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Materials Science and Engineering B 168 (2010) 237–241

Contents lists available at ScienceDirect

Materials Science and Engineering B

journa l homepage: www.e lsev ier .com/ locate /mseb

imulation of curing of a slab of rubber

.M. Abhilasha, K. Kannana,∗, Bijo Varkeyb

Department of Mechanical Engineering, IIT Madras, Chennai 600036, IndiaAdvanced Design Department, MRF Ltd., Chennai 600019, India

r t i c l e i n f o

rticle history:eceived 7 August 2009eceived in revised form2 November 2009ccepted 6 December 2009

a b s t r a c t

The objective of the present work is to predict the degree of curing for a rectangular slab of rubber, whichwas subjected to non-uniform thermal history. As the thermal conductivity of rubber is very low, thetemperature gradient across a slab is quite large, which leads to non-uniform vulcanization, and hencenon-uniform mechanical properties—an inhomogeneous material. Since curing is an exothermic reaction,heat transfer and chemical reactions are solved in a coupled manner. The effect of heat generation on

eywords:atural rubberulcanizationing’s chemical kineticseat conductionrosslink density

curing is also discussed.© 2009 Elsevier B.V. All rights reserved.

eat generation

. Introduction

Rubber is a sticky, liquid-like substance at room temperature,nd cannot be used for making rubber products in its naturalorm. Natural rubber is blended with many chemicals such as sul-hur, a crosslinking agent, accelerator, which increases the rate ofhemical reactions, activators such as zinc oxide, retarders suchs phthalimide, which delay the onset of crosslinking reactions,nd fillers such as carbon black and silica for improving mechan-cal properties of the vulcanized rubber products. The blended

ixture which is liquid-like, is vulcanized at about 140 ◦C. A vul-anized material cannot be processed in an extruder, mixer, or anyevice, which requires the material to flow. Therefore, the vul-anization is done after the blended mixture has taken its finalhape or form. Vulcanization process triggers a cascade of chemi-al reactions, which result in the formation of chemical crosslinks,three-dimensionally networked polymer with suitable materialroperties.

The current developmental procedure for a new tire involverriving at the amount of various chemicals (initial concentration of

atural rubber, sulphur, accelerator and retarder) and maintainingrequired temperature history at the boundary of a tire in a cur-

ng press by a trial and error process until a product with requiredechanical properties is achieved. Any change in mix design and

∗ Corresponding author. Tel.: +91 44 2257 4708; fax: +91 44 2257 4652.E-mail address: krishnakannan@iitm.ac.in (K. Kannan).

921-5107/$ – see front matter © 2009 Elsevier B.V. All rights reserved.oi:10.1016/j.mseb.2009.12.035

mold temperature will affect the chemical reactions, and thereforethe final density of crosslinking, and hence the mechanical proper-ties. In other words, by controlling the amount of chemicals in therubber mixture along with temperature at the surface of a mold, onecould control the mechanical properties of a cured product. A trialand error process is both expensive and time-consuming because itinvolves many full-scale experiments. Modeling and simulation ofthis process will cut down on product development cost and time.To that end, one needs to develop appropriate equations describingthe chemical reactions (refer to [1] for a detailed description). Mod-eling of chemical kinetics developed by Ding et al. [2] is simplistic,in that, it is a lumped model and cannot be used for designing thecomposition of chemicals. However for a fixed chemical composi-tion, the model developed by Ding and Leonov [3] could be used fordesigning the curing process (boundary condition for temperatureand the time for completion of curing process). Therefore, due tosimplicity, we resort to chemical kinetics to that developed by Dingand Leonov [3].

The modeling of this process is extremely important for anyrubber industry, and particularly for a tire industry. As the ther-mal diffusivity of rubber is very low, temperature gradient acrossthe cross-section of a tire is quite large. This leads to non-uniformvulcanization. One needs to determine the curing time in such a

way that the required distribution of degree of cure is achieved. Assimulating the progression of cure for a tire is quite complicated,we chose a much simpler geometry, namely, a large rectangularslab. Such a simpler geometry can still provide insight into theprogression of cure.

2 ce and Engineering B 168 (2010) 237–241

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Table 1Ingredients used in natural rubber mixture. All values are in part perhundred (phr) grams natural rubber. Accelerator used in the mixture is N-tert-butyl-benzothiazole-2-sulphenamide (TBBS) and the retarder used isN-cyclohexylthio pthalimide (CTP).

Ingredient Concentration (phr)

Natural rubber 100Carbon black (N330) 50Sulphur (rhombic, S8) 1.58Accelerator (TBBS) 1.4Retarder (CTP) 0.2Zinc oxide 5Anti-oxidant 3

Table 2Run-time for homothermal tests.

Temperature in ◦C Time in minutes

128 140

38 P.M. Abhilash et al. / Materials Scien

. Experimental

Oscillating disc rheometry (ODR) is a technique for measuringhe shear modulus of a material that is structurally changing. Usinguitable theories, one could estimate the crosslink density of a rub-er sample using the measured shear modulus. The equipmentsed for the ODR runs is Rubber Process Analyzer (RPA)-2000 madey Alpha technologies. One could vary the amplitude and frequencyf oscillation of the upper disc, and temperature of the sample as aunction of time. The basic construction of RPA-2000 is describedelow: the bottom die is fixed and the top die oscillates. The shapef the pair of dies is in such a way that the sample takes the shapef a bi-conical disc—the cone angle is very small. Such a geometrynsures that the strain undergone be a material at any radius fromhe center of disc is constant.

The blended natural rubber mixture is set in the die cavity, ands heated to a required curing temperature. The top die oscillates atpredefined amplitude and frequency, and the torque is measured.he natural rubber mixture behaves like a viscoelastic liquid. The-ry of linear viscoelasticity is used obtain the storage and loss shearodulus. This information is used to calculate the storage and the

oss shear modulus of the material during the curing process. As theubber sample cures, the torque required to bring about the samelastic deformation increases, which results in increase of storagehear modulus, i.e., the more the density of crosslink, the more thetorage modulus. From this point onwards, we shall refer to storagehear modulus as shear modulus.

Carbon black (filler) is an essential ingredient for most rub-er formulations. In automobile tires, carbon black improveset traction amongst other properties. The addition of carbon

lack in the powder form (30–100 nm) to a rubber mixtureignificantly improves many mechanical properties of vulcan-zed rubber products. This is a result of adsorption of polymeric

olecules of rubber on to the surface of carbon black particlesfiller–rubber interactions) [4,5]. Further, carbon filler forms aetwork (filler–filler interactions), which resists deformation. Tostimate the degree of chemical crosslinking in carbon black filledubber mixtures, the measured shear modulus has to be cor-ected for filler–rubber and filler–filler interactions. Medalia [6]uggested that the volume fraction of filler in the familiar Guth-old equation should be replaced by effective volume fraction ofller, i.e.:

g = (1 + 2.5Veff + 14.1Veff2)Gm, (1)

here Veff represents effective volume fraction of filler, Gg repre-ents equivalent shear modulus of gum rubber (without filler) andm represents the storage modulus (shear) of filled rubber mix-

ure. One can determine the effective volume fraction of filler asollows: a rubber mixture was prepared without adding filler andhe sample was cured at 138◦C, and shear modulus is measureds the sample cures. Thus the gum-state shear modulus is deter-ined. Using the shear modulus data for filled and unfilled rubberixture, using Eq. (1), and by only using the plateau shear mod-

lus of gum-state rubber mixture, the effective volume fraction isetermined to be 0.28. The actual volume fraction of added filler

s only about 0.18. Further, we assume that the effective volumeraction remains unchanged at all temperatures. Once the equiva-ent shear modulus of gum rubber is estimated, Gaussian statisticalheory of rubber network [7] is invoked, assuming crosslinks to beetra-functional in nature (For details, see Lee et al. [8]), to estimatehe density of crosslinks, i.e.:

uchem = Gg − Gog

2RT, (2)

here Gg is the shear modulus of gum rubber, Gog is shear modulus

f uncrosslinked rubber mixture, Vuchem is the number of crosslinks

138 70148 70158 60

in mole/m3, R is the universal gas constant and T is the absolutetemperature of the sample.

2.1. Preparation of sample

The samples were prepared by mixing natural rubber and theother chemicals together in a Branbury mixer (see Table 1). Themixing is done until the mixture reaches a temperature of about100 ◦C. Then, the blended sample is rolled into sheets in a rollingmill—eight to ten passes are required. The sheets are immersed inwater to bring the temperature of the sheets to about room tem-perature. The density of the blended sample is 1130 kg/m3. Thesesamples are used in determining storage and loss modulus using adisc rheometer.

2.2. Experiments at homothermal condition

A portion of a rolled sample of blended rubber is placed inbetween the pair of dies. The temperature is brought to the requiredtemperature, and is maintained at that temperature. For all theseexperiments, the top die is oscillated at a strain amplitude of 7%at the periphery of the disc, and at a frequency of 1.6 Hz. Table 2illustrates the conditions at which the tests were performed. Thestorage modulus and loss modulus were recorded during curing ofa sample, and the degree of cure is calculated, after correcting forfiller interactions, as a function of time using the procedure outlinedearlier in this section.

3. Results and discussion

3.1. Modeling of chemical reactions

The cascade of chemical reactions that occur during vul-canization is very complicated. Ghosh et al. [1] developed acomprehensive model taking into account 111 reactants and prod-ucts. In this article, we only consider chemical reaction mechanismsin a simplistic fashion. These models do not include all the reac-tants and products in the reaction scheme. Only the most importantspecies are retained and some of the species may be lumped, and

the number of moles of certain species produced is chosen in sucha way that the mass-balance is satisfied. Coran [9] proposed sucha model for vulcanization of rubber. In this model, Coran consid-ered four species, namely, a sulphurating species, A, vulcanizationprecursor, B, an activated vulcanization precursor, B∗, and the vul-

ce and Engineering B 168 (2010) 237–241 239

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P.M. Abhilash et al. / Materials Scien

anizate, Vu. The reaction scheme is as follows:

k1→Bk2→B∗ k3→˛Vu (3a)

nd

+ B∗ k4→ˇB. (3b)

eaction Eq. (3a) represents the basic steps that occur during vul-anization. It has been observed experimentally that the entireccelerator is consumed before the formation of crosslinks. Reac-ion Eq. (3b) was proposed by Coran to emulate the experimentalbservations. For a sufficiently large rate parameter k4, one couldnsure that all of accelerator is consumed before the forma-ion of crosslinks (induction period). Ding et al. [2] introduced aompetitive reaction Eq. (3c) to account for the decrease in theoncentration of the vulcanizate at higher temperatures, i.e.:

∗ k5→�D (3c)

ater, Ding and Leonov [3] added a reaction to account for thehenomenon of reversion, which was observed during the vulcan-

zation of natural rubber, i.e.:

uk6→�D, (3d)

hich results in five non-linear coupled differential equations, i.e.:

d[A]dt

= −k1[A] − k4[A][B∗], (4a)

d[B]dt

= k1[A] − k2[B] + ˇk4[A][B∗], (4b)

d[B∗]dt

= k2[B] − (k3 + k5)[B∗] − k4[A][B∗], (4c)

d[Vu]dt

= ˛k3[B∗] − k6[Vu], (4d)

nd

d[D]dt

= �(k5[B∗] + k6[Vu]), (4e)

here [A], [B], [B∗], [Vu] and [D] are the concentrations of appro-riate species. The coefficients ˛, ˇ and � are stoichiometricoefficients, introduced as a result of lumping of various chemicalpecies. Using mass-balance, these coefficients were determinedo be ˛=1, ˇ=2 and �=1 (refer to Ding and Leonov [3]). The fiveon-linear, coupled, differential equations are solved with non-ero initial condition for species A and zero initial conditions forhe rest of the species. Referring to Ghosh et al. [1], it is clear thathe lumped species A referred to in Coran, Ding and Leonov anding et al. [2,3,9] represents active sulphurating agent, which is

ormed as a result of interaction of accelerator and elemental sul-hur. Such an interaction is ignored in the lumped models underonsideration. Since one does not know the amount of active sul-

hurating agent, we assume that the initial concentration of activeulphurating agent is equal to that of initial concentration of sul-hur. Accordingly, using the density of the rubber mixture and themount of sulphur (refer to Table 1), the initial concentration ofpecies A is determined to be 250 mole/m3. The above equations

able 3eaction constants obtained after optimization procedure.

Rate constant (unit) 128 ◦C 138 ◦C

k1 (min−1) 4.1730e −002 7.5308ek2 (min−1) 1.0174e −001 2.2832ek3 (min−1) 1.6333e+000 4.3531ek4 (m3/mol/min) 5.0684e −001 9.3045ek5 (min−1) 5.6595e+000 1.6462ek6 (min−1) 3.3123e −005 1.7960e

Fig. 1. Prediction of Ding and Leonov’s model at various temperatures.

are solved using a stiff equation solver in MATLAB. In order to findthe values of the rate constants through k1, k2, k3, k4, k5 and k6 thefollowing objective function is minimized:

� = 100

√√√√ 1N

N∑i=1

{(Vui

exp − Vuimodel

)

Vuiexp

}2

, (5)

where � is the objective function, N is the number of experimentalpoints, Vui

exp is the i th experimental value and Vuimodel

is the i thpredicted value by the model. The objective function representsrms error in percentage. An optimization algorithm (Levenberg-Marquardt) is invoked for obtaining the numerical values for thereaction constants. Table 3 lists the value of rate constants at varioustemperatures.

Prediction of the model at four different temperature is shownin Fig. 1. In order to predict the density of crosslinks for an arbitrarytemperature history, one needs to know how the rate parametersare dependent on temperature. To that end, one may appeal tocollision theory, which predicts that k = A0 exp−Ea/RT , where theconstants A0 and Ea are collision frequency (or pre-exponential)and activation energy, respectively, R is gas constant and T is abso-lute temperature. Further, to determine the constants A0 and Ea,curing experiments at homothermal conditions are sufficient, andare obtained from the intercept and slope of the plots ln(k) versus1/T , respectively. Fig. 2 shows the plot of ln(k) versus 1/T and A0and Ea are obtained from the best fit, which are listed in Table 4.

3.1.1. Modeling of heat transferThe heat conduction process in a slab is approximated as a one-

dimensional process. Accordingly, assuming Fourier law of heat

148 ◦C 158 ◦C

−002 1.3215e −001 2.2591e −001−001 4.9308e −001 1.0275e+000+000 1.1074e+001 2.6976e+001−001 1.6525e+000 2.8637e+000+001 4.5518e+001 1.2005e+002−004 8.9865e −004 4.1729e −003

240 P.M. Abhilash et al. / Materials Science and Engineering B 168 (2010) 237–241

F

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Ewio

T

wzio

T

3

t

TV

ig. 2. A plot used to determine collision frequency, A0 and activation energy, Ea .

onduction, one-dimensional form of heat equation is solved, i.e.:

1˛

∂T

∂t= ∂2T

∂x2+ Q̇

K, (6)

here ˛ is the thermal diffusivity of the rubber material, Q̇ is theate of heat released per unit volume and K is the thermal conduc-ivity of the rubber sample. We shall assume that the rate of heateleased per unit volume is of the following form:

˙ = qdVu

dt, (7)

here q is the rate of heat released per unit volume per unit rate ofhange of total crosslink density. We have assumed that only therosslinking reactions contribute significantly to the heat source.urther, please note that Vu is a function of T and time t, i.e., Vu =u(T(x, t), t).

To obtain the distribution of crosslink density, one need to solveqs. (4a) through (4e), together with the partial differential Eq. (6),hich are coupled in nature. To solve these equations, apart from

nitial conditions for Eqs. (4a)–(4e), one need both initial conditionsf the form:

(x, 0) = f (x), (8)

here f (x) is the temperature distribution of the flat plate at timeero, and boundary conditions at the two boundaries of the slab,.e., at x = a and x = b. We assumed Dirichlet boundary conditionsf the following form:

(a, t) = Ta and T(b, t) = Tb. (9)

.2. Simulation

Let us consider the case of a flat slab of thickness 2 cm, ini-ially at 50 ◦C, and maintained at 140 ◦ C on either side for 50 min.

able 4alues of A0 and Ea obtained using Fig. 2.

Rate constant A0 Ea in Joules

k1 1.4399e+009 9.7300e+003k2 2.7267e+013 1.3322e+004k3 5.1313e+017 1.6156e+004k4 3.2312e+010 9.9762e+003k5 6.4780e+019 1.7598e+004k6 4.9494e+025 2.7861e+004

Fig. 3. Distribution of temperature and corresponding crosslink density. (a) Distri-bution of temperature in a flat slab made of rubber and (b) distribution of crosslinkdensity corresponding to (a).

The initial and boundary conditions are not compatible only attime t = 0. We assume that thermal diffusivity and conductiv-ity of rubber does not depend on temperature and is taken tobe approximately 4.2 × 10−2 cm2/min and 0.12 J/min/K (refer todata for HAF black in Hands and Horsfall [10]). The rate of heatgenerated per unit volume unit rate of change of crosslink den-sity (q) is taken as 1.98 × 105 J/mol. Such a value for q will makesure that the volumetric heat generated is around 2 × 104 W/m3

[11], which is assumed to be a typical value for rubber vulcan-ization. Notice that the heat conduction equation, i.e., Eq. (6) andvulcanization reactions, i.e., Eqs. (4a)–(4e), are coupled with eachother.

First, the heat equation is solved by assuming that the heatsource is zero. Second, the temperature history corresponding toevery node, associated with the slab, is extracted and is used forsolving equations representing vulcanization reactions. Recall that

an Arrhenius expression is used for reaction rates, i.e., k1 through k6,which are function of temperature. Thus, the density of crosslinksrealized at the end of the reaction is a function of temperature his-tory. Third, the rate of change of crosslink density, associated withevery node, computed in the second step, is used to re-calculate

P.M. Abhilash et al. / Materials Science and

Fc

tlhtp(M

1958.[8] S. Lee, H. Pawlowski, A. Coran, Rubber Chemistry and Technology 67 (1994)

ig. 4. The effect of heat source on temperature distribution and density ofrosslinks. (a) Temperature profile and (b) total crosslink density.

he heat source term. Cubic splines are used to obtain interpo-ated values for dVu

dt at arbitrary x and t. Fourth, using the known

eat source term, new temperature field is computed. Finally,he entire procedure is repeated until one obtains identical tem-erature and crosslink density field. The heat equation, i.e., Eq.6) is solved using a suitable initial and boundary condition in

ATLAB.

[[

Engineering B 168 (2010) 237–241 241

The temperature distribution of the flat slab as function of timeis shown in Figs. 3(a) and 4(a), and its corresponding crosslink dis-tribution is shown in Figs. 3(b) and 4(b), respectively. If one wantsto obtain uniform crosslink density, it is obvious from Fig. 4(b) thata curing time of about 50 min is required. After 50 min, temperaturereaches about 140 ◦ C throughout the slab. However, crosslink den-sity shows a slight non-uniformity because the temperature historyassociated with every point is different. Further, as expected, a heatsource term in the heat equation causes a higher temperature dis-tribution to exist in a slab compared to that of a heat equation(with the same material parameters) without a heat source (seeFig. 4(a)). Referring to Fig. 4(b), at 40 and 45 min, the degree of cureis slightly higher compared to that of a model for curing withouta heat source and, however, at 50 min, the trend is reversed. Thereason for this behavior is attributed to a large heat source, whichcauses reversion.

4. Summary and conclusions

The distribution of crosslink density for a flat slab was pre-dicted using Ding and Leonov’s model. It was found that this modeldescribes the curing behavior adequately. The presence of a heatsource term in the heat equation did not alter the results signif-icantly. Using such a model, one could estimate the curing timeand study the possibility of reduction of curing time, and therebyenhancing the productivity. Should one be interested in predictingthe subsequent mechanical response of a cured rubber product,such as in the case of an automobile tire, one could use this model,for a given temperature history, to estimate the distribution ofcrosslink density. In other words, the inhomogeneity can be pre-dicted.

Acknowledgment

P.M. Abhilash and K. Kannan thank MRF for sponsoring researchin this area.

References

[1] P. Ghosh, S. Katare, P. Patkar, J. Caruthers, V. Venkatasubramanian, RubberChemistry and Technology 76 (2003) 592–694.

[2] R. Ding, A. Leonov, A. Coran, Rubber Chemistry and Technology 69 (1996) 81–91.[3] R. Ding, A. Leonov, Journal of Applied Polymer Science 61 (1996) 455–463.[4] J. Diani, B. Fayolle, P. Gilormini, European Polymer Journal 45 (2009) 601–612.[5] J.L. Leblanc, Progress in polymer science 27 (2002) 627–687.[6] A.I. Medalia, Rubber Chemistry and Technology 46 (1973) 877–896.[7] L.R.G. Treloar, The Physics of Rubber Elasticity, Oxford University Press, Oxford,

854–864.[9] A. Coran, Rubber Chemistry and Technology 37 (1964) 689–697.10] D. Hands, F. Horsfall, Rubber Chemistry and Technology 11 (1977) 253–265.11] M. Juma, M. Bafrnec, Chemical Papers 58 (2004) 29–32.