# Hugoniot States and Mie–Grüneisen Equation of State of Iron Estimated Using Molecular Dynamics

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## Abstract

**:**

_{c}), the cold energy (E

_{c}), the Grüneisen coefficient (γ), and the melting temperature (T

_{m}) are discussed. The error between SC and NC iron results was found to be less than 1.5%. Interestingly, the differences in Hugoniot state (P

_{H}) and the internal energy between SC and NC iron were insignificant, which shows that the effect of grain size (GS) under high pressures was not significant. The P

_{c}and E

_{c}of SC and NC iron calculated based on the Morse potential were almost the same with those calculated based on the Born–Mayer potential; however, those calculated based on the Born–Mayer potential were a little larger at high pressures. In addition, several empirical and theoretical models were compared for the calculation of γ and T

_{m}. The Mie–Grüneisen EOSs were shown on the 3D contour space; the pressure obtained with the Hugoniot curves as the reference was larger than that obtained with the cold curves as the reference.

## 1. Introduction

_{m}), is vital [4,5].

_{0}·ρ

_{0}= γ·ρ = const in the pressure range.

_{m}is related to γ

_{0}. Based on the Lindemann melting criterion and the volume dependence of the Gruneisen coefficient γ, Yang et al. [5] investigated the melting characteristics of single and polycrystalline Al. Errandonea [34] used a laser-heated diamond-anvil cell to study the melting characteristics of Cu, Ni, Pd, and Pt in high-pressure conditions; it was shown that Simon equations could be used to describe the melting curves. Anzellini et al. [35] conducted static laser-heated diamond anvil cell experiments to investigate the melting point of iron under pressures up to 200 GPa. The results indicated that the temperature of iron at the inner core boundary was 6230 ± 500 K. Bouchet et al. [36] calculated the melting curve, and EOS of hcp and bcc iron under pressures up to 1500 GPa; the melting temperature was 11,000 K at the highest pressure.

_{m}. The time scale and model size in NEMD and EMD simulations are restricted by computing resources; as the time scale and model size increase, the computational time will significantly increase. The first-principle method can fundamentally calculate the molecular structure and material properties, but the model size is too small in the shockwave propagation simulation. To solve the computational time and model size shortcomings in the above methods, Reed et al. [40] proposed a multiscale simulation method, called the multiscale shock technique (MSST), to study the shockwave propagation in materials. Compared with the above methods, the MSST method can efficiently save computational time and guarantee simulation accuracy with an acceptable model size of the MD system [41,42]. Thus, in the present study, the MSST method is applied to obtain the shock Hugoniot data in SC and nanocrystalline (NC) iron.

_{m}are significant physical properties that must be examined to study the behavior of metal materials under high-pressure conditions. There are two issues involved in the relation between shock Hugoniots and EOS: one regards obtaining the EOS based on the shock Hugoniots, and the other concerns the determination of the shock Hugoniots based on the EOS. In addition, the Grüneisen parameter and melting temperature can be calculated from the shock Hugoniot data. The investigation path of this work firstly utilizes MD simulation to obtain the shock Hugoniots of SC and NC iron; thus, the Hugoniot curve can be obtained. Therefore, based on the Hugoniot data, γ and T

_{m}can be determined. Finally, the EOS of SC and NC iron can be presented.

## 2. Methodology

#### 2.1. Hugoniot Pressure (P_{H}) and Internal Energy (E_{H})

_{s}, u

_{p}, P

_{H}, E

_{H}, and V (=1/ρ) are the density, shockwave velocity, particle velocity, pressure, and internal energy per unit mass and specific volume of the shocked material, respectively. The shock wave velocity in the MD simulation was set to 6–11 km/s, and the particle velocity under different loading conditions was obtained by varying the loading velocity. The subscript “0” denotes the qualities in the initial state without shock, and “H” represents the quantities in the Hugoniot state. For example, P

_{H}represents the pressure in the Hugoniot state.

_{s}and the particle velocity u

_{p}is approximately linear [11]:

_{0}and λ are the volume sound speed at zero pressure and the fitting parameter, respectively. From MD simulation results, C

_{0}and λ of SC and NC iron can be determined. Once C

_{0}and λ is determined, it can be used to calculate the Hugoniot curve P

_{H}(V) and the internal energy E

_{H}(V).

_{0}= 0, the Hugoniot curve P

_{H}(V) and the internal energy E

_{H}(V) can be derived:

_{v}and T

_{0}are the specific heat at constant volume and the initial temperature, respectively. If the E

_{H}is much larger than E

_{0}, E

_{0}can usually be ignored.

#### 2.2. Cold Pressure (P_{c}) and Cold Energy (E_{c})

_{0K}, ρ

_{0K}, ${C}_{0}^{\prime}$, and λ′ can be obtained as follows [43]:

_{0}) is the Grüneisen coefficient at room temperature (usually 300 K); α

_{v}(T) is the volume expansion coefficient, which is approximately constant from 0 K to 300 K; and ${C}_{0}^{\prime}$, and λ′ are parameters of u

_{s}and u

_{p}relations at 0 K.

_{c}and cold energy E

_{c}can be expressed as follows:

_{0K}/V; and ρ

_{0K}is the material density at 0.

_{H}, isentropic P

_{s}, and P

_{c}are almost equal [5,46]. Thus, we have:

_{0}and λ from MD simulations, and according to Equations (5) and (10)–(15), Q, q, A, and B can be calculated as follows:

#### 2.3. Grüneisen Coefficient γ

_{s}(V); t = 1 represents the Dugdale–MacDonald model [28] γ

_{DM}(V); and t = 2 represents the free-volume model [29] γ

_{f}(V).

_{0}, Equation (22) approximately satisfies the isentropic P

_{s}(V), and, consequently, Equation (22) can be expressed as follows:

_{0}, the first-order and second-order derivatives of P

_{s}(V) and P

_{H}(V) are identical. In addition, ${P}_{H}^{\prime}\left({V}_{0}\right)$ and ${P}_{H}^{\u2033}\left({V}_{0}\right)$ can be derived from P

_{H}obtained via MD simulations or experiments; therefore, we have:

#### 2.4. Melting Temperature (T_{m})

_{m}(V) is a significant material characteristic. The Lindemann melting criterion is given by [50]:

_{0}is determined from Equations (30)–(32), T

_{m}

_{0}is the melting temperature at 0 GPa, and is 1811 K for iron [51].

#### 2.5. Mie–Grüneisen Equation of State

_{H}, E

_{H}, P

_{c}, E

_{c}, and γ(V), the Mie–Grüneisen EOS could be obtained.

## 3. MD Simulation

^{3}with 54,000 atoms. The model sizes of NC iron were about 10.9 × 10.9 × 10.9 nm

^{3}with 111,195 atoms and 54.7 × 54.7 × 54.7 nm

^{3}with 8,268,098 atoms. There were two grain sizes (GSs): 5 nm and 25 nm, which were chosen to study the GS effect in the shocking behavior.

## 4. Results and Discussion

#### 4.1. Shock Hugoniot Pressure and Internal Energy

_{s}ranged from 5.3 to 11.5 km/s at about 40–440 GPa. To ensure the stability of the MD simulation and evaluate the model validity, the velocity of u

_{s}was set in the range of 6 to 11 km/s in this study. Figure 2 shows the relationship between u

_{s}and u

_{p}for SC iron obtained via the MSST method; there is a good agreement between the MD simulation results and experimental data. Moreover, the linear relationship between u

_{s}and u

_{p}is in excellent accordance with the data in Figure 2. The linear relationship is also consistent with the study by Prieto and Renero [54], which indicates that the relationship between u

_{s}and u

_{p}of iron below 5 megabars can be described by a linear relationship. For SC iron, the relationship between u

_{s}and u

_{p}is given as u

_{s}= 4.071 + 1.538u

_{p}. The relationship between u

_{s}and u

_{p}from the experimental data is expressed as u

_{s}= 3.935 + 1.578u

_{p}. The relative errors of C

_{0}and λ obtained from the MSST method and experiment were 3.3% and 2.6%, respectively, which proves the accuracy of the MD simulation.

_{s}–u

_{p}relationship for NC iron, MD simulation was conducted at the same shock velocity as that of SC iron. The results are shown in Figure 3. The results of NC iron with different GS values are close. These discrete data are characterized by u

_{s}= 4.011 + 1.552u

_{p}and u

_{s}= 4.01 + 1.553u

_{p}for GS 5 nm and GS 25 nm, respectively. The relative error of C

_{0}and λ between GS 5 nm and GS 25 nm was less than 0.1%, and compared with the SC iron data, C

_{0}and λ were approximately equal to an error of <1.5%. The results indicate that the GS effect had little influence on the linear relationship between u

_{s}and u

_{p}for SC and NC iron, and it could be neglected in the case of NC iron.

_{0}in SC iron is higher than that in NC iron; however, the fitting parameter λ in SC iron is lower than that in NC iron, and the same characteristics are also observed in SC and NC Al [5]. The maximum relative error of C

_{0}between the experiment and SC iron is 3.34%, and the maximum relative error of NC iron is 1.89%. The difference of C

_{0}between SC and NC iron is ~1.5%, which is less than the difference with experiment data. The same characteristics in λ can be observed; the maximum relative error between the experiment and SC iron is 2.60%, and the maximum relative error of NC iron is 1.67%. The difference of λ between SC and NC iron is ~0.96%. The results indicate that a shock response difference between SC and NC iron does exist, but the difference is not very significant.

_{0}> 0.6, the curves obtained through the MSST method agree well with the results of Brown et al. [11]; when V/V

_{0}< 0.6, the curves are approximately equal to the experiment results, and the deviation is less than 3%. Considering the insignificant error of C

_{0}and λ for SC and NC iron, there is no noticeable difference between the three curves in Figure 4a. As shown in Figure 4b, the internal energies obtained in the MD simulation and experiment were almost the same; when V

_{0}/V = 1.7, the internal energy of SC iron was smaller than that of NC iron, and the deviation at V

_{0}/V = 1.7 was 0.5%. The results not only prove the reliability of the linear relationship between u

_{s}and u

_{p}but also show the EAM potential for iron in the MD simulation. In addition, the results signify that the MD results are appropriate for describing the cold pressure, the cold energy, the melting temperature, and the Mie–Grüneisen EOS.

#### 4.2. Cold Pressure and Cold Energy

_{0}and λ obtained from the Hugoniot data, the material constants for calculating cold pressure and cold energy in Equations (18)–(21) could be obtained. The values are presented in Table 3.

_{c}and cold energy E

_{c}between SC and NC iron, with the experimental data as a reference. The solid and dashed curves represent P

_{c}and E

_{c}obtained using the Born–Mayer and Morse potentials, respectively. The results from the Born–Mayer and Morse potentials presented the same increasing trend: P

_{c}and E

_{c}increased with volume compression (V

_{0K}/V), which presented a sharp increment at higher compression.

_{0K}/V less than 1.3, the curves of P

_{c}agree well with each other. The result shows that the cold energies described by the Born–Mayer and Morse potentials are basically the same in the low-compression region. When the V

_{0K}/V is above 1.3, the P

_{c}curves gradually deviate from each other, and the P

_{c}curves obtained by the Born–Mayer potential gradually become larger. The relative error of P

_{c}was about 10% when V

_{0K}/V was 2.0. Irrespective of the potential function used, the cold pressure curve of SC iron was above that of the NC iron. This indicates that the cold pressure of SC iron was higher than that of NC iron, although the difference was quite small.

_{0K}/V of less than 1.4, the cold energy curves agree well with each other, as the cold pressures obtained by the Born–Mayer and Morse potentials in the low-compression region are approximately equal. Under V

_{0K}/V of above 1.4, the E

_{c}curves gradually deviate from each other. The cold energy obtained by the Born–Mayer potential is larger than that from the Morse potential, and the same trend was found for the cold pressure. The relative error of E

_{c}is about 7% when V

_{0K}/V is 2.0. Furthermore, the cold energy in the SC iron is larger than that of the NC iron in both Born–Mayer and Morse potentials.

#### 4.3. Grüneisen Coefficient

_{0}of the Slater, Dugdale–Macdonald, and free-volume models through Equations (30)–(32). Subsequently, γ

_{0}(V) was substituted into Equations (34)–(38) to obtain γ(V), and the experimental data were used for comparison. Figure 7 shows the results of γ(V) from different empirical and theoretical models.

_{0}/V increased. At high compression, the difference between SC and NC results became smaller; this indicates that the microstructure has little effect on γ(V) under high pressure. Equation (38) is suitable for low-compression data, that is, V

_{0}/V up to 1.5, and is free with λ; thus, different models do not affect their results. With Equation (38) as a reference, when V

_{0}/V = 1.0, the initial value of γ

_{0}is γ

_{f}(V

_{0}) < γ

_{DM}(V

_{0}) < γ

_{S}(V

_{0}). For all three models, the γ(V) obtained by the experimental data (solid curve) is at the top followed by that for the NC iron (dash curve), and that for the SC iron (dot curve) is at the bottom due to the difference between the λ values. Compared with the results from Equations (35)–(38), the results from Equation (34) (black curves) show a more significant downward trend with the increase in V

_{0}/V. This shows that the γ(V) calculated using Equation (34) has the maximum change rate with the increase in V

_{0}/V. The results from Equation (35)–(38) present a similar trend in predicting γ(V).

_{0}/V up to 1.5 [49]. The γ(V) obtained using the free-volume model has the smallest difference under V

_{0}/V up to 1.5. When V

_{0}/V is 1.5, the γ(V) obtained by Equation (37) (cyan curve) intersects with the magenta curve. This indicates that Equation (27) is applicable to characterize the γ(V) of iron at higher compression compared with other expressions. This has also been proved by Jacobs and Schmid-Fetzer [48]. In Figure 7c, the green curve is closer to the cyan curve than other curves. The result shows that Equation (36) (green curve) is more suitable than other expressions for describing γ(V). The relative error of the green curve is 7.8% at V

_{0}/V = 2.0.

#### 4.4. Melting Temperature

_{m}can be obtained. In this study, γ

_{0}is calculated via the free-volume model with Equation (26), and T

_{m}is predicted by Equations (40)–(44). The relationship between T

_{m}and pressure is illustrated in Figure 8, together with experimental data [35,55,56,57,58,59,60,61,62,63] and theoretical data [64,65,66,67,68].

_{m}shows an increasing trend as the pressure increases. The results from Equations (41), (42) and (44) present a similar and close trend; the red curve at the top corresponds to Equation (41), while the green curve at the bottom corresponds to Equation (43), and the maximum error is 29.5% at 450 GPa. Moreover, the melting temperature in SC iron is a little lower than that in NC iron; the maximum error is 2.5% at 450 GPa.

_{m}calculated by Equations (40) and (44) agrees well with DAC experimental data in [57], and T

_{m}calculated by Equation (43) agrees well with DAC experimental data in [59,60,62]. T

_{m}calculated by Equations (41) and (42) is little higher than that obtained from DFT calculations [64,65,66]. However, the difference between the results gradually increased at higher pressure (>100 GPa). The increasing trend indicates that in the low-compression area, the effect of γ

_{0}on T

_{m}is insignificant; the effect of γ

_{0}on T

_{m}becomes more significant with increasing pressure. This feature shows the opposite trend compared with the γ(V) results. In Figure 8, it can be found that the DAC experimental data of Anzellini et al. [35], the theoretical data obtained by MD calculations conducted by Zhang et al. [58], and DFT calculations conducted by Alfè et al. [64,65,66] are in good agreement. Thus, the orange melting curve obtained by the DAC experiment up to ~350 GPa was chosen as the main reference in this study. The results obtained from Equations (40)–(44) are similar to the orange line, although there are some differences in the results. At lower pressure (<100 GPa), the black line obtained by Equation (40) is the closest to the orange line, and the green line (Equation (43)) and purple line (Equation (44)) slightly deviate from the orange line. This indicates that Equation (40) can be used to approximately predict T

_{m}under 100 GPa. In the entire pressure range, the orange line is within the calculation result from Equation (44) at pressure less than 300 GPa; the result obtained from Equation (44) is lower than the orange line above 300 GPa. This result shows that the calculation results under 300 GPa can cover the experimental data, and the results obtained at higher pressures are lower than the experimental data.

#### 4.5. Mie–Grüneisen Equation of State

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Configurations of (

**a**) SC iron and (

**b**) NC iron with GS of 5 nm and (

**c**) NC iron with GS of 25 nm. Separate grains are distinguished by different colors.

**Figure 2.**The relation between u

_{s}and u

_{p}for SC at 300 K (the linear fitting curve is applied to the MD results; experimental data are shown for comparison).

**Figure 3.**The relationship between u

_{s}and u

_{p}for NC iron at 300 K: (

**a**) NC iron with GS 5 nm (

**b**) NC iron with GS 25 nm (the linear fitting curve is applied to the MD results; experimental data are shown for comparison).

**Figure 7.**The relationship between the Grüneisen coefficient γ(V) and V

_{0}/V. (

**a**) γ(V) calculated from the Slate model, (

**b**) γ(V) calculated from the Dugdale-MacDonald model, (

**c**) γ(V) calculated from the Free-volume model.

**Figure 9.**Mie–Grüneisen EOS in P-V-E form with reference from the cold curve (

**a**) and the Hugoniot curve (

**b**).

**Table 1.**Hugoniot data for SC and NC iron determined by MD simulations at 300 K. An initial density of 7.85 g/cm

^{3}is used.

Material | u_{p} (km/s) | u_{s} (km/s) | P (GPa) | ρ (g/cm^{3}) |
---|---|---|---|---|

SC iron | 1.25 | 6.00 | 60.84 | 9.92 |

1.90 | 7.00 | 102.92 | 10.78 | |

2.55 | 8.00 | 158.50 | 11.53 | |

3.20 | 9.00 | 224.72 | 12.19 | |

3.85 | 10.00 | 301.71 | 12.77 | |

4.50 | 11.00 | 389.62 | 13.29 | |

NC iron (GS 5 nm) | 1.27 | 6.00 | 62.36 | 9.98 |

1.92 | 7.00 | 105.82 | 10.82 | |

2.57 | 8.00 | 161.41 | 11.56 | |

3.21 | 9.00 | 227.10 | 12.21 | |

3.85 | 10.00 | 302.92 | 12.78 | |

4.50 | 11.00 | 388.85 | 13.29 | |

NC iron (GS 25 nm) | 1.28 | 6.00 | 62.34 | 9.98 |

1.92 | 7.00 | 105.74 | 10.82 | |

2.56 | 8.00 | 161.22 | 11.56 | |

3.21 | 9.00 | 226.75 | 12.20 | |

3.85 | 10.00 | 302.34 | 12.77 | |

4.50 | 11.00 | 387.98 | 13.28 |

**Table 2.**C

_{0}and λ obtained from the linear relationship between u

_{s}and u

_{p}for SC and NC iron at 300 K. The initial density is 7.85 g/cm

^{3}in both the experiment and MD simulation.

Material | Shock Velocity | C_{0} (km/s) | λ | Relative Error | Reference | |||||
---|---|---|---|---|---|---|---|---|---|---|

SC | NC (GS 5 nm) | NC (GS 25 nm) | ||||||||

C_{0} | λ | C_{0} | λ | C_{0} | λ | |||||

Commercial iron | 5.3–11.5 km/s | 3.935 | 1.578 | 3.34% | 2.60% | 1.89% | 1.67% | 1.87% | 1.61% | [11] |

SC | 6–11 km/s | 4.071 | 1.538 | - | - | 1.49% | 0.90% | 1.50% | 0.96% | This work |

NC (GS 5 nm) | 6–11 km/s | 4.011 | 1.552 | 1.49% | 0.90% | - | - | 0.02% | 0.06% | This work |

NC (GS 25 nm) | 6–11 km/s | 4.010 | 1.553 | 1.50% | 0.96% | 0.02% | 0.06% | - | - | This work |

Material Constants | V_{0} (g/cm^{3}) | ${\mathit{C}}_{0}^{\prime}(\mathbf{km}/\mathbf{s})$ | λ′ | Q (GPa) | q | A (GPa) | B |
---|---|---|---|---|---|---|---|

Experiment | 0.127 | 4.010 | 1.581 | 41.245 | 11.192 | 87.662 | 4.325 |

SC iron | 0.127 | 4.094 | 1.540 | 46.473 | 10.716 | 97.319 | 4.162 |

NC iron | 0.127 | 4.093 | 1.541 | 44.238 | 10.894 | 93.170 | 4.223 |

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**MDPI and ACS Style**

Wang, Y.; Zeng, X.; Chen, H.; Yang, X.; Wang, F.; Ding, J.
Hugoniot States and Mie–Grüneisen Equation of State of Iron Estimated Using Molecular Dynamics. *Crystals* **2021**, *11*, 664.
https://doi.org/10.3390/cryst11060664

**AMA Style**

Wang Y, Zeng X, Chen H, Yang X, Wang F, Ding J.
Hugoniot States and Mie–Grüneisen Equation of State of Iron Estimated Using Molecular Dynamics. *Crystals*. 2021; 11(6):664.
https://doi.org/10.3390/cryst11060664

**Chicago/Turabian Style**

Wang, Yuntian, Xiangguo Zeng, Huayan Chen, Xin Yang, Fang Wang, and Jun Ding.
2021. "Hugoniot States and Mie–Grüneisen Equation of State of Iron Estimated Using Molecular Dynamics" *Crystals* 11, no. 6: 664.
https://doi.org/10.3390/cryst11060664