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Oxidation kinetic method in studying the defect structure and transport properties of non-stoichiometric manganous sulphide
Journal of Physics and Chemistry of Solids 62 (2001) 1967±1976
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Oxidation kinetic method in studying the defect structure and transport properties of non-stoichiometric manganous sulphide Z. Grzesik*, S. Mrowec Department of Solid State Chemistry, Faculty of Materials Science and Ceramics, University of Mining and Metallurgy, al. A. Mickiewicza 30, 30-059 Krakow, Poland Received 27 July 2000; accepted 19 January 2001
Abstract Thermodynamics and kinetics of point defects in metal-de®cient manganous sulphide (Mn12yS) have been studied using Rosenburg's two-stage kinetic method. Sulphidation rate measurements have been carried out in a new microthermogravimetric apparatus, enabling the mass changes of the sulphidized sample to be determined as a function of time with the accuracy of the order of 10 27 g. In one series of two-stage kinetic experiments the concentration of cation vacancies and their mobility in Mn12yS have been determined as a function of temperature (1073±1273 K) and sulphur activity (10 21 2 10 4 Pa). In agreement with the literature data it has been found that cation vacancies in this sulphide are doubly ionized and their concentration is the 00 following function of equilibrium sulphur pressure and temperature: VMn 5:41 z 1022 p1=6 S2 exp 242kJ=mol=RT. The chemiÄ cal diffusion coef®cient, in turn, has been found to be pressure independent and is the following function of temperature: D 3.88 z 10 22 exp((281.5 kJ/mol)/RT). Recalculation of this coef®cient into a defect diffusion coef®cient, which is a direct measure of their mobility, yields the following empirical equation: DV 1.29 z 10 22 exp((281.5 kJ/mol)/RT). Using these data self-diffusion coef®cient of cations in Mn12yS has been calculated as a function of temperature and sulphur activity: DMn 6:98 z 1024 ps1=6 exp 2123:5 kJ=mol=RT). Enthalpies and entropies of formation and migration of point defects in metalde®cit manganous sulphide have been reported. q 2001 Elsevier Science Ltd. All rights reserved. Keywords: A. Chalcogenides; C. Thermogravimetric analysis (TGA); D. Diffusion; D. Thermodynamic properties
1. Introduction Defect and transport properties of non-stoichiometric transition metal oxides and sulphides have been extensively studied for years because of their fundamental importance in the description of reactivity and semi-conducting properties of these materials. Information on defect structure is obtained from non-stoichiometry and electrical conductivity data, and that on transport properties from chemical and self-diffusion measurements. However, all these data could not be obtained on the same specimen in one experimental series. As a consequence, it is impossible to avoid at least trace differences in the impurity content of particular samples, as well as to carry out different experiments under exactly the same thermodynamic conditions, and this is the reason why the results obtained in different laboratories are * Corresponding author. Fax: 148-12-6172493. E-mail address: [email protected] (Z. Grzesik).
not in full agreement. About 40 years ago Rosenburg [1] published a novel, fascinating method enabling the determination of the concentration and the mobility of point defects in a given non-stoichiometric metal oxide or sulphide as a function of temperature and oxidant activity in one series of relatively simple thermogravimetric experiments. As modern microthermogravimetric equipment now offer very high accuracy in determining very small mass changes of a given sample under strictly de®ned thermodynamic conditions, this method creates much better possibilities for studying the thermodynamics and kinetics of point defects in a given material. This method has successfully been applied to several oxide systems [2±5] but not to the sulphides, so far, because of much greater experimental dif®culties in studying the kinetics of solid-sulphur interaction at high temperatures. The application of Rosenburg's method depends, namely, not only on the accuracy of weight gain measurements as a function of time, but also on the possibility of rapid changes of the oxidant pressure in the
0022-3697/01/$ - see front matter q 2001 Elsevier Science Ltd. All rights reserved. PII: S 0022-369 7(01)00038-5
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Z. Grzesik, S. Mrowec / Journal of Physics and Chemistry of Solids 62 (2001) 1967±1976
reaction chamber. In the case of oxygen atmosphere this fundamental requirement may readily be ful®lled, but in this respect heavy sulphur vapour creates considerable experimental dif®culties. In addition, thermogravimetric equipments currently utilized in sulphidation studies offer much lower accuracy because of limited possibilities of the quartz spiral thermobalances being used for this atmosphere. Being active in this area of research for more than 30 years we have been able to develop a new generation of microthermogravimetric apparatus which makes it possible to measure not only the kinetics of solid-sulphur interaction with very high precision, but also to suddenly change the sulphur vapour pressure in the reaction chamber from one constant value to another [6,7]. The possibility has, then, been created to apply the Rosenburg's method for studying the defect and transport properties of transition metal sulphides. The aim of the present paper is an attempt to apply this interesting method to the study of the thermodynamics and kinetics of point defects in metal-de®cit manganous sulphide (Mn12yS). The choice of the Mn-MnS-S2 system for the ®rst application of Rosenburg's method to sulphides results from the fact that the defect structure and transport properties in this system have been most extensively studied by different authors using various experimental techniques [8±15]. 2. Description of the method The method is based on the Wagner's theory of metal oxidation [16,17] and thus, the main assumptions of this theory have to be ful®lled, i.e., the layer of the reaction product (scale) must be compact and adhere well to the substrate, and its growth rate has to be governed by volume diffusion of cations or anions through point defects in the scale (parabolic kinetics). From Wagner's theory it follows that if the defect diffusion coef®cient is independent of their concentration i.e., when the defect concentration is low and the defects do not interact, the parabolic rate constant of metal oxidation (or sulphidation) is the product of the concentration of defects at the interface where they are created and the chemical diffusion coef®cient: 0
kp Nd z D~ 0
1
where kp is the parabolic rate constant of metal oxidation, expressed in cm 2 s 21, Nd Ðdefect concentration in site fracÄ Ðthe chemical diffusion coef®cient. tion, and D Chemical diffusion is the ambipolar process of defect migration under their concentration gradient [18,19] and Ä is a direct measure of the rate of defect diffuconsequently, D sion in a given solid under non-equilibrium conditions. The defect diffusion coef®cient, in turn, is a measure of defect mobility under thermodynamic equilibrium and consequently, it is the measure of the random walk of defects. Thus, the only difference between these two diffusion coef®cients results from accelerating effect of the much more mobile electronic
defects. If the concentration of defects (non-stoichiometry) is Ä very low and the defects do not interact, the relation between D and Dd is given by [20,21]: D~ 1 1 upuDd
2
where Dd denotes the defect diffusion coef®cient and `p'Ðthe effective charge of ionic defects. As the self-diffusion coef®cient of ions is the product of the self-diffusion coef®cient of defects and their concentration expressed in site fraction: Di Dd Nd
3
Eq. (1) can be extended to the following form: 0
kp D~ z Nd 1 1 upu z Dd Nd 1 1 upu z Di
4
where Dd is the self-diffusion coef®cient of cations or anions, depending on whether the cation or anion sublattice is predominantly disordered. As can be seen, Wagner's approach enables the calculation of the product of Nd and Dd when the degree of defect ionization is known, and thereby the self-diffusion coef®cient of ions, but not the separation of Nd from Dd which have fundamental signi®cance. Rosenburg [1] proposed a method for separating these two important quantities by an alternative, yet simple thermogravimetric technique. It consists in the interruption of the oxidation process at a given temperature by removing of the oxidant from reaction chamber and, after equilibrating of the metallic phase with the reaction product, continuing the oxidation by readmission of the oxidizing gas to the reaction space. In the ®rst stage of oxidation the process follows familiar parabolic kinetics with the linear concentration gradient of diffusing defects in the growing scale, as shown schematically in Fig. 1. If the thickness of the scale reaches a required value the oxidation process is interrupted by removing the oxidizing gas and the M-MX-X2 system goes gradually to equilibrium. In the case of p-type semi-conducting scales (M12yX or MX11y) equilibrium means that there is a minimum in the concentration of cation vacancies or interstitial anions at a given temperature, and in case of n-type scales (M11yX or MX12y) the maximum concentration of interstitial cations or anion vacancies (Fig. 1). If now the oxidant pressure is suddenly raised to its value prior to the interruption, the diffusion of defects is resumed and their original concentration distribution is gradually re-established (Fig. 2). Two stages of this process may be distinguished. Immediately after re-admission of the oxidant, the reaction proceeds much faster than before the interruption, as a result of rapid changes of defect concentration in deeper and deeper parts of the scale, and this stage lasts the longer the thicker is the scale formed prior the interruption. It should be noted, that in this stage the scale layer is only `oxidized' (nonstoichiometry is changing) and consequently, in terms of diffusion theory this process proceeds in semi-in®nitive system. The reaction rate gradually decreases and after restoration of the linear concentration gradient of defects
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Fig. 1. Defect distribution (schematic) in the growing p- and n-type scales under steady-state conditions (dotted lines) and in M-MX-X2 system at the thermodynamic equilibrium (solid lines).
Fig. 2. Defect concentration changes (schematic) in the growing pand n-type scales after re-admission of oxidant to the reaction chamber.
in the scale, oxidation process proceeds again with parabolic kinetics observed before the interruption. However, if the increase in the scale thickness during these two stages of reaction may be neglected, at the beginning a `new' parabolic kinetics should be observed (diffusion in semi-in®nitive system) and later on the course of the process may be approximated by the straight line (steady state diffusion problem). Thus, the course of the reaction in these two stages constitutes the paralinear kinetic curve (Fig. 3). Considering appropriate boundary conditions for Fick's second law Rosenburg [1] derived a rather complicated differential equation which he solved numerically obtaining two simple limiting solutions: pp Dm 1:128C0 D~ t when t ! X02 =D~ 5 A 1 and ~ 0 Dm X C DC z t 1 0 0 when t . X02 =2D~ A 2 X0 3
6
where (Dm/A)1 denotes the weight gains of the oxidized sample per unit surface area in the ®rst stage of oxidation after readmission of the oxidant to the reaction chamber, and (Dm/A)2 Ðweight gains in the second stage of the reaction, C0 is the concentration of defects in the scale at the interface where they are created, X0 Ðthe scale thickness obtained before the break of the reaction (X0 @ DX, where X is the
Fig. 3. Rosenburg's paralinear kinetic curve (schematic).
increase in the scale thickness during the whole period of oxidation after readmission of the oxidant to the reaction chamber). Eq. (5) describes the initial, parabolic section of schematic curve shown in Fig. 3 with an accuracy of about 1% when the condition t ! X02 =D~ is ful®lled, and Eq. (3) characterises the linear course of the reaction for times longer than ~ These equations can be expressed in a slightly different X02 =2D. form, more suitable for graphical interpretation: p Dm kp t 1 C p 7 A 1 and Dm k1 t 1 C 1 A 2
8
where kp and kl are parabolic and linear rate constants in successive stages of the reaction, expressed in gcm 22s 20.5 and gcm 22s 21, respectively. Both these constants can readily be obtained by plotting the results obtained in parabolic and linear systems of co-ordinates, respectively. Cp characterizes deviations from a parabolic course of the reaction at its very beginning, resulting from not yet constant oxidant pressure after its readmission to the reaction chamber. Cl, in turn, is directly given by extrapolation of the straight line in Fig. 3 to the y-axis. Since kp and kl , as well as Cp and Cl can be determined in one two-stage oxidation experiment, two unknown
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Ä , and Cd, can separately be calculated using the quantities, D following relationships: !2 1:128k1 X0 D~ 9 kp and Cd
kp 1:128 k1 X0
2 10
The concentration of defects can also be obtained from the following equation: Cd
3 C1 2 Cp X0
11
The two-stage oxidation kinetic experiments discussed above can be repeated several times at different temperatures and oxidant pressures without removing the sample from the reaction chamber. Thus, in one series of such experimentsÐcarried out on one and the same materialÐ the concentration and the mobility of point defects can be determined as a function of temperature and oxidant activity, enabling the enthalpy and entropy of their formation and migration, to be calculated. It should be noted, that all these data may be obtained with very high precision, dif®cult or even impossible to be achieved with other conventional experimental techniques. In Rosenburg's method, namely, this precision depends only on the accuracy of weight gain measurements of a given sample as a function of time, which in modern automatic microthermogravimetric equipment it is possible to follow continuously with the accuracy of the order of 10 27 g. As a consequence, the whole Rosenburg's paralinear kinetic curve (Fig. 3) can be determined with very high precision, enabling the concentration of defects and their mobility to be calculated with the corresponding accuracy. The following conditions must be ful®lled in order to obtain the correct results. 1. The scale layer formed on the metal surface in preliminary and subsequent stages of the reaction must be compact and adhere well to the substrate. 2. The overall growth rate of the scale in every stage of oxidation must be diffusion controlled. 3. The homogenization time after preliminary oxidation ~ must exceed X02 =D. 4. The parabolic rate constant, kp, must be determined at a time much shorter than X02 =D~ and the linear rate constant, ~ In order to properly estikl, at times longer than X02 =2D. mate the required scale thickness X0, it is necessary to Ä . If this value is know at least the order of magnitude of D unknown the required thickness X0 must be determined in preliminary two stage oxidation experiments. 5. During the determination of the whole paralinear kinetic
curve the increase of the scale thickness, DX, must be much smaller than the initial thickness X0 (DX ! X0).
3. Materials and experimental procedure Sulphidation rate measurements of spectrally pure manganese plates with the total impurity content not exceeding 100 ppm and the total surface area of about 2 cm 2 and the thickness of 0.1 cm, have been carried out in the apparatus shown schematically in Fig. 4. One of the main features of this novel microthermogravimetric equipment consists in the possibility of determining the mass changes of a given sample, occurring as a result of solidsulphur interaction, with an accuracy of the order of 10 27 g. This has been achieved owing to the replacement of the helix balance, currently used in such equipments, by an automatic electronic microthermobalance, which is protected against sulphur attack by the stream on inert gas (helium) ¯owing through the balance chamber and preventing the condensation of sulphur vapour on the cool upper parts of the apparatus (Fig. 4). A second gas stream passes through one of two liquid-sulphur reservoirs (A or B) carrying its vapours to the reaction chamber, where both helium ¯uxes join together and, after having passed through the apparatus leave, it through one of two liquid-nitrogen sulphur traps (4). The desired partial pressure of sulphur vapour in the reaction chamber is obtained by suitable mixing of this vapour with the carrier gas, the ¯ow rate of which together with the temperature of liquid-sulphur bath (A or B) determines the activity of sulphur in the He-S2 gas mixture of the total pressure 10 5 Pa. The second, most important advantage of this apparatus, equipped with two sulphur reservoirs, consists in the possibility of making rapid changes of sulphur vapour pressure in the reaction chamber, which is absolutely necessary in applying twostage Rosenburg's kinetic method. The details of this
Fig. 4. The scheme of the microthermogravimetric apparatus; (1) gas puri®er, (2) ¯owmeter, (3) microbalance, (4) sulphur trap, (5) thermoelement, (6) reaction furnace, (7) sample, (8) rotation pump, (9) bulb, (10) solid electrolyte probe, (A, B) sulphur baths, (C, D) valves.
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novel thermogravimetric assembly and the experimental procedure in studying the kinetics of metal-sulphur and sulphide±sulphur interaction, have been described elsewhere [6,7]. 4. Results and discussion In contrast with the majority of transition metal sulphides, manganous sulphide shows very low deviations from stoichiometry, and thereby defect concentration of the order of that in nickel oxide [22]. Based on extensive non-stoichiometry and electrical conductivity data, as well as on selfdiffusion measurements, the thermodynamics and kinetics of point defects in this sulphide have been described [11,23±25]. It has been shown that over the major part of the phase ®eld, corresponding to higher sulphur activities, a-MnS, having rocksalt structure, is a metal-de®cit p-type semi-conductor with predominant defects being doubly ionized cation vacancies and electron holes (Mn12yS). It has been found also that the mobility of point defects does not depend on their concentration, clearly indicating that these defects do not interact and are randomly distributed in the crystal lattice [12]. Finally, it has been demonstrated that the slowest step determining the overall growth rate of MnS scale on manganese is the outward diffusion of cations (parabolic kinetics) and the pressure and temperature dependence of this rate is in agreement with that calculated from self-diffusion coef®cients of Mn in Mn12yS and non-stoichiometry data [26]. As can be seen, the Mn-Mn12yS-S2 system is very convenient for the veri®cation of all these results using Rosenburg's two-stage sulphidation method. The application of this method became possible owing to a vary high sensitivity of weight gain measurements of the sulphidized Mn-sample, as well as due to the possibility of rapid changes of sulphur vapour pressure in our new microthermogravimetric apparatus [6,7]. Sulphidation rate measurements have been carried out at temperatures ranging from 973 to 1273 K and in the sulphur pressure interval 10 21 ±10 4 Pa. The experimental procedure was as follows. A manganese plate was suspended on the balance in the reaction chamber and the furnace (6) was moved down below the reaction tube. The whole apparatus was then evacuated and ¯ushed for several hours with spectrally pure helium by passing the helium through all three ¯owmeters (2) with both valves C and D open. The carrier gas, before entering the apparatus, passed through the gas puri®er (1) with iron wool heated to 900 K, as a result of which the partial pressure of oxygen was reduced from 10 21 down to 10 212 Pa. This partial pressure was continuously monitored in the outlet gas during the whole of every experiment with the use of calcia stabilized zirconia solid electrolyte probe (10). Subsequently, the valve C was closed and the sulphur bath A was heated to the desired temperature. Simultaneously, the reaction furnace (6) was heated to the temperature at which the Mn-sample was to be sulphidized.
1971
When the desired partial pressure of sulphur vapour in He-S2 gas mixture was reached, the furnace (6) was raised to the reaction position (see Fig. 4) and the He-S2 gas was let into the reaction chamber by appropriate changing the status of valves C and D (C-open and D closed). As a result of this operation He-S2 gas, previously leaving the apparatus through trap E, started to ¯ow in the opposite direction, i.e., through the reaction chamber, and left the apparatus through the trap F. It has been found that the steady state conditions in the reaction chamber were established during about 60 s [7]. This rather surprisingly short time in which the pressure of heavy sulphur vapour reached a constant value results from the laminar (without any turbulence) ¯ow of He-S2 gas from sulphur reservoir to the reaction chamber. It should be stressed once again, that it was due to this possibility in particular that Rosenburg's method could be successfully utilized in studying the kinetics and thermodynamics of point defects in non-stoichiometric manganous sulphide. The rate of sulphidation was continuously followed by an automatic microthermobalance (3) with the accuracy of 10 27 g. Maximum ¯uctuations of temperature in the reaction chamber and in liquid sulphur reservoir did not exceed ^1 K. It has been found in agreement with the literature data [26±28] that the sulphidation process strictly follows the parabolic rate law at all the temperatures and sulphur pressures, and the reproducibility of the kinetic results was excellent. For illustration, some of these results are shown in Fig. 5 in a parabolic plot. When, at a given temperature and sulphur pressure, a desired thickness of the scale, X0, estimated from available self-diffusion data, was reached, the He-S2 gas mixture in the reaction chamber was replaced by a pure helium gas stream and the system Mn-Mn12yS went gradually to equilibrium. After thermodynamic equilibrium was reached, as manifested by the constant weight of the sample, He-S2 gas stream was again let into the reaction chamber and weight gain measurements were resumed. Ä This process was continued until the condition t . X0/2D
Fig. 5. The kinetics of manganese sulphidation for several temperatures in parabolic plot.
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Fig. 6. Paralinear kinetic curves of manganese sulphidation for several temperatures.
Fig. 8. Parabolic plots of early stages of manganese sulphidation for several temperatures.
was ful®lled, however, in a limited period of time, such that the increase in the scale thickness was negligible (DX ! X0). Subsequently, the whole equilibration-sulphidation procedure was repeated at different temperatures and sulphur pressures in order to obtain Rosenburg's paralinear kinetic curves enabling the concentration and the mobility of defects to be calculated as a function of both these parameters. In every subsequent experiment the increase in the scale thickness, X0, has been taken into account. For illustration, several kinetic curves obtained at different temperatures and sulphur pressures are shown in Figs. 6 and 7. As can be seen, in all cases two stages of the reaction are clearly visible, manifesting themselves by initial parabolic and subsequent linear kinetics. In order to calculate the parabolic rate constants, the results of initial stages of sulphidation were plotted once again in a parabolic system of co-ordinates (Figs. 8 and 9). As predicted, the parabolic rate law in this stage of sulphidation was obeyed strictly, enabling kp and Cp values to be calculated. In turn kl and Cl valves were obtained from the linear parts of the kinetic
curves presented in Figs. 6 and 7. Using all these data and Eqs. (9±11) the concentration of cation vacancies and the chemical diffusion coef®cient in Mn12yS have been calculated as functions of temperature and sulphur activity. The results of these calculations are shown in Figs. 10±13 in double logarithmic and Arrhenius plots, respectively. From the diagrams presented in Figs. 10 and 11 it follows, that in agreement with non-stoichiometry data reported by other authors [8,13] the concentration of cation vacancies in Mn12yS increases with increasing sulphur activity with the slope 1/6 and this dependence does not change with temperature. Thus, the obtained results can be presented in the form of the following empirical equation:
This result con®rms the conclusion that the predominant point defects in metal-de®cit manganous sulphide are doubly ionized cation vacancies end electron holes. Applying
Fig. 7. Paralinear kinetic curves of manganese sulphidation for two different sulphur vapour pressure.
Fig. 9. Parabolic plots of early stages of manganese sulphidation for two different sulphur vapour pressures.
00
VMn 5:41 z 1022 p1=6 s2 exp
42 kJ=mol RT
12
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1973
Fig. 10. The dependence of non-stoichiometry, y, in Mn12yS on sulphur vapour pressure for several temperatures.
Fig. 12. The dependence of the chemical diffusion coef®cient on sulphur vapour pressure for several temperatures.
the mass action00 law and the appropriate electro-neutrality condition 2VMn hz to the defect equilibrium:
in Mn12yS does not depend on sulphur activity and thereby on defect concentration. Thus, the temperature dependence of this coef®cient can be expressed by the following empirical equation: 81:5 kJ=mol D~ 3:88 z 1022 exp 2 15 RT
00 1 S VMn 1 2hz 1 SS 2 2
the following theoretical relationship is obtained: 00 DSf DHf exp exp 2 VMn 0:63 z p1=6 S2 3R 3RT
13
14
Introducing this empirical relationship into theoretical Eq. (2) and taking p 22 one obtains the following equation describing the temperature dependence of vacancy (defect) diffusion coef®cient in metal-de®cit manganous sulphide: 81:5 kJ=mol DV 1:29 z 1022 exp 2 16 RT
where DSf and DHf are entropy and enthalpy of defect formation, respectively (KroÈger-Vink notation [29] of defects is used throughout this paper). From the comparison of empirical Eq. (12) with theoretical Eq. (14) it follows that DSf 261.2 J/mol deg and DHf 126 kJ/mol. These values are compared in Table 1 with those obtained by Rau [8] and Grzesik and Mrowec [13] from non-stoichiometry data. From Figs. 12 and 13, in turn, it clearly follows that, in agreement with other data obtained from re-equilibration rate measurements [12], the chemical diffusion coef®cient
The fact that the mobility of point defects in Mn12yS does not depend on their concentration is not surprising, because this concentrationÐeven at very high temperatures and sulphur activitiesÐis very low [8,13] and consequently, the defects do not interact and are randomly distributed in the crystal lattice. Thus, the behaviour of point defects in
Fig. 11. The comparison of non-stoichiometry data obtained in this work with those reported by Rau [8] and Grzesik et al. [13].
Fig. 13. The dependence of the chemical diffusion coef®cient in Mn12yS on temperature, in Arrhenius plot.
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Table 1 Thermodynamic parameters of formation and migration of defects in Mn12yS. DHf, kJ/mol
DSf, J/mol z deg
DHm, kJ/mol
DSm, J/mol z deg
References
Remarks
124.5 123 ± 126 ±
264.4 266.2 ± 261.2 ±
± ± 83.4 81.5 79.7
± ± 12.9 9.52 11.0
Rau [8] Mrowec et al. [13] Grzesik et al. [12] This work Danielewski [26]
Calculated from non-stoichiometry Determined from non-stoichiometry Determined from re-equilibrium kinetics Rosenburg method From MnÐsulphidation rate measurements
Mn12yS can be treated in terms of point defect thermodynamics and their transport properties should follow the rules of solid state diffusion theory based on simple point defect model [20,21]. The temperature dependence of the defect diffusion coef®cient for metal oxides and sulphides with such a simple defect structure is described by the following relationship [20,21]: DSm DHm exp 2 17 DV aa20 kv exp R RT where is the geometrical factor, kÐthe transmission coef®cient, the vibration frequency, and a0 the jump distance from the lattice site to the neighbouring vacancy. By comparing this relationship with empirical Eq. (16) the activation entropy (DSm) and enthalpy (DHm) of defect diffusion can be calculated. From this comparison it follows directly that DHm 81.5 kJ/mol. Assuming, in turn, a k 1, and Ê [30], as well as calculating the vibration a0 5.30 A frequency of cations according to Ref. [31]: s 2 DHm 1:46 z 1012 s21 18 n pa0 M where M is the atomic weight of manganese, one obtains by combining Eqs. (13 and 14) DSm 9.52 J/mol deg. These DHm and DSm values are compared in Table 1 with analogous data reported by other authors. Empirical Eqs. (12) and (16) can be utilized for the calculation of the self-diffusion coef®cient of cations in metalde®cit manganous sulphide as a function of temperature and sulphur activity, using Eq. (3): 123:5 kJ=mol DMn 6:98 z 1024 p1=6 19 S2 exp 2 RT It seems again reasonable to compare the results obtained by the two-stage kinetic method with other data available in the literature. Because of considerable experimental dif®culties in studying the high temperature reactions in sulphur containing atmosphere, DMn values in Mn12yS have only been estimated from Mn-sulphidation rate measurements [11]: 121 kJ=mol DMn 6:7 z 1024 p1=6 20 S2 exp 2 RT 54
During last few years, in very careful experiments using Mn radioactive manganese isotope as a tracer, the self-
diffusion coef®cient of 54Mn in Mn12yS has ®nally been determined as a function of temperature (1073±1373 K) and sulphur activity (10±10 4 Pa) [23,24]. The tracer diffusion coef®cient, DT, reported in this work, have been recalculated into `real' self-diffusion coef®cients, DMn, by taking into account the correlation effects. The results of these calculations can be expressed by the following empirical relationship [24]: 115 kJ=mol 21 DMn 3:38 z 1024 p1=6 exp 2 S2 RT In addition, owing to the development of the novel microthermogravimetric apparatus [7] used in the present work, the chemical diffusion coef®cient [12] and deviations from stoichiometry [13] in Mn12yS have recently been determined, and basing on these results DMn in Mn12yS has been calculated as a function of temperature and sulphur activity [32]: 124:4 kJ=mol 22 exp 2 DMn 8:73 z 1024 p1=6 S2 RT From the comparison of Eqs. (19±22) it follows clearly that the results obtained in one series of rather simple, twostage sulphidation experiments (Eq. 19) are in excellent agreement with those derived from much more dif®cult and time-consuming studies. This agreement is better visible in collective plot presented in Fig. 14. However, the scattering of tracer data, inevitably observed in such experiments, results in much higher standard deviation of DMn values than in the case of kinetic rate measurements. If one considers also the fact that in Rosenburg's method the concentration and mobility of defects are determined in one series of kinetic experiments, it becomes obvious that the most reasonable and accurate information on defect and transport properties of a given sulphide and oxide may be obtained by this method. 5. Conclusions The results obtained in the present work allow the following conclusions to be formulated. Thermodynamics and kinetics of point defects in nonstoichiometric metal sulphides, like in oxides, can be successfully studied using the two-stage kinetic method proposed by Rosenburg [1], provided that the
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1975
Fig. 14. The comparison of available data in the literature [11,12,24] of self-diffusion coef®cient of manganese in metal de®cient manganous sulphide with those obtained in the present work.
thermogravimetric apparatus offers suf®ciently high accuracy of weight gain measurements as a function of time, as well as makes it possible to rapidly change the sulphur partial pressure in the reaction chamber. One of the most important advantages of this procedure consists in the possibility of determining the chemical diffusion coef®cient and the concentration of defects in a given sulphide in one series of sulphidation rate measurements without the necessity of removing the metal sample from the reaction chamber. As a consequence, the possible in¯uence of trace impurities on defect and transport properties of a given material may be easier explained or neglected. Finally, all this information can be obtained in a rather simple experimental procedure, in contrast with other, much more dif®cult and time-consuming methods. The application of Rosenburg's method to metal sulphides has been proved on the Mn-Mn12yS-S2 system, and it has been shown that the obtained results are in satisfactory agreement with the available literature data. As, however, all the results concerning defect concentration and their mobility, reported in this paper, have been obtained on the same material, with very high precision, they may be considered as most reliably describing the defect and transport properties of metal-de®cit manganous sulphide. Acknowledgements This work was supported by the Polish State Committee for Scienti®c Research (project no. 7T08A03815). References [1] A.J. Rosenburg, J. Electrochem. Soc. 107 (1960) 795. [2] E. Fryt, S. Mrowec, T. Walec, Oxid. Met. 7 (1973) 117.
[3] P. Kofstad, J. Phys. Chem. Solids 44 (1983) 879. [4] B. OberlaÈnder, P. Kofstad, I. Kvernes, Mat. Wiss. Und Werkstofftechnik 19 (1988) 190. [5] T. Smith, J. Electrochem. Soc. 112 (1965) 560. [6] Z. Grzesik, S. Mrowec, T. Walec, J. Dabek, J. Therm. Anal. Cal. 59 (2000) 985. [7] Z. Grzesik, S. Mrowec, Solid State Phenomena 72 (2000) 47. [8] H. Rau, J. Phys. Chem. Solids 39 (1978) 339. [9] H. Le Brusq, J.P. Delmaire, Rev. Int. Hautes Temp. Refract. 11 (1974) 193. [10] J. Rasner, D. Carton, C.R. Acad. Sci. (Paris), Ser. II 290 (1981) 405. [11] M. Danielewski, S. Mrowec, Solid State Ionics 17 (1985) 29. [12] Z. Grzesik, S. Mrowec, T. Walec, J. Dabek, J. Phys. Chem. Solids 61 (2000) 809. [13] S. Mrowec, Z. Grzesik, Solid State Phenomena 72 (2000) 69. [14] M. Danielewski, S. Mrowec, Solid State Ionics 17 (1985) 41. [15] M. Danielewski, S. Mrowec, Solid State Ionics 17 (1985) 319. [16] S. Mrowec, An Introduction to the Theory of Metal Oxidation, National Bureau of Standards and National Science Foundation, Washington, DC, 1982, p. 172. [17] P. Kofstad, High Temperature Corrosion, Elsevier, Amsterdam, 1988, p. 423. [18] P. Child, L.B. Laub, J.B. Wagner, Chemical diffusion in nonstoichiometric compounds, Mass Transport in Nonmetallic Solids, Proc. Brit. Ceram,. Soc. 19 (1971) 29. [19] S. Mrowec, K. Hashimoto, J. Mater. Sci. 30 (1995) 4801. [20] P. Kofstad, Nonstoichiometry, Diffusion and Electrical Conductivity in Binary Metal Oxides, J. Wiley Interscience, New York, London, Sydney, Toronto, 1972. [21] S. Mrowec, Defects and Diffusion in Solids, Elsevier, Amsterdam, Oxford, New York (1980) 174. [22] S. Mrowec, J. Janowski, Similarities and Differences in Defect Dependent Properties of Transition Metal Sulphides and Oxides, in: O. Johannesen, A.G. Andersen (Eds.), Selected Topics in High Temperature Chemistry. Defect Chemistry of Solids, Elsevier, Amsterdam, Oxford, New York, 1989, p. 55. [23] J. Gilewicz-Wolter, M. Danielewski, S. Mrowec, Physical Rev. B. 56 (1997) 8695.
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[24] J. Gilewicz-Wolter, M. Wolter, Physical Rev. (2001). [25] S. Mrowec, M. Danielewski, J. Gilewicz-Wolter, Solid State Ionics 117 (1999) 65. [26] M. Danielewski, Oxid. Met. 25 (1986) 54. [27] F.A. Erlefaie, W.W. Smeltzer, Oxid. Met. 16 (1981) 267. [28] P. Papaiacovou, H. Schmidt, H. Erhard, H.J. Grabke, Werkstoffe und Korrosion 38 (1987) 498.
[29] F. KroÈger, The Chemistry of Imperfect Crystals, North Holland, Amsterdam, 1964. [30] S. Furuseth, A. Kjekshus, Acta. Chem. Scand. 19 (1965) 1405. [31] M. O'Keefe, W.J. Moore, J. Chem. Phys. 36 (1962) 3009. [32] Z. Grzesik, S. Mrowec, Solid State Ionics (2001).
www.elsevier.com/locate/jpcs
Oxidation kinetic method in studying the defect structure and transport properties of non-stoichiometric manganous sulphide Z. Grzesik*, S. Mrowec Department of Solid State Chemistry, Faculty of Materials Science and Ceramics, University of Mining and Metallurgy, al. A. Mickiewicza 30, 30-059 Krakow, Poland Received 27 July 2000; accepted 19 January 2001
Abstract Thermodynamics and kinetics of point defects in metal-de®cient manganous sulphide (Mn12yS) have been studied using Rosenburg's two-stage kinetic method. Sulphidation rate measurements have been carried out in a new microthermogravimetric apparatus, enabling the mass changes of the sulphidized sample to be determined as a function of time with the accuracy of the order of 10 27 g. In one series of two-stage kinetic experiments the concentration of cation vacancies and their mobility in Mn12yS have been determined as a function of temperature (1073±1273 K) and sulphur activity (10 21 2 10 4 Pa). In agreement with the literature data it has been found that cation vacancies in this sulphide are doubly ionized and their concentration is the 00 following function of equilibrium sulphur pressure and temperature: VMn 5:41 z 1022 p1=6 S2 exp 242kJ=mol=RT. The chemiÄ cal diffusion coef®cient, in turn, has been found to be pressure independent and is the following function of temperature: D 3.88 z 10 22 exp((281.5 kJ/mol)/RT). Recalculation of this coef®cient into a defect diffusion coef®cient, which is a direct measure of their mobility, yields the following empirical equation: DV 1.29 z 10 22 exp((281.5 kJ/mol)/RT). Using these data self-diffusion coef®cient of cations in Mn12yS has been calculated as a function of temperature and sulphur activity: DMn 6:98 z 1024 ps1=6 exp 2123:5 kJ=mol=RT). Enthalpies and entropies of formation and migration of point defects in metalde®cit manganous sulphide have been reported. q 2001 Elsevier Science Ltd. All rights reserved. Keywords: A. Chalcogenides; C. Thermogravimetric analysis (TGA); D. Diffusion; D. Thermodynamic properties
1. Introduction Defect and transport properties of non-stoichiometric transition metal oxides and sulphides have been extensively studied for years because of their fundamental importance in the description of reactivity and semi-conducting properties of these materials. Information on defect structure is obtained from non-stoichiometry and electrical conductivity data, and that on transport properties from chemical and self-diffusion measurements. However, all these data could not be obtained on the same specimen in one experimental series. As a consequence, it is impossible to avoid at least trace differences in the impurity content of particular samples, as well as to carry out different experiments under exactly the same thermodynamic conditions, and this is the reason why the results obtained in different laboratories are * Corresponding author. Fax: 148-12-6172493. E-mail address: [email protected] (Z. Grzesik).
not in full agreement. About 40 years ago Rosenburg [1] published a novel, fascinating method enabling the determination of the concentration and the mobility of point defects in a given non-stoichiometric metal oxide or sulphide as a function of temperature and oxidant activity in one series of relatively simple thermogravimetric experiments. As modern microthermogravimetric equipment now offer very high accuracy in determining very small mass changes of a given sample under strictly de®ned thermodynamic conditions, this method creates much better possibilities for studying the thermodynamics and kinetics of point defects in a given material. This method has successfully been applied to several oxide systems [2±5] but not to the sulphides, so far, because of much greater experimental dif®culties in studying the kinetics of solid-sulphur interaction at high temperatures. The application of Rosenburg's method depends, namely, not only on the accuracy of weight gain measurements as a function of time, but also on the possibility of rapid changes of the oxidant pressure in the
0022-3697/01/$ - see front matter q 2001 Elsevier Science Ltd. All rights reserved. PII: S 0022-369 7(01)00038-5
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reaction chamber. In the case of oxygen atmosphere this fundamental requirement may readily be ful®lled, but in this respect heavy sulphur vapour creates considerable experimental dif®culties. In addition, thermogravimetric equipments currently utilized in sulphidation studies offer much lower accuracy because of limited possibilities of the quartz spiral thermobalances being used for this atmosphere. Being active in this area of research for more than 30 years we have been able to develop a new generation of microthermogravimetric apparatus which makes it possible to measure not only the kinetics of solid-sulphur interaction with very high precision, but also to suddenly change the sulphur vapour pressure in the reaction chamber from one constant value to another [6,7]. The possibility has, then, been created to apply the Rosenburg's method for studying the defect and transport properties of transition metal sulphides. The aim of the present paper is an attempt to apply this interesting method to the study of the thermodynamics and kinetics of point defects in metal-de®cit manganous sulphide (Mn12yS). The choice of the Mn-MnS-S2 system for the ®rst application of Rosenburg's method to sulphides results from the fact that the defect structure and transport properties in this system have been most extensively studied by different authors using various experimental techniques [8±15]. 2. Description of the method The method is based on the Wagner's theory of metal oxidation [16,17] and thus, the main assumptions of this theory have to be ful®lled, i.e., the layer of the reaction product (scale) must be compact and adhere well to the substrate, and its growth rate has to be governed by volume diffusion of cations or anions through point defects in the scale (parabolic kinetics). From Wagner's theory it follows that if the defect diffusion coef®cient is independent of their concentration i.e., when the defect concentration is low and the defects do not interact, the parabolic rate constant of metal oxidation (or sulphidation) is the product of the concentration of defects at the interface where they are created and the chemical diffusion coef®cient: 0
kp Nd z D~ 0
1
where kp is the parabolic rate constant of metal oxidation, expressed in cm 2 s 21, Nd Ðdefect concentration in site fracÄ Ðthe chemical diffusion coef®cient. tion, and D Chemical diffusion is the ambipolar process of defect migration under their concentration gradient [18,19] and Ä is a direct measure of the rate of defect diffuconsequently, D sion in a given solid under non-equilibrium conditions. The defect diffusion coef®cient, in turn, is a measure of defect mobility under thermodynamic equilibrium and consequently, it is the measure of the random walk of defects. Thus, the only difference between these two diffusion coef®cients results from accelerating effect of the much more mobile electronic
defects. If the concentration of defects (non-stoichiometry) is Ä very low and the defects do not interact, the relation between D and Dd is given by [20,21]: D~ 1 1 upuDd
2
where Dd denotes the defect diffusion coef®cient and `p'Ðthe effective charge of ionic defects. As the self-diffusion coef®cient of ions is the product of the self-diffusion coef®cient of defects and their concentration expressed in site fraction: Di Dd Nd
3
Eq. (1) can be extended to the following form: 0
kp D~ z Nd 1 1 upu z Dd Nd 1 1 upu z Di
4
where Dd is the self-diffusion coef®cient of cations or anions, depending on whether the cation or anion sublattice is predominantly disordered. As can be seen, Wagner's approach enables the calculation of the product of Nd and Dd when the degree of defect ionization is known, and thereby the self-diffusion coef®cient of ions, but not the separation of Nd from Dd which have fundamental signi®cance. Rosenburg [1] proposed a method for separating these two important quantities by an alternative, yet simple thermogravimetric technique. It consists in the interruption of the oxidation process at a given temperature by removing of the oxidant from reaction chamber and, after equilibrating of the metallic phase with the reaction product, continuing the oxidation by readmission of the oxidizing gas to the reaction space. In the ®rst stage of oxidation the process follows familiar parabolic kinetics with the linear concentration gradient of diffusing defects in the growing scale, as shown schematically in Fig. 1. If the thickness of the scale reaches a required value the oxidation process is interrupted by removing the oxidizing gas and the M-MX-X2 system goes gradually to equilibrium. In the case of p-type semi-conducting scales (M12yX or MX11y) equilibrium means that there is a minimum in the concentration of cation vacancies or interstitial anions at a given temperature, and in case of n-type scales (M11yX or MX12y) the maximum concentration of interstitial cations or anion vacancies (Fig. 1). If now the oxidant pressure is suddenly raised to its value prior to the interruption, the diffusion of defects is resumed and their original concentration distribution is gradually re-established (Fig. 2). Two stages of this process may be distinguished. Immediately after re-admission of the oxidant, the reaction proceeds much faster than before the interruption, as a result of rapid changes of defect concentration in deeper and deeper parts of the scale, and this stage lasts the longer the thicker is the scale formed prior the interruption. It should be noted, that in this stage the scale layer is only `oxidized' (nonstoichiometry is changing) and consequently, in terms of diffusion theory this process proceeds in semi-in®nitive system. The reaction rate gradually decreases and after restoration of the linear concentration gradient of defects
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Fig. 1. Defect distribution (schematic) in the growing p- and n-type scales under steady-state conditions (dotted lines) and in M-MX-X2 system at the thermodynamic equilibrium (solid lines).
Fig. 2. Defect concentration changes (schematic) in the growing pand n-type scales after re-admission of oxidant to the reaction chamber.
in the scale, oxidation process proceeds again with parabolic kinetics observed before the interruption. However, if the increase in the scale thickness during these two stages of reaction may be neglected, at the beginning a `new' parabolic kinetics should be observed (diffusion in semi-in®nitive system) and later on the course of the process may be approximated by the straight line (steady state diffusion problem). Thus, the course of the reaction in these two stages constitutes the paralinear kinetic curve (Fig. 3). Considering appropriate boundary conditions for Fick's second law Rosenburg [1] derived a rather complicated differential equation which he solved numerically obtaining two simple limiting solutions: pp Dm 1:128C0 D~ t when t ! X02 =D~ 5 A 1 and ~ 0 Dm X C DC z t 1 0 0 when t . X02 =2D~ A 2 X0 3
6
where (Dm/A)1 denotes the weight gains of the oxidized sample per unit surface area in the ®rst stage of oxidation after readmission of the oxidant to the reaction chamber, and (Dm/A)2 Ðweight gains in the second stage of the reaction, C0 is the concentration of defects in the scale at the interface where they are created, X0 Ðthe scale thickness obtained before the break of the reaction (X0 @ DX, where X is the
Fig. 3. Rosenburg's paralinear kinetic curve (schematic).
increase in the scale thickness during the whole period of oxidation after readmission of the oxidant to the reaction chamber). Eq. (5) describes the initial, parabolic section of schematic curve shown in Fig. 3 with an accuracy of about 1% when the condition t ! X02 =D~ is ful®lled, and Eq. (3) characterises the linear course of the reaction for times longer than ~ These equations can be expressed in a slightly different X02 =2D. form, more suitable for graphical interpretation: p Dm kp t 1 C p 7 A 1 and Dm k1 t 1 C 1 A 2
8
where kp and kl are parabolic and linear rate constants in successive stages of the reaction, expressed in gcm 22s 20.5 and gcm 22s 21, respectively. Both these constants can readily be obtained by plotting the results obtained in parabolic and linear systems of co-ordinates, respectively. Cp characterizes deviations from a parabolic course of the reaction at its very beginning, resulting from not yet constant oxidant pressure after its readmission to the reaction chamber. Cl, in turn, is directly given by extrapolation of the straight line in Fig. 3 to the y-axis. Since kp and kl , as well as Cp and Cl can be determined in one two-stage oxidation experiment, two unknown
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Ä , and Cd, can separately be calculated using the quantities, D following relationships: !2 1:128k1 X0 D~ 9 kp and Cd
kp 1:128 k1 X0
2 10
The concentration of defects can also be obtained from the following equation: Cd
3 C1 2 Cp X0
11
The two-stage oxidation kinetic experiments discussed above can be repeated several times at different temperatures and oxidant pressures without removing the sample from the reaction chamber. Thus, in one series of such experimentsÐcarried out on one and the same materialÐ the concentration and the mobility of point defects can be determined as a function of temperature and oxidant activity, enabling the enthalpy and entropy of their formation and migration, to be calculated. It should be noted, that all these data may be obtained with very high precision, dif®cult or even impossible to be achieved with other conventional experimental techniques. In Rosenburg's method, namely, this precision depends only on the accuracy of weight gain measurements of a given sample as a function of time, which in modern automatic microthermogravimetric equipment it is possible to follow continuously with the accuracy of the order of 10 27 g. As a consequence, the whole Rosenburg's paralinear kinetic curve (Fig. 3) can be determined with very high precision, enabling the concentration of defects and their mobility to be calculated with the corresponding accuracy. The following conditions must be ful®lled in order to obtain the correct results. 1. The scale layer formed on the metal surface in preliminary and subsequent stages of the reaction must be compact and adhere well to the substrate. 2. The overall growth rate of the scale in every stage of oxidation must be diffusion controlled. 3. The homogenization time after preliminary oxidation ~ must exceed X02 =D. 4. The parabolic rate constant, kp, must be determined at a time much shorter than X02 =D~ and the linear rate constant, ~ In order to properly estikl, at times longer than X02 =2D. mate the required scale thickness X0, it is necessary to Ä . If this value is know at least the order of magnitude of D unknown the required thickness X0 must be determined in preliminary two stage oxidation experiments. 5. During the determination of the whole paralinear kinetic
curve the increase of the scale thickness, DX, must be much smaller than the initial thickness X0 (DX ! X0).
3. Materials and experimental procedure Sulphidation rate measurements of spectrally pure manganese plates with the total impurity content not exceeding 100 ppm and the total surface area of about 2 cm 2 and the thickness of 0.1 cm, have been carried out in the apparatus shown schematically in Fig. 4. One of the main features of this novel microthermogravimetric equipment consists in the possibility of determining the mass changes of a given sample, occurring as a result of solidsulphur interaction, with an accuracy of the order of 10 27 g. This has been achieved owing to the replacement of the helix balance, currently used in such equipments, by an automatic electronic microthermobalance, which is protected against sulphur attack by the stream on inert gas (helium) ¯owing through the balance chamber and preventing the condensation of sulphur vapour on the cool upper parts of the apparatus (Fig. 4). A second gas stream passes through one of two liquid-sulphur reservoirs (A or B) carrying its vapours to the reaction chamber, where both helium ¯uxes join together and, after having passed through the apparatus leave, it through one of two liquid-nitrogen sulphur traps (4). The desired partial pressure of sulphur vapour in the reaction chamber is obtained by suitable mixing of this vapour with the carrier gas, the ¯ow rate of which together with the temperature of liquid-sulphur bath (A or B) determines the activity of sulphur in the He-S2 gas mixture of the total pressure 10 5 Pa. The second, most important advantage of this apparatus, equipped with two sulphur reservoirs, consists in the possibility of making rapid changes of sulphur vapour pressure in the reaction chamber, which is absolutely necessary in applying twostage Rosenburg's kinetic method. The details of this
Fig. 4. The scheme of the microthermogravimetric apparatus; (1) gas puri®er, (2) ¯owmeter, (3) microbalance, (4) sulphur trap, (5) thermoelement, (6) reaction furnace, (7) sample, (8) rotation pump, (9) bulb, (10) solid electrolyte probe, (A, B) sulphur baths, (C, D) valves.
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novel thermogravimetric assembly and the experimental procedure in studying the kinetics of metal-sulphur and sulphide±sulphur interaction, have been described elsewhere [6,7]. 4. Results and discussion In contrast with the majority of transition metal sulphides, manganous sulphide shows very low deviations from stoichiometry, and thereby defect concentration of the order of that in nickel oxide [22]. Based on extensive non-stoichiometry and electrical conductivity data, as well as on selfdiffusion measurements, the thermodynamics and kinetics of point defects in this sulphide have been described [11,23±25]. It has been shown that over the major part of the phase ®eld, corresponding to higher sulphur activities, a-MnS, having rocksalt structure, is a metal-de®cit p-type semi-conductor with predominant defects being doubly ionized cation vacancies and electron holes (Mn12yS). It has been found also that the mobility of point defects does not depend on their concentration, clearly indicating that these defects do not interact and are randomly distributed in the crystal lattice [12]. Finally, it has been demonstrated that the slowest step determining the overall growth rate of MnS scale on manganese is the outward diffusion of cations (parabolic kinetics) and the pressure and temperature dependence of this rate is in agreement with that calculated from self-diffusion coef®cients of Mn in Mn12yS and non-stoichiometry data [26]. As can be seen, the Mn-Mn12yS-S2 system is very convenient for the veri®cation of all these results using Rosenburg's two-stage sulphidation method. The application of this method became possible owing to a vary high sensitivity of weight gain measurements of the sulphidized Mn-sample, as well as due to the possibility of rapid changes of sulphur vapour pressure in our new microthermogravimetric apparatus [6,7]. Sulphidation rate measurements have been carried out at temperatures ranging from 973 to 1273 K and in the sulphur pressure interval 10 21 ±10 4 Pa. The experimental procedure was as follows. A manganese plate was suspended on the balance in the reaction chamber and the furnace (6) was moved down below the reaction tube. The whole apparatus was then evacuated and ¯ushed for several hours with spectrally pure helium by passing the helium through all three ¯owmeters (2) with both valves C and D open. The carrier gas, before entering the apparatus, passed through the gas puri®er (1) with iron wool heated to 900 K, as a result of which the partial pressure of oxygen was reduced from 10 21 down to 10 212 Pa. This partial pressure was continuously monitored in the outlet gas during the whole of every experiment with the use of calcia stabilized zirconia solid electrolyte probe (10). Subsequently, the valve C was closed and the sulphur bath A was heated to the desired temperature. Simultaneously, the reaction furnace (6) was heated to the temperature at which the Mn-sample was to be sulphidized.
1971
When the desired partial pressure of sulphur vapour in He-S2 gas mixture was reached, the furnace (6) was raised to the reaction position (see Fig. 4) and the He-S2 gas was let into the reaction chamber by appropriate changing the status of valves C and D (C-open and D closed). As a result of this operation He-S2 gas, previously leaving the apparatus through trap E, started to ¯ow in the opposite direction, i.e., through the reaction chamber, and left the apparatus through the trap F. It has been found that the steady state conditions in the reaction chamber were established during about 60 s [7]. This rather surprisingly short time in which the pressure of heavy sulphur vapour reached a constant value results from the laminar (without any turbulence) ¯ow of He-S2 gas from sulphur reservoir to the reaction chamber. It should be stressed once again, that it was due to this possibility in particular that Rosenburg's method could be successfully utilized in studying the kinetics and thermodynamics of point defects in non-stoichiometric manganous sulphide. The rate of sulphidation was continuously followed by an automatic microthermobalance (3) with the accuracy of 10 27 g. Maximum ¯uctuations of temperature in the reaction chamber and in liquid sulphur reservoir did not exceed ^1 K. It has been found in agreement with the literature data [26±28] that the sulphidation process strictly follows the parabolic rate law at all the temperatures and sulphur pressures, and the reproducibility of the kinetic results was excellent. For illustration, some of these results are shown in Fig. 5 in a parabolic plot. When, at a given temperature and sulphur pressure, a desired thickness of the scale, X0, estimated from available self-diffusion data, was reached, the He-S2 gas mixture in the reaction chamber was replaced by a pure helium gas stream and the system Mn-Mn12yS went gradually to equilibrium. After thermodynamic equilibrium was reached, as manifested by the constant weight of the sample, He-S2 gas stream was again let into the reaction chamber and weight gain measurements were resumed. Ä This process was continued until the condition t . X0/2D
Fig. 5. The kinetics of manganese sulphidation for several temperatures in parabolic plot.
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Fig. 6. Paralinear kinetic curves of manganese sulphidation for several temperatures.
Fig. 8. Parabolic plots of early stages of manganese sulphidation for several temperatures.
was ful®lled, however, in a limited period of time, such that the increase in the scale thickness was negligible (DX ! X0). Subsequently, the whole equilibration-sulphidation procedure was repeated at different temperatures and sulphur pressures in order to obtain Rosenburg's paralinear kinetic curves enabling the concentration and the mobility of defects to be calculated as a function of both these parameters. In every subsequent experiment the increase in the scale thickness, X0, has been taken into account. For illustration, several kinetic curves obtained at different temperatures and sulphur pressures are shown in Figs. 6 and 7. As can be seen, in all cases two stages of the reaction are clearly visible, manifesting themselves by initial parabolic and subsequent linear kinetics. In order to calculate the parabolic rate constants, the results of initial stages of sulphidation were plotted once again in a parabolic system of co-ordinates (Figs. 8 and 9). As predicted, the parabolic rate law in this stage of sulphidation was obeyed strictly, enabling kp and Cp values to be calculated. In turn kl and Cl valves were obtained from the linear parts of the kinetic
curves presented in Figs. 6 and 7. Using all these data and Eqs. (9±11) the concentration of cation vacancies and the chemical diffusion coef®cient in Mn12yS have been calculated as functions of temperature and sulphur activity. The results of these calculations are shown in Figs. 10±13 in double logarithmic and Arrhenius plots, respectively. From the diagrams presented in Figs. 10 and 11 it follows, that in agreement with non-stoichiometry data reported by other authors [8,13] the concentration of cation vacancies in Mn12yS increases with increasing sulphur activity with the slope 1/6 and this dependence does not change with temperature. Thus, the obtained results can be presented in the form of the following empirical equation:
This result con®rms the conclusion that the predominant point defects in metal-de®cit manganous sulphide are doubly ionized cation vacancies end electron holes. Applying
Fig. 7. Paralinear kinetic curves of manganese sulphidation for two different sulphur vapour pressure.
Fig. 9. Parabolic plots of early stages of manganese sulphidation for two different sulphur vapour pressures.
00
VMn 5:41 z 1022 p1=6 s2 exp
42 kJ=mol RT
12
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1973
Fig. 10. The dependence of non-stoichiometry, y, in Mn12yS on sulphur vapour pressure for several temperatures.
Fig. 12. The dependence of the chemical diffusion coef®cient on sulphur vapour pressure for several temperatures.
the mass action00 law and the appropriate electro-neutrality condition 2VMn hz to the defect equilibrium:
in Mn12yS does not depend on sulphur activity and thereby on defect concentration. Thus, the temperature dependence of this coef®cient can be expressed by the following empirical equation: 81:5 kJ=mol D~ 3:88 z 1022 exp 2 15 RT
00 1 S VMn 1 2hz 1 SS 2 2
the following theoretical relationship is obtained: 00 DSf DHf exp exp 2 VMn 0:63 z p1=6 S2 3R 3RT
13
14
Introducing this empirical relationship into theoretical Eq. (2) and taking p 22 one obtains the following equation describing the temperature dependence of vacancy (defect) diffusion coef®cient in metal-de®cit manganous sulphide: 81:5 kJ=mol DV 1:29 z 1022 exp 2 16 RT
where DSf and DHf are entropy and enthalpy of defect formation, respectively (KroÈger-Vink notation [29] of defects is used throughout this paper). From the comparison of empirical Eq. (12) with theoretical Eq. (14) it follows that DSf 261.2 J/mol deg and DHf 126 kJ/mol. These values are compared in Table 1 with those obtained by Rau [8] and Grzesik and Mrowec [13] from non-stoichiometry data. From Figs. 12 and 13, in turn, it clearly follows that, in agreement with other data obtained from re-equilibration rate measurements [12], the chemical diffusion coef®cient
The fact that the mobility of point defects in Mn12yS does not depend on their concentration is not surprising, because this concentrationÐeven at very high temperatures and sulphur activitiesÐis very low [8,13] and consequently, the defects do not interact and are randomly distributed in the crystal lattice. Thus, the behaviour of point defects in
Fig. 11. The comparison of non-stoichiometry data obtained in this work with those reported by Rau [8] and Grzesik et al. [13].
Fig. 13. The dependence of the chemical diffusion coef®cient in Mn12yS on temperature, in Arrhenius plot.
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Table 1 Thermodynamic parameters of formation and migration of defects in Mn12yS. DHf, kJ/mol
DSf, J/mol z deg
DHm, kJ/mol
DSm, J/mol z deg
References
Remarks
124.5 123 ± 126 ±
264.4 266.2 ± 261.2 ±
± ± 83.4 81.5 79.7
± ± 12.9 9.52 11.0
Rau [8] Mrowec et al. [13] Grzesik et al. [12] This work Danielewski [26]
Calculated from non-stoichiometry Determined from non-stoichiometry Determined from re-equilibrium kinetics Rosenburg method From MnÐsulphidation rate measurements
Mn12yS can be treated in terms of point defect thermodynamics and their transport properties should follow the rules of solid state diffusion theory based on simple point defect model [20,21]. The temperature dependence of the defect diffusion coef®cient for metal oxides and sulphides with such a simple defect structure is described by the following relationship [20,21]: DSm DHm exp 2 17 DV aa20 kv exp R RT where is the geometrical factor, kÐthe transmission coef®cient, the vibration frequency, and a0 the jump distance from the lattice site to the neighbouring vacancy. By comparing this relationship with empirical Eq. (16) the activation entropy (DSm) and enthalpy (DHm) of defect diffusion can be calculated. From this comparison it follows directly that DHm 81.5 kJ/mol. Assuming, in turn, a k 1, and Ê [30], as well as calculating the vibration a0 5.30 A frequency of cations according to Ref. [31]: s 2 DHm 1:46 z 1012 s21 18 n pa0 M where M is the atomic weight of manganese, one obtains by combining Eqs. (13 and 14) DSm 9.52 J/mol deg. These DHm and DSm values are compared in Table 1 with analogous data reported by other authors. Empirical Eqs. (12) and (16) can be utilized for the calculation of the self-diffusion coef®cient of cations in metalde®cit manganous sulphide as a function of temperature and sulphur activity, using Eq. (3): 123:5 kJ=mol DMn 6:98 z 1024 p1=6 19 S2 exp 2 RT It seems again reasonable to compare the results obtained by the two-stage kinetic method with other data available in the literature. Because of considerable experimental dif®culties in studying the high temperature reactions in sulphur containing atmosphere, DMn values in Mn12yS have only been estimated from Mn-sulphidation rate measurements [11]: 121 kJ=mol DMn 6:7 z 1024 p1=6 20 S2 exp 2 RT 54
During last few years, in very careful experiments using Mn radioactive manganese isotope as a tracer, the self-
diffusion coef®cient of 54Mn in Mn12yS has ®nally been determined as a function of temperature (1073±1373 K) and sulphur activity (10±10 4 Pa) [23,24]. The tracer diffusion coef®cient, DT, reported in this work, have been recalculated into `real' self-diffusion coef®cients, DMn, by taking into account the correlation effects. The results of these calculations can be expressed by the following empirical relationship [24]: 115 kJ=mol 21 DMn 3:38 z 1024 p1=6 exp 2 S2 RT In addition, owing to the development of the novel microthermogravimetric apparatus [7] used in the present work, the chemical diffusion coef®cient [12] and deviations from stoichiometry [13] in Mn12yS have recently been determined, and basing on these results DMn in Mn12yS has been calculated as a function of temperature and sulphur activity [32]: 124:4 kJ=mol 22 exp 2 DMn 8:73 z 1024 p1=6 S2 RT From the comparison of Eqs. (19±22) it follows clearly that the results obtained in one series of rather simple, twostage sulphidation experiments (Eq. 19) are in excellent agreement with those derived from much more dif®cult and time-consuming studies. This agreement is better visible in collective plot presented in Fig. 14. However, the scattering of tracer data, inevitably observed in such experiments, results in much higher standard deviation of DMn values than in the case of kinetic rate measurements. If one considers also the fact that in Rosenburg's method the concentration and mobility of defects are determined in one series of kinetic experiments, it becomes obvious that the most reasonable and accurate information on defect and transport properties of a given sulphide and oxide may be obtained by this method. 5. Conclusions The results obtained in the present work allow the following conclusions to be formulated. Thermodynamics and kinetics of point defects in nonstoichiometric metal sulphides, like in oxides, can be successfully studied using the two-stage kinetic method proposed by Rosenburg [1], provided that the
Z. Grzesik, S. Mrowec / Journal of Physics and Chemistry of Solids 62 (2001) 1967±1976
1975
Fig. 14. The comparison of available data in the literature [11,12,24] of self-diffusion coef®cient of manganese in metal de®cient manganous sulphide with those obtained in the present work.
thermogravimetric apparatus offers suf®ciently high accuracy of weight gain measurements as a function of time, as well as makes it possible to rapidly change the sulphur partial pressure in the reaction chamber. One of the most important advantages of this procedure consists in the possibility of determining the chemical diffusion coef®cient and the concentration of defects in a given sulphide in one series of sulphidation rate measurements without the necessity of removing the metal sample from the reaction chamber. As a consequence, the possible in¯uence of trace impurities on defect and transport properties of a given material may be easier explained or neglected. Finally, all this information can be obtained in a rather simple experimental procedure, in contrast with other, much more dif®cult and time-consuming methods. The application of Rosenburg's method to metal sulphides has been proved on the Mn-Mn12yS-S2 system, and it has been shown that the obtained results are in satisfactory agreement with the available literature data. As, however, all the results concerning defect concentration and their mobility, reported in this paper, have been obtained on the same material, with very high precision, they may be considered as most reliably describing the defect and transport properties of metal-de®cit manganous sulphide. Acknowledgements This work was supported by the Polish State Committee for Scienti®c Research (project no. 7T08A03815). References [1] A.J. Rosenburg, J. Electrochem. Soc. 107 (1960) 795. [2] E. Fryt, S. Mrowec, T. Walec, Oxid. Met. 7 (1973) 117.
[3] P. Kofstad, J. Phys. Chem. Solids 44 (1983) 879. [4] B. OberlaÈnder, P. Kofstad, I. Kvernes, Mat. Wiss. Und Werkstofftechnik 19 (1988) 190. [5] T. Smith, J. Electrochem. Soc. 112 (1965) 560. [6] Z. Grzesik, S. Mrowec, T. Walec, J. Dabek, J. Therm. Anal. Cal. 59 (2000) 985. [7] Z. Grzesik, S. Mrowec, Solid State Phenomena 72 (2000) 47. [8] H. Rau, J. Phys. Chem. Solids 39 (1978) 339. [9] H. Le Brusq, J.P. Delmaire, Rev. Int. Hautes Temp. Refract. 11 (1974) 193. [10] J. Rasner, D. Carton, C.R. Acad. Sci. (Paris), Ser. II 290 (1981) 405. [11] M. Danielewski, S. Mrowec, Solid State Ionics 17 (1985) 29. [12] Z. Grzesik, S. Mrowec, T. Walec, J. Dabek, J. Phys. Chem. Solids 61 (2000) 809. [13] S. Mrowec, Z. Grzesik, Solid State Phenomena 72 (2000) 69. [14] M. Danielewski, S. Mrowec, Solid State Ionics 17 (1985) 41. [15] M. Danielewski, S. Mrowec, Solid State Ionics 17 (1985) 319. [16] S. Mrowec, An Introduction to the Theory of Metal Oxidation, National Bureau of Standards and National Science Foundation, Washington, DC, 1982, p. 172. [17] P. Kofstad, High Temperature Corrosion, Elsevier, Amsterdam, 1988, p. 423. [18] P. Child, L.B. Laub, J.B. Wagner, Chemical diffusion in nonstoichiometric compounds, Mass Transport in Nonmetallic Solids, Proc. Brit. Ceram,. Soc. 19 (1971) 29. [19] S. Mrowec, K. Hashimoto, J. Mater. Sci. 30 (1995) 4801. [20] P. Kofstad, Nonstoichiometry, Diffusion and Electrical Conductivity in Binary Metal Oxides, J. Wiley Interscience, New York, London, Sydney, Toronto, 1972. [21] S. Mrowec, Defects and Diffusion in Solids, Elsevier, Amsterdam, Oxford, New York (1980) 174. [22] S. Mrowec, J. Janowski, Similarities and Differences in Defect Dependent Properties of Transition Metal Sulphides and Oxides, in: O. Johannesen, A.G. Andersen (Eds.), Selected Topics in High Temperature Chemistry. Defect Chemistry of Solids, Elsevier, Amsterdam, Oxford, New York, 1989, p. 55. [23] J. Gilewicz-Wolter, M. Danielewski, S. Mrowec, Physical Rev. B. 56 (1997) 8695.
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