Progress on Equation of State of Hydrogen and Deuterium
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摘要: 针对近二十多年的氢氘物态方程理论研究工作进行了综述分析，结合本课题组改进的自由能模型、直接量子蒙特卡洛和量子分子动力学方法的模拟结果，对多个研究小组采用不同方法获得的氢氘宽区物态方程数据进行了定量评估分析。结果表明：在当前理论框架下，仅基于第一原理数值模拟得到的氢氘物态方程能够描述的热力学相空间有限；多模型集成的H-REOS.3数据库在105 K以下温度与数值模拟结果的相对差别较大，且数据稀少，二者均不能满足工程应用需求。建议采用基于半经验模型的宽区域物态方程研究方法，即结合高精度的实验研究、数值模拟和解析模型，构建满足工程应用需求的氢氘宽区实用物态方程。Abstract: This paper reviews the research on the equation of state (EOS) for hydrogen and deuterium in the last twenty years.A quantitative analysis of the wide-range EOS for hydrogen and deuterium is given by combining the modified chemical free energy model and the modern computational techniques such as Quantum Molecular Dynamic (QMD) and Quantum Monte Carlo (QMC).It is demonstrated that the wide-range EOS obtained from ab initio calculations only covers a limited region of density-temperature space, and the H-REOS.3 database, which is obtained by integrating different models, exhibits disagreement with the modern computational results below the temperature of 105 K and its data points distribution is not dense enough.EOS of deuterium is not transformed from the H-REOS.3.Therefore, both the ab initio calculation and the H-REOS.3 are not sufficient for the engineering application, and it is necessary to construct a wide range EOS database for hydrogen and its isotopes, which combines the high-precision experiments, the modern computational techniques, and the analytical or semi-analytical models.
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金属柱壳在内部爆炸加载条件下的动态变形、破碎特性有着重要的理论意义和应用价值。在军事领域，由于柱壳结构在战斗部中较为常见，其在爆炸载荷下的力学响应及终点效应一直是关注的热点[1-2]；在公共安全领域，为有效限制爆炸冲击波及爆轰产物的作用范围，对人员和设备实现有效保护，抗爆容器可用于紧急处理疑似爆炸物品[3-4]；在石油化工领域，压力容器经常在高温和高压下运行，在突发故障下内部易燃易爆介质可能发生爆炸，产生碎片引发多米诺事故也是不可避免的[5-6]。
国内外早期的研究主要集中在柱壳膨胀后期裂纹形成的物理机制、断裂应变分析和破片统计分布规律。Gurnery[7]根据能量平衡模型估算了柱壳外爆膨胀断裂后的破片速度。Taylor[8]基于拉伸断裂假设及弹塑性理论，提出柱壳内部的应力状态与断裂判据。Hoggatt等[9]观察到，随着爆轰压力的增加，柱壳的断裂模式由拉伸型向剪切型转变。Mott[10]和Grady[11]分别从统计学方法和能量守恒观点出发，系统地研究了柱壳破碎后形成大量破片的统计分布规律，并给出相应的破片分布模型。近些年随着计算机技术的进步，数值仿真成为广泛使用的一种研究方法。Hiroe等[12]基于实验与数值模拟结合的方法，研究了不同材料、缺陷结构、爆炸能量及起爆位置对柱壳变形及破碎行为的影响规律。Tanapornraweekit等[13]通过数值仿真研究了柱壳自然破碎后其破片初速度、飞散角和质量分布特征，并分析了不同材料性能对柱壳破碎性能的影响。Kong等[14]利用光滑粒子数值法计算了柱壳在内爆载荷下的动态断裂行为，破片分布及飞散速度与实验结果基本一致。由以上可见，系统研究不同条件下壳体破碎特性的报道并不多见[15]。针对这类强动载柱壳结构，国内外也未形成一种通用的破碎性评估准则。
本工作以圆柱钢壳为研究对象，分析不同装药条件对壳体膨胀断裂行为的影响，探讨破碎特性和装药与壳体质量比C/M(C为装药质量，M为壳体质量)的内在关系，得出一种新的适用于圆柱壳体的破碎性评估准则。
1. 实验
1.1 实验方法
试验材料为某典型高强合金钢，其主要化学成分包括0.6%C、1.27%Si、1.15%Mn、0.018%S、0.027%P。装药为B炸药，密度为1.67 g/cm3。所有圆柱钢壳在水井中爆破，且壳体周围预留足够空气区域供其膨胀断裂，按标准方法将破片回收，回收率均在97%以上。按质量分组并记录各组破片的质量和数量，其中质量小于0.04 g的破片只统计总质量。
1.2 破碎性表征
一般认为，壳体破片的质量分布符合修正Payman直线方程[16]
lgP=2−C0mM (1) 式中:m为规定的破片分组质量范围的下限; M为回收破片的总质量; P为质量大于m的破片质量百分数(0＜P≤100);直线的斜率C0称为修正Payman破碎参数，可用来表征壳体的破碎性能。一般来说，C0值越大，壳体破片的质量分布越合理，破碎性越好。
研究还发现，修正Payman破碎参数C0与壳体长度L成正比，即
Cu=C0/L (2) 式中：Cu即为单位长度修正Payman破碎参数，单位cm-1。Cu与壳体长度无关，只与壳体材料、装药类型及C/M比值有关。
2. 结果分析与讨论
2.1 壳体破碎性能评估准则
为研究壳体的破碎特性与C/M比值的关系，设计了如图 1所示的圆柱壳体，具体尺寸见表 1。为消除端部效应，药柱两端均突出壳体约20 mm。共进行5组试验，每组3发壳体。
表 1 壳体尺寸及装药条件Table 1. Dimensions of shell and chargeNo. D/mm d/mm L/mm C/M 1 30.64 20.64 70 0.18 2 41.99 30.13 70 0.23 3 40.13 30.13 70 0.28 4 50.87 39.52 70 0.33 5 49.52 39.52 70 0.38 破片回收后按质量分拣，并根据(1)式分析其质量分布规律，可得出修正Payman破碎参数C0与C/M值的关系。如图 2所示，随着C/M比值的增加，C0变大，壳体的破碎性能提高，该现象与壳体断裂及破片形成机制有关，与Reid等[17]的实验结果相符。
图 2 破碎参数C0与C/M值的关系Figure 2. Variation of Payman fragmentation parameter C0 with C/M ratio对每组C0取平均值并除以壳体长度，可得到单位长度修正Payman破碎参数Cu。如图 3所示，对Cu与C/M比值进行线性拟合，拟合直线方程可表示为
Cu=19.1+92.8C/M (3) 
图 3 破碎参数Cu与C/M比值的关系Figure 3. Variation of normalized Payman fragmentation parameter Cu with C/M ratio因此可认为，内爆炸加载条件下，任何圆柱钢壳与炸药组合一般都存在这类线性关系，即
Cu=a+bC/M (4) 式中：a和b被称为壳体材料与炸药组合的特征常数。对于给定的材料与炸药组合，a和b为定值。根据壳体破碎的相似性原则，利用(4)式可以评估在相同壳体材料及炸药条件下任何尺寸圆柱壳体的破碎参数，具有广泛的用途。由图 3还可以看出，当C/M比值为0.33和0.38时，破碎参数Cu值偏低。这是因为这两组壳体的直径较大，毛坯热锻后冷却特别慢，温度不均匀导致组织中铁素体含量增多，并呈块状形态，阻碍了裂纹扩展与贯通，最终造成壳体整体的破碎性降低。
2.2 端部效应对破碎性的影响
为研究端部效应对破碎性的影响，设计了如图 4所示的圆柱钢壳，其特点是起爆端药柱与壳体齐平。所有壳体分为两组：一组壁厚t固定，C/M比值变化，考察C/M比值对壳体破碎性的影响；另一组C/M比值保持不变，壁厚变化，考察壁厚对壳体破碎性的影响。壳体尺寸如表 2所示，得到的破碎性结果如图 5所示。可以看出, 当存在端部效应时，壳体破碎性随C/M比值的增大而提高，单位长度修正Payman破碎参数Cu与C/M比值同样可拟合成直线关系，其方程可表示为
Cu=4.7+93.1C/M (5) 表 2 考虑端部效应的壳体尺寸及装药条件Table 2. Dimensions of shell with end effect and chargeNo. D/mm d/mm L/mm t/mm C/M 1 30.64 20.64 60 5 0.18 2 40.13 30.13 60 5 0.28 3 49.52 39.52 60 5 0.38 4 58.84 48.84 60 5 0.48 5 32.11 24.11 60 4 0.28 6 48.16 36.16 60 6 0.28 7 56.19 42.19 60 7 0.28 
图 5 端部效应对壳体破碎参数Cu的影响Figure 5. Influence of end effect on the normalized Payman fragmentation parameter Cu图 5中虚线为(3)式拟合直线，表示壳体不存在端部效应时，其破碎参数Cu与C/M比值的变化关系。由此可以得出，无论C/M比值如何变化，端部效应将导致壳体破碎性能降低。这是因为该条件下装药起爆不能立刻达到最高压力，导致壳体起爆端不能及时断裂形成破片，整体破碎性能变小。需要注意的是，图 5中两条直线的斜率分别为92.8和93.1，近似相等，意味着不管C/M比值多大，由端部效应引起破碎参数Cu的降低量是个常数。
如果忽略两条直线斜率的微小差别，将(3)式与(5)式相减，即可得到如下关系式
Cu_end=Cu−34a (6) 式中：Cu_end和Cu分别为相同C/M比值条件下存在端部效应和无端部效应的破碎参数，a为材料与炸药组合的特性常数。
当C/M比值保持不变时，破碎参数Cu随壳体壁厚的变化趋势如图 6所示。可以看出，Cu与壁厚的关系基本为一条水平直线，具有很小的正斜率，表明端部效应与壁厚无关。
图 6 壁厚对壳体破碎参数Cu的影响Figure 6. Influence of wall thickness on the normalized Payman fragmentation parameter Cu2.3 无装药部分对破碎性的影响
为考察无装药部分对壳体破碎性的影响，设计了如图 7所示的圆柱钢壳。圆柱钢壳共分为3组：(1)壁厚t及无装药部分长度L′固定，C/M比值变化；(2)壁厚及C/M比值固定，无装药部分长度变化；(3) C/M比值及无装药部分长度固定，壁厚变化。壳体尺寸及装药条件见表 3。
表 3 考虑无装药部分的壳体尺寸及装药条件Table 3. Dimensions of shell with charge vacancy and chargeNo. D/mm d/mm L/mm L′/mm t/mm C/M 1 30.64 20.64 60 30 5 0.18 40.13 30.13 60 30 5 0.28 49.52 39.52 60 30 5 0.38 58.84 48.84 60 30 5 0.48 2 40.13 30.13 60 10 5 0.28 40.13 30.13 60 20 5 0.28 40.13 30.13 60 30 5 0.28 40.13 30.13 60 40 5 0.28 3 32.11 24.11 60 30 4 0.28 40.13 30.13 60 30 5 0.28 48.16 36.16 60 30 6 0.28 56.19 42.19 60 30 7 0.28 壳体装药部位及无装药部位的单位长度修正Payman破碎参数Cu与C/M比值的关系如图 8所示。可以看出，无装药部分的破碎参数Cu较小且变化不大，而壳体装药部分的破碎参数Cu与C/M比值之间依然存在线性关系，并且可用下式表示
Cu=4.8+91.7C/M (7) 
图 8 壳体破碎参数Cu与C/M比值的关系Figure 8. Variation of normalized Payman fragmentation parameter Cu with C/M ratio将(7)式与(5)式做对比发现，两者的截距及斜率几乎相等，因此可以近似认为壳体装药部分的破碎参数与C/M比值的关系和具有端部效应的壳体完全相同，即无装药部分的存在对装药部分的破碎性没有任何影响。对回收的破片仔细观察发现，壳体破碎时主要沿装药与无装药的连接处断裂，不产生既包括装药部分又包括无装药部分的破片，因此装药部分的破碎性与具有端部效应的壳体是等同的。
当装药部分的C/M比值和无装药部分长度L′保持不变时，壳体的破碎参数Cu随壁厚的变化关系如图 9所示。随着壳体壁厚的增加，装药部分及无装药部分的破碎性几乎保持不变，表明壳体壁厚改变时无装药部分对装药部分的破碎性无影响。对于无装药部分长度不同的壳体，其破碎性参数变化如图 10所示，可以看出，当无装药部分长度增加时，装药部分的破碎参数几乎不变。因此可得出，无装药部分长度不影响壳体装药部分的破碎性能。
图 9 壳体破碎参数Cu与壁厚的关系Figure 9. Variation of normalized Payman fragmentation parameter Cu with wall thickness
图 10 壳体破碎参数Cu与无装药部分长度的关系Figure 10. Variation of normalized Payman fragmentation parameter Cu with length of charge vacancy综合图 8、图 9和图 10还可以得出，无装药部分壳体的破碎参数事实上很小。当装药部分壳体向外膨胀时，无装药部分受影响而被“撕裂”成破片。由于两部分壳体向外膨胀的加速度相差悬殊，所以沿两部分的连接处断裂。因此可近似认为，装药部分壳体等效于存在端部效应的壳体。无装药部分的破碎性则主要取决于壳体材料本身性质，C/M比值即使有影响也不会很明显，而长度和壁厚的变化虽然会影响壳体质量和破片大小，但总破碎参数C0近似相等，因为C0是经过壳体质量(回收破片总质量)修正过的参数。
3. 结论
针对内爆炸载荷作用下金属壳体的动态碎裂问题，通过圆柱钢壳内爆加载试验研究了不同装药条件对破片质量分布的影响规律，得到如下主要结论。
(1) 对于不同的给定材料与炸药组合，壳体的单位长度修正Payman破碎参数Cu与C/M比值均存在线性关系，即Cu=a+bC/M，其中a、b为与壳体材料和炸药组合有关的特征常数；
(2) 无论壁厚和C/M比值如何变化，端部效应使壳体的破碎参数Cu降低至一个定值3a/4；
(3) 无装药部分对壳体的破碎性影响很小，当无装药部分长度或壁厚发生变化时，壳体的破碎参数Cu近似不变，且Cu与C/M比值的线性关系同样成立。
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图 1 QMD模拟氘物态方程的压强相对偏差
(下标i分别指代Gali2000、Desjarlais2003、Liu2004、Bonev2004、Danel2016、Lenosky2000, 以下类似)
Figure 1. Relative differences of pressure from QMD simulation for deuterium
(The subscript i represents Gali2000, Desjarlais2003, Liu2004, Bonev2004, Danel2016, Lenosky2000, respectively.Similarly hereinafter.)
图 2 QMD模拟氢物态方程的压强相对偏差
Figure 2. Relative differences of pressure from QMD simulation for hydrogen
图 3 不同泛函计算的氘物态方程压强相对差别
Figure 3. Relative differences of pressure from QMD simulation for deuterium in different exchange-correlation analysis
图 4 PIMC模拟氘物态方程的压强相对偏差
Figure 4. Relative differences of pressure from PIMC simulation for deuterium
图 5 PIMC模拟氢物态方程的压强相对偏差
Figure 5. Relative differences of pressure from PIMC simulation for hydrogen
图 6 氢的MP2011与REOS物态方程的相对偏差
Figure 6. Relative differences of equation of state for hydrogen between MP2011 and REOS
表 1 氢氘物态方程的QMD模拟工作
Table 1. Works of QMD simulation for equation of state of hydrogen and deuterium
Isotope Time Firstauthor Ref. Method Code Organization Temperature/K Density/(g·cm-3) D 1997 T. J. Lenosky [135] TBMD LANL 3 000-31 250 0.58-1.47 D 2000 G. Galli [136] CPMD GP code LLNL 10 000 0.67，1.0 D 2000 S. Bagnier [137] BOMD VASP CEA The effect of the local-spin-density-approximation functional D 2000 T. J. Lenosky [138] BOMD VASP LLNL，LANL 2 000-31 500 0.506-0.851 D 2002 F. Gygi [139] CPMD GP code LLNL The compressibility is determined by shock-induced electronic excitations D 2003 M. P. Desjarlais [140] BOMD VASP SNL 3 800-50 105 0.553-1.756 D 2004 S. A. Bonev [141] CPMD GP code LLNL 20-19 860 0.171-0.779 D 2004 H. F.Liu [149] BOMD VASP IAPCM 1 000-10 000 0.506-1.0 D 2016 J. F. Danel [142] QMD，OFWMD ABINIT，VAAQP CEA 11 604-116 045 0.2-20 D 2016 V. V. Karasiev [143] QMD，OFMD PROFESS@Q-ESPRESSO，ABINIT Universityof Florida 2 000-250 000 0.2-10, No data list D 2017 M. D. Knudson [12] QMD VASP，Diff Exc SNL, WashingtonState University H 2001 L. A. Collins [146] BOMD VASP LANL，LLNL 5 000-30 000 0.334-0.525 H 2008 B. Holst [147] BOMD VASP Institut für Physik，SNL 500-20 000 0.5-5.0 H 2010 M. A. Morales [126] BOMD，QMC QBOX，CEIMC University of Illinois，University of L'Aquila 2 000-10 000 0.724-2.329 H 2011 L. Caillabet [148] QMD，CEIMC PIMC, QMD, CEIMC CEA -116 045 0.2-5 H 2013 C. Wang [151] QMD，OFMD ABINIT IAPCM 1.564×104-5.004441×107 9.82×10-3-1.347×103 
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表 2 氢氘物态方程的QMC数值模拟
Table 2. Works of QMC simulation for equation of state of hydrogen and deuterium
Isotope Time Firstauthor Ref. Method Organization Temperature/K Density/(g·cm-3) D 2000 B. Militzer [114] PIMC(VDM) University of Illinois at Urbana-Champaign 105-106 0.674，0.838 D 2004 V. Bezkrovniy [155] DPIMC Universität Greifswald, Domstrasse, Germany；Russian Academy of Science 15 625-1 000 000 0.674，0.838，1.097 D 2011 S. X. Hu [46] RPIMC University of Rochester，University of California 15 665-63 822 000 0.002-1 596 H 20012003 V. S. Filinov [117]
[116]DPIMC Russian Academy of Sciences 31 250-106 0.419 H 2001 B. Militzer [114] RPIMC LLNL，University of Illinois at Urbana-Champaign 5 000-250 000 9.833×10-4-0.153 H 2004 C. Pierleoni [121] CEIMC Universita' of L'Aquila, Via Vetoio, University of Illinois at Urbana-Champaign, Université Pierre et Marie Curie 300-10 000 5.267，2.697，1.561 H 2010 M. A. Morales [126] CEIMC University of Illinois at Urbana-Champaign, University of L'Aquila, Italy 2 000-10 000 0.724-2.329 H 2011 L. Caillabet [148] QMD，CEIMC CEA 116 045 0.2-5 H 2015 Q. L. Zhang [150] DPIMC IAPCM 116 045 0.98×10-3-1 346.1
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