3.1. General Arrhenius Equation
The characteristic equation of the reaction rate constant
k used for the Arrhenius equation is generally expressed as Equation (2).
In this case,
P represents the characteristic value of the rubber,
P0 represents an initial characteristic value,
t represents time, and
k is a reaction rate constant. If the lifetime of the rubber is defined as the time until the characteristic value becomes
P from Equation (2), the lifetime (
t) can be expressed by Equation (3) at that time.
The reaction rate constant
k from Equation (2), a value indicating a degradation reaction, can be expressed by Equations (4) and (5) using the Arrhenius equation. At time
t,
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 absolute temperature.
The lifetime
t in Equation (3) can be calculated using the empirical Arrhenius equation (Equation [
4]) because the equations represent the relationship between lifetime and temperature. Thus, the lifetime can be converted into temperature. With the characteristic value of
P, lifetime
t1 is derived at temperature
T1, and lifetime
t2 is derived at temperature
T2. Consequently, Equation (6) can be obtained.
For each aging temperature (70, 80, 90,= and 100 °C), the relationship between aging time and characteristic values is shown in
Figure 6, and the reaction rate constant (
k) of the characteristic equation is derived and listed in
Table 3. When the temperature is low, the reaction rate constant
k is small and the characteristic value changes gradually. However, as the temperature increases, the reaction rate increases and the characteristic value changes sharply. In this study, the aging time (
t) is calculated at 15, 25, 35, and 45% reductions from the characteristic values using the characteristic equation. The results are plotted in
Figure 7. The condition at a 25% decrease in characteristic value is derived as a function of temperature and time in Equation (7), and compared with the actual value.
The results of calculating the equivalent aging time, converted from the aging temperature using Equation (7), are shown in
Table 4. The mean deviation between the predicted characteristic value using the characteristic equation and the actual experimental value (
Figure 6) is 42% or more. Therefore, if we convert the characteristic values into aging time by applying the general Arrhenius equation, a large difference occurs in the resulting values [
17]. For example, when the degradation condition caused by decreasing the characteristic value by 25% is calculated with the Arrhenius equation, it becomes nine days at 100 °C (
Table 4). However, as the actual specimen is equivalent to a specimen aged at 100 °C 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.
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
Figure 8. 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,
27,
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
Figure 9a. 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 the permeation of oxygen [
29,
30], as shown in
Figure 9b, and the reaction of the molecules is inhibited over time. Therefore, we suggest a modified characteristic equation and nonlinear Arrhenius equation, which is expressed as a function of time.
3.3. Modified Arrhenius Equation
In a general characteristic equation, the properties decrease linearly in proportion to the reaction rate constant and time in accordance with Equation (2). However, in the case of the actual test results on the rubber composites shown in
Figure 6, as aging time increases, the reduction rate of the characteristic value decreases. Thus, if an existing characteristic equation is used, a large error occurs. Because of the above reasons, other researchers have used the non-Arrhenius equation by substituting the accelerative shift factors for the individual activation energy [
25,
31]. However, it is only a numerical conversion approach using the test results. Further, the fundamental parameters of the degradation 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.
where
.
By integrating Equation (8), we can obtain Equation (9) as follows:
If the initial value at aging time 0 is substituted for
t0, the modified characteristic equation is derived as Equation (10).
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).
when Equation (10) is substituted into the modified Arrhenius equation (Equation (11)),
k* is eliminated and the result is expressed as Equation (12).
where
,
,
.
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).
The modified characteristic equation (Equation (10)) is applied to derive the reaction rate constant, and the results are presented in
Table 5. 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
Figure 12 and
Figure 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).
3.4. Verification and Application of Modified Arrhenius Equation
In this study, to verify the modified Arrhenius equation, an additional tensile test was conducted on specimens aged at room temperature for one year. Room temperature aging was conducted in a laboratory with a temperature distribution of 8 °C to 25 °C, and the test specimen was stored in the shade without exposure to sunlight. Even in the room temperature aging test, the rubber composites become hardened and the stress increased in a manner equivalent to the accelerated aging test results. It indicated the same tendency that the strain value decreases at the same SED, as shown in
Figure 14 and
Figure 15. Finally, the errors were analyzed by comparing the characteristics of the rubber composites obtained experimentally and the calculated values of the modified Arrhenius equation; the results are shown in
Table 6. The average error between the experimental values and the predicted values for nine degradation conditions indicate a high accuracy of 3%; however, in the case of the room temperature aging test, an error of 5.3% occurred. This is because we used 17 °C as room temperature, and this value was the average value of the temperature distribution (8 °C to 25 °C), to calculate the characteristic value obtained from the modified Arrhenius equation. It is expected that if the temperature of the laboratory during the room temperature aging test is maintained constant and an accurate temperature is used in the modified Arrhenius equation, the error will decrease. In this study, the method to obtain the degradation rate of rubber composites under all conditions was presented by Equation (17). Additionally, an equivalent degradation conversion formula, which can convert short-term high-temperature aging into long-term low-temperature aging, was derived using Equation (13). The accelerated conditions at room temperature for one year are presented in
Table 7.