Prediction of the stress relaxation property of diene rubber composites by artificial neural network approaches

Abstract

An artificial neural network (ANN) based on an improved radial basis function (RBF) was established to predict the stress relaxation property of diene rubber composites during ozone aging. The compression ratio, ozone concentration, ozonation time, and rubber type were used as the input vectors, and the stress relaxation coefficient (F/F₀) was used as the output vector. Regularization was introduced into the RBF ANN, and the average prediction accuracy of the established ANN for three types of diene rubber was 98.1%. Compared with back-propagation (BP) ANN, the proposed model displays better prediction performance. Moreover, a sensitivity analysis based on the analytical expression generated by the RBF indicates that compression ratio is the most important factor affecting the stress relaxation response. The RBF prediction shows that rubber under a high compression ratio possesses a higher F/F₀ than that under a low compression ratio in the range of 25% to 35%, indicating that highly compressed rubber experiences a low degree of ozone attack. Analysis of the relationship between the microstructure and the stress relaxation property of the polymer reveals that application of a high compression ratio reduces the free volume and degree of cracks and that rubber under a high compression ratio has high ozone resistance. This finding is in agreement with the above RBF prediction. Comparisons between the RBF model predictions and the experimental data indicate that the well-trained model provides a deep understanding of the stress relaxation property and has the potential to guide development of materials highly resistant to ozonation.

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1. Introduction

Diene rubber, including natural rubber (NR), butadiene rubber (BR), and chloroprene rubber (CR), is an important class of industrial materials because of its processability and permanent set resistance. Diene rubbers are widely used in critical service applications such as seals that are designed to withstand stresses, and material failure can be mostly attributed to polymer aging.^1–5 Ozone concentrations have gradually increased in recent years, especially in industrial areas.² Ozone concentrations of less than 1 pphm can severely attack the C=C bonds in strained diene rubber.³ Therefore, ozonation is an important contributor to rubber failure. Since the 1950s, the stress relaxation property has considerably drawn the attention of researchers seeking to quantitatively characterize the ozone aging of rubber.^1,6,7 The stress relaxation property is one of the most important physico-chemical properties that determines the sealing capability and service life of rubber. To design highly ozone-resistant polymers, it is necessary to obtain data on the stress relaxation behavior of these materials. Because experimental methods are time-consuming and costly,^8,9 many researchers have tried to predict the stress relaxation property by theoretical methods, such as the time–temperature-superposition principle, Arrhenius theory, and Hertz contact theory, etc.^10–12 However, the nonlinear nature of the stress relaxation property means that conventional theoretical methods are insufficient to meet application requirements. Curro et al. studied the influence of temperature on the stress relaxation property and presented a theoretical model to describe this behavior.¹³ However, this theoretical model was not suitable to quantitatively predict stress relaxation property. Le et al. studied chemical and physical stress relaxations. They proposed a quantitative model to describe the relationship between the physical stress relaxation property and time, but their model did not consider the chemical stress relaxation property.¹⁴ Zhao et al. investigated the stress relaxation property of nitrile rubber. They found that the stress relaxation mechanism under different conditions was the comprehensive result of thermal rearrangement, chain scission, and crosslinking.¹⁵ Razumovskii et al. investigated the stress relaxation mechanism of polyisoprene vulcanizates responding to ozone aging.¹ Their study showed that the influence of ozone concentration, ozonation time, and stress relaxation ratio on the stress relaxation property was not a simple linear relationship. Therefore, establishing an accurate model to quantitatively predict the influence of ozonation on rubber material lifetimes via the stress relaxation property is useful for practical applications.

As a heuristic computational modeling approach to simulate complex and nonlinear relationships with good self-learning ability, artificial neural networks (ANNs) have been successfully applied to multiple scientific and engineering endeavors in recent decades.^16–18 Comparisons between equation of state (EOS) and ANN were presented by Yousefi¹⁹ and Normandin,²⁰ and their studies indicated that ANNs were powerful models with higher accuracy. Demirhan et al. investigated the physical properties of styrene–butadiene rubber composites by ANN technology.²¹ Koch, presented an ANN model to study chemical reactions on potential energy surfaces.²² Wu et al. proposed an ANN model for the fatigue life of NR.²³ Karaağaç et al. predicted the optimum curing time of rubber by ANN technology.²⁴ Currently, the radial basis function (RBF) has received considerable attention because of its potential to approximate nonlinear behavior. Li et al.²⁵ successfully proposed an RBF model for prediction of polymer solubility. The above examples illustrate the effectiveness of neural computing methods in describing the physico-chemical properties of polymers, especially rubbers. However, very few studies on quantitatively predicting and understanding the effects of ozone aging on the stress relaxation property of rubber materials have been reported in the scientific literature.

As motived and inspired by the aforementioned studies, an ANN based on an improved RBF was proposed to predict the stress relaxation property of diene rubber composites. Regularization²⁶ was then introduced into the RBF model to guarantee the reasonability of parameters. Detailed comparisons between RBF and BP ANNs revealed that the proposed RBF ANN outperformed the BP ANN. Moreover, a sensitive analytical method,^27,28 based on the partial derivatives of the output vector to the input vectors, was implemented to determine which input variable had the greatest influence on the output. We deduced the most critical factor affecting the stress relaxation property. To interpret the macroscopic stress relaxation property, detailed microstructural analyses on crosslink density, cracks, and free volume were performed. The present study aimed to use an ANN-based approach to investigate the influence of ozonation factors on the stress relaxation property, thereby providing a deep understanding of this relationship as it applies to diene rubber composites.

2. Simulation

2.1 Radial basis function artificial neural network (RBF ANN)

The RBF-ANN, also termed RBF, is a to local approximation artificial neural network, that is widely used in supervised learning.^25,29 Differing from universal approximation neural networks, such as a multilayer perceptron, RBF employs specific radial basis functions in hidden layer nodes to realize nonlinear mapping. Accordingly, this feature leads to two advantages in training:^25,29–31 the prior knowledge learned from experimental data can be directly used to tune the structure and parameters of the RBF, and few weights and parameters of the RBF need to be updated when new training data become available. Both advantages result in a training time for the RBF that is significantly shorter than those of networks based on the error back propagation method.³² Meanwhile, due to convex quadratic programming theory, RBF-training avoids the problem of local minimums that is common in BP-training.^33,34

A typical three-layer feedforward structure of an RBF is shown in Fig. 1. In feedforward computation, an unweighted input vector x ∈ [Doublestruck R]^N is sent to every hidden layer node through the input layer connections. The number of hidden layer nodes M is larger than the dimension of input vector N. Here, for each hidden node, an RBF was assigned, wherever the Gaussian function was used in this paper. Under our experimental setting, the Gaussian functions in different hidden layer nodes possess different center vectors c_i ∈ [Doublestruck R]^N, but have an identical width parameter σ ∈ [Doublestruck R]. The output h_i in the ith hidden node is given as follows:

where the squared norm ‖⋅‖₂² is defined as ‖x‖₂² = x^Tx. All of the elements h_i compose the output vector of the hidden layer, h ∈ [Doublestruck R]^M. Next, the vector h is multiplied by a linear weight matrix W ∈ [Doublestruck R]^M×P through the output layer connections, where P is the dimension of the output data. The final computational results y ∈ [Doublestruck R]^P are then obtained as the output of the RBF. Above all, the analytical expression for the RBF input–output relationship is given as follows:where w_ik is the ith-row kth-column element in the matrix W.

Fig. 1 Structure of RBF artificial neural network.

Previous studies^{25,29,30,33,35} indicated that empirical formula and heuristics were generally adopted to optimize and determine the number of hidden layer nodes. Therefore, a batch operation parameter training method was used to train the RBF in this study. First, the center vectors c and an initial width parameter σ₀ were assigned to all the hidden nodes. A hidden layer output matrix H̃ ∈ [Doublestruck R]^M×S was calculated through the sequential feedforward computation of the training input matrix X̃ ∈ [Doublestruck R]^M×S, where S is the number of samples, and each row of X̃ is corresponded to each row of H̃ with the input–output relationship. Now, the only parameters to determine are the components of the linear weight matrix W. For a given training output matrix Ỹ ∈ [Doublestruck R]^P×S, a convex optimization problem is formulated as

where the second term β‖W‖₂² aims to minimize the squared norm of W, is to realize L2-norm regularization, and β is a tradeoff weight. The solution of (3) is the optimal W* both to minimize the training errors and to enhance the robustness of the RBF. According to the convex optimization theory, the unique W* can be solved by the ridge regression as follows:

W* = (H̃^TH̃ + βI)⁻¹H̃^TỸ (4)

where I ∈ [Doublestruck R]^M×M is the identity matrix. The RBF-training process contains only the above two steps: calculating H̃ by feedforward computation and obtaining W* by eqn (4). Therefore, the training time is far less than that involved in the numerous iterations necessary for the error back propagation method, and we can use a 1-D search to obtain the optimal width parameter σ* by repeatedly calculating the cost function in eqn (4). However, σ has low sensitivity to the performance of RBF. The main task of the RBF in this paper is to create an analytical function and further to bridge the gap between experimental data and theoretical analysis.

2.2 Sensitivity coefficient derivation

The subtask of the RBF is to derive the mathematical representation for the sensitivity coefficient. The sensitivity coefficient is defined as the partial derivative of the output variable y_j with respect to the input variable x_i, and it reflects how significantly this input variable can affect the corresponding output.

A typical three-layer feedforward structure of the RBF is shown in Fig. 1. An RBF is constructed for this subtask. For accurate results, the experimental input–output data x̃_i and ỹ_j should be normalized as x̄_i and ȳ_j, respectively, into the range of [0, 1] by using the following representation:³⁶

where k is the number of data. x̄_i and ȳ_j are used to train the RBF parameters by the batch operation parameter training method. Finally, a complete analytical function of the RBF is obtained. Therefore, the partial derivative of the output variable y_j with respect to the input variable x_i can be obtained as the following differentiation chain:

By the form of eqn (2), we can directly solve the differentiation chain in parentheses in eqn (6) as follows:

From eqn (5) and its inverse functions, we have

Substitution of eqn (7) and (8) into (6) leads to the following representation for the partial derivative of y_j with respect to x_i:

This representation, derived through RBF, can be used as an analytical function for the sensitivity coefficient.

3. Experimental

3.1 Materials

The diene rubber composites under investigation were NR (Cloud Standard, Changli Rubber Trading Co., Ltd, China), BR (BR 9000, Yanshan Petrochemical Co., Ltd, China), and CR (CR322, Synthetic Rubber Group Co., Ltd, China). Zinc oxide, magnesium oxide, stearic acid, sulfur and accelerators were supplied by Beijing Additives Institute (China). Carbon black N234 was supplied by Cabot Co., Ltd, (American). The conventional industrial formulations of the rubber composites are shown in Table 1.

Table 1 Formulations of rubber composites

Ingredient	Formulation 1 (phr)^c	Formulation 2 (phr)	Formulation 3 (phr)
a N-Cyclohexyl-2-truxene-thiazole-sulfonamide.b Ethylene thiourea.c phr stands for parts per hundred parts of rubber by weight.
Cloud standard NR	100	—	—
BR9000	—	100	—
CR322			100
Zinc oxide	5	5	5
Magnesium oxide	—	—	1.5
Stearic acid	2	2	2
Accelerator CZ^a	0.7	0.7	—
Accelerator NA-22^b	—	—	0.7
Sulfur	1.5	1.5	—
Carbon black N234	25	25	25
Total	134.2	134.2	134.2

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3.2 Sample preparation

NR, BR, and CR composites were prepared with the following procedures: (1) the raw rubber was kneaded on a Φ 150 mm two-roll mill at room temperature for 5 min; (2) the rubber was then blended with given amounts of zinc oxide, stearic acid, carbon black, accelerators, and crosslinking agents by kneading on the two-roll mill; (3) the optimum vulcanization time for each composite was obtained using a disc rheometer (P355C2, Huanfeng Chemical Technology and Experimental Machine Co., Ltd, China); and (4) the composites were vulcanized on a hot press at the curing temperature (NR: 145 °C, BR: 150 °C, CR: 160 °C) under the pressure of 15 MPa to form samples for the stress relaxation experiments.

3.3 Stress relaxation

The compression stress relaxation mode in the compression ratio range of 25% to 35% was selected because compression ratios in this range are widely encountered in industrial applications. Stress relaxation measurements were carried out on a compression stress relaxometer (MZ-0419, Pearl Test Machinery Co., Ltd, China) according to ISO 3384: 2005MOD. The relaxation samples with 25%, 30%, and 35% compression ratios were placed in an ozone accelerated aging test box (GT-7005-C, Cotech Co., Ltd, China) for testing at elevated ozone concentrations of 75 pphm, 100 pphm, and 125 pphm. Then, measurements of stress relaxation as a function of time were performed. For comparison, the stress relaxation coefficient (F/F₀), where F₀ is the initial stress and F is the stress measured in real time, was generally used to evaluate the stress relaxation property.

3.4 Characterization

The crosslink density of samples under a compression ratio of 25% and an ozone concentration of 75 pphm was analyzed using a magnetic resonance crosslink density spectrometer (XLDS-15, Dr Kuhn Innovative Imaging Co., Ltd, Germany) at room temperature. The surfaces of samples were observed with a high-resolution scanning electron microscope (SEM) (S-4800, Hitachi, Japan). A 1 × 1 mm area was chosen in the center of each sample to gather statistics for the average width of the longest crack. Positron annihilation spectroscopy (PALS) measurements were performed at room temperature to study the fractional free volume (FFV) using a conventional fast–slow ORTEC system³⁷ with a time resolution of 198 ps (full width at half-maximum for the ⁶⁰Co prompt γ-rays). A 27μCi²²Na positron source was sandwiched between two identical samples (with a thickness of 1.5 ± 0.05 mm and a diameter of 15 ± 0.05 mm). A total of 2 × 10⁶ counts were accumulated for each spectrum to reduce the statistical error in the calculation of lifetimes. The routine LT9.0 (ref. 38) with three lifetime components was used to analyze the lifetime spectra.

4. Results and discussion

4.1 Modeling results

In our experiment, the computation was calculated using MATLAB 2013b. A 4-81-1 structure of the RBF was trained. Four variables including ozone concentration, compression ratio, ageing time and rubber type were chosen as the input variables, and the stress relaxation coefficient was chosen as the output variable. After a normalization transformation, 81 centers in hidden layer nodes were arranged in a 3 × 3 × 3 × 3 grid over a 4 dimensional input variable space, which included {0, 0.5, 1} for each dimension. The regularization parameter was selected as 10⁻⁶. Considering that the width parameter σ has the most significant contribution to the performance of the RBF, we tested training and predicted performance for different values of σ. The errors between the true values and the output of the RBF were measured by the mean squared error (MSE). First, the parameter for the RBF was to be determined. A total of 270 samples were randomly and equally divided into the training set and the predicted set. For 10 independent runs, the MSEs and the standard deviations for training and prediction under different width parameter σ were calculated, as shown in Fig. 2. We observe that the training MSE increases with an increase of σ and then levels off, whereas the predicted MSE decreases with an increase of σ. Three routines help select an appropriate value of σ: (1) minimize the difference between the training and the predicted MSEs; (2) minimize the predicted MSE; and (3) minimize the standard deviation. Considering the above requirements, we select σ = 1.5 for training and predicting.

Fig. 2 Variations of training and predicted MSEs under width parameter (σ) of RBF-ANN.

Fig. 3 compares the measured and predicted values of the stress relaxation coefficient F/F₀. The multiple correlation coefficient (MCC) between the measured and predicted values was calculated. As is shown in Fig. 3 (the values are normalized as identical as the input and output of RBF, to compare with the results in the literatures), the designed ANN is capable of modeling and predicting the stress relaxation coefficient for three types of diene rubber. The output displays a linear relationship between measured and predicted values with a high accuracy of 98.1%. In a previous study,²³ the BP ANN was used to study the fatigue life of NR with a MCC of 97.3%. The results using the RBF ANN are more accurate than that from the existing model using BP ANN.

Fig. 3 Comparisons between measured and predicted stress relaxation coefficients, F/F₀, of diene rubber composites.

4.2 Comparison of the proposed model against BP

To verify the efficiency and validity of the proposed computational model, a BP model is employed as a basis of comparison.

The theory and algorithm of BP have been illustrated in the literaturs.^32,35 To make a comparison with the RBF, a BP network with the identical 4-81-1 structure as the proposed RBF network is trained. The sigmoid and linear functions are used in hidden and output layers, respectively.

In our study, both RBF and BP independently run for twenty times. The training MSE, the predicted MSE, and the computation time of both methods are averaged of all runs as the reported value. Obviously, the superiority of the RBF to the BP can be seen in Table 2. This is especially the case for the training and the predicted MSEs, where the proposed model could predict the stress relaxation property of diene rubber with high accuracy. The RBF also consumed less computation time. Therefore, the RBF is better at predicting the stress relaxation property of diene rubber composites during ozonation.

Table 2 Statistical parameters of the comparison models

Model	Training MSE	Predicted MSE	Running time (s)
RBF ANN	2.77 × 10^−4	1.97 × 10^−3	0.029
BP ANN	6.28 × 10^−4	6.56 × 10^−3	2.17

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4.3 Sensitivity analysis

To observe the sensitivity of the input vectors, we calculated the partial derivative of the two inputs, compression ratio and ozone concentration, at different ozonation times. The larger the partial derivative, the greater the effect of the input vector on the stress relaxation property. Fig. 4 shows the absolute values of sensitivity, as the partial derivative values of the input ozone concentration at different ozonation times in Fig. 4 are negative. This was done to compare the magnitudes of the effect of the compression ratio and the ozone concentration on the stress relaxation property rather than the direction. As seen, the absolute sensitivity of the compression ratio is much higher than that of the ozone concentration. It can be inferred from these results that (1) the compression ratio has a higher influence on the stress relaxation property than does the ozone concentration; (2) the ozone concentration affects the stress relaxation property in the form of a negative correlation, i.e., the higher the ozone concentration, the lower F/F₀; and (3) the compression ratio and the ozone concentration significantly affect the stress relaxation property of NR and BR in the first two days and affect the stress relaxation property of CR in the final days.

Fig. 4 Absolute value of sensitivity at different ozonation times for (a) NR, (b) BR, and (c) CR.

4.4 Stress relaxation prediction

Fig. 5 shows the predicted evolutions of F/F₀ vs. compression ratio and ozonation time. The experimental values under different ozonation conditions are also shown in Fig. 5. The validity of the predicted values is demonstrated by their good agreement with the experimental values. The F/F₀ of NR and BR decreases with time in the first two days, which suggests that the effects of ozone aging are more prominent at the beginning. The F/F₀ of CR decreases slightly with time, suggesting that CR has better ozone resistance than NR and BR. This could be attributed to highly polar C–Cl bonds in CR, which is not favorable for the ozone attack of double bonds by electrophilic addition.³⁹ Zhao et al.¹⁵ studied the stress relaxation property of NBR aged in air. They found that the smaller the compression ratio, the higher the value of F/F₀. According to their stress relaxation mechanism, high stresses lead to chain scission and molecular chain rearrangement. More interestingly, the stress relaxation coefficients of diene rubber composites were higher during ozonation under a compression ratio of 35% than under a compression ratio of 25%. In other words, the ozone attack on diene rubber composites had less of an impact under a compression ratio of 35% than under a compression ratio of 25%. This interesting finding was not accidental. For rubbers in an ozone atmosphere, Raab et al. found a remarkable deviation from the basic trend that macroscopic failure of materials decreases monotonically with increasing stress.⁴⁰ Their study showed that the degree of ozonation and the cracking degree of NR increased with an increase of compression ratio in the range of 0% to 20%, but the opposite was found for compression ratios between 20% and 40%. The implication is that there is a change in the mechanism of stress relaxation in the compression ratio range 20% to 40%. According to the physical relaxation mechanism,¹⁵ molecular chain rearrangement leads to a decrease of F/F₀ under a high compression ratio. However, diene rubber composites have isolated double bonds that can easily be attacked on the rubber surface by electrophilic addition. Ozone molecules have high a polarity and strongly interact with polymers. For ozone-induced destruction of a polymer to occur, the ozone has to first permeate into the polymer and then fracture the polymer chains resulting in a decrease of F/F₀. According to the previous reports,⁴¹ a material with a high free volume possesses a high gas permeability coefficient. With increasing compression ratios, the free volumes of the diene rubber composites reduce, which leads to a lower ozone permeability and minimizes ozone destruction. This mechanism will be further analyzed and confirmed in the following sections.

Fig. 5 Predicted evolutions of F/F₀ vs. compression ratio and ozonation time of (a) NR, (b) BR, and (c) CR at ozone concentration of 75 pphm ozone (solid dots denote experimental values).

4.5 Crosslink density analysis

Crosslink density tests were performed to obtain chemical information on the rubber networks at an ozone concentration of 75 pphm. Fig. 6 shows the changes in crosslink density for the diene rubber composites as a function of ozonation time while under compression ratios of 25% and 35%. Ozone-induced chain scission in the stressed state could lead to a decrease in crosslink density.¹⁵ The value for day zero represented samples in air. The crosslink density of NR and BR declines rapidly in the first 2 days because of chain scission. With an increase of ozonation time, the crosslink density of NR and BR declines slowly because crosslinking reactions occur at the same time.¹³ The crosslink density of CR declines slowly in the first few days. This phenomenon indicates that CR has a higher resistance to ozone oxidation than NR and BR. For the same type of rubber, the crosslink density declines faster (Fig. 6) and thus results in a lower equilibrium value of F/F₀ (Fig. 5) under a compression ratio of 25% than under a compression ratio of 35%.

Fig. 6 Crosslink density of stressed NR, BR, and CR samples versus ozonation time.

4.6 Microscopic morphology analysis

NR, BR, and CR samples were placed in an ozone accelerated aging test box under an ozone concentration of 75 pphm. Fig. 7 shows the micrographs of the sample surfaces on the fifth day of aging. The average crack widths of Fig. 7a–f is 3.42 μm, 1.74 μm, 0.92 μm, 0.86 μm, 0.43 μm, and 0.40 μm, respectively. These surfaces are not homogeneous because they were attacked by ozone and the cracks are generated perpendicular to the loading direction. For the same type of rubber, the average crack width is larger under a compression ratio of 25% than under a compression ratio of 35%. These results are in agreement with the results of Raab et al.⁴⁰ For a given compression ratio, the lower crack width of CR indicates that it exhibits better ozone resistance than does BR and NR, whereas NR displays the highest degree of cracking because it has the weakest resistance to ozone. These microscopic results also explain the behaviors of F/F₀ predicted by the RBF model (Fig. 5).

Fig. 7 SEM micrograph showing a typical crack at room temperature in NR (a and b), BR (c and d), and CR (e and f) after an ozone exposure of 5 d under the compression ratios of 25% (a, c and e) and 35% (b, d and f).

4.7 Verification of free volume by PALS analysis

Previous studies^41–43 showed that the free volume change in a polymer greatly affected the structure and permeability of the polymer. The lifetime of a positron is sensitive to structural inhomogeneity because the formation and annihilation of positronium (Ps, bound state of e⁺ and e⁻) occur in nanoscale and sub-nanoscale holes.⁴⁴ An ortho-positronium (o-Ps, the triple bound state of Ps) probe with a relatively small diameter (0.106 nm) has a repulsive nature compared with the atoms, therefore PALS can obtain information about free volume. Based on these facts, a semi-empirical relation between the o-Ps lifetime τ₃ and free volume hole radius R was established:⁴⁵where τ₃ and R are expressed in the units of ns and Å. The spherical potential of radius R₀ = R + ΔR, where ΔR (=1.656 Å) is the thickness of the fitted empirical electron layer.

The mean hole free volume V_f can be obtained from the following equation:

The probability of o-Ps formation is the intensity I₃ that has been used to estimate the fractional free volume fraction (FFV):

where C is a constant. The FFV of the rubber can be obtained when C is equal to 1.³⁸

The measured τ₃, I₃, and FFV for the diene rubber composites are shown in Table 3; the values are representative of samples after five days of ozone aging under 75 pphm of ozone concentration and compression ratios of 25% and 35%. The diene rubber composites have higher FFVs under a compression ratio of 25% than under a compression ratio of 35%. The increase in FFV corresponds to an increase of free volume concentration, which agrees well with the interpretation based on chain scission.⁴⁶ The motion of the polymer chains can influence FFV.⁴⁷ From the results of PALS, compression of the diene rubber composites plays an important role in lowering the free volume during ozonation. Under a compression ratio of 35%, the molecular chains rearrange to reduce the conformation entropy, the concentration of molecular chains increases, and the FFV decreases. This decrease in FFV reduces the permeability of ozone and then reduces ozone destruction. NR has methyl functional groups, and the steric effect of these groups creates a larger free volume than hydrogen atoms. Therefore, NR has a higher FFV than BR. CR exhibits a lower FFV than NR and BR because the highly polar C–Cl bonds in CR interact strongly with ozone. The PALS results explain the higher ozone resistance of CR, compared with that of NR and BR, at the level of microstructure and validate the results predicted by the RBF model.

Table 3 Positron lifetime and relative intensities for diene rubber composites after 5 d of ozonation under the compression ratios of 25% and 35%

Sample	Compression ratio	I3 (%)	τ3 (ns)	FFV (%)
NR	25%	2.526 ± 0.012	13.89 ± 0.33	2.071
NR	35%	2.486 ± 0.020	14.11 ± 0.11	2.045
BR	25%	2.674 ± 0.011	11.94 ± 0.24	1.973
BR	35%	2.653 ± 0.020	12.06 ± 0.24	1.959
CR	25%	2.344 ± 0.026	7.29 ± 0.23	0.951
CR	35%	2.268 ± 0.012	7.40 ± 0.14	0.909

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5. Conclusions

An improved RBF ANN model consisting of three ozonation factors (compression ratio, ozone concentration, and ozonation time) and rubber type was established to predict the stress relaxation property of diene rubber composites. The average predicted accuracy of this model was 98.1%. A sensitive analytical model based on the partial derivatives of the output vector with respect to the input vectors was developed to quantitatively understand the influence of the input vectors on the stress relaxation property. The sensitivity analysis showed that the compression ratio was the most important factor affecting the stress relaxation property. An interesting finding offered by the RBF model was that the diene rubber composites stressed at 35% displayed a higher F/F₀ than those stressed at 25%, indicating weaker ozonation at 35% than at 25%. Crosslink density analysis and microstructure characterization using SEM and PALS showed that application of a 35% compression ratio to diene rubber composites resulted in (1) reducing the free volume and ozone molecule penetration and (2) weakening the ozonation of rubber. The RBF model can be used as a quantitative tool to provide a deep understanding of the stress relaxation property of the diene rubber composites and can served as a guide for the design of ozonation-resistant materials.

Acknowledgements

The authors gratefully acknowledge the financial supports of the National Natural Science Foundation of China (Grant No. 51473012 and 51320105012) and the Ministry of Science and Technology of China (Grant No. 2014BAE14B01).

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