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Viscoelastic relaxation time is an important concept to characterize the viscoelastic response of materials, which is directly related to the interactions among the microscopic atoms of materials. Few studies have focused on the methods of characterizing viscoelastic relaxation time. To investigate how to represent viscoelastic relaxation time effectively, the viscoelastic relaxation times of the monoatomic Lennard-Jones system on 22 conditions in a range of
$ T^{ *} $ = 0.85–5, ρ* = 0.85–1, ε = 0.97–1, and σ = 0.8–1.3 are discussed from a microscopic perspective by the equilibrium molecular dynamics methods. Static viscoelasticity (viscosity η*, high-frequency shear modulus$ G_{\infty}^* $ ) is calculated by the Green-Kubo formula, and the Fourier transform is applied to the calculation of dynamic viscoelasticity (storage modulus$ G'^* $ and loss modulus$ G''^* $ ). On this basis, the viscoelastic characteristic relaxation time ($ \tau _{{\rm{MD}}}^*$ ), Maxwell relaxation time ($ \tau _{{\rm{Maxwell}}}^*$ ) and the lifetime of the state of local atomic connectivity ($ \tau _{{\rm{LC}}}^*$ ) are calculated. The viscoelastic characteristic relaxation time$ \tau _{{\rm{MD}}}^*$ , defined when the two responses crossover, is the key measure of the period of such a stimulus when the storage modulus (elasticity) equals the loss modulus (viscosity). Maxwell relaxation time$ \tau _{{\rm{Maxwell}}}^* = {\eta ^*}/G_\infty ^*$ , where η* is the static viscosity under infinitely low stimulus frequency (i.e., zero shear rate),$ G_{\infty}^* $ is the instantaneous shear modulus under infinitely high stimulus frequency, and$ \tau _{{\rm{LC}}}^*$ is the time it takes for an atom to lose or gain one nearest neighbor. The result is observed that$ \tau _{{\rm{LC}}}^*$ is closer to$ \tau _{{\rm{MD}}}^*$ than$ \tau _{{\rm{Maxwell}}}^*$ . But the calculation of$ \tau _{{\rm{LC}}}^*$ needs to take into count the trajectories of all atoms in a certain time range, which takes a lot of time and computing resources. Finally, in order to characterize viscoelastic relaxation time more easily, Kramers’ rate theory is used to describe the dissociation and association of atoms, according to the radial distribution functions. And a method of predicting the viscoelasticity of the monoatomic Lennard-Jones system is proposed and established. The comparison of all the viscoelastic relaxation times obtained above shows that$ \tau _{{\rm{Maxwell}}}^*$ is quite different from$ \tau _{{\rm{MD}}}^*$ at low temperature in the monoatomic Lennard-Jones system. Compared with$ \tau _{{\rm{Maxwell}}}^*$ ,$ \tau _{{\rm{LC}}}^*$ is close to$ \tau _{{\rm{MD}}}^*$ . But the calculation of$ \tau _{{\rm{LC}}}^*$ requires a lot of time and computing resources. Most importantly, the relaxation time calculated by our proposed method is closer to$ \tau _{{\rm{MD}}}^*$ . The method of predicting the viscoelastic relaxation time of the monoatomic Lennard-Jones system is accurate and reliable, which provides a new idea for studying the viscoelastic relaxation time of materials.-
Keywords:
- Lennard-Jones system /
- molecular simulation /
- viscoelasticity /
- relaxation time /
- prediction method
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Google Scholar
He P S, Yang H Y, Zhu P P 2004 Chem. Bull. 9 705
Google Scholar
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Jin R G, Hua Y Q 2007 Polymer Physics (Beijing: Chemical Industry Press) p123 (in Chinese)
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[4] Sunthar P 2010 Rheology of Complex Fluids (New York: Springer-Verlag) pp171−191
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Google Scholar
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Google Scholar
[12] Ashwin J, Sen A 2015 Phys. Rev. Lett. 114 055002
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Chen Y 2006 M. S. Thesis (Lanzhou: Northwest Normal University) (in Chines)
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[16] Plimption S 1995 J. Comput. Phys. 117 1
Google Scholar
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Google Scholar
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Google Scholar
[19] Frenkel D, Smit B 2002 Understanding Molecular Simulation, Second Edition: From Algorithms to Applications (New York: Academic Press) p589
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Google Scholar
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Google Scholar
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Google Scholar
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图 1 L-J体计算系统
Figure 1. L-J fluid simulation system.
图 2 不同温度下L-J体的黏度验证
Figure 2. Verification of viscosity of L-J fluid at different temperatures.
图 3 L-J体的径向分布函数验证 (a)
${\rho }^{*}=1,\; {T}^{*}=0.9$ ; (b)${\rho }^{*}=1, \;{T}^{*}=4.0$ Figure 3. Verification of the radial distribution function of L-J fluid: (a)
${\rho }^{*}=1, \;{T}^{*}=0.9$ ; (b)${\rho }^{*}=1, \;{T}^{*}=4.0$ .
图 4
${\rho }^{*}=1, \;\varepsilon =1,\; \sigma =1$ 时不同温度下L-J体的径向分布函数Figure 4. Radial distribution function of L-J fluid at
${\rho }^{*}=1, \varepsilon =1,\; \sigma =1$ with different temperatures.
图 5 储能模量和损耗模量曲线 (a)不同温度条件写的储能模量和损耗模量曲线; (b)粘弹性特征弛豫时间
$ {\tau }_{\rm{MD}}^{*} $ Figure 5. Storage modulus and loss modulus: (a) Storage modulus and loss modulus at different temperatures; (b) viscoelastic characteristic relaxation time
$ {\tau }_{\rm{MD}}^{*} $ .
图 6 不同温度、密度以及L-J势参数下
$ {\tau }_{\rm{Maxwell}}^{*} $ ,$ {\tau }_{\rm{LC}}^{*} $ ,$ {\tau }_{\rm{MD}}^{*} $ ,$ {\tau }_{{\rm{m}}{\rm{o}}{\rm{d}}{\rm{e}}{\rm{l}}}^{*} $ (a)不同温度条件; (b)不同密度条件; (c)不同势能为0时的原子距离条件; (d)不同势能阱深度条件Figure 6.
$ {\tau }_{\rm{Maxwell}}^{*} $ ,$ {\tau }_{\rm{LC}}^{*} $ ,$ {\tau }_{\rm{MD}}^{*} $ ,$ {\tau }_{{\rm{m}}{\rm{o}}{\rm{d}}{\rm{e}}{\rm{l}}}^{*} $ under different T,$ \rho $ , ε and σ: (a) Different temperatures; (b) different densities; (c) different distances between atoms when potential is 0; (d) different potential well depths.
图 7
$\rho^* = 1,\; T^* = 1$ , ε = 1, σ = 1条件下L-J体原子扩散或汇聚时间分布Figure 7. Distribution of the dissociation or association time of L-J fluid at
$\rho^* = 1,\; T^* = 1$ , ε = 1, σ = 1
图 8 Kramers逃逸速率理论示意图
Figure 8. Schematic diagram of Kramers’ rate theory.
图 9 A原子和A—A分子的数量变化曲线 (a)不同温度条件; (b)不同密度条件; (c)不同势能为0时的原子距离条件; (d)不同势能阱深度条件
Figure 9. The quantity curves of A and A—A: (a) Different temperatures; (b) different densities; (c) different distances between atoms when potential is 0; (d) different potential well depths.
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[1] 何平笙, 杨海洋, 朱平平 2004 化学通报 9 705
Google Scholar
He P S, Yang H Y, Zhu P P 2004 Chem. Bull. 9 705
Google Scholar
[2] 金日光, 华幼卿 2007 高分子物理 (北京: 化学工业出版社) 第123页
Jin R G, Hua Y Q 2007 Polymer Physics (Beijing: Chemical Industry Press) p123 (in Chinese)
[3] Winter H H 1987 Polym. Eng. Sci. 27 1698
Google Scholar
[4] Sunthar P 2010 Rheology of Complex Fluids (New York: Springer-Verlag) pp171−191
[5] Agrawal V, Holzworth K, Nantasetphong W, Amirkhizi V A, Oswald J, Nemat-Nasser 2016 J. Polym. Sci., Part B: Polym. Phys. 54 797
Google Scholar
[6] Hattkamp R, Daivids P J, Todd B D 2013 Phys. Rev. E 87 032155
Google Scholar
[7] Green M S 1954 J. Chem. Phys. 22 398
Google Scholar
[8] Kudo R, Toda M, Hashitsume N 2008 Statistical Physics II (Berlin: Springer-Verlag) pp109−145
[9] Guillaud E, Merabia S, De Ligny D, Joly L 2017 Phys. Chem. Chem. Phys. 19 2124
Google Scholar
[10] Iwashita T, Egami T 2012 Phys. Rev. Lett. 108 196001
Google Scholar
[11] Iwashita T, Nicholson D M, Egami T 2013 Phys. Rev. Lett. 110 205504
Google Scholar
[12] Ashwin J, Sen A 2015 Phys. Rev. Lett. 114 055002
Google Scholar
[13] Maxwell J C 1867 Philos. Trans. R. Soc. A 157 49
Google Scholar
[14] 陈艳 2006 硕士学位论文 (兰州: 西北师范大学)
Chen Y 2006 M. S. Thesis (Lanzhou: Northwest Normal University) (in Chines)
[15] Meier K, Laesecke A, Kabelac S 2004 J. Chem. Phys. 121 3671
Google Scholar
[16] Plimption S 1995 J. Comput. Phys. 117 1
Google Scholar
[17] Verlet L 1967 Phys. Rev. 159 98
Google Scholar
[18] Takahashi K, Yasuoka K, Narumi T 2007 J. Chem. Phys. 127 044107
Google Scholar
[19] Frenkel D, Smit B 2002 Understanding Molecular Simulation, Second Edition: From Algorithms to Applications (New York: Academic Press) p589
[20] Frantz Dale B, Plimpton S J, Shephard M S 2010 Eng. Comput. 26 205
Google Scholar
[21] Boon J P, Yip S, Percus J K 1980 Molecular Hydrodynamics (New York: Dover Publications) p285
[22] Hattemer G D, Arya G 2015 Macromol. 48 1240
Google Scholar
[23] Barnes S C, Lawless B M, Shepherd D E T, Espino D M, Bicknell G R, Bryan R T 2016 J. Mech. Behav. Biomed. 61 250
Google Scholar
[24] Heyes D M, Dini D, Brańka A C 2015 Phys. Status Solidi B 252 1514
Google Scholar
[25] Abidine Y, Laurent V M, Michel R 2015 Eur. Phys. J. Plus 130 202
Google Scholar
[26] Michalet, Xavier 2010 Phys. Rev. E 82 041914
Google Scholar
[27] Galliéro, Guillaume, Boned C, Baylaucq A 2005 Ind. Eng. Chem. Res. 44 6963
Google Scholar
[28] Morsali A, Goharshadi E K, Mansoori G A, Abbaspour M 2005 Chem. Phys. 310 11
Google Scholar
[29] Peter Hänggi, Talkner P, Borkovec M 1990 Rev. Mod. Phys. 62 251
Google Scholar
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