Research progress in numerical simulation of rubber vulcanization

doi:10.11949/0438-1157.20200400

Abstract

Abstract:

Vulcanization, as a key process in the molding process of rubber products, determines the performance and dimensional accuracy of the products. Through simulation technology, it helps to achieve accurate control of the vulcanization process and reduce energy consumption. In this paper, the research progress of numerical simulation of rubber vulcanization is reviewed from two aspects of micro scale and macro scale, and the development status of theoretical and empirical vulcanization reaction kinetic model is mainly analyzed. Finally, the development trend of numerical simulation of rubber vulcanization process is pointed out.

Key words: rubber products, vulcanization, polymer processing, reaction kinetics, numerical simulation

摘要：

硫化作为橡胶制品成型过程中的关键工序，决定着制品的使用性能与尺寸精度，通过模拟仿真技术有助于实现对硫化过程的精确控制以及减少能量消耗。基于此，从微观尺度和宏观尺度两个层面综述了橡胶硫化过程数值模拟的研究进展，重点分析了理论与经验硫化反应动力学模型的发展现状。最后，指出了橡胶硫化过程数值模拟的发展方向与趋势。

关键词: 橡胶制品, 硫化, 聚合物加工, 反应动力学, 数值模拟

CLC Number:

TQ 336

Fei GUO,Zhaoxiang ZHANG,Wei SONG,Xiaohong JIA. Research progress in numerical simulation of rubber vulcanization[J]. CIESC Journal, 2020, 71(8): 3393-3402.

郭飞,张兆想,宋炜,贾晓红. 橡胶硫化过程数值模拟研究进展[J]. 化工学报, 2020, 71(8): 3393-3402.

Figures/Tables 7

Fig.1 Mechanism of NBR cross-linking using sulfur[4]

Fig.2 Mechanical properties of vulcanizates with different crosslinking degree[5]

Fig.3 Diagram of common rubber vulcanization curve

Fig.4 Schematic diagram of the determination process of a kinetic model for sulfur vulcanization of SBR rubber[11]

Fig.5 Schematic diagram of solving process of a phenomenological kinetic model parameters

Table 1 Several typical phenomenological reaction kinetic models

模型名称	模型形式	参考文献
Piloyan	$d α d t = k (1 - α) n$	[38]
Kamal-Sourour	$d α d t = k α m (1 - α) n$	[35]
Leroy	$d α d t = (A 1 + A 2 α m) (1 - α) n e - E a / R T$	[27,39]
Rafei	$α = k n k n + t n$	[40]
Yeoh	$d α d t = (α 1 - α) 1 / p (1 k) 1 / p e x p (1 R T) 1 / p$	[41]
Jia	$α i = α i - 1 + k 0 Δ t e x p - E a R T 1 + k 0 Δ t e x p - E a R T$	[42]

Table 1 Several typical phenomenological reaction kinetic models

模型名称	模型形式	参考文献
Piloyan	$d α d t = k (1 - α) n$	[38]
Kamal-Sourour	$d α d t = k α m (1 - α) n$	[35]
Leroy	$d α d t = (A 1 + A 2 α m) (1 - α) n e - E a / R T$	[27,39]
Rafei	$α = k n k n + t n$	[40]
Yeoh	$d α d t = (α 1 - α) 1 / p (1 k) 1 / p e x p (1 R T) 1 / p$	[41]
Jia	$α i = α i - 1 + k 0 Δ t e x p - E a R T 1 + k 0 Δ t e x p - E a R T$	[42]

Fig.6 Diagram of vulcanization of cylindrical rubber section at different times

References 76

1	Trautner S, Lackner J, Spendelhofer W, et al. Quantification of the vulcanizing system of rubber in industrial tire rubber production by laser-induced breakdown spectroscopy (LIBS)[J]. Anal. Chem., 2019, 91(8): 5200-5206.
2	Qu L, Huang G, Wu J, et al. Damping mechanism of chlorobutyl rubber and phenolic resin vulcanized blends[J]. J. Mater. Sci., 2007, 42(17): 7256-7262.
3	Xiang C, Guo F, Liu X, et al. Numerical algorithm for fluid–solid coupling in reciprocating rod seals[J]. Tribo. Int., 2020, 143:106078.
4	Passador F R, Rodolfo A, Pessan L A, et al. In situ dynamic vulcanization of poly(vinyl chloride)/acrylonitrile-butadiene rubber blends[J]. J. Macromo. Sci. B., 2009, 48(2): 282-298.
5	Nakao T, Kohjiya S. Computer simulation of network formation in natural rubber (NR)[M]//Kohjiya S. Chemistry, Manufacture and Applications of Natural Rubber. London: Woodhead Publishing, 2014: 216-246.
6	Huang Y, Wang Z, Gao L, et al. The deterioration of foamed silicone rubber in humid and hot environments[J]. J. Cell. Plast., 2019, 55(6): 633-654.
7	Wei Y, Wu H, Weng G, et al. Effect of interface on bulk polymer: control of glass transition temperature of rubber[J]. J. Polym. Res., 2018, 25(8):173.
8	Ghoreishy M H R. A state-of-the-art review on the mathematical modeling and computer simulation of rubber vulcanization process[J]. Iran. Polym. J. Eng. Edu., 2016, 25(1): 89-109.
9	Milani G, Milani F. Parabola-hyperbola P-H kinetic model for NR sulphur vulcanization[J]. Polym. Test., 2017, 58:104-115.
10	戈明亮, 阚长华, 易玉华, 等. 橡胶硫化反应动力学研究概况[J]. 橡胶工业, 2004, 51(10):631-635.
	Ge M L, Kan C H, Yi Y H, et al. A survey of the kinetics of rubber vulcanization[J]. China Rubber Industry, 2004, 51(10):631-635.
11	Nozu S, Tsuji H, Itadani M, et al. Study of cure process of thick solid rubber[J]. J. Mater. Process. Technol., 2008, 201(1/2/3): 720-724.
12	Coran A Y. Chemistry of the vulcanization and protection of elastomers: a review of the achievements[J]. J. Appl. Polym. Sci., 2003, 87(1): 24-30.
13	Coran A Y. Vulcanization（Part VI）： A model and treatment for scorch delay kinetics[J]. Rubber. Chem. Technol., 1964, 37(3): 689-697.
14	Mitraa A, Leonov A I. On rheology, cure kinetics and chemorheology of gum rubbers[J]. Polym. Sci. Ser. A Chem. Phys., 2010, 52(11): 1114-1123.
15	Espósito L H, Marzocca A J. Effect of electron-beam irradiation on the thermal vulcanization of a natural rubber compound[J]. J. Appl. Polym. Sci., 2018, 136(13): 1-12.
16	Ding R, Leonov A I. A kinetic model for sulfur accelerated vulcanization of a natural rubber compound[J]. J. Appl. Polym. Sci., 1996, 61(3): 455-463.
17	Ding R, Leonov A I, Coran A Y. A study of the vulcanization kinetics of an accelerated-sulfur SBR compound[J]. Rubber Chem. Technol., 1996, 69(1): 81-91.
18	杨昌金, 罗勇悦, 陈帮乾, 等. 环氧化天然橡胶硫化反应动力学参数的拟合 [J]. 广州化工, 2014, 42(9): 42-44.
	Yang C J, Luo Y Y, Chen B Q, et al. Fitting of cure reaction kinetics parameters based on epoxidized natural rubber[J]. Guangzhou Chemical Industry, 2014, 42(9): 42-44.
19	Fan R L, Zhang Y, Huang C, et al. Simulation and verification for sulfur accelerated vulcanization of gum natural rubber compound[J]. Rubber Chem. Technol., 2002, 75(2): 287-297.
20	龚蓬, 张祥福, 张隐西. NR硫化返原动力学研究[J]. 橡胶工业, 1997, 44(4): 195-200.
	Gong P, Zhang X F, Zhang Y X. Kinetic study on the reversion of NR vulcanization[J]. China Rubber Industry, 1997, 44(4):195-200.
21	黄琛, 范汝良, 张勇, 等. NR硫化返原过程的动力学研究[J]. 橡胶工业, 2000, 47(4): 195-200.
	Huang C, Fan R L, Zhang Y, et al. Kinetic study on the reversion process of NR vulcanization[J]. China Rubber Industry, 2000, 47(4): 195-200.
22	Jeong J H, Moon C W, Leonov A I, et al. Cure kinetics for silane coupled silica filled SBR compounds[J]. Rubber Chem. Technol., 2002, 75(1): 93-109.
23	Han I S, Chung C B, Kang S J, et al. A kinetic model of reversion type cure for rubber compounds[J]. Polymer (Korea), 1998, 22(2): 223-230.
24	Han I S, Chung C B, Lee J W. Optimal curing of rubber compounds with reversiontype cure behavior[J]. Rubber Chem. Technol., 2000, 73(1): 101-113.
25	Sun X, Isayev A I. Cure kinetics study of unfilled and carbon black filled synthetic isoprene rubber[J]. Rubber Chem. Technol., 2009, 82(2):149-169.
26	Milani G, Milani F. Iterative robust numerical procedure for the determination of kinetic constants in Han􀆳s model for NR cured with sulphur[J]. J. Math. Chem., 2015, 53(6): 1363-1379.
27	Leroy E, Souid A, Deterre R. A continuous kinetic model of rubber vulcanization predicting induction and reversion[J]. Polym. Test., 2013, 32(3): 575-582.
28	Milani G, Hanel T, Donetti R, et al. Combined experimental and numerical kinetic characterization of NR vulcanized with sulphur, N-terbutyl, 2 benzothiazylsulphenamide and N,N-diphenyl guanidine[C]//Cueto E, Chinesta F. International Conference on Numerical Analysis and Applied Mathematics. New York, America: American Institute of Physics, 2016, 1738: 480014.
29	Milani G, Leroy E, Milani F, et al. Mechanistic modeling of reversion phenomenon in sulphur cured natural rubber vulcanization kinetics[J]. Polym. Test., 2013, 32(6): 1052-1063.
30	Milani G, Milani F. EPDM accelerated sulfur vulcanization: a kinetic model based on a genetic algorithm[J]. J. Math. Chem., 2011, 49(7): 1357-1383.
31	Milani G, Hanel T, Donetti R, et al. A closed form solution for the vulcanization prediction of NR cured with sulphur and different accelerators[J]. J. Math. Chem., 2015, 53(4): 975-997.
32	Ghosh P, Katare S, Patkar P, et al. Sulfur vulcanization of natural rubber for benzothiazole accelerated formulations: from reaction mechanisms to a rational kinetic model[J]. Rubber Chem. Technol., 2003, 76(3): 592-693.
33	Likozar B, Krajnc M. Kinetic and heat transfer modeling of rubber blends􀆳 sulfur vulcanization with N-t-butylbenzothiazole-sulfenamide and N,N-di-t-butylbenzothiazole-sulfenamide[J]. J. Appl. Polym. Sci., 2007, 103(1):293-307.
34	Barghamadi M, Ghoreishy M H R, Karrabi M, et al. Investigation on the kinetics of cure reaction of acrylonitrile-butadiene rubber (NBR)/polyvinyl chloride (PVC)/graphene nanocomposite using various models[J]. J. Appl. Polym. Sci., 2020, 137(18): 48632.
35	Kamal M R, Sourour S. Kinetics and thermal characterization of thermoset cure[J]. Polym. Eng. Sci., 1973, 13(1): 59-64.
36	Isayev A I, Deng J S. Nonisothermal vulcanization of rubber compounds[J]. Rubber Chem. Technol., 1988, 61(2): 340-361.
37	Isayev A I, Sobhanie M, Deng J S. Two-dimensional simulation of injection molding of rubber compounds[J]. Rubber Chem. Technol., 1988, 61(5): 906-937.
38	Kissinger H E. Reaction kinetics in differential thermal analysis[J]. Anal. Chem., 1957, 29(11): 1702-1706.
39	Leroy E, Souid A, Sarda A, et al. A knowledge based approach for elastomer cure kinetic parameters estimation[J]. Polym. Test., 2013, 32(1): 9-14.
40	Rafei M, Ghoreishy M H R, Naderi G. Development of an advanced computer simulation technique for the modeling of rubber curing process[J]. Comp. Mater. Sci., 2009, 47(2): 539-547.
41	Yeoh O H. Mathematical modeling of vulcanization characteristics[J]. Rubber Chem. Technol., 2012, 85(3): 482-492.
42	贾玉玺, 孙胜, 季忠, 等. 硅橡胶硫化反应场的数值模拟[J]. 化学学报, 2002, 60(8): 1368-1373.
	Jia Y X, Sun S, Ji Z, et al. Numerical simulation of vulcanization field of silicone rubber[J]. Acta Chimica Sinica, 2002,60(8): 1368-1373.
43	Tw C, Gd S, Al I. Reduced time approach to curing kinetics(Part I): Dynamic rate and master curve from isothermal data[J]. Rubber Chem. Technol., 1994, 66(94): 849-864.
44	Šesták J, Berggren G. Study of the kinetics of the mechanism of solid-state reactions at increasing temperatures[J]. Thermochim. Acta, 1971, 3(1): 1-12.
45	李咏金. 橡胶硫化曲线变化的数学模型拟合(Ⅰ): 有返原现象的硫化曲线[J]. 合成橡胶工业, 1995, 18(1): 29-31.
	Li Y J. A mathematical model of rubber vulcanization curve change (Ⅰ): Vulcanization curve for reversion characteristic[J]. China Synthetic Rubber Industry, 1995, 18(1): 29-31.
46	李咏金. 橡胶硫化曲线变化的数学模型拟合(Ⅱ): 转矩上升的硫化曲线[J]. 合成橡胶工业, 1995, 18(3): 168-170.
	Li Y J. A mathematical model of rubber vulcanization curve change(Ⅱ): Vulcanization curve for rotative moment always rising [J]. China Synthetic Rubber Industry, 1995, 18(3): 168-170.
47	李咏金. 橡胶硫化曲线变化的数学模型拟合(Ⅲ): 平坦型的硫化曲线[J]. 合成橡胶工业, 1997, 20(2): 108-111.
	Li Y J. A mathematical model of rubber vulcanization curve change (Ⅲ): Vulcanization curve for smooth type[J]. China Synthetic Rubber Industry, 1997, 20(2): 108-111.
48	Russell R, Smith D A, Welding G N. Kinetics of thiazole-accelerated sulfur vulcanization of natural rubber[J]. Rubber Chem. Technol., 1963, 36(3):835-843.
49	Ambelang J C, Prentice G A. Digital method of calculating the flow of heat through a tire during vulcanization[J]. Rubber Chem. Technol., 1972, 45(5): 1195-1201.
50	Prentice G A, Williams M C. Numerical evaluation of the state of cure in a vulcanizing rubber article[J]. Rubber Chem. Technol., 1980, 53(5): 1023-1031.
51	Abdul M, Rochette B, Sadr A, et al. Effect of variation in the values of the thermal properties of rubber on the vulcanization of thin and thick sheets[J]. Thermochim. Acta, 1984, 80(2): 287-296.
52	Chataing G, Chater M, Vergnaud J M. Determination of heat flux and profiles of temperature and state of cure in isothermal calorimetry with reaction of high enthalpy[J]. Thermochim. Acta, 1985, 89: 107-118.
53	Khouider A, Bouzon J, Vergnaud J M. Modelization of the DSC technique by considering heat transfer and kinetics of reaction in the case of rubber cure[J]. Thermochim. Acta, 1986, 98: 285-299.
54	Khouider A, Vergnaud J M. Effect of temperature of motionless air on the cure of vulcanizates after removal from the mold[J]. J. Appl. Polym. Sci., 1986, 32(6): 5301-5313.
55	贾玉玺, 孙胜, 刘莉莉, 等. 模具硅橡胶材料热硫化工艺的有限元模拟[J]. 化工学报, 2003, 54(9): 1300-1304.
	Jia Y X, Sun S, Liu L L, et al. Finite element simulation of hot vulcanization process of moldable silicone rubber material[J].Journal of Chemical Industry and Engineering (China), 2003, 54(9): 1300-1304.
56	王雪峰. 轮胎硫化过程的模拟和无探针微机控制[D]. 北京: 清华大学, 1989.
	Wang X F. Simulation of tire curing process and microprobe free microcomputer control[D]. Beijing: Tsinghua University, 1989.
57	Marzocca A J. Finite element analysis of cure in a rubber cylinder[J]. Polymer, 1991, 32(8): 1456-1460.
58	刘柏兵, 温泰斗, 卜继玲, 等. 仿真分析技术在橡胶弹簧硫化时间设计中的应用[J]. 橡胶工业, 2018, 65(9): 965-970.
	Liu B B, Wen T D, Pu J L, et al. Application of simulation analysis technology in the design of vulcanization time of rubber spring[J]. China Rubber Industry, 2018, 65(9): 965-970.
59	杨俊, 陈平, 付金伦, 等. 叠层橡胶隔震支座硫化工艺有限元仿真与试验研究[J]. 橡胶工业, 2018, 65(11): 1278-1283.
	Yang J, Chen P, Fu J L, et al. Finite element simulation and experimental study on vulcanization process of laminated rubber bearing[J]. China Rubber Industry, 2000, 65(11): 1278-1283.
60	张金云, 刘肖英, 王伯刚, 等. 轮胎硫化温度场数值模拟[J]. 橡塑技术与装备, 2019, 45(11): 30-36.
	Zhang J Y, Liu X Y, Wang B G, et al. Numerical simulation of tire vulcanization temperature field[J]. China Rubber/Plastics Technology and Equipment, 2019, 45(11): 30-36.
61	Han I S, Chung C B, Kim J H, et al. Dynamic simulation of the tire curing process[J]. Tire Science and Technology, 1996, 24(1): 50-76.
62	Tong J, Yan X. Finite element analysis of tire curing process[J]. J. Reinf. Plast. Compos., 2003, 22(11): 983-1002.
63	Yan X. A numerical modeling of dynamic curing process of tire by finite element[J]. Polym. J., 2007, 39(10): 1001-1010.
64	El Labban A, Mousseau P, Bailleul J L. et al. Numerical natural rubber curing simulation, obtaining a controlled gradient of the state of cure in a thick-section part[C]//Cueto E, Chinesta F. 10th ESAFORM Conference on Material Forming. New York, America: American Institute of Physics, 2007, 907: 921-926.
65	Wu J, Su B, Liu Q, et al. Research on vulcanization process simulation of butyl rubber based on a new characterization model of curing degree[J]. Prog. Rubber Plast. Recycling Technol., 2014, 30(4): 237-248.
66	Ghoreishy M H R, Rafei M, Naderi G. Optimization of the vulcanization process of a thick rubber article using an advanced computer simulation technique[J]. Rubber Chem. Technol., 2012, 85(4): 576-589.
67	Zhang J, Tang W X, Zhao X L, et al. Simulation of rubber piston curing process based on hybrid kinetic model[J]. Mater. Sci. Tech-Lond., 2012, 20(2): 16-22.
68	Su B, Wu J, Cui Z, et al. Modeling of truck tire curing process by an experimental and numerical method[J]. Iran. Polym. J. Eng. Edu., 2015, 24(7): 583-593.
69	Vafayan M, Ghoreishy M H R, Abedini H, et al. Development of an optimized thermal cure cycle for a complex-shape composite part using a coupled finite element/genetic algorithm technique[J]. Iran. Polym. J. Eng. Edu., 2015, 24(6): 459-469.
70	Shi F, Dong X. Three-dimension numerical simulation for vulcanization process based on unstructured tetrahedron mesh[J]. J. Manuf. Process., 2016, 22: 1-6.
71	Gough J. Calculation of times and temperatures for press vulcanization of thick rubber pads[J]. Rubber Chem. Technol., 2017, 90(1): 89-107.
72	Xu X, Zhou Q, Song N, et al. Kinetic analysis of isothermal curing of unsaturated polyester resin catalyzed with tert-butyl peroxybenzoate and cobalt octoate by differential scanning calorimetry[J]. J. Therm. Anal. Calorim., 2017, 129(2): 843-850.
73	Lucio B, De La Fuente J L. Kinetic and chemorheological modelling of the polymerization of 2,4-toluenediisocyanate and ferrocene-functionalized hydroxyl-terminated polybutadiene[J]. Polymer, 2018, 140: 290-303.
74	Yeo H. Curing kinetics of liquid crystalline 4,4’-diglycidyloxybiphenyl epoxy cured with 4,4’-diaminodiphenylsulfone[J]. Polymer, 2018, 159: 6-11.
75	Rabiei S, Shojaei A. Vulcanization kinetics and reversion behavior of natural rubber/styrene-butadiene rubber blend filled with nanodiamond — the role of sulfur curing system[J]. Eur. Polym. J., 2016, 81: 98-113.
76	Jin K, Heath W H, Torkelson J M. Kinetics of multifunctional thiol-epoxy click reactions studied by differential scanning calorimetry: effects of catalysis and functionality[J]. Polymer, 2015, 81: 70-78.

[1]	Zhanyu YE, He SHAN, Zhenyuan XU. Performance simulation of paper folding-like evaporator for solar evaporation systems [J]. CIESC Journal, 2023, 74(S1): 132-140.
[2]	Yifei ZHANG, Fangchen LIU, Shuangxing ZHANG, Wenjing DU. Performance analysis of printed circuit heat exchanger for supercritical carbon dioxide [J]. CIESC Journal, 2023, 74(S1): 183-190.
[3]	Zhiguo WANG, Meng XUE, Yushuang DONG, Tianzhen ZHANG, Xiaokai QIN, Qiang HAN. Numerical simulation and analysis of geothermal rock mass heat flow coupling based on fracture roughness characterization method [J]. CIESC Journal, 2023, 74(S1): 223-234.
[4]	Jiahao SONG, Wen WANG. Study on coupling operation characteristics of Stirling engine and high temperature heat pipe [J]. CIESC Journal, 2023, 74(S1): 287-294.
[5]	Siyu ZHANG, Yonggao YIN, Pengqi JIA, Wei YE. Study on seasonal thermal energy storage characteristics of double U-shaped buried pipe group [J]. CIESC Journal, 2023, 74(S1): 295-301.
[6]	Cheng CHENG, Zhongdi DUAN, Haoran SUN, Haitao HU, Hongxiang XUE. Lattice Boltzmann simulation of surface microstructure effect on crystallization fouling [J]. CIESC Journal, 2023, 74(S1): 74-86.
[7]	Song HE, Qiaomai LIU, Guangshuo XIE, Simin WANG, Juan XIAO. Two-phase flow simulation and surrogate-assisted optimization of gas film drag reduction in high-concentration coal-water slurry pipeline [J]. CIESC Journal, 2023, 74(9): 3766-3774.
[8]	Lei XING, Chunyu MIAO, Minghu JIANG, Lixin ZHAO, Xinya LI. Optimal design and performance analysis of downhole micro gas-liquid hydrocyclone [J]. CIESC Journal, 2023, 74(8): 3394-3406.
[9]	Linzheng WANG, Yubing LU, Ruizhi ZHANG, Yonghao LUO. Analysis on thermal oxidation characteristics of VOCs based on molecular dynamics simulation [J]. CIESC Journal, 2023, 74(8): 3242-3255.
[10]	Xiaosong CHENG, Yonggao YIN, Chunwen CHE. Performance comparison of different working pairs on a liquid desiccant dehumidification system with vacuum regeneration [J]. CIESC Journal, 2023, 74(8): 3494-3501.
[11]	Wenzhu LIU, Heming YUN, Baoxue WANG, Mingzhe HU, Chonglong ZHONG. Research on topology optimization of microchannel based on field synergy and entransy dissipation [J]. CIESC Journal, 2023, 74(8): 3329-3341.
[12]	Rui HONG, Baoqiang YUAN, Wenjing DU. Analysis on mechanism of heat transfer deterioration of supercritical carbon dioxide in vertical upward tube [J]. CIESC Journal, 2023, 74(8): 3309-3319.
[13]	Chen HAN, Youmin SITU, Bin ZHU, Jianliang XU, Xiaolei GUO, Haifeng LIU. Study of reaction and flow characteristics in multi-nozzle pulverized coal gasifier with co-processing of wastewater [J]. CIESC Journal, 2023, 74(8): 3266-3278.
[14]	Jie LIU, Lisheng WU, Jinjin LI, Zhenghong LUO, Yinning ZHOU. Preparation and properties of polyether-based vinylogous urethane reversible crosslinked polymers [J]. CIESC Journal, 2023, 74(7): 3051-3057.
[15]	Mengmeng ZHANG, Dong YAN, Yongfeng SHEN, Wencui LI. Effect of electrolyte types on the storage behaviors of anions and cations for dual-ion batteries [J]. CIESC Journal, 2023, 74(7): 3116-3126.