rheological models of rubber-carbon black compounds: low interaction viscoelastic models and high...

Journal of Non-Newtonian Fluid Mechanics, 49 (1993) 277-298

Elsevier Science Publishers B.V., Amsterdam

277

Rheological models of rubber-carbon black compounds: low interaction viscoelastic models and high interaction thixotropic-plastic-viscoelastic models

Sergio Montes ’ and James L. White *

Institute of Polymer Engineering, University of Akron, Akron, OH 44325-0301 (USA)

(Received November 5, 1992; in revised form March 23, 1993)

Abstract

The rheological behavior of a gum elastomer (natural rubber) and its carbon black compounds is represented in terms of a series of interrelated models. A distinction is made between those compounds with low interaction (large particles or low loadings of small particles) and those with high interaction (large loadings of small particles). The low interaction compounds are treated as viscoelastic materials with filler loading modified properties. The high interaction compounds are represented as ‘thixotropic- plastic-viscoelastic’ materials. The basic non-linear viscoelastic model used in this paper is that due to Bogue, but other non-linear viscoelastic formulations could be used.

Keywords: carbon black; gum elastomer; rubber; thixotropic-plastic-viscoelastic materials

1. Introduction

The rheological properties of suspensions of small particles in Newtonian fluids have been studied since the 1920s and before [l-4]. The occurrence of both yield values and thixotropy in concentrated suspensions of small particles in Newtonian fluid matrices has been well established since the 1930s [5-71. Similar observations of yield values and thixotropic behavior have been made for rubber carbon black compounds [8- 151. The yield value and thixotropic character of these systems seem to be similarly intertwined. Similar behavior is found with other small particulates such as

’ Present address: Industrias Conelec S.A. DE C.V., 15 Norte 5602, Puebla, Pue. 72080, Mexico. * Corresponding author.

0377-0257/93/$06.00 ci 1993 - Elsevier Science Publishers B.V. All rights reserved

278 S. Montes und J.L. White 1 J. Non-Newtonian Fluid Mech. 49 (1993) 277-298

calcium carbonate [ 16- 191 and titanium dioxide [ 16,19,20]. For all of these particulates and matrices, the yield stress/thixotropic behavior regimes only occur above a critical loading characteristic of the particle and matrix, similar to that associated with ‘percolation’ models. Below this concentra- tion, the behavior is similar to that of the matrix.

It is our purpose in this paper to present a new formulation of the representation of the rheological behavior of compounds of small particles in polymer matrices, including the influence of loading and particle size. We consider that there are two regimes of behavior, which we characterize as ‘low interaction’ and ‘high interaction’ compounds. The former involves large particles or low loadings of small particles, the latter consists of systems with high loadings of small particles. This classification is based on many studies in the literature dating to McMillen [5] and Freundlich and Jones [7]. It is clearly seen in recent studies from our laboratories involving calcium carbonate [ 171 and carbon black [ 14,151. A second purpose of this paper is to develop an improved model of ‘high interaction’ compounds which exhibit yield values and thixotropy. We begin with a critique of earlier efforts at theories on high interaction compounds.

2. Critique of earlier work on modeling ‘high interaction systems’ with yield values and thixotropy

2.1 Early one-dimensional modeling

The perception of yield values and development of rheological models based on their occurrence has a long history dating to 1890 when Schwedoff [ 1,211, studying geletin suspensions, presented a one-dimensional plastic viscoelastic version of a Maxwell model, i.e.

do dy 1 dt=G-d-~(~-Y) o= > Y, (la)

and the form in steady state shear flow, i.e.

a=Y+rGdl’ dt

Q > Y. (lb)

Below stress Y, there is no flow. Shear flow behavior suggesting yield values was rediscovered by Bingham [2] a generation later from capillary experiments. The form of eqn. ( lb) with a coefficient ( l/p) replacing zG was used by Buckingham [22] to analyze flow in a tube. Bingham published a monograph [ 31 in 1922 which was widely read. Fluids following eqn. ( 1 b) began to be called Bingham plastics.

Herschel and Bulkley [23], using capillary extrusion data on polymer solutions, suggested that these materials had a yield value and that power

S. Mantes and J.L. White 1 J. Non-Newtonian Fluid Mech. 49 (1993) 277-298 279

law behavior exists beyond the yield value. This was probably not correct for their system which should have exhibited a zero shear viscosity. Subse- quently, Scott [24] proposed the specific equation

fJ = Y+Kj” (2)

to represent the shear stress in unvulcanized rubber filled with small particles and applied it to analyze flow in a compressional rheometer. Scott clearly associated the existence of the yield value with an elastomer containing a high level of small particles.

The phenomenon of time-dependent viscosities of suspensions which we call thixotropy grew out of the work of Freundlich in the 1920s. By the 1930s McMillen [5] and Freundlich and coworkers [6,7] were clearly identifying yield values and thixotropy as aspects of the same problem. Goodeve and Whitfield [25] and Goodeve [26] developed the first rheological model for thixotropy and recognized this connection, but their modeling does not explicitly include it. Rather they write kinetic equations for the rate of agglomeration and disintegration of agglomerates and associate the viscosity level with specific agglomerate structure levels. It seems not until the work of Slibar and Pasley [27] in the 1960s that thixotropic phenomena were associated with time-dependent yield values.

2.2 Three-dimensional modeling of plastic viscous fluids

Three-dimensional models of plastic fluids grew out of efforts in the theory of plasticity to develop yield surfaces that would represent torsional as well as uniaxial extension and combined stress fields. This development may be followed in monographs such as those by Nadai [28,29], Hill [30] and Prager and Hodge [31]. The earliest efforts in this area involved the concept of using the maximum shear stress. In 1913 von Mises [ 321 suggested that for isotropic solids the yield surface should be represented in terms of invariants of the stress tensor, preferably of the deviatoric stress tensor Z defined as

a=i(tro)Z+ T. (3)

T is defined so that tr T is zero. The invariants which arise beyond the external pressure are tr T2 and tr T3. One of the simple proposals of von Mises was the form

tr T’=2Y2, (4)

which was close to the earlier hypothesis of a critical shear stress. It was noted that if the velocity gradients were proportional to the deviatoric stress

280 S. Montes und J.L. White 1 J. Non-Newtoniun Fluid Mech. 49 (1993) 277-298

tensor components when yielding occurred. then

Tzd

T=&-$d. (5)

Equation (5) was used as a mathematical model for deformations of materials obeying eqn. (4).

The three-dimensional Bingham plastic grew out of the efforts of Hohen- emser and Prager [33] in 1932 to represent strain hardening of metals. They wrote

T = $f& T f 2w’,

where ylB of eqn (6) corresponds to zG of eqn. (lb). Subsequently Oldroyd [ 341 in 1947 redeveloped eqn.

(&?%$Y)‘=4qgtrd2.

He showed that eqn. (6) was

> d.

Oldroyd [ 351 later generalized expressing !jB as a function of tr d2. non-Newtonian viscosity functions (see e.g. Ref. 36).

(6) noting that

equivalent to

(7)

(8)

eqn. (6) to non-Newtonian fluids by Specific forms of eqn. (6) using different have been discussed by various authors

The development of a fluid mechanics of fluids with yield values has always caused problems for investigators seeking to derive velocity fields and positions of solid plug which the von Mises criterion prescribes. The situation with numerical simulations is no better (see, e.g. the discussion of O’Donovan and Tanner [37] and later authors [38,39]). This has led to the proposal of constitutive models which resemble, but are not true, plastic fluids. O’Donovan and Tanner [37] have proposed a ‘biviscosity theory’ where the fluid has a very high Newtonian viscosity say y10 at low determina- tion ratios d and when at a critical deformation rate

2trd’=fz, (9)

the material takes on a new shear viscosity (y, + Y/d=). From eqn. (7) there is an equivalence of tr T’ and tr d2 and these correspond to equivalent yield surfaces in the case of a rigid core y. + W. The procedure of O’Dona- van and Tanner is in reality a useful, valid hydrodynamic asymptote similar in utility perhaps to the hydrodynamic lubrication/boundary layer argu-

S. Montes and J.L. White 1 J. Non-Newtonian Fluid Mech. 49 (1993) 277-2998 281

ments of the Navier-Stokes fluids. Another approach is given by Papanas- tasiou [ 381 who takes eqn. (8) to be valid over the entire flow but accommodates low stress region rigidity by taking Y to depend upon tr d*. This is more difficult to interpret.

2.3 Three dimensional plastic -viscoelastic fluids

Plastic viscous fluid models are inherently unable to represent the behavior of particle filled polymer systems with their inherent complex memories. The first efiort to develop a three-dimensional form of a plastic fluid exhibiting viscoelastic behavior beyond the yield surface is found to the work of Hohenemser and Prager [33] who suggested models with Voigt and Maxwellian behavior. More than 35 years later, the problems was reconsid- ered by White [40] in 1979 who wrote

(10)

where H is a general memory functional. Equation (7) was shown to be equivalent to

T=&H+H. (11)

The viscoelastic contribution is specified by H. A specific simple form for H was proposed by White [40] for the purpose

of illustrating the characteristics of eqn ( 11). Particular detailed forms of H were subsequently used by White and Tanaka [41] and White and Lobe [42] to compare with experimental data on filled thermoplastics and elastomers. White and Tanaka [41] represented Has a single integral constitutive equation with a Maxwellian relaxation modulus function. Specifically they took

H= s

“G - ep=“e~c-’ - $( tr c-‘)Zj dz, (12)

0 rtzff

where zCff was a relaxation time of the form

70 z -

eff - 1 + aT(2) 1/2 ’ (13)

the superposed bars indicate a time averaging. This formulation of H was suggested by Bogue and his coworkers [43-451. White and Lobe [42] presumed that the relaxation modulus function was a series of exponentials and used five terms to represent their data. Specifically

(14)

282 S. Mantes and J.L. White 1 J. Non-Newtoniart Fluid Mech. 49 (1993) 277-298

More recently Isayev and Fan [46] have developed a plastic-viscoelastic fluid model using Leonov’s viscoelastic model [47] for H.

A different formulation viscoelastic-plastic fluids was given by Beverly and Tanner [39] in 1989. These authors were concerned again about the yield surface of eqn. (4) and the difficulty it introduces into numerical simulation procedures. They propose

tr d2 < ljc ts = -pI + 2y,d,

tr d2 > yc 6= -pI+ (1%

where P is an extra stress determined from a non-linear viscoelastic constitutive model for which they choose a differential model formulation. Totally aside from choices of P in eqn. (15) and H in eqn. (1 l), the consistency of eqns. ( 11) and ( 15) is unclear. It is not as straightforward as the O’Dona- van-Tanner approach to plastic viscous fluids.

2.4 Thixotropic plastic -viscoelastic fluids

As noted earlier McMillen [ 51 and Freundlich and coworkers [6,7] and others had associated thixotropy with yield values. In 1964 Slibar and Pasley [27] proposed one-dimensional three-dimensional thixotropic-plastic-viscous fluids of forms

T = 2 Y(t, tr d2) J_ +2flB d

1 (16)

and showed that this equation at least qualitatively agreed with suspension behavior. A specific form of Y(t, tr d’) was presented and discussed.

Subsequently, White [40] had suggested that Y of eqns. (IO) and ( 11) should similarly be considered to depend upon time and deformation history. The formulation of White and Tanaka [41] and White and Lobe [42] which used eqns. (10) and (11) seemed to be better in the steady states than in transient flows. This led Suetsugu and White [IS] to propose using eqn. (11) with

U

X’ Y(t, &) = r,(&) - cx e-an~‘e dz][ Y,( II;“) - Y, 1 , (17)

0

where Yr is the steady state yield value and Yi the initial yield value. y, was taken to be

Y = Y,+ /XI;‘“, (18)

where IId is 2 tr d’. Suetsugu and White represented H as a single integral constitutive equation.


An alternate formulation of ‘thixotropic-plastic-viscoelastic’ materials has recently been published by Leonov [48]. This is based on thermody- namic arguments similar to his viscoelastic constitutive equation. The total stress is the sum of a particle stress trP and a matrix stress cr,,,

d = Cm + err, (19)

both of which derive from strain energy functions.

3. New model

3.1 Approach

The earlier literature on constitutive models for small-particle filled thermoplastics and unvulcanized elastomers gives the impression of two classes of behavior, (i) viscoelastic models, determined by molecular structure, for neat resins and (ii) thixotropic plastic viscoelastic models for highly filled systems exhibiting yield values.

We classify the behavior of particle filled compounds into three rather than two categories these are

(i) Neat resins/gum elastomers-represented by viscoelastic models determined by polymer structure;

(ii) Lightly filled or large-particle filled polymer melts-represented by modified viscoelastic models where matrix behavior is modified by the volume fraction of filler;

(iii) Highly filled small-particle filled compounds -represented by a thixotropic-plastic-viscoelastic fluid.

In this paper we shall describe constitutive equations which fulfill these requirements. For neat resins/gum elastomers, we choose the form

s

CC d = -pl+ m (z, polymer structure)c - ’ dz, (20)

0

where 4 is the volume fraction filler. For lightly filled/large-particle systems, the form we take is

s

Cc C= -pz+ uf, (z, polymer structure, 4)~’ dz,

0

(21)

For the highly filled small-particle compounds

tT= -prz+

where

s

CrU H= uf,(z, 4)[c-’ - +(tr c-‘)Zj dz,

0

pI = -f tr b.

VW

284 S. Mantes und J.L. White 1 J. Non-Newtonian Fluid Mech. 49 (1993) 277-298

Extensive experimental investigations [ 14,491 have been carried out by the authors on the influence of carbon black on the steady state and transient rheological properties of gum elastomers. We shall use specific forms of eqns. (lo)-( 22) to fit the data on natural rubber and its carbon black

compounds.

4. Gum elastomer

4.1 Formulation

For small strains, the constitutive equation of a viscoelastic material reduces to the Boltzmann superposition equation which may be alterna- tively expressed as

s

,X Cr= -pz+2 @(s)e(~) d,7 (23a)

0

J II = -pz+2 G(z)d( z) d,_, (23b)

0

the relaxation modulus functions O(t) and G(t) may be expressed in terms of series of exponentials. Specificially

I = 1,1

Q(t) = 2 f! e-‘l’z, I I

(24b)

(24b)

The relaxation modulus functions G(t) and Q(t) are related to the zero shear viscosity q through

yO= % s

G(s) ds, 0

(25)

For the non-linear region, we use eqn. (20) with

(26)

where rjcff is the effective relaxation time based on the formulation of Bogue and his coworkers [43-451. Specifically this is

11 z rrf =

1 + az,s ’ (27a)

S. Montes and J.L. White 1 J. Non-Newtonian Fluid Mech. 49 (1993) 277-298 285

where

~ 1 = n112 = -

s II;‘” dz (27b)

= 0

and IId is the second invariant of the rate of deformation tensor, i.e.

fId = 2 tr d’. (27~)

The steady state shear viscosity in a steady shear flow, y(y), is non-Newto- nian. q(T) is given by

(28)

The transient shear viscosity at the beginning of flow of a virgin material is

where Ei is the exponential integral function which is defined by

s

71 Ei(x) = ec5 d In 5.

r (29b)

Consider a long-duration flow in which the shear rate is suddenly set equal to zero at time t = 0. The stress decay of the model of this section is according to

vl(Y9 t>l relaxation = w [ r~,eR (1 - avt) ecr”i + g ealtEi

We may also consider ‘flow-rest-flow’ transients. Let a material be sheared at constant shear rate v until the steady state is

achieved (t = t,). The flow is then halted. The stress then decays towards zero. At time t2 = t, + At we reimpose shear rate y, i.e.

t -=I t, cl, = i’x* 212 = u3 = 0,

t, < t < t2 U] =u2=ug=o

t2 < t u1 = 3x2 tlz = u3 = 0.

We now compute y(j, t, At) for times t* = t - t2:

(31)

(32)

286 S. Mantes and J.L. White 1 J. Non-Newtonian Fluid Mech. 49 (1993) 277-298

4.2 Comparison with experiment

Relaxation modulus functions have been determined for natural rubber at 100°C by Montes et al. [ 14,491 using stress relaxation measurements. The parameters of eqn (24) are summarized in Table 1.

In Fig. 1, we compare the experimental steady state shear viscosity of the natural rubber with the prediction of eqn. (28). In Fig. 2, we compare the transient viscosity in the startup flow with the prediction given in eqn. (29). A similar comparison of theory and experiment for stress relaxation at the end of flow is given in Fig. 3, where we compare experimental data on the same natural rubber sample with eqn. (30). Some samples which had undergone partial stress relaxation were subjected to a second shearing flow. A comparison of their behavior with the predictions of eqn. (32) is shown in Fig. 4.

TABLE 1

Relaxation moduli and relaxation times of natural rubber from stress relaxation experiments

Term, i G, (Pa) 71 (s)

m 2 199 230

In - 1 9 166 37.7 m-3 33 290 7.31 m - 3 59 200 0.925 ??I -4 59 000 0.095

lE-4 lE-3 lE-2 lE-1 1-1 ‘0 100 1000 lE4 ?(sec >

Fig. 1. Comparison of theory of this paper with steady state shear viscosity for natural rubber sample (data of Montes et al. [ 141).

S. Montes and J.L. White / J. Non-Newtonian Fluid Mech. 49 (1993) 277-298 287

lE6 0.0227

lE5 :

a=0.7 -Theory NR 100°C

lE4' ..,...’ “....’ .‘..“.’ ...... lE-1 1 10 100 1000

time(sec)

Fig. 2. Comparison of theory of this paper with stress transients at startup of flow for natural rubber sample (data of Montes et al. [14]).

lE-1 1 10 100 1000 time(sec)

Fig. 3. Comparison of the theory of this paper with stress relaxation following the flow of the natural rubber sample (data of Montes et al. [14]).

The agreement shown between theory and experiment for the steady state shear viscosity is excellent in Fig. 1. The agreement between theory and experiment for the transients at the startup of flow shown in Fig. 2 is also excellent with the stress overshoot being accurately predicted. The transient stress relaxation data following flow in Fig. 3 agree well at low shear rates, but at higher shear rates, the predicted stresses fall off more rapidly than experiment. For the transient viscosity following various rest periods shown in Fig. 4, the agreement seems good with the suppression of overshoot being

288 S. Mantes and J.L. White 1 J. Non-Newtonian Fluid Mech. 49 (1993) 277-298

lE6

0 2.6 l 30 A 250 A Ref.

lE5

lE4 lE-1

At(sec.)

a=O.7 NR 100 ‘C - Theory

1 1 10 50

time(sec)

Fig. 4. Comparison of the theory of this paper with ‘flow-rest-flow’, second flow transient data, for natural rubber with various rest periods (data of Montes et al. [14]).

predicted for short times. However, at long times, the model predicts a more rapid buildup of the tendency to overshoot than is observed.

5. Compounds with low interparticle interaction

5.1 Formulation

For small strains, we consider the constitutive equation for filled materials with low interactions to be of the form of Eqn. (21) i.e.

a=pz+2 s

z Qf,(z)e(z) dz 0

(334

= -pz+2 s oi G,, (z)d(z) dz,

with relaxation moduli Qr, (t)

I = ,,1

Qf,(t) = C z ectirfl, ’ I

,=m

Gf, (t) = 1 Gj ectirn, I

where

Tf, =f(4h

and Gf, (t) expressible as

(33b)

(34a)

(34b)

( 34c)

f(4) is a monotonically increasing function of volume loading 4 and z, the relaxation time for the gum elastomer.

S. Mantes and J.L. White / J. Non-Newtonian Fluid Mech. 49 (1993) 277-298 289

It is important to note that we see the effect of volume loading much more in the relaxation rates than in the modulus and in our modeling we place the loading effect in the relaxation times.

The zero shear viscosity is

ylr,= CC s

X Gf, (s) ds =

0 s s %, 6) ds, (35a)

0

We now turn to the non-linear region. In compounds with low interparticle interactions we shall use the form of eqn. (21) and select the relaxation function pf,(t, 4) to be similar to m(t):

pf, (t, 4) = C G, e’l’fefT(4), I cd&P>

with

The quantity 4fr(+) is taken as

(374

(37)

where z, are the linear viscoelastic relaxation times and f(4) again is a monotonically increasing function of volume loading. We repeat that the formulation of eqns. (36) and (37) should be seen to imply that we represent the effect of filler loading by increasing the relaxation times, lowering the rate of stress relaxation, rather than by increasing the modulus.

The steady state shear viscosity for the low interaction particle compounds is of the form

m G,z, S(4) T 1 + a[ 1 +f(+)zli]

(38)

290 S. Montes and J.L. White 1 J. Non-Newtonian Fluid Mech. 49 (1993) 277-298

The absolute magnitude of the zero shear viscosity qfO and the non-Newto- nian character are both increased through thef(4) function. Equation (35b) is the low shear rate asymptote of eqn. (38).

The transient shear viscosity at the start of flow is again of the form of eqn. (29) but with ra(4) defined by eqn. (37b), in place of rL, and rfetr defined by eqn. (37a) in place of z,,~.


In Fig. 5 we compare the theory of this section with the shear viscosity of two rubber compounds. These are compounds with 0.2 volume fraction N990 carbon black and 0.1 volume fraction N326 carbon black. The details of the carbon blacks used in these experiments are summarized in Table 2 and those of the compounds in Table 3. The agreement is quite good. Transient shear stresses at the startup of flow are presented for these rubber compounds in Figs. 6(a) and 6(b) in comparison with theory. Again the

lE8

lE7

lE6

lE5

lE4

1000

100

lE-51E-41E-31E-21E-1 1 10 j(tec-1)

100 1000 lE4

Fig. 5. Comparison of the theory of this paper with steady state shear viscosity for two low interaction natural rubber-carbon black compounds containing 0.1 volume fraction N326 and 0.2 volume fraction N990 (data of Montes et al. [ 141).

TABLE 2

Characteristics of carbon blacks involved in this study

Grade BET surface area

(m2 8-l)

Average Particle size (h&m)

Supplier

N990 8 N326 80 NllO 140

0.450 Huber 0.027 Phillips 0.020 Cabot

S. Mantes and J.L. White / J. Non-Newtonian F&id Meek 49 (1993) 277-298 291

TABLE 3

Natural rubber-carbon black compounds involved in this study

Carbon black Volume fraction Bet surface area ( m2 g-*)

N326 0.1 80

N990 0.2 8

1E7

NR/O. 1 N326

lE4 TIC-1 1 10 100 1000

time(sec)

Fig. 6. Comparison of shear stress transient at the startup of flow with low interaction rubber compounds investigated in this paper (data from Montes et al. [ 141). (a) 0.1 volume fraction N326 in natural rubber; (b) 0.2 volume fraction N990 in natural rubber.

lE7 r 1

- Theory f

lE5 :

tE4 ;

f NR,‘O.l N326

1000’ - .. ‘.I ......’ .. ‘. * -.-J lE-1 1 10 100 1000

time(3ec)

Fig. 7. Comparison of shear stress relaxation following flow data obtained on low interaction rubber-carbon black compounds with models developed in this paper (data from Montes et al. [ 141).

292 S. Mantes and J.L. White I J. Non-Newtonian Fluid Mech. 49 (1993) 277-298

lE6

NR/O.l N326 100 ‘C

lE4

0=0.7 A Ref.

lE-1 1 10 100 time(sec)

Fig. 8. Comparison of theory of this paper with ‘shear flow-rest-shear flow’ experiments (second flow) for low interaction rubber-carbon black compounds (data from Montes et al. [ 141). (a) 0.1 volume fraction N326 in natural rubber; (b) 0.1 volume fraction N990 in natural rubber.

agreement is quite good. Stress relaxation following steady shear flow has also been investigated and compared to the model of this section. The results are shown in Fig. 7. Here the agreement is not so good. The stress relaxation is much slower, especially following high shear rates, than the prediction of the theory. A similar tendency, but not a pronounced, was found with the gums. We also consider ‘shear flow-rest-flow’ shear stress transient studies of the type carried out with the gums. The predicted and experimental shear stresses for the second runs are shown in Figs. 8(a) and (b). The agreement is reasonably good.

6. Highly filled high interaction compounds

6.1 Formulation

For highly filled high interaction compounds, we used the formulation of eqn. (22) with a thixotropic yield surface and deviatoric functional H. We use a form different from, but basically similar to, Suetsugu and White’s [ 171 eqn. ( 19). We again assume

Y(t, II,) = Y, (&) - [s

x a epn”@ dz [Y, (HA”) - Y,]. 1 (394 0

However, now we take

(39b)

The p(t) function is taken as being the same as in eqns. (36) and (37).


( P a

S

1

t.+i2-

1.+11--

1*+10--

i.+os--

1m+00--

1a+o7--

i.+oo--

i.+os--

i.+o.--

1.+03--

s.+oT! ! I I I I I I I I

-m -7 -6 -B -4 -3 -2 -I 0 I 2 3

log (shear rate). l/set

Fig. 9. Comparison of thixotropic plastic viscoelastic theory of this paper with steady state shear viscosity for 0.2 N326 carbon black-natural rubber sample (data from Montes et al.

Ll41).

~0000000

V i S C 0 ,oooooo

S

i

t

Y

( P iooooo a

10000 c

, I

0.0277 i/set

d 0.768

1.0

8 , iO.0 100.0 ,000.o 10'

time (set)

I.0

Fig. 10. Comparison of thixotropic plastic viscoelastic theory of this paper with transient shear stress data on 0.2 N326 carbon black-natural rubber sample (data from Montes et al. [ 141).

294 S. Mantes and J.L. White / J. Non-Newtonian Fluid Mech. 49 (1993) 277-298

Certain characteristics of Y(t, II,) in shearing flow are of interest. For a flow which begins at t = 0, Y is

Y = Yr + (B1 1; + p21j2) e-“?‘. (40)

At t = 0, this has the asymptotic value

Y= Yr+D,i +B2j2, (41a)

and at t = co, it has the asymptote

Y= Y,. (41b)

For a sequential shear flow, it has the value of eqn. (40) during the first stage of flow, during stress relaxation following flow it is Yr and in the second flow it is

Y = Y, + (Fiji + P21j2)[e-Wt _ e-zg(f+Af)],

where At indicates the rest period. In evaluating eqn. (22a) we must know tr H2. This is

(42)

tr H2 = H:, + Hf2 + H& + 2HT2 + 2HT3 + 2H$,,

( P a

(434

100 iooo

time ksec)

Fig. 11. Comparison of thixotropic plastic viscoelastic model of this paper with stress relaxation behavior following shear flow of 0.2 N326 carbon black-natural rubber compounds (data of Montes et al. [14]).


” 1000000 I I i S I I

S

100000 I 1

10000 I

I I

0.1 1.0 10.0 100.0 1000.0 10000.0

time (set)

0 ref l 3.4 set I3 13.2 n 6900 A 53940

Fig. 12. Comparison of thixotropic plastic viscoelastic model of this paper with stress relaxation behavior following shear flow of 0.2 N326 carbon black-natural rubber compound (data of Montes et al. [14]).

and in a shearing flow from eqn. (22b):

[S

CC

1 [S Xi

1 2

itrH’= /L&~(Z) d- ’ +; &-‘)Y ‘(z) dz > 0 0

(43b)

where y(z) is the shear strain measured from the present instant. For a long-duration steady shear flow with constant shear rate jr we have (where we use eqn. ( 14))

(43c)

More complex expressions exist for the transients.


We have contrasted the model of this section to a compound of 0.2 N326 carbon black which has a much smaller particle size than the N990 compound discussed in the previous section. In order to proceed we need to fit parameters. The values for the pf,(t, II,) of eqn. (21b) were fit as for the low interaction compound. The value of Y, was determined from steady state shear viscosity data. It was found to have a value of 5000 Pa. The

296 S. Montes and J.L. White 1 J. Non-Newtonian Fluid Mech. 49 (1993) 277-298

V i s 10000000-1 , c 0.0277 l/set 0 000 0 000

s

i ioooooo--

t Y

1 P a

ioooo I 8 I I r 0.1 1.0 10.0 ioo.0 1000.0 10000.0

; time bet)

- Proposed Model ---------- PLASTIC VISC.

Fig. 13. Comparison of predictions of plasticPviscoelastic and thixotropicPplasticPvis- coelastic models and experiment.

values of a, /3, and /?? were determined from fitting the transient shear stress data. They were found to be

X = 0.001,

p, = 2.5 x lo5 Pa.s,

j$ = 1.0 x lo5 Pa.?.

The predictions for the steady shear viscosity are compared in Fig. 9 with viscosity data on the 0.2 N326 carbon black-natural rubber compound. The transient buildup of stresses at the beginning of flow is shown in Fig. 10 together with the predictions of this section. The agreement is again good except that problems seem to be developing at high shear rates. A comparison between theory and experiment for stress relaxation following flow is given in Fig. 11. Here the agreement is not so good, with the experimental data indicating a much slower rate of relaxation than is predicted. This seems to be an inherent problem of the Bogue viscoelastic model rather than of the thixotropic-plastic-viscoelastic formulation. A comparison of predictions of the ‘flow-rest-flow’ history with experiment is shown in Fig. 12. The agreement here is good.

We have also made comparisons of our new ‘thixotropic-plastic-viscoelastic’ model with predictions of the associated ‘plastic-viscoelastic’ model possessing the steady state yield surface, i.e. Y = Y, in transient flow problems. The results are shown in Fig. 13. The addition of thixotropy into the model improves agreement with experiment.

S. Mantes and J.L. White / J. Non-Newtonian Fluid Mech. 49 (1993) 277-298 291

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rheological models of rubber-carbon black compounds: low interaction viscoelastic models and high...

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