The Relative Viscosity of Concentrated Rubber Suspensions and Viscoelastic Modulus of 3D Сross-Linked Elastomers Filled with Solid Particles

doi:10.4236/ojapps.2013.38057

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Open Journal of Applied Sciences, 2013, 3, 477-481

Published Online December 2013 (http://www.scirp.org/journal/ojapps)

http://dx.doi.org/10.4236/ojapps.2013.38057

Open Access OJAppS

The Relative Viscosity of Concentrated Rubber

Suspensions and Viscoelastic Modulus of 3D Сross-Linked

Elastomers Filled with Solid Particles

Alexandr Ermilov, Ergasch Nurullaev

Perm National Research Polytechnic University, Perm, Russia

Email: [email protected]

Received July 29, 2013; revised September 1, 2013; accepted September 8, 2013

Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is

a guardian.

ABSTRACT

For the first time, this paper describes the concentration dependence of the relative dynamic viscosity coefficient of

rubber suspensions and the initial viscoelastic modulus of 3D cross-linked elastomers on the maximum volume filling

with solid polydisperse particles. It allows to predict the rheological and mechanical properties of the polymer composi-

tions being developed now. In this paper, we present the first experimental study of the pole of the concurrent lines of

the concentration dependence in the coordinates of the linear form. The pole validates the invariant value of the constant

of the developed equation and allows the experimental determination of the maximum volume filling of polymer bind-

ers filled with separate fractions or polydisperse mixtures. The results of the study are recommended for use in devel-

oping new polymer composite materials.

Keywords: Viscosity; Mechanical Destruction; Elastomeric Composites with Dispersed Fillers; Rheology; Rubbers;

Polymeric Binders; Asphalt Coatings

1. Introduction

Engineering prediction of the coefficient of dynamic

viscosity (



) and the initial viscoelastic modulus (E) of

low-molecular-weight rubbers and 3D cross-linked elas-

tomers based on themin relation to the maximum (or lim-

iting) volume filling (φ/φm) is of great importance in the

development and manufacture of new polymer composite

materials.

The degree of influence of the volume fraction of solid

particles in the filler (φ) is known to be related to the

maximum filler volume fraction determined by the shape

and fractional composition of the particles, as well as their

physicochemical interaction with the molecules of the

polymer binder, optionally containing a plasticizer [1].

Rutgers analyzed about 100 empirical formulas of the

concentration dependence of the coefficient of dynamic

viscosity of various suspensions [2].

Let us note the most frequently used equations that

adequately describe the dependencies for suspensions:

Eilers [3] proposed a formula for the highly-concen-

trated suspensions on the basis of experimental data:















where the parameters and k

depend on the compo-

sition of the compound and the physical and chemical

nature of the components, the superscripts o and

refer to unfilled and filled elastomeric binders, respec-

tively. The value of varied in the range of 0.75 - 1.25,

the value of

for the spherical particles ranged from

1.20 to 1.35. Further studies confirmed the relation





, where m



Mooney [4] used his own formula:

is the maximum volume filling.

exp ,















in which the same parameters ( and

) are deter-

mined by the physical and chemical features of the filled

suspensions.

Chong et al. [5] and Fedors [6] summarized the ex-

А. ERMILOV, E. NURULLAEV

478

periments with various polymer suspensions and 3D

cross-linked filled elastomers and got an empirical for-

mula for the coefficient of dynamic viscosity and the

initial viscoelastic modulus:

11.25 ,

ff m

oo m

















(1)

where the relations 1.25 m



 and 1m



 are ob-

served.

The aim of this paper is to provide a theoretical justi-

fication of the empirical Equation (1) for the dependence

of the relative viscosity of concentrated rubber suspend-

sions and the viscoelastic modulus of 3D cross-linked

elastomers filled with solid particles. We also aim to

check whether the equation agrees with the experimental

data and apply it for the determination of the maximum

volume filling of the polymer binder with separate frac-

tions and polydisperse mixtures of solid particles.

2. Theoretical Study

We will consider solid particles of arbitrary shapes and

fractional composition distributed randomly but uni-

formly in the matrix, for example, low-molecular-weight

rubber, optionally containing a plasticizer. In the mixing

process, at a higher volume concentration smaller parti-

cles are pushed into the gaps between the larger particles.

This increases the effect of the packing density of the

particles of the initial poly fractional dispersed filler on

the average size layer of the polymer binder between the

particles at a constant value of



Let us denote the coefficient of increasing dynamic

viscosity of the initial rubber or the initial viscoelastic

modulus of the 3D cross-linked elastomer as

fof o

REE





In the first approximation, the value of

R is propor-

tional to the ratio of the filler volume fraction to the

volume fraction of the polymer binder—







.

However, due to the finite particle size the coefficient of

strengthening tends to infinity at m



, rather than at



 Therefore, the argument for the function

should be the ratio







 



Let the pa-

rameter be a new variable of the given dependence,

which allows to generate a differential equation describ-

ing the concentration rate of variation of the coefficient:

z

dR CCz

dz  (2)

where C1 and C2 are the unknown coefficients of the lin-

ear part of the equation in a general form.

While m



varies from 0 to 1, the value of as a

linear fractional function varies from 0 to ∞, which cor-

responds to the experimental data. In its turn, the indefi-

nite integration of the Equation (2):





RCC

z

leads us to the corresponding algebraic expression:

0.5

RCCz Cz  (3)

Taking into account the boundary conditions





1 when 0

CR z





and the preliminary conditions

2 and 2CKC K

( is a combined parameter), we can use the expression

(3) to obtain a quadratic equation as a polynomial of the

second degree:



121 ,

RKzKzKz  (4)

which formally corresponds to the empirical formulas

proposed by Chong et al. (1971)-(5) and Fedors (1979)-

(6):















(5)

The value of Kas a combined parameter was deter-

mined on the basis of the rheological experimental data.

To do this, the Formula (5) was used in the linear form:

fKK



 

 (6)

The results of the experiments obtained by using the

expression (6) are shown in Figure 1 as













with respect to the relative coefficient of dynamic viscos-

ity







 for the compositions based on low-mo

lecular-weight polybutadiene SKD-KTR grade rubber

with terminal carboxyl groups filled with silica (SiO2) of

varying degree of fineness: 1—1 µm; 2—5 µm; 3—15

µm; 4—240 µm; 5—600 µm; 6—a mixture of 2 or 3

fractions taken in the optimal ratio in terms of the mini-

mum porosity for increasing the maximum volume filling

of the composition (m



Correlation analysis shows that the most probable

value of the combined parameter K (a) at the

pole of the concurrent lines in the given coordinates

equals to 1.25, whereas 1. The presence of

the “pole” is indicative of the invariant nature of the

concentration dependence. It is interesting to note that

the Formulas (1) and (5) can be transformed into the

wellknown theoretical Einstein’s formula for viscosity of

dilute suspensions [7]:

t P0.95

2.52CK





12.5 at1

fo m

 





A theoretical justification of the dependence of the

relative viscosity of concentrated rubber suspensions and

Open Access OJAppS

А. ERMILOV, E. NURULLAEV 479

Figure 1. The dependence



-1 -1

-1 =



for the

compositions based on low-molecular-weight polybutadiene

SKD-KTR grade rubber with terminal carboxyl groups

filled with silica (SiO2) of varying degree of fineness: —1

µm; —5 µm; —15 µm; —240 µm; —600 µm;

—a mixture of 2 or 3 fractions taken in the optimal

ratio in terms of the minimum porosity for increasing the

maximum volume filling of the composition







the viscoelastic modulus of 3D cross-linked elastomers

filled with solid particles has been provided for the first

time. We have also detected the pole which validates the

invariant value of the constant of the developed equa-

tion.

Thus, the theoretical dependence (5) corresponds to

the Formula (1), which (alongside with our experimental

data and the results of Chong et al. [5] and Fedors [6] is

recommended for use in calculating the coefficient of

reinforced rubber suspensions and the viscoelastic modu-

lus of 3D cross-linked elastomers filled with solid parti-

cles.

3. Experimental Study

Ermilov et al. [8] verified the Formula (1) in terms of

enhancement of the initial viscoelastic modulus of 3D

cross-linked elastomers for developing a frost-resistant

waterproofing rolled material for asphalt coatings.

Various low-molecular-weight rubbers (polyethylene

butyl formal, polyesterurethane, polybutadiene, polydie-

neepoxy urethane) with terminal functional groups





-SH, -OH, -COOH, -CHOCH



and high-molecular-weight unsaturated rubbers (polyiso-

prene butyl, polyisoprenedivinyl) three-dimensionally

crossed-linked by means of the functional groups, in-

cluding double bonds (-CH=CH-), were used as polymer

binders.

The cross-linking agents were compounds having the

antipodal functional groups capable of forming the nec-

essary chemical bonds. For example, the reaction be-

tween the isocyanate group and the hydroxyl one results

in the urethane group which links two molecules of the

polymer. Some high-molecular-weight and low-molecu-

lar-weight rubbers contained a plasticizer as a component

of the polymer binder.

It should be noted that the formation of the elastic and

viscous components of the initial modulus at a uniaxial

stretching (elongation) of the specimens was as follows.

The concentration of the transverse chemical bonds in

the 3D cross-linked polymer binder determined the elas-

tic component of the modulus, whereas the degree of

polarity of molecular groups of the binder components

(including plasticizer) determined the viscous component

of the modulus. In contrast to the elastic component, the

viscous component of the modulusis in inverse propor-

tion to temperature and strain rate due to the nature of

intermolecular interactions [1].

The dispersed filler was a three-fraction mixture of

silica (river quartz sand) with the optimal size grading

(600:240:15) µm = (50:30:20)%. The fraction of 15 µm

(as well as the other fractions 1 µm and 5 µm) was ob-

tained by grinding quartz sand in the vacuum grinder.

Figure 2 shows the calculated dependence:

Figure 2. shows the calculated dependence







rfo

EEE f





and the experimental data for

filled elastomers based on these rubbers:

1) () polyethylene butyl formal 2) () polyester

urethane, 3) () polybutadiene, 4) () polyisoprene

butyl, 5) () polyisoprenedivinyl, 6) () polydi-

ene epoxy urethane filled with polydisperse silica.

Open Access OJAppS

А. ERMILOV, E. NURULLAEV

480



rfo m

EEE f





and the experimental data for filled elastomers based on

these rubbers. It is evident that the result of the experi-

mental verification of the theoretical approach is in

agreement with the data of Fedors [6] and Chong et al. [5]

both in terms of the dynamic viscosity coefficient (Fig-

ure 1) and the relative initial viscoelastic modulus of

different polymer compositions (Figure 2).

4. Engineering Applications

The formula of reinforcement of rubbers and 3D cross-

linked elastomers (1) allows, alongside with the calcula-

tion methods proposed by Ermilov and Fedoseev [9], to

determine the value of the maximum filling of the poly-

mer binder with separate filler fractions or polydisperse

fractions based on them and to minimize the number of

the experiments. Indeed, the transformation of the Equa-

tion (1) in relation to m



results in the following for-

mula:



1.25 1

 





 (7)

which allows to estimate the value of m



by aviscomet-

ric method. This provides a more accurate estimation of

the maximum filling compared to determination of the

packing density of the particles in bulk since it “auto-

matically” takes into account the intermolecular interac-

tion at the filler-binder interface. Therefore, the most

appropriate binder is low-molecular-weight polybutadi-

ene SKD-KTR grade rubber with terminal carboxyl

groups of medium polarity in terms of the immobilizing

effect of the dispersed filler on the molecular mobility of

the binder. The coefficients of dynamic viscosity of the

binders and filled compositions based on them were de-

termined using Heppler consistometer at one of the mod-

erate values of



and 3 - 5 parallel measurements. The

levels of the gradients of shear rate and experimental

temperatures were selected in compliance with practical

needs and interests of the particular branches of science

and industry.

Table 1 presents the results of viscosimetric determi-

nation of the maximum filling of polybutadiene rubber

with various fractions of silica used in the experiments.

It is clear that with increase in the degree of mechani-

cal grinding (fractions 1, 5, 15 µm) the value of m



decreases significantly, apparently due to the more angu-

lar particles. The fractions 240 µm and 600 µm screened

out from river quartz sand feature particles of more

rounded shapes, which are more favorable in shear flows

of the compositions.

Nielsen [1] studied the physical properties and the

chemical structure of the polymeric base of the binder

and plasticizer and concluded that they affect the inter-

molecular interaction at the filler-binder interface. As a

result of the immobilizing effect, the mobility of the

molecules at the filler-binder interface decreases, which

reduces the value of the maximum volume filling, allo-

ther conditions being equal. Therefore, less polar mole-

cules of the binder components show higher values of



Table 2 presents the immobilizing effect of the physi-

cal and chemical properties of the polymeric binder on its

maximum filling with a three-fraction mixture of silica

with the optimal size grading (600:240:15) µm = (50:30:

20)%.

5. Conclusions

1) For the first time, we have presented a theoretical

justification and an experimental validation of the for-

mula (1) which invariantly describes the concentration

dependence of reinforcement of rubbers and 3D cross-

linked elastomers. We have also first located the “pole”

of the experimental data of the Equation (5) in the coor-

dinates of the linear form(6) that substantiates the value

of the constant (K = 1.25).

2) On the basis of the obtained dependence we have

developed a viscometric method of determining the maxi-

mum volume filling of polymer binders (7) with solid

particles of varying fractional composition.

Table 1. The maximum filling of low-molecular-weight po-

lybutadiene SKD-KTR grade rubber with various fractions

of silica.

Weight-average particle size

of silica fractions in the mixture

with SKD-KTR rubber, µm

1 5 15 240600

The maximum filling of

polybutadiene SKD -KTR grade

rubber with various fractions

of silica, m



0.530 0.547 0.614 0.6990.640

Table 2. The influence of physicochemical features of the

polymer binder on its maximum filling with polydisperse-

silica.

Polymer binder filled with a three-fraction

mixture of silica

Maximum volume

filling, m



P-9А grade polyester 0.770

NVTS-2 grade polyethylene butyl

formal 0.810

Polydieneepoxy urethane PDI-3B grade rubber

plasticized by dioctylsebacate (30%) 0.830

BK grade polyisoprene butyl plasticized by

transformer oil (80%) 0.910

SKID-L grade polyisoprenedivinyl plasticized

by transformer oil (70%) 0.940

Open Access OJAppS

А. ERMILOV, E. NURULLAEV

Open Access OJAppS

481

The use of computational methods for the determina-

tion of the maximum filling of the polymer binder with

polydisperse mixtures as the initial viscometric values of

the maximum filling with separate fractions significantly

increased accuracy.

3) The experiments showed that less polar molecules

of the binder components provide the value of the maxi-

mum filling of the elastomeric binder with three-frac-

tional mixtures of silica at the level of 0.95, all other

conditions being equal. This reduces the coefficient of

dynamic viscosity of the filled elastomer and its initial

viscoelastic modulus in a 3D cross-linked state.

4) The results of the studies are recommended for use

in the accelerated development and optimization of new

formulations of polymeric composite materials using

mathematical methods, for example, the simplex-lattice

design for experiments.

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http://dx.doi.org/10.1007/BF01976051

[3] H. Eilers, “Die Zahigkeiteinekonzentrieren Suspensionen,”

Kolloid-Zeitschrift, Vol. 97, No. 3, 1941, pp. 313-317.

http://dx.doi.org/10.1007/BF01503023

[4] M. Mooney, “Rheology of Concentration Suspensions,”

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156. http://dx.doi.org/10.1016/0095-8522(51)90036-0

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2007-2024.

http://dx.doi.org/10.1002/app.1971.070150818

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Number 2473581, Russian Federation, 2011.

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