聚合物流体微观和介观尺度下的模拟研究
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摘要
粘度是聚合物流体的一个重要表征参数,也是聚合物加工成型性能的重要影响因素。在进行传统的聚合物流动有限元模拟过程中,获得揭示材料物性的本构方程(粘度函数)已经成为了模拟的关键。通过微观模拟或介观模拟的方法来获得一定条件下的应力应变关系,从而避免封闭本构方程是实现复杂流体模拟的重要方向。聚合物流体由于分子链的长链结构,尺度跨度比较大,单尺度模拟往往难以实现。
在本文中,我们采用了将微观的分子动力学模拟和介观的耗散粒子动力学模拟相结合的方法进行了聚乙烯熔体的非平衡态模拟,考察了在我们建立的模拟体系中的粘度随剪切速率的变化情况。本文的主要工作如下：
1)建立了450K、latm下40条链长为96的聚乙烯体系的分子模型,并进行了分子动力学的模拟。通过径向分布函数、均方回转半径、密度、以及均方位移来对聚乙烯模型进行表征,获得平衡态下该模型结构和动力学信息。经过与实验结果的对比分析,发现在所给的条件下,所建立的分子模型及模拟方法是合理的。
2)采用基于径向分布函数的方法建立了耗散粒子动力学模拟保守势能,实现了从分子动力学模拟到耗散粒子动力学模拟的贯通连接。考察了几个关键模拟参数对模拟结果的影响,将DPD模拟结果与分子动力学模拟结果进行对比,选择了合适的参数作为进一步模拟的参数。
3)建立了平板拖拽流和反转泊萧叶流两种不同的模型对聚合物流体的流动进行了模拟,通过计算得到了流动过程中的速度分布、剪切应力的分布以及剪切粘度,并将所得的结果与理论分析的结果进行了对比分析,发现两种模型的速度分布以及剪切应力的分布都具有与理论分析的结果相同的变化趋势。
Viscosity is one of the most important characteristics of the polymer fluids, and it is also a key influence factor of polymer material processing.Getting the constitutive equation that reveals material property has become the key to caryy out polymer processing simulation with traditional finite element simulation. It has become an important reserch direction to obtain the stress-strain relationship of complex fluid under certain conditions through the microscopic simulation or mesoscopic simulation method, so as to avoid the constitutive equation.
The simulation of polymer fluid is often difficult to achieve only through single scaling simulation,due to its long chains of chain structure and big scale span.
In this paper,we conduct a nonequilibrium simulation about polymer fluid through a connection of microscale molecular dynamic simulation and mesoscale dissipative particle dynamics simulation.And we focus on the viscosity in our simulation mold. The main work in this paper are as follows:
1) We set up the molecular simulation model of polyethylene, and implement the molecular dynamics simulation with the software lammps. Structure information, thennodynamic properties, and transport properties are analyzed in detail under the condition of450K,1atm,with the system of96PE chains,one of which contain96union atoms. The radial distribution function, mean square rotary radius, density, and orientation of polyethylene are used to characterize the model.
2) Through the method that based on the radial distribution function,we set up DPD simulation of conservative potential energy, and realizes the connection between molecular dynamics simulation and dissipation particle dynamics simulation. Several key parameters have been investigate to obtain the proper simulation parameters through comparing with the result of MD simulation.
3) We set up two kinds of different models of polymer fluid flow, and calculated the velocity distribution, shear stress distribution and shear viscosity in the process of flow, then the result with the simulation are compared with results of the theoretical analysis, which show that both of the models indicate the same variation tendency in velocity distribution and stress distribution.
在本文中,我们采用了将微观的分子动力学模拟和介观的耗散粒子动力学模拟相结合的方法进行了聚乙烯熔体的非平衡态模拟,考察了在我们建立的模拟体系中的粘度随剪切速率的变化情况。本文的主要工作如下：
1)建立了450K、latm下40条链长为96的聚乙烯体系的分子模型,并进行了分子动力学的模拟。通过径向分布函数、均方回转半径、密度、以及均方位移来对聚乙烯模型进行表征,获得平衡态下该模型结构和动力学信息。经过与实验结果的对比分析,发现在所给的条件下,所建立的分子模型及模拟方法是合理的。
2)采用基于径向分布函数的方法建立了耗散粒子动力学模拟保守势能,实现了从分子动力学模拟到耗散粒子动力学模拟的贯通连接。考察了几个关键模拟参数对模拟结果的影响,将DPD模拟结果与分子动力学模拟结果进行对比,选择了合适的参数作为进一步模拟的参数。
3)建立了平板拖拽流和反转泊萧叶流两种不同的模型对聚合物流体的流动进行了模拟,通过计算得到了流动过程中的速度分布、剪切应力的分布以及剪切粘度,并将所得的结果与理论分析的结果进行了对比分析,发现两种模型的速度分布以及剪切应力的分布都具有与理论分析的结果相同的变化趋势。
Viscosity is one of the most important characteristics of the polymer fluids, and it is also a key influence factor of polymer material processing.Getting the constitutive equation that reveals material property has become the key to caryy out polymer processing simulation with traditional finite element simulation. It has become an important reserch direction to obtain the stress-strain relationship of complex fluid under certain conditions through the microscopic simulation or mesoscopic simulation method, so as to avoid the constitutive equation.
The simulation of polymer fluid is often difficult to achieve only through single scaling simulation,due to its long chains of chain structure and big scale span.
In this paper,we conduct a nonequilibrium simulation about polymer fluid through a connection of microscale molecular dynamic simulation and mesoscale dissipative particle dynamics simulation.And we focus on the viscosity in our simulation mold. The main work in this paper are as follows:
1) We set up the molecular simulation model of polyethylene, and implement the molecular dynamics simulation with the software lammps. Structure information, thennodynamic properties, and transport properties are analyzed in detail under the condition of450K,1atm,with the system of96PE chains,one of which contain96union atoms. The radial distribution function, mean square rotary radius, density, and orientation of polyethylene are used to characterize the model.
2) Through the method that based on the radial distribution function,we set up DPD simulation of conservative potential energy, and realizes the connection between molecular dynamics simulation and dissipation particle dynamics simulation. Several key parameters have been investigate to obtain the proper simulation parameters through comparing with the result of MD simulation.
3) We set up two kinds of different models of polymer fluid flow, and calculated the velocity distribution, shear stress distribution and shear viscosity in the process of flow, then the result with the simulation are compared with results of the theoretical analysis, which show that both of the models indicate the same variation tendency in velocity distribution and stress distribution.
引文
[1]Allen M P, Tildesley D J. Computer simulation of liquids [M]. Oxford university press, 1989.
[2]Frenkel D, Smit B. Understanding molecular simulation:from algorithms to applications [M]. Academic Press, Inc.,1996.
[3]Leach A, Kier L B. Molecular modeling:principles and applications [M]. Prentice Hall, 2001.
[4]施良利,胡汉杰,高分子.高分子科学的今天与明天[M].化学工业山版社,1994.
[5]刘欣,顾宜.高分子科学中的计算机模拟[J].高分子材料科学与工程,2000,16(6)：28-31.
[6]杨小震.高分子的计算机模拟研究进展[J].计算机与应用化学,1999,16(5)：321-324.
[7]He Y D, Qian H J, Lu Z Y, et al. Polymer translocation through a nanopore in mesoscopic simulations [J]. Polymer,2007,48(12):3601-3606.
[8]Liu H, Xue Y H, Qian H J, et al. A practical method to avoid bond crossing in two-dimensional dissipative particle dynamics simulations [J]. The Journal of Chemical Physics,2008,129(2):024902-1-024902-8.
[9]Qian H J, Chen L J, Lu Z Y, et al. Surface diffusion dynamics of a single polymer chain in dilute solution [J]. Physical Review Letters,2007,99(6):68301-68304.
[10]Wilson S. Chemistry by computer:An overview of the applications of computers in chemistry [M]. Kluwer Academic Publishers,1986.
[11]Liu T W. Flexible polymer chain dynamics and rheological properties in steady flows [J]. The Journal of Chemical Physics,1989,90(10):5826-5842.
[12]Doyle P S, Shaqfeh E S G, Gast A P. Dynamic simulation of freely draining flexible polymers in steady linear flows [J]. Journal of Fluid Mechanics,1997,334(1):251-291.
[13]Grassia P, Hinch E. Computer simulations of polymer chain relaxation via Brownian motion [J]. Journal of Fluid Mechanics,1996,308(1):255-288.
[14]Zylka W, Ottinger H C. A comparison between simulations and various approximations for Hookean dumbbells with hydrodynamic interaction [J]. The Journal of Chemical Physics, 1989,90(1):474-480.
[15]Chen S, Doolen G D. Lattice Boltzmann method for fluid flows [J]. Annual review of fluid mechanics,1998,30(1):329-364.
[16]Owens R G, Phillips T N. Computational rheology [M]. Imperial College Press,2002.
[17]Reith D, Meyer H, Muller-Plathe F. Mapping atomistic to coarse-grained polymer models using automatic simplex optimization to fit structural properties [J]. Macromolecules,2001, 34(7):2335-2345.
[18]Meyer H, Biennann O, Faller R, et al. Coarse graining of nonbonded inter-particle potentials using automatic simplex optimization to fit structural properties [J]. The Journal of Chemical Physics,2000,113(15):6264-6275.
[19]Lyubartsev A P, Laaksonen A. Calculation of effective interaction potentials from radial distribution functions:A reverse Monte Carlo approach [J]. Physical Review E,1995,52(4): 3730-3737.
[20]郭洪霞.高分子粗粒化分子动力学模拟进展[J].高分子通报,2011,10)：154-163.
[21]Kremer K, Grest G S, Carmesin I. Crossover from Rouse to reptation dynamics:A molecular-dynamics simulation [J]. Physical Review Letters,1988,61(5):566-569.
[22]Carmesin I, Kremer K. Static and dynamic properties of two-dimensional polymer melts [J]. Journal de Physique,1990,51(10):915-932.
[23]Duering E R, Kremer K, Grest G S. Dynamics of model networks:the role of the melt entanglement length [J]. Macromolecules,1993,26(12):3241-3244.
[24]Schlijper A, Hoogerbrugge P, Manke C. Computer simulation of dilute polymer solutions with the dissipative particle dynamics method [J]. Journal of Rheology,1995, 39(3):567-579.
[25]Kong Y, Manke C, Madden W, et al. Effect of solvent quality on the conformation and relaxation of polymers via dissipative particle dynamics [J]. The Journal of Chemical Physics, 1997,107(2):592-602.
[26]Kong Y, Manke C, Madden W, et al. Modeling the rheology of polymer solutions by dissipative particle dynamics [J]. Tribology Letters,1997,3(1):133-138.
[27]Groot R D, Warren P B. Dissipative particle dynamics:Bridging the gap between atomistic and mesoscopic simulation [J]. The Journal of Chemical Physics,1997,107(11):4423-4435.
[28]Groot R D, Madden T J, Tildesley D J. On the role of hydrodynamic interactions in block copolymer microphase separation [J]. The Journal of Chemical Physics,1999,110(19)礴9739-9749.
[29]Ten Bosch B. On an extension of Dissipative Particle Dynamics for viscoelastic flow modelling [J]. Journal of Non-Newtonian Fluid Mechanics,1999,83(3):231-248.
[30]Spenley N. Scaling laws for polymers in dissipative particle dynamics [J]. EPL (Europhysics Letters),2000,49(4):534-540.
[31]Van Der Vegt N F A, Peter C, Kremer K. Structure-Based Coarse-and Fine-Graining in Soft Matter Simulations [M]. CRC Press,2008.
[32]Verlet L. Computer" experiments" on classical fluids. I. Thermodynamical properties of Lennard-Jones molecules [J]. Physical Review,1967,159(1):98-103.
[33]Gear C W. Numerical initial value problems in ordinary differential equations [M]. Prentice HallPTR,1971.
[34]Hockney R W. Potential calculation and some application [J]. Methods Comput Phys, 1970,9(1):135-211.
[35]Koelman J, Hoogerbrugge P. Dynamic simulations of hard-sphere suspensions under steady shear [J]. EPL (Europhysics Letters),1993,21(3):363-368.
[36]Hoogerbrugge P, Koelman J. Simulating microscopic hydrodynamic phenomena with dissipative particle dynamics [J]. EPL (Europhysics Letters),1992,19(3):155-160.
[37]Shardlow T. Splitting for dissipative particle dynamics [J]. SIAM Journal on Scientific Computing,2003,24(4):1267-1282.
[38]Lowe C. An alternative approach to dissipative particle dynamics [J]. EPL (Europhysics Letters),1999,47(2):145-151.
[39]Van Swygenhoven H, Farkas D, Caro A. Grain-boundary structures in polycrystalline metals at the nanoscale [J]. Physical Review B,2000,62(2):831-838.
[40]Chen D. Structural modeling of nanocrystalline materials [J]. Computational materials science,1995,3(3):327-333.
[41]曾凡林,孙毅,周玉.有机材料的分子模拟模型(Ⅰ)：单分子链的构建[J].哈尔滨工业大学学报,2009,010：95-99.
[42]Soper A. Empirical potential Monte Carlo simulation of fluid structure [J]. Chemical physics, 1996,202(2-3):295-306.
[43]Lees A, Edwards S. The computer study of transport processes under extreme conditions [J]. Journal of Physics C:Solid State Physics,1972,5(15):1921-1929.
[44]Evans D J, Morriss G P. Statistical mechanics of nonequilibrium liquids [M]. Cambridge Univ Pr,2008.
[45]Fan X, Phan-Thien N, Chen S, et al. Simulating flow of DNA suspension using dissipative particle dynamics [J]. Physics of Fluids,2006,18(6):063102-1-063102-10.
[46]Kong Y, Manke C, Madden W, et al. Simulation of a confined polymer in solution using the dissipative particle dynamics method [J]. International journal of thermophysics,1994,15(6): 1093-1101.
[47]Revenga M, Zuniga I, Espanol P. Boundary model in DPD [J]. Int J Mod Phys,1998, 9(8):1319-1328.
[48]Karayiannis N C, Giannousaki A E, Mavrantzas V G, et al. Atomistic Monte Carlo simulation of strictly monodisperse long polyethylene melts through a generalized chain bridging algorithm [J]. The Journal of Chemical Physics,2002,117(11):5465-5479.
[49]Guerrault X, Rousseau B, Farago J. Dissipative particle dynamics simulations of polymer melts. I. Building potential of mean force for polyethylene and cis-polybutadiene [J]. The Journal of Chemical Physics,2004,121(13):65386546.
[50]Ashbaugh H S, Patel H A, Kumar S K, et al. Mesoscale model of polymer melt structure: Self-consistent mapping of molecular correlations to coarse-grained potentials [J]. The Journal of Chemical Physics,2005,122(10):104908-1-104908-5.
[51]陈正隆,徐为人,汤立达.分子模拟的理论与实践[M].化学工业出版社,2007.
[52]Lahmar F, Rousseau B. Influence of the adjustable parameters of the DPD on the global and local dynamics of a polymer melt [J]. Polymer,2007,48(12):3584-3592.
[53]吴其晔,巫静安.高分子材料流变学[M].高等教育出版社,2002.
[54]Heinz H, Paul W, Binder K. Calculation of local pressure tensors in systems with many-body simulation of polydisperse linear polyethylene melts [J]. Macromolecules,1998,31(22): 7934-7943.
[56]Depa P K, Maranas J K. Dynamic evolution in coarse-grained molecular dynamics simulations of polyethylene melts [J]. Journal of Chemical Physics,2007,126(5): 54903-1-54903-8.
[57]周志平,颜德岳.关于聚乙烯分子链均方回转半径的进一步讨论[J].化学学报,1992,50(4)：313-319.
[58]闵志宇,曹伟,中长雨等.FENE珠-簧链聚合物分子模型流变性质Brown动力学模拟[J].高分子材料与工程,25(2)：171-174.
[2]Frenkel D, Smit B. Understanding molecular simulation:from algorithms to applications [M]. Academic Press, Inc.,1996.
[3]Leach A, Kier L B. Molecular modeling:principles and applications [M]. Prentice Hall, 2001.
[4]施良利,胡汉杰,高分子.高分子科学的今天与明天[M].化学工业山版社,1994.
[5]刘欣,顾宜.高分子科学中的计算机模拟[J].高分子材料科学与工程,2000,16(6)：28-31.
[6]杨小震.高分子的计算机模拟研究进展[J].计算机与应用化学,1999,16(5)：321-324.
[7]He Y D, Qian H J, Lu Z Y, et al. Polymer translocation through a nanopore in mesoscopic simulations [J]. Polymer,2007,48(12):3601-3606.
[8]Liu H, Xue Y H, Qian H J, et al. A practical method to avoid bond crossing in two-dimensional dissipative particle dynamics simulations [J]. The Journal of Chemical Physics,2008,129(2):024902-1-024902-8.
[9]Qian H J, Chen L J, Lu Z Y, et al. Surface diffusion dynamics of a single polymer chain in dilute solution [J]. Physical Review Letters,2007,99(6):68301-68304.
[10]Wilson S. Chemistry by computer:An overview of the applications of computers in chemistry [M]. Kluwer Academic Publishers,1986.
[11]Liu T W. Flexible polymer chain dynamics and rheological properties in steady flows [J]. The Journal of Chemical Physics,1989,90(10):5826-5842.
[12]Doyle P S, Shaqfeh E S G, Gast A P. Dynamic simulation of freely draining flexible polymers in steady linear flows [J]. Journal of Fluid Mechanics,1997,334(1):251-291.
[13]Grassia P, Hinch E. Computer simulations of polymer chain relaxation via Brownian motion [J]. Journal of Fluid Mechanics,1996,308(1):255-288.
[14]Zylka W, Ottinger H C. A comparison between simulations and various approximations for Hookean dumbbells with hydrodynamic interaction [J]. The Journal of Chemical Physics, 1989,90(1):474-480.
[15]Chen S, Doolen G D. Lattice Boltzmann method for fluid flows [J]. Annual review of fluid mechanics,1998,30(1):329-364.
[16]Owens R G, Phillips T N. Computational rheology [M]. Imperial College Press,2002.
[17]Reith D, Meyer H, Muller-Plathe F. Mapping atomistic to coarse-grained polymer models using automatic simplex optimization to fit structural properties [J]. Macromolecules,2001, 34(7):2335-2345.
[18]Meyer H, Biennann O, Faller R, et al. Coarse graining of nonbonded inter-particle potentials using automatic simplex optimization to fit structural properties [J]. The Journal of Chemical Physics,2000,113(15):6264-6275.
[19]Lyubartsev A P, Laaksonen A. Calculation of effective interaction potentials from radial distribution functions:A reverse Monte Carlo approach [J]. Physical Review E,1995,52(4): 3730-3737.
[20]郭洪霞.高分子粗粒化分子动力学模拟进展[J].高分子通报,2011,10)：154-163.
[21]Kremer K, Grest G S, Carmesin I. Crossover from Rouse to reptation dynamics:A molecular-dynamics simulation [J]. Physical Review Letters,1988,61(5):566-569.
[22]Carmesin I, Kremer K. Static and dynamic properties of two-dimensional polymer melts [J]. Journal de Physique,1990,51(10):915-932.
[23]Duering E R, Kremer K, Grest G S. Dynamics of model networks:the role of the melt entanglement length [J]. Macromolecules,1993,26(12):3241-3244.
[24]Schlijper A, Hoogerbrugge P, Manke C. Computer simulation of dilute polymer solutions with the dissipative particle dynamics method [J]. Journal of Rheology,1995, 39(3):567-579.
[25]Kong Y, Manke C, Madden W, et al. Effect of solvent quality on the conformation and relaxation of polymers via dissipative particle dynamics [J]. The Journal of Chemical Physics, 1997,107(2):592-602.
[26]Kong Y, Manke C, Madden W, et al. Modeling the rheology of polymer solutions by dissipative particle dynamics [J]. Tribology Letters,1997,3(1):133-138.
[27]Groot R D, Warren P B. Dissipative particle dynamics:Bridging the gap between atomistic and mesoscopic simulation [J]. The Journal of Chemical Physics,1997,107(11):4423-4435.
[28]Groot R D, Madden T J, Tildesley D J. On the role of hydrodynamic interactions in block copolymer microphase separation [J]. The Journal of Chemical Physics,1999,110(19)礴9739-9749.
[29]Ten Bosch B. On an extension of Dissipative Particle Dynamics for viscoelastic flow modelling [J]. Journal of Non-Newtonian Fluid Mechanics,1999,83(3):231-248.
[30]Spenley N. Scaling laws for polymers in dissipative particle dynamics [J]. EPL (Europhysics Letters),2000,49(4):534-540.
[31]Van Der Vegt N F A, Peter C, Kremer K. Structure-Based Coarse-and Fine-Graining in Soft Matter Simulations [M]. CRC Press,2008.
[32]Verlet L. Computer" experiments" on classical fluids. I. Thermodynamical properties of Lennard-Jones molecules [J]. Physical Review,1967,159(1):98-103.
[33]Gear C W. Numerical initial value problems in ordinary differential equations [M]. Prentice HallPTR,1971.
[34]Hockney R W. Potential calculation and some application [J]. Methods Comput Phys, 1970,9(1):135-211.
[35]Koelman J, Hoogerbrugge P. Dynamic simulations of hard-sphere suspensions under steady shear [J]. EPL (Europhysics Letters),1993,21(3):363-368.
[36]Hoogerbrugge P, Koelman J. Simulating microscopic hydrodynamic phenomena with dissipative particle dynamics [J]. EPL (Europhysics Letters),1992,19(3):155-160.
[37]Shardlow T. Splitting for dissipative particle dynamics [J]. SIAM Journal on Scientific Computing,2003,24(4):1267-1282.
[38]Lowe C. An alternative approach to dissipative particle dynamics [J]. EPL (Europhysics Letters),1999,47(2):145-151.
[39]Van Swygenhoven H, Farkas D, Caro A. Grain-boundary structures in polycrystalline metals at the nanoscale [J]. Physical Review B,2000,62(2):831-838.
[40]Chen D. Structural modeling of nanocrystalline materials [J]. Computational materials science,1995,3(3):327-333.
[41]曾凡林,孙毅,周玉.有机材料的分子模拟模型(Ⅰ)：单分子链的构建[J].哈尔滨工业大学学报,2009,010：95-99.
[42]Soper A. Empirical potential Monte Carlo simulation of fluid structure [J]. Chemical physics, 1996,202(2-3):295-306.
[43]Lees A, Edwards S. The computer study of transport processes under extreme conditions [J]. Journal of Physics C:Solid State Physics,1972,5(15):1921-1929.
[44]Evans D J, Morriss G P. Statistical mechanics of nonequilibrium liquids [M]. Cambridge Univ Pr,2008.
[45]Fan X, Phan-Thien N, Chen S, et al. Simulating flow of DNA suspension using dissipative particle dynamics [J]. Physics of Fluids,2006,18(6):063102-1-063102-10.
[46]Kong Y, Manke C, Madden W, et al. Simulation of a confined polymer in solution using the dissipative particle dynamics method [J]. International journal of thermophysics,1994,15(6): 1093-1101.
[47]Revenga M, Zuniga I, Espanol P. Boundary model in DPD [J]. Int J Mod Phys,1998, 9(8):1319-1328.
[48]Karayiannis N C, Giannousaki A E, Mavrantzas V G, et al. Atomistic Monte Carlo simulation of strictly monodisperse long polyethylene melts through a generalized chain bridging algorithm [J]. The Journal of Chemical Physics,2002,117(11):5465-5479.
[49]Guerrault X, Rousseau B, Farago J. Dissipative particle dynamics simulations of polymer melts. I. Building potential of mean force for polyethylene and cis-polybutadiene [J]. The Journal of Chemical Physics,2004,121(13):65386546.
[50]Ashbaugh H S, Patel H A, Kumar S K, et al. Mesoscale model of polymer melt structure: Self-consistent mapping of molecular correlations to coarse-grained potentials [J]. The Journal of Chemical Physics,2005,122(10):104908-1-104908-5.
[51]陈正隆,徐为人,汤立达.分子模拟的理论与实践[M].化学工业出版社,2007.
[52]Lahmar F, Rousseau B. Influence of the adjustable parameters of the DPD on the global and local dynamics of a polymer melt [J]. Polymer,2007,48(12):3584-3592.
[53]吴其晔,巫静安.高分子材料流变学[M].高等教育出版社,2002.
[54]Heinz H, Paul W, Binder K. Calculation of local pressure tensors in systems with many-body simulation of polydisperse linear polyethylene melts [J]. Macromolecules,1998,31(22): 7934-7943.
[56]Depa P K, Maranas J K. Dynamic evolution in coarse-grained molecular dynamics simulations of polyethylene melts [J]. Journal of Chemical Physics,2007,126(5): 54903-1-54903-8.
[57]周志平,颜德岳.关于聚乙烯分子链均方回转半径的进一步讨论[J].化学学报,1992,50(4)：313-319.
[58]闵志宇,曹伟,中长雨等.FENE珠-簧链聚合物分子模型流变性质Brown动力学模拟[J].高分子材料与工程,25(2)：171-174.