differential dynamic modulus of carbon black filled

ISSN 1349-7308 29http://www.jstage.jst.go.jp/browse/ejsm

Introduction

Rubber materials are highly deformable and recoverablesoft materials. The rubber materials only can be expectedto show such rubber elasticity. This means we cannotreplace rubber materials with alternatives. In addition torubber elasticity, high mechanical strength is required forindustrial products such as automotive tire, seismicisolation bearing and automotive engine mount.Remarkable enhancement in mechanical strength has beenobtained through vulcanization and filler reinforcement.However, introduction of filler brings complicatedmechanical behaviors. In almost practical purposes, rubbermaterials undergo large deformations or large loads. Filledrubbers show remarkable nonlinear viscoelasticity even atsmall strain of a few percent or lower, although unfilledrubbers show linear or quasi-linear viscoelasticity atmoderate strain of five or ten percent1–4). For example,automotive tire undergoes ca. 40% strain in carcass by airpressure load. Tire treads may be strained at ca. 300%5).Filled rubbers suffer clearly very high strain inmanufacture processing. Hence it is very important to

understand nonlinear viscoelastic behaviors of filledrubbers in order to design and to manufacture high-qualityand high-performance rubber products.

Numerous investigators have studied viscoelasticproperties on filled rubbers, but their main targets havebeen filled, cured rubbers because of their purposes beingoriented to performance improvement of rubber products.However, processing is, of course, important as well asproduct itself. In general, rubbers are processed in threestages; 1) mixing of rubber with filler and chemicals, 2)processing of compounds such as compression molding,injection molding, extrusion, and so on, and 3) curing.Almost rubber products are made through processing offilled, uncured rubbers to which large deformations aregiven. Filled, uncured rubbers show marked nonlinearviscoelasticity as well as filled, cured rubbers. Thebehaviors of pure rubbers are fairly simple because theyare determined basically by interchain interaction only.However, those of filled rubbers are complicated due tomany interactions, e.g. polymer–filler and filler–fillerinteractions in addition to polymer–polymer one. Suchcomplicated behaviors sometimes cause production defects

Differential Dynamic Modulus of Carbon Black Filled, Uncured SBR in Single-Step Large Shearing Deformations

Youji SATOH1), Shuji FUJII1), Seiichi KAWAHARA1), Yoshinobu ISONO*,1), and Shigeru KAGAMI2)

1) Department of Chemistry, Nagaoka University of Technology, Nagaoka, Niigata 940–2188, Japan

2) The Yokohama Rubber Co., Ltd., Hiratsuka, Kanagawa 254–8601, Japan* Corresponding author: [email protected]

Received February 15, 2007; Accepted February 20, 2007© 2007 The Society of Rubber Industry, Japan

Abstract Correspondence between nonlinear viscoelastic properties and change in various networks in carbon black(CB) filled, uncured SBRs has been studied by using combined measurements of relaxation modulus, differentialdynamic modulus, and volume resisitivity in wide range of filler concentrations at various shear strains. Volumeresistivity at no deformation showed step-off like change which can be explained by the percolation theory. Thisindicates formation of contact filler network at high filler loading. In addition, change in volume resistivity showed clearcorrespondence with linear-nonlinear transition in viscoelasticity. By the use of simple three-network model,contributions of contact filler, bridged filler, and entanglement networks to relaxation modulus were estimated. It wasfound that contact filler and bridged filler networks were dominant at lower and at higher filler concentrations,respectively. It was proposed, furthermore, that differential dynamic modulus can be used as the probes for changes incontact filler and bridged filler networks, respectively.

Keywords Rubber, Carbon black, Filler, Network, Dynamic modulus, Viscoelasticity, Volume resistivity.

Regular Article

e-Journal of Soft Materials, Vol. 3, pp. 29–40 (2007)

in processing, e.g. under- or over- shrinkage of rubbersheets after extrusion. Generally speaking, the rubber sheetshrinkage consists of two processes having short and longrelaxation times. Basically they may be understood asstress relaxation phenomena. Unfortunately, however, therelaxations cannot be predicted quantitatively. It may beimportant to shed light on the nonlinear viscoelasticproperties and the relaxation mechanisms of filled, uncuredrubbers as well as filled, cured rubbers from not only theviewpoint of academia but also that of industry. In thepresent study, we deal with uncured rubber filled withcarbon black which is one of the typical fillers.

Recently existence of filler network in filled rubbers hasbeen recognized experimentally not only in macroscopicscale by rheometry in reversing double-step deformations6)

but also in microscopic scale by in-situ microscopy7,8) and3D-TEM9–12). Let us think over various networks existingin filled, uncured rubbers and other factors giving effectson the aimed viscoelasticity in somewhat detail. Filledrubbers consist of polymer and filler. So there should existthree networks at least due to three different interactions;polymer–polymer, polymer–filler, and filler–fillerinteractions. The first is entanglement network due totopological interchain interaction13). The second is fillernetwork composed of carbon black particles that are joinedby bridging polymer chains attached more or less firmly tothe particles through e.g. entanglement trapped at thefiller–polymer interface2,4,14–17) (hereafter abbreviated asbridged filler network). The third is filler network by directparticle contact2,14,18) (abbreviated as contact fillernetwork). We may recognize contact filler network wherethe perimeter distance between the nearest aggregates isca. 3 nm which is observed by 3D-TEM12). Bridged fillernetwork has not been confirmed by experiment. However,it may be plausible, because of the next points. The fillergel2,4) surrounding carbon black filler particle has beenaccepted on the basis of the evidence of insoluble fractionof polymers. Its adiabatic compressibility has beenestimated to be half of the matrix SBR and four times ashigh as carbon black19). Some observations have beenreported on nonlinear viscoelasticity due to bonding anddebonding between polymer chains and filler surface20) andcomplex slip between polymers and flat surface due todisentanglement21). Recently, furthermore, the polymerfield theory and the off-lattice computer simulation havepredicted the existence and behavior of polymer bridgedgeletaion22). Various networks and filler gel may contributeto filler reinforcement.

Features of viscoelastic properties of filled rubbersappear typically in strain amplitude dependence of

absolute value of dynamic modulus, so-called Payneeffect1,6), in the next manner; (a) modulus enhancement inlinear viscoelasticity by filler loading, (b) strain amplitudedependent drop in modulus, and (c) existence of lowerlimit in modulus to be close to the one having no filler. Asdiscussed above, filler gel may be one of the reasons forthe modulus enhancement in linear viscoelasticity.However, it may not be applicable to the explanation of thefacts (b) and (c), because it may be difficult that filler geldiminish partly at small strain of a few percent and quickgel-sol transition occurs even at large strain of onehundred percent. On the other hand, changes in bridgedand/or contact filler networks may be reasonable for theexplanation of the nonlinear viscoelasticity. Strain-inducedsoftening may occur with rupture of networks. If almostcomplete rupture is attained, the lower limit in modulusmay appear. Such speculations imply changes in fillernetworks are main factors for the nonlinear viscoelasticityof filled, uncured rubbers.

As explained above, our target is carbon black filled,uncured rubbers. So the nonlinear viscoelasticity mayoriginate with strain-induced chain anisotropy, chaindisentanglement, and rupture of bridged and/or contactfiller networks. However, chain anisotropy may disappearshortly with relaxation of stress, if we adopt stressrelaxation after imposition of step strain as deformationhistory. Then we should focus our concern on the latterthree effects. Here it should be noted that the three effectsmay occur simultaneously. We need additional probe inorder to distinguish them separately. We like to proposethe complementary use of volume resistivity18) anddifferential dynamic moduous6,14,23) in addition torelaxation modulus.

Polymers are non-conductive materials, but carbonblack is well known conductor. So volume resistivity canbe used as the probe for the formation of contact fillernetwork18). In addition, the strain dependence of dynamicproperties can be examined not by varying the dynamicstrain, as customarily studied2–4,15,24–28), where the strainhistory goes through complicated cycles, but by theconceptually simpler experiment of imposing smalloscillating deformations on a large static strain.Differential dynamic modulus may be effective to discusschange in networks not only for conductive filler such ascarbon black but also for non-conductive filler such assilica.

In the previous studies6,18), we have used the filledrubber samples having carbon black filler from 0 to 50 phrcovering percolation threshold concentration, because wehad liked to observe linear-nonlinear transition in

30 e-J. Soft Mater.

viscoelasticity and to get the evidence of the existence offiller network. However, in order to discuss nonlinearviscoelasticity form both academic and practicalviewpoints, it is better to use the samples having widerrange of filler concentrations.

The purpose of this work is to discuss nonlinearviscoelastic properties of carbon black filled, uncuredSBRs from the viewpoints of change in various networksby simultaneous measurements of relaxation modulus,differential dynamic modulus, and volume resistivity withthe use of SBRs having no filler and carbon black fillers ofwide range of concentrations.

Experimental

MaterialsThe polymer used was styrene-butadiene random

copolymer (Nipol SBR-1502, styrene content 23.5%,M–

W�4.3�105, Tg��52°C, Zeon Corporation). Carbonblack (CB) used was N330 (primary particle size; 31 nm,N2(BET) surface area; 79 m2/g, Mitsubishi ChemicalCorporation). Other additives were zinc oxide (ShodoChemical), antioxidant, N-(1,3-dimethylbutyl)-N�-phenyl-p-phenylenediamine (SANTOFLEX 6PPD, Flexsys), andstearic acid (Beads Stearic Acid YR, NOF Corporation).Details on compound formulation are shown in Table 1.CB master batches were prepared in 1.7 L Banbury mixer(Type BB-2, Kobe-Steel Ltd.) for 4.5 min at 333 K in thebeginning (at ca. 403 K in the final stage). Subsequentsheeting on a two-roll mill was performed to achieve gooddispersion of carbon black.

MeasurementsThe apparatus used was biaxial rheometer made by

ourselves29–31). It is equipped with two independent drivingsystems. Axial motion was driven by an oil hydraulicactuator (JT Toshi). Axial force and displacement weredetected by a load cell (Nihon Tokushu Sokki, LRM-50K)and a cantilever spring type displacement transducer(Tokyo Sokki, CE-10). Special air bearings were equippedto prevent rotation of the actuator axis. Transversal

rotation around axis was driven by a stepping motorequipped with harmonic gear of 1/100 gear ration andmicro-step control system (Oriental Motor, UPD556HG2-A2, minimum step angle 1.8�10�5 degree). Rotationaltorque and angle were detected by a non-rotational typetorque meter (Nihon Tokushu Sokki, TCG-01K) and anon-contact type Laser Feed Monitor (Keyence, FC-2000).

Figure 1 shows a sketch of the experimental procedureto measure shear relaxation modulus G(g ; t) anddifferential dynamic modulus E* (w , g ; t) in axialdeformations. Before shearing, dynamic modulus

Vol. 3. pp. 29–40 (2007) 31

Table 1. Sample formulations (unit in phr)

Sample No. 1 2 3 4 5 6 7

SBR (1502) 100 100 100 100 100 100 100

Carbon black (N330) 0 20 35 50 65 80 95

Antioxidant* 1 1 1 1 1 1 1

Stearic acid 2 2 2 2 2 2 2

Zinc oxide 3 3 3 3 3 3 3

* N-(1,3-dimethylbutyl)-N�-phenyl-p-phenylenediamine (6PPD).

Figure 1. Schematic illustration of the experimental procedures.See texts for detail.

E* (w , 0), where 0 denotes zero static strain, wasmeasured. A step-like shear g was given in torsion at timezero. At intervals, small oscillatory deformations wereimposed along axis. The frequency and the strainamplitude of the small oscillations were 0.2 Hz (1.26 rad/s)and 0.003, respectively.

A specimen used in this work was a solid circularcylinder block (19.8 mm in diameter and 2 mm inthickness). Electrical resistance was measuredsimultaneously with rheometry by using DigitalElectrometer R8340A (ADVANTEST) that permits us tomeasure wide range of electrical resistance from 10 to3�1016 W with accuracy of 0.25 to 0.8%. In order to takeelectrical contact, the sample was fixed to the disk-shapedelectrodes of 19.8 mm in diameter made of stainless steelplates in a Teflon mold by fusion at 443 K for 30 min. Noslippage between sample and plates and uniform shearstrain at the outer edge of the disk were confirmed fromthe check of the fiducial ink line drawn on the outside ofthe sample disk18).

The apparent shear relaxation modulus after step shearGa(gR; t) was calculated from Eq. (1).

(1)

where T(t) is the torque at time t, b(�pR 4/ 2h) is the formfactor, a is the angle in torsion, R and h are the radius andthe height of the sample disk, respectively, and gR is theshear strain at the radius R. When such a disk is deformedin torsion, the strain is proportional to the radial distancefrom the center. The relaxation modulus G(gR; t) at theperiphery at each time can be obtained through Eq. (2)from a set of experimental values of the pair(Ga(gR; t),gR)33–35).

(2)

Here it should be noted again that there was no slippagebetween the sample specimen and the stainless steel plates,and the uniform shear strain was given to the samplespecimen at the periphery at least. Volume resistivity rwas calculated from the electrical resistance � byr��A /L, where A is the sectional area, and L the distancebetween two parallel plates. Relaxation modulus,differencial dynamic modulus, and volume resistivity weremeasured simultaneously at 313 K.

Results

Figure 2 shows CB filler concentration dependencies ofvolume resistivity r (0) and tensile storage modulusE�(w , 0) of unfilled, uncured and filled, uncured SBRs atno deformation. The value of r (0) for unfilled, uncuredSBR was very high value of 1.5�1010 Wm. At lowconcentration of 20 phr, filled, uncured SBR showedalmost the same resistivity of 1.6�1010 Wm as the unfilledSBR, indicating no formation of contact filler network.With the change in CB loading from 20 to 35 phr, however,resistivity showed decrease by around six orders ofmagnitude, followed by gradual decrease with increase inCB concentration. The data indicate that percolationthreshold exists between 20 and 35 phr. Above thethreshold, CB concentration dependence of storagemodulus became higher, indicating that CB contact fillernetwork was formed.

Figures 3(a)–9(a) show the curves of relaxation modulusG(g ; t) for seven samples at various shear strains rangedfrom 0.001 to 0.5. The curves of relaxation modulus forthe unfilled SBR show linear or quasi-linear behaviors inviscoelasticity. The filled SBRs also show linear or quasi-linear behaviors at very small strains, but did clearnonlinear behaviors at large strains. The higher the CBconcentration is, the clearer the nonlinearity in relaxationmodulus is.

Figures 3(b)–9(b) and 3(c)–9(c) show the change indifferential storage modulus E�(w , g ; t) and differentialloss tangent tand (w , g ; t) observed simultaneously withthe measurement of relaxation modulus for the sevensamples at various shear strains, respectively. Figures3(d)–9(d) denote the change in volume resistivity r (g ; t).In the panels (b)–(d), broken lines denote the values of

G t G tG t

R a Ra R

R

γ γ∂ γ

∂ γ; ;

ln ;

ln( ) ( ) ( )

� �1

1

4

G tT t

b

T t R

bha RR

γα γ

;( ) ( )( )� �

32 e-J. Soft Mater.

Figure 2. CB filler concentration dependencies of volumeresistivity r(0) and tensile storage modulus E�(w ,0) of unfilled,uncured and filled, uncured SBRs at no deformation.

E�(w , 0), tand (w , 0), and r (0) before shearing,respectively. For the unfilled sample, all curves inE�(w , g ; t), tand (w , g ; t), and r (g ; t) overlap the brokenlines and show practically no change irrespective of shearstrain. For the SBR having 20 phr of CB, relaxationmodulus showed nonlinear behavior at large strains, whiledifferential storage modulus, differential loss tangent, andvolume resistivity showed almost no changes. Especiallyno change in volume resistivity may indicate that CBcontact filler network was not formed in the sample.However, different behaviors in volume resistivity wereobserved for other samples having CB of 35 phr or higher.

These samples showed fairly low or low values of r (0),1.3�104�9�101 Wm. In addition, the values of r (g ; t) didnot change in the range of strains where linear or quasi-linear behaviors in G(g ; t) were found, but increased withincrease in strain corresponding to nonlinearviscoelasticity. The values of r (g ; t) increased afterimposition of strain followed by continuous decrease withelapse of time. The relative increase in r (g ; t) from theinitial value r (0) was the largest at 35 phr, and was leveleddown with increase in CB concentration. Similar behaviorswere found in differential storage modulus, although thechanges from E�(w , 0) were not so large. However, therelative changes in differential loss tangent from the initialvalue tand (w , 0) increased with increase in CBconcentration.

Vol. 3. pp. 29–40 (2007) 33

Figure 3. Double logarithmic plots of (a) relaxation modulus, (b)differential storage modulus, (c) differential loss tangent, and (d)volume resistivity against time for unfilled, uncured SBR (sample 1)at various shear strains. The magnitudes of shear strains are indicatedin the panel. The broken lines in the panels (b)–(d) denote thecorresponding original values measured before straining. The solidlines in the panel (b) denote the values of differential storage moduluscalculated from the relaxation modulus and original value of storagemodulus with the BKZ theory. The meanings of the colors of solidlines correspond to those of symbols. The measurements were madeat 313 K.

Figure 4. Double logarithmic plots of (a) relaxation modulus, (b)differential storage modulus, (c) differential loss tangent, and (d)volume resistivity against time at 313 K for uncured SBR having 20phr of carbon black (sample 2) at various shear strains. The meaningsof the symbols and the lines are the same as in Figure 3.

Discussion

As being found by Kojiya and Katoh12), the perimeterdistance between the nearest aggregates for the CB filledrubber percolated was ca. 3 nm irrespective of CBconcentration. So decrease in resistivity at high CBconcentration may imply increase in number of contactfiller network strands which leads to the increase in storagemodulus. The CB concentration dependencies of resistivityand storage modulus shown in Figure 2 may support suchspeculation.

Percolation threshold volume fraction jC may dependon filler particle form and spatial distribution4). Under anassumption of simple cubic lattice of spherical filler, jC isgiven by36)

(3)

where d is filler density and n specific void space ofrandom dense packed filler. Use of the values d�1.8g cm�3 and n�0.98 cm3 g�1 for N330 carbon black36) givesjC�0.124 (28 phr), which is not only close to the valuesreported4) but also in agreement with the value observed inthe present study. According to percolation theory, thevolume resistivity at no deformation r(0) should decreasewith the net CB concentration j�jC of carbon blackaccording to a power law4).

(4)

where r° is the limiting volume resistivity for j�1 and mis called percolation exponent. The plot of log r(0) againstlog (j�jC) gives a line having slope of �3.7 as shown inFigure 10. The fitted line in Figure 11 can be obtained with

ρ ρϕ ϕ

ϕ

µ

( )01

��

�

�

°

C

C

ϕ νC d� �1 1 4/ ( )

34 e-J. Soft Mater.



the values of r°�2.6�10�2 Wm, m�3.7 and jC�0.124.The value of r° and m obtained are close to the valuesreported4). The curves in Figures 3 and 4 and the values ofjC, r°, and m indicate that the change in volume resistivityoriginates with percolation and contact filler network isformed above the CB concentration threshold.

Strain dependent relaxation modulus curves in Figures3(a)–9(a) showed parallel drop, the features of which weresimilar to the old results for CB filled vulcanizates14,37).This means that nonlinear viscoelasticity of uncured, filledrubbers can be discussed by using degree of damping inthe G(g ; t) curves at a time specified arbitrarily. Theunfilled, uncured SBR showed linear viscoelasticity up toshear strain 0.1, which agrees with the onset value ofdisentanglement predicted theoretically38) and foundexperimentally for polystyrene39–44) and

polyisobutylene23,29,30,45,46). Therefore we may consider thatthe nonlinear behavior of the sample 1 (unfilled, uncuredSBR) originates with disentanglement.

The sample 2 showed no change in volume resistivityeven at large shear strain of 0.5, while other samples 3–7having CB filler concentration higher than the thresholdshowed increase in resistivity at large strain. The rise inresistivity may be due to rupture of contact filler network.Therefore we may consider that the damping in G(g ; t) ofthe sample 2 did not originate with rupture of contact fillernetwork, but with that of bridged filler network.

Three kinds of networks, viz. chain entanglement,bridged and contact filler networks, may hold stress. If weexpress their contributions as sCeN, sBFN, and sCFN,respectively, the total stress s may be written by addition.

Vol. 3. pp. 29–40 (2007) 35



s �sCeN�sBFN�sCFN (5)

The first term sCeN may depend on number of chainentanglement in unit volume of polymer, and the secondterm sBFN on number of trapped entanglement at theinterface between filler aggregate and matrix polymer inunit volume. So relaxation modulus for the sample of CBvolume fraction j at shear strain g and time t may bewritten approximately as18).

G(g ; t ; j)�(1�j)GCeN(g ; t ; j�0)�jGBFN(g ; t ; j�1)�GCFN(g ; t ; j) (6)

The G(g ; t ) curve for the unfilled, uncured sample 1depends on the first term only in Eq. (6). The curve for thesample 2 depends on the first and the second terms, andthose for other samples 3–7 on all terms. Therefore thecontributions of the three terms can be estimated by

successive substitutions.Figure 12 shows the contributions of the three terms to

the total relaxation modulus for the filled, uncured samples2–7. In Figure 12(f) for the sample 2, there is nocontribution of CFN term and contributions of two CeNand BFN terms are comparable. In Figure 12(e) for thesample 3, the BFN term has larger contribution than theCFN and CeN terms, because the contact filler network hasnot been developed enough. However, after the percolationwas attained enough as was the case for the samples 4–7,the modulus is dominated by CFN term. As the CBconcentration increases, CFN contribution becomes much

36 e-J. Soft Mater.


Figure 10. Power law dependence of volume resistivity r(0) forfilled, uncured SBRs on j�jC above the percolation threshold,where jC was assumed to be 0.124.

Figure 11. Volume fraction dependence of volume resistivity r(0)for filled, uncured SBRs where only the filled symbols are assumed tolie above the percolation threshold. The solid line are least square fitsaccording equation 4 with r°�2.6�10�2 Wm, m�3.7, and jC�0.124.

larger. BFN term can be distinguishable even at high CBconcentration region. But its relative contribution becomessmall. The contribution of CeN term does not vanish, butis negligibly small at high concentration. It can be pointedout that the bridged filler network is important at low CBconcentration, and the contact filler network is at highconcentration. Contact filler network is considered to beweak in comparison with bridged filler network. So thedominance of contact filler network may lead to strainsensitive nature in viscoelasticity.

From Figure 12, it was found qualitatively at least thatthat large drop in G(g ; t ) with increasing g in the presentstudy is attributed to damage of the contact filler network.If there is healing of damage during the course of stressrelaxation14,47), it does not appear in G(g ; t ) because thenew structures are formed in configurations that do notcontribute to the stress. It may appear, however, in

E�(w , g ; t ).If we follow the BKZ constitutive equation48), our aimed

differential storage modulus can be represented asfollows14).

(7)

where G(0 ; t ) is the relaxation modulus in linearviscoelasticity. If G(g ; t ) is separable into strain-dependentand time-dependent terms as G(g ; t )�h(g)G(0 ; t ), Eq. (7)can be written with h(g).

(8)

In the linear viscoelasticity, E�(w , g ; t ) is reduced toE�(w , 0 ). Equation 7 predicts that E�(w , g ; t ) is smallerthan E�(w , 0 ) at finite times and attains E�(w , 0 ) at infinitetime, in qualitative agreement with the presentexperimental results. The values of differential storagemodulus predicted form strain-dependent relaxationmodulus with BKZ constitutive equation agreequantitatively with the values observed for the polymerdisentanglement system23). So comparison of the observedand the predicted values of differential storage modulusmay give additional information on damage and healingprocess of contact filler network.

Here it should be noted that there exist straindistribution to be proportional to the radial distance formthe center in the sample disk. Therefore we must comparethe values of E�(w , g ; t ) observed with those averagedover the radial distance from the center to radius R.Average of E�(w , g ; t )BKZ can be taken approximately asfollows.

(9)

In the present study, strain-dependent relaxation moduluscan be expressed by the product of linear relaxationmodulus G(0 ; t ) and strain-dependent term h(g). Hence

(10)

Solid lines in Figures 3(b)–9(b) show the calculated valuesof E�an , BKZ(w , g ; t ) for four different shear strainsmeasured, g�0.001, 0.01, 0.1, and 0.5, respectively. Theagreement between the calculated and observed values isnot so bad for the unfilled sample in Figure 3(b). However,

G tR

h rdr ER

� � � �γ ω( ; ) ( ) ( , )06

1 02

0

[ ]∫E ta BKZ� ν ω γ, ( , ; )

rR

G t G t E rdrR

� � � �γ ω( ; ) ( ; ) ( , )2

3 3 0 02

0∫[ ]

E tR

E t rdra BKZ BKZ

R

� � �ν ω γ ω γ, ( , ; ) ( , ; )2

20∫

E t G t h E� � � � �( , ; ) ( ; ) ( ) ( , )ω γ γ ω3 0 1 0[ ]

E t G t G t E� � � � �( , ; ) ( ; ) ( ; ) ( , )ω γ γ ω3 0 0[ ]

Vol. 3. pp. 29–40 (2007) 37

Figure 12. Separation of effects of contact filler, bridged filler, andchain entanglement networks on the strain dependent relaxationmodulus at t�5 s. Panels (a)–(f) correspond to the estimation forsamples 7 (90 phr), 6 (85 phr), 5 (65 phr), 4 (50 phr), 3 (35 phr), and 2(20 phr), respectively.

the agreement becomes poorer as increasing CBconcentration, especially at large shear strain. The drop inE�an , BKZ(w , g ; t ) predicted by the theory is too large. Theinitial drop in E� below the original value of E�(w , 0 ) canbe interpreted as due to damage of contact filler networkand its subsequent rise to a healing process. What happensin the first 5 seconds after imposition of stress isunfortunately not available form our experiments, so wecannot see whether some healing takes place in this period.However, E�(w , g ; t ) was found to be essentially constant,so the healing may occur at the initial stage faster than 5seconds.

We can find another feature in CB concentrationdependencies of drop in E�(w , g ; t ). At large strain, thedrop in E�(w , g ; t ) below E�(w , 0 ) shows minimum valueat 35 or 50 phr followed by very slight upturn withincreasing concentration as shown typically in Figure13(b) where relative values of E�(w , g ; t ) to the originalvalue E�(w , 0 ) is taken as ordinate. The feature found forthe change in E�(w , g ; t )observed shows contrast with that forthe change in E�(w , g ; t )BKZ calculated from relaxationmodulus shown in Figure 13(c). To the contrary, thefeature found for E�(w , g ; t )observed resembles that forvolume resistivity as shown in Figure 13(a). The curves ofr(g ; t) /r(0) and E�(w , g ; t ) /E�(w , 0 ) at shear straing�0.5 and t�5 s were plotted against CB concentration inFigure 14(a) and 14(b), respectively. The values ofE�(w , g ; t ) predicted from relaxation modulus by the BKZtheory shows continuous decrease with increasing CBconcentration. If there is no healing of damage of thecontact filler network, E�(w , g ; t ) should show continuousdecrease with increasing CB concentration as shown bythe blue line in Figure 14(b). In fact, however, there existsfast healing process of contact filler network during stressrelaxation. The quick healing may be mainly due to densecontact filler network. For example, suppose two CBcontact aggregates on two adjacent shear planes. Whenshear deformation is given, one aggregate gets away fromthe other. In dense CB system, the aggregate may comeinto collision with new aggregate with high possibility.Such the probability increases with increase in CBconcentration. In addition, interaggregate bridging chainmay contribute to reformation of contact filler network dueto its entropic force. This speculation may be supported bythe result that bridged filler network is still alive even atlarge deformation which is indicated by relatively weakstrain dependence of its contribution to stress as shown inFigure 12.

Differential loss tangent showed initial rise followed bygradual recovery with time elapsed. This behavior was

similar to that observed for differential storage modulusexcept opposite sign in the deviations from the initialvalues. Its concentration dependence was, however,different from that for differential storage modulus. Theinitial rise in tand(w , g ; t ) was monotonically increasingfunction of CB concentration. Similar time dependence oftand(w , g ; t ) has been observed for polyisobutylene inlarge shearing deformations23,49,50). The behavior has been

38 e-J. Soft Mater.

Figure 13. Strain and carbon black concentration dependence of (a)volume resistivity, (b) differential storage modulus E�, and (c) E�

predicted from G(g ; t) and E�(w , 0) with the BKZ theory.

attributed to initial chain extension followed byconformational relaxation. The discussion has beensupported by another experiment for polyisobutylene thatstorage and loss modulus showed monotonic increase anddecrease, respectively, in stress growth after onset ofsteady shear flow51). Hence the present change intand(w , g ; t ) from the initial value may be due toextension of interaggregate bridging chain followed by itsconformational relaxation. This may mean a possible useof differential loss tangent to characterize bridged fillernetwork. The point will be studied in a subsequent paper.

Conclusions

Features and mechanisms on the nonlinearviscoelasticity of filler–rubber composites have beenexplored. Percolation behavior has been confirmed forvolume resistivity of carbon black filled rubber, whichshowed formation of contact filler network in highly filledrubbers. The simultaneous measurement of relaxationmodulus and volume resistivity showed clearly thatincrease in volume resistivity with increasing straincorresponds to linear-nonlinear transition in viscoelasticityof filled rubbers. In addition, it was found that change instrain-dependent differential storage modulus corresponds

not to change in relaxation modulus but to that in volumeresistivity. The application of simple three-network modelto strain dependent relaxation modulus allowed us toseparate the contributions of contact filler, bridged filler,and entanglement networks to stress. The contribution ofentanglement network does not vanish, but is negligiblysmall at high CB concentration. The bridged filler networkis dominant at low CB concentration, and the contact fillernetwork is at high concentration. Dominance of contactfiller network leads to strain sensitive nature inviscoelasticity of filled rubbers.

Volume resistivity and differential storage modulusshowed minimum and maximum changes in CBconcentration dependencies, respectively, at 35 or 50 phrwhich was slightly higher than the percolation threshold.Comparison of the observed and the predicted values ofdifferential storage modulus with BKZ theory indicatedexistence of quick healing mechanism in the rubberspercolated. The quick healing in contact filler network mayresult in the similarity between strain dependencies ofvolume resistivity and differential storage modulus, whichimply possible use of differential storage modulus tocharacterize contact filler network. On the other hand,differential loss tangent showed monotonically increasingdeviations from the initial values with increase in CBconcentration. The change may originate with extensionalconformation of interaggreagte bridging chain, which mayimply possible characterization of bridged filler networkwith differential loss tangent.

Acknowledgement

This work was supported in part by Grant-in-Aid forScientific Research (B) (No. 14350359), the 21st CenturyCOE Program for Scientific Research from the Ministry ofEducation, Science, Sports and Culture, Japan, andcollaborative project with the Yokohama RubberCompany.

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