Viscoelastic relaxation time of the monoatomic Lennard-Jones system
Authors: Wang Yang , Zhao Ling-Ling ,   Key Laboratory of Energy Thermal Conversion and Control of Ministry of Education, School of Energy and Environment, Southeast University, Nanjing 210096, China
Published: December 2020

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 <inline-formula><tex-math id="Z-20200527053943-1">\begin{document}$ T^{ *} $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="12-20200138_Z-20200527053943-1.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="12-20200138_Z-20200527053943-1.png"/></alternatives></inline-formula> = 0.85–5, <i>ρ</i><sup>*</sup> = 0.85–1, <i>ε</i> = 0.97–1, and <i>σ</i> = 0.8–1.3 are discussed from a microscopic perspective by the equilibrium molecular dynamics methods. Static viscoelasticity (viscosity <i>η</i><sup>*</sup>, high-frequency shear modulus <inline-formula><tex-math id="Z-20200609112430-1">\begin{document}$ G_{\infty}^* $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="12-20200138_Z-20200609112430-1.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="12-20200138_Z-20200609112430-1.png"/></alternatives></inline-formula>) is calculated by the Green-Kubo formula, and the Fourier transform is applied to the calculation of dynamic viscoelasticity (storage modulus <inline-formula><tex-math id="Z-20200609112836-1">\begin{document}$ G'^* $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="12-20200138_Z-20200609112836-1.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="12-20200138_Z-20200609112836-1.png"/></alternatives></inline-formula> and loss modulus <inline-formula><tex-math id="Z-20200609112836-2">\begin{document}$ G''^* $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="12-20200138_Z-20200609112836-2.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="12-20200138_Z-20200609112836-2.png"/></alternatives></inline-formula>). On this basis, the viscoelastic characteristic relaxation time (<inline-formula><tex-math id="M5">\begin{document}$ \tau _{{\rm{MD}}}^*$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="12-20200138_M5.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="12-20200138_M5.png"/></alternatives></inline-formula>), Maxwell relaxation time (<inline-formula><tex-math id="M6">\begin{document}$ \tau _{{\rm{Maxwell}}}^*$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="12-20200138_M6.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="12-20200138_M6.png"/></alternatives></inline-formula>) and the lifetime of the state of local atomic connectivity (<inline-formula><tex-math id="M7">\begin{document}$ \tau _{{\rm{LC}}}^*$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="12-20200138_M7.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="12-20200138_M7.png"/></alternatives></inline-formula>) are calculated. The viscoelastic characteristic relaxation time <inline-formula><tex-math id="M8">\begin{document}$ \tau _{{\rm{MD}}}^*$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="12-20200138_M8.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="12-20200138_M8.png"/></alternatives></inline-formula>, defined when the two responses crossover, is the key measure of the period of such a stimulus when the storage modulus (elasticity) equals the loss modulus (viscosity). Maxwell relaxation time <inline-formula><tex-math id="M9">\begin{document}$ \tau _{{\rm{Maxwell}}}^* = {\eta ^*}/G_\infty ^*$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="12-20200138_M9.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="12-20200138_M9.png"/></alternatives></inline-formula>, where <i>η</i><sup>*</sup> is the static viscosity under infinitely low stimulus frequency (i.e., zero shear rate), <inline-formula><tex-math id="Z-20200609112519-1">\begin{document}$ G_{\infty}^* $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="12-20200138_Z-20200609112519-1.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="12-20200138_Z-20200609112519-1.png"/></alternatives></inline-formula> is the instantaneous shear modulus under infinitely high stimulus frequency, and <inline-formula><tex-math id="M10">\begin{document}$ \tau _{{\rm{LC}}}^*$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="12-20200138_M10.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="12-20200138_M10.png"/></alternatives></inline-formula> is the time it takes for an atom to lose or gain one nearest neighbor. The result is observed that <inline-formula><tex-math id="M11">\begin{document}$ \tau _{{\rm{LC}}}^*$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="12-20200138_M11.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="12-20200138_M11.png"/></alternatives></inline-formula> is closer to <inline-formula><tex-math id="M12">\begin{document}$ \tau _{{\rm{MD}}}^*$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="12-20200138_M12.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="12-20200138_M12.png"/></alternatives></inline-formula> than <inline-formula><tex-math id="M13">\begin{document}$ \tau _{{\rm{Maxwell}}}^*$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="12-20200138_M13.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="12-20200138_M13.png"/></alternatives></inline-formula>. But the calculation of <inline-formula><tex-math id="M14">\begin{document}$ \tau _{{\rm{LC}}}^*$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="12-20200138_M14.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="12-20200138_M14.png"/></alternatives></inline-formula> needs to take into count the trajectories of all atoms in a certain time range, which takes a lot of time and computing resources. Finally, in order to characterize viscoelastic relaxation time more easily, Kramers’ rate theory is used to describe the dissociation and association of atoms, according to the radial distribution functions. And a method of predicting the viscoelasticity of the monoatomic Lennard-Jones system is proposed and established. The comparison of all the viscoelastic relaxation times obtained above shows that <inline-formula><tex-math id="M15">\begin{document}$ \tau _{{\rm{Maxwell}}}^*$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="12-20200138_M15.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="12-20200138_M15.png"/></alternatives></inline-formula> is quite different from <inline-formula><tex-math id="M16">\begin{document}$ \tau _{{\rm{MD}}}^*$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="12-20200138_M16.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="12-20200138_M16.png"/></alternatives></inline-formula> at low temperature in the monoatomic Lennard-Jones system. Compared with <inline-formula><tex-math id="M17">\begin{document}$ \tau _{{\rm{Maxwell}}}^*$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="12-20200138_M17.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="12-20200138_M17.png"/></alternatives></inline-formula>, <inline-formula><tex-math id="M18">\begin{document}$ \tau _{{\rm{LC}}}^*$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="12-20200138_M18.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="12-20200138_M18.png"/></alternatives></inline-formula> is close to <inline-formula><tex-math id="M19">\begin{document}$ \tau _{{\rm{MD}}}^*$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="12-20200138_M19.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="12-20200138_M19.png"/></alternatives></inline-formula>. But the calculation of <inline-formula><tex-math id="M20">\begin{document}$ \tau _{{\rm{LC}}}^*$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="12-20200138_M20.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="12-20200138_M20.png"/></alternatives></inline-formula> requires a lot of time and computing resources. Most importantly, the relaxation time calculated by our proposed method is closer to <inline-formula><tex-math id="M21">\begin{document}$ \tau _{{\rm{MD}}}^*$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="12-20200138_M21.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="12-20200138_M21.png"/></alternatives></inline-formula>. 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.