Abstract
High-pressure third-order elastic constants of materials have rarely been investigated experimentally and theoretically to date, so the predictive ability of the method of the volume-conserving, homogeneous deformations based on the first-principles total-energy calculations is tested for the first time in this work. Using this approach, the high-pressure third-order elastic constants
1 Introduction
In the theory of nonlinear elasticity, the third-order elastic constants (TOECs) are important quantities for characterising the nonlinear elastic properties of materials, and numerous studies [1], [2], [3], [4], [5], [6] have been carried out to investigate them. Most of these studies, to the best of our knowledge, are restricted to the TOECs of materials at zero pressure, and little research on the high-pressure TOECs has been performed till now. Nevertheless, knowledge of the high-pressure TOECs of materials can assist us in predicting and understanding the interatomic forces, mechanical stability [7], [8], dynamic fracture [9], earthquakes, and the internal structures of the Earth. Therefore, investigations of the high-pressure TOECs are very necessary and interesting.
Experimentally, the subtle TOECs are quite sensitive to the experimental conditions and usually exhibit distinct divergence between the values measured by different teams (see [10], [11]). Up to now, experimental determination of the zero-pressure TOECs still presents significant difficulties [2], [3], to say nothing of the measurement of the high-pressure TOECs. However, the accuracy and reliability of theoretical calculations have substantially increased recently, and the computed results have been accepted by the research community almost at the same level of trust as the measured data. Naturally, theoretical calculations have become an important approach to predict the high-pressure TOECs of materials. At present, several theoretical methods [12], [13], [14], [15], [16] have been developed to calculate the zero-pressure TOECs of materials, such as the approach of homogeneous deformation based on empirical or first-principles total-energy calculations [12], the empirical interatomic force constant models [13], molecular dynamics simulation using the fluctuation formula [14], and other quantum methods [15], [16]. Among these approaches, the method of homogeneous deformation based on the first-principles total-energy calculations seems to be the most promising one and it has been successfully employed in the literature [1], [4], [5], [6]. Nevertheless, this approach has not been used to calculate the high-pressure TOECs of materials yet.
In this work, we take the MgO single crystal as an example and try to obtain its high-pressure TOECs by using the method of the homogeneous deformation based on first-principles total-energy calculations. The reasons for choosing the MgO single crystal are as follows: First, MgO, as one of the major earth-forming minerals, has great geophysical importance. Prediction of its high-pressure TOECs is very important for interpreting the seismically observed velocity variations in the Earth’s mantle and for constructing reliable mineralogical modes of the Earth’s interior [17]. Second, MgO, as one of the most stable oxides, can retain its simple NaCl-type structure under very high pressure (>227 GPa) [18], thereby making it an ideal candidate for the calculations of its high-pressure TOECs. Third, the MgO single crystal belongs to the cubic structure, which has the least number of independent TOECs among all single crystals; accordingly, the number of the homogeneous deformations required for obtaining the complete set of high-pressure TOECs is the least. Last but not least, many experimental and other theoretical results on its structural and elastic properties are available and they can be employed to verify the accuracy of the present calculations. The rest of this paper is organised as follows: In Section 2, the nonlinear elastic theory, special deformations required for the calculations of the high-pressure TOECs, and computational details are described briefly. Subsequently, the calculated results and discussion are given in Section 3. Finally, conclusions are drawn in Section 4.
2 Nonlinear Elastic Theory and Computational Details
2.1 Nonlinear Elasticity Theory
In the theory of nonlinear elasticity, we suppose that the un-deformed state X corresponding to a certain pressure P moves to the deformed state x when a crystal undergoes a finite deformation. Here, we denote the un-deformed coordinates of a point in configuration X by (X1, X2, X3) and represent those in configuration x by (x1, x2, x3). Then, the deformation gradient matrix can be defined as
where the subscripts i and j vary from 1 to 3. The Lagrangian strain
in which
so the condition describing the conservation of volume is
In addition, the corresponding change of the Helmholtz free energy F and the internal energy U per unit mass in the configuration x are [19], [20]
and
Then the corresponding isothermal and adiabatic stress and the second-order and third-order elastic constants at X state can be defined as
and
where V(X) is the volume of the system at X state, and the subscript
In the Voigt convention (
where
and
If we apply a series of proper, homogeneous deformations to the X state and calculate the total energies of the X and x states using the first-principles method, we can easily obtain the corresponding elastic constants by using (9)–(11). In the following, the specific deformations and computational details of the total energies are presented.
2.2 Deformation Types
For a cubic crystal, the calculation of the high-pressure TOECs is similar to that of the high-pressure SOECs. Specifically, when the SOECs of a cubic crystal under a certain pressure P are calculated, we first use the two volume-conserving homogeneous deformations displayed as matrixes (30) and (32) in [21] to obtain the two shear elastic constants
To determine the elastic constant C111 − 3C112 + 2C123, we adopt the volume-conserving homogeneous deformation
which increases the length of all vectors parallel to [100] and uniformly reduces that of all vectors perpendicular to [100]. According to (2), we calculate the corresponding Lagrangian strain
where η is the strain magnitude, F(η) and F(0), respectively, represent the total energies of the strained and unstrained unit cells, V is the volume of unstrained unit cell, and P denotes the pressure applied on the reference state (state X).
With the purpose of calculating the elastic constant C456, the deformation (14) is used:
It represents that all vectors parallel to [111] are increased while all vectors perpendicular to [111] are uniformly reduced. And the corresponding expression for the energy is
With the view to obtaining the elastic constant
and its related energy expansion
Finally, we apply the uniform and isotropic (hydrostatic) deformation (18) to calculate
Accordingly, the total energy of the strained state can be expressed as
To the best of our knowledge, only the four homogeneous deformations (12), (14), (16), and (18) could be employed to calculate the high-pressure TOECs of the MgO single crystal so far. Consequently, we can only obtain
2.3 Computational Details
2.3.1 Density Functional Theory (DFT) Calculations
It is well known that the NaCl-type (B1) structure of the MgO single crystal will transform into the CsCl-type (B2) structure under high pressure. And knowledge of the structural stability of the B1 structure is a precondition for the calculations of the high-pressure TOECs, so the structural optimisations and calculations of the total energies for the two structures B1 and B2 are performed using the Vienna ab initio Simulation Package (VASP) [23]. In particular, the generalised gradient approximation (GGA) in the Perdew-Burke-Ernzerhof (PBE) parameterisation [24] is used to treat the exchange-correlation effects. Simultaneously, the projector augmented-wave (PAW) potential [25] is employed to deal with the ion-electron interactions. In addition, careful convergence tests (tolerance for the total-energy difference in the self-consistent-field calculation is 1.0 × 10−9 eV and tolerance for the maximum force in the geometry optimisation is 1.0 × 10−5 eV/Å) are performed to determine the cut-off energy and the number of k-points. And the reasonable cut-off energy (700 eV for B1 and 650 eV for B2 structures) and rational k-points (17 × 17 × 17 for B1 and 15 × 15 × 15 for B2 structures) are ascertained. Afterwards, the total energies of various unit cell volumes in both B1 and B2 structures are calculated and a series of energy-volume data are obtained. Then, we obtain their equation of states (EOS) at 0 K, equilibrium volumes V0, bulk modulus B0, and their pressure derivatives
2.3.2 Determination of the Third-Order Elastic Constants
To calculate the TOECs of the single-crystal MgO in B1 structure at a certain pressure, we first ascertain the equilibrium state (un-deformed state) under this pressure by relaxing the cell parameters and then obtain the deformed state by using the equation
Since TOECs are very sensitive to the cut-off energy, k-points, the maximum strain magnitude
Based on the parameters ascertained above, the relations of η and
3 Results and Discussion
3.1 Structural Properties
To verify the accuracy of our DFT calculations, we list the computed structural parameters (equilibrium lattice parameter a, bulk modulus B0, and its pressure derivations B0′ and B0′′) for both the B1 and B2 structures of the MgO in Table 1, and compare them with the available experimental results [31], [29], [35], [39], [37] and other theoretical data [32], [34], [38], [28], [30], [33], [42], [43], [36]. Table 1 clearly shows that our structural parameters are in good agreement with the corresponding experimental as well as previous theoretical values. Experimentally, for the B1 structure of MgO, the calculated lattice constant 4.256 Å is slightly larger than the experimental data of 4.19 Å [29] and 4.213 Å [31], whereas the computed bulk modulus B0 (155.5 GPa) is somewhat smaller than the measured values (160–168.8 GPa) [29], [31], [35], [37]. Similarly, when the present value 4.14 of the B0′ is compared with the corresponding experimental ones of 4.15 [31] and 4.5 [39], we find that it is also underestimated by about 0.2–8.0 %. Such phenomena can be attributed to the following two aspects. On one hand, the theoretical data is calculated at 0 K while the experimental results are usually obtained at 300 K. On the other hand, GGA calculations typically overestimate the equilibrium lattice parameters and underestimate the bulk modulus of materials at zero pressure. Theoretically, all the computed structural properties of the B1 structure are well consistent with the previous theoretical values. As regards the structural parameters of the B2 structure, though there is no corresponding experimental results for comparison at present, all the data obtained in this work show excellent agreement with the previous theoretical predications [30], [42], [43].
Present work | Theoretical references | Experimental references | |||
---|---|---|---|---|---|
NaCl-type structure | |||||
a (Å) | 4.256 | 4.250 | [28] | 4.19 | [29] |
4.240 | [30] | 4.213 | [31] | ||
4.273 | [32] | ||||
4.259 | [33] | ||||
B0(GPa) | 155.5 | 158 | [34] | 168.8 | [35] |
153.9 | [28] | 160 | [31] | ||
158.7 | [36] | 163 | [37] | ||
153 | [32] | 164.6 | [29] | ||
160 | [33] | ||||
B0′ | 4.14 | 4.18 | [38] | 4.15 | [31] |
4.26 | [34] | 4.5 | [39] | ||
3.95 | [36] | ||||
4.00 | [30] | ||||
B0′′ (GPa−1) | −0.028 | −0.025 | [30], [38] | – | |
−0.026 | [34] | ||||
−0.024 | [36] | ||||
Ptr (GPa) | 507 | 490 | [30] | >227 | [18] |
510 | [40] | ||||
509 | [28] | ||||
515 | [41] | ||||
512 | [32] | ||||
CsCl-type structure | |||||
a (Å) | 3.352 | 3.32 | [30] | – | |
B0 (GPa) | 137.493 | 140.30 | [42] | – | |
134.33 | [43] | ||||
B0′ | 4.129 | 4.10 | [42] | – | |
4.24 | [43] | ||||
B0′′ (GPa−1) | −0.03 | – | – |
In addition, the pressure dependence of our volume ratio (V/V0, where V0 is the equilibrium unit cell volume of the B1 structure) is also compared with that obtained experimentally [18], [44], [45], [46], [47], [48], [49] and in previous calculations [38], [28], [50], [51], [52], [53], [54], which is illustrated in Figure 6a and b, respectively. From Figure 6a, we see that our results are slightly smaller than those measured by Mao and Bell [44] in the pressure range 50–95 GPa. This may be induced by the existence of the non-hydrostatic stresses in their experiment. However, it is delightful that our theoretical results are consistent with the other experimental values [18], [45], [46], [47], [48], [49]. Besides, comparison between our data and previous theoretical results [28], [38], [50], [51], [52], [53], [54] shown in Figure 6b also shows satisfactory agreement. These good consistencies indicate that the choice of the PAW pseudo-potential and the GGA-PBE approximation is reasonable for the current study and that the present calculations are reliable.
To ascertain the highest pressure of TOECs of the B1 structure of the MgO single crystal, we calculate the enthalpies H = E + PV of the B1 and B2 structures at 0 K and display them in Figure 7. For the sake of clarity, we also give their enthalpy differences as a function of pressure in the insert of Figure 7. It is clear from Figure 7 that the phase transition (B1⟶B2) of the MgO single crystal occurs at about 507 GPa. For comparison, we list it together with the available experimental results [18] and other theoretical predications [28], [30], [32], [40], [41] in Table 1. Results show that our predicted value 507 GPa is very close to the previous data of 509 GPa [28], 490 GPa [30], 512 GPa [32], 510 GPa [40], and 515 GPa [41]. The only small differences (about 0.4–3.5 %) between them may originate from the different computational procedures and forms of exchange correlation functional used by different groups. That is to say, the transition pressure predicted in this study is reliable. Therefore, the high-pressure elastic constants of the MgO single crystal in the B1 structure are investigated only up to 500 GPa in the following.
3.2 Elastic Constants
Since the experimental and previous theoretical values of the high-pressure SOECs of MgO in the B1 structure are available, and they can be used to test the accuracy of our high-pressure elastic constants, the high-pressure SOECs are considered first before the high-pressure TOECs are discussed. In Figure 8, pressure dependence of B44 (
Having established the accuracy of the elastic constants calculations, we now turn to the TOECs of the MgO single crystal. Among the six independent TOECs, namely C111, C112, C123, C144, C155, and C456, only C456 is calculated directly through the four homogeneous deformations (12), (14), (16), and (18). So the zero-pressure value 106.49 GPa of C456 is first compared with the existing experimental data of 147 GPa [39] and other theoretical values of 307.9 GPa [56] and 356 GPa [57]. Comparisons show that our C456 agrees much better with the experimental data than the previous theoretical results. And the difference between the present result and the experimental data may be caused by the following two factors. First, it is rather difficult to measure TOECs experimentally, which leads to considerable uncertainties in the reported experimental results inevitably. Second, the theoretical result is calculated at 0 K while the experimental values are often determined in conditions that are far from the ideal case of 0 K. Particularly, the temperature effects on the TOECs of the MgO are distinguishable and should not be ignored (see [29]). Therefore, the deviation between them is reasonable and understandable.
With regard to the high-pressure C111 − 3C112 + 2C123, C111/2 + 3C112 + C123, C144 − C155, and C456, there is no related experiment or calculation till now, so our predications first obtained by using the method of the volume-conserving homogeneous deformation based on first-principles total-energy calculations are expected to serve as a valuable guidance or reference for further related investigations. As shown in Figure 9, the absolute values of
Finally, it is worth noting that though the six independent high-pressure TOECs C111, C112, C123, C144, C155, and C456 of the MgO single crystal could not be calculated separately in the present work, the first attempt to calculate the high-pressure TOECs of crystals by using the method of the volume-conserving homogeneous deformation based on first-principles total-energy calculations has been successful and encouraging. We hope that the present work would make a useful contribution to the development of nonlinear elasticity at high pressures.
4 Conclusions
The predictive ability of the method of the volume-conserving homogeneous deformation based on the first-principles total-energy calculations for computing the high-pressure TOECs of single crystals was tested successfully for the first time in the present work. Using this approach, the pressure dependence of
Funding source: National Natural Science Foundation of China
Award Identifier / Grant number: 11747062
Award Identifier / Grant number: 11747110
Award Identifier / Grant number: 11504035
Funding statement: This work was supported by the National Natural Science Foundation of China (Funder Id: 10.13039/501100001809, Grant Nos. 11747062, 11747110 and 11504035), the Science and Technology Tackling Project of the Education Department of Henan Province (Grant No. 172102210072), the Key Scientific Research Project of Higher Education of Henan Province (No. 17A140014), and the Science and Technology Research Project of the Chongqing Education Committee (No. KJ1703062).
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