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A diffusion model for dissolution phenomena in oxide powder mixtures
Powder Technology, 79 (1994) 11-16
11
A diffusion model for dissolution phenomena mixtures* N.S. Srinivasan,
A. Jakobsson
in oxide powder
and S. Seetharaman
Department of Theoretical Metallurgy, Royal Institute of Technology, S-100 44 Stockholm (Sweden) (Received
February 12, 1993; in revised form December 7, 1993)
Abstract A diffusion model is proposed for studying interdiffusion in binary oxide mixtures. The model has been applied to data obtained by dynamic galvanic cell measurements on NiO-MgO and COO-MgO systems. Cationic interdifksivities derived from EMF data are in reasonable agreement with those reported in the literature using the conventional diffusion couple technique. The dynamic nature of the present method offers scope for continuous monitoring of solid state reactions and related processes.
Introduction The study
7 =[l
- (1 -x)ln]*
(1)
where r,, is the particle radius, k is a constant and D is the diffusion coefficient which is assumed to be invariant with time. The following assumptions are implied in Jander’s model: (1)Nucleation and surface difIusion are possible at temperatures lower than that needed for bulk diffusion. A coherent product layer is therefore present when bulk diffusion takes place. (2) Reaction takes place in one of the components which is completely and continuously covered by the other component. (3) The activities of the reactants remain the same on either side of the interface. The assumption of a constant ditfusion coefficient restricts the applicability of Jander’s equation. Kroger and Ziegler [5] assumed that the diffusion coefficient of the transported species varies inversely with time and modified Jander’s equation as follows:
Assuming that the activity of the reacting substance is proportional to the fraction of unreacted material (1 -x), Zhuravlev et al. [6] modified the Jander relation as:
0032-5910/94/$07.00 0 1994 Elsevier Sequoia. All rights reserved SSDI 0032-5910(94)02808-2
2 (3)
Ginstling and Brounshtein [7] pointed out that the parabolic rate law considers the reaction interface to be constant, although it actually decreases for spherical particles during the course of reaction. By analogy with heat transfer through a spherical shell, they arrived at the following relationship: 2kLlt =l-
ir-(l-X)m
ro2 Carter [S] corrected the above relationship for large differences in volume between the product and the reactants. He introduced a factor z which represents the product volume per unit volume of reactant consumed (eqn. (5)). However, it has been shown by Geiss [9] that the correction is significant only when z exceeds a value of 2. 2kDt -= ro2
z-(z-1)(1-x)~3-[1+(z-1~]~ z-l
(5)
analysis, they determined the chemical diffusion coefficient as a function of composition. This relationship is of the form: rS = fiO exp[ - bX,,,]
(8)
where b is a constant and &, refers to X,,o=O. Co0 and MgO form a continuous series of solid solutions at temperatures above N 1175 K and for a wide range of oxygen partial pressures. The system shows near-ideal mixing behaviour at these temperatures. The pure oxides and the solid solution have NaCl-type structure. Pure Co0 is a metal-deficit semiconductor and its defect structure is characterised mainly by cation vacancies and electron holes. Yurek and Schmalzried developed a rigorous theoretical approach to interpret their experimental data. The cations and the anions are assumed to occupy different sublattices. The large oxygen ions are relatively immobile compared to the cations. Relative to the oxygen ion sublattice, the fluxes of cobalt ions, magnesium ions and the electron holes (h’) can be expressed as:
Assuming that the diffusion coefficient varies inversely with time, eqn. (5) can be modified as: -2k In t = z-(z-l)(l-x)zn-[1+(z-lb]u3 z-l rlJ2
(6)
According to Hulbert [16], eqn. (6) represents the rate of formation of many spinels. For reactants with low defect concentrations, eqn. (5) is to be preferred at high temperatures while eqn. (6) provides a good representation for highly reactive components at low temperatures. Based on a solution to Fick’s second law for diffusion into or out of a sphere, Dunwald and Wagner [lo] derived a relationship for solid-solid reactions. In terms of fractional conversion of the reactants, Serin and Ellickson [ 1l] expressed the Dunwald-Wagner relationship as:
r12D
-In
r0”
6 II”( 1 -x)
Diffusion in binary monovalent ionic crystals has been studied by several workers. Mtiller and Schmalzried [12] studied diffusion in the KCl-RbCl system and derived an expression for the chemical diffusion coefficient based on Darken’s theory [13]. This approach has been extended to ternary systems containing a single anion by de Lamotte and Kirkaldy [14]. Yurek and Schmalzried [15] have studied interdiffusion in (A, B)O-type solid solutions. They employed diffusion couples of pure Co0 and MgO single crystals and studied interditfusion in the Coo-MgO system (1573 K, air atmosphere). Using the Boltzman-Matano
where i=Co2+, Mg2+, h’. The diffusion potential arising out of the interdiffusion of cations would be negligible if electron hole conduction in the oxide solid solution is much greater than ionic conduction. Under these conditions, due to the electroneutrality constraint, there is no coupling between the cationic fluxes. Further, assuming local thermodynamic equilibrium to prevail in the diffusion couple, the chemical diffusion coefficient can be calculated from the flux equations. The conventional diffusion couple technique for the determination of interdiffusivities is quite reliable, but imposes stringent restrictions on experimental possibilities. For example, the analysis is strictly valid for planar geometry. Diffusion in powder mixtures cannot be studied using this technique. This is a serious limitation, as high temperature applications often involve reactions taking place in mixtures of solids and agglomerates containing spherical particles. Further, the conventional technique is often tedious, time-consuming and requires sophisticated equipment like SEM-EDS for obtaining reliable results. Even the relatively simpler tracer methods for diffusion studies require a good deal of sophistication. Hence, there is a need for a simple, reliable and continuous method to follow the kinetics of solid-solid reactions and to measure interditfusivities in ceramic systems. Recently, a galvanic cell method has been proposed [17] for kinetic studies of solution phenomena in solid oxide systems. It is a dynamic method which uses a
13
solid electrolyte cell to study dissolution kinetics in powder mixtures. As pointed out earlier, diffusion models proposed in the literature so far are not directly applicable to such studies. It is the purpose of this work to develop a simple approach to interdiffusion in oxide powder mixtures which can be used to interpret the kinetic data from galvanic cell measurements.
of electron holes at the diffusion temperatures is quite high relative to that of a cation vacancy [18]. The rapid diffusion of electron holes leads to electrical neutrality in the diffusion sample. With the above background, Fick’s second law for diffusion into a spherical BO particle of radius r, can be written as:
Theoretical development For analytical purposes, a mixture of oxide particles A0 and BO of known particle size distribution is envisaged. The oxides A0 and BO are assumed to mix ideally and form a continuous series of solid solutions. No phase separation occurs on interditfusion. The solid solution (A, B)O which is formed has the same crystal structure as the pure oxides. Other assumptions include: (1) The reacting particles have spherical geometry. (2) The anions and cations are contined to separate lattices. The anions have low mobility at the diffusion temperatures. (3) Surface diffusion is rapid and each particle is surrounded at any instant by a film of (A,B,_,O) solid solution. The composition of this film would correspond to that of the bulk oxide composition. (4) The reaction rate is limited by cationic diffusion and, in particular, the diffusion of A*’ in the solid solution A,B, _,O is rate-limiting. (5) The diffusion problem can be set up simply in terms of concentration gradients. The anion (O’-) has limited mobility in the oxide lattice due to its large size relative to the cations. The assumption that the oxygen ions in the lattice are relatively immobile is therefore justifiable. For example, Yurek and Schmalzried [15] report that the ratios of the anion and cation diffisivities in pure Co0 and MgO at 1573 K are: (D,/D,)=10-4 and (Do/ D&J= 10-z to 10-3. The assumption that each particle is surrounded by a film of oxide solid solution is solely for the treatment of diffusion and may be justified at high temperatures by the rapid diffusion of cations on the surface. However, it lacks physical meaning in the case of the galvanic cell measurement. Pure A0 is assumed to be a metal-deficit semiconductor with cation vacancies and electron holes as the major defects at high temperatures and oxygen partial pressures less than the ambient. It is assumed that the defect concentration in pure BO is much lower than in A0 and the (A,B,_,)O solid solution is also metal deficit over a wide composition range. Cationic vacancies and electron holes characterise the defect structure of the solid solution. For the (Co, Ni)O solid solution, it has been reported in the literature that the mobility
The initial and boundary conditions are: C=O for O
t=O
C=C, for r=rO, t>O ac r
=0 at r=O, t>O
Assuming fi is a constant, eqn. (10) can be solved and the fraction of reaction completed &) can be expressed as: K=l-
6 m 1 -mzl 2 exp7?
n2r2h -
[
6
1
(11)
where fi is the cation interdfisivity. Equation (11) refers to the fraction of particles in the mixture having an average particle size r,. Taking into account the particle size variation, the overall conversion (FL,) for the mixture can be calculated from:
where Wi is the weight fraction of particles in the ith size interval with average radius ri and fty is the corresponding value of the fraction of dissolution completed at time 1. The experimental data from the galvanic cell measurements are obtained as EMF values from which the oxide activities can be computed. For the dissolution of AO, we can write: - nFE = pAo - - &
= RT In aAO
(13)
where n is the number of electrons taking part in the electrode reaction, E the cell EMF in volts and F the faraday constant. p,+o and &, are the chemical potentials of the oxide20 in the solid solution as well as in the pure component, R the gas constant, T the temperature and aAO the activity of the oxide A0 in the solid solution, the standard state being pure oxide A0 at the temperature of measurement. At zero time, aAO and aBO are equal to unity. As dissolution proceeds, there is a decrease in activity till
14
the equilibrium value is attained. The overall fractional conversion (J&J determined experimentally at any given time t can be written as:
(14) where Y& is the mole fraction of A0 at time t and ~2% is the mole fraction of A0 at equilibrium. Assuming ideal mixing behaviour, the overall conversion of solids can be calculated from the galvanic cell data using eqns. (13) and (14). If the mixing is non-ideal, conversion of solids can be calculated using thermodynamic activities obtained from the EMF data. It may be noted that the above definition (eqn. (14)) is also valid for FL. At a given temperature and oxygen partial pressure, with a knowledge of the overall conversion corresponding to a given time, eqns. (11) and (12) can be used to calculate the cation interdiffusivity D by a differential fitting technique. The flow scheme for the diffusivity calculations is given in the appendix.
Results
and discussion
Using the galvanic cell method, dissolution kinetics in the oxide systems NiO-MgO [17] and Coo-MgO [19] have been studied in our laboratories. Figure 1 shows typical EMF VS. time plots obtained at 1373 K for the NiO-MgO system. The equilibrium values corresponding to various electrode compositions were obtained by separate experiments using the conventional heating and grinding method. These values are also indicated in Fig. 1. It is seen from the figure that as the NiO content increases from 30 to 70 mol.%, there is an apparent decrease in dissolution rate. From a statistical analysis, it can be shown that the total number of contact points between NiO and MgO decreases
with increasing NiO content, thereby explaining the decrease in reaction rate. However, overall fractional conversions calculated from cell voltages show minimal variations with composition in the range 40 to 60 mol.% NiO. For example, the fractional conversion after 1 h, for the case with 40 mol.% NiO, is 0.4686 and the corresponding value for 60 mol.% NiO is 0.4692. At the extreme compositions, the values are N 0.62 and -0.30 respectively. This shows that for the particle characteristics considered, the effect of contact points is not significant around the equimolar composition. For NiO contents much less than or much greater than 50 mol.%, the assumption of rapid surface diffusion may introduce some errors. In this work, emphasis has been placed on demonstrating a simple method for determining interdiffusivities. Restricting the calculations to the initial stages of the reaction, interdiffusivity values have been calculated for the NiO-MgO system from the EMF data obtained at different temperatures for different electrode compositions. Particle size distribution data for these calculations have been taken from Table 1. Figure 2 presents interdiffusivities corresponding to low conversions (MgO-12 mol.% NiO solid solution) at different temperatures. Within the range studied, there seems to be some scatter in the data, but the order of magnitude TABLE
1. Particle size distribution
NiO
in oxide materials Mgo
Size (pm)
Cumulative undersize (%)
Size (pm)
Cumulative undersize (%)
30 20 10 5
93 73 18 3
100 50 30 10
90 75 40 10
,iLog(D) [m’s_‘]
17-
0
70%NiO
l
hO%NiO
X
40°A.Ni0
n
50%NiO
A
30%NiO
Ih-
Oi.,.,‘,‘,‘,‘,‘,.,‘,‘,‘,‘I 0
2
4
6
8
10 12 14 16 18 20 22 24
Time(h) Fig. 1. Effect of composition on the rate of dissolution of nickel oxide in a MgO-NiO mixture [17], T= 1373 K, (1) xNio=0.3, (2) .rNio= 0.4, (3) xnio = 0.6, (4) xNio = 0.7.
7.0
7.2
74
7.6
7.8
8.0
l/T [K] x lo4 Fig. 2. Effect of temperature and initial composition of the oxide mixture on cation interdiffusivities in the MgO-12 mol.% NiO solid solution [17].
of the interdiffusivities compare well with the values reported for similar systems [15]. Cation interdiffusivities for the system NiO-MgO has not been reported in the literature. However, Yurek and Schmalzried [15] have determined the interdiffusivities in the system Coo-MgO in air at 1573 K. These authors report the value of D&,,,,,, as 3.46 x lo-l6 m2 s-l which is comparable with the diffusivity values given in Fig. 2. In a separate study [19], interdiffusion in Coo-MgO powder mixtures has also been studied in our laboratory using the galvanic cell technique. Figure 3 presents interdiffusivity values calculated at different conversions from the EMF data obtained at 1074 K. Extrapolation of diffusivity data to zero conversion yields D&cMgOj as N 1.0 x lo-l4 cm2 s-l. This value is in good agreement with the data of Yurek and Schmalzried [15], extrapolated to 1074 K (1.87~10-‘~ cm2 s-l) as shown in Fig. 3. The variation of interdifhisivity with progress in conversion and the magnitude of its variation are also in agreement with reported data. In addition to the galvanic cell measurements, high temperature X-ray studies have also been carried out to study interdiffusion in NiO-MgO powder mixtures. Typical results obtained are shown in Fig. 4. There is a continuous decrease in the intensity of the nickel oxide peak during the dissolution reaction. Further, the position of the nickel oxide peak is constant during the reaction while there is a continuous shift in the diffraction angle (28) corresponding to the magnesium oxide peak. These observations strongly suggest that it is the diffusion of Ni’+ ions which is rate-limiting, as assumed. The simple approach used in the present work indicates that it is possible to derive difhrsivity data from galvanic cell measurements. However, one should be careful in drawing general conclusions from the results. The assumptions involved in the present theoretical approach and the nature of the experiment limit the ,6-lag
I
COO-40%
I.
4
I.,
6
I
I.,
8
10
12
14
Time(h) Fig. 4. Results from high temperature X-ray studies on interdiffusion in equimolar NiO-MgO powder mixtures at 1373 K.
extent to which the results can be interpreted at the present stage. Efforts are underway to redesign the galvanic cell so that the geometry may be defined better and the theoretical analysis can be improved further. However, the present work has demonstrated that solid state reactions can be followed continuously and this opens up possibilities for continuous monitoring and control of similar processes.
Conclusions
A diffusion model has been developed for studying dissolution kinetics in binary oxide mixtures. The model has been applied to interdiffusion data in NiO-MgO and Coo-MgO systems, obtained by dynamic galvanic cell measurements. Cationic interdiffusivities derived from EMF data using the model are in reasonable agreement with those reported in the literature using the conventional diffusion couple technique. The present method offers the advantage of continuous measurement of reaction rates and the possibility of computer control of solid state reactions and related processes.
MgO,
T= IO74
K
aio I
b
C r) Di Di(jo)* F,” F talc o’25 Fm. conv. o’s’ Fig. 3. Interdiffusion K [15, 191.
I,
2
Symbols
(6, [rt? d] 60%
1, 0
data in COO-MgO powder mixtures at 1074
F =P
thermodynamic activity of the ith oxide (-) constant in eqn. (8) concentration of the diffusing species (kg mol mP3) cationic interdiffusivity (m2 s-l) diffusivity of ionic species i (m2 s-‘) ditfusivity of cationic species i in the jth oxide (m’ s-l) overall conversion for all particle sizes (-) calculated overall conversion for all particle sizes (same as F,,) (-) experimentally determined overall conversion for all particle sizes (-)
16
G n
r0
R t
T wi
x,! xeq
z Zi
Gibbs free energy (J mol-‘) number of electrons taking part in the cell reaction (-) particle radius (m) gas constant (J mol-’ K-l) time (s) temperature (K) weight fraction of particles in the ith size interval mole fraction of ith component in solid solution at time t mole fraction of ith component in solid solution at equilibrium product volume per unit volume of reactant consumed (-) charge on ith ionic species
10 H. Dunwald and C. Wagner, Z. Phys. Chem., (Leipzig), 824 (1934) 53. 11 B. Serin and R.T. Ellickson, J. Chem. Phys., 9 (1941) 742. 12 W. Miiller and H. Schmaizried, Z. Phys. Chem. N. F., 57 (1968) 203. 13 L.S. Darken, Trans. AIME, 175 (1948) 184. 14 E. de Lamotte and J.S. Kirkaldy, Z. Phys. Chem. N. F., 67 (1969) 31. 15 G.J. Yurek and H. Schmalzried, Ber. Busenges., 78 (1974) 1379. 16 S.F. Hulbert, J. Br. Ceram Sot., 6 (1969) 11. 17 A. Jakobsson, N.S. Srinivasan and S. Seetharaman, presented at theAnnual Con& Inst. Ceramics, Nanoceramics, Cambridge, April 8-10, 1992. 18 J.J. Stiglich, Jr, J.B. Cohen and D.H. Whitmore,J. Am Ceram Sot., 56 (1973) 119. 19 A. Ljubinkovic, M.Sc. Thesis, Department of Theoretical Metaiiurgy, The Royal Institute of Technology, Stockholm, 1993.
Greek letters 8 angle of diffraction
(degree) chemical potential of ith component electrical potential
Pi
@
Appendix
Subscripts
av talc exp f max (s)
average calculated experimental film maximum solid
I_/?;)
supersc?ipts
t *
J
I
calculate No
References J.W. Christian, 77re Theory of Transformations in MetaLFand Alloys, Pergamon Press, New York, 1965, p. 471. J.W. Christian, in R.W. Cahn (ed.), Phase Transformations in Physical Metallurgy, North-Holland, Amsterdam, 1965. J. Laidler, Chemical Kinetics, McGraw-Hill, New York, 1965, pp. 316-318. W. Jander, 2. Anorg Chem., 163 (1927) 1. C. Kroger and G. Ziegler, Glastech. Ber., 26 (1953) 346. V.F. Zhuvarlev, I.G. Lesokhin and R.G. Tempel’man, Glastech. Ber., 26 (1953). A.M. Ginstling and B.J. Brounshtein, .I. Appl. Chem., USSR, 23 (1950) 1327. R.E. Carter, J. Chem. Phys., 34 (1961) 2010. E.A. Geiss, J. Am. Ceram. Sot., 46 (1963) 374.
eq.
Diffusivity calculations (t) ’ , Assume D = lo“* m*s“
equilibrium refers to standard state value at tune t refers to cationic diffusion in a pure oxide
eq
0
/
calculate fraction converte
f,,t (eq.11)
1, 1 alculation complete for all size intervals 1 JYeS calculate F,,’ (eq. 12) 4 change D value calculate AE (t) = f ( F,z-F,‘)
T
I
Yes -----+s
4
I
tsf,?J 1 No stop
Flow scheme for difhisivity calculations.
11
A diffusion model for dissolution phenomena mixtures* N.S. Srinivasan,
A. Jakobsson
in oxide powder
and S. Seetharaman
Department of Theoretical Metallurgy, Royal Institute of Technology, S-100 44 Stockholm (Sweden) (Received
February 12, 1993; in revised form December 7, 1993)
Abstract A diffusion model is proposed for studying interdiffusion in binary oxide mixtures. The model has been applied to data obtained by dynamic galvanic cell measurements on NiO-MgO and COO-MgO systems. Cationic interdifksivities derived from EMF data are in reasonable agreement with those reported in the literature using the conventional diffusion couple technique. The dynamic nature of the present method offers scope for continuous monitoring of solid state reactions and related processes.
Introduction The study
7 =[l
- (1 -x)ln]*
(1)
where r,, is the particle radius, k is a constant and D is the diffusion coefficient which is assumed to be invariant with time. The following assumptions are implied in Jander’s model: (1)Nucleation and surface difIusion are possible at temperatures lower than that needed for bulk diffusion. A coherent product layer is therefore present when bulk diffusion takes place. (2) Reaction takes place in one of the components which is completely and continuously covered by the other component. (3) The activities of the reactants remain the same on either side of the interface. The assumption of a constant ditfusion coefficient restricts the applicability of Jander’s equation. Kroger and Ziegler [5] assumed that the diffusion coefficient of the transported species varies inversely with time and modified Jander’s equation as follows:
Assuming that the activity of the reacting substance is proportional to the fraction of unreacted material (1 -x), Zhuravlev et al. [6] modified the Jander relation as:
0032-5910/94/$07.00 0 1994 Elsevier Sequoia. All rights reserved SSDI 0032-5910(94)02808-2
2 (3)
Ginstling and Brounshtein [7] pointed out that the parabolic rate law considers the reaction interface to be constant, although it actually decreases for spherical particles during the course of reaction. By analogy with heat transfer through a spherical shell, they arrived at the following relationship: 2kLlt =l-
ir-(l-X)m
ro2 Carter [S] corrected the above relationship for large differences in volume between the product and the reactants. He introduced a factor z which represents the product volume per unit volume of reactant consumed (eqn. (5)). However, it has been shown by Geiss [9] that the correction is significant only when z exceeds a value of 2. 2kDt -= ro2
z-(z-1)(1-x)~3-[1+(z-1~]~ z-l
(5)
analysis, they determined the chemical diffusion coefficient as a function of composition. This relationship is of the form: rS = fiO exp[ - bX,,,]
(8)
where b is a constant and &, refers to X,,o=O. Co0 and MgO form a continuous series of solid solutions at temperatures above N 1175 K and for a wide range of oxygen partial pressures. The system shows near-ideal mixing behaviour at these temperatures. The pure oxides and the solid solution have NaCl-type structure. Pure Co0 is a metal-deficit semiconductor and its defect structure is characterised mainly by cation vacancies and electron holes. Yurek and Schmalzried developed a rigorous theoretical approach to interpret their experimental data. The cations and the anions are assumed to occupy different sublattices. The large oxygen ions are relatively immobile compared to the cations. Relative to the oxygen ion sublattice, the fluxes of cobalt ions, magnesium ions and the electron holes (h’) can be expressed as:
Assuming that the diffusion coefficient varies inversely with time, eqn. (5) can be modified as: -2k In t = z-(z-l)(l-x)zn-[1+(z-lb]u3 z-l rlJ2
(6)
According to Hulbert [16], eqn. (6) represents the rate of formation of many spinels. For reactants with low defect concentrations, eqn. (5) is to be preferred at high temperatures while eqn. (6) provides a good representation for highly reactive components at low temperatures. Based on a solution to Fick’s second law for diffusion into or out of a sphere, Dunwald and Wagner [lo] derived a relationship for solid-solid reactions. In terms of fractional conversion of the reactants, Serin and Ellickson [ 1l] expressed the Dunwald-Wagner relationship as:
r12D
-In
r0”
6 II”( 1 -x)
Diffusion in binary monovalent ionic crystals has been studied by several workers. Mtiller and Schmalzried [12] studied diffusion in the KCl-RbCl system and derived an expression for the chemical diffusion coefficient based on Darken’s theory [13]. This approach has been extended to ternary systems containing a single anion by de Lamotte and Kirkaldy [14]. Yurek and Schmalzried [15] have studied interdiffusion in (A, B)O-type solid solutions. They employed diffusion couples of pure Co0 and MgO single crystals and studied interditfusion in the Coo-MgO system (1573 K, air atmosphere). Using the Boltzman-Matano
where i=Co2+, Mg2+, h’. The diffusion potential arising out of the interdiffusion of cations would be negligible if electron hole conduction in the oxide solid solution is much greater than ionic conduction. Under these conditions, due to the electroneutrality constraint, there is no coupling between the cationic fluxes. Further, assuming local thermodynamic equilibrium to prevail in the diffusion couple, the chemical diffusion coefficient can be calculated from the flux equations. The conventional diffusion couple technique for the determination of interdiffusivities is quite reliable, but imposes stringent restrictions on experimental possibilities. For example, the analysis is strictly valid for planar geometry. Diffusion in powder mixtures cannot be studied using this technique. This is a serious limitation, as high temperature applications often involve reactions taking place in mixtures of solids and agglomerates containing spherical particles. Further, the conventional technique is often tedious, time-consuming and requires sophisticated equipment like SEM-EDS for obtaining reliable results. Even the relatively simpler tracer methods for diffusion studies require a good deal of sophistication. Hence, there is a need for a simple, reliable and continuous method to follow the kinetics of solid-solid reactions and to measure interditfusivities in ceramic systems. Recently, a galvanic cell method has been proposed [17] for kinetic studies of solution phenomena in solid oxide systems. It is a dynamic method which uses a
13
solid electrolyte cell to study dissolution kinetics in powder mixtures. As pointed out earlier, diffusion models proposed in the literature so far are not directly applicable to such studies. It is the purpose of this work to develop a simple approach to interdiffusion in oxide powder mixtures which can be used to interpret the kinetic data from galvanic cell measurements.
of electron holes at the diffusion temperatures is quite high relative to that of a cation vacancy [18]. The rapid diffusion of electron holes leads to electrical neutrality in the diffusion sample. With the above background, Fick’s second law for diffusion into a spherical BO particle of radius r, can be written as:
Theoretical development For analytical purposes, a mixture of oxide particles A0 and BO of known particle size distribution is envisaged. The oxides A0 and BO are assumed to mix ideally and form a continuous series of solid solutions. No phase separation occurs on interditfusion. The solid solution (A, B)O which is formed has the same crystal structure as the pure oxides. Other assumptions include: (1) The reacting particles have spherical geometry. (2) The anions and cations are contined to separate lattices. The anions have low mobility at the diffusion temperatures. (3) Surface diffusion is rapid and each particle is surrounded at any instant by a film of (A,B,_,O) solid solution. The composition of this film would correspond to that of the bulk oxide composition. (4) The reaction rate is limited by cationic diffusion and, in particular, the diffusion of A*’ in the solid solution A,B, _,O is rate-limiting. (5) The diffusion problem can be set up simply in terms of concentration gradients. The anion (O’-) has limited mobility in the oxide lattice due to its large size relative to the cations. The assumption that the oxygen ions in the lattice are relatively immobile is therefore justifiable. For example, Yurek and Schmalzried [15] report that the ratios of the anion and cation diffisivities in pure Co0 and MgO at 1573 K are: (D,/D,)=10-4 and (Do/ D&J= 10-z to 10-3. The assumption that each particle is surrounded by a film of oxide solid solution is solely for the treatment of diffusion and may be justified at high temperatures by the rapid diffusion of cations on the surface. However, it lacks physical meaning in the case of the galvanic cell measurement. Pure A0 is assumed to be a metal-deficit semiconductor with cation vacancies and electron holes as the major defects at high temperatures and oxygen partial pressures less than the ambient. It is assumed that the defect concentration in pure BO is much lower than in A0 and the (A,B,_,)O solid solution is also metal deficit over a wide composition range. Cationic vacancies and electron holes characterise the defect structure of the solid solution. For the (Co, Ni)O solid solution, it has been reported in the literature that the mobility
The initial and boundary conditions are: C=O for O
t=O
C=C, for r=rO, t>O ac r
=0 at r=O, t>O
Assuming fi is a constant, eqn. (10) can be solved and the fraction of reaction completed &) can be expressed as: K=l-
6 m 1 -mzl 2 exp7?
n2r2h -
[
6
1
(11)
where fi is the cation interdfisivity. Equation (11) refers to the fraction of particles in the mixture having an average particle size r,. Taking into account the particle size variation, the overall conversion (FL,) for the mixture can be calculated from:
where Wi is the weight fraction of particles in the ith size interval with average radius ri and fty is the corresponding value of the fraction of dissolution completed at time 1. The experimental data from the galvanic cell measurements are obtained as EMF values from which the oxide activities can be computed. For the dissolution of AO, we can write: - nFE = pAo - - &
= RT In aAO
(13)
where n is the number of electrons taking part in the electrode reaction, E the cell EMF in volts and F the faraday constant. p,+o and &, are the chemical potentials of the oxide20 in the solid solution as well as in the pure component, R the gas constant, T the temperature and aAO the activity of the oxide A0 in the solid solution, the standard state being pure oxide A0 at the temperature of measurement. At zero time, aAO and aBO are equal to unity. As dissolution proceeds, there is a decrease in activity till
14
the equilibrium value is attained. The overall fractional conversion (J&J determined experimentally at any given time t can be written as:
(14) where Y& is the mole fraction of A0 at time t and ~2% is the mole fraction of A0 at equilibrium. Assuming ideal mixing behaviour, the overall conversion of solids can be calculated from the galvanic cell data using eqns. (13) and (14). If the mixing is non-ideal, conversion of solids can be calculated using thermodynamic activities obtained from the EMF data. It may be noted that the above definition (eqn. (14)) is also valid for FL. At a given temperature and oxygen partial pressure, with a knowledge of the overall conversion corresponding to a given time, eqns. (11) and (12) can be used to calculate the cation interdiffusivity D by a differential fitting technique. The flow scheme for the diffusivity calculations is given in the appendix.
Results
and discussion
Using the galvanic cell method, dissolution kinetics in the oxide systems NiO-MgO [17] and Coo-MgO [19] have been studied in our laboratories. Figure 1 shows typical EMF VS. time plots obtained at 1373 K for the NiO-MgO system. The equilibrium values corresponding to various electrode compositions were obtained by separate experiments using the conventional heating and grinding method. These values are also indicated in Fig. 1. It is seen from the figure that as the NiO content increases from 30 to 70 mol.%, there is an apparent decrease in dissolution rate. From a statistical analysis, it can be shown that the total number of contact points between NiO and MgO decreases
with increasing NiO content, thereby explaining the decrease in reaction rate. However, overall fractional conversions calculated from cell voltages show minimal variations with composition in the range 40 to 60 mol.% NiO. For example, the fractional conversion after 1 h, for the case with 40 mol.% NiO, is 0.4686 and the corresponding value for 60 mol.% NiO is 0.4692. At the extreme compositions, the values are N 0.62 and -0.30 respectively. This shows that for the particle characteristics considered, the effect of contact points is not significant around the equimolar composition. For NiO contents much less than or much greater than 50 mol.%, the assumption of rapid surface diffusion may introduce some errors. In this work, emphasis has been placed on demonstrating a simple method for determining interdiffusivities. Restricting the calculations to the initial stages of the reaction, interdiffusivity values have been calculated for the NiO-MgO system from the EMF data obtained at different temperatures for different electrode compositions. Particle size distribution data for these calculations have been taken from Table 1. Figure 2 presents interdiffusivities corresponding to low conversions (MgO-12 mol.% NiO solid solution) at different temperatures. Within the range studied, there seems to be some scatter in the data, but the order of magnitude TABLE
1. Particle size distribution
NiO
in oxide materials Mgo
Size (pm)
Cumulative undersize (%)
Size (pm)
Cumulative undersize (%)
30 20 10 5
93 73 18 3
100 50 30 10
90 75 40 10
,iLog(D) [m’s_‘]
17-
0
70%NiO
l
hO%NiO
X
40°A.Ni0
n
50%NiO
A
30%NiO
Ih-
Oi.,.,‘,‘,‘,‘,‘,.,‘,‘,‘,‘I 0
2
4
6
8
10 12 14 16 18 20 22 24
Time(h) Fig. 1. Effect of composition on the rate of dissolution of nickel oxide in a MgO-NiO mixture [17], T= 1373 K, (1) xNio=0.3, (2) .rNio= 0.4, (3) xnio = 0.6, (4) xNio = 0.7.
7.0
7.2
74
7.6
7.8
8.0
l/T [K] x lo4 Fig. 2. Effect of temperature and initial composition of the oxide mixture on cation interdiffusivities in the MgO-12 mol.% NiO solid solution [17].
of the interdiffusivities compare well with the values reported for similar systems [15]. Cation interdiffusivities for the system NiO-MgO has not been reported in the literature. However, Yurek and Schmalzried [15] have determined the interdiffusivities in the system Coo-MgO in air at 1573 K. These authors report the value of D&,,,,,, as 3.46 x lo-l6 m2 s-l which is comparable with the diffusivity values given in Fig. 2. In a separate study [19], interdiffusion in Coo-MgO powder mixtures has also been studied in our laboratory using the galvanic cell technique. Figure 3 presents interdiffusivity values calculated at different conversions from the EMF data obtained at 1074 K. Extrapolation of diffusivity data to zero conversion yields D&cMgOj as N 1.0 x lo-l4 cm2 s-l. This value is in good agreement with the data of Yurek and Schmalzried [15], extrapolated to 1074 K (1.87~10-‘~ cm2 s-l) as shown in Fig. 3. The variation of interdifhisivity with progress in conversion and the magnitude of its variation are also in agreement with reported data. In addition to the galvanic cell measurements, high temperature X-ray studies have also been carried out to study interdiffusion in NiO-MgO powder mixtures. Typical results obtained are shown in Fig. 4. There is a continuous decrease in the intensity of the nickel oxide peak during the dissolution reaction. Further, the position of the nickel oxide peak is constant during the reaction while there is a continuous shift in the diffraction angle (28) corresponding to the magnesium oxide peak. These observations strongly suggest that it is the diffusion of Ni’+ ions which is rate-limiting, as assumed. The simple approach used in the present work indicates that it is possible to derive difhrsivity data from galvanic cell measurements. However, one should be careful in drawing general conclusions from the results. The assumptions involved in the present theoretical approach and the nature of the experiment limit the ,6-lag
I
COO-40%
I.
4
I.,
6
I
I.,
8
10
12
14
Time(h) Fig. 4. Results from high temperature X-ray studies on interdiffusion in equimolar NiO-MgO powder mixtures at 1373 K.
extent to which the results can be interpreted at the present stage. Efforts are underway to redesign the galvanic cell so that the geometry may be defined better and the theoretical analysis can be improved further. However, the present work has demonstrated that solid state reactions can be followed continuously and this opens up possibilities for continuous monitoring and control of similar processes.
Conclusions
A diffusion model has been developed for studying dissolution kinetics in binary oxide mixtures. The model has been applied to interdiffusion data in NiO-MgO and Coo-MgO systems, obtained by dynamic galvanic cell measurements. Cationic interdiffusivities derived from EMF data using the model are in reasonable agreement with those reported in the literature using the conventional diffusion couple technique. The present method offers the advantage of continuous measurement of reaction rates and the possibility of computer control of solid state reactions and related processes.
MgO,
T= IO74
K
aio I
b
C r) Di Di(jo)* F,” F talc o’25 Fm. conv. o’s’ Fig. 3. Interdiffusion K [15, 191.
I,
2
Symbols
(6, [rt? d] 60%
1, 0
data in COO-MgO powder mixtures at 1074
F =P
thermodynamic activity of the ith oxide (-) constant in eqn. (8) concentration of the diffusing species (kg mol mP3) cationic interdiffusivity (m2 s-l) diffusivity of ionic species i (m2 s-‘) ditfusivity of cationic species i in the jth oxide (m’ s-l) overall conversion for all particle sizes (-) calculated overall conversion for all particle sizes (same as F,,) (-) experimentally determined overall conversion for all particle sizes (-)
16
G n
r0
R t
T wi
x,! xeq
z Zi
Gibbs free energy (J mol-‘) number of electrons taking part in the cell reaction (-) particle radius (m) gas constant (J mol-’ K-l) time (s) temperature (K) weight fraction of particles in the ith size interval mole fraction of ith component in solid solution at time t mole fraction of ith component in solid solution at equilibrium product volume per unit volume of reactant consumed (-) charge on ith ionic species
10 H. Dunwald and C. Wagner, Z. Phys. Chem., (Leipzig), 824 (1934) 53. 11 B. Serin and R.T. Ellickson, J. Chem. Phys., 9 (1941) 742. 12 W. Miiller and H. Schmaizried, Z. Phys. Chem. N. F., 57 (1968) 203. 13 L.S. Darken, Trans. AIME, 175 (1948) 184. 14 E. de Lamotte and J.S. Kirkaldy, Z. Phys. Chem. N. F., 67 (1969) 31. 15 G.J. Yurek and H. Schmalzried, Ber. Busenges., 78 (1974) 1379. 16 S.F. Hulbert, J. Br. Ceram Sot., 6 (1969) 11. 17 A. Jakobsson, N.S. Srinivasan and S. Seetharaman, presented at theAnnual Con& Inst. Ceramics, Nanoceramics, Cambridge, April 8-10, 1992. 18 J.J. Stiglich, Jr, J.B. Cohen and D.H. Whitmore,J. Am Ceram Sot., 56 (1973) 119. 19 A. Ljubinkovic, M.Sc. Thesis, Department of Theoretical Metaiiurgy, The Royal Institute of Technology, Stockholm, 1993.
Greek letters 8 angle of diffraction
(degree) chemical potential of ith component electrical potential
Pi
@
Appendix
Subscripts
av talc exp f max (s)
average calculated experimental film maximum solid
I_/?;)
supersc?ipts
t *
J
I
calculate No
References J.W. Christian, 77re Theory of Transformations in MetaLFand Alloys, Pergamon Press, New York, 1965, p. 471. J.W. Christian, in R.W. Cahn (ed.), Phase Transformations in Physical Metallurgy, North-Holland, Amsterdam, 1965. J. Laidler, Chemical Kinetics, McGraw-Hill, New York, 1965, pp. 316-318. W. Jander, 2. Anorg Chem., 163 (1927) 1. C. Kroger and G. Ziegler, Glastech. Ber., 26 (1953) 346. V.F. Zhuvarlev, I.G. Lesokhin and R.G. Tempel’man, Glastech. Ber., 26 (1953). A.M. Ginstling and B.J. Brounshtein, .I. Appl. Chem., USSR, 23 (1950) 1327. R.E. Carter, J. Chem. Phys., 34 (1961) 2010. E.A. Geiss, J. Am. Ceram. Sot., 46 (1963) 374.
eq.
Diffusivity calculations (t) ’ , Assume D = lo“* m*s“
equilibrium refers to standard state value at tune t refers to cationic diffusion in a pure oxide
eq
0
/
calculate fraction converte
f,,t (eq.11)
1, 1 alculation complete for all size intervals 1 JYeS calculate F,,’ (eq. 12) 4 change D value calculate AE (t) = f ( F,z-F,‘)
T
I
Yes -----+s
4
I
tsf,?J 1 No stop
Flow scheme for difhisivity calculations.