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The viscosities of matters under extreme conditions, i.e. warm dense matter (WDM) and hot dense matter (HDM), have significant applications in various fields, such as the design of inertial confinement fusion targets, the astrophysical structure evolution, and the interfacial instability and mixing development under extreme conditions. Since the temperature and pressure ranges accessible by experimental techniques for viscosity measurement are very limited, the acquisition of viscosity data under extreme conditions mainly relies on theoretical calculations. This work introduces a variety of molecular dynamics (MD) methods and models for calculating the viscosities of WDM and HDM, they being quantum MD (QMD), orbital-free MD (OFMD), average atom model combined with hypernetted chain (AAHNC), effective potential theory combined with average atom model (EPT+AA), hybrid kinetics MD (KMD), integrated Yukawa viscosity model (IYVM), Stanton-Murillo transport model (SMT), pseudo-ion in jellium (PIJ), one-component plasma model (OCP), and random-walk shielding-potential viscosity model (RWSP-VM). Simultaneously, the viscosities of various elements obtained by these methods are shown, ranging from low to high atomic number (Z), i.e., H, C, Al, Fe, Ge, W, and U. The accuracy and the applicability of each method are analyzed in detail by comparison. RWSP-VM, which is based on physical modeling and independent of MD data, has comparable accuracy to simulation data over a wide range of temperature and pressure, and is an efficient method of obtaining viscosity data of WDM and HDM. This work will pave the way for calculating the shear viscosities under extreme conditions, and may play an important role in promoting the relevant applications. The data calculated from RWSP-VM in this work are openly available at
https://doi.org/10.57760/sciencedb.j00213.00180 .-
Keywords:
- shear viscosity /
- molecular dynamics /
- ab initio calculations /
- kinetic theory
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图 1 H的粘性. (a)—(d) 密度分别为0.1, 1, 10, 100 g/cm3. 黑色实线, 红色虚线, 蓝色点线和, 绿色虚点线和青色点点线分别为模型RWSP-VM, SMT, OCP, IYVM和PIJ的由公式计算的结果. 黑色十字, 红色星号, 蓝色方块和绿色圆圈分别为AAHNC, KMD, EPT+AA和OFMD的计算结果, 数据来源文献[54]
Figure 1. Shear viscosity of H. (a)–(d) stand for the densities of 0.1, 1, 10, 100 g/cm3, respectively. Black solid, red dashed, blue dotted, and green dash-dot curves stand for the results of RWSP-VM, SMT, OCP, IYVM, and PIJ respectively. Black crosses, red stars, blue squares, and green diamonds stand for the results of AAHNC, KMD, EPT+AA, and OFMD, respectively, which are from Ref.[54].
图 3 Al的粘性. (a)—(d) 密度分别为0.27, 2.7, 8.1, 27 g/cm3. 黑色实线, 红色虚线, 蓝色点线, 绿色虚点线和青色虚点点线分别为模型RWSP-VM, SMT, OCP, IYVM和PIJ的由公式计算的结果. 黑色十字为AAHNC (CMD)的计算结果, 数据来源文献[47]
Figure 3. Shear viscosity of Al. (a)–(d) stand for the densities of 0.27, 2.7, 8.1, and 27 g/cm3, respectively. Black solid, red dashed, blue dotted, green dash-dot, and cyan dash-dot-dot curves stand for the results of RWSP-VM, SMT, OCP, IYVM, and PIJ, respectively. Black crosses stand for the results of AAHNC (CMD), which are from Ref.[47].
图 4 Fe的粘性. (a)—(f) 密度分别为1.6, 4.0, 7.9, 16, 32, 40 g/cm3. 曲线图例与图 3的一致, 除了橙色实线代表EPT+AA[27]. 红色圆圈, 蓝色方块和黑色十字分别为OFMD1[55], OFMD2[27]和AAHNC (CMD)[39]的计算结果
Figure 4. Shear viscosity of Fe. (a)–(f) stand for the densities of 1.6, 4.0, 7.9, 16, 32, and 40 g/cm3, respectively. The legends of the curves are the same as Fig. 3. Red circles, blue squares, and black crosses stand for the results of OFMD1[55], OFMD2[27], and AAHNC (CMD)[39], respectively.
图 7 U的粘性. (a)—(i) 密度分别为$ \rho_0 $的0.1, 1, 2, 3, 4, 5, 6, 8, 10倍, 其中$ \rho_0 = 18.93\ {\rm{g}}/{\rm{cm}}^3 $. 曲线的图例与图 3一致. 黑色圆圈, 红色十字和蓝色叉分别为OFMD[57], AAHNC (CMD)和AAHNC (LMD)[48]的计算结果. (j), (k), (l)分别为(a), (b), (f)的放大图
Figure 7. Shear viscosity of U. (a)–(i) stand for the densities of (0.1, 1, 2, 3, 4, 5, 6, 8, and 10)$ \rho_0 $, where $ \rho_0 = 18.93\ {\rm{g}}/{\rm{cm}}^3 $. The legends of the curves are the same as Fig. 3. Black circles, red crosses (+), and blue crosses (×) stand for the results of OFMD[57], AAHNC (CMD), and AAHNC (LMD)[48], respectively. (j), (k), (l) represent the zoom of (a), (b), (f), respectively.
表 1 约化的碰撞积分(28)—(29)式在(2, 2)阶的拟合系数[36]
Table 1. Coefficients for Eqs. (28)–(29) of the reduced collision integrals at index pair of (2, 2)[36]
$ a_1 $ $ a_2 $ $ a_3 $ $ a_4 $ $ a_5 $ 0.85401 –0.22898 –0.60059 0.80591 –0.30555 $ b_0 $ $ b_1 $ $ b_2 $ $ b_3 $ $ b_4 $ 0.43475 –0.21147 0.11116 0.19665 0.15195 
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$ a_0 $ $ a_1 $ $ a_2 $ $ a_3 $ 0.794811 0.0425698 0.00205782 7.03658×10–5 $ b_0 $ $ b_1 $ $ b_2 $ $ b_3 $ $ b_4 $ 0.862151 0.0429942 –0.000270798 3.25441×10–6 –1.15019×10–8
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表 B1 本文使用的基本物理常量以及物质的一些基本物理量
Table B1. Fundamental physical constants and some basic physical quantities of the materials used in this work.
符号 物理量 表达式 e 元电荷 $ k_{\rm{B}} $ 玻尔兹曼常量 $ \hbar $ 约化普朗克常量 $ m_e $ 电子质量 $ \varepsilon_0 $ 真空电导率 $ a_{\rm{ws}} $ Wigner-Seitz半径 $ (4\pi n/3)^{-1/3} $ $ E_{\rm{F}} $ 费米能量 $ \hbar^2(3 \pi^2 n_e)^{2/3}/(2 m_e) $ m 离子质量 n 离子数密度 $ n_{\mathrm{e}} $ 电离的电子数密度 $ \overline{Z}n $ $ q^2 $ 电荷平方 $ (\overline{Z}e)^2/(4 \pi \varepsilon_0) $ T 温度 Z 原子序数 $ \overline{Z} $ 平均电离度 $ \overline{Z^2} $ 电离度方均值 Γ 耦合参数 $ q^2/(a_{\rm{ws}} k_{\rm{B}} T) $ θ 电子简并参数 $ k_{\rm{B}} T/E_F $ κ 屏蔽参数 $ a_{\rm{ws}}/\lambda $ λ 屏蔽距离 根据模型需要取值 $ \omega_{\rm{p}} $ 等离子体频率 $ \sqrt{4\pi q^2 n/m} $
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