# On the evaluation of the carbon dioxide solubility in polymers using gene expression programming

Scientific Reports volume 13, Article number: 12505 (2023) Cite this article

91 Accesses

Metrics details

Evaluation, prediction, and measurement of carbon dioxide (CO2) solubility in different polymers are crucial for engineers in various chemical applications, such as extraction and generation of novel materials. In this paper, correlations based on gene expression programming (GEP) were generated to predict the value of carbon dioxide solubility in three polymers. Results showed that the generated correlations could represent an outstanding efficiency and provide predictions for carbon dioxide solubility with satisfactory average absolute relative errors of 9.71%, 5.87%, and 1.63% for polystyrene (PS), polybutylene succinate-co-adipate (PBSA), and polybutylene succinate (PBS), respectively. Trend analysis based on Henry’s law illustrated that increasing pressure and decreasing temperature lead to an increase in carbon dioxide solubility. Finally, outlier discovery was applied using the leverage approach to detect the suspected data points. The outlier detection demonstrated the statistical validity of the developed correlations. William’s plot of three generated correlations showed that all of the data points are located in the valid zone except one point for PBS polymer and three points for PS polymer.

In the recent years, application of different polymers has become an attractive issue in various industries including the petroleum industry. The fluid adsorption process in different polymers is a vital circumstance in the oil industry concepts such as enhanced oil recovery (EOR)1,2,3, gas separation, imbibition of additives, and foaming processes4,5. Carbon dioxide (CO2) is one of the most significant gases, which plays a noteworthy role in polymers’ structure, polymer foams, and production properties4,6. Also, CO2 and supercritical carbon dioxide (SCCO2), (a supercritical carbon dioxide is described as a fluid for which both temperature and pressure are higher than critical values) have become one of the most conventional green materials, that have been extensively used in solvent, anti-solvent or a solute in numerous field processing including material synthesis, material modification, foaming processes, polymerization and particle production7,8,9. SCCO2 is potentially appealing as a solvent that shows properties that are a mixture of those commonly combined with liquids or gases. CO2 solubility is the maximum CO2 quantity that can solute in different solutions. Evaluation, prediction, and measurement of CO2 solubility in different biodegradable polymers has become notable technology for engineers in various chemical applications such as extraction and generation of novel materials10,11,12,13,14. Biodegradable polymers are a particular type of polymers that collapse by bacterial dissolution process to eventuate in natural fluids such as CO2 and N2. Poly butylene succinate (PBS) and polybutylene succinate-co-adipate (PBSA) are two applicable biodegradable polymers that have been generated by Showa Highpolymer Co. Ltd. and Showa Denko K.K15,16.

In order to predict gas solubilities in polymers, especially CO2, various experimental, empirical, and theoretical approaches were investigated since 1986. In 1986 and 1993, Shah et al.17,18 measured solubility of different gases including CO2 in silicone polymers at pressures up to 26 atmosphere and temperature values of 10, 35, and 55 °C. In 1994, Li et al.19 predicted the solubility of CO2 in amine systems. They considered binary and ternary mixtures containing three solvents, namely mono-ethanolamine (MEA), methyl-diethanolamine (MDEA), and water (H2O). They used temperature in a range of 0–225 °C. They modeled CO2 solubility in amine mixtures as a function of temperature. Two years later, Sato et al.20 investigated solubility of CO2 and N2 in polystyrene under high pressure and temperature conditions. They measured gas solubility at pressures up to 20 MPa and temperatures from 373.2 to 453.2 K. In 1998, Aubert21 calculated CO2 solubility at pressures up to 9.65 MPa using quartz crystal microbalance technique. Next year, Webb et al.22 and Sato et al.23 evaluated diffusion and solubility of CO2 in polymers under high pressures and temperatures. According to their research, the solubilities increased by increasing pressure and decreased by increasing temperature. In 2000, Sato et al.15 suggested empirical relations to determine solubility and diffusion coefficient of CO2. They considered pressure and temperature as the dependent variables in the range of 1.025–20.144 MPa and 323.15–453.15 K, respectively. They achieved that solubility of CO2 in molten state polymers increases by increasing pressure and decreasing temperature. A year later, Hilic et al.24 measured solubility of N2 and CO2 in polystyrene, which considered pressure from 3.05 to 45 MPa and temperature from 338 to 402 K. In addition, an experimental technique with a vibrating-wire force sensor was applied. They got a linear relationship between increasing solubility with increasing pressure and decreasing temperature. In the same year, Sato et al.25 calculated solubilities of CO2 at the temperature range of 313.15–373.15 K and pressures up to 17.5 MPa. In 2002, Park et al.26 studied about CO2 solubility in alkanolamine solutions in the values of 40, 60 and 80 °C for temperature and 0.1–50 psia for pressure. They represented a vapor–liquid equilibrium of CO2 in these solutions. In the same year, Sato et al.27 examined CO2 solubility in poly (2,6-dimethyl-1,4-phenylene ether) (PPO) and PS at temperatures of 373.15, 427.15, and 473.15 K and pressures up to 20 MPa. They obtained that solubility of CO2 increases with increasing PPO concentration. A year later, in 2003, Hamedi et al.28 predicted the adsorption of CO2 in various polymers based on a group contribution equation of state (EoS) with input ranges of 283–453 K and 1–200 bar for temperature and pressure, respectively. Their best result was an average absolute relative error (AARE) of 5.5% for polystyrene. In 2006, Li et al.29 measured gas solubilities and diffusivities in polylactide at a temperature of 180–200 K and pressures up to 28 MPa using a magnetic suspension balance (MSB). Furthermore, they adopted a theoretical model based on Fick’s second law to extract diffusion coefficients of N2 and CO2 in polylactide. They obtained that CO2 exhibited lower diffusivity than N2 at the same temperature. At that year, Nalawade et al.9 used SCCO2 as a green solvent for processing polymer melts. They earned SCCO2 is applicable in many polymerization processes due to its high solubility in polymers. In 2007, Lei et al.30 generated buoyancy correlations and Sanchez and Lacombe equation of state to estimate CO2 swelling degree, crystallinity, and solubility in polypropylene. They achieved CO2 solubility first decreased and then increased with temperature. Two years later, Khajeh et al.31 developed intelligent model based on adaptive neuro fuzzy inference system (ANFIS) to predict solubility of CO2 in polymers. They used up to 37 data points for different polymers. In 2011, Xu et al.32 investigated a theoretical study of solubility correlations of CO2 in ether and carbonyl groups of polymers, namely poly(ethylene oxide) (PEO), poly(propylene oxide) (PPO), poly(vinyl acetate) (PVAc), poly(ethylene carbonate) (PEC) and poly(propylene carbonate) (PPC). They showed that the CO2 solubility in PPC is higher than other polymers used in their study. Next year, Han et al.13 developed continuous reactions and considered economical concepts in SCCO2 applications. In 2013, Li et al.33 developed an artificial neural network (ANN) to estimate gas solubilities in polymers. Their research demonstrated good agreement between experimental and predicted data using their correlation. At the same year, Minelli and Sarti34 measured solubility and permeability of CO2 in various glassy polymers by considering diffusion coefficient as a kinetic factor. In 2015, different mathematical and theoretical approaches by Ting and Yuan10, Li et al.7 and Quan et al.12 were studied to estimate CO2 properties including solubility. All of them showed that the CO2 solubility has direct relation with pressure and reverse relation with temperature. Two years later, Mengshan et al.8,35 developed an artificial neural network and artificial intelligence technique based on diffusion theory to predict solubility of CO2 and SCCO2 in polymers. In 2019, Soleimani et al.4 developed decision tree (DT) based smart model for estimating solubility of CO2. They used 515 data points with a range of 306–483.7 K for temperature and 1.025–44.41 MPa for pressure. One year later, Li et al.36 investigated a comprehensive review of CO2 polymer system. They used two types of multi-scaled methods, namely thermodynamic-calculation model and computer simulation to measure CO2 solubility in polymers. Their developed model can be utilized in chemistry and chemical industries, such as phase rheological property and polymer self-assembly. In 2022, various experimental, theoretical, and modeling researches have been done in order to measure solubility of CO2 and other gases in water-polymer systems. Sun et al.37 measured CO2 solubility in oil-based and water-based drilling fluids using the sample analysis approach. Their results indicated that the salting-out effect of electrolyte on gas solubility can be increased with increasing the molar concentration of ions. Their study also showed that the errors of CO2 solubility in the oil-based and water-based drilling fluids are 6.75% and 3.47%, respectively. Besides, Ushiki et al.38 evaluated CO2 solubility and diffusivity in polycaprolactone (PCL) performing perturbed-chain statistical associating fluid theory (PC-SAFT) and free volume methods. According to their work, CO2 solubility was recognized to conform with Henry’s law, and the PC-SAFT EoS sufficiently described the solubility. Also, Kiran et al.39 assessed diffusivity and solubility of CO2 and N2 in polymers. They used Sanchez-Lacombe EoS in modeling solubility. Furthermore, Ricci et al.40 provided a comprehensive theoretical framework for the supercritical sorption and transport of CO2 in polymers. In their study, CO2 sorption was modelled utilizing data available across the critical region, at different temperatures and pressures up to 18 MPa.

The present research mostly focuses on generating accurate correlations for CO2 solubility prediction considering the pressure and temperature of the polymer as input variables. The generated correlations are based on gene expression programming (GEP) technique. A comprehensive databank including of 53 data points for PBS, 43 data points for PBSA and 92 data points for PS polymer is collected15,20,24,25. After generating correlations, statistical and graphical error tests are applied to assess the accuracy of the correlations. Likewise, the capability of the represented correlations in predicting the real trend of the CO2 solubility with the change of pressure and temperature is appraised. Lately, the leverage approach is performed to detect the outlier data points in the dataset.

In this research, GEP algorithm was implemented to predict the amount of CO2 solubility in three different polymers, namely PBS, PBSA, and polystyrene (PS). For this aim, 53 data points for PBS, 43 data points for PBSA, and 92 data points for PS polymer were collected15,20,24,25. In this work, pressure and temperature of carbon dioxide were considered as input parameters. A summary of the gathered data points is shown in Table 1. As pointed up in Table 1, extensive ranges of temperature and pressure of CO2 are supplied in this study.

In order to generate CO2 solubility correlations, Gene expression programming (GEP) evolutionary algorithm has been applied. GEP which was firstly proposed by Ferreira in 200141, is a normally comprehensive phenotype technique in which the chromosomes form a correctly inseparable, operative entity42. This technique is extensively used in computer programming and modeling applications43,44,45,46. Gene expression programming algorithms are complicated tree-based structures that coordinate by changing their shape, composition and sizes. By encoding trees as vectors of symbols and transforming them into them just in order to assess their fitness, this technique can indirectly produce trees47. This soft computing technique is strong predictive algorithm that is widely used for various field application purposes. Commonly, the GEP technique has two components, namely chromosome and the expression trees (ETs). The possible solutions are encoded by the chromosomes and is regarded as the linear string with particular length, hence these solutions will be decoded into the real candidate solution termed expression tree48. After producing of chromosomes of first-production individuals and choosing them based on fitness function to re-generate with modifications, new generation individuals were presented to the developmental operation of selection environment confrontation, genome expression, and modified reproduce49. Additionally, gene expression programming automatically creates algebraic expressions to answer nonlinear problems50. The schematic flowchart of GEP procedure is depicted in Fig. 1.

The schematic framework of the gene expression programming (GEP).

In the present study, gene expression programming tree-based soft computing approach was carried out to develop accurate correlations for predicting CO2 solubility in different polymers. The developed correlations consider CO2 solubility as a function of pressure and temperature of corresponding polymer and use them as input variables. To generate accurate and user-friendly correlations, an exhaustive databank consists of 53 data points for PBS polymer, 43 data points for PBSA polymer and 92 data points for PS polymer was collected from previous literature. Table 2 represents the GEP parameters utilized in this research.

Using the aforementioned approach, the final formulas for the determination of CO2 solubility based on gene expression programming technique, are listed below:

where P and T denote pressure and temperature of aforenamed polymers, respectively. In the above correlations, the units of P and T are MPa and K, respectively. The generated correlations in this study are applicable for CO2 solubility prediction in various ranges of temperature and pressure of the mentioned polymers.

In order to show and compare the precision of the generated correlations, some important statistical parameters including root mean square error (RMSE), standard deviation (SD), coefficient of determination (R2), the average relative error (ARE) and the average absolute relative error (AARE) were applied51. These terms are given below:

where \(Ei\) is the partial deviation that is described as:

where n, S (exp), S (cal) and S (avg) are the number of data, actual CO2 solubility value, calculated CO2 solubility value, and the average of the actual data points, respectively. The prementioned statistical parameters for the three generated correlations are detailed for the training, testing, and whole datasets in Table 3. As described in this table, the AARE of the correlation for the PBS polymer is lower than other two correlations generated in this work. Results demonstrate that generated correlation for the PBS polymer has the lowest standard deviation (0.028) and RMSE (0.00178). However, the correlations developed for the other two polymers also have acceptable accuracy. As presented in Table 3, the AARE values for PBS and PBSA polymers were obtained less than AARE for PS polymer, which was due to the nature of the experimental data related to PS polymer. It is obvious that the generated correlations are reliable and sometimes, due to the nature of the experimental data values of different materials (like polymers), different error values may be obtained.

This section represents a graphical description of the comparison among the results of the generated correlations and the actual data. The predicted CO2 solubility values in PBS polymer are sketched versus actual ones in Fig. 2a. Likewise, the predicted CO2 solubility values in PBSA and PS polymers are depicted versus experimental data in Fig. 2b,c, respectively. The closer the sketched data points to the 45° line, the greater the uniformity of the correlations is. According to these plots, it is apparent that the results of the generated user-friendly correlations illustrate satisfactory agreement around the ideal line. Additionally, the relative error curves of the developed correlations of the CO2 solubility in PBS polymer, PBSA polymer, and PS polymer are presented in Fig. 3a–c, respectively.

Cross plots of the predicted and experimental CO2 solubility values in (a) PBS, (b) PBSA, (c) PS polymers.

Relative error distribution curves of the generated correlation of the CO2 solubility in (a) PBS, (b) PBSA, (c) PS polymers.

Furthermore, to show the accuracy of the presented correlations in different ranges of pressure and temperature, the correlations’ performances in terms of AARE were sketched against five sets of pressure and three sets of temperature. Figure 4 demonstrates the AARE of the correlations in different ranges of input parameters. For various ranges of pressure, the correlation of CO2 solubility in PBS polymer clarifies a steady performance and its AARE is lower than 2.9% in all ranges. Besides, a reliable performance can be perceived from the correlation of CO2 solubility in PBS polymer up to the last temperature range. This figure validates the efficiency of the developed correlation of CO2 solubility in PBS polymer over other developed correlations in the present study.

AARE for the different correlations performed in this research for the three polymers in various inputs ranges. (a,b) PBS; (c,d) PBSA; (e,f) PS.

Afterwards, the cumulative frequency analysis of the absolute percent relative error (APRE) for the generated correlations in this work is shown in Fig. 5. According to the results of this figure, the correlation of CO2 solubility in PBS polymer could estimate more than 90% of CO2 solubility values with an APRE of less than 5%, and also more than 98% of the CO2 solubility values by the correlation for PBS polymer have an AARE of less than 10%.

Cumulative frequency plot of generated correlations in this study.

Additionally, absolute relative error comparison among generated correlations was carried out. Figure 6 describes the AARE comparison between the prementioned correlations. According to this figure, the developed correlation of CO2 solubility in PBS polymer revealed the highest accuracy and the lowest AARE between other correlations generated in this research.

Comparison among AARE values of the implemented correlations.

Trend analysis is a well-known applicable technique to visualize the output variation with the change of input variables52,53. The predictions of the CO2 solubility correlations are depicted versus temperature and pressure in Fig. 7 to investigate the capability of the generated correlations in following the actual expected trends of CO2 solubility values with the change of pressure and temperature. According to Henry’s law, it is evident that CO2 solubility increases with decreasing temperature and increasing pressure54. Carbon dioxide has a propensity, namely plasticizing effect55. It means that the molecules of CO2 are pressured in the chains of the polymer as a consequence of increasing pressure, which results in an extension of the pore space within the molecules and, then, for this reason, an addition of their movement56,57. This causes it feasible to absorb more gas molecules. Likewise, by decreasing the temperature the CO2 molecules obtain lower kinetic energy and they do not have a tendency for releasing from the solution and for staying in a condition with more independence58. As a consequence, the solubility would increase.

Comparison of the CO2 solubility variation for the generated correlations in this work with actual data. (a) CO2 solubility change with pressure; (b) CO2 solubility change with temperature.

Outlier discovery plays an important role to identify data that may vary from other data points exist in a dataset59. The leverage technique is a trustworthy method for outlier discovery which concerns with the values of the standardized residuals and a matrix, namely the Hat matrix made of the actual and the predicted values obtained from the correlations60. According to this approach, if most of the data points located in the ranges of − 3 ≤ R ≤ 3 (R denotes the standardized residual) and 0 ≤ Hi ≤ H*, it illustrates that the results of the generated correlations are dependable and valid61,62,63. Figures 8, 9 and 10 represent William plots of the generated correlations of CO2 solubility in PBS, PBSA, and PS polymers, respectively. For PBS polymer it is obvious that all of the data points placed in a valid zone except one. Also, the results of the generated correlation of PBSA polymer show that all of the data points located in a valid region. At the end, Fig. 10 presents a William plot of CO2 solubility correlation in PS polymer, showing that among whole dataset consists of 92 data points used for this polymer, only 3 data points are recognized as out of leverage data points.

The William plot of the generated correlation for PBS polymer.

The William plot of the generated correlation for PBSA polymer.

The William plot of the generated correlation for PS polymer.

The present research aimed to predict CO2 solubility as a strong effective parameter in polymerization processes. PBS, PBSA, and PS were three polymers, which were utilized in this work. For this purpose, gene expression programming (GEP) technique was applied. To this aim, a widespread dataset was gathered from previous literature. Results showed that the generated correlation of CO2 solubility for PBS polymer could present the highest accuracy in predicting solubility of CO2 with an AARE of 1.63%, SD of 0.028, and RMSE of 0.001. The sketched CO2 solubility curves using the trend analysis demonstrated that all three generated correlations in this study could exactly fit the actual trends of CO2 solubility variation. The simple generated correlations can be performed in wide ranges of pressures and temperatures and represent high accuracy. The leverage approach showed that all the data points seem to be reliable and valid except four, which were placed in a lower suspected and out of leverage zones. In order to precisely simulate CO2 solubility in polymers in a future works, it is recommended to generate new correlations, and also develop intelligent schemes.

All the data are available from the corresponding author on reasonable request.

Sheng, J. J. Enhanced Oil Recovery Field Case Studies (Gulf Professional Publishing, 2013).

Google Scholar

Thomas, S. Enhanced oil recovery—an overview. Oil Gas Sci. Technol. Rev. l’IFP 63, 9–19 (2008).

Article CAS Google Scholar

Divandari, H., Amiri-Ramsheh, B. & Zabihi, R. Steam flooding (steam drive). Thermal Methods 20, 47 (2023).

Article Google Scholar

Soleimani, R. et al. Evolving an accurate decision tree-based model for predicting carbon dioxide solubility in polymers. Chem. Eng. Technol. 43, 514–522 (2020).

Article CAS Google Scholar

Li, D.-C., Liu, T., Zhao, L. & Yuan, W.-K. Solubility and diffusivity of carbon dioxide in solid-state isotactic polypropylene by the pressure-decay method. Ind. Eng. Chem. Res. 48, 7117–7124 (2009).

Article CAS Google Scholar

Zheng, H., Mahmoudzadeh, A., Amiri-Ramsheh, B. & Hemmati-Sarapardeh, A. Modeling viscosity of CO2–N2 gaseous mixtures using robust tree-based techniques: Extra tree, random forest, GBoost, and LightGBM. ACS Omega 8, 13863–13875 (2023).

Article CAS PubMed PubMed Central Google Scholar

Li, M. et al. Solubility prediction of supercritical carbon dioxide in 10 polymers using radial basis function artificial neural network based on chaotic self-adaptive particle swarm optimization and K-harmonic means. RSC Adv. 5, 45520–45527 (2015).

Article ADS CAS Google Scholar

Mengshan, L. et al. Prediction of supercritical carbon dioxide solubility in polymers based on hybrid artificial intelligence method integrated with the diffusion theory. RSC Adv. 7, 49817–49827 (2017).

Article ADS Google Scholar

Nalawade, S. P., Picchioni, F. & Janssen, L. Supercritical carbon dioxide as a green solvent for processing polymer melts: Processing aspects and applications. Prog. Polym. Sci. 31, 19–43 (2006).

Article CAS Google Scholar

Ru-Ting, X. & Xing-Yuan, H. Predictive calculation of carbon dioxide solubility in polymers. RSC Adv. 5, 76979–76986 (2015).

Article ADS Google Scholar

Zhang, Q., Vanparijs, N., Louage, B., De Geest, B. G. & Hoogenboom, R. Dual pH-and temperature-responsive RAFT-based block co-polymer micelles and polymer–protein conjugates with transient solubility. Polym. Chem. 5, 1140–1144 (2014).

Article CAS Google Scholar

Quan, S. et al. A bio-inspired CO2-philic network membrane for enhanced sustainable gas separation. J. Mater. Chem. A 3, 13758–13766 (2015).

Article CAS Google Scholar

Han, X. & Poliakoff, M. Continuous reactions in supercritical carbon dioxide: Problems, solutions and possible ways forward. Chem. Soc. Rev. 41, 1428–1436 (2012).

Article CAS PubMed Google Scholar

Chandra, R. & Rustgi, R. Biodegradable polymers. Prog. Polym. Sci. 23, 1273–1335 (1998).

Article CAS Google Scholar

Sato, Y. et al. Solubility and diffusion coefficient of carbon dioxide in biodegradable polymers. Ind. Eng. Chem. Res. 39, 4813–4819 (2000).

Article CAS Google Scholar

Nishioka, M., Tuzuki, T., Wanajyo, Y., Oonami, H. & Horiuchi, T. Studies in Polymer Science Vol 12, 584–590 (Elsevier, 1994).

Google Scholar

Yampolskii, Y. & Paterson, R. Solubility of gases in polymers. Exp. Determin. Solubil. 6, 151–171 (2003).

CAS Google Scholar

Shah, V., Hardy, B. & Stern, S. Solubility of carbon dioxide, methane, and propane in silicone polymers. Effect of polymer backbone chains. J. Polym. Sci. Part B Polym. Phys. 31, 313–317 (1993).

Article ADS CAS Google Scholar

Li, Y.-G. & Mather, A. E. Correlation and prediction of the solubility of carbon dioxide in a mixed alkanolamine solution. Ind. Eng. Chem. Res. 33, 2006–2015 (1994).

Article CAS Google Scholar

Sato, Y., Yurugi, M., Fujiwara, K., Takishima, S. & Masuoka, H. Solubilities of carbon dioxide and nitrogen in polystyrene under high temperature and pressure. Fluid Phase Equilib. 125, 129–138 (1996).

Article CAS Google Scholar

Aubert, J. H. Solubility of carbon dioxide in polymers by the quartz crystal microbalance technique. J. Supercrit. Fluids 11, 163–172 (1998).

Article CAS Google Scholar

Webb, K. F. & Teja, A. S. Solubility and diffusion of carbon dioxide in polymers. Fluid Phase Equilib. 158, 1029–1034 (1999).

Article Google Scholar

Sato, Y., Fujiwara, K., Takikawa, T., Takishima, S. & Masuoka, H. Solubilities and diffusion coefficients of carbon dioxide and nitrogen in polypropylene, high-density polyethylene, and polystyrene under high pressures and temperatures. Fluid Phase Equilib. 162, 261–276 (1999).

Article CAS Google Scholar

Hilic, S., Boyer, S. A., Pádua, A. A. & Grolier, J. P. E. Simultaneous measurement of the solubility of nitrogen and carbon dioxide in polystyrene and of the associated polymer swelling. J. Polym. Sci. Part B Polym. Phys. 39, 2063–2070 (2001).

Article ADS CAS Google Scholar

Sato, Y., Takikawa, T., Takishima, S. & Masuoka, H. Solubilities and diffusion coefficients of carbon dioxide in poly (vinyl acetate) and polystyrene. J. Supercrit. Fluids 19, 187–198 (2001).

Article CAS Google Scholar

Park, S. H., Lee, K. B., Hyun, J. C. & Kim, S. H. Correlation and prediction of the solubility of carbon dioxide in aqueous alkanolamine and mixed alkanolamine solutions. Ind. Eng. Chem. Res. 41, 1658–1665 (2002).

Article CAS Google Scholar

Sato, Y., Takikawa, T., Yamane, M., Takishima, S. & Masuoka, H. Solubility of carbon dioxide in PPO and PPO/PS blends. Fluid Phase Equilib. 194, 847–858 (2002).

Article Google Scholar

Hamedi, M., Muralidharan, V., Lee, B. & Danner, R. P. Prediction of carbon dioxide solubility in polymers based on a group-contribution equation of state. Fluid Phase Equilib. 204, 41–53 (2003).

Article CAS Google Scholar

Li, G., Li, H., Turng, L., Gong, S. & Zhang, C. Measurement of gas solubility and diffusivity in polylactide. Fluid Phase Equilib. 246, 158–166 (2006).

Article CAS Google Scholar

Lei, Z., Ohyabu, H., Sato, Y., Inomata, H. & Smith, R. L. Jr. Solubility, swelling degree and crystallinity of carbon dioxide–polypropylene system. J. Supercrit. Fluids 40, 452–461 (2007).

Article CAS Google Scholar

Khajeh, A., Modarress, H. & Rezaee, B. Application of adaptive neuro-fuzzy inference system for solubility prediction of carbon dioxide in polymers. Expert Syst. Appl. 36, 5728–5732 (2009).

Article Google Scholar

Xu, M., Chen, J., Zhang, C., Du, Z. & Mi, J. A theoretical study of structure–solubility correlations of carbon dioxide in polymers containing ether and carbonyl groups. Phys. Chem. Chem. Phys. 13, 21084–21092 (2011).

Article CAS PubMed Google Scholar

Li, M. et al. Prediction of gas solubility in polymers by back propagation artificial neural network based on self-adaptive particle swarm optimization algorithm and chaos theory. Fluid Phase Equilib. 356, 11–17 (2013).

Article CAS Google Scholar

Minelli, M. & Sarti, G. C. Permeability and solubility of carbon dioxide in different glassy polymer systems with and without plasticization. J. Membr. Sci. 444, 429–439 (2013).

Article CAS Google Scholar

Mengshan, L., Wei, W., Bingsheng, C., Yan, W. & Xingyuan, H. Solubility prediction of gases in polymers based on an artificial neural network: A review. RSC Adv. 7, 35274–35282 (2017).

Article ADS Google Scholar

Li, M. et al. Models for the solubility calculation of a CO2/polymer system: A review. Mater. Today Commun. 25, 101277 (2020).

Article CAS Google Scholar

Sun, X. et al. Experiments and modeling of CO2 solubility in water-based and oil-based drilling fluids. J. Petrol. Sci. Eng. 212, 110336 (2022).

Article CAS Google Scholar

Ushiki, I., Kawashima, H., Kihara, S.-I. & Takishima, S. Solubility and diffusivity of supercritical CO2 for polycaprolactone in its molten state: Measurement and modeling using PC-SAFT and free volume theory. J. Supercrit. Fluids 181, 105499 (2022).

Article CAS Google Scholar

Kiran, E., Sarver, J. A. & Hassler, J. C. Solubility and diffusivity of CO2 and N2 in polymers and polymer swelling, glass transition, melting, and crystallization at high pressure: A critical review and perspectives on experimental methods, data, and modeling. J. Supercrit. Fluids 185, 105378 (2022).

Article CAS Google Scholar

Ricci, E., De Angelis, M. G. & Minelli, M. A comprehensive theoretical framework for the sub and supercritical sorption and transport of CO2 in polymers. Chem. Eng. J. 435, 135013 (2022).

Article CAS Google Scholar

Ferreira, C. Gene expression programming: A new adaptive algorithm for solving problems. arXiv preprint cs/0102027 (2001).

Umar, A. A., Saaid, I. M., Sulaimon, A. A. & Pilus, R. M. Predicting the viscosity of petroleum emulsions using gene expression programming (GEP) and response surface methodology (RSM). J. Appl. Math. 20, 20 (2020).

Google Scholar

Zhong, J., Feng, L. & Ong, Y.-S. Gene expression programming: A survey. IEEE Comput. Intell. Mag. 12, 54–72 (2017).

Article Google Scholar

Amar, M. N., Larestani, A., Lv, Q., Zhou, T. & Hemmati-Sarapardeh, A. Modeling of methane adsorption capacity in shale gas formations using white-box supervised machine learning techniques. J. Petrol. Sci. Eng. 208, 109226 (2022).

Article Google Scholar

Amar, M. N. Prediction of hydrate formation temperature using gene expression programming. J. Nat. Gas Sci. Eng. 89, 103879 (2021).

Article Google Scholar

Amar, M. N., Ghriga, M. A., Seghier, M. E. A. B. & Ouaer, H. Predicting solubility of nitrous oxide in ionic liquids using machine learning techniques and gene expression programming. J. Taiwan Inst. Chem. Eng. 128, 156–168 (2021).

Article Google Scholar

Baniasadi, H., Kamari, A., Heidararabi, S., Mohammadi, A. H. & Hemmati-Sarapardeh, A. Rapid method for the determination of solution gas-oil ratios of petroleum reservoir fluids. J. Nat. Gas Sci. Eng. 24, 500–509 (2015).

Article Google Scholar

Rostami, A., Arabloo, M., Kamari, A. & Mohammadi, A. H. Modeling of CO2 solubility in crude oil during carbon dioxide enhanced oil recovery using gene expression programming. Fuel 210, 768–782 (2017).

Article CAS Google Scholar

Mirzaie, M. & Tatar, A. Modeling of interfacial tension in binary mixtures of CH4, CO2, and N2-alkanes using gene expression programming and equation of state. J. Mol. Liq. 320, 114454 (2020).

Article CAS Google Scholar

Traore, S., Luo, Y. & Fipps, G. Gene-expression programming for short-term forecasting of daily reference evapotranspiration using public weather forecast information. Water Resour. Manage 31, 4891–4908 (2017).

Article Google Scholar

Sarapardeh, A. H., Larestani, A., Menad, N. A. & Hajirezaie, S. Applications of Artificial Intelligence Techniques in the Petroleum Industry (Gulf Professional Publishing, 2020).

Google Scholar

Kirk, H., Haynes, F. & Monroe, R. Application of trend analysis to horticultural field trials. J. Am. Soc. Hortic. Sci. 20, 20 (1980).

Google Scholar

Amiri-Ramsheh, B., Safaei-Farouji, M., Larestani, A., Zabihi, R. & Hemmati-Sarapardeh, A. Modeling of wax disappearance temperature (WDT) using soft computing approaches: Tree-based models and hybrid models. J. Petrol. Sci. Eng. 208, 109774 (2022).

Article CAS Google Scholar

Kumełan, J., Kamps, A.P.-S., Tuma, D. & Maurer, G. Solubility of CO2 in the ionic liquid [hmim][Tf2N]. J. Chem. Thermodyn. 38, 1396–1401 (2006).

Article Google Scholar

Minelli, M., Oradei, S., Fiorini, M. & Sarti, G. C. CO2 plasticization effect on glassy polymeric membranes. Polymer 163, 29–35 (2019).

Article CAS Google Scholar

Khoshraftar, Z. & Ghaemi, A. Prediction of CO2 solubility in water at high pressure and temperature via deep learning and response surface methodology. Case Stud. Chem. Environ. Eng. 7, 100338 (2023).

Article CAS Google Scholar

Messabeb, H., Contamine, F., Cézac, P., Serin, J. P. & Gaucher, E. C. Experimental measurement of CO2 solubility in aqueous NaCl solution at temperature from 323.15 to 423.15 K and pressure of up to 20 MPa. J. Chem. Eng. Data 61, 3573–3584 (2016).

Article CAS Google Scholar

Thibault, Y. & Holloway, J. R. Solubility of CO 2 in a Ca-rich leucitite: Effects of pressure, temperature, and oxygen fugacity. Contrib. Miner. Petrol. 116, 216–224 (1994).

Article ADS CAS Google Scholar

Rousseeuw, P. J. & Van Zomeren, B. C. Unmasking multivariate outliers and leverage points. J. Am. Stat. Assoc. 85, 633–639 (1990).

Article Google Scholar

Rousseeuw, P. J. & Leroy, A. M. Robust Regression and Outlier Detection (Wiley, 2005).

MATH Google Scholar

Goodall, C. R. 13 Computation using the QR decomposition. (1993).

Gramatica, P. Principles of QSAR models validation: Internal and external. QSAR Comb. Sci. 26, 694–701 (2007).

Article CAS Google Scholar

Amiri-Ramsheh, B., Zabihi, R. & Hemmati-Sarapardeh, A. Modeling wax deposition of crude oils using cascade forward and generalized regression neural networks: Application to crude oil production. Geoenergy Sci. Eng. 224, 211613 (2023).

Article CAS Google Scholar

Download references

Department of Petroleum Engineering, Shahid Bahonar University of Kerman, Kerman, Iran

Behnam Amiri-Ramsheh & Abdolhossein Hemmati-Sarapardeh

Département Etudes Thermodynamiques, Division Laboratoires, Sonatrach, Boumerdes, Algeria

Menad Nait Amar

Department of System Engineering, École de Technologie Supérieur, Montreal, QC, Canada

Mohammadhadi Shateri

State Key Laboratory of Petroleum Resources and Prospecting, China University of Petroleum (Beijing), Beijing, China

Abdolhossein Hemmati-Sarapardeh

You can also search for this author in PubMed Google Scholar

You can also search for this author in PubMed Google Scholar

You can also search for this author in PubMed Google Scholar

You can also search for this author in PubMed Google Scholar

B.A.-R.: investigation, visualization, writing-original draft, M.N.A.: methodology, visualization, data analysis, M.S.: supervision, writing-review and editing. A.H.-S.: supervision, writing-review and editing.

Correspondence to Mohammadhadi Shateri or Abdolhossein Hemmati-Sarapardeh.

The authors declare no competing interests.

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

Reprints and Permissions

Amiri-Ramsheh, B., Nait Amar, M., Shateri, M. et al. On the evaluation of the carbon dioxide solubility in polymers using gene expression programming. Sci Rep 13, 12505 (2023). https://doi.org/10.1038/s41598-023-39343-8

Download citation

Received: 11 February 2023

Accepted: 24 July 2023

Published: 02 August 2023

DOI: https://doi.org/10.1038/s41598-023-39343-8

Anyone you share the following link with will be able to read this content:

Sorry, a shareable link is not currently available for this article.

Provided by the Springer Nature SharedIt content-sharing initiative

By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.