement implies a polymerchain contribution due mainly to physical chain entanglements. Inthe concentrated regime, the reinforcement directly depends onthe particleeparticle interactions and consequently on the rigidityof the filler network.

Then, the present review is focused on the time evolution of thefiller structure during annealing (or rest time) upon shear flowcessation. In fact, the viscous forces are the dominant ones inmolten nanocomposites based on entangled polymer matrix. As aresult, the nanofillers are submitted to strong orientation underflow resulting in the break-up of the nanofiller network. However,the filler structure upon the cessation of the flow is not stable andmost of the time this nanostructure was observed to change withtime. The measurement of the complex shear modulus is then usedto follow this time evolution generally consisting of the build up ofthe filler network. Actually, this build up process is really a recoveryexperiment from a rheological viewpoint. The most striking ques-tion is: what is the relevant mechanism of this recovery process?

Cassagnau and Mélis [84] performed recovery tests, subse-quently to strain sweep (non-linear deformation), of the complexshear modulus in the case of PP filled with fumed silica. Theseexperiments showed evidence that the initial equilibrium networkstructure cannot completely restore within several hours as shownin Fig. 14. It is clearly seen that this modulus evolution obeys a two-step process. However the authors did not exploit these results.Such recovery phenomenonwas also reported on organoclay-basednanocomposites [85e89] and fumed silica nanocomposites [90,91].Bailly et al. [91] showed that the aggregation in melt nano-composites (ethylene-octene copolymer (EOC)/nanosilica) dependson silica surface treatment. Rest time experiments showed a pro-nounced tendency towards aggregation for the EOC-g-VTES-basednanocomposites (VTES: vinyltriethylsilane). In contrast, strongpolymerefiller interactions between EOC-g-VTEOS (VTEOS: vinyl-triethoxysilane) and SiO2 modified with octylsilane showed a time-stable response. This latter nanocomposite is characterised bystronger adhesion between the silica filler and the EOC matrix.More recently, Kim and Macosko [92] observed that the storagemodulus of polycarbonate/graphene nanocomposites increaseswith annealing time over a few hours. The authors explained thisphenomenon by the increased effective volume of rotating disc andthe restoration of an elastic network, which was broken down bythe squeezing flow during sample loading. This phenomenon oftime evolution in nanocomposites has recently been studied indetail by Zouari et al. [93] on organoclay/polypropylene nano-composites. The authors suggested to follow this time evolutionfrom the time variation of the melt yield stress. Lertwimolnun andVergnes [94] used a Carreau-Yasuda lawwith a yield stress to fit theabsolute complex viscosity of a melt nanocomposite.

2

100

101

102

103

104

105

Time (s)

G'

G"

T=200°C

complex shear modulus recovery versus time (u1 rad/s and g ¼ 0:1%). Silica: 16%wt,

Fig. 15. Variation of the yield stress of organoclay PP nanocomposite (5% clay) fordifferent temperatures. From Zouari et al. [93], with permission from J. Rheol.

Fig. 16. Variation of the storage modulus versus frequency for different rest timesupon flow cessation after a short shear deformation. From Alig et al. [99] withpermission from Elsevier.

P. Cassagnau / Polymer 54 (2013) 4762e47754772

���h*ðuÞ��� ¼ s0u

þ h0�1þ ðluÞa�n�1

a (18)

Where s0 is the melt yield stress, h0 is the zero shear viscosity, g isthe time constant (i.e. relaxation time), a is the Yasuda exponentand n is the power law exponent.

Fig. 15 shows the variation of the melt yield stress versus re-covery time at different temperatures. These experiments are

Fig. 17. TEM pictures for polycarbonate/MWNT nanocomposites (0.6vol% MWNT). (a) Theaggregates after annealing at T ¼ 300 �C. From Alig et al. [99] with permission from Elsevi

subsequent to a steady shearflowof 1 s�1 for 155 s.Many commentscan be addressed from this figure. First, it is observed that the yieldstress evolution follows a two-step process. The first kinetics(t < tcrit,s0 a t1/2) is related to the disorientation of clay tactoids andthe second one to the aggregation of these tactoids and layers into anetwork (t> tcrit, s0 a t1). Second, these kinetics are acceleratedwithtemperature as expected. However, the network structure of thenanocomposites does not seem to stabilise even at long rest times.Third, the kinetics transition time tcrit can be compared to the timeofdisorientation that can be theoretically calculated from the rotarydiffusivity Dr0 (equation (6)). Then, Zouari et al. [93] calculated thatthe time from the rotary diffusion (1/Dr0) is much shorter than thetime corresponding to the end of the first kinetics associated withplatelet disorientation by Brownian motion. However, this calcula-tion was made assuming a dilute concentration of clay platelets.Actually, this hypothesis is over-simplified as these platelets are in asemi-dilute regime a priori with a slow down of the diffusion pro-cess. Note that Kim andMacosko [92] showed in the case of graphitenanocomposite that the rotational relaxation time is in the order ofmagnitude of the experimental time of recovery. To conclude, thefirst kinetic can be associated with the disorientation of nanofillersby Brownian motion and the second one to the aggregation ofclusters into a network.

In the sameway, Cao et al. [95], Tan et al. [96] and Song et al. [97]studied the influence of annealing time on carbon-black-basednanocomposites. They showed that annealing over a few hoursleads to a liquid-to-solid like transition. As previously explained,they came to the conclusion that the variation of the viscoelasticproperties (and electrical resistivity) comes from the aggregation ofcarbon black. However, they suspected that the interfacial tensionbetween polymer (polystyrene in the present case) and carbonblack plays an important role in the aggregation mechanism ofcarbon black.

Regarding carbon nanotubes in polymer melts, the break up andformation of the network was extensively studied by the groups ofAlig and Pötschke [98e100]. One of the striking features in polymermelt is the dispersion of nanotubes [41]. The first challenge is toseparate the carbon nanotubes from their initial agglomerate as-semblies (bundle). Assuming that the nanotubes have been finallywell dispersed as individual entities, Huang [41] showed that at lowconcentration, probably in the dilute regime, the nanotube arequite stable against re-aggregation provided that the viscosity ishigh enough to suppress Brownian motion. From our own study onnanotubes of L/d ¼ 160, the rotary relaxation time (1/Dr0, equation(5)) in polymer of viscosity h ¼ 1000 Pa s is w60 h. Consequently,the Brownian motion of nanotubes in viscous polymer can beneglected. A fortiori, the Brownian motion of nanotubes in the

nanotubes are well dispersed in PC matrix after shearing. (b, c) The nanotubes former.

P. Cassagnau / Polymer 54 (2013) 4762e4775 4773

semi-dilute regime can be totally neglected as the rotary diffusionis considerably slowed downwith the concentration (equation (9)).However, aggregation of nanotubes in melt polymers is not a fictionas has been reported in the literature. For example, Alig et al. [99]clearly showed from linear viscoelasticity, electrical conductivityand transmission electronic microscopy (TEM) that aggregation isan effective process during annealing without the application ofany shear forces. Fig. 16 shows that the storage modulus increaseswith the annealing time and Fig. 17 shows (TEM observation) thataggregates have been formed during annealing.

If Brownian motion can be assumed in suspensions of low vis-cosity as previously discussed, another process must be consideredin molten polymers of high viscosity (typically h > 100 Pa s) for there-aggregation of nanotubes during annealing. Alig et al. [99]assumed that the driving force for the re-aggregation of CNTs inmelt polymers is a combination of strong dispersive interactionsbetween nanotubes and depletion interactions between adjacentnanotubes.

Finally, themain striking behaviour of nanocomposites based onthermoplastic polymers filled with fillers of high aspect ratio is thecontrol of the anisotropy of these materials. Indeed from a pro-cessing point of view, the fillers are submitted to non linear flowresulting in a strong difficulty to control the anisotropy andconsequently the difficulty to control the mechanical or electricalproperties for example. In other words, we experience the pro-cessing conditions rather than control.

5. Conclusion

The objective of the present paper was to review the linearviscoelasticity of suspensions and of polymers filled with nano-sizeparticles of different aspect ratios and structuration. It is ofimportance to point out that the viscoelastic behaviour of nano-filled system must be discussed from the knowledge of thedispersion of nanoparticles in terms of dispersion at different scales(from primary to aggregate particles) and anisotropy (privilegedorientation of non spherical particles).

First, we reported and discussed the linear viscoelasticity ofsuspensions (fluid of low viscosity) filled with well-dispersed andstabilised particles (PS latex, silica, carbon nanotubes and PSnanofibres). For these suspensions, the Brownian motion is thedominant mechanism of relaxation. For the spherical particles, thesemi-empirical equation of KriegereDougherty is themost relevantone to predict the viscosity of these suspensions. Carbon nano-tubes, cellulose whiskers and polymer nanofibres in dilute andsemi-dilute regimes obey a universal diffusion process by Brownianmotion according to the DoieEdwards theory providing that thesenanofillers are well dispersed and stabilized in the suspendingliquid.

The second part of the present review was devoted to theviscoelastic behaviour of fractal filler suspensions. Two categoriesof fractal fillers can be distinguished: nanofillers such as fumedsilica and carbon black due to their native specific structure fromnano- to micro-scale and exfoliated fillers such as organoclays,carbon nanotubes and more recently graphite oxide and graphene.These fillers can be classified as fractal fillers due to the long-rangeconnectivity that arises from interparticle physical interactions. Theparticular rheological behaviour of these suspensions arises fromthe presence of the network structure (interparticle interaction),which leads to a drastic decrease in the percolation threshold fc

(typically fc < 0.05) at which the zero shear viscosity diverges toinfinity. Scaling concepts, based on a fractal dimension, are gener-ally used to study the effect of interparticle forces on the elasticityof aggregated suspensions. Fractal exponents are then derived andrelated to the structure of the aggregate clusters. It must be pointed,

that the aggregation behaviour is governed by the Brownian mo-tion and local particle rearrangements under linear deformation insteady or dynamic (frequency) conditions.

In the third and last part, we investigated the viscoelastic anddynamic behaviours of nanocomposites, i.e nanoparticles dispersedin entangled polymers. As the viscous forces are undoubtedly thedominant ones, the nanofillers are submitted to strong orientationunder flow resulting in the network break-up. It is generallyobserved from linear viscoelastic measurements that the fillerstructure is not stable upon the cessation of the flow. In the case ofplatelet nanocomposite (organoclays, graphite oxide) a two-stepprocess of recovery is generally reported: the first one is thedisorientation of the fillers followed by a second phenomenon ofaggregation/clustering. In the case of carbon nanotubes only theaggregation is observed, at least the two phenomena cannot bediscerned by rheology. A priori the disorientation of nanofillers canbe assumed to be controlled by the Brownian motion. However,other mechanisms such as dispersive interactions and depletionforces must be considered for the aggregation of fillers in a physicalnetwork.

Acknowledgements

Prof B. Charleux is thanked for her stimulating discussion on thedynamic of polymer fibres and for kindly reading the presentmanuscript.

References

[1] Pan W, Caswell B, Em Karniadakis G. Rheology, microstructure and migrationin Brownian colloidal suspensions. Langmuir 2010;26:133e42.

[2] Krieger IM, Dougherty TJ. A mechanism for non-Newtonian flow in sus-pensions of rigid spheres. Trans Soc Rheol 1959;3:137e52.

[3] Quemada D. Rheology of concentrated disperse systems and minimum en-ergy dissipation. Rheol Acta 1977;16:82e94.

[4] Bicerano J, Douglas JF, Douglas AB. Model for the viscosity of particle dis-persions. J Macromol Sci Rev Macromol Chem Phys 1999;C39(4):561e642.

[5] Smith WE, Zukoski CF. Flow properties of hard structured particle suspen-sions. J Rheol 2004;48:1375e88.

[6] Barnes HA. Shear thickenning ‘dilatancy’ in suspensions of non aggregatingsolid particles dispersed in Newtonian fluids. J Rheol 1989;33:329e66.

[7] Pishvaei M, Graillat C, McKenna TF, Cassagnau P. Experimental investigationand phenomenological modeling of the viscosity-shear rate of bimodal highsolid content latex. J Rheology 2007;51:51e69.

[8] Genovese DB. Shear rheology of hard-sphere, dispersed, and aggregatedsuspensions and filler-matrix composites. Adv Colloid Interface Sci2012;171e172:1e16.

[9] Galindo-Rosales FJ, Moldenaers P, Vermant J. Assessment of the dispersionquality in polymer nanocomposites by rheological methods. MacromolMater Eng 2011;296:331e40.

[10] Ray SS. Rheology of polymer/layered silicate nanocomposites. J Indust EngChem 2006;12:811e42.

[11] Cassagnau P. Melt rheology of organoclays and fumed silica nanocomposites.Polymer 2008;49:2183e96.

[12] Kim H, Abdala A, Macosko CW. Graphene/polymer nanocomposites. Mac-romolecules 2010;43:6515e30.

[13] Choi HJ, Ray SS. A review on melt-state viscoelastic properties of polymernanocomposites. J Nanosci Nanotechnol 2011;11:8421e849.

[14] Filippone G, Salzano de Luna M. A unifying approach for the linear visco-elasticity of polymer nanocomposites. Macromolecules 2012;45:8853e60.

[15] Larson RG. The structure and rheology of complex fluids. Oxford, UniversityPress; 1999. p. 263e323. [Chapter 6]: Particulate Suspensions.

[16] Shikata T, Pearson DS. Viscoelastic behaviour of concentrated sphericalsuspensions. J Rheology 1994;38:601e16.

[17] Doi M, Edwards SF. Dynamics of rod-like macromolecules in concentratedsolution. Part 1. J Chem Soc Faraday Trans 1978;2(74):560e70.

[18] Doi M, Edwards SF. Dynamics of rod-like macromolecules in concentratedsolutions. Part 2. J Chem Soc Faraday Trans 1978;2(74):918e32.

[19] Mori Y, Ookubo N, Hayakawa R. Microstructural regimes of colloidalrod suspensions, gels, and glasses. J Polym Sci Part B Polym Phys 1982;20:2111e24.

[20] Doi M, Edwards SF. The theory of polymer dynamics. London: Oxford Press;1986.

[21] Teraoka I, Hayakawa R. Theory of dynamics of entangled rod-like polymersby use of a mean-field green function formulation. II. Rotational diffusion.J Chem Phys 1989;91:2643e8.

P. Cassagnau / Polymer 54 (2013) 4762e47754774

[22] Bitsanis I, Davis HT, Tirrell M. Brownian dynamics of nondilute solutionsof rodlike polymers. 2. High concentrations. Macromolecules 1990;23:1157e65.

[23] Marceau S, Fulchiron R, Dubois Ph, Cassagnau P. Viscoelasticity of Browniancarbon nanotubes in PDMS semidilute regime. Macromolecules 2009;42:1433e8.

[24] Zhang W, Charleux B, Cassagnau P. Viscoelastic properties of water sus-pensions of polymer nanofibers synthesized via RAFT-mediated emulsionpolymerization. Macromolecules 2012;45:5273e80.

[25] van der Werff JC, de Kruif CG, Blom C, Mellema J. Linear viscoelasticbehaviour of dense hard-sphere dispersions. Phys Rev 1989;A39:795e807.

[26] Beenaker CW. The effective viscosity of a concentrated suspension of spheres(and its relation to diffusion). Physica A 1984;128:48e81.

[27] Beenaker CW, Mazur P. Diffusion of spheres in a concentrated suspension II.Physica A 1984;126:349e70.

[28] Pishvaei M, Graillat C, McKenna TF, Cassagnau P. Rheological behavior ofpolystyrene latex near the maximum packing fraction of particles. Polymer2005;46:1235e44.

[29] Winter HH, Mours M. Rheology of polymers near liquid-solid transitions.Adv Polym Sci 1997;134:165e234.

[30] Hobbie EK. Shear rheology of carbon nanotube suspensions. Rheol Acta2010;49:323e34.

[31] Lin-Gibson S, Pathak JA, Grulke EA, Wang H, Hobbie EK. Elastic flow insta-bility in nanotube suspensions. Phy Rev Lett 2008;92(4):048302.

[32] Moreira L, Fulchiron R, Seytre G, Dubois P, Cassagnau P. Aggregation ofcarbon nanotubes in semidilute suspension. Macromolecules 2010;43:1467e72.

[33] Fan Z, Advani SG. Rheology of multiwall carbon nanotube suspensions.J Rheol 2007;51:585e604.

[34] Chapartegui M, Markaide N, Florez S, Elizetxea C, Fernandez M,Santamaria A. Specific rheological and electrical features of carbon nanotubesuspensions in an epoxy matrix. Compos Sci Technol 2010;70:879e84.

[35] Schulz SC, Bauhofer W. Shear influenced network dynamics and electricalconductivity recovery in carbon nanotube/epoxy suspensions. Polymer2010;51:5500e5.

[36] Khalkhal F, Carreau PJ. Scaling behavior of the elastic properties of non-diluteMWCNT-epoxy suspensions. Rheol Acta 2011;50:717e28.

[37] Khalkhal F, Carreau PJ, Ausias G. Effect of flow history on linear viscoelas-ticity properties and the evolution of the structure of multiwalled carbonnanotube suspension in an epoxy. J Rheol 2011;55:153e75.

[38] Yearsley KM, Mackley MR, Chinesta F, Leygue A. The rheology of multiwalledcarbon nanotube and carbon black suspensions. J Rheol 2012;56:1465e90.

[39] Ma WKA, Chinesta F, Mackley MR. The rheology and modelling of chemicallytreated carbon nanotubes suspensions. J Rheology 2008;52:1311e30.

[40] Derakhshandeh B, Petekidis G, Sabet SS, Hamad WY, Hatzikiriakos SG.Ageing, yielding and rheology of nanocrystalline cellulose suspensions.J Rheology 2013;57:131e48.

[41] Huang YY, Ahir SV, Terentjev EM. Dispersion rheology of carbon nanotubesin a polymer matrix. Phys Rev B 2006;73:125422.

[42] Hobbie EK, Fry DJ. Non equilibrium phase diagram of sticky nanotube sus-pensions. Phys Rev Lett 2006;97:036101.

[43] Fry D, Langhorst B, Wang H, Becker ML, Bauer BJ, Grulke EA, et al. Rheo-optical studies of carbon nanotube suspensions. J Chem Phys 2006;124:054703.

[44] Ma WKA, Chinesta F, Ammar A, Mackley MR. The rheology and modelingof chemically treated carbon nanotubes suspensions. J Rheology 2009;53:547e73.

[45] Petekidis G, Vlassopoulos D, Fytas G, Kountourakis N, Kumar S. Associationdynamics in solutions of hairy-rod polymers. Macromolecules 1997;30:919e31.

[46] Boissé S, Rieger J, Belal K, Di-Cicco A, Beaunier P, Li MH, et al. Amphiphilicblock copolymer nano-fibers via RAFT-mediated polymerization in aqueousdispersed system. Chem Commun 2010;46:1950e2.

[47] Boissé S, Rieger J, Pembouong G, Beaunier P, Charleux B. Influence of thestirring speed and CaCl2 concentration on the nano-object morphologiesobtained via RAFT-mediated aqueous emulsion polymerization in the pres-ence of a water-soluble macroRAFT agent. J Polym Sci Part A Polym Chem2011;49:3346e54.

[48] Zhang X, Boissé S, Zhang W, Beaunier P, D’Agosto F, Rieger J, Charleux B.Polymerization-induced self-assembly: from solublemacromolecules to blockcopolymer nano-objects in one step. Macromolecules 2011;44:4149e58.

[49] Zhang W, Charleux B, Cassagnau P. Dynamic behavior of crosslinkedamphiphilic block copolymer nanofibers dispersed in liquid poly(ethyleneoxide) below and above their glass transition temperature. Soft Matter2013;9:2197e205.

[50] Wierenga AM, Philipse AP. Low-shear viscosity of isotropic dispersions of(Brownian) rods and fibres; e review of theory and experiments. ColloidsSurf A Physicochem Eng Aspects 1998;137:355e72.

[51] Bercea M, Navard P. Shear dynamics of aqueous suspensions of cellulosewhiskers. Macromolecules 2000;33:6011e6.

[52] Cassagnau P, Zhang W, Charleux B. Universal dynamics of nanorods (carbonnanotubes, cellulose whiskers, stiff polymers and polymer fibers) suspen-sions 2013, submitted for publication.

[53] Philipse AP, Wierenga AM. On the density and structure formation in gelsand clusters of colloidal rods and fibers. Langmuir 1998;14:49e54.

[54] Ramakrishman S, Zukoski CF. Microstructure and rheology of thermorever-sible nanoparticle gels. Langmuir 2006;22:783e7842.

[55] Raghavan SR, Walls HJ, Khan SA. Rheology of silica dispersions in organicliquids: new evidence for solvation forces dictated by hydrogen-bonding.Langmuir 2000;16:7920e30.

[56] Yziquel F, Carreau PJ, Tanguy PA. Non-linear viscoelaslic behavior of fumedsilica suspensions. Rheol Acta 1999;38:14e25.

[57] Khan SA, Zoeller NJ. Dynamic rheological behavior of flocculated fumed silicasuspensions. J Rheol 1993;37:1225e35.

[58] RuebCJ.Zukoski, viscoelasticpropertiesofcolloidalgels. JRheol1997;41:197e218.[59] Nordström J, Matic A, Sun J, Forsyth M, Doug R. MacFarlane, Aggregation,

ageing and transport properties of surface modified fumed silica dispersion.Soft Matter 2010;6:2293e9.

[60] Wu XJ, Wang Y, Yang W, Xie BH, Yang MB, Dan W. a rheological study ontemperature dependent microstructural changes of fumed silica gels indodecane. Soft Matter 2012;8:10457e63.

[61] Amiri A, Bye G, Sjöblom J. Influence of pH, high salinity and particle con-centration on stability and rheological properties of aqueous suspensions offumed silica. Colloids Surf A Physicochem Eng Asp 2009;349:43e54.

[62] Bartholome C, Beyou E, Bourgeat-Lami E, Cassagnau P, Chaumont P, David L,et al. Viscoelastic properties and morphological characterization of silica/polystyrene nanocomposites synthesized by nitroxide-mediated polymeri-sation. Polymer 2005;46:9965e73.

[63] Mall S, Russel WB. Effective medium approximation for an elastic networkmodel of flocculated suspensions. J Rheol 1987;31:651e81.

[64] Chen M, Russel WB. Characteristics of flocculated silica dispersions. J Coll IntSci 1991;141:564e77.

[65] Wolthers W, van de Ende D, Breedveld V, Duits MHG, Potanin AA,Wientjes RHW, et al. Linear viscoelastic behavior of aggregated colloidaldispersions. Phys Rev E 1997;56:5726e33.

[66] Piau JM, Dorget M, Palierne JF, Pouchelon A. Shear elasticity and yield stressof silica silicone physical gels: fractal approach. J Rheol 1999;43:305e14.

[67] Paquien JN, Galy J, Gérard JF, Pouchelon A. Colloids Surf A Physicochem EngAsp 2005;260:165e72.

[68] Shih WH, Shih WY, Kim SI, Liu J, Aksay IA. Scaling behavior of the elasticproperties of colloidal gels. Phys Rev B 1990;42:4772e9.

[69] Klüppel M. Elasticity of fractal filler network in elastomers. Macromol Symp2003;194:39e45.

[70] Trappe V, Weitz DA. Scaling of the viscoelasticity of weakly attractive par-ticles. Phys Rev Lett 2000;85:449e52.

[71] Selimovic S, Maynard SM, Hu Y. Aging effects of precipitated silica in pol-y(dimethylsiloxane). J Rheol 2007;51:325e40.

[72] Guimont A, Beyou E, Martin G, Sonntag Ph, Cassagnau P. Viscoelasticityof graphite oxide-based suspensions in PDMS. Macromolecules 2011;44:3893e900.

[73] Chen Q, Saltiel C, Manickavasagam S, Schalder LS, Siegel RW, Yang H. J CollInt Sci 2004;280:91e7.

[74] Saltiel C, Manickavasagam S, Mengüc MP, Andrews R. Light-scattering anddispersion behavior of multiwalled carbon nanotubes. J Opr Soc Am A2005;22:1546e54.

[75] Lisunova MO, Lebovka N, Melezhyk OV, Boiko YP. Stability of the aqueoussuspensions of nanotubes in the presence of nonionic surfactant. J Coll InterSci 2006;299:740e6.

[76] Litchfield DW, Baird DG. The rheology of high aspect ratio nanoparticlesfilled liquids. Rheology Rev 2006:1e60.

[77] Cassagnau P. Payne effect and shear elasticity of silica-filled polymers inconcentrated solutions and in molten state. Polymer 2003;44:2455e62.

[78] Jouault N, Vallat P, Dalmas F, Said S, Jestin J, Boué F. Well-dispersed fractalaggregates as filler in polymer-silica nanocomposites: long-range effects inrheology. Macromolecules 2009;42:2031e40.

[79] Jancar J, Douglas JF, Starr FW, Kumar SK, Cassagnau P, Lesser AJ, et al. Currentissues in research on structure-property relationships in polymer nano-composites. Polymer 2010;51:3321e43.

[80] Song Y, Zheng Q. Linear viscoelasticity of polymer mets filled with nano-sized fillers. Polymer 2010;51:3262e8.

[81] Moll JF, Akcora P, Rungta A, Gong S, Colby RH, Benicewicz BC, et al. Me-chanical reinforcement in polymer melts filled with polymer grafted nano-particles. Macromolecules 2011;44:7473e7.

[82] Chevigny C, Dalmas F, Di Cola E, Gigmes D, Bertin D, Boué F, et al. Polymer-grafted-nanoparticles nanocomposites: dispersion, grafted chain conforma-tion, and rheological behavior. Macromolecules 2011;44:122e33.

[83] Jouault N, Dalmas F, Boué F, Jestin J. Multiscale characterization of fillerdispersion and origins of mechanical reinforcement in model nano-composites. Polymer 2012;53:761e5.

[84] Cassagnau P, Mélis F. Non-linear viscoelastic behaviour and modulus re-covery in silica filled polymers. Polymer 2003;44:6607e15.

[85] Ren J, Casanueva BF, Mitchell CA, Krishnamoorti R. Disorientation linetics ofaligned polymer layered silicate nanocomposites. Macromolecules 2003;36:4188e94.

[86] Treece MA, Oberhauser P. Soft glassy dynamics in polypropylene-claynanocomposites. Macromolecules 2007;40:571e82.

[87] Treece MA, Oberhauser P. Ubiquity of soft glassy dynamics in polypropylene-clay nanocomposites. Polymer 2007;48:1083e95.

[88] Mobuchon C, Carreau PJ, Heuzey MC. Structural analysis of non-aqueouslayered silicate suspensions subjected to shearflow. J Rheol 2009;53:1025e48.

P. Cassagnau / Polymer 54 (2013) 4762e4775 4775

[89] Eslami H, Grmela M, Bousmina M. Structure build-up at rest in polymernanocomposites: flow reversal experiments. J Polym Sci Part B Polym Phys2009;47:1728e41.

[90] Filippone G, Romeo G, Acierno D. Viscoelasticity and Structure of poly-styrene/fumed silica nanocomposites: filler network and hydrodynamicscontributions. Langmuir 2010;26:2714e20.

[91] Bailly M, Kontopoulou M, El Mabrouk K. Effect of polymer/filler interactionson the structure and rheological properties of ethylene-octene/nanosilicacomposites. Polymer 2010;51:5506e15.

[92] Kim H, Macosko CW. Processing-property relationships of polycarbonate/graphene composites. Polymer 2009;50:3797e809.

[93] Zouari R, Domenech T, Vergnes B, Peuvrel-Disdier E. Time evolution of thestructure of organoclay/polypropylene nanocomposites and application ofthe time-temperature superposition principle. J Rheol 2012:56.

[94] Lertwimolnun W, Vergnes B. Influence of compatibilizer and processingconditions on the dispersion of nanoclay nanocomposites in a twin screwextruder. Polymer 2005;46:3462e71.

[95] Cao Q, Song Y, Tan Y, Zheng Q. Conductive and viscoelastic behaviors ofcarbon black filled polystyrene during annealing. Carbon 2010;48:4268e75.

[96] Tan Y, Yin X, Chen M, Song Y, Zheng Q. Influence of annealing on lineaviscoelasticity of carbon black filled polystyrene and low-density poly-ethylene. J Rheol 2011;55:965e79.

[97] Song Y, Cao Q, Zheng Q. Annealing-induced rheological and electric resis-tance variations in carbon black-filled polymer melts. Colloid Polym Sci2012;290:1837e42.

[98] Pegel S, Pötschke P, Petzold G, Alig I, Dudkin SM, Lellinger D. Dispersion,agglomeration, and network formation of multiwalled carbon nanotubes inpolycarbonate melts. Polymer 2008:974e84.

[99] Alig I, Skipa T, Lellinger D, Pötschke P. Destruction and formation of a carbonnanotube network in polymer melts: rheology and conductivity spectros-copy. Polymer 2008;49:3524e32.

[100] Alig I, Pötschke P, Lellinger D, Skipa T, Pegel S, Kasaliwal GR, et al. Estab-lishment, morphology and properties of carbon nanotube networks inpolymer melts. Polymer 2012;53:4e28.