A. I. GUSEV: Structural Vacancies in Nonstoichiometric Compounds 159

phys. stat. sol. (a) 86, 169 (1984)

Subject classification: 10.1; 8; 12.1; 21.1

Institute of Chemistry, Academy of Sciences of the USSR, Ural Scientific Centre, Sverdlovskl)

Structural Vacancies in Nonstoichiometric Compounds at High Pressure Thermodynamic Model

BY A. I. GUSEV

A thermodynamic model is proposed which describes the effect of high pressure on the equilibrium structural vacancy concentration of nonstoichiometric compounds possessing wide regions of homogeneity. The implicit pressure and temperature dependence of the equilibrium structural vacancy concentration is found in general form for any nonequivalent crystal sublattice of EI com- pound.

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1. Introduction High-melting NaC1-structure compounds such as transition metal carbides, nitrides, and oxides posses wide regions of homogeneity. Within these regions they may have structural vacancies in either the nonmetallic or metallic sublattice, or in both a t a time. For some of the compounds concerned (for example, such as vanadium and zirco- nium carbides and vanadium monoxide), no stoichiometric compositions with all the sublattices being free from structural vacancies have been synthesized so far. A possible way to produce practically defect-free compounds (or compounds with crystal sublattice defect relations incapable of realization using conventional synthesis methods) is to apply high pressures in combination with high temperatures. (A defect- free compound is one containing no structural vacancies ; the defect concentration, or imperfection, is the structural vacancy content of the crystal).

Since temperature and pressure involved in the formula for the free energy of formation of a compound are opposite in sign, an increase in pressure, just as a decrease in temperature, should in the general case lead to a lower equilibrium structural vacancy concentration for the conditions given. However, the synthesis of practically defect-free (devoid of structural vacancies) compounds in the region of temperatures close to absolute zero appears to be unfeasible due to primarily kinetic reasons. TO synthesize a compound with a crystal lattice containing no structural vacancies it is obviously an acceptable procedure to use simultaneously high temperatures assur- ing sufficiently high rates of diffusion processes in the solid state and high pressures reducing the structural vacancy Concentration. It should be noted that the theoretical

l) Pervomaiskaya 91, 620219 Sverdlovsk, USSR.

160 A. I. GUSEV

equilibrium structural vacancy concentration depends on temperature and pressure in an exponential manner, i.e. may be as small as desired but never goes to zero. In this context it is necessary to consider the possibility of synthesizing under certain condi- tions not absolutely defect-free compounds but compounds possessing a structural vacancy concentration as small as desired, i.e. practically defect-free ones.

Depending on the relation of sublattice defects in the basic material and on which relation between defect concentrations one wishes to obtain, the synthesis may be performed with or without the addition of some components of a given basic material. If the synthesis involving the combined use of pressure and temperature is effected without adding components the formation of a defect-free (with respect to one or several sublattices) compound will occur due to the migration of a certain number of atoms of one component or another from the surface of the crystal and due to the collapse of vacancies in the corresponding sublattices. In this case conditions can also be established under which not all of the vacancies vanish completely. It is thus possible to synthesize a compound with crystal sublattice vacancy contents which are not realized under normal conditions. If relevant components are added to a de- fective compound subject to compression, the formation of a defect-free compound occurs due to the migration of the atoms of the components added inside the crystal and due to these atoms filling the vacancies in the corresponding sublattices. Like in the previous case, it is possible here to create conditions under which not all the vacancies will be filled.

A number of experimental and theoretical papers are available which deal with the effect of high pressure on materials possessing structural vacancies (e.g., on compounds such as TiO, VO, and NbO which contain structural vacancies in the nonmetallic and metallic sublattices simultaneously). Thus, it has been demonstrated experimen- tally [l, 21 that when titanium monoxide is compressed under a pressure of 60 kbar and more and a t temperatures above 1500 K the structural vacancy concentration can be decreased substantially. An investigation of the effect of high pressure on the crystal structure of extreme compositions in the region of homogeneity of TiOz a t T = 300 K has revealed a very insignificant lattice parameter variation [3]. A statis- tical thermodynamical analysis of the equilibrium vacancy concentration as a func- tion of pressure P and temperature T, performed in [a], has shown that, subject to the conditions specified in [l], no defect-free Ti0 should arise. However, that analysis was carried out involving a number of arbitrary assumptions. The Griineisen constant y was taken to be negative, although the value of y is positive for all solid substances. The values of the formation energy of a vacancy in the titanium and oxygen sublattices were taken to be equal and composition-independent [a], although i,he behaviour of the Ti- and 0-sublattice defects in the region of homogeneity of TiOz indicates that these energies differ in magnitude and vary with composition. A calculation of the formation energy of a vacancy in the titanium and oxygen sublattices of titanium monoxide has confirmed their dependence on composition TiOz and their substantial difference in magnitude [5].

The present paper proposes a thermodynamic model to describe nonstoichiometric compounds containing structural vacancies. This model permits the calculation of the pressure and temperature required to synthesize, under thermodynamic-equilibrium conditions, similar compounds with any predetermined crystal sublattice defect relation, including practically defect-free compounds.

2. Thermodynamic Model Examine a defective compound possessing nonequivalent crystal sublattices, in each of which the relative number of structural vacancies (defect concentration) per

Structural Vacancies in Nonstoichiometric Compounds at High Pressure 161

Lcmolecule” is equal to n, = Nni /Nt , and the relative number of occupied sites is mi = M , / N , , where NDi + M t = N t is the total number of i-sublattice sites; hence nt + mi = 1. Let the Gibbs free energy of this defective compound be equal to AG(n). Then the variation of the Gibbs free energy in the case of passing from a defec- tive compound to a compound with a given crystal sublattice defect relation (cccrys- tal sublattice defect relation” is a particular combination of the relative numbers of structural vacancies present in each crystal sublattices, i.e. a particular set of values nl, n2 ... ni, characterizing the compound under investigation) may be represented as

AG(n) = AG + C, J AH, dnt - TAS, + C J J AVi dn, d P + A F , (1 ) where AG is the Gibbs free energy of a crystal with a given sublattice defect relation, AHi the formation energy of vacancies in the i-sublattice, AS, the change in the configurational entropy, AVi the change in volume during the formation of structural vacancies in the i-sublattice, A F the change in the vibrational free energy of the crystal (AHi and AV, are continuous functions of ni).

If some components are added to a defective compound subject to compression, the formation of a compound with a lower structural vacancy content or of a totally defect-free compound occurs due to the added-component atoms migrating inside the crystal and filling the vacancies in the corresponding sublattices. The change in volume here will be

C J A V , d n i = T / ‘ + Vadd-vf , (2)

where V is the volume of the defective crystal, V,,, the volume of the component added, Vf the volume of the crystal produced as a result of P-T treatment. The volume will vary not only due to a change in defect concentration but also due to the compression of the material under pressure, as well as due to thermal expansion. Let u, be the composition-independent volume of the i-component of the compound, voi the volume of an i-sublattice vacancy, bi a proportionality coefficient indicating which portion of the structural vacancies of the i-sublattice should be filled as a result of the P-T treatment, z = V p / V , is the compression ( V , is the volume of the material when P = 0, V , is the volume of the compressed material), a the thermal expansion coefficient.

Calculation in terms of one “molecule” in this case yields

V = c [%(I - nt) f I “mi dntl Vadd = C bt’hni >

Vf = z(1 + 3aT) C [w, - v4nt(l - b,) + (1 - b,) J vni dn, . (3) Por solid substances the reduced pressure P / K (with K being the bulk modulus) dependence of compression x is described, according to [6], by the expression

where A = (3K’ - 7 ) (K‘ = BK/BP when P = 0). According to [el, for various solid materials (aK/aP)p,o varies over a quite narrow interval and the mean value of this quantity amounts to 4.8, the respective value for A being 7.4. With these values of the parameters (4) gives a good fit to the compression of various solid materials (except for solid inert gases) with PIK 5 2 . For PIK 4 0.25 (4) may be shown to transform to a linear function of compression versus reduced pressure

(5) P K z = 1 - 0.8--.

11 phpica (a) 85/1

162 A. I. GUSEV

In view of (5 ) the total change in the volume of the crystal as calculated in terms of one ,,molecule” of the compound will be

C J AVC dnt = C [ ~ t ( l - nt + + I uoi dnil -

- (1 - 0.8;) (1 + 3aT) C [v, - vCnC(l - b,) + + (1 - bt) I V U i dntl * (6)

If the compression is effected without adding any component the number of crystal- line atoms remains unchanged, the volume of the crystal upon P-T treatment is

and the change in the volume of the crystal as a result of compression is

C I Art dni = C [ u d l - fit) + I uai dnt] -

- (1 - 0.8;) (1 + 3aT) C [vi( l - nr) + + (1 - bt) I uni dntl - (8)

Expression (6) differs from (8) in that it contains an additional term, which is equal to

The change in vibrational free energy may be written in the form

AF = AE - T A X . (10)

To describe the thermal vibrations one may adopt the Einstein model and represent the crystal as a set of harmonic linear oscillators with the mean energy E,

where Z is the statistical sum of states, k the Boltzniann constant, h the Planck con- stant. In this case the change in energy AE in a three-dimensional crystal a t the transi- tion from a defective state to a defect-free one will be

where mi and my stand for the number of atoms in a defective crystal and in a crystal resulting from P-T treatment, respectively, vn and v are the atomic vibration fre- quencies for a defective crystal and for a crystal resulting from P-T treatment.

Structural Vacancies in Nonstoichiometric Compounds at High Pressure 163

The total change in the vibrational non-configuration entropy A S may be found, as has been shown in [7], in the form

A S = C J ASi dnl , (13)

where AS, is the change in the non-configuration entropy during the formation of i-sublattice vacancies.

The change in configuration entropy is determined by the equation

to be subsequently expanded according to the Stirling approximation. When an equilibrium concentration of structural vacancies in the i-sublattice for a

given pressure and for a given temperature is reached, the Gibbs free energy AG(n) will be minimum and, accordingly, aAG(n)/an, = 0. Proceeding from this situation it is possible to determine the temperature and pressure dependence of the equili- brium structural vacancy concentration.

Substitution of (6), (lo), (12) to (14) or (8), (lo), (12) to (14) into (1) and differentia- tion of the latter with respect to ni yields

where

8 C JJ AV5 an, dn,dP = ~ v o i [ 1 - ( 1 - 0 . 8 ~ ) ( l + 3 a T ) ( l - - b , ) 1 d P -

Allowance for hvn = ke and

'where 8 is the characteristic Einstein temperature, gives

11.

164 A. I. GUSEV

Substituting (16), (17), (20) into (15) and solving i t for In - we find n, m4

(1 + 3aT) (1 - b,)] d P +

+ & / wi(1 - b,) [ 1 - (1 - 0.8 g) (1 + 3aT) d P + 1

for the case when some components are added to a defective compound subject to P-T treatment. If the latter is performed without adding any components the last but one term in (21) should be written as

& J w i [ l - (1 -0.8;)(1 + 3 a T ) I d P .

Other parameters being known, the derived equation (21) admits a self-consistent solution for n,.

3. Numerical Calculations As an example, we will calculate the thermodynamic-equilibrium conditions of syn- thesis for zirconium and vanadium carbides, which are close to stoichiometry. Referred to one c‘moleculeyy and suitable for the 1500 to 3000 K temperature range, the struc- tural vacancy formation energy in the carbon sublattice AH = (0.524 - 0.520n) eV and the dependence of the vibrational entropy on the defect concentration of the carbon: lattice A S = (0.88 - 31.84%) x lod5 eVK-’, which are required for this calculation, were determined earlier for ZrC, -n [7]. According to the most trustworthy phase diagram for the Zr-C system [8], the zirconium carbide composition which is closest to stoichiometry is ZrCo.98. For a zirconium carbide close to stoichiometry the characteristic Einstein temperature 8 is equal to 750 K [9]. The value of the Griin- eisen constant y for most of the carbides ranges between 1 and 2 [lo], in the present calculation the mean y value of 1.5 has been adopted. To estimate the volume of a car- bon vacancy in zirconium carbide the following assumptions have been made. The volume of one “moleculey’ of stoichiometric ZrC with NaCl structure equals, on the one hand, one fourth of the volume of the unit cell and, on the other hand, the sum of the volumes of a zirconium atom, wZr, and of a carbon atom, wc. The volume of an atom is proportional to R3, where R is the atomic radius. Hence, if a is the period of a unit cell of a stoichiometric compound and RZ,{Rc = q, then

For some composition ZrCl -An 3

an 4 V = VZ, + (1 - An) wc + Amnc = wZr + wc + An(wDc - wC) = -. (24)

Since wZr + wc = a3/4 then wnc - wc = (a,: - a,3)/4; in the limit An -. 0, a, + a, therefore wUc - wc = a(a:)/4 an and

Structural Vacancies in Nonstoichiometric Compounds at High Pressure 165

Using t,he data [ll] on the variation of the lattice period of zirconium carbide in the region of homogeneity we have arrived at the result

v;: = (2.494 - 2.684n) x m3.

For vanadium carbide the upper boundary of the region of homogeneity is vcO.87 [ll]. The temperature dependence of the Gibbs free energy ACl, = AHi - TAX, for the formation of structural vacancies in the carbon sublattice of vanadium carbide for a temperature interval from 300 to 1300 K, as calculated for one “molecule”, has the form [12]

AGyC = 0.415 + 1.574 x 1O-V + 7.553 x 10-sT2 - - 0.17 lg T - 0.458 x lO-3T lg T + 4.525T-1 eV .

For the vanadium carbide Vc0.87 the characteristic Einstein temperature 8 is equal to 1050 K [9]. The volume of a carbon vacancy in vanadium carbide has been calculat- ed with the use of structure data [13, la], just as is the case with zirconium carbide, and amounts to

vVc = (3.586 - 19 .45~ - 0 . 5 6 ~ ~ + 6.95~3) x 10-30m3. 00

The values of the bulk modulus K for the carbides VCo,87 and Zfl0 .98 are equal to 3.98 x 1014 and 2.65 x 1014 Pa, respectively [14, 151, and the respective values of the mean linear thermal expansion coefficient for these carbides are 9 x lo-“ and 7 X x K-l [15].

For the structural vacancies to be filled the synthesis of defect-free zirconium and vanadium carbides should obviously be performed involving a supply of the carbon deficit in these carbides.

Assuming reasonable values of the quantity n and temperature T, calculations based on (21) were carried out to determine the values of the pressure required to synthesize the compositions ZrCl-,, and VCl-,, when using ZfiO.98 and vc0.87 as initial carbides (see Table 1).

Table 1 Pressure P required to synthesizeZrC1-, andVCl-, with a given n at a given tempera- ture T under thermodynamic-equilibrium conditions

?& zirconium carbide ZrCl-, vanadium carbide VCl-,, filling pressure P (lW1 Pa) filling pressure P (1W1 Pa) factor T (9) factor T (K) b 1500 2000 2500 800 1000 1300

5 x 10-a 0.616 30 70 120 1 x 10-8 0.500 130 280 410 0.923 80 110 160 1 x lo-$ 0.960 310 520 730 0.992 140 190 260 1 x lo-* 0.995 500 770 1050 1.0 210 270 370 1 x 10-5 1.0 690 1030 1370 1.0 280 360 480 1 x 10-6 1.0 880 1280 1680 1.0 35 3 450 600

A s the calculation results show, with increasing pressure the structural vacancy concentration decreases quickly, asymptotically approaching zero. The initial va- cancy concentrations are equilibrium concentration for P = 0 and for the temperatures chosen, i.e. the upper boundaries of the region of homogeneity coincide practically

166 A. I. GUSEV: Structural Vacancies in Nonstoichiometric Compounds

with the corresponding phase boundaries on the phase diagrams of the Zr-C and V-C- systems [8, 131. For example, calculation according to (21) at P = 0 and T = 1300 K for the vanadium carbide yields n = 0.135, i.e. in accordance with the calculation, vanadium carbide VCo.8G6 is in thermodynamic equilibrium under these conditions, which is practically in agreement with the phase diagram.

It follows from (21) that the pressure is determined chiefly by the relation between AHi and ASi, i.e. by the quantity AGi since i3AE18ni is small in absolute magnitude compared to AGi. Taking account of the fact that the free energies of formation of structural vacancies in transition metal carbides and nitrides are close in magnitude [7, 12, 161, it niay be expected that approximately equal conditions will be required for the synthesis of these compounds in a practically defect-free state. Indeed, com- parison of the results obtained shows that the synthesis of zirconium and vanadium carbides possessing equal crystal lattice imperfections requires approximately equal conditions: for ZrC1-, with n = 10-3 the equilibrium temperature and the equilibrium pressure are equal to 1500 K and 3.07 x 1013 Pa, respectively, whereas for VC1-, with n = a pressure of 2.58 x 1013 Pa is required a t T = 1300 K.

Thus, simultaneous application of high pressures and high temperatures permits synthesis of compounds with crystal sublattice defect relations that are not realized under normal conditions. The teniperature and pressure dependence, found in general form (taking into account the volume variation of a substance due to a change in defect content, due to compression under pressure and due to thermal expansion), of the equilibrium vacancy concentration of any nonequivalent sublattice of a com- pound, makes it possible to calculate the conditions of P-T treatment for any com- pound containing structural vacancies.

References

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[3] Y. SYONO, T. GOTO, 5. NAKAI, Y. NAKAOAWA, and H. IWASAKI, J. Phys. SOC. Japan 37, 442

[4] H. IWASAKI, Japan. J. appl. Phys. 10, 1149 (1971). [5] A. I. GUSEV, Zh. fiz. Khim. 64, 773 (1980). [6] F. F. VORONOV, Dokl. Akad. Nauk SSSR 22G, 1052 (1976). [7] A. I. GUSEV, Teplofiz. vysok. Temp. li, 1232 (1979). [8] R. V. SARA, 5. Amer. Ceram. SOC. 48, 243 (1965). [9] A. S. BOLGAR, A. G. TURCHANIN, and V. V. FESENKO, Termodinamicheskie svoistva karbidov,

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Izd. Naukova Dumka, Kiev 1973. [lo] C. JUN and P. SHAFFER, J. Icss-common Metals 23, 367 (1971). [ll] H. BITTNER and H. GORETZKY, Monatsh. Chem. 93, 1000 (1962). [12] ,4. I. GUSEV, Zh. fiz. Khim. 6 i , 1382 (1983). [13] E. K. STORMS and R. J. MCNEAL, J. phys. Chem. 88, 1402 (1962). [14] G. V. SAMSONOV, G. SH. UPADHAYA, and V. S. NESHPOR, Fizicheskoe materialovedenie karbi-

[15] A. I. GUSEV, N. A. IVANOV, G. P. SHVEIKIN, and P. V. GELD, Izv. Akad. Nauk SSSR, Ser.

[16] A. I. GUSEV, Zh. neorg. Khim. 26, 629 (1980).

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(Received April 6 , 1984)