StructuralandelasticpropertiesofγTiAlunderhighpressureintheatomiclevelinvestigations.doc

structural and elastic properties of tial under high pressure in the atomic level investigationsfu hongzhi 1*，li dehua 2，peng feng 1，gao tao 3，cheng xinlu 31. college of physics and electronic information，luoyang normal college，luoyang，shanxi（471022）2. institute of physics and electronic engineering，sichuan normal university，chengdu（610065）3. institute of atomic and molecular physics，sichuan university，chengdu （610065）e-mail：abstractwe have investigated the structural and elastic properties of tial under high pressures using thenorm-conserving pseudopotentials within the local density approximation (lda) in the frame of density functional theory. the calculated pressure dependence of the elastic constants is in excellent agreement with the experimental results. the elastic constants and anisotropy as a function of applied pressure are presented. through the quasi-harmonic debye model, we also investigate the thermodynamic properties of tial.pacs: 62.20.dc; 65.40.-b; 62.50.+p; 71.15.apkeywords：a. tial；d. elastic constants；d. debye temperature；d. anisotropy1. introductionbecause of the most promising high temperature structural applications, chemical stability, low density, and good oxidation resistance1-4, tial has been extensively investigated in the last 15 years. tial is an intermetallic compound with an l10-type structure. this type structure is based on an ordered fcc tetragonal cell in which the ti and al atoms occupy alternating 002 planes.for a long time, there has been an attempt to correlate the mechanical properties with their electronic structures. in theory, ref5 and ref6 have calculated the distribution of the integrated charge density of tial alloy. ref7 and ref8 have investigated its the density of states, and ref9 and ref10 investigated the preferred site of v, cr in tial alloy, etc. furthermore, morinaga et al.11 and fu et al.12 have studied the electronic structural effect of alloying elements and structural defects on the ductility. on the other hand, song et al.13 have discussed the bonding character in tial and fcc titanium. jund et al.14 and zhou et al.15 have calculated the phase stability and modulus of elasticity of the tial. with regard to computational investigations, many researchers concentrated on the effect of alloying elements, the phase stability 5-11, the dislocation configurations and the phase transition 12,13 in titanium-aluminum alloys. in our case, we focus on investigating the eos（equations of state）and the elastic properties of the tial in the range of 060 gpa by the plane-wavepseudopotential density functional theory method through the cambridge serial total energy package (castep) program 16,17 and by the quasi-harmonic debye model 18, which allows us to obtain all thermodynamics in the atomic level.elastic properties, which are closely related to many fundamental solid-state properties, such as equation of state, specific heat, thermal expansion, debye temperature, grneisen parameter, melting point, and so on, are important in fields ranging from geophysics to materials research, chemistry and physics. the knowledge of elastic constants is essential for many practical applications relatedto the mechanical properties of a solid: load deflection, thermoelastic stress, internal strain, sound velocities and fracture toughness 19.2. methodologyall the calculations were performed using the castep code 16,17, which is based on the- 15 -implementation of the density functional theory with the electronic density described by a plane-wave basis. here, we used the norm-conserving pseudopotentials by lee 20. the 3s,3p al orbitals and3d,4s ti orbitals are treated as valence states. the lda functional was employed for the determination of the exchange-correlation energy, as parametrized by perdew and zunger21 from the numerical results of ceperley and alder22. this choice is based on the description of the l10-type structure. with norm-conserving pseudopotentials, this exchange-correlation functional gives a 0.10% better value of the cell parameter than the gradient corrected functional. the size of the plane-wave basis set and the sampling of the brillouin zone were carefully tested. a monkhorst-pack k-point grid23 of666 was used with which the total energy is within 0.02 mev per atom compared to that calculated for 888 grid. the l10-type structure has been optimized within the tp4 space group and by using the bfgs algorithm24, for hydrostatic pressures ranging from 0 to 60 gpa. the total energies are converged to 2.010-5 ev per atom and the forces to 5.010-2 ev/. a very small energy tolerance has also been chosen for the electronic minimization (i.e., 1.010-7 ev) in order to achieve a sufficiently well converged set of ground-state wave functions. the density mixing method has been used for the electronic minimization. and the atomic displacement amplitude was tested and finally taken as 5.3 10-3 .to investigate the thermodynamic properties of tial, we here apply the quasi-harmonic debye model 18, in which the non-equilibrium gibbs function g*(v; p, t) takes the form of*g (v; p,t ) = e (v ) + pv + avib (v );t ),(1)where e(v) is the total energy per unit cell for -tial, (v) is the debye temperature, and the vibrational helmholtz free energy avib can be written as 25,26a(;t ) = nkt 9 + 3ln(1 e /t ) d(/t ),(2)vib8 twhere d(/t) represents the debye integral and n is the number of atoms per formula unit. for an isotropic solid, is expressed by 26, = = 6 2v 1 / 2 1 / 3 f ( )kbs ,m(3)where m is the molecular mass per formula unit, bs is the adiabatic bulk modulus approximated by the static compressibility 18 d 2 e (v ) bs b(v ) = v dv 2(4)and f() is given by refs. 27,28.therefore, the nonequilibrium gibbs function g*(v; p, t) as a function of(v; p, t) can be minimized with respect to volume v as g *(v; p,tv) p, t= 0 .(5)by solving eq. (6), one can get the thermal equation of state (eos) v(p,t). the isothermal bulk modulus bt is given by 18 2g *b ( p,t ) = v (v; p,t ) ,(6)tv 2 p, tthe heat capacity cv and the thermal expansion() are expressed asvc= 3nk 4d( / t ) 3/te /t 1, = cvbt v,(7)where is the grneisen parameter defined as = - dln(v )dlnv3. results and discussion(8)we carry out total energy electronic structure calculations over a wide range of primitive cell volumes v, i.e., from 0.7v0 to 1.2v0, where v0 is the zero pressure equilibrium primitive cell volume. no constraints are imposed on the c/a ratio, i.e., both lattice parameters a and c are optimized simultaneously. at each volume v, we determine the corresponding equilibrium ratio c/a by performing the total energy electronic structure calculations on a series of different values of c/a. for a fixed value of c/a, a series of different values of a are set to calculate the total energies e, which are minimized as a function of the c/a ratio for a given v. through these calculations, we can obtain the equilibrium lattice parameters a and c as well as the corresponding equilibrium ratio c/a for any pressure. the energy-volume (e-v) curve can be obtained by fitting the calculated e-v results to thebirchmurnaghan eos29, in which the pressure-volume relationship expanded asp = 3b f(1 + 2 f) 5 / 2 + 3 (b 4) f+ 3 bb + (b 4)(b 3) + 35 f 2 ,0 e 1 v0e122 / 3e 02db0e 9 (9)wheref e =( )2v 1,b, b0=and b0 are hydrostatic bulk modulus, thedppressure derivative of the bulk modulus and zero pressure bulk modulus, respectively. b is givenby30b = 1 (3 b)(4 b) + 35 (10)9.b0 let us consider the tial crystal isotropically compressed to the density 1=1/v1, where v1 is thedistorted volume due to the strain tensor =ij with small lattice distortion ij( i , j=1,2,3), which takes every bravais lattice point r of the undistorted lattice to a new position r in the strained lattice, ri=j(ij+ij) rj. for a homogeneous strain the parameters ij are simply constants withij=ji, where thesubscripts i, j indicate cartesian components and each ranges over three values; ij is the kronecherdelta. since tp4 crystal structure possesses six independent elastic constants, i.e., c11, c12, c13, c33 ,c44andc66, we thus use the six independent strains listed in table 1. all these stains are not volume conserving. the atomic positions are optimized at all strains. for each strain, a number of small values of are taken to calculate the total energies e for the strained crystal structure. the calculated e- points are then fitted to a second-order polynomial e(1,), and the second-order derivatives of e(1,) with respect to are obtained. the detailed calculation method has been reported in ref. 31.table.1. strains used to calculate the elastic constants of tial at zero pressurestrainparameters(unlisted eij=0)1 2 e (v ,0)0v 2 =01e1 = c12112e3 = 1c2333e4 = 22c 444e1 = 2 ,e2 = e3 = 21 (542)c11 2c12 c13 + c 335e1 = e2= , e3 = 2(c11 + c12 4c13 + 2c 33 )6e1 = e2 = , e3 = 2 , e6 = 2(c11 + c12 4c13 + 2c 33 + 2c 66 )the obtained compression dependences of the elastic constants at t=0 are presented in table.2, together with the six independent elastic constants by others at zero pressure. obviously, the calculated results are well consistent with the experimental data32-34and the other theoretical results35-41.table.2. the elastic constants cij (in gpa), bulk modulus b, youngs modulus e, shear modulusg and poissons ratio , together with the experimental data and other theoretical results.tialexperimentpresent workother calculations a() 3.997a 4.001 4.003d, 3.989k c/a1.02a1.0121.014d, 1.011kc11(gpa)186b, 183c164183e, 187g, 190h, 188i, 170kc12(gpa) 72b,74.1c 85.574.1e, 74.8g, 105h, 98i, 79k c13(gpa) 74b,74.4c 81.04 74.4e, 74.8g, 90h, 96i, 78k c33(gpa)176b,178c178.57178e, 182g,185h,190i, 178k c44(gpa)101b,105c 109.6105e, 109g,120h,126i, 113k c66(gpa) 77b,78.4c 72.678.4e, 81.2g, 50h, 100i, 73kbulk modulus b(gpa)109.78b,113.29c110.69110e, 112dyoungs moduluse (gpa) shear modulus g (gpa) poissons ratio debye temperature (k)182.9b,160.32c170.50184.7e, 173j74.8b,63.41c68.5775.7e, 70j0. 22b,0.22c0.260.220e, 0.234j584b583587earef32 bref33 cref34 dref35 eref36 gref37 href38 iref39 jref40 kref41the ab initio computations have been also used to calculate the bulk modulus of the tial, being equal to 110.69 gpa, which is close to others literature value and experiment result. furthermore, the agreements among them are also good.these elastic stiffness coefficients of tetragonal tial are shown in table.2. they satisfy thegeneralized elastic stability criteria for tetragonal crystals under hydrostatic pressures42,43, showingthe tetragonal cell mechanically stable.c11 c12 0,(c11 + c 33 2c13 ) 0(12)2c11 + 2c12 + c 33 + 4c13 0(13)c11 0,c 33 0,c 44 0,c 66 0(14)on the other hand, the calculated elastic constants are shown as functions of pressure in fig.1. the present results agree well with ref44. in the calculated pressure range, it is surprising that c66, followed by c44 increases monotonically with increasing pressure, but their rates of increase are very moderate. however, the increasing rates of pressure dependence of the present c33, keeping in step with c12 and c13 above 5gpa is always larger than that of c11, in which the rapid increase under pressure is not clearly seen.fig.1. elastic constants of tial as functions of pressure.we have investigated the total energy as a function of primitive cell volume for the l10 crystal structure of tial. it is found that the most stable structure (i.e., the normalized volume vn= v/v0= 1.0, where v0 is the equilibrium volume at zero pressure) corresponds to the ratio c/a of about 1.012, and the equilibrium lattice parameters a and c are about 4.001 and 4.071 , respectively, which are consistent with experimental data32-34 and other theoretical results35-41. the equilibrium ratio and the corresponding normalized lattice parameters a/a0, c/c0 and the normalized volume v/v0 as a function of the applied pressure are plotted in fig.2, where a0, c0, and v0 are their values at t = 0 and p=0, respectively.fig.2 the variations about ratio of the normalized lattice parameters a/a0, c/c0 and the normalized volume v/v0 with the applied pressure of tial.by fitting the calculated data to the second-order polynomial, we obtain their relationships at the temperature t = 0 k,a/a0=0.99914-2.5310-3p+1.42510-5p2(15)c/c0=0.9918-2.1710-3p+1.4013810-5p2(16)v/v0=0.99586-6.9510-3p+3.08610-4p2(17)it is shown that, as pressure increases, the ratio of a/a0, c/c0 and v/v0 decreases. the compression along c axis is much smaller than that along the a axis, consistent with the comparatively stronger (ti-al) bonds that determine the c axis length 13. when pressure increases, the atoms in the interlayers become closer, and their interactions become stronger.the negligible changes of ti-ti and ti-al bond distances with the applied pressures are observed in fig.3. this result is important since to a first approximation the vibrational frequencies are dependent on bond distances. as expected experimentally and theoretically 45,46, the ti-ti and ti-al bond lengths decrease with pressure. we can also note that the ti-al bonds shorten slightly faster than the ti-ti bonds(figure3), which shows that there exist a high apb energy and a fairly large elastic shear anisotropy along 001 direction of tial45,46. the high apb energy is attributed to the directional bonding between ti d and al p electrons. the charge distribution about the ti site has been shown to be highly non-spherical, and the orbital character is dominated by dxy, dxz and dyz components. there is a pz-type charge polarization about the al layers, indicating strong cohesion between ti and al layers45,46. the directional bonding between ti d and al p electrons manifested in large c44 (109.6gpa) value, while a relatively low value of c66(72.6gpa) implies a more isotropic charge distribution in pure al layerstable.2.fig.3.the changes of ti-ti and ti-al bond distances with the applied pressures.to qualify the mechanical anisotropy of tial, one can define the bulk modulus along the a axis (ba)and the c axis (bc) as follows47:b = a dp =ada22 + （18）b = c dp = ba（19）cdc = 2(c11+ c12) + 4c13 + c 33 （20） = c11 + c12 2c13c 33 c13（21）the calculated values of the bulk modulus of ba and bc at the zero pressure are 327.93gpa and341.60gpa, respectively. we find that the variations of the contours of ba and bc nearly increase with pressure by trajectory and straight line, respectively. the value of ba/bc has a trend of gradual decline as the normalized volume v/v0 decreases. it is shown that the mechanical behavior of tial under zero pressure is of large anisotropy.it is well known that microcracks are induced in alloys owing to the anisotropy of the coefficient of thermal expansion as well as elastic anisotropy48. hence it is important to calculate elastic anisotropy in structural intermetallics in order to understand these properties and hopefully find mechanisms which will improve their durability. essentially all the known crystals are elastically anisotropic, and a proper description of such an anisotropic behavior has, therefore, an important implication in engineering science as well as in crystal physics. the shear anisotropic factors provide a measure of the degree of anisotropy in the bonding between atoms in different planes. the shear anisotropic factorsa100 in the 100 plane and a001 in the 001 plane take the following forms, respectively31.a=4c 44,(22)100c11+ c 33 2c13a001=2c66c11 c12(23)fig.4. the changes of anisotropic factors a100 and a001 with applied pressures.in fig.4, we illustrate the calculated pressure dependencies of the anisotropies of the shear anisotropic factors a100 as well as the shear anisotropic factors a001. unfortunately, there is no experimental data to check our calculated results at high pressure. the shear anisotropic factors a100 increases slowly initially(from 0gpa to 30gpa),then more rapidly(from 30gpa to 50gpa) with increasing pressure, eventually arriving at the maximum at the pressure of about 50gpa and decreasing rapidly above 50gpa. however, the shear anisotropic factors a001 changes slightly in the wide range of applied pressure and becomes moderate with the pressure rise. at about 31.3gpa, the a100 and a100 become equal, reading about 4.56, respectively. these results are similar to a