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

Polymer 54 (2013) 4762e4775

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Feature article

Linear viscoelasticity and dynamics of suspensions and molten polymers filled with nanoparticles of different aspect ratios Philippe Cassagnau Université 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

a b s t r a c t

Article history: Received 15 February 2013 Received in revised form 1 May 2013 Accepted 9 June 2013 Available online 18 June 2013

In the present review, we report the linear viscoelasticity of suspensions and polymers filled with nanosize particles of different aspect ratios and structuration. The viscoelastic behaviour of liquid suspension filled with well-dispersed and stabilised particles proves that the Brownian motion is the dominant mechanism 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 DoieEdwards theory. Regarding spherical particles, the KriegereDougherty equation is generally successfully used to predict the zero shear viscosity of these suspensions. Regarding fractal fillers, two categories can be considered: nanofillers such as fumed silica and carbon black due to their native structure; and secondly exfoliated fillers such as organoclays, carbon nanotubes, graphite oxide and graphene. The particular 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 the zero shear viscosity diverges to infinity. Fractal exponents are then derived from scaling concepts and related to the structure of the aggregate clusters. In the case of melt-filled polymers, the viscous forces are obviously the dominant ones and the nanofillers are submitted to strong orientation under flow. It is generally observed from linear viscoelastic measurements that the network structure is broken up under flow 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 is generally reported: disorientation of the fillers followed by re-aggregation. Disorientation can be assumed to be governed by the Brownian motion; however, other mechanisms are responsible for the reaggregation process. Ó 2013 Elsevier Ltd. All rights reserved.

Keywords: Suspension Nanofillers Viscoelasticity

1. Introduction The rheology of the suspensions started with the famous work of Einstein in 1905 and 1911 on the prediction of the viscosity of hard-sphere suspensions at low particle concentrations. In this dilute regime, the hydrodynamic disturbance of the flow field induced by the particles in the suspending liquid leads to an increase in the energy dissipation and an increase in the relative viscosity, hr, according to the equation (1):

hr ¼

h ¼ 1 þ 2:5f hs

(1)

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

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constant 2.5 is the intrinsic viscosity ð½h ¼ lim ðh  hs =fhs ÞÞ, and f/0 theoretically for rigid spheres [h] ¼ 2.5. In the dilute regime of hard particles, the interparticle forces are negligible compared with hydrodynamic forces and Brownian diffusion. In other words, there are no attractions between the particles but only an excluded volume effect. At higher concentrations the probability of particle collisions increases so that the hydrodynamic interactions become the dominant ones. As a result the Einstein law fails since a significant positive deviation of the relative viscosity is observed. Furthermore, a shear thinning behaviour of the viscosity is observed with increasing volume fraction of particles. There are numerous models available for the description of the rheology of suspensions of spherical spheres, which is still an open domain of theoretical investigations [1]. However, the semiempirical equation of Krieger and Dougherty [2] for monodispersed suspensions is one of the most used and developed in the literature. The concentration dependence of the zero shear viscosity is expressed as:

P. Cassagnau / Polymer 54 (2013) 4762e4775




h0 ¼ hs 1 

f fm

½hfm (2)

Where fm is the maximum packing fraction of particles. At this concentration, the zero shear viscosity rises to infinity and the suspension exhibits a yield stress behaviour. The KriegereDougherty equation can be simplified to the Quemada equation [3] h0 ¼ hs ð1  ðf=fm ÞÞ2 since it generally observed that [h]fm ¼ 2 in most usual suspensions. In model systems of monodispersed hard spheres, the critical volume fraction (fm z 0.63) reported by Bicerano et al. [4] and Smith and Zukoski [5] for the approach of the zero shear viscosity to infinity is near the random close packing value of 0.64. The right particle fraction at which the zero shear viscosity diverges to infinity is still an open debate as most of the experimental results are far below the theoretical packing value. The KriegereDougherty model is generally used as a mathematical equation, where fm and [h] are two fitting parameters, to numerically fit the concentration dependence of the zero shear viscosity. To give a physical meaning to both parameters, a common way is to derive fm as an effective maximum packing fraction fm,eff and [h] is related to the deviation from ideal spheres in terms of aspect ratio of the particles. The value of fm,eff is no more a universal value as it has to be determined for each suspension. In fact, fm,eff includes deviation from ideal particles, as deformable spheres for example. On the other hand, aggregate particles can be expressed from an effective volume fraction feff. Regarding the particle aspect ratio, Barnes [6] proposed the following formula for the intrinsic viscosity ½h ¼ 0:07ðL=dÞ5=3 and ½h ¼ 0:3ðL=dÞ for rod-like and disc-like particles respectively (L: longest length and d ¼ diameter or thickness respectively). Even without any physical meaning, the KriegereDougherty equation is a useful tool to visualise at a glance the rheological trend of suspensions. In some cases (latex for example), it is of importance for useful development from the point view of process-engineering applications to have the lowest viscosity for the highest solid content. From equation (2) and keeping constant the viscosity of the suspending liquid, the only way is to increase the packing fraction fm. This is generally achieved by using a bi-modal distribution of particles [7]. In contrast, in some applications like suspensions filled with conductive fillers, the lowest percolation threshold is required in terms of formulation costs and weight saving. The Kriegere Dougherty equation shows that particles with high aspect ratio (an increase of [h] and/or with aggregation (feff > f) including fractal particles) are required for such applications. Consequently, the physic of suspensions and its extension to molten filled polymers is extremely complicated as the simple case of inert and rigid spheres at low concentration is generally far from real life. The suspensions and filled polymers are the world of colloidal particles as they offer many possibilities of material developments. In the colloidal size domain, the Brownian forces, direct interparticle forces and viscous hydrodynamic forces are all of comparable magnitude. In some cases, the particleeparticle interactions play a dominant role in these systems and results in aggregation or flocculation with possible fractal organisation of such clusters. In fact, the balance of the different interaction can be tuned from a judicious chemical modification of the filler surface and then open a marvellous world for the control of the target rheological behaviours. Therefore, the rheology of colloidal dispersions exhibits a rare diversity and has been the subject of several publications and reviews. The last one of greatest and practical interest was published by Genovese [8]. This paper reviews the shear rheology of suspensions and matrix polymers filled with microscopic and colloid particles. The shear rheology of the suspensions has been discussed from the Kriegere Dougherty equation as it is well known, with some simple modifications taking into account the deviation from ideal cases, to

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effectively predict the concentration dependence of many types of suspensions. In the case of aggregated suspensions, the authors demonstrated the analogy between several theoretical models devoted to yield stress and elastic modulus of these gels. If the shear rheology (steady shear flow experiments) is well described in the literature for suspensions, the linear viscoelasticity under oscillatory measurements is more limited in terms of publications. However, the linear viscoelasticity has been extremely used to characterise the viscoelastic properties of polymer melts filled with nano-sized fillers (nanocomposites). A direct consequence of filler incorporation in molten polymers is the significant change in viscoelastic behaviour as they are sensitive to the structure, concentration, particle size, shape (aspect ratio) and surface modification of the fillers. As a result, rheological methods are useful and suitable to assess the quality of filler dispersion [9]. In recent years, nearly all types of nano-fillers have been used for the preparation of nanocomposites: organoclays, carbon black, fumed and colloid silica, carbon nanotubes, cellulose whiskers, metallic oxide, etc. and more recently graphite oxide and graphene. From a literature survey, thousands of papers and reviews have been published in this broad scientific area and consequently we can only cite the most recent reviews [10e14]. From a physical point of view, nanoparticles in suspending fluid are submitted to particleeparticle forces, particleefluid interactions, viscous forces under flow and finally Brownian forces. The Brownian motion arises from thermal randomising forces that lead to the dispersion of the nanoparticles. Consequently, Brownian motion is ever present even in highly viscous systems (entangled polymer melts). The aim of the present paper is to review some linear viscoelastic behaviours of suspensions and molten polymer filled with nanoparticles of different aspect ratios such as spheres, platelets and nanotubes (or nanofibres). Actually, the viscoelastic and dynamic behaviours have been discussed for each system taking into account the dispersion at different scales of nanoparticles as it was reported in the corresponding papers. Finally, this review is addressed in terms of a comprehensive study of the viscoelastic behaviour and modelling from the Brownian diffusion of nanoparticles. 2. Particle diffusivity In the dilute regime of suspensions, the particles can rotate about their centre of mass. The particles are then able to rotate freely without any interference interaction with neighbouring ones. The particle diffusivity in this dilute regime is controlled by the Brownian forces (wkBT, kB is the Boltzmann constant) in the suspending liquid (viscosity hs), which exerts the Stokes friction (w6phsR) on the particle (radius of the particle R) and is written as:

D0 ¼

kB T 6phs R



m2 s1



(3)

This equation is also known as the StokeseEinstein law. The rotary particle diffusivity (Unity: s1) has been derived for nonspherical particles as following [15]: Particles of nearly spherical shape (diameter d):

Dr0 ¼

kB T

(4)

phs d3

Spheroid particles of the longest length L:

Dr0 ¼

    3kB T ln 2 dL  0:5

phs L3

Platelet or circular disc-like particle of diameter d:

(5)

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Dr0 ¼

P. Cassagnau / Polymer 54 (2013) 4762e4775

3kB T 4hs d3

(6)

Rods or nanofibres of length L and diameter d:

Dr0 ¼

    3kB T ln dL  0:8

phs L3

polymers (typically hs < 1 Pa s) and in entangled molten polymers (typically hs > 100 Pa s). 3. Liquid suspensions

(7)

3.1. Hard-spheres

In the semi-dilute regime, the particles are no longer able to diffuse or rotate freely without any interaction from surrounding ones. For rigid spheres or disc-like particles, this transition is not generally well defined. However, the onset of the semi-dilute regime could be the concentration at which the concentration dependence of the viscosity deviates from the Einstein law (w3%). Consequently, the diffusivity is expected to be slowed down as proved by Shikata and Pearson [16]. For rods, this transition (dilute/semi-dilute) occurs when the rod concentration, expressed as n the number of objects per unit volume ðf ¼ ðp=4Þd2 LnÞ, reaches a value proportional to 1/L3. However, scaling developments [17,18], showed that the rotational interference becomes significant when:

The first experiments on the linear viscoelastic properties of hardsphere suspensions were reported first by van der Werff et al. [25] and Shikata and Pearson [16]. These results have been reviewed by Larson in his book [15]. A terminal relaxation zone (G0 au2 and G00 au1) is well depicted for the suspensions (Silica particles of 60, 125 and 225 nm) in a mixture of ethylene glycol and glycerol investigated in these works. Furthermore, a master curve on concentration depenfrequency uaTsw was dence of G0 and G00  h0N uaT versus the reduced . successfully plotted. h0N ðfÞ ¼ lim G00 u is derived from the high u/N frequency behaviour. sw is the longest relaxation time, sw z 0.5sp. From Shikata and Pearson [16], the particles cannot move without generating lubrication forces with other neighbouring particles. The relaxation time, assimilated to the diffusion time, is expected to be inversely proportional to the particle diffusivity:

nL3 zb

sp ðfÞ ¼

(8)

where b is a dimensionless constant. Experimentally, this constant was found [19] to be roughly close to b z 30. The molecular dynamics of straight rigid rods in suspension have been theoretically investigated by Doi and Edwards [20]. Regarding the rotary diffusivity Dr of a rod in an isotropic suspension, the DoieEdwards theory for semi-dilute regime predicts:

 2 Dr ¼ ADr0 nL3

(9)

where A is a dimensionless constant whose values are generally large. From theoretical [21] and simulation [22] considerations, the values of this constant have been found to be roughly w1000 for perfectly rigid rods. Experimentally, lower values of A have been observed 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 that the rod diameter d is large enough compared to the distance between two consecutive neighbouring rods. Consequently, the excluded volume of interaction can no longer be neglected. Just above the onset of this concentrated regime, the solution remains isotropic at equilibrium (concentrated isotropic regime). At higher concentration (n > 4.2/dL2), the rods can spontaneously orient into a nematic crystalline phase (nematic regime). The molecular dynamic of the rods is no longer Brownian. To conclude, the diffusivity of particles depends on their shape and volume concentration in the suspension. In fact, the diffusivity is considerably slowed down in the semi-dilute regime of the suspensions. On the other hand, from the StokeseEinstein law (equation (3)), the diffusivity is inversely proportional to the viscosity of the suspending liquid. Consequently, the particle diffusivity is strongly dependent of the suspending liquid. From a magnitude point of view, the diffusivity can be one-million-times lower in a melt polymer (hs w 103 Pa s) than in water (hs ¼ 103 Pa s). A priori, the Brownian motion of nanoparticles can be neglected in molten polymer. However, we will show that this Brownian motion can be at the origin of some aggregation mechanisms at long rest time in highly viscous polymer melts. The following parts of the present paper are devoted to the linear viscoelasticity of suspensions in low viscous fluids like water (latex for example) or non-entangled

R2 6Ds ðfÞ

(10)

Note that the diffusion time for a particle, i.e. the time to diffuse a distance equal to its radius, is given by equation (3) tp ¼ ð6phs R=kB TÞ. In other words and as explained by Larson, the elasticity of these suspensions is produced by the Brownian motion of the particles, which tends to restore the isotropic particle configuration under flow. By extending the StokeseEinstein law to the high frequency domain, Shikata and Pearson [16] postulated:

Ds ðfÞ ¼

kB T : 6pRh0N ðfÞ

(11)

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

Ds ðfÞ=D0 ¼

hs h0N

(12)

with

h’N 5 ¼ 1 þ fð1  fÞ1 hs 2 We reported in one of our previous works [28] the viscoelastic properties of two concentrated polystyrene lattices with different electrostatic properties around the high critical concentration. Two PS lattices have been then synthesised and named PS and PSS respectively. The major difference between the two lattices is the presence of strong acid groups on the surface of PSS. These acid groups change the degree of the hydrophilicity of the particle surface. The diameter of these particles is nearly d w 200 nm. From an experimental point of view, the complex shear modulus cannot be measured at particle concentrations lower than 35%. The variation of the complex shear modulus (G*(u) ¼ G0 (u)þjG00 (u)) versus frequency for PS lattices is shown in Fig. 1. At lower concentrations (f < 55.3%), the terminal zone of relaxation is clearly depicted, and G0 and G00 are proportional to u2 and u1 respectively. The crossover frequency uc where G0 ¼ G00 is representative of a characteristic relaxation time, sp ¼ 1/uc, which corresponds to the time for a particle to diffuse a distance equal to its radius. This mean

P. Cassagnau / Polymer 54 (2013) 4762e4775

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

relaxation time is thus generally observed to be inversely proportional to the particle diffusivity Ds(f) according to equation (10). The values of diffusion time and particle diffusivity are reported in Table 1. As expected, the particle diffusivity is considerably slowed down with increasing particle concentration. The maximum packing fraction can be defined as the solid content at which the system begins to behave like an elastic solid, i.e. with G0 > G00 over the entire experimentally accessible frequency range, and where both moduli are nearly constant (i.e. independent of frequency). At this critical concentration, one can say that the particle diffusion time tends to infinity (at least in the experimentally accessible frequency domain). This critical concentration can be assumed to be the maximum packing fraction fm. In the present system, fm must therefore be near 55.3%. Surprisingly, a quite different trend was observed for the PSS latex and these results contrast with those reported for PS samples. Fig. 2a, b (PS and PSS respectively) describe the viscoelastic

behaviour of the lattices far below the critical concentration, around this critical concentration, where the viscoelastic behaviour drastically changes, and far beyond the critical concentration. For PSS suspensions, a liquidesolid transition is observed in Fig. 2a at this critical concentration as the complex shear modulus behaves as G0 (u)fG00 (u)fun with n ¼ 0.5. There is no evidence for such a transition from liquid-like to solid-like behaviour for PS suspensions. This result means that the fractal criterion of self-similar relaxation as defined by Winter and Mours [29] can be applied to the PSS suspension. For the PS suspensions, the network elasticity resulting from the particle interactions appears to be temporarily elastic-like in entangled polymers. Actually, there is no arrangement of the particles in terms of particle aggregation and clustering. These different viscoelastic behaviours observed for PSS and PS can be attributed in large part to the fact that PSS has more electrically charged groups on its surface, therefore in an aqueous medium it has more long-range interactions leading to a fractal behaviour. 3.2. Nanotubes or nanofibres

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

f (%)

sp (s)

Ds (m2 s1)

Ds/Do

48.6 53.3 54.7 55. 1

0.008 0.02 0.5 1.0

8.3 3.3 1.3 6.7

1013 1013 1014 1015

0.75 0.30 0.01 0.006

   

As pointed out by Hobbie [30] in his recent review on shear rheology of carbon nanotubes (CNT), the main feature of semidilute suspensions of CNT under weak shear is a flow-induced aggregation and finally macroscopic agglomeration [31]. It can be assumed that the driving mechanism is due to the attractive interparticle interactions that favour aggregation. In fact, under

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P. Cassagnau / Polymer 54 (2013) 4762e4775

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.

Brownian motion and collisions induced by a weak shear flow, the nanotubes collide and stick to form at the end of the process macroscopic clusters with a size comparable to the confining cell gap. When the nanotubes have finally formed a physical network, the thermal Brownian forces are insufficient to overcome these physical contacts. Only shear deformation above a critical intensity can break up the network and disperse the nanotubes in the suspending liquid. The final isotropic dispersion of nanotubes is reached by their Brownian motion. In other words, this striking phenomenon of CNT is totally reversible and can be controlled by controlling the rest time and the shear intensity. It must be pointed out that this reversibility can only be possible in low-viscosity fluids for which the Brownian diffusion time is short enough, i.e. in the same order of magnitude as the experimental time. From an experimental point of view, a reasonable experiment time, including rest time, is one or two days. This discussion will be developed later in highly viscous polymers. This aggregation/clustering phenomenon was clearly evidenced in our recent work [32]. Simultaneous time-resolved measurements of electrical conductivity and dynamic shear modulus under different strains of untreated CNTs (L z 1.5 mm, d z 10 nm), in a low-viscosity PDMS (h ¼ 5 Pa s, not entangled chains) matrix were investigated as shown in Fig. 3. The concentration of nanotubes (f ¼ 0.2%) was chosen in order to be in the semi-dilute regime of the suspensions. First of all, it is assumed that the nanotube dispersion at t ¼ 0 of the experiment is isotropic. At low shear rate the CNT aggregation rate increases with the shear dynamic rate until a critical value (gu0 w 1). Beyond this critical value, the level of network

entanglement and/or contacts decrease as the shear rate increases. This behaviour can be explained in terms of the changing structure such as clusterecluster collision and CNT network formation [33e 38]. The network structure is given by the balance of agglomerate growth and destruction under shear deformation as quantitatively demonstrated by Ma et al. [39]. This phenomenon of CNT cluster aggregation under low shear was simply proved by optical microscopy (Fig. 4a, b). After a period at rest of a few hours, the aggregation (clustering mechanism) of the CNTs is clearly seen. It can be pointed out that the scale of this clustering is relatively macroscopic since it can be shown by simple optical microscopy. It must be pointed out that such phenomenon on the influence of shear on network formation or break up has been recently reported in the case of nanocrystalline cellulose suspensions. Indeed, Derakhshandeh et al. [40] observed that the increase of the viscosity with time was promoted at lower shear rate or under linear oscillatory shear deformation. They attributed this behaviour to an increasing effect of Brownian motion under small deformation which facilitates fibres rearrangements. Finally, these studies make the statements of Huang et al. [41] about the questioning aspect arising from some literature studies about the state of nanotube dispersion of many rheological experiments even more relevant. Fig. 5 shows that the viscoelastic behaviour strongly depends on nanotube dispersion. When the nanotubes are supposed to be well dispersed in an isotropic configuration, the variation of the complex shear modulus (under 5% strain) shows a rheological behaviour close to the viscoelastic behaviour of the PDMS matrix, i.e. a liquid viscoelastic behaviour. After the aggregation process, the viscoelastic behaviour of the

Fig. 4. Optical observation of the aggregation of CNTs. a) the CNTs are well dispersed in PDMS matrix (only micron-impurities are seen), b) the CNTs aggregate to build a network (Dt ¼ 24 h). From Moreira et al. [32], with permission from Macromolecules.

P. Cassagnau / Polymer 54 (2013) 4762e4775

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

composite shows a solid-like behaviour, at least in the frequency window used in the present study, due to the formation of the CNT network from dynamic CNT aggregation. In the same way, Hobbie and Fry [41] reported a universal phase diagram (mean cluster size versus stress) for the evolution from a solid-like network to flowing. Finally, it can be concluded that it is quite difficult to measure the linear viscoelasticity of finely dispersed carbon nanotubes as they are submitted to aggregation and clustering during measurement even under linear deformation. For example, the deviation of the rotary diffusivity to the Doi-Edward theory in the semi-dilute regime is related to a clustering phenomenon of CNTs that produces additional slowing of the rotational motion [23]. Obviously, introducing functional groups onto the CNT surface is one of the most effective methods to enhance dispersion of CNT and to prevent their aggregation in suspending liquid. Consequently, the DoieEdwards theory has been rarely applied to semi-dilute solution of carbon nanotubes. This is mainly due to the difficulties to access the terminal relaxation zone and to proceed with fine dispersion of CNT in solvent as previously explained. However, the pioneering works of Hobbie and Fry [42] and Fry et al. [43] on rheooptical studies of carbon nanotube suspensions under shear flow showed that the DoieEdwards theory can be successful at low _ 2 ). In fact, Peclet number (Pe < 200, Pe ¼ ð4=pÞ2 ððL=dÞ4 =ADro Þgf their results are in agreement with the theory prediction as they observed a leading order scaling relation for the optical anisotropy

1000

Suspension A

13.3%

100

9.7%

7.3%

10

6.2% 5.5%

1

13.3%

10

9.7%

1 G" (Pa)

G' (Pa)

such as Pe0.25. However, Ma et al. [44] proved that a mild elasticity is still observed despite CNT functionalisation. This elasticity is due to the presence of a weakly interconnected CNT network. In other words, it seems that CNT cannot be dispersed at the scale of the primary entity in the semi-dilute regime. At least, the nanotubes behave 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-phenylene) with flexible dodecyl side chains) in toluene. They observed from static and dynamic light scattering that the stiff polymer molecules are assembled into small aggregates (three to four molecules in aggregate) in dilute suspension and form larger clusters in semi-dilute regime. Actually, the molecules association into aggregates or clusters is governed by the balance of polymere polymer and polymeresolvent interactions. The conclusion is finally that toluene is not a good solvent at room temperature of these hairy-rod polymers. Fortunately, new nanoparticles of high aspect ratio such as nanorods or nanofibres based on polymeric materials have been successfully developed [46e48]. For example, the synthesis of polystyrene (PS) nanofibre aqueous dispersions was achieved via polymerisation-induced self-assembly. The method leads to selfassembled amphiphilic block copolymer nanofibres directly in water at high concentrations (>20 wt%). On the other hand, selfstabilised polystyrene nanofibres were again obtained [49] via polymerisation-induced self-assembly in water from a P(MAA-coPEOMA) hydrophilic precursor and their hydrophobic core was crosslinked by the addition of a difunctional comonomer, di(ethylene glycol) dimethacrylate (DEGDMA), at >85% of conversion of styrene. Typically, nanofibres of diameter and length nearly d z 50 nm and L z 3.5 mm were synthesised. Due to their hydrophilic surface, these nanofibres are finely and well dispersed in water. As a result, aggregation and/or clustering are prevented in the semi-dilute regime. Actually, PS fibres behave as rigid nanorods submitted to their Brownian motions in the water suspending liquid. More precisely and as shown in Fig. 6, the suspensions exhibit a clear terminal zone (G0 au2 and G00 au1) of the relaxation times for f < 6.2%. These experimental results prove that a Brownian motion of nanofibres is the dominant mechanism of relaxation. As expected, this terminal relaxation zone is shifted towards lower frequencies with the increase of the nanofibre concentrations. Indeed, the diffusion time of the nanofibres is slowed down due to the neighbouring nanofibres. At high concentration, typically f > 7.3%, the terminal relaxation cannot longer be measured from the investigated frequency range. The dominance of G0 over G00 indicates that the nanofibre relaxation

100

Suspension A

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7.3% 6.2%

0.1

5.5%

3.7%

0.1

2.2%

3.7%

0.01

2.2% 0.4%

0.4%

0.01 0.01

0.1

1

Frequency

10 (rad.s-1)

100

0.001 0.01

0.1

1

10

100

Frequency (rad.s-1)

Fig. 6. Overlay of the variation of the storage modulus G0 (u) (left) and loss modulus G00 (u) (right) versus frequency for the suspension A. From Zhang et al. [24], with permission from Macromolecules.

P. Cassagnau / Polymer 54 (2013) 4762e4775

time is longer than experimentally accessible. However, it seems that the highly concentrated suspensions are in the isotropic concentrated regime where the Brownian motions of the nanofibres cannot exist anymore. The network elasticity resulting from the nanofibre interactions appears to be temporarily elastic-like in entangled polymers. Note that such behaviour was previously observed for hard-sphere PS suspensions. Fig. 6 show clearly this argument as there is no evidence for a transition from liquid-like to solid-like behaviour. Finally, the temporary elasticity, associated with the Brownian motion of nanofibres, appears to be the relevant viscoelastic behaviour. In terms of Brownian motion, the rods are able to rotate freely in the dilute regime without any interference interaction with neighbouring rods. According to the DoieEdwards theory, the rotary diffusivity Dr0 behaves as Dr0ff0 so that the concentration dependence of the zero shear viscosity for the dilute regime (f < f*) obeys the following scaling law:

h0 ¼ hs þ

2nkB T 15Dr0

h0  hs ff1

(13)

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

h0 ¼

nkB T 10Dr



h0 ff3 h0 =hs f nL3

3

(14)

It must be noted that in semi-dilute regime, rod flexibility, Brownian motion and van der Waals attractions strongly influence the zero shear viscosity of the suspensions [50]. However, the Doie Edwards theory can be applied to any nanorod or nanofibre suspension provided that these particles exhibit a nanometric scale in order 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) and Fig. 7) and of the rotary diffusivity (equation 9 and Fig. 8) should both obey a master curve. In other words, the Brownian motion of nanorods (or nanofibres) must be universal and hence independent of their nature. The dynamic behaviour of different nanorod dispersions (carbon nanotubes [23], cellulose whiskers [51]), polymer

6 5

log( η0 /ηs )

4

3

3 2

log(ν L3) 0

1

2

3

4

0 -0,5

log(Dr / Dr0 )

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-1 -1,5 -2

-2

-2,5 -3 -3,5 -4 -4,5

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

nanofibres [24,49], was recently discussed [52]. Interestingly, in Figs. 7 and 8, it can be observed that, indeed, CNTs, cellulose whiskers and polymer nanofibres obey a master curve in agreement with the DoieEdwards theory. This universal result means that the dynamic of these different rods are dominated by the Brownian forces. 3.3. Fractal fillers The native specific structure of fractal fillers (from nano- to micro-scale) such as fumed silica and carbon black, gives specific viscoelastic properties to their derivative suspensions. Exfoliated fillers such as organoclays, carbon nanotubes and more recently graphene can also be classified in fractal fillers due to the long-range connectivity that arises from interparticle physical interactions. For example, the exfoliation of organoclay layers increases the number of frictional interactions between the layers. The cluster and/or aggregate structures can be viewed as an assembly of primary particles in a structure having a fractal dimension [53]. Due to their fractal structure and their high specific area, these fillers are subjected to form a network of connected or interacting particles in the suspending liquid. Compared with colloidal fillers without aggregation effect (our previous discussion), a direct consequence of the incorporation of fractal fillers in suspensions is the significant change in the rheological and linear viscoelastic properties [54e58]. Theoretically, the percolation threshold, fc, for a spherical sphere without any interaction should be obtained at the close packing volume fraction of fm ¼ 0.64. From a practical point of view, this particular rheological behaviour arises from the presence of the network structure (interparticle interaction), which leads to a drastic decrease in fc, typically fc < 0.05 compared with fm ¼ 0.64. Above fc the storage modulus exceeds the viscous modulus and is almost frequency-independent G0 ¼ lim G0 ðuÞ. However, it is well u/0

1 0 0

1

2

3

4

log( νL3) Fig. 7. Variation of the reduced viscosity (equation (14)) versus the concentration of nanorods. :: CNTs in PDMS, from Marceau et al. [23], A: Cellulose Whisker, from Bercea 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].

known that this gel structure is significantly altered by the surface modification of silica particles as recently discussed [59,60]. Generally speaking, the surface of native fumed silica consists in part of silanol groups that [55,61]. The balance of these interactions results in the stability of the suspension and the gelation behaviour. Consequently, the viscoelastic properties can be tuned by controlling the balance of the hydrogen interactions as the silanol surface groups can easily be modified with different functional groups of different molecular lengths from monomer to polymer [62]. From the theoretical viewpoint, the scaling concept, based on a fractal dimension, is generally used to study the effect of

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interparticle forces on the elasticity of aggregated suspensions [63e66]. To model the size dependence, the aggregates of nanoparticles are described as a fractal structure with a characteristic size x, which is the radius of the smallest particle containing N primary particles of radius a.


df

NðxÞz

x

(15)

a

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


df 3

fz

x

(16)

a

Finally, the variation of the equilibrium storage modulus G0 versus the volume fraction of silica can be expressed according to the characteristics of the non-fluctuating fractal structure [65] as follows:

Go fF5=3df

(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 unmodified silica/PDMS composites. Furthermore, they demonstrated that the fractal dimension is very sensitive to the surface silica modification as df can decrease to 1.4. Note that Shih and co-workers [68] 3þx

derived a close power laws for tactoids fillers: G0 f43df where x is an exponent related to the filler volume fraction and the aggregate structure. The exponent x depends on the number of layer particles per aggregate and is thus related to the degree of exfoliation. Klüppel [69] proposed the scaling behaviour 3þdB f

G0 f4 3df where dBf is the fractal dimension of the cluster backbone dBf z1:3. In fact, all of these fractal power laws are close to each other. On the other hand, Wu et al. [60] suggested that temperature is another variable that can control the microstructure of fumed silica suspensions. For that purpose, they used hydrophobic and hydrophilic 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 hydrophilic fumed silica (6 wt.%) in dodecane at different temperatures. g ¼ 0.2% and u ¼ 1 rad/s. From Wu et al. [60], with permission from Soft Matter.

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

storage modulus G0 with rising temperature (Fig. 9) while hydrophobic silica showed only low variation. This variation is believed to be associated to the internal restructuring of aggregates. In terms of fractal dimension, the temperature tends to increase the fractal dimension df resulting in a more compact cluster and a denser network as schematically shown in Fig. 10. Interestingly, Trappe and Weitz [70] and more recently Selimovic et al. [71] reported the transformation from viscoelastic gels to viscoelastic liquid (Fig. 11) within an ageing time of 101e103 h in suspensions of precipitated silica in silicone oil. The aging process is slowed down by increasing the silica particle concentration or by decreasing the molecular weight of PDMS. A priori the PDMS chains are in the Rouse regime of the viscosity and are not entangled. The breakdown of the particle network can be explained qualitatively by the PDMS chain adsorption on silica particle surfaces. As the PDMS chains are not entangled, the PDMS chains bound onto silica surface cannot create gel chains as generally observed in rubber (bound rubber) causing the interparticle hydrogen bonding to decrease and then to transform the initial gel to viscous fluid.

Fig. 11. Variation of the complex shear modulus (G0 : full symbol and G00 : open symbols) 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.

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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.

Regarding layer fillers, we studied [72] the linear viscoelastic behaviour of graphite oxide (GO) and graphite oxide functionalised with 3-(acryloxypropyl) trimethoxysilane (GO-M) dispersed in low molar mass PDMS (no entangled chains). The transmission electron microscopy and X-ray diffraction showed an intercalated morphology of graphite oxide potentially exfoliated in PDMS suspension. The PDMS suspension filled with the graphite oxide showed a drastic change in the viscoelastic properties for weight fractions up to 6.5 wt%. In fact, for GO concentrations higher than 1.5 wt%, the viscoelastic behaviour does not show any terminal flow zone and the elastic character of this suspension becomes dominant at low frequencies with the appearance of a secondary plateau and a shear thinning behaviour (rh*(u)rfu1) as shown in Fig. 12a. Furthermore, the low percolation threshold of GO sheets is attributed to their macro-scale aggregation/agglomeration driven by their Brownian motion in these low viscosity PDMS suspensions. It was further concluded that the surface modification of graphite oxide by 3-(acryloxypropyl) trimethoxysilane prevents the aggregation of GO-M sheets resulting in a Newtonian viscous behaviour of the suspensions as shown in Fig. 12b. The surface functionalisation with 3-(acryloxypropyl) trimethoxysilane of the GO layers prevents their aggregation by screening the depletion forces. The dominant force in the present case is the repulsion force between these submicron particles. On the other hand, the visual observation of droplets deposited on a horizontal glass slide leads to some relevant and additional discussions about these suspensions. After a 24 h rest period, it can be observed in Fig. 13b that the GO-M/PDMS droplet spreads over the glass slide whereas the GO/PDMS one shows a yield stress liquid behaviour (Fig. 13a). Finally, the low percolation threshold of GO sheets is due to their aggregation at a macro-scale under the driving force of their Brownian motion. Such a mechanism is the dominant one in low viscosity suspensions in the dilute regime for which the relaxation Brownian time is in the same order of the characteristic time of the experiment. In the present case, the rotational relaxation time of GO sheets was estimated according to Eq (6) at srz15 s meaning that GO particles are able to recover their random orientation in a few seconds after their anisotropic orientation under flow. As previously discussed, the carbon nanotubes in suspension are not fractal if they are well stabilised. However, these ideal cases are scarce as the nanotubes tend to aggregate under Brownian motion combined with weak flow. Actually, many works report a fractal rearrangement of carbon nanotubes in suspensions. From a variety of light-scattering experiments Chen et al. [73] and Saltiel et al. [74] showed that carbon nanotubes in suspension formed networks with 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 rest time. At the initial time of measurement df as-received ¼ 2.27 and df treated ¼ 2.5 for as received (no treatment or at least unknown treatment) and treated (acid treatment based on a mixture of HNO3 and H2SO4) SWNT respectively. This means that the treated SWNT form a compact structure at small-length scale. On the other hand, as-received nanotubes showed a stable network structure over few days as the fractal dimension is quite constant while treated SWCNT showed a significant variability over the days. The fractal dimension was also observed to be dependent on the surfactant used to stabilise the suspension. From electrical experiments, Lisunova et al. [76] showed that the fractal dimension continuously increased with surfactant concentration from df ¼ 2.5 to df ¼ 3. In fact, a more homogenous network is formed with increasing the CNT dispersion. However, the number of physical contacts between nanotubes for the electrical conductivity decreases. Finally, Khalhkal and Carreau [36] showed from the scaling behaviour of

Fig. 13. Visual observation of PDMS-based droplets (upper pictures: t ¼ 0 h) after a 24 h rest period: (a) Graphite oxide, (b) Graphite oxide decorated with 3-(acryloxypropyl) trimethoxysilane. The scheme is a representation of the filler morphology.

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the linear viscoelastic properties that the CNT suspensions have a fractal dimension df ¼ 2.15 corresponding to a weakly flocculating network. Furthermore, this fractal structure was observed to be slightly dependent on the flow history due to an initial compact structure of the CNT network. To conclude, carbon nanotubes in suspension generally behave as fractal nanofillers due to the long-range connectivity that arises from interparticle physical interactions. It is then generally accepted that the association of CNT into a fractal structure seems more relevant for low percolation threshold than the random distribution of isolated particles finely dispersed and stabilised in the liquid medium. On the other hand, it is extremely difficult to compare literature results as most of the time the CNT suspensions are quite different in terms of CNT nature and suspending liquid. Furthermore, the determination of the fractal dimension from the CNT concentration dependence of the equilibrium modulus (and/or the transition between linear and non-linear regime) is questionable. Precise criteria for the equilibrium modulus have been defined for crosslinked systems [29]. A priori these criteria can be extended to physical systems; however it seems that these criteria are not systematically used in most of the studies. To conclude, the Brownian forces are the dominant ones at least in the dilute and the semi-dilute regime of the suspensions. However, the interparticle interactions generally lead to the formation of fractal network. The anisotropy of the materials in coating or latexes applications for example can be then favoured and controlled. Taking for example a painting process, the anisotropy of the film can be controlled by combining the paint flow process (shear or elongational) with the kinetics of painting cure or filmification. 4. Highly viscous suspensions: molten filled polymers The use of nanoparticles has been of considerable interest in recent years to improve the mechanical, electrical and barrier properties of polymers. The last ten years have seen the emergence and the developments of nanocomposites. A lot of papers and reviews e too many to cite them all e have addressed the rheology of nanocomposites. For example, reviews were addressed by Litchfield and Baird [76] on the rheology of high aspect ratio nanoparticles filled liquids and by Cassagnau [11] on melt rheology of organoclays and fumed silica nanocomposites. Many authors have discussed the connection that can be made between the filler morphology (structure, particle size) and the melt viscoelastic properties of polymeric materials. Many papers have been devoted to the mechanisms of polymer reinforcement [62,77e83]. Even the polymer reinforcement is still an open debate in the literature, the last work of Jouault et al. [83] on model nanocomposites seems to have proved the relevant mechanisms. First of all, two regimes of

4771

filler concentration must be considered: dilute and concentrated regimes. In the dilute regime, the reinforcement implies a polymer chain contribution due mainly to physical chain entanglements. In the concentrated regime, the reinforcement directly depends on the particleeparticle interactions and consequently on the rigidity of the filler network. Then, the present review is focused on the time evolution of the filler structure during annealing (or rest time) upon shear flow cessation. In fact, the viscous forces are the dominant ones in molten nanocomposites based on entangled polymer matrix. As a result, the nanofillers are submitted to strong orientation under flow resulting in the break-up of the nanofiller network. However, the filler structure upon the cessation of the flow is not stable and most of the time this nanostructure was observed to change with time. The measurement of the complex shear modulus is then used to follow this time evolution generally consisting of the build up of the filler network. Actually, this build up process is really a recovery experiment from a rheological viewpoint. The most striking question is: what is the relevant mechanism of this recovery process? Cassagnau and Mélis [84] performed recovery tests, subsequently to strain sweep (non-linear deformation), of the complex shear modulus in the case of PP filled with fumed silica. These experiments showed evidence that the initial equilibrium network structure cannot completely restore within several hours as shown in Fig. 14. It is clearly seen that this modulus evolution obeys a twostep process. However the authors did not exploit these results. Such recovery phenomenon was also reported on organoclay-based nanocomposites [85e89] and fumed silica nanocomposites [90,91]. Bailly et al. [91] showed that the aggregation in melt nanocomposites (ethylene-octene copolymer (EOC)/nanosilica) depends on silica surface treatment. Rest time experiments showed a pronounced tendency towards aggregation for the EOC-g-VTES-based nanocomposites (VTES: vinyltriethylsilane). In contrast, strong polymerefiller interactions between EOC-g-VTEOS (VTEOS: vinyltriethoxysilane) and SiO2 modified with octylsilane showed a timestable response. This latter nanocomposite is characterised by stronger adhesion between the silica filler and the EOC matrix. More recently, Kim and Macosko [92] observed that the storage modulus of polycarbonate/graphene nanocomposites increases with annealing time over a few hours. The authors explained this phenomenon by the increased effective volume of rotating disc and the restoration of an elastic network, which was broken down by the squeezing flow during sample loading. This phenomenon of time evolution in nanocomposites has recently been studied in detail by Zouari et al. [93] on organoclay/polypropylene nanocomposites. The authors suggested to follow this time evolution from the time variation of the melt yield stress. Lertwimolnun and Vergnes [94] used a Carreau-Yasuda law with a yield stress to fit the absolute complex viscosity of a melt nanocomposite.

107 PP

G’ and G” (Pa)

106

T=200°C

105

G'

104

G"

3

10

10-2

10-1 100 Strain (%)

101

102 100

101

102 103 Time (s)

104

105

Fig. 14. Variation of the complex shear modulus versus strain amplitude (u ¼ 1 rad/s) and complex shear modulus recovery versus time (u1 rad/s and g ¼ 0:1%). Silica: 16%wt, T ¼ 200  C. From Cassagnau et al. [84], with permission from Elsevier.

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

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

   s0   a n1 þ h0 1 þ ðluÞ a h*ðuÞ ¼

u

(18)

Where s0 is the melt yield stress, h0 is the zero shear viscosity, g is the time constant (i.e. relaxation time), a is the Yasuda exponent and n is the power law exponent. Fig. 15 shows the variation of the melt yield stress versus recovery time at different temperatures. These experiments are

subsequent to a steady shear flow of 1 s1 for 155 s. Many comments can be addressed from this figure. First, it is observed that the yield stress evolution follows a two-step process. The first kinetics (t < tcrit,s0 a t1/2) is related to the disorientation of clay tactoids and the second one to the aggregation of these tactoids and layers into a network (t > tcrit, s0 a t1). Second, these kinetics are accelerated with temperature as expected. However, the network structure of the nanocomposites does not seem to stabilise even at long rest times. Third, the kinetics transition time tcrit can be compared to the time of disorientation that can be theoretically calculated from the rotary diffusivity Dr0 (equation (6)). Then, Zouari et al. [93] calculated that the time from the rotary diffusion (1/Dr0) is much shorter than the time corresponding to the end of the first kinetics associated with platelet disorientation by Brownian motion. However, this calculation was made assuming a dilute concentration of clay platelets. Actually, this hypothesis is over-simplified as these platelets are in a semi-dilute regime a priori with a slow down of the diffusion process. Note that Kim and Macosko [92] showed in the case of graphite nanocomposite that the rotational relaxation time is in the order of magnitude of the experimental time of recovery. To conclude, the first kinetic can be associated with the disorientation of nanofillers by Brownian motion and the second one to the aggregation of clusters into a network. In the same way, Cao et al. [95], Tan et al. [96] and Song et al. [97] studied the influence of annealing time on carbon-black-based nanocomposites. They showed that annealing over a few hours leads to a liquid-to-solid like transition. As previously explained, they came to the conclusion that the variation of the viscoelastic properties (and electrical resistivity) comes from the aggregation of carbon black. However, they suspected that the interfacial tension between polymer (polystyrene in the present case) and carbon black plays an important role in the aggregation mechanism of carbon black. Regarding carbon nanotubes in polymer melts, the break up and formation of the network was extensively studied by the groups of Alig and Pötschke [98e100]. One of the striking features in polymer melt is the dispersion of nanotubes [41]. The first challenge is to separate the carbon nanotubes from their initial agglomerate assemblies (bundle). Assuming that the nanotubes have been finally well dispersed as individual entities, Huang [41] showed that at low concentration, probably in the dilute regime, the nanotube are quite stable against re-aggregation provided that the viscosity is high enough to suppress Brownian motion. From our own study on nanotubes 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 be neglected. A fortiori, the Brownian motion of nanotubes in the

Fig. 17. TEM pictures for polycarbonate/MWNT nanocomposites (0.6vol% MWNT). (a) The nanotubes are well dispersed in PC matrix after shearing. (b, c) The nanotubes form aggregates after annealing at T ¼ 300  C. From Alig et al. [99] with permission from Elsevier.

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semi-dilute regime can be totally neglected as the rotary diffusion is considerably slowed down with the concentration (equation (9)). However, aggregation of nanotubes in melt polymers is not a fiction as has been reported in the literature. For example, Alig et al. [99] clearly showed from linear viscoelasticity, electrical conductivity and transmission electronic microscopy (TEM) that aggregation is an effective process during annealing without the application of any shear forces. Fig. 16 shows that the storage modulus increases with the annealing time and Fig. 17 shows (TEM observation) that aggregates have been formed during annealing. If Brownian motion can be assumed in suspensions of low viscosity as previously discussed, another process must be considered in molten polymers of high viscosity (typically h > 100 Pa s) for the re-aggregation of nanotubes during annealing. Alig et al. [99] assumed that the driving force for the re-aggregation of CNTs in melt polymers is a combination of strong dispersive interactions between nanotubes and depletion interactions between adjacent nanotubes. Finally, the main striking behaviour of nanocomposites based on thermoplastic polymers filled with fillers of high aspect ratio is the control of the anisotropy of these materials. Indeed from a processing point of view, the fillers are submitted to non linear flow resulting in a strong difficulty to control the anisotropy and consequently the difficulty to control the mechanical or electrical properties for example. In other words, we experience the processing conditions rather than control.

that the aggregation behaviour is governed by the Brownian motion and local particle rearrangements under linear deformation in steady or dynamic (frequency) conditions. In the third and last part, we investigated the viscoelastic and dynamic behaviours of nanocomposites, i.e nanoparticles dispersed in entangled polymers. As the viscous forces are undoubtedly the dominant ones, the nanofillers are submitted to strong orientation under flow resulting in the network break-up. It is generally observed from linear viscoelastic measurements that the filler structure is not stable upon the cessation of the flow. In the case of platelet nanocomposite (organoclays, graphite oxide) a two-step process of recovery is generally reported: the first one is the disorientation of the fillers followed by a second phenomenon of aggregation/clustering. In the case of carbon nanotubes only the aggregation is observed, at least the two phenomena cannot be discerned by rheology. A priori the disorientation of nanofillers can be assumed to be controlled by the Brownian motion. However, other mechanisms such as dispersive interactions and depletion forces must be considered for the aggregation of fillers in a physical network.

5. Conclusion

References

The objective of the present paper was to review the linear viscoelasticity of suspensions and of polymers filled with nano-size particles of different aspect ratios and structuration. It is of importance to point out that the viscoelastic behaviour of nanofilled system must be discussed from the knowledge of the dispersion of nanoparticles in terms of dispersion at different scales (from primary to aggregate particles) and anisotropy (privileged orientation of non spherical particles). First, we reported and discussed the linear viscoelasticity of suspensions (fluid of low viscosity) filled with well-dispersed and stabilised particles (PS latex, silica, carbon nanotubes and PS nanofibres). For these suspensions, the Brownian motion is the dominant mechanism of relaxation. For the spherical particles, the semi-empirical equation of KriegereDougherty is the most relevant one to predict the viscosity of these suspensions. Carbon nanotubes, cellulose whiskers and polymer nanofibres in dilute and semi-dilute regimes obey a universal diffusion process by Brownian motion according to the DoieEdwards theory providing that these nanofillers are well dispersed and stabilized in the suspending liquid. The second part of the present review was devoted to the viscoelastic behaviour of fractal filler suspensions. Two categories of fractal fillers can be distinguished: nanofillers such as fumed silica and carbon black due to their native specific structure from nano- 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-range connectivity that arises from interparticle physical interactions. The particular 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 fc (typically fc < 0.05) at which the zero shear viscosity diverges to infinity. Scaling concepts, based on a fractal dimension, are generally used to study the effect of interparticle forces on the elasticity of aggregated suspensions. Fractal exponents are then derived and related to the structure of the aggregate clusters. It must be pointed,

Acknowledgements Prof B. Charleux is thanked for her stimulating discussion on the dynamic of polymer fibres and for kindly reading the present manuscript.

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