A Study on the Modified Arrhenius Equation Using the Oxygen

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Article A Study on the Modiﬁed Arrhenius Equation Using the Oxygen Permeation Block Model of Crosslink Structure

Byungwoo Moon , Namgyu Jun, Soo Park, Chang-Sung Seok * and Ui Seok Hong

Department of Mechanical Engineering, Sungkyunkwan University, Suwon-si, Gyeonggi-do 16419, Korea; [email protected] (B.M.); [email protected] (N.J.); [email protected] (S.P.); [email protected] (U.S.H.) * Correspondence: [email protected]; Tel.: +82-31-290-7477

Received: 2 December 2018; Accepted: 11 January 2019; Published: 14 January 2019

Abstract: Polymers are widely used in various industries because of their characteristics such as elasticity, abrasion resistance, fatigue resistance and low temperature. In particular, the tensile characteristic of rubber composites is important for the stability of industrial equipment because it determines the energy absorption rates and vibration damping. However, when a product is used for a long period of time, polymers become hardened owing to the changes in characteristics because of aging, thereby reducing the performance and increasing the possibility of accidents. Therefore, accurately predicting the mechanical properties of polymers is important for preventing industrial accidents while operating a machine. In general reactions, the linear Arrhenius equation is used to predict the aging characteristics; however, for rubber composites, it is more accurate to predict the aging characteristics using nonlinear equations rather than linear equations. However, the reason that the characteristic equation of the polymer appears nonlinear is not well known, and studies on the change in the characteristics of the natural and butadiene rubber owing to degradation are still lacking. In this study, a tensile test is performed with different aging temperatures and aging time to evaluate the aging characteristics of rubber composites using strain energy density. We propose a block effect of crosslink structure to express the nonlinear aging characteristics, assuming that a limited reaction can occur owing to the blocking of reactants in the rubber composites. Consequently, we found that a relationship exists between the crosslink structure and aging characteristics when the reduction in crosslink space owing to aging is represented stochastically. In addition, a modiﬁed Arrhenius equation, which is expressed as a function of time, is proposed to predict the degradation rate for all aging temperatures and aging times, and the formula is validated by comparing the degradation rate obtained experimentally with the degradation rate predicted by the modiﬁed Arrhenius equation.

Keywords: modiﬁed Arrhenius equation; degradation rate; crosslink density; oxygen permeation; nonlinear characteristics equation

1. Introduction Rubber composites are widely used in industrial components because of their good energy-absorbing properties, resilience, elasticity and high extensibility. In particular, in a vibrating machine, the role of rubber that supports the body of the structure and absorbs vibrations and shocks generated from the ground is crucial. [1,2] In addition, the tensile characteristic of rubber composites determines the energy absorption rates and vibration damping, which is an important factor for the stability of an industrial equipment. However, when the product is used for a long period, rubber composites become hardened [3,4] owing to the changes in the characteristics because of degradation,

Polymers 2019, 11, 136; doi:10.3390/polym11010136 www.mdpi.com/journal/polymers Polymers 2019, 11, 136 2 of 15 thereby reducing the energy absorption rate and increasing the probability of accidents. Therefore, the evaluation of the mechanical properties of aged rubber composites has become important to prevent accidents while operating a machine [5]. Generally, the room-temperature aging test is adopted as a method of evaluating the degradation properties of rubber composites; however, this test takes a long time to evaluate the degradation properties. Hence, many researchers have conducted accelerated tests to evaluate the mechanical properties of rubber using the Arrhenius equation [6–8]. Kim et al. used the Arrhenius equation by substituting the compression set of ethylene propylene diene monomer (EPDM) rubber, which is used for motor fans for its characteristic value [9]. Han et al. analyzed the results obtained using the tensile strain of rubber hoses from manufacturing engine radiators as the characteristic value of the Arrhenius equation [10]. However, the parameters for the degradation characteristics of rubber composites appropriate for use in the Arrhenius equation are not well known; hence, research on these parameters should be performed in terms of reliability. In this study, we attempt to evaluate the degradation characteristics of rubber composites by performing tensile tests [11] on the rubbers at various aging temperatures and aging times. For this purpose, strain energy densities (SED) [12,13] were used and the tensile properties and SED values of rubber materials with varied degradation conditions were compared and analyzed [14]. In addition, we assumed that a limited reaction can occur owing to the blocking of the reactants in the rubber composite, and proposed an oxygen permeation block model of the crosslinked structure to express nonlinear aging characteristics. Consequently, a relationship between the crosslink structure and aging characteristics was found when the reduction in crosslink space owing to aging was represented stochastically. Thus, we suggest a modified characteristic equation and nonlinear Arrhenius equation, which is expressed as a function of time. By applying the modified Arrhenius equation, we derived the relationship between short-term high-temperature aging and long-term low-temperature aging, and suggest a method for predicting the degradation rate of rubber composites under all aging conditions. Finally, the accuracy of the formula was demonstrated by comparing the actual experimental results obtained using an aged specimen with the calculated degradation rates predicted by the modified Arrhenius equation.

2. Evaluation of Tensile Properties of Aged Specimen

2.1. Tensile Tests The primary materials of the polymer used in this study were a containing 50% natural rubber (NR) and 50% butadiene rubber (BR). The shape of the specimens used for the tensile test was a dumbbell type No. 3, according to KS M 6518. The specimen was made by die cutting a 2 mm rubber sheet. A blade meeting the KS standard was used for cutting. The cut specimens were degraded at high temperatures using an environmental chamber (within a 1 ◦C temperature error). The conditions for the degradation procedure are listed in Table1.

Table 1. Accelerated degradation conditions for rubber composites.

Degradation Condition Aging temp. (◦C) 23 70 80 90 100 Aging time (Day) - 17, 35 17, 35 17 3, 7, 17

An electro hydraulic universal testing machine was used for the tensile test, and a 200 N load cell, suitable for a load of a rubber tensile test, was installed and tested. The overall testing process was performed according to the ASTM D412-a [11] tensile test speciﬁcations for vulcanized rubbers and thermoplastic elastomers. Before starting this test, the strain range to be tested was repeated 30 times to stabilize the rubber molecular structure, and Mullin’s effect [15] was removed. Polymers 2019, 11, 136 3 of 15

Polymers 2018, 10, x FOR PEER REVIEW 3 of 15 Polymers2.2. Results 2018, of10, the x FOR Tensile PEER Test REVIEW 3 of 15 Tensile tests were conducted to obtain stress–strain curves with four strain values (strain: 0.69, Tensile tests were conducted to obtain stress–strain curves with four strain values (strain: 0.69, 0.69, 0.93, 1.16, 1.39) of aged rubber composites, and the results are plotted in Figures 1 and 2. The results 0.93, 1.16, 1.39) ofof agedaged rubberrubber composites,composites, and and the the results results are are plotted plotted in in Figures Figures1 and 1 and2. The 2. The results results of of the tensile test, based on the aging time and aging temperature, indicate that as the degradation ofthe the tensile tensile test, test, based based on on the the aging aging time time and and aging aging temperature, temperature, indicate indicate that that as as the the degradation degradation of of the rubber progressed, it hardened and the stress increased under the same strain condition. ofthe the rubber rubber progressed, progressed, it hardened it hardened and and the stressthe stre increasedss increased under under the same the same strain strain condition. condition. Thus, Thus, we conclude that the unaged rubber composites specimens are the softest, and the Thus,we conclude we conclude that the that unaged the rubberunaged composites rubber comp specimensosites specimens are the softest, are the and softest, the degradation and the degradation specimens become stiffer as aging time and temperature increased. degradationspecimens become specimens stiffer become as aging stiffer time as and aging temperature time and increased.temperature increased.

Figure 1. Tensile curves according to aging time (Strain: 0.69, 0.93, 1.16, 1.39). FigureFigure 1. TensileTensile curves according to aging time (Strain: 0.69, 0.93, 1.16, 1.39).

FigureFigure 2. TensileTensile curves according to agingaging temperature (Strain: 1.39). Figure 2. Tensile curves according to aging temperature (Strain: 1.39). 2.3. Derivation of SED-strain relationship 2.3. Derivation of SED-strain relationship 2.3. Derivation of SED-strain relationship To determine the strain energy density, the lower area of the stress-strain curve must be calculated. To determine the strain energy density, the lower area of the stress-strain curve must be Therefore,To determine Simpson’s the rule strain [16 ],energy which density, is suitable the forlower calculating area of the areastress-strain of a stress-strain curve must curve be calculated. Therefore, Simpson's rule [16], which is suitable for calculating the area of a stress-strain calculated.exhibiting nonlinearTherefore, behavior, Simpson's was rule applied [16], which to the is equation.suitable for The calculating SED-strain the equation area of a (as stress-strain shown by curve exhibiting nonlinear behavior, was applied to the equation. The SED-strain equation (as curveEquation exhibiting (1)) for thenonlinear aged specimens behavior, was was determined applied to by the fitting equation the calculated. The SED-strain SED value. equation In this case, (as shown by Equation (1)) for the aged specimens was determined by fitting the calculated SED value. shownε represents by Equation strain of (1)) the for rubber, the ageda represents specimens coefficient was determined number, by and fittingb is the an exponentialcalculated SED term value. [14]. In this case, 𝜀 represents strain of the rubber, a represents coefficient number, and b is an InThe this coefficients, case, 𝜀 accordingrepresents to strain aging of conditions, the rubber, are listeda represents in Table2 ;coefficient the strain valuesnumber, at theand same b is SED an exponential term [14]. The coefficients, according to aging conditions, are listed in Table 2; the strain exponential(1 MJ/m3) were term obtained [14]. The as coefficients, shown in Figures according3 and to4 .aging conditions, are listed in Table 2; the strain values at the same SED (1 MJ/m3) were obtained as shown in Figures 3 and 4. values at the same SED (1 MJ/m3) were obtained as shown in Figures 3 and 4. b SED𝑆𝐸𝐷= a × =( 𝑎ε) ×(𝜀) (1)(1) 𝑆𝐸𝐷 = 𝑎 ×(𝜀) (1) The comprehensive results of the evaluation of rubber composites are as follows. As The comprehensivecomprehensive results results of theof evaluationthe evaluation of rubber of compositesrubber composites are as follows. are as As follows. degradation As degradation progresses, the rubber composites become hardened and the slope of the tensile degradationprogresses, theprogresses, rubber composites the rubber become composites hardened become and thehardened slope of and the tensilethe slope strength of the increases. tensile strength increases. In addition, the strain tends to decrease at the same3 SED (1 MJ/m3), depending strengthIn addition, increases. the strain In addition, tends to decreasethe strain at tends the same to decrease SED (1 at MJ/m the same), depending SED (1 MJ/m on the3), depending increase in on the increase in aging temperature and time (Figure 5). Therefore, we regard the strain value for on the increase in aging temperature and time (Figure 5). Therefore, we regard the strain value for this SED (1 MJ/m3) as the characteristic value and the degradation rate of rubber composites as the this SED (1 MJ/m3) as the characteristic value and the degradation rate of rubber composites as the strain reduction rate. strain reduction rate.

Polymers 2019, 11, 136 4 of 15

3 agingPolymers temperature 2018, 10, x FOR and PEER time REVIEW (Figure 5). Therefore, we regard the strain value for this SED (1 MJ/m4 of) as15 Polymersthe characteristic 2018, 10, x FOR value PEER and REVIEW the degradation rate of rubber composites as the strain reduction rate.4 of 15

Figure 3. Strain energy density (SED)-strain curves according to aging time. Figure 3. Figure 3. StrainStrain energy energy density (SED)-strain curves according to aging time.

FigureFigure 4.4. SED-strain curves accordingaccording toto agingaging temperature.temperature. Figure 4. SED-strain curves according to aging temperature. Table 2. Coefﬁcients of SED strain curves. Table 2. Coefficients of SED strain curves. ◦ Table 2. Coefficients of SED strain curves. AgingAging (°C, ( C, day) day) Coefﬁcient Coefficient Number, aa ExponentialExponential Term, Term,b b AgingUnagedUnaged (°C, day) Coefficient 0.8020.802 Number, a Exponential 1.647 1.647 Term, b Unaged70,70, 17 17 1.0880.802 1.088 1.685 1.647 1.685 70,70,70, 1735 35 1.2381.088 1.238 1.835 1.685 1.835 70,80,80, 3517 17 1.2441.238 1.244 1.706 1.835 1.706 80,80,80, 1735 35 1.4621.244 1.462 1.836 1.706 1.836 80,90,90, 3517 17 1.5301.462 1.530 1.709 1.836 1.709 90,100,100, 17 3 3 1.2891.530 1.289 1.816 1.709 1.816 100,100,100, 37 7 1.5151.289 1.515 1.764 1.816 1.764 100,100, 177 17 1.6391.515 1.639 1.700 1.764 1.700 100, 17 1.639 1.700

PolymersPolymers2019 2018,,11 10,, 136 x FOR PEER REVIEW 55 ofof 1515

Figure 5. Decrease in strain on aging. Figure 5. Decrease in strain on aging. 3. Application of Modiﬁed Arrhenius Equation 3. Application of Modified Arrhenius Equation 3.1. General Arrhenius Equation 3.1. GeneralThe characteristic Arrhenius Equation equation of the reaction rate constant k used for the Arrhenius equation is generally expressed as Equation (2). The characteristic equation of the reaction rate constant k used for the Arrhenius equation is generally expressed as Equation (2). dP P − = kP, ln = −kt (2) d𝑃 dt P0 𝑃 − = 𝑘𝑃 , ln =−𝑘𝑡 (2) d𝑡 𝑃 In this case, P represents the characteristic value of the rubber, P0 represents an initial characteristic value,Int representsthis case, time,P represents and k is a reactionthe characteristic rate constant. value If theof lifetimethe rubber, of the P rubber0 represents is deﬁned an asinitial the timecharacteristic until the characteristicvalue, t represents value time, becomes and kP isfrom a reaction Equation rate (2), constant. the lifetime If the (lifetimet) can be of expressed the rubber by is Equationdefined as (3) the at time that time.until the characteristic value becomes P from Equation (2), the lifetime (t) can be expressed by Equation (3) at that time. P t = − ln /k (3) P0 𝑃 𝑡 = − ln /𝑘 (3) The reaction rate constant k from Equation (2), a value𝑃 indicating a degradation reaction, can be expressedThe reaction by Equations rate constant (4) and (5) k usingfrom theEquation Arrhenius (2), a equation. value indicating At time t ,aA degradationand C are constants, reaction,E acanis · thebe activationexpressed energyby Equations (J/mol), (4)R is and a gas (5) constant using (8.314the Arrhenius J/mol K), equation. and T is theAt absolutetime t, temperature.A and C are constants, Ea is the activation energy (J/mol), R is a gas constant (8.314 J/mol∙K), and T is the k = A·e−Ea/RT (4) absolute temperature.

E / ln𝑘=k =𝐴−∙ea + C (4)(5) RT

The lifetime t in Equation (3) can be calculated𝐸 using the empirical Arrhenius equation ln 𝑘 = − +𝐶 (5) (Equation [4]) because the equations represent the𝑅𝑇 relationship between lifetime and temperature. Thus,The the lifetimelifetime cant in be Equation converted (3) into can temperature. be calculated With using the characteristic the empirical value Arrhenius of P, lifetime equationt1 is derived(Equation at temperature[4]) because theT1, andequations lifetime representt2 is derived the relationship at temperature betweenT2. Consequently, lifetime and temperature. Equation (6) can be obtained. Thus, the lifetime can be converted into temperature. With the characteristic value of P, lifetime t1 is t1 Ea 1 1 derived at temperature T1, and lifetimeln t2 is derived= at− temperature T2. Consequently, Equation (6)(6) t2 R T1 T2 can be obtained. For each aging temperature (70, 80, 90,= and 100 ◦C), the relationship between aging time and 𝑡 𝐸 1 1 characteristic values is shown in Figureln6, and= the reaction− rate constant ( k) of the characteristic(6) 𝑡 𝑅 𝑇 𝑇 equation is derived and listed in Table3. When the temperature is low, the reaction rate constant k is smallFor each and aging the characteristic temperature value (70, 80, changes 90,= and gradually. 100 °C), the However, relationship as the between temperature aging increases, time and thecharacteristic reaction rate values increases is shown and the in characteristic Figure 6, and value the changesreaction sharply. rate constant In this study,(k) of the agingcharacteristic time (t) equation is derived and listed in Table 3. When the temperature is low, the reaction rate constant k

Polymers 2018, 10, x FOR PEER REVIEW 6 of 15 isPolymers small 2018 and, 10 the, x FOR characteristic PEER REVIEW value changes gradually. However, as the temperature increases,6 of the 15 reaction rate increases and the characteristic value changes sharply. In this study, the aging time (t) Polymersis calculatedsmall2019 and, 11 theat, 136 15, characteristic 25, 35, and 45%value reductions changes gradually. from the characteristicHowever, as valuesthe temperature using the increases,characteristic6 of the 15 equation.reaction rate The increases results are and plotted the characteristic in Figure 7. valueThe condition changes atsharply. a 25% Indecrease this study, in characteristic the aging time value (t) is calculatedderived as at a 15,function 25, 35, ofand temperature 45% reductions and timefrom inthe Equation characteristic (7), and values compared using the with characteristic the actual is calculated at 15, 25, 35, and 45% reductions from the characteristic values using the characteristic value.equation. The results are plotted in Figure 7. The condition at a 25% decrease in characteristic value equation.is derived The as resultsa function are plottedof temperature in Figure 7and. The time condition in Equation at a 25% (7), decrease and compared in characteristic with the value actual is derivedvalue. as a function of temperature and ln(𝑡) time in = Equation6472.6/𝑇 (7), − 15.2 and compared with the actual value.(7)

ln ln(𝑡)(t) = 6472.6/ = 6472.6/𝑇T − 15.2 − 15.2 (7) (7)

Figure 6. Characteristic equation according to aging time by temperature.

FigureTable 6. 3.Characteristic Reaction rate equation constant according (k) depending toto agingaging on aging timetime by bytemperature. temperature.temperature.

TableAging 3. Reaction(°C) rate constant ( 70k) depending80 on aging temperature.90 100 Table 3. Reaction rate constant (k) depending on aging temperature. Reaction rate constant (k) 0.00801 0.01115 0.02253 0.03430 Aging (◦C) 70 80 90 100 Aging (°C) 70 80 90 100 ReactionReaction rate rate constant constant ((kk)) 0.00801 0.00801 0.011150.01115 0.022530.02253 0.034300.03430

Figure 7. General Arrhenius plot according toto thethe characteristiccharacteristic value.value.

TheThe resultsresults of of calculatingFigure calculating 7. General the the equivalent Arrhenius equivalent agingplot agin according time,g time, converted to convertedthe characteristic from from the aging value.the aging temperature temperature using Equationusing Equation (7), are (7), shown are inshown Table 4in. TheTable mean 4. The deviation mean betweendeviation the between predicted the characteristic predicted characteristic value using thevalue characteristicThe using results the characteristic equationof calculating and equation the the actual equivalent and experimental the actuaginalg value experimentaltime, (Figureconverted6 )value is 42%from (Figure or the more. aging 6) Therefore,is 42% temperature or more. if we convertTherefore,using Equation the if characteristic we (7),convert are shown the values characteristic in into Table aging 4. valuesThe time me by intoan applying deviationaging time the between generalby applying Arrheniusthe predicted the general equation, characteristic Arrhenius a large differencevalue using occurs the characteristic in the resulting equation values and [17]. the For actu example,al experimental when the degradationvalue (Figure condition 6) is 42% caused or more. by decreasingTherefore, if the we characteristic convert the valuecharacteristic by 25% isvalues calculated into aging with thetime Arrhenius by applying equation, the general it becomes Arrhenius nine ◦ ◦ days at 100 C (Table4). However, as the actual specimen is equivalent to a specimen aged at 100 C Polymers 2019, 11, 136 7 of 15 for approximately three days, the difference is almost a factor of three. Depending on the difference in these results, ISO 11346 [18] suggests using ﬁtting functions of the logarithmic scale or establishing characteristic equations as suitable expressions. However, hitherto, standard characteristic equations for rubber are few. Thus, most studies pertaining to the degradation life of rubber composites used general characteristic equations with large deviations. Other researchers have substituted certain characteristic values regardless of the degradation effect and acquired an Arrhenius equation by analyzing the relationship between aging temperature and aging time [19,20]. In this study, we attempt to determine mathematical expressions that can predict the degradation rate of rubber composites under all aging conditions, using the modiﬁed characteristic equation and the Arrhenius equation.

Table 4. Aging days conversion results.

Property Equivalent Degradation Conditions (P/P0) T(◦C) Days 23 786 70 39 0.75 80 23 90 13 100 9

3.2. Oxygen Permeation Block Model As a result of predicting the aging properties by applying the linear Arrhenius equation, the difference between the experimental and theoretical values was significant. Therefore, when developing products using linear Arrhenius equations, it is difficult to predict changes owing to aging, and product characteristics can often vary when used for long periods of time. To solve this nonlinear [21] aging behavior problem, researchers have analyzed the aging behavior of the polymer by dividing the aging period of the polymer into linear sections of 2–3 stages. However, because polymer aging requires a wide temperature range and high test sensitivity, empirical extrapolation could not solve the essential problem. In 2000, Dakin’s kinetic equation shows the reaction of the polymer and the calibrated activation energy that varies with the deterioration of the material [22]. Using this, an individual Ea was set in the reaction at a high temperature and a low temperature. However, it is only a numerical conversion approach using the test results. In 2005, nonlinear Arrhenius [23,24] behavior was studied to accurately predict the aging characteristics of polymers. Celina et al. presented a nonlinear Arrhenius equation with two reactions, assuming that the reaction rate exhibits temperature dependence [25]. This method does not require complex kinetic modeling, easily determines the individual activation energies and demonstrates excellent compatibility by representing at least two reactions. However, the primary reason that activation energy appears nonlinearly is not suggested. Many studies have emphasized the importance of changes in mechanical properties and activation energies, but they have not yet found the fundamental cause of the nonlinear reaction rate constant of characteristic values. The oxidative hardening reaction of the polymer owing to aging, and the increase in the crosslinking density are shown in Figure8. At this time, Bernstein et al. showed that the aging rate is related to the consumption of oxygen by analyzing the relation between the oxygen consumption measurement and the reaction rate of the polymer [26–28]. Thus, the reason that the reaction rate of the polymer decreases as the aging progresses is because the probability of the rubber reacting with oxygen is reduced. In this study, the reason that the reaction rate decreases as the aging progresses, and the difference between the linear Arrhenius equation and the nonlinear Arrhenius equation are suggested as follows. In the case of gas and liquid, oxygen molecules diffuse freely between reactants, as shown in Figure9a. Therefore, the reaction rate constant is represented by a linear equation that is independent of time (t) by a continuous reaction, and a general Arrhenius equation is established. However, in the case of rubber molecules, the crosslinking structure increased by aging interferes with Polymers 2018, 10, x FOR PEER REVIEW 8 of 15 Polymers 2018, 10, x FOR PEER REVIEW 8 of 15 is established. However, in the case of rubber molecules, the crosslinking structure increased by Polymers 2019, 11, 136 8 of 15 isaging established. interferes However, with the inpermeation the case of of rubberoxygen molecules, [29,30], as shownthe crosslinking in Figure 9(b),structure and theincreased reaction by of agingthe molecules interferes iswith inhibited the permeation over time. of Therefore, oxygen [29, we30], suggest as shown a modified in Figure characteristic 9(b), and the equation reaction andof thenonlinear permeationmolecules Arrhenius is of inhibited oxygen equation, [ 29over,30 ],time.which as shown Therefore, is expressed in Figure we as 9suggestb, a andfunction the a modified reaction of time. of characteristic the molecules equation is inhibited and nonlinearover time. Arrhenius Therefore, equation, we suggest which a modified is expressed characteristic as a function equation of time. and nonlinear Arrhenius equation, which is expressed as a function of time.

Figure 8. Crosslinking mechanism. Figure 8. Crosslinking mechanism. Figure 8. Crosslinking mechanism.

(a) (a)

(b) (b) FigureFigure 9. 9.( a(a)) General General aging aging reaction; reaction; ( b(b)) Aging Aging reaction reaction according according to to the the oxygen oxygen permeation permeation block block Figureeffecteffect of of9. crosslink crosslink(a) General structure. structure. aging reaction; (b) Aging reaction according to the oxygen permeation block 3.3. Modiﬁedeffect of crosslink Arrhenius structure. Equation 3.3. Modified Arrhenius Equation In a general characteristic equation, the properties decrease linearly in proportion to the reaction 3.3. ModifiedIn a general Arrhenius characteristic Equation equation, the properties decrease linearly in proportion to the rate constant and time in accordance with Equation (2). However, in the case of the actual test results on reaction rate constant and time in accordance with Equation (2). However, in the case of the actual the rubberIn a general composites characteristic shown in Figure equation,6, as aging the proper time increases,ties decrease the reduction linearly rate in ofproportion the characteristic to the test results on the rubber composites shown in Figure 6, as aging time increases, the reduction rate reactionvalue decreases. rate constant Thus, and if antime existing in accordance characteristic with Equation equation (2). is used, However, a large in errorthe case occurs. of the Because actual of the characteristic value decreases. Thus, if an existing characteristic equation is used, a large error testof the results above on reasons, the rubber other composites researchers shown have in used Figure the 6, non-Arrhenius as aging time equationincreases, by the substituting reduction rate the occurs. Because of the above reasons, other researchers have used the non-Arrhenius equation by ofaccelerative the characteristic shift factors value for decreases. the individual Thus, activationif an existing energy characteristic [25,31]. However, equation itis isused, only a a large numerical error occurs. Because of the above reasons, other researchers have used the non-Arrhenius equation by conversion approach using the test results. Further, the fundamental parameters of the degradation

Polymers 2019, 11, 136 9 of 15 characteristics for rubber composites in the Arrhenius equation are not well known [32]. In this study, we formulated the relationship that the reaction rate constant is inversely proportional to the time based on the oxygen permeation block model. Therefore, a modiﬁed characteristic equation is expressed as Equation (8), and the modiﬁed Arrhenius equation is derived as follows by substituting the time term for the characteristic equation.

dP − = k∗ P (8) dt n 0 ∗ ∗ where k n = k /(t + 1). By integrating Equation (8), we can obtain Equation (9) as follows:

Z P Z t 1 ∗ 1 P ∗ − dP = k dt = 1 − k × [ln(t + 1) − ln(t0 + 1)] (9) Po P0 t0 (t + 1) P0

If the initial value at aging time 0 is substituted for t0, the modiﬁed characteristic equation is derived as Equation (10). P = 1 − k∗ × ln(t + 1) (10) P0 In addition, the modiﬁed Arrhenius equation in which the activation energy and the constant are expressed as a function of the characteristic value is presented as Equation (11).

∗ k∗ = A∗·e−Ea /RT (11) when Equation (10) is substituted into the modiﬁed Arrhenius equation (Equation (11)), k* is eliminated and the result is expressed as Equation (12).

P E∗ ln 1 − = − a + C + ln(t) (12) P0 RT where A∗ = f t, P , C = f p , E∗ = f p . P0 1 p0 a 2 p0 Finally, in the case of the characteristic value P, time T1 at temperature T1 can be represented as equal to time t2 at temperature T2, which is expressed as Equation (13).

t E∗ 1 1 ln 1 = a − (13) t2 R T1 T2

The modified characteristic equation (Equation (10)) is applied to derive the reaction rate constant, and the results are presented in Table5. As a result of the derivation, the k* value of the reaction rate increases with increasing temperature, and the characteristic value decreases significantly at the same aging time. As the aging time increases, the reduction rate of the characteristic value decreases. These results are shown in Figure 10. An analysis of the data shows that the mean deviation of the values predicted by the experiment and the modified characteristic equation decrease significantly to within 17%. In addition, the difference between the two results is less than 4% when the upper usage temperature limits (100 ◦C) for the NR compound is excluded. The temperatures and times required for the characteristic value to decrease by 15, 25, 35, and 45% are shown in Figure 11. By applying the modified Arrhenius equation (Equation (11)), the activation energy (Ea*) and constant (C) were obtained. Further, the results based on the characteristic values, are represented in Figures 12 and 13. The activation energy (Ea*) and the constant (C) both indicated a linear relation to the characteristic value, and regression analysis was used to derive their respective Equations, i.e., (14) and (15). Finally, the modified Arrhenius equation was derived as Equation (16). The Arrhenius expression with the characteristic value is expressed as Equation (17). Polymers 2019, 11, 136 10 of 15

Polymers 2018, 10, x FOR PEER REVIEW 10 of 15 Polymers 2018, 10, x FOR PEER REVIEW 10 of 15 p p C = f 𝑝 = −87.73 × 𝑝 + 93.13 (14) 𝐶=1𝑓 p = −87.73 × p +93.13 (14) 𝑝0𝑝 𝑝𝑝0 𝐶=𝑓 = −87.73 × +93.13 (14) 𝑝 𝑝 ∗ ∗ p𝑝 𝑝p E𝐸a ==f2𝑓 == −296.7−296.7 ×× ++ 311.86311.86 (15) (15) ∗ p𝑝0𝑝 𝑝p𝑝0 𝐸 = 𝑓 = −296.7 × + 311.86 (15) 𝑝 p𝑝 𝑝 𝑝p ln(𝑡) = 𝑓 /𝑅𝑇−𝑓 (16) ln(t) = f2 𝑝𝑝 /RT − f1 𝑝𝑝 (16) p0 p0 ln(𝑡) = 𝑓 /𝑅𝑇−𝑓 (16) 𝑝 𝑝 𝑃P 𝑅𝑇RT·( ∙ln (ln[[𝑡t]] + 93.13)93.13) − 311.86 F𝐹(𝑇,(T, t) = 𝑡) = == (17)(17) 𝑃𝑃 𝑅𝑇87.73 ∙ (ln[𝑡] + ∙· 𝑅𝑇 93.13)− −− 296.7 311.86 𝐹(𝑇, 𝑡) = P0 = 87.73 RT 296.7 (17) 𝑃 87.73 ∙ 𝑅𝑇 − 296.7

Table 5. Reaction rate constant (k*) depending on aging temperature. AgingTable (°C)5. Reaction rate constant 70 (k*) depending 80 on aging temperature. 90 100 Aging (◦C) 70 80 90 100 ReactionAging rate constant (°C) (k*) 0.060 70 0.080 80 0.110 90 0.135 100 ReactionReaction rate constant constant (k (*)k *) 0.0600.060 0.0800.080 0.1100.110 0.135 0.135

Figure 10. Modified characteristic equation according to aging time by temperature. FigureFigure 10.10. ModiﬁedModified characteristiccharacteristic equationequation accordingaccording toto agingaging timetime byby temperature.temperature.

Figure 11. Arrhenius plot according to characteristic values. Figure 11. Arrhenius plot according to characteristic values.

PolymersPolymers 20192018,, 1110,, 136x FOR PEER REVIEW 11 ofof 1515 Polymers 2018, 10, x FOR PEER REVIEW 11 of 15

Figure 12. Coefficient term according to characteristic value. Figure 12.12. CoefﬁcientCoefficient termterm accordingaccording toto characteristiccharacteristic value.value.

Figure 13. Activation energy according to the characteristic value.value. Figure 13. Activation energy according to the characteristic value. 3.4. VerificationVerification andand ApplicationApplication ofof ModifiedModified ArrheniusArrhenius EquationEquation 3.4. Verification and Application of Modified Arrhenius Equation InIn thisthis study, study, to verifyto verify the modifiedthe modified Arrhenius Arrhen equation,ius equation, an additional an additional tensile test tensile was conducted test was In this study, to verify the modified Arrhenius equation, an additional tensile test was onconducted specimens on agedspecimens at room aged temperature at room fortemperat one year.ure Roomfor one temperature year. Room aging temperature was conducted aging was in a conducted on specimens aged at room temperat◦ ure for◦ one year. Room temperature aging was laboratoryconducted within a laboratory a temperature with distribution a temperature of 8 distributionC to 25 C, andof 8 the°C to test 25 specimen °C, and the was test stored specimen in the conducted in a laboratory with a temperature distribution of 8 °C to 25 °C, and the test specimen shadewas stored without in the exposure shade towithout sunlight. exposure Even into the sunlig roomht. temperatureEven in the agingroom temperature test, the rubber aging composites test, the was stored in the shade without exposure to sunlight. Even in the room temperature aging test, the becomerubber hardenedcomposites and become the stress hardened increased and in athe manner stress equivalent increased to in the a accelerated manner equivalent aging test results.to the rubber composites become hardened and the stress increased in a manner equivalent to the Itaccelerated indicated theaging same test tendency results. thatIt indicated the strain the value same decreases tendency at that the samethe strain SED, asvalue shown decreases in Figures at the 14 accelerated aging test results. It indicated the same tendency that the strain value decreases at the andsame 15 SED,. Finally, as shown the errors in Figures were analyzed 14 and by15. comparing Finally, the the errors characteristics were analyzed of the rubberby comparing composites the same SED, as shown in Figures 14 and 15. Finally, the errors were analyzed by comparing the obtainedcharacteristics experimentally of the rubber and comp the calculatedosites obtained values experimentally of the modified and Arrhenius the calculated equation; values the resultsof the characteristics of the rubber composites obtained experimentally and the calculated values of the aremodified shown Arrhenius in Table6. Theequation; average the error results between are sh theown experimental in Table 6. valuesThe average and the error predicted between values the modified Arrhenius equation; the results are shown in Table 6. The average error between the forexperimental nine degradation values conditionsand the predicted indicate values a high accuracyfor nine ofdegradation 3%; however, conditions in the caseindicate of the a roomhigh experimental values and the predicted values for nine degradation conditions◦ indicate a high temperatureaccuracy of 3%; aging however, test, an errorin the of case 5.3% of occurred. the room Thistemperature is because aging we used test, 17an Cerror as room of 5.3% temperature, occurred. accuracy of 3%; however, in the case of the room temperature aging test, an◦ error of◦ 5.3% occurred. andThis thisis because value was we theused average 17 °C as value room of temperat the temperatureure, and distributionthis value was (8 theC to average 25 C), value to calculate of the This is because we used 17 °C as room temperature, and this value was the average value of the thetemperature characteristic distribution value obtained (8 °C to from 25 °C), the modifiedto calculate Arrhenius the characteristic equation. value It is expected obtained that from if thethe temperature distribution (8 °C to 25 °C), to calculate the characteristic value obtained from the temperaturemodified Arrhenius of the laboratory equation. duringIt is expected the room that temperature if the temperature aging test of is the maintained laboratory constant during andthe modified Arrhenius equation. It is expected that if the temperature of the laboratory during the anroom accurate temperature temperature aging is test used is inmaintained the modified constant Arrhenius and equation,an accurate the temperature error will decrease. is used Inin thisthe room temperature aging test is maintained constant and an accurate temperature is used in the study,modified the Arrhenius method to equation, obtain the the degradation error will ratedecrea ofse. rubber In this composites study, the under method all conditions to obtain wasthe modified Arrhenius equation, the error will decrease. In this study, the method to obtain the presenteddegradation by Equationrate of rubber (17). Additionally, composites anunder equivalent all conditions degradation was conversion presented formula,by Equation which (17). can degradation rate of rubber composites under all conditions was presented by Equation (17). Additionally, an equivalent degradation conversion formula, which can convert short-term Additionally, an equivalent degradation conversion formula, which can convert short-term

PolymersPolymers2019 2018, 11, 10, 136, x FOR PEER REVIEW 1212 of of 15 15 Polymers 2018, 10, x FOR PEER REVIEW 12 of 15 high-temperature aging into long-term low-temperature aging, was derived using Equation (13). convert short-term high-temperature aging into long-term low-temperature aging, was derived using high-temperatureThe accelerated conditions aging into at roomlong-term temperat low-temperatureure for one year aging, are presentedwas derived in Table using 7. Equation (13). EquationThe accelerated (13). The conditions accelerated at room conditions temperat at roomure for temperature one year are for presented one year in are Table presented 7. in Table7.

Figure 14. Tensile curves for specimen aged for one year. FigureFigure 14.14. TensileTensile curvescurves for specimen aged for for one one year. year.

FigureFigure 15. 15.SED-strain SED-strain curvescurves forfor specimenspecimen aged for for one one year. year. Figure 15. SED-strain curves for specimen aged for one year. Table 6. Results of degradation conversion. Table 6. Results of degradation conversion. Aging ConditionTable 6. Results Normalizedof degradation Property conversion. Value Aging Condition Normalized Property Value ◦ Error Error T (°C)Aging ConditionT ( C) Days Days Experimental Experimental Normalized Property Predicted Value Predicted Error T (°C)70 70 Days17 17 Experimental 0.832 0.832 0.824 Predicted 0.824 1.0% 1.0% 7070 701735 35 0.832 0.779 0.779 0.779 0.824 0.779 0% 1.0% 0% 7080 803517 17 0.779 0.770 0.770 0.773 0.779 0.773 0.4% 0.4% 0% 8080 801735 35 0.770 0.711 0.711 0.719 0.773 0.719 1.1% 0.4%1.1% 8090 903517 17 0.711 0.682 0.682 0.699 0.719 0.699 2.5% 1.1%2.5% 10090 100173 3 0.682 0.761 0.761 0.802 0.699 0.802 5.4% 2.5%5.4% 100100 10037 7 0.761 0.691 0.691 0.695 0.802 0.695 0.6% 5.4%0.6% 100100 100177 17 0.691 0.654 0.654 0.582 0.695 0.582 11.0% 11.0%0.6% 17 (Room)100 17 (Room) 36517 365 0.654 0.907 0.907 0.859 0.582 0.859 5.3% 11.0% 5.3% 17 (Room) 365 0.907 0.859 5.3%

TableTable 7. 7.Equivalent Equivalent degradationdegradation conversion results. Table 7. Equivalent degradation conversion results. The Equivalent Degradation Conversion Table The Equivalent Degradation Conversion Table T (°C) 17 The Equivalent40 Degradation Conversion60 Table 80 100 TDay (°C) 36517T (◦C) 174065 40 6017.560 80 100805.4 1001.9 Day 365 Day 36565 65 17.517.5 5.4 1.95.4 1.9

Polymers 2019, 11, 136 13 of 15

4. Conclusions The aim of this study is to determine the fundamental parameters for the degradation properties of rubber compounds, suitable for the Arrhenius equation. In this study, an oxygen permeation blocking model was proposed to predict the degradation rate of rubber composites. In this process, a tensile test was performed with the different aging conditions to evaluate the degradation characteristics using strain. Consequently, it was confirmed that as degradation progressed, the rubber composites become hardened, and the strain decreased at the same SED (1 MJ/m3). Here, we assumed that the crosslinked structure interfered with the reaction of oxygen and rubber as the aging progressed. Thus, the modified Arrhenius equation was derived by adding the degradation time term to the reaction rate constant of the characteristic equation. Finally, the validity of the formula was verified by comparing the degradation rate obtained by experimentally with the degradation rate predicted by the modified Arrhenius equation, and the following conclusions were obtained:

1 In a general characteristic equation, the properties decreased linearly in proportion to reaction rate constant and time. However, in most cases, the reaction rate of the characteristic value decreased as the aging time increased. The reason that the reaction rate of the polymer decreased as the aging progressed was because the probability of the rubber reacting with oxygen was reduced. In the case of rubber molecules, the crosslinking structure increased by aging interfered with the permeation of oxygen, and the reaction of the molecules was inhibited over time. Therefore, we formulated a relationship where the reaction rate constant was inversely proportional to time based on the experimental results. The modified characteristic equation was proposed as a function of time, and the modified Arrhenius equation was derived by substituting the time function for the characteristic equation. 2 In the case of the general Arrhenius equation, the resulting average deviation between the calculated and experimental values was 42% or more. However, in the case of a modified characteristic equation as a function of time, we observed that the average deviation in the experimental and calculated value decreased considerably to within 17%. Consequently, comparisons of the nine experimental values obtained with different degradation conditions with the predicted values indicated that the accuracy of the modified Arrhenius equation was relatively high, with an average error of 3%. Thus, using a modified Arrhenius equation derived from an oxygen permeation block model could predict the aging behavior of rubber materials accurately. 3 By applying the modified Arrhenius equation, we derived the relationship between short-term high-temperature aging and long-term low-temperature aging, and suggested a method for predicting the degradation rate of rubber composites under all aging conditions. Therefore, it was possible to accurately predict changes in the characteristics of the rubber composites by performing the acceleration test, and the energy absorption rate and stability that changed with degradation rate could be considered quickly in the design stage.

Author Contributions: Conceptualization, B.M., and C.-S.S.; Data curation, B.M., N.J. and U.S.H.; Formal analysis, B.M., N.J. and U.S.H.; Funding acquisition, C.-S.S. and U.S.H.; Methodology, B.M., S.P. and C.-S.S.; Project administration, B.M. and C.-S.S.; Supervision, C.-S.S.; Validation, B.M., S.P. and C.-S.S.; Visualization, B.M.; Writing–original draft, B.M.; Writing–review & editing, B.M., S.P. and C.-S.S. Funding: This research was supported by the Hyundai Motor Group. This work has supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. 2018R1A2A1A05077886). Acknowledgments: This research was supported by the Hyundai Motor Group. This work has supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. 2018R1A2A1A05077886). Conﬂicts of Interest: The authors declare no conﬂict of interest. Polymers 2019, 11, 136 14 of 15

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