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Kinetic methods in studying the defect concentration and their mobility in non-stoichiometric metal sulphides
Solid State Ionics 141–142 Ž2001. 493–498 www.elsevier.comrlocaterssi
Kinetic methods in studying the defect concentration and their mobility in non-stoichiometric metal sulphides Z. Grzesik ) , S. Mrowec Department of Solid State Chemistry, Faculty of Materials Science and Ceramics, UniÕersity of Mining and Metallurgy, al. A. Mickiewicza 30, 30-059 Krakow, Poland
Abstract The chemical diffusion coefficient and point defect concentration in metal-deficit manganous sulphide ŽMn 1yy S. has been studied using re-equilibration kinetics and Rosenburg’s method. Measurements have been carried out in a microthermogravimetric apparatus of a new generation, enabling to make rapid changes of sulphur vapour pressure in the reaction chamber and determine the mass changes of a given sample with the high accuracy, up to 10y7 g. Using this device deviations from stoichiometry, chemical diffusion in metal-deficit manganous sulphide was investigated, as a function of temperature Ž973–1273 K. and sulphur activity Ž10y2 –10 4 Pa.. From obtained data, the self-diffusion coefficient of cations in Mn 1yy S has been calculated as a function of temperature and sulphur activity. q 2001 Elsevier Science B.V. All rights reserved. Keywords: Re-equilibration kinetics; Rosenburg’s method; Chemical diffusion; Defect concentration; Chemical compounds, Mn 1y y S
1. Introduction Because of considerable experimental difficulties in studying the high-temperature reactions in sulphur-containing atmospheres, the physico-chemical properties of transition metal sulphides are less known than those of corresponding oxides. Such a situation results mainly from the fact that sulphur is not gaseous under normal conditions and its vapours are extremely aggressive at high temperatures, attacking all the metals including gold and platinum w1,2x. Consequently, all standard thermogravimetric and other equipments currently used in oxidation studies are not applicable under such conditions. ) Corresponding author. Tel.: q48-12-6172-467; fax: q48-126172-493. E-mail address: [email protected] ŽZ. Grzesik..
Thus, in all thermogravimetric equipments utilized in sulphidation studies, a quartz spiral plays the role of a thermobalance. As a consequence, the weight of the sample is very limited and the accuracy of mass change measurements does not exceed 50 mg. In addition, because the sulphidation experiments are carried out in static conditions, the constant sulphur pressure in the reaction chamber can only be established after a certain period of time: the longer the period of time, the lower is the temperature of the liquid sulphur reservoir. Thus, it is impossible to study the early stages of any solid–sulphur interaction, especially at low sulphur pressures, and equilibration and re-equilibration kinetic measurements are entirely excluded. Being active in this area of research since many years, we have been able to develop a microthermogravimetric apparatus of the new generation w3x, which makes it possible to fol-
0167-2738r01r$ - see front matter q 2001 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 7 - 2 7 3 8 Ž 0 1 . 0 0 7 7 9 - 2
Z. Grzesik, S. Mrowecr Solid State Ionics 141–142 (2001) 493–498
494
low the kinetics of solid–sulphur interactions with very high precision and also to change suddenly the sulphur vapour pressure in the reaction chamber from one constant value to another. As a consequence, this novel thermogravimetric assembly allows to study the concentration and the mobility of point defects in non-stoichiometric metal sulphides. These very important physico-chemical properties of metal sulphides can only be determined from two gravimetric methods, i.e. from the kinetics of nonstoichiometry changes of a given sulphide when going from one thermodynamic equilibrium state to another Žthe re-equilibration kinetics. and from Rosenburg’s method w4x. In the present work, the re-equilibration kinetics and Rosenburg’s method have been applied for the determination of the defect concentration and their mobility in Mn 1y y S.
2. Description of methods
where D m t denotes the weight change of the sulphide sample after time t, D m k is the total weight change when the new equilibrium state is established, l is one-half of the sample thickness, and D˜ is the chemical diffusion coefficient. It should be noted that Eq. Ž1. is equally valid for oxidation and reduction runs, since the boundary conditions in the solution of Fick’s second law are identical for both cases. As a consequence, the same values of D˜ should be obtained in both types of experiments carried out under the same experimental conditions. As already mentioned, this procedure offers important possibility for verifying the applicability of the discussed re-equilibration kinetics method. It is convenient to express the exponential Eq. Ž1. in logarithmic form, so that the chemical diffusion coefficient can be calculated from the slope of the straight line obtained by plotting the results of reequilibration kinetics in semi-logarithmic system of coordinates:
2.1. The re-equilibration kinetics
log 1 y
To determine the chemical diffusion coefficient from re-equilibration kinetics, a given metal should be completely sulphidized to obtain the sulphide sample of a required composition. After thermodynamic equilibrium had been reached, manifesting itself by a constant weight of sulphide sample, the sulphur pressure was suddenly changed to a lower value, and the re-equilibration kinetics was continuously followed by determining the mass losses of the specimen as a function of time, until a new equilibrium state was obtained. After this reduction run, the sulphur pressure was suddenly raised to a higher value and the kinetics of oxidation process was followed by continuous recording of weight gains of the sample until a new equilibrium was again reached in metal sulphide–S 2 system. In order to calculate the chemical diffusion coefficient from re-equilibration rate measurements, the following solution of Fick’s second law has been applied w5x: 1y
D mt D mk
8 s
p
ž
exp y 2
D˜ p 2 t 4l2
/
Ž 1.
D mt D mk
s log
D˜ p 2 t
8
p2
y
2.303 = 4 = l 2
Ž 2.
2.2. The Rosenburg’s method The method proposed by Rosenburg w4x has been used for determination of defect concentration and chemical diffusion coefficient in Mn 1y y S by a rather simple thermogravimetric technique, which consists of a two-stage sulphidation rate measurements of a manganese metal sample under isothermal conditions at a given sulphur pressure. After formation on a metal surface of a manganese sulphide layer Žscale. of sufficient thickness, the process of sulphidation is interrupted by changing the ambient sulphur pressure to the value of the dissociation pressure of manganous sulphide. Then, as a consequence, the metal–metal sulphide–S 2 system proceeds gradually to equilibrium and the defect concentration becomes uniform through the entire scale, which manifests itself by no further sulphidation. In the second stage of sulphidation, sulphur vapour under desired pressure is readmitted to the reaction chamber and the further growth of the sulphide layer is measured. At the beginning of the second stage of sulphidation, the manganous
Z. Grzesik, S. Mrowecr Solid State Ionics 141–142 (2001) 493–498
sulphide may be considered as a semi-infinite system and later the process takes place under steady-state conditions. The solution of Fick’s second law for the case under discussion makes it possible to determine separately the defect concentration and the chemical diffusion coefficient from one series of two-stage sulphidation rate measurements in the studied sulphide. The details of the procedure of these calculation are described elsewhere w6x. It is enough to mention here that two-stages of sulphidation kinetics are described by the following kinetic equations: Dm
ž / A
s k p't q Cp
Ž 3.
s k l t q Cl
Ž 4.
1
and Dm
ž / A
2
where k p and k l are parabolic and linear rate constants in successive stages of the reaction, expressed in g cmy2 sy0 .5 and g cmy2 sy1 , respectively. Both these constants can readily be determined by plotting the obtained results in parabolic and linear systems of coordinates, respectively. The first equation describes the parabolic course of the reaction, observed just after readmission of sulphur vapour to the reaction chamber, when the scale behaves as a semi-infinitive system and consequently, the parabolic kinetic is observed. The second equation illustrates the later stages of the reaction, when the linear concentration gradient of defects in the scale is re-established Žsteady-state conditions.. Since k p and k l , as well as Cp and C l can be determined in a two-stage oxidation experiment, two unknown quantities, D˜ and Cd , can be calculated separately using the following relationships: D˜ s
ž
1.128 k l X 0 kp
2
/
Ž 5.
and
Cd s
ž
kp 1.128 k l X0
2
/
Ž 6.
495
The concentration of defects can also be obtained from the following equation: Cd s
3 Ž C l y Cp . X0
Ž 7.
The discussed two-stage oxidation kinetic experiments 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 experiments—carried out on one and the same material—the concentration and the mobility of point defects can be determined as a function of temperature and oxidant activity. It should be noted that all these data may be obtained with very high precision, difficult or even impossible to be reached 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 equipments is possible to follow continuously with the accuracy of the order of 10y7 g. As a consequence, the whole Rosenburg’s paralinear kinetic curve can be determined with very high precision, enabling the concentration of defects and their mobility to be calculated with the corresponding accuracy.
3. Results and discussions In the present work, the chemical diffusion coefficient and defect concentration in Mn 1y y S have been determined in an apparatus described elsewhere w3x. Re-equilibration rate measurements have been carried out at temperatures ranging from 973 up to 1273 K and sulphur pressure interval 10y1 –10 4 Pa. After the manganese samples were completely sulphidized at a desired temperature and sulphur vapour pressure and the thermodynamic equilibrium had been reached Žno further mass increase of the sulphide sample has been observed., the sulphur pressure was suddenly changed and the re-equilibration kinetics was continuously determined, until a new equilibrium state was obtained. The results of these experiments are shown in Fig. 1. As can be seen, both the oxidation and
Z. Grzesik, S. Mrowecr Solid State Ionics 141–142 (2001) 493–498
496
Fig. 1. The re-equilibration kinetics of manganous sulphide.
reduction runs are fully reproducible with the maximum error not exceeding "5%. In addition, no hysteresis was observed between oxidation and reduction relaxation curves, clearly indicating that the re-equilibration kinetics of Mn 1y y S was diffusioncontrolled. Analogous results have been obtained in other temperatures and sulphur pressure intervals. According to obtained results, chemical diffusivities calculated from oxidation and reduction runs are virtually the same, clearly indicating that the rate-determining step in the overall re-equilibration process is the solid state diffusion. Thus, the fundamental assumption of the method is fulfilled, and thereby D˜ values correctly represent the rate of cation vacancy migration in Mn 1y y S under non-equilibrium conditions. Secondly, the chemical diffusion coefficient does not depend on sulphur activity and thereby on defect concentration. Thus, the temperature dependence of D˜ for all sulphur pressures constitutes one straight line only in Arrhenius plot, as shown in Fig. 2, enabling the chemical diffusion coefficient in Mn 1y y S to be expressed by the following empirical equation: D˜ s 5.9 = 10y2 exp y
ž
83.4 kJrmol RT
/
Ž 8.
Nonstoichiometry, y, of Mn 1y y S has been calculated from re-equilibration data through the following empirical formula: ys1y
m Mn = MS m S = M Mn
Ž 9.
where m Mn denotes the mass of manganese plate, M Mn is the atomic mass of manganese, m S is the mass of sulphur reacted with the sample, and MS is
Fig. 2. The dependence of the chemical diffusion coefficient in Mn 1y y S on temperature obtained from re-equilibration kinetic measurements, in Arrhenius plot.
Z. Grzesik, S. Mrowecr Solid State Ionics 141–142 (2001) 493–498
497
the atomic mass of sulphur. Thus, the deviation from stoichiometry can be presented in the following form: y s 4.43 = 10y2 p 1r6 S 2 exp y
ž
41.0 kJrmol RT
/
Ž 10 .
The chemical diffusion coefficient and defect concentration in Mn 1y y S have also been calculated from a two-stage sulphidation rate measurement using Rosenburg’s method. For illustration, several kinetic curves obtained at different temperatures and sulphur pressures are shown in Figs. 3 and 4. As can be seen, in all cases, two stages of the reaction are clearly visible, manifesting themselves by initial parabolic and subsequent linear kinetics. Analogous results have been obtained for different temperatures and sulphur pressures. Using all these data, the concentration of cation vacancies and chemical diffusion coefficient in Mn 1y y S have been calculated as a function of temperature and sulphur activity: D˜ s 3.88 = 10y2 exp y
ž
w
Y V Mn
y2
x s 5.41 = 10
81.5 kJrmol
p 1r6 S 2 exp
RT
ž
/
42 kJrmol y RT
Fig. 4. The Rosenburg’s method, paralinear curves of manganese sulphidation for several temperatures.
mined from re-equilibration kinetics and Rosenburg’s method, respectively: D Mn s 8.73 = 10y4 p 1r6 S 2 exp y
ž
124.4 kJrmol RT
/ Ž 13 .
Ž 11 . D Mn s 6.98 = 10y4 p 1r6 S 2 exp y
ž
/ Ž 12 .
123.5 kJrmol RT
/ Ž 14 .
The results of nonstoichiometry and chemical diffusion data can finally be utilized for calculation of self-diffusion coefficient of cations in Mn 1y y S as a function of temperature and sulphur activity, deter-
From the comparison of the above equations, it follows that the agreement between that determined from re-equilibration kinetics and Rosenburg’s method, D Mn in Mn 1yy S, is quite satisfactory ŽFig. 5..
Fig. 3. The Rosenburg’s method, parabolic plot of early stages of manganese sulphidation for several temperatures.
Fig. 5. The comparison of self-diffusion coefficient of manganese in Mn 1y y S.
498
Z. Grzesik, S. Mrowecr Solid State Ionics 141–142 (2001) 493–498
Acknowledgements This work was supported by the Polish State Committee for Scientific Research ŽProject no. 7T08A03815..
References w1x P. Kofstad, Nonstoichiometry, Diffusion and Electrical Conductivity in Binary Metal Oxides, Wiley-Interscience, New York, 1972, pp. 117–360.
w2x S. Mrowec, Defects and Diffusion in Solids, Elsevier, Amsterdam, 1980, p. 174. w3x Z. Grzesik, S. Mrowec, T. Walec, J. Dabek, J. Therm. Anal. Calorim. 59 Ž2000. 985. w4x A.J. Rosenburg, J. Electrochem. Soc. 107 Ž1960. 795. w5x J.B. Wagner, Diffusion coefficients for some nonstoichiometric metal oxides,Mass Transport in Oxides,, 1967, p. 65, NBS Special Publ. No. 296. w6x Z. Grzesik, S. Mrowec, Oxidation kinetic method in studying the defect structure and transport properties of nonstoichiometric manganous sulphide, J. Phys. Chem. Sol., in press.
Kinetic methods in studying the defect concentration and their mobility in non-stoichiometric metal sulphides Z. Grzesik ) , S. Mrowec Department of Solid State Chemistry, Faculty of Materials Science and Ceramics, UniÕersity of Mining and Metallurgy, al. A. Mickiewicza 30, 30-059 Krakow, Poland
Abstract The chemical diffusion coefficient and point defect concentration in metal-deficit manganous sulphide ŽMn 1yy S. has been studied using re-equilibration kinetics and Rosenburg’s method. Measurements have been carried out in a microthermogravimetric apparatus of a new generation, enabling to make rapid changes of sulphur vapour pressure in the reaction chamber and determine the mass changes of a given sample with the high accuracy, up to 10y7 g. Using this device deviations from stoichiometry, chemical diffusion in metal-deficit manganous sulphide was investigated, as a function of temperature Ž973–1273 K. and sulphur activity Ž10y2 –10 4 Pa.. From obtained data, the self-diffusion coefficient of cations in Mn 1yy S has been calculated as a function of temperature and sulphur activity. q 2001 Elsevier Science B.V. All rights reserved. Keywords: Re-equilibration kinetics; Rosenburg’s method; Chemical diffusion; Defect concentration; Chemical compounds, Mn 1y y S
1. Introduction Because of considerable experimental difficulties in studying the high-temperature reactions in sulphur-containing atmospheres, the physico-chemical properties of transition metal sulphides are less known than those of corresponding oxides. Such a situation results mainly from the fact that sulphur is not gaseous under normal conditions and its vapours are extremely aggressive at high temperatures, attacking all the metals including gold and platinum w1,2x. Consequently, all standard thermogravimetric and other equipments currently used in oxidation studies are not applicable under such conditions. ) Corresponding author. Tel.: q48-12-6172-467; fax: q48-126172-493. E-mail address: [email protected] ŽZ. Grzesik..
Thus, in all thermogravimetric equipments utilized in sulphidation studies, a quartz spiral plays the role of a thermobalance. As a consequence, the weight of the sample is very limited and the accuracy of mass change measurements does not exceed 50 mg. In addition, because the sulphidation experiments are carried out in static conditions, the constant sulphur pressure in the reaction chamber can only be established after a certain period of time: the longer the period of time, the lower is the temperature of the liquid sulphur reservoir. Thus, it is impossible to study the early stages of any solid–sulphur interaction, especially at low sulphur pressures, and equilibration and re-equilibration kinetic measurements are entirely excluded. Being active in this area of research since many years, we have been able to develop a microthermogravimetric apparatus of the new generation w3x, which makes it possible to fol-
0167-2738r01r$ - see front matter q 2001 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 7 - 2 7 3 8 Ž 0 1 . 0 0 7 7 9 - 2
Z. Grzesik, S. Mrowecr Solid State Ionics 141–142 (2001) 493–498
494
low the kinetics of solid–sulphur interactions with very high precision and also to change suddenly the sulphur vapour pressure in the reaction chamber from one constant value to another. As a consequence, this novel thermogravimetric assembly allows to study the concentration and the mobility of point defects in non-stoichiometric metal sulphides. These very important physico-chemical properties of metal sulphides can only be determined from two gravimetric methods, i.e. from the kinetics of nonstoichiometry changes of a given sulphide when going from one thermodynamic equilibrium state to another Žthe re-equilibration kinetics. and from Rosenburg’s method w4x. In the present work, the re-equilibration kinetics and Rosenburg’s method have been applied for the determination of the defect concentration and their mobility in Mn 1y y S.
2. Description of methods
where D m t denotes the weight change of the sulphide sample after time t, D m k is the total weight change when the new equilibrium state is established, l is one-half of the sample thickness, and D˜ is the chemical diffusion coefficient. It should be noted that Eq. Ž1. is equally valid for oxidation and reduction runs, since the boundary conditions in the solution of Fick’s second law are identical for both cases. As a consequence, the same values of D˜ should be obtained in both types of experiments carried out under the same experimental conditions. As already mentioned, this procedure offers important possibility for verifying the applicability of the discussed re-equilibration kinetics method. It is convenient to express the exponential Eq. Ž1. in logarithmic form, so that the chemical diffusion coefficient can be calculated from the slope of the straight line obtained by plotting the results of reequilibration kinetics in semi-logarithmic system of coordinates:
2.1. The re-equilibration kinetics
log 1 y
To determine the chemical diffusion coefficient from re-equilibration kinetics, a given metal should be completely sulphidized to obtain the sulphide sample of a required composition. After thermodynamic equilibrium had been reached, manifesting itself by a constant weight of sulphide sample, the sulphur pressure was suddenly changed to a lower value, and the re-equilibration kinetics was continuously followed by determining the mass losses of the specimen as a function of time, until a new equilibrium state was obtained. After this reduction run, the sulphur pressure was suddenly raised to a higher value and the kinetics of oxidation process was followed by continuous recording of weight gains of the sample until a new equilibrium was again reached in metal sulphide–S 2 system. In order to calculate the chemical diffusion coefficient from re-equilibration rate measurements, the following solution of Fick’s second law has been applied w5x: 1y
D mt D mk
8 s
p
ž
exp y 2
D˜ p 2 t 4l2
/
Ž 1.
D mt D mk
s log
D˜ p 2 t
8
p2
y
2.303 = 4 = l 2
Ž 2.
2.2. The Rosenburg’s method The method proposed by Rosenburg w4x has been used for determination of defect concentration and chemical diffusion coefficient in Mn 1y y S by a rather simple thermogravimetric technique, which consists of a two-stage sulphidation rate measurements of a manganese metal sample under isothermal conditions at a given sulphur pressure. After formation on a metal surface of a manganese sulphide layer Žscale. of sufficient thickness, the process of sulphidation is interrupted by changing the ambient sulphur pressure to the value of the dissociation pressure of manganous sulphide. Then, as a consequence, the metal–metal sulphide–S 2 system proceeds gradually to equilibrium and the defect concentration becomes uniform through the entire scale, which manifests itself by no further sulphidation. In the second stage of sulphidation, sulphur vapour under desired pressure is readmitted to the reaction chamber and the further growth of the sulphide layer is measured. At the beginning of the second stage of sulphidation, the manganous
Z. Grzesik, S. Mrowecr Solid State Ionics 141–142 (2001) 493–498
sulphide may be considered as a semi-infinite system and later the process takes place under steady-state conditions. The solution of Fick’s second law for the case under discussion makes it possible to determine separately the defect concentration and the chemical diffusion coefficient from one series of two-stage sulphidation rate measurements in the studied sulphide. The details of the procedure of these calculation are described elsewhere w6x. It is enough to mention here that two-stages of sulphidation kinetics are described by the following kinetic equations: Dm
ž / A
s k p't q Cp
Ž 3.
s k l t q Cl
Ž 4.
1
and Dm
ž / A
2
where k p and k l are parabolic and linear rate constants in successive stages of the reaction, expressed in g cmy2 sy0 .5 and g cmy2 sy1 , respectively. Both these constants can readily be determined by plotting the obtained results in parabolic and linear systems of coordinates, respectively. The first equation describes the parabolic course of the reaction, observed just after readmission of sulphur vapour to the reaction chamber, when the scale behaves as a semi-infinitive system and consequently, the parabolic kinetic is observed. The second equation illustrates the later stages of the reaction, when the linear concentration gradient of defects in the scale is re-established Žsteady-state conditions.. Since k p and k l , as well as Cp and C l can be determined in a two-stage oxidation experiment, two unknown quantities, D˜ and Cd , can be calculated separately using the following relationships: D˜ s
ž
1.128 k l X 0 kp
2
/
Ž 5.
and
Cd s
ž
kp 1.128 k l X0
2
/
Ž 6.
495
The concentration of defects can also be obtained from the following equation: Cd s
3 Ž C l y Cp . X0
Ž 7.
The discussed two-stage oxidation kinetic experiments 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 experiments—carried out on one and the same material—the concentration and the mobility of point defects can be determined as a function of temperature and oxidant activity. It should be noted that all these data may be obtained with very high precision, difficult or even impossible to be reached 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 equipments is possible to follow continuously with the accuracy of the order of 10y7 g. As a consequence, the whole Rosenburg’s paralinear kinetic curve can be determined with very high precision, enabling the concentration of defects and their mobility to be calculated with the corresponding accuracy.
3. Results and discussions In the present work, the chemical diffusion coefficient and defect concentration in Mn 1y y S have been determined in an apparatus described elsewhere w3x. Re-equilibration rate measurements have been carried out at temperatures ranging from 973 up to 1273 K and sulphur pressure interval 10y1 –10 4 Pa. After the manganese samples were completely sulphidized at a desired temperature and sulphur vapour pressure and the thermodynamic equilibrium had been reached Žno further mass increase of the sulphide sample has been observed., the sulphur pressure was suddenly changed and the re-equilibration kinetics was continuously determined, until a new equilibrium state was obtained. The results of these experiments are shown in Fig. 1. As can be seen, both the oxidation and
Z. Grzesik, S. Mrowecr Solid State Ionics 141–142 (2001) 493–498
496
Fig. 1. The re-equilibration kinetics of manganous sulphide.
reduction runs are fully reproducible with the maximum error not exceeding "5%. In addition, no hysteresis was observed between oxidation and reduction relaxation curves, clearly indicating that the re-equilibration kinetics of Mn 1y y S was diffusioncontrolled. Analogous results have been obtained in other temperatures and sulphur pressure intervals. According to obtained results, chemical diffusivities calculated from oxidation and reduction runs are virtually the same, clearly indicating that the rate-determining step in the overall re-equilibration process is the solid state diffusion. Thus, the fundamental assumption of the method is fulfilled, and thereby D˜ values correctly represent the rate of cation vacancy migration in Mn 1y y S under non-equilibrium conditions. Secondly, the chemical diffusion coefficient does not depend on sulphur activity and thereby on defect concentration. Thus, the temperature dependence of D˜ for all sulphur pressures constitutes one straight line only in Arrhenius plot, as shown in Fig. 2, enabling the chemical diffusion coefficient in Mn 1y y S to be expressed by the following empirical equation: D˜ s 5.9 = 10y2 exp y
ž
83.4 kJrmol RT
/
Ž 8.
Nonstoichiometry, y, of Mn 1y y S has been calculated from re-equilibration data through the following empirical formula: ys1y
m Mn = MS m S = M Mn
Ž 9.
where m Mn denotes the mass of manganese plate, M Mn is the atomic mass of manganese, m S is the mass of sulphur reacted with the sample, and MS is
Fig. 2. The dependence of the chemical diffusion coefficient in Mn 1y y S on temperature obtained from re-equilibration kinetic measurements, in Arrhenius plot.
Z. Grzesik, S. Mrowecr Solid State Ionics 141–142 (2001) 493–498
497
the atomic mass of sulphur. Thus, the deviation from stoichiometry can be presented in the following form: y s 4.43 = 10y2 p 1r6 S 2 exp y
ž
41.0 kJrmol RT
/
Ž 10 .
The chemical diffusion coefficient and defect concentration in Mn 1y y S have also been calculated from a two-stage sulphidation rate measurement using Rosenburg’s method. For illustration, several kinetic curves obtained at different temperatures and sulphur pressures are shown in Figs. 3 and 4. As can be seen, in all cases, two stages of the reaction are clearly visible, manifesting themselves by initial parabolic and subsequent linear kinetics. Analogous results have been obtained for different temperatures and sulphur pressures. Using all these data, the concentration of cation vacancies and chemical diffusion coefficient in Mn 1y y S have been calculated as a function of temperature and sulphur activity: D˜ s 3.88 = 10y2 exp y
ž
w
Y V Mn
y2
x s 5.41 = 10
81.5 kJrmol
p 1r6 S 2 exp
RT
ž
/
42 kJrmol y RT
Fig. 4. The Rosenburg’s method, paralinear curves of manganese sulphidation for several temperatures.
mined from re-equilibration kinetics and Rosenburg’s method, respectively: D Mn s 8.73 = 10y4 p 1r6 S 2 exp y
ž
124.4 kJrmol RT
/ Ž 13 .
Ž 11 . D Mn s 6.98 = 10y4 p 1r6 S 2 exp y
ž
/ Ž 12 .
123.5 kJrmol RT
/ Ž 14 .
The results of nonstoichiometry and chemical diffusion data can finally be utilized for calculation of self-diffusion coefficient of cations in Mn 1y y S as a function of temperature and sulphur activity, deter-
From the comparison of the above equations, it follows that the agreement between that determined from re-equilibration kinetics and Rosenburg’s method, D Mn in Mn 1yy S, is quite satisfactory ŽFig. 5..
Fig. 3. The Rosenburg’s method, parabolic plot of early stages of manganese sulphidation for several temperatures.
Fig. 5. The comparison of self-diffusion coefficient of manganese in Mn 1y y S.
498
Z. Grzesik, S. Mrowecr Solid State Ionics 141–142 (2001) 493–498
Acknowledgements This work was supported by the Polish State Committee for Scientific Research ŽProject no. 7T08A03815..
References w1x P. Kofstad, Nonstoichiometry, Diffusion and Electrical Conductivity in Binary Metal Oxides, Wiley-Interscience, New York, 1972, pp. 117–360.
w2x S. Mrowec, Defects and Diffusion in Solids, Elsevier, Amsterdam, 1980, p. 174. w3x Z. Grzesik, S. Mrowec, T. Walec, J. Dabek, J. Therm. Anal. Calorim. 59 Ž2000. 985. w4x A.J. Rosenburg, J. Electrochem. Soc. 107 Ž1960. 795. w5x J.B. Wagner, Diffusion coefficients for some nonstoichiometric metal oxides,Mass Transport in Oxides,, 1967, p. 65, NBS Special Publ. No. 296. w6x Z. Grzesik, S. Mrowec, Oxidation kinetic method in studying the defect structure and transport properties of nonstoichiometric manganous sulphide, J. Phys. Chem. Sol., in press.