Reinforcement of elastomers

Current Opinion in Solid State and Materials Science 6 (2002) 195–203

Reinforcement of elastomers b ¨ Gert Heinrich a , *, Manfred Kluppel , Thomas A. Vilgis c a

b

Materials Research, Continental AG, D-30001 Hannover, Germany ¨ Kautschuktechnologie, D-30519 Hannover, Germany Deutsches Institut f ur c MPI for Polymer Research, D-55021 Mainz, Germany

Received 26 March 2002; received in revised form 2 April 2002; accepted 2 April 2002

Abstract The review describes recent research about reinforcement in elastomers where the main intention is to gain insight into the relationship between disordered filler structure on different length scales and reinforcement and to microscopic mechanisms of strain enhancement. Several theoretical concepts will be discussed together with very recent experimental findings related to hydrodynamic reinforcement, rigid filler aggregates with fractal structure and polymer adsorption on heterogeneous filler surfaces. Based on the new concepts, we present several recent efforts to understand typical effects in filled rubbers that have an extraordinary practical importance (for example, stress softening of carbon black filled rubbers during repeated large strain stretching and during small strain dynamic excitations).  2002 Published by Elsevier Science Ltd. Keywords: Elastomer reinforcement; Disordered filler structure; Microscopic mechanisms; Strain enhancement; Hydrodynamic reinforcement; Rigid filler aggregates; Fractal structure; Polymer adsorption

1. Introduction Rubber elasticity has a long history. Ancient Mesoamerican peoples were processing rubber by 1600 B.C. [1] which predated development of the vulcanization process by 3500 years. They made solid rubber balls, solid and hollow rubber human figurines, wide rubber bands to haft stone ax heads to wooden handles, and other items. The use of fillers—especially, carbon black—together with accelerated sulfur vulcanization, has remained the fundamental technique for achieving the incredible range of mechanical properties required for a great variety of modern rubber products. Increased reinforcement of the rubber material has been defined as increased stiffness, modulus, rupture energy, tear strength, tensile strength, cracking resistance, fatigue resistance, and abrasion resistance [2]. Accordingly, a practical definition of reinforcement is the improvement in the service life of rubber articles that fail in a variety of ways, one of the most important being rupture failure accelerated by fatigue processes, such as occurs during the wear of a tire tread. The main intention of the present review is to gain further inside into the relationship between disordered filler *Corresponding author. E-mail address: [email protected] (G. Heinrich).

structures and the reinforcement of elastomers which is discussed mainly for the static and dynamic (shear or tensile) modulus. We will recognize that the classical approaches to (filled) rubber elasticity are not sufficient to describe the physics of such disordered systems. Instead, different theoretical methods have to be employed to deal with the various interactions and, consequently, reinforcing mechanisms on different length scales. In the case of a dilute suspension of spherical inclusions the increase of the shear modulus of filled rubbers is [3]

S

D

Gi 15s1 2 nmd 1 2 ] G0 G f 5 ] 5 1 2 ]]]]]]]w G0 Gi 7 2 5nm 1 2s4 2 5nmd] G0

(1)

where w is the volume fraction of the inclusions, nm is the Poisson ratio of the matrix, and the subscripts i and 0 refer to the inclusions and the matrix, respectively. With the assumption of perfectly rigid inclusions (Gi 4 G0 ) and an incompressible matrix, nm 51 / 2, Eq. (1) becomes the Einstein–Smallwood equation 5 f 5 1 1 ]w 2

(2)

Guth and Gold [4] later extended the relationship in Eq.

1359-0286 / 02 / $ – see front matter  2002 Published by Elsevier Science Ltd. PII: S1359-0286( 02 )00030-X

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(2) to higher concentrations taking interparticular disturbances into account. They found 5 f 5 1 1 ]w 1 14.1w 2 2

(3)

However, for typical loadings of fillers up to volume fraction w ¯ 0.35, a Pade´ approximation of the expansion of f up to second order in the volume fraction, 5 2.5w f ¯ 1 1 ]w 1 5.0w 2 1 ? ? ? ¯ 1 1 ]] 2 1—2w

(4)

turned out to be a suitable and theoretically founded expression for f [3,5]. If the hypothesis of spherical particles is released, Eqs. (3) and (4) do not have anymore a univocal formulation. For active fillers, however, f no longer depends on the simple filler volume fraction w but on some effective volume fraction weff . Medalia [6] added ‘occluded rubber’ volume to the actual carbon black filler volume to obtain the ‘effective’ volume of the rigid phase. ‘Occluded rubber’ was defined as the rubber part of the elastomeric matrix which penetrated the void space of the individual carbon aggregates, partially shielding it from deformation. Mullins and Tobin [7] have recommended the use of the same reinforcing factor of the modulus, f, to estimate the effective strain, L, in the filled rubber matrix:

L 2 1 5 f( l 2 1)

Fig. 1. Variation of the microscopic overstrain factors f as a function of the filler volume fraction w and two strains (after Ref. [**8]). The measurements are based on an SANS approach. The lower solid line is the behavior predicted from continuum according to the Pade´ approximation (Eq. (4)). The upper curves are obtained after substitution of an effective filler volume fraction weff in Eq. (4).

(5)

2. Microscopic mechanisms of strain enhancement Very recently, Westermann et al. gave the first direct microscopic insights into the mechanisms of strain enhancement in reinforced networks [**8]. They investigated the matrix chain deformation by small-angle neutron scattering (SANS) using a special designed filler–matrix system, a triblock copolymer of the type PI–PS–PI with a polystyrene middle block of wPS 50.18 and two symmetric polystyrene wings. Due to the repulsive interaction between the PS and PI blocks, this block copolymer undergoes a thermodynamically driven microphase separation; i.e., for this composition spherical PS domains are formed that can be considered as model fillers. The degree of the in situ filling was adjusted by blending the PI–PS–PI starlike micelles with a PI homopolymer matrix as the soft rubbery phase. The microscopic matrix chain deformation in the reinforced network was evaluated using a tube-like model of rubber elasticity [9] which has been proved recently to successfully describe the chain deformation of unfilled networks [10,11]. Fig. 1 displays the experimental values for the amplification factor f versus the volume fraction of filler for two different deformation ratios l. The lower line corresponds to the Pade´ approximation according to Eq.

(4); i.e., the experimental overstrain factors are underestimated if the volume fraction of filler is entirely given by the chemical composition of the triblock copolymer. However, if one introduces an effective (and strain-dependent) volume fraction [12], weff 5 ws1 1 ds ldd, one obtains the upper curves in Fig. 1 after substitution of the effective volume fraction weff for w in Eq. (4). The dependence of s1 1 ds ldd as a function of the macroscopic sample deformation was found to be linear (s1 1 ds ldd 5 1.1 1 0.12l) and very close to the experimental variation of the domain radius from small-angle X-ray scattering (SAXS), R 5 R 0s1 1 0.11ld where R 0 is the isotropic domain size [11]. The extrapolation of the SANS data to the isotropic state, 1 1 ds l → 1d ¯ 1.22, confirms, indirectly, the presence of a more or less hard transition layer between filler and rubbery matrix. This finding of the existence of a diffuse ˚ around the PS PS–PI boundary layer (thickness d | 5 A) ˚ could also domain with a mean filler radius of about 84 A be concluded from the synchrotron data [11]. Vieweg et al. [13] estimated an interface thickness of d|1.5 nm for polymeric fillers (microgels) consisting of cured polybutadiene with a glass transition temperature (shear loss factor maximum at frequency 1 Hz) of about 100 8C. The filler particles diameters were in between |24 and 75 nm; and the sample matrices were commercial statistical styrene–butadiene emulsion copolymers vulcanized with dicumylperoxide. The method they used in Ref.

G. Heinrich et al. / Current Opinion in Solid State and Materials Science 6 (2002) 195–203

[13] was to find the lengths of immobilized interfacial modes from optimum collapsing of (dynamic shear) data points in a scaling procedure; i.e., reinforcement in the dynamical experiments is explained by loss of mobility for longer polymer modes caused by too short lengths available. Comparing these mode lengths with the bulk characteristic length of glass transition and network lengths in the rubbery matrix, a dispersion law (mode length versus frequency: v | d k ) of the network modes in the shear relaxation zone between main transition and network– rubbery plateau zone could be determined with an unexpected large exponent k ¯ 2 12.5 [13]. This large exponent indicates that the network lengths affect the ‘network mode’ mobility very effective: large, steeply increasing times are needed to rearrange the crosslinks and entanglements in length scales comparable with their mutual mean distance. So far we reported some experimental determinations of the overstrain factor and filler–matrix interfaces for spherical model fillers. As already noted, active fillers like carbon black or silica are of special interest in rubber industry. First direct results of SANS measurements of the form factor of matrix chains in cross-linked, silica-filled elastomers in the isotropic state were reported, very recently, by Botti et al. [28]. Preliminary results show the influence of reinforcing agent, Si 69 in this case, on the reinforcing factor. Si 69 is a bifunctional polysulfidic organosilane for the rubber industry defined chemically as bis(3-triethoxysilylpropyl)tetrasulfane. It is used to improve the reinforcing capacity of fillers with silanol groups on their surface. Using this technology of precipitated silica and reinforcing agent, together with special solution polymerized statistically styrene–butadiene copolymers (SSBR) as hydrocarbon polymer matrix, the green tire tread could be developed. The green tire technology dramatically improves fuel economy (rolling resistance) and overall performance (especially, anti-braking system supported wet skid behaviour) over conventional tires. Preliminary simulations of anisotropic scattering from aggregates of small hard spherical particles embedded in an elastic polymer matrix were reported from Oberdisse et al. [29]. Imposing an affine displacement inside the matrix to the fillers, two-dimensional scattering spectra are shown which reproduce several experimentally observed isointensity curves. The filler particles are organized in several types of aggregates (crystalline, amorphous compact, fractals). Mark et al. developed models which enable the estimation of the effect of the excluded volume of oriented prolate filler particles in a cubic matrix on elastomeric properties (Ref. [36] and Refs. therein). Calculations of reinforcement due to the presence of the filler particles were performed using the rotational isomeric state method combined with a simulation approach for non-Gaussian networks. The obtained results underscore the sensitivity

197

of the stress–strain isotherms to the anisotropic effects of the oriented prolate particles.

3. Reinforcement of fractal fillers As known, carbon black consists of spherical particles with a rough and energetically disordered surface forming rigid aggregates in the |50–100-nm range with a disordered (fractal like) structure. These aggregates are the smallest entities in the rubber matrix. Agglomerations of the aggregates on a larger scale lead to the formation of filler clusters and even filler network at high enough concentration [14]. In the case of disordered aggregates, the strain amplification factor f is predicted to scale according to a power law j a , where a depends on the fractal dimension of the aggregates of mean size j [*15]. One obtains for the excess shear modulus the expression

j ]D S b f 21| S]bj D

df 2 ] 22d f 11 D df 2 ] 2d f D

w

2 ]

w 32d f

without overlap, w , wc (6) with overlap, w . wc

Here, the aggregate structure is characterized by the fractal exponent d f (mass fractal dimension) and D (spectral dimension as a measure of aggregate connectivity). The exponent 2d f /D is equal to the fractal dimension d w of a random walker that stumbles around on the fractal object. The dimension d w determines how time t scales with the rms displacement from the local origin where the walker started. The length b is the size of a carbon black sub-unit (primary particle) which constitutes the aggregates. The overlap condition w . wc 5s j /bd d f 23 is a function of aggregate structure (Fig. 2). If the aggregates are assumed to have no branches, i.e., D 5 1 and d f 5 2, the results for the case of random walk-like aggregate structure are recovered. For small filler concentrations (i.e., no aggregate overlap), a linear dependence of the reinforcement on filler concentration is found, as expected. At high concentration, however, the relation sensitively depends on the aggregate structure. For the case of a realistic modeling of carbon black aggregates by diffusion-limited aggregation (DLA) clusters [*15] with d f ¯ 2.5 and D ¯ 4 / 3, one finds

j ]D S b f 21| S]bj D

21 / 4

w

for

w , wc (7)

25 / 4

w

4

for

w . wc

which means a strong deviation from the linear w -dependence at large volume fraction. At even higher fractal dimension d f , the overlap condition would be increasingly hard to meet, because no overlap is possible in the limiting case of compact filler particles, d f 5 3.

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SOl 3

DS D SO D

Te 23 ? 12] ne Gc m 51 WR 5 ] ? ]]]]]]] 3 2 Te 12 ] l2 2 3 n e m 51 m

5

2 m

F SO

Te 1 ln 1 2 ] ne

3

l 2m 2 3

m 51

DG 6

SO 3

1 2Ge ?

l m21 2 3

m 51

D

(8) The first term describes the cross-link constraints of the polymer network, whereby the modulus Gc is proportional to the cross-link density mc . The second term is the result of tube constraints, whereby the modulus Ge is proportional to the entanglement density me of the rubber. The parenthetical expression in the first term takes into account the finite chain extensibility [24]. Here, l m is the deformation ratio in spatial direction m, n e is the number of chain segments between two entanglements and T e is the trapping factor (0,T e ,1), which characterises the portion of elastically active entanglements. T e increases as the crosslink density increases, whereas n e and me —as terms that are specific to polymer—are to a great extent independent of cross-link density. For uniaxial extensions l1 5 l; l2 5 l3 5 l 21 / 2 , the following relation for the nominal stress sR, m 5 ≠WR / ≠l m relative to the initial cross-section can be derived from Eq. (8): 1 2 T e /n e sR,1 5 Gcs l 2 l 22d ]]]]]]]]2 Te 2 1 2 ]s l 1 2 /l 2 3d ne

5S

T e /n e 2 ]]]]]] Te 2 1 2 ]s l 1 2 /l 2 3d ne Fig. 2. Schematic view of flocculated filler particles in elastomers (a) below and (b) above the gel-point wc of the filler network.

4. Stress-softening and fractal filler cluster breakdown Expression (7) has been used to simulate rubber reinforcement and stress softening during rubber deformation simultaneously [20–22]. Stress-softening in rubber testing is of extraordinary importance in rubber technology and often termed as Mullins effect. Usually, for technical carbon black- or silica-filled rubbers the condition w . wc is fulfilled. In a deformed state the cluster size j becomes deformation dependent. The polymer network contribution of the total free energy density is based on the non-Gaussian tube model with non-affine tube deformation [19–23]:

D

6

1 2Ge ( l

21 / 2

2l

22

)

(9) By fitting experimental data to this function, the network parameters Gc , Ge and n e /T e can be determined. Besides the action of the polymer network the effect of reinforcing fillers, e.g., the pronounced stress-softening, has to be considered for a proper description of real elastomer materials. Following the line of previous papers [19–22], this is obtained by a hydrodynamic strain amplification factor f in L 2 1 5 fs l m 2 1d ? ( l m 2 1) 5 fs´md ? ´m , which describes the excessive strain of the polymer chains due to the presence of rigid filler particles and clusters. The dependency of the amplification factor to the deformation f 5 f(´m ) is different for the first deformation of a virgin sample and for subsequent deformations, leading to the characteristic stress-softening: On the first deformation, f(´m ) decreases gradually as the deformation increases, whereby on the subsequent deformations a constant amplification factor f 5 fmax takes effect. It is achieved at

G. Heinrich et al. / Current Opinion in Solid State and Materials Science 6 (2002) 195–203

maximum deformation ´m ,max during the first extension, i.e. fmax 5 f(´m ,max ). This concept of a stress induced irreversible breakdown of filler clusters has been successfully formulated and applied for uniaxial extensions [19–21]. For finite element engineering applications a constitutive formulation for any triaxial deformations is required. An obvious concept for the generalisation of the previous model refers to an averaging of the uniaxial model over all directions in space. This results in a scalar deformation variable E, which is dependent on the first deformation invariant I1 : ]] I1 (´m ) E(´m ): 5 ]] 2 1 (10) 3

œ

where

O s1 1 ´ d 3

I1 (´m ): 5

m

2

(11)

m 51

The generalisation consists of a substitution of the uniaxial extension variable ´ with the scalar function E(I1 ), which is dependent on the deformation ´m in all directions of space. By applying the Huber–Vilgis amplification factor Eq. (6) in the case of rigid fractal filler clusters, the following amplification factor results:

S D

jsEd f(´m ) ; f(E) 5 1 1 const. ? ]] b

d w 2d f

2 ]

? w 32d f

(12)

Here, j (E) is the deformation-dependent cluster size. For this function j (E) two different empirical approaches were introduced [19–21]:

S

D

(13)

The amplification factor f(E) is expressed then as follows, using Eq. (12): fsEd 5 X` 1sX0 2 X`d ? e 2 a ?E

(14)

with a 5 g (d w 2 d f ). The following abbreviations are used here:

S D

j0 X0 5 1 1 const. ? ] b X` 5 1 1 const. ? w

d w 2d f

2 ]

? w 32d f

2 ] 32d f

(15)

(ii) power-law cluster breakdown:

S

D

j jsEd ]] 5 ]0 2 1 ?s1 1 Ed 2 b 1 1 b b

(16)

In this case, the following results for the amplification factor f(E): fsEd 5 X` 1sX0 2 X`d ?s1 1 Ed 2y with y 5 b (d w 2 d f ).

Based upon the two approaches, a fit procedure has been applied for the stress–strain data obtained with solution polymerised statistical styrene–butadiene copolymer (SSBR) vulcanizates filled with carbon black and silica with 40 and 60 phr filler content. Figs. 3 and 4 show examples of the adaptation results for the first and third uniaxial stretching cycles of the specimens with various pre-strain. First, the third stretching are adapted for various prestrains ´max with one single set of polymer parameters Gc , Ge and n e /T e . The different pre-strains are considered by various strain independent amplification factors Xmax . The resulting fit parameters are documented in the legends for Figs. 3 and 4. The quality of the adaptations (solid lines) is quite good across the entire extension area, as was already the case for the carbon black filled specimens tested previously [19]. For the calculation of the first extension without prestrain, the resulting amplification factors fmax are plotted in Figs. 5 and 6 for four different specimens against the deformation variables E(´max ) according to the postulated behaviour of Eqs. (14) and (17), respectively (discrete symbols). The solid lines are corresponding fit curves. The obtained fit parameters X0 and X` , and a and y, respectively, and are specified in the legends of Figs. 5 and 6. Using these fit curves for f(E), simulations of the first extensions (Eq. (9)) are depicted in Figs. 3 and 4. It becomes obvious that the power-law cluster breakdown provides a better adaptation. An almost ideal simulation is achieved especially for large extensions.

5. Filler networking effects and small-strain dynamic mechanical excitations

(i) exponential cluster breakdown:

j jsEd ]] 5 ]0 2 1 ? e 2g ?E 1 1 b b

199

(17)

Similar to large rubber sample deformations the role of filler cluster structure and its breakdown becomes important for the understanding of the small-strain dynamical deformation. This effect leads to a amplitude-dependence of the dynamic viscoelastic properties of filled rubbers and is often referred as the Payne-effect (for a very recent review, see Ref. [14]). When a sinusoidal strain is imposed on a linear viscoelastic material, e.g., unfilled rubbers, a sinusoidal stress response will result and the dynamic mechanical properties depend only upon temperature and frequency, independent on the type of deformation (constant strain, constant stress, or constant energy). However, the situation changes in the case of filled rubbers. The presence of filler introduces, in addition, a dependence of the dynamic mechanical properties upon dynamic strain amplitude. This is the reason why filled rubbers are considered as non-linear viscoelastic materials. Considerable progress has been obtained in the past in relating the typical dynamical behaviour at low strain amplitudes to a cyclic breakdown and reagglomeration of physical filler–filler bonds in typical clusters of varying size, including the infinite filler network. Common features

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Fig. 3. Uniaxial stress–strain data and fits for an S-SBR sample filled with 40 phr carbon black. Fit parameters are shown in the insert. Moduli Gc and Ge in MPa.

between the phenomenological agglomeration / deagglomeration Kraus approach [*25] and very recent semi-microscopical networking approaches (two aggregate VTG model [26], links–nodes–blobs model [27], kinetical cluster–cluster aggregation [*16]) are discussed recently [14]. All semi-microscopical models contain the assumption of

geometrical arrangements of sub-units (aggregates) in particular filler network structures, resulting for example from percolation or kinetical cluster–cluster aggregation. These concepts predict some features of the Payne effect that are independent of the specific types of filler. These features are in good agreement with experimental studies.

Fig. 4. Uniaxial stress–strain data and fits for an S-SBR sample filled with 40 phr silica. Fit parameters are shown in the insert. Moduli Gc and Ge in MPa.

G. Heinrich et al. / Current Opinion in Solid State and Materials Science 6 (2002) 195–203

Fig. 5. Variation of the amplification factor according to the assumption of exponential cluster breakdown Eq. (14).

For example, the shape exponent of the storage modulus, G9, drop with increasing deformation is determined by the structure of the cluster network. Another example is a scaling relation predicting a specific power law behaviour of the elastic modulus as a function of the filler volume fraction. The exponent reflects the characteristic structure of the fractal filler clusters and of the corresponding filler network: G9(GP w

31d f,B ]] 32d f

for w . wc

(18)

Eq. (18) predicts a power law behaviour G9 | w 3.5 for the elastic modulus if the exponent (3 1 d f,B ) /(3 2 d f ) ¯ 3.5 reflects the characteristic structure of the fractal heterogeneity of the filler network consisting of CCA–clusters (cluster–cluster–aggregation). The fractal dimensions d f and d f,B characterise the fractal structure and their backbone, respectively, of the filler clusters. The predicted power law behaviour is confirmed by the several results

Fig. 6. Variation of the amplification factor according to the assumption of power-law cluster breakdown Eq. (17).

201

shown in Refs. [*16,17] where the small strain storage modulus of a variety of carbon black filled rubbers is plotted against carbon black loading in a double logarithmic manner. It is important to note here that several electron microscopy studies on elastically stretched nanoparticle chainlike aggregates of inorganic oxides provided recently direct experimental support for the filler networking concept, i.e., reinforcing mechanism based on the energetics resulting from the breaking of physical bonds holding kinked particle chain segments together [*30]. These nano-springs display complimentary dynamic elastic behaviour to elastomers. New nanostructured materials with tunable Hookean characteristics and interfacial chemistry have been made available through the development of synthesis of nanoparticles with closely controlled size and structure using flame to room temperature processes (for example Ref. [31]). Eq. (18) is a scaling invariant relation for the concentration dependency of the elastic modulus of highly filled rubbers, i.e. the relation is independent of filler particle size. This scaling invariance disappears if the action of the immobilised rubber layer around the filler particles is considered. A representation of the increased solid volume due to an immobilised rubber layer is shown schematically in Fig. 7. The effect of a hard, glassy layer of immobilised polymer on the elastic modulus of CCA clusters leads to the following power law dependency of the elastic modulus G9 on filler concentration w, particle size d and layer thickness D [14]:

S

(d 1 2D)3 2 6dD 2 G9(GP ]]]]] w d3

D

31d f,B ]] 32d f

(19)

This equation predicts a strong impact of the layer thickness D on the elastic modulus G9. Furthermore, the

Fig. 7. Schematic representation of the increased solid volume due to an immobilised rubber layer on a filler cluster of spherical colloid particles ( j , cluster size; d, particle size; D, layer thickness).

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influence of particle size d becomes apparent. Obviously, the value of G9 increases significantly if d becomes smaller, i.e., if the specific surface of the filler increases.

6. Effects of filer surface roughness on polymer–filler interaction In carbon black-filled elastomers, the polymer adsorption on the filler surfaces and the formation of an immobilised polymer layer is substantial as there is a strong binding of the chains, leading to the layer of localised polymers, the ‘bound rubber’. Physical adsorption and chemisorption of polymer on filler from bulk is known to result in a partial loss of polymer solubility (‘bound rubber’). The existing models of this phenomenon were critically examined by Meissner [*32] and the random adsorption model suggested was found to provide an explanation of available experimental data like the dependence on filler concentration and surface area and on the average molar mass and molar mass distribution of the polymer. Differences in the polymer–filler interactions between carbon black- and silica-filled rubbers were studied by analysing microstructures of the bound rubbers with pyrolysis–gas chromatography [35]. The interactions of the cis-1,4- and trans-1,4-units were stronger with carbon black than with silica, whereas the 1,2-units interacted more strongly with silica than with carbon black. The ‘bound rubber’ variation during storage can be understood by considering a slow replacement of short rubber chains initially adsorbed on filler particles by larger ones, as demonstrated by Leblanc [38]. The role of the fractal nature of the filler surface has been studied by Heinrich and Vilgis [33]. The carbon black surfaces are known to be rough and even fractal on many decades of their size, down to the molecular size range, as well as highly energetically heterogeneous: the distribution of interaction strengths can be characterised by high energy sites surrounding a relatively low energy background. Experimentally, the geometric roughness and the energetic heterogeneity of different carbon black grades have been examined by static gas adsorption [34]. By using different experimental conditions and theoretical models, it has been demonstrated that this technique is a very useful tool in the examination of the primary carbon black particle surface structure on atomic length scales, e.g., the surface activity of fillers. The surface roughness of the examined fillers has been quantified by the surface fractal dimension Ds . This parameter has been evaluated with two independent methods, e.g., the fractal FHH theory and the yardstick method. According to the results all original carbon black grades have a very rough surface with a universal Ds ¯2.6 beyond a length scale of z # 6 nm. Only a slight dependence on the particle size has been found. The universal result Ds ¯2.6 has been explained on the

basis of a model that assumes ballistic deposition and surface growth during carbon black formation [37]. Parallel, the energetic surface heterogeneity has been described by the energy site distribution function f(Q). All examined original carbon black fillers have an energetic heterogeneous surface structure. At least four discrete interaction sites I–IV have been identified (interaction energies: (I) Q 5 16 kJ / kmol; (II) Q 5 20 kJ / mol; (III) Q 5 25 kJ / mol; (IV) Q 5 30 kJ / mol). The results show that the advantage of the determination method used is not only in discovering distinct energetic sites but also merely in the estimation of the portions of each site type (I–IV) at the surface [34].

7. Conclusions Collectively these examples are a snapshot of the current state of the art in studying reinforcement of rubber-like networks filled with heterogeneous fillers on different length scales. Analytical concepts based on fractal filler characterization have been introduced in several engineering applications as demonstrated in Section 4. Highsophisticated experimental (e.g., SANS, gas adsorption techniques, etc.) and modeling techniques (e.g., Monte Carlo simulation) are targeted to explore reinforcement on molecular length scales. It is obvious that this field of research is increasing rapidly. We expect to find wide applications in rubber science and technology, in addition to the few examples presented above. Furthermore, we expect stimulation for solving open questions connected with filler reinforcement. Important points are the cooperative and competitive effects of many polymer chains close to disordered and rough filler surfaces. In connection with this point the dynamics of adsorbed and bound chains at such surfaces is to be investigated. We expect a significant slowing down of the dynamics of these chains, leading also to an increase of the glass transition temperature of the bound rubber layer.

Acknowledgements Partial financial support was supplied by the German Rubber Society (DKG e.V.). GH and MK acknowledge a grant from Columbian Chemicals Co. and Continental AG.

References Papers of particular interest, published within the annual period of review, have been highlighted as: * of special interest; ** of outstanding interest.

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