The changes in physical, chemical, or molecular properties characterize material aging, which leads to reduced material quality, performance and safety. Research on material aging and anti-aging has become crucial for industries. One significant concern to the polymer industries is that of yellowing discoloration phenomenon. Yellowing is often associated with the breakdown of polymer chains, and hence it is an indication of chemical changes occurring in the material. These changes can weaken the molecular structure of the material, making it more brittle and prone to damage.

The aging process of a polymer material is a complex phenomenon that occurs over time and is influenced by various environment factors, such as light, heat, humidity, and chemical substances [1] [2] . To investigate effects of time and environmental conditions, aging tests can be either accelerated or natural. Accelerated artificial aging techniques involve subjecting products to controlled conditions in a laboratory such as high temperatures, or chemical treatments to speed up the aging process. Due to the use of more stringent conditions than the actual environment, artificial aging may not fully replicate the environmental conditions in real life. As a result, the aging effects obtained may not accurately reflect the actual performance. However, natural aging is a complex phenomenon that occurs naturally over time. These natural environmental factors are difficult to control, making it challenging to predict and manage the aging process accurately.

The yellowing phenomenon can be caused by oxidation, exposure to light, heat, chemical contamination and biological activity [3] [4] . In the natural environment, sunlight is a major contributor to the yellowing of many materials. Long-term exposure to sunlight can induce photo-oxidation reactions. High temperatures can also cause direct thermal degradation. Materials stored or used in high-temperature environments are more likely to yellow. There are several predictive models for aging of polymer materials, such as regression models and Bayesian methods [5] - [7] . They proposed linear or polynomial regression models for a giving exposure time and temperature [5] [6] . For prediction of aging of multiplex polymer systems under conditions, a physical-chemical model with complicated nonlinear dependency on exposure time was constructed, and Bayesian method was applied to estimate the unknown parameters of the model [7] . However, predictive models on the yellowing are very few [3] . A predictive model under accelerated exposures was built, and the change in yellow index was modeled using linear regression model under predefined accelerated exposure conditions for a given exposure time and photo dosage of light [8] . In this paper, we focus on the natural yellowing phenomenon and investigate the non-linear combination effect of solar radiation and temperature in the natural aging process of polystyrene (PS) materials.

With observed $\left\{Y\left(t_{k1}\right),Y\left(t_{k2}\right),\cdots ,Y\left(t_{k,12}\right)\right\}$ and more densely environmental measurements $\left\{R\left(s_{k1}\right),T\left(s_{k1}\right),R\left(s_{k2}\right),T\left(s_{k2}\right),\cdots ,R\left(s_{k,365}\right),T\left(s_{k,365}\right)\right\}$, for station GZ ( $k=1$) and QH ( $k=2$) in training dataset, the discrete version of the model is given by

$\frac{Y\left(t_{ki}\right)}{100-Y\left(t_{ki}\right)}={\displaystyle {\sum }_{s_{kj}\le t_{ki}}R}\left(s_{kj}\right)\text{exp}\left({\beta }_0-{\beta }_T\frac{296.15}{T\left(s_{kj}\right)+273.15}\right),k=1,2;i=1,2,\cdots ,12.$

Adopt least squares method to estimate the parameters $\left({\beta }_0,{\beta }_T\right)$, minimizing the sum of squared residuals ${\displaystyle {\sum }_{k=1}^2{\displaystyle {\sum }_{i=1}^{12}{\left(Y\left(t_{ki}\right)-\hat{Y}\left(t_{ki}\right)\right)}^2}}$, we obtain ${\hat{\beta }}_0=-5.035$, ${\hat{\beta }}_T=5.146$ which results in $R^2=0.980$. Applied to the trained model to testing dataset, the model also achieved good performance with $R^2=0.985$. The prediction results for training and testing datasets are illustrated in Figure 3 .

In this study, we did outdoor weathering trials and had no stringent control of the exposure condition. We tracked the yellow indexes of PS materials and monitored outdoor climatic data. Yellowing discoloration is a complex process caused by processes of photo-degradation and thermal degradation. The yellowing index primarily depends on cumulative radiation, but the effect may vary under different temperature conditions. We established a model combined with an Arrhenius law for the temperature dependence to predict the change in yellowing index. The model provided a very high performance under natural weathering exposures. The results show the yellowing proceeds faster under increased radiation and temperature. To prevent yellowing, the PS materials are suggested to be stored in a place that is shielded from direct sunlight and has a relatively low temperature.

We only consider the effects of main contributors, solar radiation and temperature. Solar radiation is the electromagnetic energy emitted by the sun, including Ultraviolet (UV) radiation, visible light (VL) and infrared radiation (IR). UV light has shorter wavelength and higher energy, and hence it might pay a major role within solar radiation to yellowing process. In future work, we can consider more potentially relevant factors, such as UV and VL radiations, temperature and humidity, in the model and use more datasets to demonstrate its generalizability.

This work was supported by Guangdong Basic and Applied Basic Research Foundation (2020B1515310007), Guangdong Province Key Laboratory of Computational Science, Sun Yat-sen University (2020B1212060032), and sub-project of National 973 Program (No. 2012CB724605). The authors would like to thank Dr. Youji Tao for his helpful suggestions and assistance.