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SINTERING ADDITIVES AND DEFECT CHEMISTRY
12 SINTERING ADDITIVES A N D DEFECT CHEMISTRY
Sintering additives are usually added to powders in an attempt to enhance the sinterability and to control the microstructure. Typical examples are the addition of Ni to W for improving sinterability, and of MgO to A1203 for suppressing abnormal grain growth and improving densification. However, for the most part the roles of sintering additives are only known empirically and their mechanisms are not well understood. This chapter considers the point defects formed by the addition of sintering additives* in ionic compounds with low defect concentrations. For a low concentration of point defects, we may assume that the matrix atoms and point defects form an ideal solution with no interaction between defects. We may assume also that the concentration of matrix atoms is 1. In this case, we can easily estimate the concentration of point defects caused by dopant addition. Therefore, in the case of lattice-diffusioncontrolled sintering, the estimation can explain the change in sinterability with dopant addition. 12.1 P O I N T DEFECTS IN C E R A M I C S
Point defects in ceramics are usually expressed using the Kr6ger-Vink notation. ~'2 According to this notation, addition (or depletion) of neutral atoms and free electrons are expressed separately. The atoms and defects are denoted by alphabetic characters, their locations by subscripts and their effective charge by superscripts; namely, in the form of A c where A means a specific atom or defect, B its location and C its effective charge. A positive effective charge is denoted as o, a negative effective charge as t, and a neutral (zero) charge as x or with nothing. Table 12.1 lists some typical point defects present in a compound MX. 3-5 Among those listed, the two most common *In ionic compounds, sintering additives are usually called dopants when their concentration is low. 173
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CHAPTER 12 SINTERINGADDITIVES AND DEFECT CHEMISTRY
Table 12.1. Various types of point defects in a compound MX Defect type
Symbols
Vacancies Interstitials Misplaced atoms Associated centres Foreign atoms Free electrons and holes
vM, Vx, v~, V'x,V'D,V'x" . . . . M~, Xi, M ' I ' , ~ ' . . . . XM, Mx ....
( VM Vx), (Xi XM) . . . . LM, LT", ~'u . . . . e l,
h"
and important types of intrinsic crystalline defects in an ionic compound are Frenkel and Schottky defects. In addition to ionic defects, electronic defects (free electrons and electron holes) are also available in ionic compounds. 12.1.1 Frenkel Defect A Frenkel defect forms when an atom (ion) goes into an interstitial site leaving behind a vacancy and therefore consists of a defect pair* of an interstitial atom and its vacancy. For a simple metal oxide compound MO with full ionization, its formation equation is expressed as MX ~ M~i" + V'~t
(12.1)
When the number of these defects is very low compared with the number of lattice points, elementary statistical mechanics or the conventional mass action law gives [M~/'][V'~t] - exp ---k-T-] -- KF
(12.2)
where [M~'~ is the concentration of interstitial atoms with an effective charge of +2, AgF the formation free energy of a Frenkel defect and KF its mass action constant. 12.1.2 Schottky Defect The Schottky defect, which is unique to ionic compounds, consists of a stoichiometric pair of cation and anion vacancies. For an MO compound with full ionization, its formation equation is expressed as
MM-t- O 0 - ~ I/~ nt- V'o* + M s + Os
(12.3)
*Here, 'pair' does not mean an interstitial-vacancy associate but the two separated conjugate defects, an interstitial and its vacancy.
12.1 POINT DEFECTS IN CERAMICS
175
Here, B denotes the place where a lattice can form, for example, the grain boundary, a surface or a dislocation. Hence, unlike the Frenkel defect, the Schottky defect creates new lattice sites. Since MM and Oo are equivalent to M8 and On, respectively, Eq. (12.3) can also be written as
null ~ V'~t + 1~o~
(12.4)
The concentrations of these defects are then expressed as [V'~t][/~o~ - exp
Ags) ----~-~-] - Ks
(12.5)
where Ags is the formation free energy of a Schottky defect and Ks its mass action constant. 12.1.3 Electronic Defect
Perfect electronic order is achieved only at a temperature of OK, where all electrons are in the lowest possible energy levels under the constraint of the Pauli exclusion principle. Any excitation of electrons from their ground state to higher energy levels results in electronic disorder. In ceramics, however, intrinsic electronic disorder refers to the formation of free electrons in the conduction band and holes in the valence band. An intrinsic electronic defect thus consists of a free electron in the conduction band and a free electron hole in the valence band. The concentrations of free electrons (e') and electron holes (h ~ are determined by the band gap and temperature. According to Fermi statistics, the probability of an electron occupying an energy level E, P(E), is expressed as P(E) =
1
1 + e x p [ ( E - EF)/kT]
(12.6)
where EF is the Fermi level. In an intrinsic insulator or semiconductor, the concentration of free electrons, n, is the same as that of free electron holes, p. Denoting Ec to be the energy level of the conduction band and Ev that of the valence band, EF is ~(Ev + Ec)/2. When the concentrations of free electrons and free electron holes are low, Eq. (12.6) gives (12.7) and
P - [h~ - Nvv - exp -
(12.8)
176
C H A P T E R12 SINTERINGADDITIVES AND DEFECT CHEMISTRY
Here, ne and nh are, respectively, the number of electrons and holes per unit volume in the conduction and valence band, and Arc and Nv are, respectively, the density of the electron state in the conduction band and that of the electron hole state in the valence band. According to elementary quantum physics, they are expressed as
*k T] 3/2 Nc-2\,{2zrm ~ j
(12.9)
{Drm*hkT~ 3/2 ~ .]
(12.10)
and
Nv-2\
where hI, is the Planck constant (6.623 x 10-34js) and m~ and m~, are respectively the effective masses of a free electron and an electron hole. When external conditions, for example, solute addition and non-stoichiometry, dominate the electronic defects (extrinsic electronic defects), the electronic energy levels change and the relative concentrations of free electrons and electron holes vary. (Non-stoichiometry means a change in stoichiometry caused by a change in external conditions, for example, atmosphere and impurities.) However, the product of the two concentrations, [e'][h~ at a given temperature is constant, namely,
np - [e'][h~ - [exp(- ~-~) ] - Ki where
Ki is the mass action constant
of the electronic defect and
(12.11)
Eg -- Ec - Ev.
12.2 F O R M A T I O N OF P O I N T DEFECTS BY ADDITIVES
The concentrations of point defects in ionic compounds vary with the concentration of dopants. The defect concentrations can be predicted using defect chemistry ~-3 where each defect species is considered to be a chemical species and the reactions among defects are expressed as chemical reaction equations. In expressing defect chemical reactions, some basic principles are applied. First, the mass action law must be satisfied between defect species. Second, the sum of effective charges on the left-hand side of any reaction equation must be the same as that on the right-hand side. Accordingly, an overall electrical neutrality condition is maintained in the sample.
177
12.2 FORMATION OF POINT DEFECTS BYADDITIVES
Third, the cation to anion site ratio must be constant even for nonstoichiometry (site relation). In other words, as long as the crystal structure of an MaXb compound is maintained, the ratio a/b is invariable. However, the absolute number of sites may vary according to defect reactions. In respect of the site relation, there are species which create lattice sites, for example, VM, Vx, MM, XM and Xx, and species which do not create lattice sites, for example, e, h, Mi and Li. From the above principles, we can predict the concentrations of various defects using (i) (ii) (iii) (iv) (v)
the equations governing the formation of ionic defects, the equation governing the formation of electronic defects, a reaction equation between material and atmosphere, a mass conservation equation of the dopant, and an electrical neutrality equation (condition) of the total defects.
The equations for (i)-(iii) are expressed as products of the defect concentrations and the equations for (iv) and (v) as sums. We can calculate the concentration of each defect from the equations exactly by using a personal computer with n equations for n concentration variables. However, the usual and simple method of calculation follows a suggestion of Brouwer. ~ Since the concentrations of defects vary drastically with external (thermodynamic) conditions over a range of several orders of magnitude, the concentrations of minor species can be ignored and only the concentration of the major defect species considered for the equations for (iv) and (v). These are then expressed in the form of concentration products and the variation of defect concentration with external condition is easily shown in a log-log scale, the Brouwer diagram. ~ Among the equations for (i)-(v), those for (i)-(iii), which follow the mass action law, hold irrespective of temperature, T, partial pressure of vapourizable species, Pa, and dopant concentration, CL. In contrast, when using equations for (iv) and (v), we first assume the defect type of the dopant in the material and the charge neutrality between the major defects. In other words, major defects in the equations for (iv) and (v) vary with T, Pa and CL; the assumption is correct only under specific conditions. Therefore, if the predicted results are not in accord with observed results, this means that the assumptions made for major defects in the equations for (iv) and (v) were incorrect. This section will consider only the effect of dopant concentration on the concentrations of other defects in systems where T and Pa are constant. Consider a case where L203 dopant is added to MO oxide. A reaction of MO with oxygen in the atmosphere occurs following
1 ~02 --+ 0~9 + V'~t + 2h"
(12.12)
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CHAPTER 12 SINTERING ADDITIVES AND DEFECT CHEMISTRY
if an oxygen atom goes into the oxide forming a new lattice site. (This is the usual case in oxides.) Then, following the mass action law, Eq. (12.13) holds. [ 0 ~ ] [ V*~/Ip 2
[ V*~]p 2
~ ~ p -~o/2-
p l/ 2 02
Kg
(12.13)
Here, Kg is the reaction constant. Therefore, the equations for (i)-(iii) are Eqs (12.2), (12.5), (12.11) and (12.13). If all of the dopant atoms L go to M sites in the lattice as donors, L203
Mo
.
,
2L M + V'M + 30Xo
(12.14)
is satisfied. From this equation, the equation for (iv) becomes [L]total
--
[L~t]
(12.15)
On the other hand, the electrical neutrality condition gives [L~t] + p + 2[1~o~ + 2[M~/~ - 2[ V'~t] + n
(12.16)
If the major defects in MO are of the Schottky type, [l~o'] ~ [V'~ ]
(12.17)
t::
g o O 03 0
/
h"
log [ L ]tot~,
Figure 12.1. Effect of the concentration of foreign atom L on the defect situation in a compound MO; Po2 = const., [V~'] > [V~] and n > p3.
12.2 FORMATION OF POINT DEFECTS BYADDITIVES
119
for the intrinsic region and [L~] ~ 2[V'~]
(12.18)
for the extrinsic region. Therefore, taking logarithmic forms of Eqs (12.5), (12.11), (12.13), (12.15), (12.17) and (12.18) gives the relations between dopant concentration and the other defect concentrations, as shown in Figure 12.1.3 This figure corresponds to a system where [V~~ and n > p satisfy the case of pure MO under a given Po2. In Figure 12.1, we observe that addition of a dopant increases the concentration of defects with opposite charges and decreases that of those with similar charges. This result is evident from the reaction equations and also from the Le Chatelier principle. If the lattice diffusion of M controls the sintering of this material, sinterability is expected to increase with the addition of L203, which increases the metal vacancy concentration [V'~t] and hence the lattice diffusion coefficient DM.
Sintering additives are usually added to powders in an attempt to enhance the sinterability and to control the microstructure. Typical examples are the addition of Ni to W for improving sinterability, and of MgO to A1203 for suppressing abnormal grain growth and improving densification. However, for the most part the roles of sintering additives are only known empirically and their mechanisms are not well understood. This chapter considers the point defects formed by the addition of sintering additives* in ionic compounds with low defect concentrations. For a low concentration of point defects, we may assume that the matrix atoms and point defects form an ideal solution with no interaction between defects. We may assume also that the concentration of matrix atoms is 1. In this case, we can easily estimate the concentration of point defects caused by dopant addition. Therefore, in the case of lattice-diffusioncontrolled sintering, the estimation can explain the change in sinterability with dopant addition. 12.1 P O I N T DEFECTS IN C E R A M I C S
Point defects in ceramics are usually expressed using the Kr6ger-Vink notation. ~'2 According to this notation, addition (or depletion) of neutral atoms and free electrons are expressed separately. The atoms and defects are denoted by alphabetic characters, their locations by subscripts and their effective charge by superscripts; namely, in the form of A c where A means a specific atom or defect, B its location and C its effective charge. A positive effective charge is denoted as o, a negative effective charge as t, and a neutral (zero) charge as x or with nothing. Table 12.1 lists some typical point defects present in a compound MX. 3-5 Among those listed, the two most common *In ionic compounds, sintering additives are usually called dopants when their concentration is low. 173
174
CHAPTER 12 SINTERINGADDITIVES AND DEFECT CHEMISTRY
Table 12.1. Various types of point defects in a compound MX Defect type
Symbols
Vacancies Interstitials Misplaced atoms Associated centres Foreign atoms Free electrons and holes
vM, Vx, v~, V'x,V'D,V'x" . . . . M~, Xi, M ' I ' , ~ ' . . . . XM, Mx ....
( VM Vx), (Xi XM) . . . . LM, LT", ~'u . . . . e l,
h"
and important types of intrinsic crystalline defects in an ionic compound are Frenkel and Schottky defects. In addition to ionic defects, electronic defects (free electrons and electron holes) are also available in ionic compounds. 12.1.1 Frenkel Defect A Frenkel defect forms when an atom (ion) goes into an interstitial site leaving behind a vacancy and therefore consists of a defect pair* of an interstitial atom and its vacancy. For a simple metal oxide compound MO with full ionization, its formation equation is expressed as MX ~ M~i" + V'~t
(12.1)
When the number of these defects is very low compared with the number of lattice points, elementary statistical mechanics or the conventional mass action law gives [M~/'][V'~t] - exp ---k-T-] -- KF
(12.2)
where [M~'~ is the concentration of interstitial atoms with an effective charge of +2, AgF the formation free energy of a Frenkel defect and KF its mass action constant. 12.1.2 Schottky Defect The Schottky defect, which is unique to ionic compounds, consists of a stoichiometric pair of cation and anion vacancies. For an MO compound with full ionization, its formation equation is expressed as
MM-t- O 0 - ~ I/~ nt- V'o* + M s + Os
(12.3)
*Here, 'pair' does not mean an interstitial-vacancy associate but the two separated conjugate defects, an interstitial and its vacancy.
12.1 POINT DEFECTS IN CERAMICS
175
Here, B denotes the place where a lattice can form, for example, the grain boundary, a surface or a dislocation. Hence, unlike the Frenkel defect, the Schottky defect creates new lattice sites. Since MM and Oo are equivalent to M8 and On, respectively, Eq. (12.3) can also be written as
null ~ V'~t + 1~o~
(12.4)
The concentrations of these defects are then expressed as [V'~t][/~o~ - exp
Ags) ----~-~-] - Ks
(12.5)
where Ags is the formation free energy of a Schottky defect and Ks its mass action constant. 12.1.3 Electronic Defect
Perfect electronic order is achieved only at a temperature of OK, where all electrons are in the lowest possible energy levels under the constraint of the Pauli exclusion principle. Any excitation of electrons from their ground state to higher energy levels results in electronic disorder. In ceramics, however, intrinsic electronic disorder refers to the formation of free electrons in the conduction band and holes in the valence band. An intrinsic electronic defect thus consists of a free electron in the conduction band and a free electron hole in the valence band. The concentrations of free electrons (e') and electron holes (h ~ are determined by the band gap and temperature. According to Fermi statistics, the probability of an electron occupying an energy level E, P(E), is expressed as P(E) =
1
1 + e x p [ ( E - EF)/kT]
(12.6)
where EF is the Fermi level. In an intrinsic insulator or semiconductor, the concentration of free electrons, n, is the same as that of free electron holes, p. Denoting Ec to be the energy level of the conduction band and Ev that of the valence band, EF is ~(Ev + Ec)/2. When the concentrations of free electrons and free electron holes are low, Eq. (12.6) gives (12.7) and
P - [h~ - Nvv - exp -
(12.8)
176
C H A P T E R12 SINTERINGADDITIVES AND DEFECT CHEMISTRY
Here, ne and nh are, respectively, the number of electrons and holes per unit volume in the conduction and valence band, and Arc and Nv are, respectively, the density of the electron state in the conduction band and that of the electron hole state in the valence band. According to elementary quantum physics, they are expressed as
*k T] 3/2 Nc-2\,{2zrm ~ j
(12.9)
{Drm*hkT~ 3/2 ~ .]
(12.10)
and
Nv-2\
where hI, is the Planck constant (6.623 x 10-34js) and m~ and m~, are respectively the effective masses of a free electron and an electron hole. When external conditions, for example, solute addition and non-stoichiometry, dominate the electronic defects (extrinsic electronic defects), the electronic energy levels change and the relative concentrations of free electrons and electron holes vary. (Non-stoichiometry means a change in stoichiometry caused by a change in external conditions, for example, atmosphere and impurities.) However, the product of the two concentrations, [e'][h~ at a given temperature is constant, namely,
np - [e'][h~ - [exp(- ~-~) ] - Ki where
Ki is the mass action constant
of the electronic defect and
(12.11)
Eg -- Ec - Ev.
12.2 F O R M A T I O N OF P O I N T DEFECTS BY ADDITIVES
The concentrations of point defects in ionic compounds vary with the concentration of dopants. The defect concentrations can be predicted using defect chemistry ~-3 where each defect species is considered to be a chemical species and the reactions among defects are expressed as chemical reaction equations. In expressing defect chemical reactions, some basic principles are applied. First, the mass action law must be satisfied between defect species. Second, the sum of effective charges on the left-hand side of any reaction equation must be the same as that on the right-hand side. Accordingly, an overall electrical neutrality condition is maintained in the sample.
177
12.2 FORMATION OF POINT DEFECTS BYADDITIVES
Third, the cation to anion site ratio must be constant even for nonstoichiometry (site relation). In other words, as long as the crystal structure of an MaXb compound is maintained, the ratio a/b is invariable. However, the absolute number of sites may vary according to defect reactions. In respect of the site relation, there are species which create lattice sites, for example, VM, Vx, MM, XM and Xx, and species which do not create lattice sites, for example, e, h, Mi and Li. From the above principles, we can predict the concentrations of various defects using (i) (ii) (iii) (iv) (v)
the equations governing the formation of ionic defects, the equation governing the formation of electronic defects, a reaction equation between material and atmosphere, a mass conservation equation of the dopant, and an electrical neutrality equation (condition) of the total defects.
The equations for (i)-(iii) are expressed as products of the defect concentrations and the equations for (iv) and (v) as sums. We can calculate the concentration of each defect from the equations exactly by using a personal computer with n equations for n concentration variables. However, the usual and simple method of calculation follows a suggestion of Brouwer. ~ Since the concentrations of defects vary drastically with external (thermodynamic) conditions over a range of several orders of magnitude, the concentrations of minor species can be ignored and only the concentration of the major defect species considered for the equations for (iv) and (v). These are then expressed in the form of concentration products and the variation of defect concentration with external condition is easily shown in a log-log scale, the Brouwer diagram. ~ Among the equations for (i)-(v), those for (i)-(iii), which follow the mass action law, hold irrespective of temperature, T, partial pressure of vapourizable species, Pa, and dopant concentration, CL. In contrast, when using equations for (iv) and (v), we first assume the defect type of the dopant in the material and the charge neutrality between the major defects. In other words, major defects in the equations for (iv) and (v) vary with T, Pa and CL; the assumption is correct only under specific conditions. Therefore, if the predicted results are not in accord with observed results, this means that the assumptions made for major defects in the equations for (iv) and (v) were incorrect. This section will consider only the effect of dopant concentration on the concentrations of other defects in systems where T and Pa are constant. Consider a case where L203 dopant is added to MO oxide. A reaction of MO with oxygen in the atmosphere occurs following
1 ~02 --+ 0~9 + V'~t + 2h"
(12.12)
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CHAPTER 12 SINTERING ADDITIVES AND DEFECT CHEMISTRY
if an oxygen atom goes into the oxide forming a new lattice site. (This is the usual case in oxides.) Then, following the mass action law, Eq. (12.13) holds. [ 0 ~ ] [ V*~/Ip 2
[ V*~]p 2
~ ~ p -~o/2-
p l/ 2 02
Kg
(12.13)
Here, Kg is the reaction constant. Therefore, the equations for (i)-(iii) are Eqs (12.2), (12.5), (12.11) and (12.13). If all of the dopant atoms L go to M sites in the lattice as donors, L203
Mo
.
,
2L M + V'M + 30Xo
(12.14)
is satisfied. From this equation, the equation for (iv) becomes [L]total
--
[L~t]
(12.15)
On the other hand, the electrical neutrality condition gives [L~t] + p + 2[1~o~ + 2[M~/~ - 2[ V'~t] + n
(12.16)
If the major defects in MO are of the Schottky type, [l~o'] ~ [V'~ ]
(12.17)
t::
g o O 03 0
/
h"
log [ L ]tot~,
Figure 12.1. Effect of the concentration of foreign atom L on the defect situation in a compound MO; Po2 = const., [V~'] > [V~] and n > p3.
12.2 FORMATION OF POINT DEFECTS BYADDITIVES
119
for the intrinsic region and [L~] ~ 2[V'~]
(12.18)
for the extrinsic region. Therefore, taking logarithmic forms of Eqs (12.5), (12.11), (12.13), (12.15), (12.17) and (12.18) gives the relations between dopant concentration and the other defect concentrations, as shown in Figure 12.1.3 This figure corresponds to a system where [V~~ and n > p satisfy the case of pure MO under a given Po2. In Figure 12.1, we observe that addition of a dopant increases the concentration of defects with opposite charges and decreases that of those with similar charges. This result is evident from the reaction equations and also from the Le Chatelier principle. If the lattice diffusion of M controls the sintering of this material, sinterability is expected to increase with the addition of L203, which increases the metal vacancy concentration [V'~t] and hence the lattice diffusion coefficient DM.