Phys Chem Minerals (2013) 40:521–530 DOI 10.1007/s00269-013-0589-1
ORIGINAL PAPER
Elastic, mechanical, and thermodynamical properties of superionic lithium oxide for high pressures Dinesh Varshney • Swarna Shriya
Received: 18 January 2013 / Accepted: 2 April 2013 / Published online: 14 April 2013 Ó Springer-Verlag Berlin Heidelberg 2013
Abstract The elastic and thermodynamic properties of Li2O for high pressures are presented. For cubic Li2O, model effective interatomic interaction potential incorporating long-range Coulomb, charge transfer interactions, covalency effect, Hafemeister and Flygare type short-range overlap repulsion extended up to the second neighbor ions and van der Waals interactions is formulated. Both charge transfer interactions and covalency effect apart from long-range Coulomb are important in revealing highpressure-induced associated volume collapse, elastic, and thermodynamical properties. The elastic constants, Debye temperature, and thermal expansion coefficient obtained are in good agreement with the available experimental data and other theoretical results. The Li2O is mechanically stiffened, thermally softened, and brittle in nature as inferred from the pressure-dependent elastic constants behavior. To our knowledge, this is the first quantitative theoretical prediction of the pressure dependence of elastic, thermal, and thermodynamical properties of Li2O and still awaits experimental confirmation. Keywords Superionic conductors Mechanical stiffening Thermal softening Brittle Debye temperature
Introduction Lithium oxide (Li2O) exhibits high ionic conductivity while in solid condition referred as superionics allowing
D. Varshney (&) S. Shriya Materials Science Laboratory, School of Physics, Vigyan Bhawan, Devi Ahilya University, Khandwa Road Campus, Indore 452001, India e-mail:
[email protected]
macroscopic movement of ions through their structure. Lithium oxide possesses several technological applications due to its high Li atomic density and high melting temperature. Its usage ranges from high power density Li ion batteries for heart pacemakers, mobile phones, laptop computers, etc., to high-capacity energy storage devices for next generation, ‘‘clean’’ electric vehicles, and blanket breeding material in nuclear fusion reactors (Keen 2002; Hu and Ruckenstein 2004). The elastic properties are of major concerned, and efforts have been made to measure the bulk modulus, elastic constants under ambient conditions of Li2O (Hull et al. 1988; Kurasawa et al. 1982; Oishi et al. 1979), and electron momentum spectroscopy (EMS) for elucidating the electronic band structure of Li2O (Mikajlo et al. 2002). Further computational approaches viz Hartree–Fock method (Dovesi et al. 1991), density functional theory (DFT) in the framework of the local-density approximation (LDA) (Rodeja et al. 2001; Weiyi et al. 2010; Zhuravlev and Obolonskaya 2011; Zhuravlev et al. 2012; Hautier et al. 2012), molecular dynamics (MD) simulation method (Goel et al. 2004), aspherical ion model (AIM) (Wilson et al. 2004), ab initio unrestricted Hartree–Fock (HF) linear combination of atomic orbital (LCAO) (Li et al. 2006), Wannier-function-based HF approach (Shukla et al. 1998), and lattice dynamical studies (Samsonov 1973; Jacobs and Vernon 1990) are employed to probe cohesive energy and elastic properties of Li2O. However, the investigations, which specially focus on mechanical, elastic, and thermal properties, are rather sparse. Among the lattice models, which have been invoked so far, to discuss the mechanical properties of several solids and alloys, interatomic interactions concentrate on Hafemeister and Flygare (1965), type overlap repulsion extended up to second neighbor ions besides short-range interactions.
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Herein, charge transfer interactions and effect of covalent nature need to be incorporated for a comprehensive study (Varshney et al. 2012a). The developed effective interatomic interaction potential includes vdW attraction, which is not explicitly accounted in band structure calculations. It is worth noting that vdW interaction appears to be effective in revealing the elastic and structural properties of binary alloys (Tosi 1964; Hafemeister and Flygare 1965). The quantum mechanical computational methods are a powerful probe; however, empirical approaches are also farsighted. For example, they allow one to calculate more complex properties such as temperature dependence, etc. Herein, we aimed at that empirical approaches as being more accurate than the first principles and hence yields more predictive results of derived quantities, such as elastic constants and higher-order elastic constants, which are basically derivatives of the total energy, can be done analytically. In the present paper, we aimed at investigating the pressure variations of second-order elastic constants and those of the third-order elastic constants. The main focus is the pressuredependent mechanical properties as ductility (brittleness), lattice softening (hardening), and sound velocity of Li2O. Computational method The cohesive energy and allied physical properties of materials are determined by different form of the potential energy functions. The idea we have in mind follows: the change in force constants is small, the short-range interactions are effective up to the second neighbor ions, and the atoms are held together with harmonic elastic forces without any internal strains within the crystal. Usually, the applications of pressures cause an increase in the overlap of adjacent ions in a crystal, and hence, charge transfer takes place between the overlapping electron shells (Varshney et al. 2012a). While developing interaction potential, the basis orbitals used in Li2O: [Li: 1s1 and O: 1s2, 2s2 2p4]. Thus, the ‘p’-like state of oxides hybridizes with Li ‘s’-like states. The transferred charges interact with other charges of lattice via Coulomb’s law leading to charge transfer interactions, which are significant in Li2O due to mobility of carriers under pressure. For cohesion, the significant contributions comes from long-range electrostatic, short-range vander Waals (vdW), overlapping due to charge transfer and polarization. The potential energy functions UB (r) is expressed as UB ¼ ðaM Ze2 =rÞ½Z þ 2nf ðrÞ Cr 6 Dr8 þ nbbij exp ri þ rj rij =q þ ðn0 b=2Þ bii exp 2ri krij =q þ bjj exp 2rj krij =q ð1Þ Here, the first two terms, the ionic charge for each atom cannot be determined uniquely, and hence, the calculation
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of the Madelung energy is modified by incorporating the covalency effects (Varshney et al. 2005). We thus need to incorporate the effective charge arose due to the polarization of a spherical-shaped dielectric in displacing the constituent positive ions. Third and fourth terms are the short-range (SR) vdW attraction and the potential energies due to long-range Coulomb, and the second term is due to charge transfer effects caused by the deformation of the electron shells of the overlapping ions and the covalency effects. The C and D are the overall vdW coefficients, which are evaluated from the variational approach (Tosi 1964). The vdW coefficients due to dipole–dipole and dipole–quadruple interactions are calculated from the Slater and Kirkwood variational approach, and the details are given elsewhere (Varshney et al. 2012a). The Li2O contains covalent bonds so that some electrons are distributed over the region between neighboring atoms; in such situation energies due to dipole–dipole and dipole– quadrupole interaction. Last term is short-range (SR) repulsive energy due to the overlap repulsion between ij, ii, and jj ions. am is the Madelung constants. bij are the Pauling coefficient defined as bij = 1 ? (Zi/ni) ? (Zj/nj) with Zi(Zj) and ni(nj) as the valence and the number of electrons in the outermost orbit. The symbol n (=4) and n0 (=6) are the numbers of the nearest unlike (n) and like (n0 ) neighbors, Ze is the ionic charge, k is being the structure factor, and b (q) is the hardness (range) parameters. The nearest-neighbor ion separations are r, respectively. The second term in Eq. (1) is an algebraic sum of central force part of the charge transfer force parameter, and the force parameter arises due to covalent nature, i.e., f(r) = fcti ? fcov. The charge transfer force parameter fcti is expressed as: fcti ¼ f0 expðr=qÞ. Here, ri (rj) are the ionic radii of ions i(j). Keeping in mind that Li2O semiconducting compound is covalent in bonding, attractive forces due to covalency are important that modifies the effective charge. The polarization effects originate from changes in covalency due to Li–Li, Li–O, and O–O interacting electric fields. The covalency term is expressed as 2 fcov ðrÞ ¼ 4e2 Vspr ½r0 Eg3 1 . Herein, Vspr is being the transfer matrix element between the outermost p orbital and the lowest excited of s state, and Eg is the transfer energy of electron from p to the s orbital. The effective charge es of the host crystal is related to the number of electrons transferred to the unoccupied orbitals from its surrounding nearest neighbor and is nc ¼ 1 es =e. Henceforth, for overlap distortion effect, es 6¼ e. Furthermore, nc =12 ffi 2 Vspr =Eg2 and the transfer matrix element Vspr and the transfer energy Eg are related to effective charge es follow2 ing: Vspr ½Eg2 1 ¼ ð1 es Þ=12. The transfer energy Eg is Eg ¼ E I þ ð2a 1Þe2 =r. Here, E is being the electron
Phys Chem Minerals (2013) 40:521–530
affinity for the non-metal atom, and I is the ionization potential of constituent metal atom (Varshney et al. 2012a). The static dielectric constant e0 and the high frequency dielectric constant e? are intimately related to Szigeti 2 effective charge es ð¼ ZeÞ as follows: e2 s ¼ 9lxTO ðe0 2 1 2 2 e1 Þ ½4pNk ðe1 þ 2Þ and e2 s =e ¼ ½9VlxTO ðe0 e1 Þ 2 1 2 ½4pe ðe1 þ 2Þ . Here, l is the reduced mass, Nk is the number of atoms present per unit cell volume i. e. Nk = 1/V, and xTO is long wavelength transverse optical phonon frequency. Thus, for Li2O, e*S deviates from e and is attributed to covalent nature of Li–Li, Li–O, and O–O bonds. In alkali metal oxides, Li2O ‘p’-like state of oxides hybridizes with Li ‘s’- ‘d’-like states. The empirical approach leads to analytical calculations assuming zero temperature, that is, the frozen ionic degrees of freedom. Although the experimental results are obtained at ambient temperature inferring a certain small temperature dependence of the transition pressures in the range of low temperatures as well on polycrystalline or bulk samples. Nevertheless, it is safe to consider the lattice model calculation results as representative of the results that would be obtained under the actual experimental conditions. At zero temperature, the thermodynamically stable phase at a given pressure is the one with lowest enthalpy, and the thermodynamical potential is the Helmholtz free energy. The Gibbs’s free energies is given by Born equation GB (r) = UB (r) ? PVB (Born and Huang 1956). Here, VB as the unit cell volume and r being the nearest-neighbor distance.
Results and discussions Knowledge of the force constants is important parameter for finding out the elastic properties at different pressures. The formalism described above is applied to Li2O belonging to the cubic crystal system. Two different factors determine the response of any crystal structure to pressure. First, changes in nearest-neighbor distances, which affect the overlaps and bandwidths of the bands. Second, changes in symmetry, which affects the hybridization and bond repulsion. For such purposes, we have then five material parameters, namely modified ionic charge, hardness, range, charge transfer, and covalency parameter [Zm, b, q, fcti and fcov]. We can then obtain these values from the equilibrium conditions (Varshney et al. 2012a). We have undertaken such elastic, thermal, and thermodynamical properties in an ordered way. To estimate the Li2O alkali metal oxide parameters, we begin by deducing the vdW coefficients C and D involved in expression (1) from the Slater–Kirkwood variational
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method (Slater and Kirkwood 1931). The estimated crystal parameters are as follows: vdW coefficients [cii (=46.59 9 10-60 erg cm6), cij (=78.438 9 10-60 erg cm6), cjj (=147.586 9 10-60 erg cm6), C (=424.749 9 10-60 erg cm6), dii (=54.352 9 10-76 erg cm8), dij (=75.578 9 10-76 erg cm8), djj (=103.421 9 10-76 erg cm8) D (=328.211 9 10-76 erg cm8)], respectively. The vdW coefficients are influenced by electronic polarizabilities and have been directly taken from least-squares fit of experimental refraction data (Tessman et al. 1953) using additive rule and a Lorentz factor of 4p/3. For further computation, we have used the experimental information on lattice constant (a) (Hull et al. 1988), the bulk modulus (BT) (Hull et al. 1988), ionic (Ze), effective charge (es*), and the second-order aggregate elastic constant (SOEC) C12 (C44) (Hull et al. 1988). While estimating the effective charge es , the values of optical dielectric constant es and the high frequency dielectric constant e?, and the long wave length transverse optical phonon frequency xTO are taken from (Jacobs and Vernon 1990) to have the covalency contribution. These enable us to deduce ˚ ), rj (=2.68 A ˚ ), optimized value of ionic radii ri (=1.25 A -9 hardness b (=1.955 9 10 erg), range q (=9.76 9 10-9 cm), charge transfer parameter f (r) (=4.564 9 10-3), and ˚ ). equilibrium distance: (r0 =4.9 A We have computed the cohesive energy as -30.305 eV for Li2O, consistent with earlier experimental value of -29.0 eV (Samsonov 1973), and theoretical value of -30.8 eV (Jacobs and Vernon 1990) -30.08 eV (Shukla et al.1998), respectively. It is noticed that the central force as charge transfer parameter f(r) is positive and is attributed to the fact that the charge transfer parameter is computed from the difference of second-order elastic constants C12 and C44. For Li2O, Cauchy energy C12–C44 is negative. We have estimated the values of relative volumes associated with various compressions following Murnaghan equation of state (Murnaghan 1944). The estimated value of pressure-dependent radius and the decrease in magnitude of volume collapse with pressure are illustrated in Fig. 1 for Li2O. We see that when pressure increases, the curve of V/V0 becomes steeper, indicating that Li2O is compressed much more easily and consistent with earlier reports (Goel et al. 2004; Li et al. 2006; Weiyi et al. 2010). We have computed the aggregate elastic constants at normal and under hydrostatic pressure by using developed effective interatomic potential (Varshney et al. 2012a) and are listed in Table 1. The second-order aggregate elastic constants Cij under hydrostatic pressure are obtained with respect to finite strain using the stress–strain coefficients and proper consideration of long-range as Coulomb and central force as charge transfer interactions and covalency effect, short-range as overlap repulsion extended up to the
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second neighbor ions and van der Waals interactions. For a cubical symmetry, there are three independent elastic constants, and Fig. 2a illustrates the variation of three independent SOEC referred to as C11, C12, and C44 with external pressure for Li2O. The second-order elastic constant C11 is usually a measure of resistance to deformation by a stress applied on (1,0,0) plane with polarization in the direction h100i, and the C44 refers to the measurement of resistance to deformation with respect to a shearing stress applied across the (100) plane with polarization in the h010i direction. Henceforth, the elastic constant C11 represents elasticity in length, and a longitudinal strain produces a change in C11. No doubt, the elastic constants C12 and C44 are intimately related to the elasticity in shape, which is a shear constant. However, a transverse strain causes a change in shape without a change in volume. Thus, the second-order aggregate elastic constants, C12 and C44, are less sensitive of pressure as compared to C11. As inferred from Fig. 2a, C11, C12, C44, and BT increase monotonically with increasing pressure, and the elastic constants under pressure are consistent, which indicates that Li2O should be stable under these pressures. Similar observations have
earlier been reported for Li2O (Li et al. 2006; Weiyi et al. 2010). According to Born criterion for a lattice to be mechanically stable states, the elastic energy density must be a positive definite quadratic function of strain. The principal minors (alternatively the Eigen values) of the elastic constant matrix should all be positive at ambient conditions. Further the mechanical stability conditions on the elastic constants in cubic crystals are BT ¼ ðC11 þ 2C12 Þ=3 [ 0; C11 ; C44 [ 0; and CS ¼ ðC11 C12 Þ=2 [ 0 (Born and Huang 1956). Here, Cij are the conventional aggregate elastic constants, and BT is the bulk modulus. The quantities C44 and CS are the shear and tetragonal moduli of a cubic crystal. Estimated values of bulk modulus (BT), shear moduli (C44), and tetragonal moduli (CS) well satisfied the above elastic stability criteria for Li2O and are listed in Table 1. The variation of aggregate bulk modulus with pressure is illustrated in Fig. 2b. It is noticed that in Li2O, the lattice is stiffened with increased in pressure and becomes stiffer at higher pressure. The mechanical stiffened bulk modulus in lithium oxide is attributed to Li–Li, O–O, and Li–O bond compression and bond strengthening due to lattice vibration. We comment that the computational details infer that the pressure-dependent mechanical induced stiffening of the lattice of Li2O is attributed to enhancement of the cohesive energy. The calculated values of pressure derivatives of SOEC’s (dBT/dP, dC44/dP and dCS/dP) are given in Table 1 and are also compared with available experimental (Hull et al. 1988) and theoretical studies (Dovesi et al. 1991; Rodeja et al. 2001; Zhuravlev and Obolonskaya 2011; Goel et al. 2004; Wilson et al. 2004; Li et al. 2006). The resulting forces (long range and short range) are only in the direction of the nearest neighbors (central force model). Usually, the Cauchy discrepancy D21 = C12 C44 - 2P is a measure of the contribution from the noncentral many-body force. However, for pure central
Table 1 Calculated aggregate second-order elastic constants (C11, C12 and C44), aggregate bulk modulus (BT), pressure derivatives of SOECs (dBT/dP, dC44/dP and dCS/dP), second-order elastic constant
anisotropy parameter (c21), isotropic shear modulus (GH), Voigt’s shear modulus (GV), Reuss’s shear modulus (GR), Young’s modulus (E), Poisson ratio (m) for lithium oxide at zero pressure
Li2O
1.0
V(P)/V(0)
0.9
0.8
0.7
0.6 0
20
40
60
80
100
P(GPa)
Fig. 1 Equation of state of lithium oxide
Property
Present
Expt.
Others
C11 (1010 Nm-2)
21.74
21.7 ± 0.4 [3]
23.37 [6], 23.8 [7], 21.3 [9], 20.2 [10], 20.2 [8], 23.65 [11]
C12 (1010 Nm-2)
2.56
2.5 ± 0.6 [3]
2.21 [6], 1.6 [7], 5.6 [9], 1.9 [10], 2.01 [8], 2.75 [11]
10
-2
6.19
6.8 ± 0.1 [3]
6.75 [6], 6.6 [7], 5.2 [9], 5.9 [10], 6.57 [8], 6.8 [11]
BT (1010 Nm-2)
8.95
CS (1010 Nm-2)
9.59
C44 (10
Nm )
dBT/dP
3.089
dC44/dP
1.018
dCS/dP c21
123
8.8 [3]
9.26 [6], 8.9 [7], 10.3 [9], 8.0 [10], 8.41 [8], 9.36 [11]
0.64 [3]
0.59 [7], 0.66 [9],0.64 [10]
-0.062 0.549
Phys Chem Minerals (2013) 40:521–530
525
6
2.5
Li 2 O
Bulk modulus
Nm )
C 11
-2
C 44
Bulk modulus (10
2
(a)
0 0
1.5
1.0
(b)
Li 2O 2
-20
-30
0.50
0.45
0.40
0.35
(c)
(d)
15
Li 2O
Li2O
2
C112
-15
-2
C111
0
11
Nm )
0
-2
Cijk (10
11 -2 Cijk (10 N m )
Li2O
0.55
Elastic Anisotropy (γ 1
-10
∇ Cauchy discripancy
2.0
11
4
C123
C144
-4
C166
-30
0
C456
-6
Li 2O
(f) Li2O
i
0
∇
Elastic anisotropy γ
3 11 -2 i (10 N m )
(e)
Cauchy discripancy
Li2O
(
2 11 -2 i (10 N m )
11 -2 Cij (10 N m )
C 12
-20
-40
-150
-300
(h)
(g) 0
30
60
90
P(GPa)
0
30
60
90
P(GPa)
Fig. 2 Elasticity of Li2O as functions of pressure. a Second-order elastic constants. b Isothermal bulk modulus (BT). c Cauchy discrepancy D. d Elastic anisotropy c. e corresponds to Cijk: C111, C112, and
C123 f corresponds to Cijk: C144, C166, and C456 g Cauchy discrepancy D. h Elastic anisotropy c
interatomic potentials, the Cauchy relation C12 = C44 ? 2P should be satisfied. At zero pressure, the Cauchy discrepancy for Li2O is about -3.627 9 1010 Nm-2 which further decreases on increasing the pressure as evident
from Fig. 2c. This might be due to the fact that for Li2O, non-central charge transfer interaction becomes significant at higher pressures. The significant deviation of D21 at different pressures is the strength of non-central many-body
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forces as charge transfer interactions and covalency effects as we dealt with. It is worth to mention that the Cauchy pressure is typically positive for metallic bonding. However, for directional bonding with angular character, the Cauchy pressure is negative, and larger negative pressure representing a more directional characteristic. The anisotropy in second-order elastic constants is of technological importance. It is known that anisotropic parameter c is unity for isotropic elasticity, but still the cubic crystal, which is isotropic in structure, has elastic anisotropy other than unity as a result of a fourth-rank tensor property of elasticity. Once second-order elastic constants are known, one can obtain the elastic anisotropic parameter c21 at various pressures (Barsch 1968). Figure 2d illustrates the pressure dependence of the elastic anisotropic parameter c21 for Li2O. It is clear from the plot that anisotropy parameter decreases with increasing pressure. The value of anisotropic parameter c21 for Li2O is given in Table 1 along with the available experimental and theoretical results at zero temperature and pressure (Hull et al. 1988; Rodeja et al. 2001; Goel et al. 2004; Wilson et al. 2004). The higher-order elastic constants are usually a measure of the anharmonicity of a crystal lattice. There are three second-order elastic constants and the six non-vanishing third-order elastic constants of cubic crystal. The thirdorder terms in the strain variables are readily obtained from elastic energy and are reported elsewhere (Varshney et al. 2012a). We thus find that aggregate elastic constants C111, C112, and C166 are negative and only C123, C144 and C456 are positive in which C123 and C144 are equal for Li2O. To the best of our knowledge, no experimental or theoretical data for third-order elastic constants of Li2O are available. Therefore, present results are a prediction study. The variation of aggregate third-order elastic constants (TOEC’s) with pressure for Li2O is discerned in Fig. 2e and f. It is noticed that the (a) values of C111, C112, and C166, show a decreasing trend with enhancing pressure while to that C456 showed a increasing trend for Li2O, (b) C123 and C144 are remarkably equal as compared to other TOEC’s, and (c) values of all aggregate elastic constants Cijk are influenced by pressure dependence. These observations imply that the aggregate elastic constants Cijk are affected by the inclusion of second-nearestneighbor interaction as well these are sensitive to the short-range interactions. Henceforth, the long-range central forces as Coulomb, charge transfer interactions and covalency are effective in lithium oxide. The pressure dependence of third-order elastic constants can have a direct means to understand interatomic forces at high pressure explicitly the short-range forces, and a balance between long- and short-range forces.
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It is also of interest to validate the Cauchy discrepancy D among third-order elastic constants. These are D31 = C112 - C166 - 2P; D32 = C123 - C456 - 2P; D33 = C144 - C456 - 2P; and D34 = C123 - C144 - 2P. It should be noted that all aggregate elastic constants Cijk contain both long-range and short-range interactions; hence, D3i are the indicators for the contribution from the non-central many-body force. These Cauchy discrepancies are plotted in Fig. 2 g as functions of pressure. The significant deviation in D3i is a natural consequence of the non-central many-body forces as charge transfer mechanism that we have emphasized. We must comment that the Cauchy discrepancy D3i among all third-order aggregate elastic constants are negative at zero as well as higher pressure and (D32 and D33) are equal at all pressure is influenced by aggregate elastic constant: As C123 and C144 are equal indicating the importance of non-central forces and anharmonic effects in Li2O. For the aggregate TOEC’s, there are three anisotropy coefficients and three isotropic coefficients (Barsch 1968). Figure 2 h illustrates the pressure dependence of the elastic anisotropic parameter c3i . It is clear from the plot that elastic anisotropy (c32 and c33) are less sensitive while to that c31 shows variation with increase in pressure. The anisotropy factor c31 shows a decreasing trend with higher pressures. Furthermore, the mechanical properties as ductility and brittleness of alkali metal oxides are significant and can be known from second-order elastic constants. We refer to Pugh (1954), who suggested a simple relationship, empirically linking the plastic properties of materials with their elastic moduli. The shear modulus GH represents the resistance to plastic deformation, while the bulk modulus BT represents the resistance to fracture. According to Pugh (1954), the ratio / ¼ BT =GH [ 1:75, the material behaves in a ductile manner, otherwise the material behaves in a brittle manner. The critical value that distinguishes the ductile and brittle nature is about 1.75. From Fig. 3a, the Pugh ratio / shows that a) at zero as well at high pressures lithium oxide is brittle. Perovskite materials usually document ductile behavior, and the ductility of a material is a measure of the extent to which a material will deform before fracture. It is argued that ductility is also used a quality control measure to assess the level of impurities and proper processing of a material. Later on, Frantsevich elaborated the ductility and brittleness of materials in terms of Poisson’s ratio (Frantsevich et al. 1983). According to Frantsevich rule, the critical value of Poisson’s ratio of a material that separates ductile and brittle nature is about 0.33. For ductile materials, the Poisson’s ratio is larger than 0.33, otherwise the material behaves in a brittle manner. Usually, the Poisson’s ratio lies in between -1.0 and 0.5 which are the lower and upper bounds. The lower bound is where the materials do not
Phys Chem Minerals (2013) 40:521–530
527 0.5
2.0
0.3 1.0 0.2 0.5 0.1
(a)
0.0
Elastic wave velocity (m sec-1)
0.4
1.5
Poisson's Ratio ( )
Pugh Ratio (BT GH)
Li2O 36000
30000
24000
18000
(b) 0.0
1400
Li2O
0.96
Li2O
vl vs
Li2O
D (K)
Gruneisen Parameter ( G
( 0.90
1200
(c) 0.84
(d) Heat capacity [Cv - Cv(0)]/Cv(0)
-0.04
T = 600 K T = 800 K T = 1200 K T = 1500 K
-0.08
0
30
(e) 60
90
P(GPa)
Thermal expansion coefficient 10-5 (K -1)
Li2O
0.00
1000
Li2O
T = 600 K 0.35
T = 800 K T = 1200 K T = 1500 K
0.30
0.25
(f) 0.20
0
30
60
90
P (GPa)
Fig. 3 Elastic and thermodynamical characteristics of Li2O as functions of pressure. a Poisson’s ratio m and ratio RBT/GH. b Elastic wave velocity vl and vs. c Gruneisen parameter (cG). d Debye
Temperature (hD). e Heat capacity (Cv) and f thermal expansion coefficient with pressure at different temperatures
change its shape, and the upper bound is where the volume remains unchanged. Thus, defining the Poisson’s ratio m in terms of bulk modulus BT and the shear modulus GH as m ¼ 0:5½ð3BT =GH Þ 2½ð3BT =GH Þ þ 11 (Vaitheeswaran et al. 2007). From Fig. 3a the Poisson’s ratio m shows that at zero as well as at high pressure, lithium oxide is brittle. Thus, both Pugh and Frantsevich rule confirm the brittle nature of lithium oxide. Furthermore, at zero pressure, the value of
m is about 0.177 for lithium oxide, and it is in good agreement with available results (Zhuravlev and Obolonskaya 2011). It is to be noted that the values of the Poisson’s ratio (m) for covalent materials are small (m * 0.1), whereas for metallic materials, m is typically 0.33. The brittle nature is also observed in face-centered cubic intermetallic compounds as Ir and Ir3X (X = Ti, Ta, Nb, Zr, Hf, V) using ab initio density functional theory (Chen et al. 2003), intermetallic MgCNi3 as brittle materials
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(Vaitheeswaran et al. 2007), brittle materials as Ga based monopnictides (Varshney et al. 2010a), Ga1-xInxP mixed valent compounds (Varshney et al. 2010b), Boron monopnictides: BY (Y = N, P, As) III–V semiconductors (Varshney et al. 2010c), III–V antimonides: YSb (Y = B, Al, Ga, and In) (Varshney et al. 2010d), for Scandium Chalcogenides (ScX) (Maachou et al. 2011), rare earth pnictides: LaY (Varshney et al. 2012a), rare earth chalcogenides: EuX (Varshney et al. 2012b), mercury chalcogenides (Varshney et al. 2012c), and diluted magnetic semiconductors (Varshney et al. 2011). Lithium oxide is promising material with wide range of applicability’s with effective mechanical properties (Keen 2002; Hu and Ruckenstein 2004). Usually, materials elastic properties are a source of valuable information where materials mechanics is significant as the knowledge of deformational characteristics of materials is essential in engineering design and construction of effective structures. The response of the elasticity is further investigated through the compressional and shear wave velocity. The pressure dependence of the longitudinal (shear) velocity is shown in Fig. 3b, respectively. It is noticed that vl and vs increase with enhanced pressure. Deduced values of longitudinal (vl = 29,660 m/s), shear (vl = 18,710 m/s), and average elastic wave velocity (vm = 20610 m/s) propagating in Li2O at zero temperature and pressure. In order to describe the anharmonic properties of a crystal, we have calculated Gru¨neisen constant cG from cG ¼ ½r0 =6½U 000 ðr0 Þ=U 00 ðr0 Þ to get a value of 0.954 and compressibility b using b ¼ ½r02 fU 00 ðr0 Þg=9V1 to get a value of 0.111. The Gru¨neisen constant as functions of pressure is plotted in Fig. 3c for Li2O. Usually, the value of Gru¨neisen constant for most of the solids is in between 1.5 and 2.5. It is evident from figure that the Gru¨neisen constant decreases linearly with increase in pressure. Further investigations are required from knowledge of the phonon frequencies as a function of the crystal volume V. The elastic constants determine the velocity of elastic waves through the lattice, and hence, one can relate the Debye temperature (hD) with the elastic constants, since hD may be estimated from the average sound velocity, Vm, using: hD ¼ ðhtm =kÞ½3nNA q=8p0:33 . Here, h is Planck’s constant, k is Boltzmann’s constant, NA is Avogadro’s number, n is the number of atoms in the molecule, M is the molecular weight, q is the density, and tm is average wave velocity. Usually, the Debye temperature is also a function of temperature and varies from technique to technique as well depends on the sample quality with a standard deviation of about 15 K. The Debye temperature as functions of pressure is plotted in Fig. 3d for Li2O. It is noticed from the figure that the hD increases with increasing in pressure for lithium oxide, and the calculated value is listed in
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Table 1 and is compared with available theoretical results (Zhuravlev and Obolonskaya 2011; Li et al. 2006). Deduced higher-order elastic constants are further useful in revealing the heat capacity and thermal expansion coefficient as functions of pressure and temperature, and we need to work with quasi-harmonic Debye model in which the non-equilibrium Gibbs’s free energy function, G*(V; P, T) takes the form G = U ? PV ? Avib. [hD(V); T]. Here, Avib. is the vibration term, which can be written using the Debye model of the phonon density of states as (Blackman 1942): Avib: ðhD ; TÞ ¼ nkB T ½ð9hD =8TÞ þ3 lnf1 expðhD =Tg DðhD =TÞ. The non-equilibrium Gibbs function, G*(V; P, T), as a function of (V; P, T) can be minimized with respect to volume V as oG ðV; P; TÞ= oV P;T ¼ 0. The heat capacity and thermal expansion coefficient at constant volume are: Cv ¼ 3nkB ½4DðhD =T Þ ½3hD =T½ehD =T 11 and ath:exp: ¼ cCv ðBT VÞ1 . The pressure dependence of Debye temperature is earlier discussed. The variations of the heat capacity at constant volume, Cv, with pressure P of lithium oxide are illustrated in Fig. 3e at T = 600, 800, 1,200, and 1,500 K. We have plotted the normalized heat capacity, [Cv - Cv (0)]/Cv(0), where Cv and Cv (0) are heat capacity at any pressure P and at zero pressure. The heat capacity at different temperatures decreases non-linearly with the applied pressures. It infers that the vibration frequency of the particles in Li2O changes with pressure as well as temperature. For higher temperatures T ? hD, the variation in heat capacity with pressure is weak. The pressure-dependent Gruneisen parameter and Bulk modulus are required apart from heat capacity at constant volume Cv to elucidate the thermal expansion coefficient. The thermal expansion coefficient (ath.exp.) describes any alteration in frequency of the crystal lattice vibration depending on the lattice’s expansion or contraction in volume as a result of variation in temperature. We have determined the pressure dependence of a as shown in Fig. 3f for Li2O. It can be seen that ath.exp decreases nonlinearly with the pressure. However, the thermal expansion coefficient in higher temperature decreases rapidly with pressure than that in lower temperature.
Conclusion The present study addresses for the first time, the high pressure-dependent, aggregate second-order elastic constants, Cauchy discrepancy, anisotropy in higher-order elastic constants, ductile (brittle) nature, propagation of compression and shear wave, Gru¨neisen constant, Debye temperature, heat capacity, and thermal expansion coefficient of superionic alkali metal oxide (Li2O) by formulating an effective
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interatomic interaction potential with emphasis on long-range Coulomb, charge transfer, and covalent effect as well as shortrange overlap repulsion extended up to the second neighbor ions and van der Waals interactions. Some of them have not been measured yet; hence, the present calculations will be serving as guidelines for future experimental work. The obtained values of free parameters allow us to predict cohesive energy and volume collapse in the pressure-dependent behavior for superionic Li2O. The consistency of the potential energy functions and pressure-dependent change in volume is attributed to the screening of the effective Coulomb potential through modified ionic charge (Zm). From the computed values of pressure-dependent Poisson’s ratio m (&0.177) and the Pugh ratio / (1.213), we comment that Li2O showed brittle nature at low as well at high pressures. The pressure-dependent isothermal bulk modulus of Li2O infers that the lattice becomes stiff at higher pressures up to 100 GPa. The mechanical stiffened bulk modulus in Li2O is attributed to Li–Li, O–O, and Li–O bond compression and bond strengthening due to lattice vibration. To our knowledge, this is the first quantitative theoretical prediction of the high-pressure dependence elastic moduli of superionic Li2O and still awaits experimental confirmations. The pressure-dependent thermal expansion coefficient ath.exp decreases non-linearly with the pressure. The decreasing trend in ath.exp confirms the mechanical softening in Li2O. Within the framework of quasi-harmonic Debye model, the heat capacity at different temperatures decreases non-linearly with applied pressures. It infers that the vibration frequency of the particles in superionic Li2O changes with pressure. For higher temperatures T ? hD, the variation in heat capacity with pressure is weak. Furthermore, Cv(T) is consistent with ideal Dulong–Petit limit of 3R, at higher temperatures as well at all pressures for superionic Li2O cannot be compared due to unavailability of data.
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