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The diffusion coefficient and the adsorption and desorption rate constants of oxygen in YBa2Cu3O6.9
Volume 7, number
4
MATERIALS
LETTERS
THE DIFFUSION COEFFICIENT AND THE ADSORPTION RATE CONSTANTS OF OXYGEN IN YBa2Cu30s.9
Received
1988
AND DESORPTION
’ and AK. MALLICK
E. RUCKENSTEIN Department
October
of Chemical
Engineering,
SUNY at Buffalo, Buffalo, NY 14260, USA
18 July 1988
A method was developed and experiments were carried out to measure the diffusion coefficient as well as the desorption and adsorption rate constants of oxygen in the lattices of the l-2-3 compound ( YBazCu106.9) and its constituent oxides (BaO, CuO and Y?O,). The method involves the measurement of the pressure build up in time when a small amount of oxide is heated in a closed vessel whose initial pressure was less than 10m4 Torr. The transport equation along with the non-linear boundary condition due to the adsorption and desorption were solved by the finite element method. The diffusion coefficient of oxygen is smaller in the l-2-3 compound than in the constituent oxides. It was also observed that the diffusion coefficient in the l-2-3 compound increases more rapidly with temperature above the phase transition temperature from orthorhombic to tetragonal. The adsorption rate constant becomes much smaller while the desorption rate constant becomes significantly larger for temperatures greater than 400°C.
1. Introduction The role of oxygen in superconductors is complex and also crucial from the view of their superconducting properties. In the case of YBa2Cus0,, it has been observed [ 1 ] that this compound manifests superconducting properties only for x2 6.3, whereas for ~~6.3 it behaves as a semiconductor. The dependence of the supercritical temperature (T,) on x was also studied extensively and it was found that it decreases continuously with decreasing x [ 11. In all the mechanisms [2-41 proposed for superconductivity, one assumes that the oxygen atom plays an important role. It will be shown here that the mobility of the oxygen atoms in the lattice is much smaller in the l-2-3 compound than in the three constituent oxides. The present paper proposes a method of measuring the diffusivity coefficient of oxygen and the desorption and adsorption coefficient of oxygen at the surface of the l-2-3 compound at different temperatures. A small amount of YBazCu3069 was heated in a closed volume whose initial pressure was lop4 Torr. The pressure build up against time was re’ To whom correspondence
122
should be addressed.
corded at several temperatures. The surface area of the sample was measured prior to heating. The data were used to calculate the different parameters involved namely, the diffusion coefficient, and the adsorption/desorption rate constants by using the finite element method. The oxygen content of the l-2-3 compound decreases progressively on heating in a closed volume (initially under vacuum) above 25O”C, but this oxygen loss is reversible upon slow cooling in oxygen. The mobility of oxygen is relatively high in this material, leading to a rapid approach to a quasi-equilibrium between the solid and gaseous O2 in about 30-45 min, for any temperature above 250°C.
2. Experimental
methods
The l-2-3 compound was synthesized from stoichiometric mixtures of CuO, BaO, and YZ03 by employing the conventional procedure [ 5 1. The surface area measurement of the samples was carried out using an ORR surface area-pore volume analyzer (model 2100D) supplied by Micromeritics Instrument Corporation. The measurement of the oxygen depletion consisted of the following steps:
0167-577x/88/$03.50 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)
B.V.
Volume 7. number
4
MATERIALS
( 1) 1 g of the sample was introduced in a sample flask which was connected to a pressure transducer through a manifold. The flask as well as the manifold was evacuated to 10e4 Torr for 15 min being maintained at liquid-nitrogen temperature, to avoid the impact of vacuum on the sample. (2 ) The sample was then left at high vacuum for 12 h and slowly brought back to room temperature, in order to desorb moisture and air. (3) The sample was then heated to the desired temperature and the pressure build up in time was recorded. (4) Finally, the surface area was measured again to verify if changes have occurred during the depletion process.
LETTERS
October
where A is the surface area of the sample, R the universal gas constant and p is the pressure. The initial conditions are c=cO, p=O,
The governing
transport
equation
can be written
at t=O
and the additionally &/3x=
0,
(4) needed boundary
condition
at x= infinity.
Now the following introduced: X=x/L,
3. Determination of the diffusion adsorption and desorption coeffkients
I988
T=td/L’,
dimensionless
P=p/p,,
is (5)
variables
are
C=C/C~,,
where L is a characteristic length and y, is a characteristic pressure. Note that at x=0. C=@. The above equations become
as 3c,W=D@c/&x2,
(1)
where c is the concentration of oxygen at the spatial coordinate x and at time t, and D is the diffusion coefficient of oxygen atoms through the lattice. At the gas-solid interface the diffusional flux has to be equal to the difference between the rates of desorption and adsorption. Hence, D&/3x=K,8”-K2p(
1 -@)n,
at x=0,
aC/aT=
8C/&X”,
(la)
aCia.~=(K,LfDco)C”-(Kzp,L/De,,)(
l-C)“P, (2a)
dP/dT=
C”la--
y( 1 - C)“P/cu,
(3a)
P=O, C=O,
at T=O,
(4a)
dC/dX=O,
at X= 1.
f5a)
(2)
where the rate of desorption is assumed to be proportional to the fraction of oxygen atoms covering the surface, 8, raised to the power n and the rate of adsorption proportional to the pressure p as well as the fraction ( 1-e) of the vacant sites raised to the power n. In the above expression, K, is the desorption rate constant and & is the adsorption rate constant. The closed volume where the oxygen gas accumulates is composed of two parts: one is at the desired temperature To and has the volume V=40 ml of the sample flask and the other, the distribution manifold, has the volume V,=28.4 ml which is maintained at the temperature T, = 307 K. Using the ideal gas law, a mass balance for the gas phase leads to
Here
(ha) Y=fc;P,lK,.
(6b)
Because the depth of penetration by diffusion is small, the concentration of oxygen is not affected after about 1000 atomic layers. For this reason L is taken in the numerical calculations as equal to 1000 x 11.7 A, where 11.7 8, is the distance between two layers. The above equations have been solved by using the finite element method (see appendix). The parameters K,, K2, and D have been obtained by the modified Simplex search method [ 91. 123
Volume 7, number
MATERIALS
4
LETTERS
October
1988
4. Experimental results Fig. 1 contains the pressure versus time curves obtained experimentally at various temperatures. Using the modified Simplex search method [9] these experimental data have been used to calculate K,, K,, and D. The calculated pressure against time curves for different temperatures are represented in fig. 1 by the continuous line. The dependence of the diffusion coefficient on temperature is shown in fig. 2. A significant change in slope is found at SOO”C, temperature at which the compound changes its structure from orthorhombic to tetragonal. The diffusion coefficients of oxygen in the individual compounds (CuO, BaO, and Y203) and l-2-3 are compared in table 1. It is apparent that the diffusion coefficients in the individual oxides are much larger than those in the l-2-3 compounds at any given temperature. Fig. 3 provides a plot of the saturation pressure, Psat, versus 1 /To. The desorption and adsorption coefficient K, and K2 are plotted versus 1/ To in fig. 4. The adsorption rate constant is found to decrease with increasing temperature, while the desorption rate constant increases. This is as expected.
r-2
" 0
I 20
. =---
Temp.mOC
q
"
" 40
I" 60
I 80
Time (min.)
Fig. I. Pressure against time at various temperatures. The line in each set of readings represents the calculated value obtained from the numerical solution. The different symbols represent the experimental values for different temperature.
124
0 0014
l/Temperature
00018
(l/K)
Fig. 2. Diffusion coefficient versus 1/To for the l-2-3 compound. A sharp change in the slope is noticed at 500°C where the phase transition (O-T) takes place.
5.
Discussion
Numerical solutions were obtained by taking the value of n=2 in eq. (2). With n=l, 3, 4 the calculated values drifted away from the experimentally obtained values. This shows that oxygen is in atomic state while desorbing and also that a pair of neighboring sites is required for the adsorption of a molecule of oxygen. A steady increase in the diffusion coefficient value of oxygen atoms in the l-2-3 compound with temperature was observed. The slope below the phase transition temperature is much smaller than the one above it. The phase transition temperature spans over the range of 480°C to 600°C as the partial pressure of oxygen varies from 0 to 1 atm [ 11. In the present experiments, the phase transition occurs at 500’ C and around 5 Torr. Santoro et al. [ 6 ] suggest that the l-2-3 compound changes from orthorhombic to tetragonal (O-T) with the elimination of the oxygen atoms in the CuOz chains extending along the b axis of the orthorhombic cell. It has also been reported that at the O-T transition the T a axis becomes nearly half the sum of the 0 a and b axes and the c axis becomes elongated resulting in a volume increase of the unit cell [ 71. The elimination of the oxygen atom from the CuOz chain during the phase transition also results in an increment of the lattice parameters a and b. The net effect
Volume 7. number Table
4
MATERIALS
LETTERS
October
1988
I Compound
I-2-3 cue BaO y&t,
I-2-3 cue BaO V,O,
Temperature (“C)
10’ ‘K, (mole/s
250 250 250 250 400 400 400 400
0.50 5.00 0.05 10.00 1.00 10.00 0.03 30.00
of the increasing oxygen vacancies and of the enlarged lattice parameters (volume increment) leads to a substantial increment of the oxygen atom diffusivity in the unit cell. As expected. the desorption rate constant K, and the adsorption rate constant K2 are found to increase and decrease, respectively, with temperature. It has been reported [ 8 ] that below 400°C there is appreciable adsorption of oxygen by the l-2-3 compound. There is a sharp decrease in the K2 value for temperatures greater than 400°C. In contrast, the desorption rate coefficient significantly increases for To greater than 400°C. This can explain why temperatures below 400°C are needed to readsorb oxygen.
lO”lK 2 (mole/s cm’ Torr)
cm’)
120.0 2.0 1000.0 I.0 50.0
3.0 50.0 3.0
lO”iZr (cm’/s) 0.5
10.0 40.0 80.0 I.0 50.0 10.0 400.0
The surface coverage at equilibrium pressure, obtained by equating the dp/dt term in eq. (3 ) to zero, is plotted in fig. 5. The temperature affects indirectly, through p, K, and K,, the values of the surface coverage at equilibrium. The diffusion coefficients of oxygen in the individual oxides (BaO, CuO, and YZ03) are found to be much greater than that of the composite material l-2-3. It is not yet clear what is the relation (if any) between this reduced mobility and superconductivity.
m
h
-141 0 0010
0 0014
l/Temperature
l/Temperature
Ftg. 3. Dependence compound.
of saturation
(l/K)
pressure
on
I /To for the I-2-3
0 0018
(l/K)
Fig. 4. Desorption and adsorption rate constants for the I-2-3 compound versus I/T,,.The desorption rate curve tncreases whereas the adsorption rate curve decreases with temperature. A significant change in slope is observed for temperatures greater than 4OO’C in both curves.
125
Volume 7, number
4
MATERIALS
LETTERS
method
5 050
L
245
\ 345
445
Temperature Fig. 5. Fraction of surface area covered ferent temperatures at saturation.
545
1 645
(C)
October
for the spatial discretization
I988
and the Crank
Nicholson technique for time integration. The concentration was approximated by non-linear lagrangian basis functions over isoparametric elements. The finite element formalism transforms the partial differential equation to a coupled set of N+ 2 ordinary differential equations for the concentration and the pressure at the boundary. Finally, the implicit timeintegration scheme reduces these equations into a set of non-linear algebraic equations which is solved iteratively at every time step using Newton’s method. Convergence was achieved in approximately 3-5 iterations. Initial trial runs indicated some oscillations in the solution. This problem was overcome by mesh refinement near the boundary X= 0. Results were obtained using different mesh size and time steps. Optimum results were achieved at time step = 0.0 1 and mesh size of 100.
by oxygen atoms at dif-
References 6. Conclusion [ 1] R.J. Cava, B. Batlogg, A.P. Ramirez, D. Werder, C.H. Chen, Measurements of the build up in time of oxygen pressure when a small amount of oxide is heated in a closed volume at various temperatures, are employed to obtain the diffusion coefficient as well as the desorption/adsorption rate constants. The experiments have been carried out for the l-2-3 superconducting compound and its constituent oxides. The diffusion of oxygen is slower in the l-2-3 compound than in the BaO, CuO and Y203.
Acknowledgement The authors would like to thank S.M. Gupte and P. Karpe for their help in numerical calculation.
Appendix: Numerical method The equations
126
were solved using the finite element
E.A. Rietman and S.M. Zahurak, Mater. Res. Sot. Symp. Proc. 99 (1987) 19. [2] R. Kanno, Y. Takeda, M. Hasegawa, 0. Yamamoto, M. Takano, Y. Ikeda and Y. Bando, Mater. Res. Bull. 22 (1987) 1525. [3] S. Miraglia, F. Beech, A. Santoro, D. Tran Quin, S.A. Sunshine and D.W. Murphy, Mater. Res. Bull. 22 ( 1987) 1733. [4] C.N.R. Rao, P. Ganguly, J. Gopalakrishnan and D.D. Sharma, Mater. Res. Bull. 22 (1987) 1159, 1163. [ 5 ] D.S. Ginley, P.J. Nigrey, E.L. Venturini, B. Morosin and J.F. Kwak, J. Mater. Res. 2 (1987) 732. [6] A. Santoro, S. Miraglia and F. Beech, Mater. Res. Bull. 22 (1987) 1007. [7] P.K. Gallagher, H.M. O’Bryan, S.A. Sunshine and D.W. Murphy, Mater. Res. Bull. 22 (1987) 995. [8] D.E. Morris,U.M. Scheven, L.C. Bourne,M.L. Cohen,M.F. Crommie and A. Zettl, Proceedings of Symposium S of the 1987 Spring Meeting of the Materials Research Society, 2 l25 April 1987, Vol. EA-11. [ 91 C.S.G. Beveridge and R.S. Schecter, Optimization: theory and practice (McGraw-Hill, New York, 1970).
4
MATERIALS
LETTERS
THE DIFFUSION COEFFICIENT AND THE ADSORPTION RATE CONSTANTS OF OXYGEN IN YBa2Cu30s.9
Received
1988
AND DESORPTION
’ and AK. MALLICK
E. RUCKENSTEIN Department
October
of Chemical
Engineering,
SUNY at Buffalo, Buffalo, NY 14260, USA
18 July 1988
A method was developed and experiments were carried out to measure the diffusion coefficient as well as the desorption and adsorption rate constants of oxygen in the lattices of the l-2-3 compound ( YBazCu106.9) and its constituent oxides (BaO, CuO and Y?O,). The method involves the measurement of the pressure build up in time when a small amount of oxide is heated in a closed vessel whose initial pressure was less than 10m4 Torr. The transport equation along with the non-linear boundary condition due to the adsorption and desorption were solved by the finite element method. The diffusion coefficient of oxygen is smaller in the l-2-3 compound than in the constituent oxides. It was also observed that the diffusion coefficient in the l-2-3 compound increases more rapidly with temperature above the phase transition temperature from orthorhombic to tetragonal. The adsorption rate constant becomes much smaller while the desorption rate constant becomes significantly larger for temperatures greater than 400°C.
1. Introduction The role of oxygen in superconductors is complex and also crucial from the view of their superconducting properties. In the case of YBa2Cus0,, it has been observed [ 1 ] that this compound manifests superconducting properties only for x2 6.3, whereas for ~~6.3 it behaves as a semiconductor. The dependence of the supercritical temperature (T,) on x was also studied extensively and it was found that it decreases continuously with decreasing x [ 11. In all the mechanisms [2-41 proposed for superconductivity, one assumes that the oxygen atom plays an important role. It will be shown here that the mobility of the oxygen atoms in the lattice is much smaller in the l-2-3 compound than in the three constituent oxides. The present paper proposes a method of measuring the diffusivity coefficient of oxygen and the desorption and adsorption coefficient of oxygen at the surface of the l-2-3 compound at different temperatures. A small amount of YBazCu3069 was heated in a closed volume whose initial pressure was lop4 Torr. The pressure build up against time was re’ To whom correspondence
122
should be addressed.
corded at several temperatures. The surface area of the sample was measured prior to heating. The data were used to calculate the different parameters involved namely, the diffusion coefficient, and the adsorption/desorption rate constants by using the finite element method. The oxygen content of the l-2-3 compound decreases progressively on heating in a closed volume (initially under vacuum) above 25O”C, but this oxygen loss is reversible upon slow cooling in oxygen. The mobility of oxygen is relatively high in this material, leading to a rapid approach to a quasi-equilibrium between the solid and gaseous O2 in about 30-45 min, for any temperature above 250°C.
2. Experimental
methods
The l-2-3 compound was synthesized from stoichiometric mixtures of CuO, BaO, and YZ03 by employing the conventional procedure [ 5 1. The surface area measurement of the samples was carried out using an ORR surface area-pore volume analyzer (model 2100D) supplied by Micromeritics Instrument Corporation. The measurement of the oxygen depletion consisted of the following steps:
0167-577x/88/$03.50 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)
B.V.
Volume 7. number
4
MATERIALS
( 1) 1 g of the sample was introduced in a sample flask which was connected to a pressure transducer through a manifold. The flask as well as the manifold was evacuated to 10e4 Torr for 15 min being maintained at liquid-nitrogen temperature, to avoid the impact of vacuum on the sample. (2 ) The sample was then left at high vacuum for 12 h and slowly brought back to room temperature, in order to desorb moisture and air. (3) The sample was then heated to the desired temperature and the pressure build up in time was recorded. (4) Finally, the surface area was measured again to verify if changes have occurred during the depletion process.
LETTERS
October
where A is the surface area of the sample, R the universal gas constant and p is the pressure. The initial conditions are c=cO, p=O,
The governing
transport
equation
can be written
at t=O
and the additionally &/3x=
0,
(4) needed boundary
condition
at x= infinity.
Now the following introduced: X=x/L,
3. Determination of the diffusion adsorption and desorption coeffkients
I988
T=td/L’,
dimensionless
P=p/p,,
is (5)
variables
are
C=C/C~,,
where L is a characteristic length and y, is a characteristic pressure. Note that at x=0. C=@. The above equations become
as 3c,W=D@c/&x2,
(1)
where c is the concentration of oxygen at the spatial coordinate x and at time t, and D is the diffusion coefficient of oxygen atoms through the lattice. At the gas-solid interface the diffusional flux has to be equal to the difference between the rates of desorption and adsorption. Hence, D&/3x=K,8”-K2p(
1 -@)n,
at x=0,
aC/aT=
8C/&X”,
(la)
aCia.~=(K,LfDco)C”-(Kzp,L/De,,)(
l-C)“P, (2a)
dP/dT=
C”la--
y( 1 - C)“P/cu,
(3a)
P=O, C=O,
at T=O,
(4a)
dC/dX=O,
at X= 1.
f5a)
(2)
where the rate of desorption is assumed to be proportional to the fraction of oxygen atoms covering the surface, 8, raised to the power n and the rate of adsorption proportional to the pressure p as well as the fraction ( 1-e) of the vacant sites raised to the power n. In the above expression, K, is the desorption rate constant and & is the adsorption rate constant. The closed volume where the oxygen gas accumulates is composed of two parts: one is at the desired temperature To and has the volume V=40 ml of the sample flask and the other, the distribution manifold, has the volume V,=28.4 ml which is maintained at the temperature T, = 307 K. Using the ideal gas law, a mass balance for the gas phase leads to
Here
(ha) Y=fc;P,lK,.
(6b)
Because the depth of penetration by diffusion is small, the concentration of oxygen is not affected after about 1000 atomic layers. For this reason L is taken in the numerical calculations as equal to 1000 x 11.7 A, where 11.7 8, is the distance between two layers. The above equations have been solved by using the finite element method (see appendix). The parameters K,, K2, and D have been obtained by the modified Simplex search method [ 91. 123
Volume 7, number
MATERIALS
4
LETTERS
October
1988
4. Experimental results Fig. 1 contains the pressure versus time curves obtained experimentally at various temperatures. Using the modified Simplex search method [9] these experimental data have been used to calculate K,, K,, and D. The calculated pressure against time curves for different temperatures are represented in fig. 1 by the continuous line. The dependence of the diffusion coefficient on temperature is shown in fig. 2. A significant change in slope is found at SOO”C, temperature at which the compound changes its structure from orthorhombic to tetragonal. The diffusion coefficients of oxygen in the individual compounds (CuO, BaO, and Y203) and l-2-3 are compared in table 1. It is apparent that the diffusion coefficients in the individual oxides are much larger than those in the l-2-3 compounds at any given temperature. Fig. 3 provides a plot of the saturation pressure, Psat, versus 1 /To. The desorption and adsorption coefficient K, and K2 are plotted versus 1/ To in fig. 4. The adsorption rate constant is found to decrease with increasing temperature, while the desorption rate constant increases. This is as expected.
r-2
" 0
I 20
. =---
Temp.mOC
q
"
" 40
I" 60
I 80
Time (min.)
Fig. I. Pressure against time at various temperatures. The line in each set of readings represents the calculated value obtained from the numerical solution. The different symbols represent the experimental values for different temperature.
124
0 0014
l/Temperature
00018
(l/K)
Fig. 2. Diffusion coefficient versus 1/To for the l-2-3 compound. A sharp change in the slope is noticed at 500°C where the phase transition (O-T) takes place.
5.
Discussion
Numerical solutions were obtained by taking the value of n=2 in eq. (2). With n=l, 3, 4 the calculated values drifted away from the experimentally obtained values. This shows that oxygen is in atomic state while desorbing and also that a pair of neighboring sites is required for the adsorption of a molecule of oxygen. A steady increase in the diffusion coefficient value of oxygen atoms in the l-2-3 compound with temperature was observed. The slope below the phase transition temperature is much smaller than the one above it. The phase transition temperature spans over the range of 480°C to 600°C as the partial pressure of oxygen varies from 0 to 1 atm [ 11. In the present experiments, the phase transition occurs at 500’ C and around 5 Torr. Santoro et al. [ 6 ] suggest that the l-2-3 compound changes from orthorhombic to tetragonal (O-T) with the elimination of the oxygen atoms in the CuOz chains extending along the b axis of the orthorhombic cell. It has also been reported that at the O-T transition the T a axis becomes nearly half the sum of the 0 a and b axes and the c axis becomes elongated resulting in a volume increase of the unit cell [ 71. The elimination of the oxygen atom from the CuOz chain during the phase transition also results in an increment of the lattice parameters a and b. The net effect
Volume 7. number Table
4
MATERIALS
LETTERS
October
1988
I Compound
I-2-3 cue BaO y&t,
I-2-3 cue BaO V,O,
Temperature (“C)
10’ ‘K, (mole/s
250 250 250 250 400 400 400 400
0.50 5.00 0.05 10.00 1.00 10.00 0.03 30.00
of the increasing oxygen vacancies and of the enlarged lattice parameters (volume increment) leads to a substantial increment of the oxygen atom diffusivity in the unit cell. As expected. the desorption rate constant K, and the adsorption rate constant K2 are found to increase and decrease, respectively, with temperature. It has been reported [ 8 ] that below 400°C there is appreciable adsorption of oxygen by the l-2-3 compound. There is a sharp decrease in the K2 value for temperatures greater than 400°C. In contrast, the desorption rate coefficient significantly increases for To greater than 400°C. This can explain why temperatures below 400°C are needed to readsorb oxygen.
lO”lK 2 (mole/s cm’ Torr)
cm’)
120.0 2.0 1000.0 I.0 50.0
3.0 50.0 3.0
lO”iZr (cm’/s) 0.5
10.0 40.0 80.0 I.0 50.0 10.0 400.0
The surface coverage at equilibrium pressure, obtained by equating the dp/dt term in eq. (3 ) to zero, is plotted in fig. 5. The temperature affects indirectly, through p, K, and K,, the values of the surface coverage at equilibrium. The diffusion coefficients of oxygen in the individual oxides (BaO, CuO, and YZ03) are found to be much greater than that of the composite material l-2-3. It is not yet clear what is the relation (if any) between this reduced mobility and superconductivity.
m
h
-141 0 0010
0 0014
l/Temperature
l/Temperature
Ftg. 3. Dependence compound.
of saturation
(l/K)
pressure
on
I /To for the I-2-3
0 0018
(l/K)
Fig. 4. Desorption and adsorption rate constants for the I-2-3 compound versus I/T,,.The desorption rate curve tncreases whereas the adsorption rate curve decreases with temperature. A significant change in slope is observed for temperatures greater than 4OO’C in both curves.
125
Volume 7, number
4
MATERIALS
LETTERS
method
5 050
L
245
\ 345
445
Temperature Fig. 5. Fraction of surface area covered ferent temperatures at saturation.
545
1 645
(C)
October
for the spatial discretization
I988
and the Crank
Nicholson technique for time integration. The concentration was approximated by non-linear lagrangian basis functions over isoparametric elements. The finite element formalism transforms the partial differential equation to a coupled set of N+ 2 ordinary differential equations for the concentration and the pressure at the boundary. Finally, the implicit timeintegration scheme reduces these equations into a set of non-linear algebraic equations which is solved iteratively at every time step using Newton’s method. Convergence was achieved in approximately 3-5 iterations. Initial trial runs indicated some oscillations in the solution. This problem was overcome by mesh refinement near the boundary X= 0. Results were obtained using different mesh size and time steps. Optimum results were achieved at time step = 0.0 1 and mesh size of 100.
by oxygen atoms at dif-
References 6. Conclusion [ 1] R.J. Cava, B. Batlogg, A.P. Ramirez, D. Werder, C.H. Chen, Measurements of the build up in time of oxygen pressure when a small amount of oxide is heated in a closed volume at various temperatures, are employed to obtain the diffusion coefficient as well as the desorption/adsorption rate constants. The experiments have been carried out for the l-2-3 superconducting compound and its constituent oxides. The diffusion of oxygen is slower in the l-2-3 compound than in the BaO, CuO and Y203.
Acknowledgement The authors would like to thank S.M. Gupte and P. Karpe for their help in numerical calculation.
Appendix: Numerical method The equations
126
were solved using the finite element
E.A. Rietman and S.M. Zahurak, Mater. Res. Sot. Symp. Proc. 99 (1987) 19. [2] R. Kanno, Y. Takeda, M. Hasegawa, 0. Yamamoto, M. Takano, Y. Ikeda and Y. Bando, Mater. Res. Bull. 22 (1987) 1525. [3] S. Miraglia, F. Beech, A. Santoro, D. Tran Quin, S.A. Sunshine and D.W. Murphy, Mater. Res. Bull. 22 ( 1987) 1733. [4] C.N.R. Rao, P. Ganguly, J. Gopalakrishnan and D.D. Sharma, Mater. Res. Bull. 22 (1987) 1159, 1163. [ 5 ] D.S. Ginley, P.J. Nigrey, E.L. Venturini, B. Morosin and J.F. Kwak, J. Mater. Res. 2 (1987) 732. [6] A. Santoro, S. Miraglia and F. Beech, Mater. Res. Bull. 22 (1987) 1007. [7] P.K. Gallagher, H.M. O’Bryan, S.A. Sunshine and D.W. Murphy, Mater. Res. Bull. 22 (1987) 995. [8] D.E. Morris,U.M. Scheven, L.C. Bourne,M.L. Cohen,M.F. Crommie and A. Zettl, Proceedings of Symposium S of the 1987 Spring Meeting of the Materials Research Society, 2 l25 April 1987, Vol. EA-11. [ 91 C.S.G. Beveridge and R.S. Schecter, Optimization: theory and practice (McGraw-Hill, New York, 1970).