- Email: [email protected]
Structural stability and cation disorder in Aurivillius phases
Materials Research Bulletin 47 (2012) 3850–3854
Contents lists available at SciVerse ScienceDirect
Materials Research Bulletin journal homepage: www.elsevier.com/locate/matresbu
Structural stability and cation disorder in Aurivillius phases M. Garcı´a-Guaderrama a,*, L. Fuentes b, A. Ma´rquez-Lucero b, O. Blanco a a b
Centro de Investigacio´n en Materiales DIP-CUCEI, Universidad de Guadalajara, Av. Revolucio´n 1500, Col. Olı´mpica, Guadalajara, Mexico Centro de Investigacio´n en Materiales Avanzados SC, Complejo Industrial Chihuahua, M. Cervantes 120, Chihuahua 31109, Mexico
A R T I C L E I N F O
A B S T R A C T
Article history: Received 23 April 2012 Received in revised form 11 July 2012 Accepted 6 August 2012 Available online 23 August 2012
A model to determine the structural stability and cation configuration of Aurivillius phases is presented. Structural stress caused by elastic coupling between the (M2O2) and (An1BnO3n+1) substructures that form the phase and the electrostatic attraction forces between them are considered. The model suggests that the cation exchanges observed in these substructures result from a balance between the cohesive electrostatic energy and the disruptive elastic coupling energy. The model explains and predicts the cation configuration of Aurivillius phases of type (Bi2xPbxO2)(PbyBi1y)n1BnO3n+1, where the cation exchange between M and A sites is of Bi3+ $ Pb2+ type. The PbBi2Nb2O9 phase is taken as a case study. This material has been studied quantifying this cation exchange and several theoretical models have been developed to explain the phenomenon. The cation equilibrium configuration determined by the present model is much more accurate than previous studies. Furthermore, a precise explanation of this phenomenon is provided. ß 2012 Elsevier Ltd. All rights reserved.
Keywords: A. Oxides D. Crystal structure D. Ferroelectricity
1. Introduction Aurivillius phases [1] present a M2An1BnO3n+3 type structure, which consist of (M2O2) fluorite-like substructures interleaved along the c-axis with (An1BnO3n+1) perovskites-like substructures. In this last expression, n is the number of BO6 octahedron, A is the cube-octahedral site occupied by 12-coordinated cations (Ca2+, Sr2+, Ba2+, Pb2+, La3+, Bi3+, etc.) and B is the octahedral site occupied by 6-coordinated cations of transition metals (Ti4+, Fe3+, Mn3+, Nb5+, Ta5+, W6+, etc.). The (M2O2) layer is described as a basal edgeshared MO4 group, where M occupies the apex of a square pyramid, with oxygen forming the basal plane. Classic studies on these materials [2,3] have considered this substructure invariant, consisting specifically of (Bi2O2)2+ layers, whereas more recent studies have quantified the partial substitution of Bi3+ in this layer by the same cations that are found in A [4–11]. Several lines of investigation have been focused on these materials, such as structural characterization [12–14], functional properties [15–17], and crystallochemical models [18–20]. One of the topics that recently has received attention and that could have incidence in all the previous studies, is the cation exchange between the M2O2 and (An1BnO3n+1) substructures. This phenomenon could be related with topics apparently disconnected such as the inexistence of phases with high n value [3,21–23], the variation of the structural parameters as a function of the thermal annealing
* Corresponding author. E-mail address: [email protected] (M. Garcı´a-Guaderrama). 0025-5408/$ – see front matter ß 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.materresbull.2012.08.038
conditions [9,10], changes of structural symmetry [24,25] or the narrow interval of cation substitutions [2,26]. In regard to last point, cations that are usually present in perovskites, such as Ho3+, Yb3+, Zr4+ or Ge4+, with ionic radii of 1.015, 0.985, 0.72 and 0.53 A˚, respectively [27], are absent in Aurivillius phases as single occupants of A or B positions. Analyzing the inclusion of these and other cations in Bi4Ti3O12 (n = 3), Armstrong and Newnham [2] proposed that the phase formation is set by the elastic coupling between (An1BnO3n+1) and (M2O2) substructures. According to these authors, in the uncoupled form, these two substructures have different ab plane dimension. The dimension of the (M2O2) layer is the feature that restricts the size of the cations that can be placed in the A and/or B site of the (An1BnO3n+1) substructure. When the size of these two substructures is modified through the coupling, elastic stresses appear between them and could be decisive on the Aurivillius phase formation. Kikuchi [3], using this idea, has developed a model that predicts the maximum energy that the Aurivillius phase can store due to elastic coupling between the (An1BnO3n+1) and (M2O2) substructures. According to Kikuchi, the volume changes generated by the coupling, generate elastic tensions, which are proportional to the disruptive energy. An important restriction to this model is that it assumes the (M2O2) layer as unalterable (M Bi3+). Experimental studies in those days did not suggest cation exchanges between this layer and the perovskite cube-octahedral sites. Only recently, this fundamental feature of Aurivillius phases has been reported in studies where the partial substitution of Bi3+ in the (M2O2) layer has been quantified for cations of alkaline earths [4–6], lanthanides [7,8] or with lone pair electrons [4,9,10].
M. Garcı´a-Guaderrama et al. / Materials Research Bulletin 47 (2012) 3850–3854
In addition to the Kikuchi model, several works have studied some features related to the stability of these materials: the absence of phases with a high number of perovskite layers [21], the reason why some cation substitutions are not allowed [26], and the origin of cation exchange between (M2O2) and (An1BnO3n+1) substructures [5,11,28,29]. The first part of our proposed model is based on the idea expressed by Armstrong and Newnham, and mathematically reformulated by Kikuchi. For this reason in order to understand our model, we present a brief review of these ideas. 2. Theory 2.1. Brief description of the Kikuchi model According to Kikuchi, aM0 , the lattice parameter of the (M2O2) layer, is significantly smaller than aP0 , the pseudo-tetragonal lattice parameter of (An1BnO3n+1)2–, when both substructures are in an uncoupled form. During the phase formation, the two substructures are coupled and their lattice parameters are equaled. This fact originates volume changes in both layers, DV M0 for (M2O2) and nDV P0 for (An1BnO3n+1), resulting in elastic tensions inside (M2O2), tM, and (An1BnO3n+1), ntP. These stresses can be expressed as:
t M ¼ K M
nt P ¼ nK P
DV M 0
(1)
V M0
DV P 0
(2)
V P0
where KM and KP are the (M2O2) and (An1BnO3n+1) bulk moduli. The elastic energies EM and EP that are stored in these substructures due to volume changes are: EM ¼
Z DV M0 0
EP ¼ n
1 2
t M dðDV M0 Þ ¼ K M
Z DV P0 0
DV M 0
2
V M0
V M0
1 DV P 0 2 t P dðDV P0 Þ ¼ nK P V P0 2 V P0
(3)
(4)
The sum of these two energies gives the disruptive energy EA, stored in the Aurivillius phase due to the coupling between the (M2O2) and (An1BnO3n+1) substructures EA ¼ EM þ EP
(5)
A relationship between KP and KM is obtained by equaling Eqs. (1) and (2)
a¼
KP 1 DV M0 =V M0 ¼ n DV P0 =V P0 KM
(6)
The disruptive energy EA can then be expressed as: EA ¼
1 K M f ðDV A Þ 2
f ðDV A Þ ¼ V M0
DV M 0 V M0
(7) 2
þ naV P0
DV P 0 V P0
2
(8)
The f(DVA) term in Eq. (8), is taken as a stability criterion of the Aurivillius phases, and can be calculated according to the following considerations. The coupling tensions appear only in the ab plane, while the lattice parameters in c plane are not modified. The relative volume changes can be expressed as a function of estimated parameters (V M0 , aM0 , V P0 and aP0 ) and the pseudo-tetragonal lattice parameter
3851
(aP) of the Aurivillius phase can be calculated according to:
DV M 0 V M0
DV P 0 V P0
¼1
¼1
aP aM 0
aP aP0
2
2
(9)
(10)
The parameter aM0 is estimated by analogy with isostructural
a-PbO material, which shows the same geometrical coordination of the (Bi2O2)2+ layers. According to Armstrong and Newnhan [2], the a-PbO lattice parameter, aPb0 , is 3.98 A˚, while the (Bi2O2)2+ lattice parameter, aM0 ¼ aBi0 , is 3.80 A˚. The volume of this last non-tensioned substructure, V M0 ¼ V Bi0 , was estimated by Kikuchi [3] as 65.4 A˚3. The volume of a perovskite unit in uncoupled state, V P0 , is V P0 ¼ ðaP0 Þ3
(11)
Also the mathematical formulation of the model, Kikuchi applies it to materials reported in the literature and some others synthesized by himself, determining as a general rule that Aurivillius phases with f(DVA) values smaller than 0.4 A˚3 are stable. Most of the materials analyzed by Kikuchi have a f(DVA) value lower than that mentioned in the paragraph above. Some others phases according to this rule should not exist, are commonly synthesized (BaBi2Nb2O9 and PbBi2Nb2O9, for example). Another material analyzed by Kikuchi (using the data reported by Ismailzade et al. [30]), is Bi9Ti3Fe5O27 (n = 8), the model predicts that the phase should be stable. However, several studies have shown that this material and other members with a high number of layers that belong to the series Bin+1Ti3Fen3O3n+3, present structural instability that results in the formation of superstructures [22,23]. 2.2. Development of a new model The model that we are proposing, based on the one developed by Kikuchi, quantifies the effect that the M $ A cation exchange type, between the (M2O2) and (An1BnO3n+1) substructures, have on structural stability. The model also incorporates two fundamentals concepts: (1) the idea expressed by Hervoches and Lightfoot [11], and Lightfoot et al. [29] that this phenomenon has the objective of reducing the coupling tensions between the substructures during the phase formation, and (2) the idea expressed by Rentchestler [31] that the cation exchanges modify the balance of electrostatic attractions between both substructures. We will only consider the case where the exchange between the sites M and A is of Bi3+ $ Pb2+ type, that is for (Bi2xPbxO2)(PbyBi1y)n1BnO3n+1 Aurivillius phases. The generalization to phases with other cation exchange types requires considerations that will not be treated quantitatively in this model. Then the (M2O2) layer is formed by (Bi2xPbxO2). The lattice parameter aM0 , of this substituted substructure, is given by x 2x (12) þ aPb0 aM0 ¼ aBi0 2 2 and its volume in coupled state V M0 , is V M0 ¼ V Bi0
aM0 aBi0
2
(13)
The aBi0 and aPb0 values considered in this model are the ones suggested by Armstrong and Newnham (3.80 and 3.98 A˚, respectively) [2], while the value of V Bi0 , is the one determined by Kikuchi (65.4 A˚3) [3].
3852
M. Garcı´a-Guaderrama et al. / Materials Research Bulletin 47 (2012) 3850–3854
In the Kikuchi model, the parameter a (Eq. (6)), is a proportionality constant that relates the changes of volume of (M2O2) and (An1BnO3n+1). This constant is calculated without taking into account cation exchange. In the model presented here, the volumes change in the substructures are a consequence of elastic coupling and cation exchange. To calculate this parameter, since a change in one substructure directly generates a change in the other one, only the part in which the change in volume that results by the coupling between substructures should be considered. The volume changes that are generated by cation exchange are not considered, because the volume increase or decrease in a substructure for the inclusion or migration of a cation, does not affect the volume of the other one. Therefore, to determine the parameter a we have 1 DV M0 =V M0 1 DV Bi0 =V Bi0 a¼ ¼ (14) n DV P0 =V P0 x¼0 n DV P0 =V P0 The lattice parameter of the (An1BnO3n+1) perovskite blocks in an uncoupled state, aP0 , is calculated with the empirical relationship estimated by Armstrong and Newnham [2]: aP0 ¼ 1:33RB þ 0:60RA þ 2:36
(15)
where RA represents the ionic radii of eight-coordinated A cation and RB is the ionic radii of six-coordinated B cation, both radii are taken from Shannon [27]. The error factor in using this equation is near 0.5% for the most reported perovskites. According to Armstrong and Newnham [2], the Bi3+ and Pb2+ ionic radii inside the perovskites, are 1.20 and 1.34 A˚ respectively, significantly different than the ones suggested by Shannon. These values will be used in later calculations. The exchange also has the effect of diminishing the cohesion energy between (M2O2) and (An1BnO3n+1) due to the difference of the Bi3+ and Pb2+ electric charge. This phenomenon was suggested by Rentchestler [31] and in our model it has been taken into account for the structural stability for the following reason. An Aurivillius phase with Bi3+ $ Pb2+ cation exchange, can be considered as an alternate stacking of substructures with electric charge Z according to Z ¼ ð2 xÞ e
(16)
where e is the electron electric charge and x is the exchange level in the layer (M2O2) = (Bi2xPbxO2). The cohesion energy in this type of system is a function of the electric charges and their distribution in the substructures. Due to the way in which the Bi3+ and Pb2+ cations are distributed inside the (Bi2xPbxO2) substructure, where half of them are located above the oxygen basal plane, while the other half are below it, two planes containing these cations exist with charge +Z/2 separated by the oxygen basal plane. Inside the (An1BnO3n+1) substructure, the A cation is absent in the cube-octahedral site of the octahedrons next to the (M2O2) layer; therefore, it may be considered that the negative charge is concentrated in the oxygen plane formed by the vertices of these octahedrons. Because the (An1BnO3n+1) substructure has two of these planes, the electric charge of each one is Z/2. The distance dMP, between these planes with charge +Z/2 and Z/2, will be considered a constant for any analyzed configuration. The electric charge surface density s, for each one of these planes is
s¼
Z=2 ðaP0 Þ2
Z=2 2e0
and the cohesion energy EC among all the (M2O2) and (An1BnO3n+1) substructures that conforms the phase can be written as " # 1 e2 ð2 xÞ2 (20) EC ¼ 2 ALR EC jMP ¼ ALR dMP 4 e0 ðaP Þ2 In this expression, ALR is a constant similar to that from Madelung and includes long range forces present among non adjacent charged planes. By integrating all the constants in a single one, KC, we have " # 1 ð2 xÞ2 (21) EC ¼ K C 2 ðaP Þ2 A structural stability criterion is proposed by taking into account the cohesive electrostatic energy EC, and the disruptive coupling energy EA. We assume that the cation equilibrium configuration is obtained when the relationship between EC and EA is maximized EC K C ð2 xÞ2 =ðaP Þ2 ¼ (22) E K f ðDV A Þ M A If the relationship KC/KM is considered as a constant for all the Aurivillius phases, we propose as a structural stability factor the expression: AA ¼
ð2 xÞ2 =ðaP Þ2 f ðDV A Þ
(23)
For any (Bi2xPbxO2)(PbyBi1y)n1BnO3n+1 Aurivillius phase type, this factor allows the calculation of relative stabilities and predicts its most stable cation configuration, starting from parameters that depend on the level of cation exchange (x). Some of these parameters (aM0 , and aP0 ) are calculated directly by knowing the value of x, while the value of the hypothetical lattice parameter in a coupled state as a function of x, aP = aP(x), can be calculated according to the following consideration: under coupling, the substructure (An1BnO3n+1) experiences a decrease in its lateral dimensions. A compression factor k, that will be assumed as a constant for any cation configurations which can take on an Aurivillius phase, is defined as: k¼
aP ðexpÞ aP0 ðxe Þ
(24)
where aP(exp) is the experimental lattice parameter, and aP0 ðxe Þ is the lattice parameter calculated at the cation exchange level xe, for the considered equilibrium configuration. This compression factor allows us to calculate aP for any value of x, knowing that aP0 ¼ aP0 ðxÞ: aP ¼ k aP 0
(25)
3. Calculation 3.1. Application of the model
(17)
and the electric field U inside the space between them is U¼
where e0 is the free space permittivity. The cohesion energy EC jMP between these planes is " # 1 e2 ð2 xÞ2 (19) dMP EC jMP ¼ U s ¼ 8 e0 ðaP Þ2
(18)
The Bi2PbNb2O9 (n = 2), was the first Aurivillius phase ferroelectrically characterized. Subbarao, in order to explain ferroelectric anomalies in this material [32,33], proposed a cation exchange of type Bi3+ $ Pb2+, but he did not show any structural evidence.
M. Garcı´a-Guaderrama et al. / Materials Research Bulletin 47 (2012) 3850–3854
The first experimental structural confirmation of cation exchange in this material (and the first one in any other Aurivillius phase) was carried out by Srikanth et al. [9] using neutron diffraction. However, the accuracy of the determinations were limited by the similar neutron coherent lengths of Pb2+ and Bi3+ cations. Ismunandar [10], using neutron diffraction and anomalous X-ray dispersion, reported a phase configuration as (Bi1.53Pb0.47O2)Pb0.53Bi0.47Nb2O7 when the material was cooled slowly after synthesis. For the same material, when cooled quickly, the phase configuration reported is (Bi1.33Pb0.67O2)Pb0.33Bi0.67Nb2O7. In the same work where the previous structures were presented, it was suggested that at high temperatures the cations Pb2+ and Bi3+ have a similar occupying preference for the A sites in (An1BnO3n+1) structure and for the M sites in (M2O2) structure; therefore a random statistical distribution is present between the two sites, which would explain, according to Ismunandar, that the slowly cooled material presents similar quantities of Pb2+ and Bi3+ in the A site. So far, in addition to this explanation, other work that tries to give a quantitative explanation of cation configurations for these phases was made by Pirovano et al. [28]. These authors, by applying computer simulation techniques, carry out energy minimization calculations for the possible PbBi2Nb2O9 cation configurations. Based on the lattice parameters values reported by Ismunandar [10], Pirovano concludes that the equilibrium for this material, (Bi2xPbxO2)Pb1xBixNb2O7, when it is cooled slowly after synthesis, is reached at a cation exchange level of x = 0.14, a value significantly far from the experimental result, x = 0.47. The same material will be analyzed according to the model that has been developed in this work. We will proceed to compare the relative stabilities of possible configurations that can adopt the phase, (Bi2xPbxO2)Pb1xBixNb2O7, for exchange level values from x = 0 until the point in which the aM0 value equals the aP value, with Dx = 0.02 intervals. The values of lattice parameters used in the following calculations are the ones reported by Ismunandar for the slowly cooled material. By considerations made in later works the parameters ALR, dMP, KM and KP are considered constants for any cation configuration and for any phase with (Bi2xPbxO2) (PbyBi1y)n1BnO3n+1 structural formula. The experimental cubic lattice parameter aP(exp) = 3.88442 A˚, at the experimental equilibrium configuration value of x(exp) = 0.47, is calculated based on the orthorhombic lattice parameters aort and bort (space group A21am), according to the equation: aP ðexpÞ ¼
aort þ bort pffiffiffi 2 2
3853
Fig. 1. Relative structural stability, expressed by the stability factor (AA), of the (Bi2xPbxO2)Pb1xBixNb2O7 Aurivillius phase as a function of the cation exchange (x). The first (a) and last (b) iteration resulting from the model are shown. xa is the cation exchange value where the iterative process started, and xe is the value where the structure achieved highest stability.
configuration to xe = 0.40. The following three iterations carry the equilibrium point to x = 0.44, 0.46 and 0.47. A sixth iteration does not move the equilibrium point (Fig. 2). Based on these calculations, we concluded that the cation configuration predicted by the model is x = 0.47, (Fig. 1b), in agreement with the experimental result.
(26)
With the values of aP(exp), ionic radii of A (Pb2+/Bi3+, 1.34/ 1.20 A˚) and B (Nb5+, 0.64 A˚), the model predicts an equilibrium configuration that comes closer to the experimental configuration by means of successive approaches. In the first step, the lattice parameter aP0 of (An1BnO3n+1) in an uncoupled state is calculated, (by Eq. (15)), for all configurations to be analyzed. These values will not be modified in the iterations. The values of aP(x) depend on the equilibrium cation exchange, x = xe, considered. In this point aP(exp) is assigned. For the first iteration, a configuration where the exchange is not present (xe = 0) is analyzed. With this value, the compression factor of (An1BnO3n+1) is calculated by Eq. (24) (k = 0.96743), which is used to calculate the hypothetical lattice parameter, aP, by Eq. (25), for any exchange value. By applying the model equations the maximum value of the stability factor AA, for this first iteration, is obtained at x = 0.29 (Fig. 1a). A second iteration, reassigning aP(exp) = 3.88442 A˚ at xe = 0.29, moves the equilibrium
Fig. 2. Equilibrium cation exchange value for each iteration. The value for the sixth iteration does not show change compared to the fifth iteration value; so, it is considered that the (Bi1.53Pb0.47O2)Pb0.53Bi0.47Nb2O7 cation configuration is the most stable, in agreement with the experimental configuration.
3854
M. Garcı´a-Guaderrama et al. / Materials Research Bulletin 47 (2012) 3850–3854
4. Discussion
5. Conclusions
Even though we consider the model proposed in this paper as correct, we also think that the remarkable concordance between the experimental result and the predicted one is to a certain extent coincidental. A great amount of physical simplifications and estimated parameters were applied. According to the main idea of the model, as the cation exchange level is increased, the (M2O2) substructure dimensions also increase, coupling with the (An1BnO3n+1) substructure, whose dimensions in turn decrease. This effect is due to the larger size of Pb2+ compared with the size of Bi3+. So, the cation exchange diminishes the disruptive energy EA, proportional to f(DVA), and arrives to a minimum value when the dimensions of (M2O2) equals (An1BnO3n+1), aP ¼ aM0 . If only this energy is considered, the cations jumping from M to A and vice versa, do not stop until the (M2O2) layer is free of elastic stresses. The mechanism that stops this cation exchange is the reduction of the electrostatic cohesion energy between the substructures, proportional to (2 x)2/(aP)2, due to the minor electric charge of Pb2+ compared with that of Bi3+. With these two mechanisms of opposed effect in the phase stability (both of them depend of the cation exchange level), a structural equilibrium is achieved when the energy relationship EC/EA is maximized. Applying the model to PbBi2Nb2O9 phase, with the lattice parameters reported by Ismunandar, it was verified that the phase acquires its maximum stability with the cation exchange level at x = 0.47. In addition to making quantitative predictions for cation exchanges in phases of the (Bi2xPbxO2)(PbyBi1y)n1BnO3n+1 type, the model can be used to explain qualitatively the inexistence of phases with a high value of n. In the isoestructural series as Pbn3Bi4TinO3n+3, for example, only phases with n 7 are reported [3,34]. In these materials, the cation exchange should contribute to the phase stability. However, considering a (Bi2O2)Pbn3Bi2TinO3n+1 structure without exchange, is evident that when the number of layers increase, the lattice parameter increase in an uncoupled state, aP0 of the (An1BnO3n+1) structure, according to Eq. (15). This increases at the same time as the disruptive energy. It is expected that for a certain value of aP0 (and therefore n), not even cation exchange is enough to achieve a stability factor AA that could allow the formation of the phase. The data reported in the literature indicates that for these particular phases the limit is n = 7. When the exchange is of type Bi3+ $ A, (A 6¼ Pb2+), it is possible to apply the model if cation A presents the same coordination environment (a lone-pair electron) as Bi3+, and the relationship between the lattice parameter of (Bi2O2)2+ and a hypothetical (A2O2) structure (both in an uncoupled state) is known. This estimation, however, has not been carried out in this investigation and is not in the literature either. In addition, if cation A does not present the lone-pair electrons, as it occurs in the most Aurivillius phases, the model cannot be used in these cases; however, it is possible that with appropriate modifications, it can be adapted to explain the cation disorder that has been reported in Aurivillius phases with this exchange type [4–8].
The parameter AA is a structural factor that results from certain logical considerations which allows us to calculate the cation configuration of Aurivillius phases of type (Bi2xPbxO2)(PbyBi1y)n1BnO3n+1. However, this parameter is not a universal indicator for the stability of Aurivillius phases since the model only considers the ionic radii and electrical charges of A cations and does not consider the electrical charges of the B cations, nor the octahedron deformation, nor Jahn-Teller effects, nor anion/cation vacancies, in the same way that the Goldschmidt equation only considers ionic radii to predict the stability of perovskites, and does not consider other factors. Experimental investigation focused on determining cation configurations, is required to confirm, modify, or discard what it is proposed here. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34]
B. Aurivillius, Ark. Kemi. 1 (1949) 463–499. R.A. Armstrong, R.E. Newnham, Mater. Res. Bull. 7 (1972) 1025–1034. T. Kikuchi, Mater. Res. Bull. 14 (1979) 1561–1569. B.J. Kennedy, Q. Zhou, Ismunandar, Y. Kubota, K. Kato, J. Solid State Chem. 181 (2008) 1377–1386. S.M. Blake, M.J. Falconer, M. McCreedy, P. Lightfoot, J. Mater. Chem. 7 (1997) 1609–1613. M.S. Haluska, S.T. Misture, J. Solid State Chem. 177 (2004) 1965–1975. S.N. Achary, S.J. Patwe, P.S.R. Krishna, A.B. Shinde, A.K. Tyagi, Pramana-J. Phys. 71 (2008) 935–940. N.C. Hyatt, J.A. Hriljac, T.P. Comyn, Mater. Res. Bull. 38 (2003) 837–846. V. Srikanth, H. Idink, W.B. White, E.C. Subbarao, H. Rajagopal, A. Sequeira, Acta Crystallogr. B 52 (1996) 432–439. Ismunandar, B.A. Hunter, B.J. Kennedy, Solid State Ionics 112 (1998) 281–289. C.H. Hervoches, P. Lightfoot, J. Solid State Chem. 153 (2000) 66–73. A.D. Rae, J.G. Thompson, R.L. Withers, A.C. Willis, Acta Crystallogr. B 46 (1990) 474–487. Ismunandar, T. Kamiyama, A. Hoshikawa, Q. Zhou, B.J. Kennedy, Y. Kubota, K. Kato, J. Solid State Chem. 177 (2004) 4188–4196. A. Snedden, D.O. Charkin, V.A. Dolgikh, P. Lightfoot, J. Solid State Chem. 178 (2005) 180–184. H. Irie, M. Miyayama, T. Kudo, J. Am. Ceram. Soc. 83 (2000) 2699–2704. K. Aizawa, Y. Ohtani, Jpn. J. Appl. Phys. 46 (2007) 6944–6947. D. Machura, J. Rymarczyk, J. Ilczuk, Eur. Phys. J. Special Topics 154 (2008) 131–134. Ph. Boullay, G. Trolliard, D. Mercurio, L. Elcoro, J.M. Perez-Mato, J. Solid State Chem. 164 (2002) 252–260. Ph. Boullay, G. Trolliard, D. Mercurio, L. Elcoro, J.M. Perez-Mato, J. Solid State Chem. 164 (2002) 261–271. R.L. Withers, J.G. Thompson, A.D. Rae, J. Solid State Chem. 94 (1991) 404–417. A. Sanson, R.W. Whatmore, J. Am. Ceram. Soc. 88 (2005) 3147–3153. G.N. Subbanna, L. Ganapathi, B. Mater. Sci. 9 (1987) 29–35. J.L. Hutchison, J.S. Anderson, N.R. Rao, Proc. R. Soc. London, Ser. A 335 (1977) 301–312. M.W. Chu, M.T. Caldes, L. Brohan, M. Ganne, A.M. Marie, O. Joubert, Y. Piffard, Chem. Mater. 16 (2003) 31–42. J. Tellier, Ph. Boullay, D.B. Jennet, D. Mercurio, Solid State Sci. 10 (2008) 177–185. A. Ramanan, J. Gopalakrishnan, C.N.R. Rao, J. Solid. State Chem. 60 (1985) 376–381. R.D. Shannon, Acta Cryst. A 32 (1976) 751–767. C. Pirovano, M.S. Islam, R.N. Vannier, G. Nowogrocki, G. Mairesse, Solid State Ionics 140 (2001) 115–123. P. Lightfoot, A. Snedden, S.M. Blake, K.S. Knight, Mat. Res. Soc. Symp. Proc. 755 (2003) 89–96. I.G. Ismailzade, V.I. Nesterenko, F.A. Mirishli, P.G. Rustamov, Sov. Phys. Cryst. 12 (1967) 400–404. T. Rentschler, Mater. Res. Bull. 32 (1997) 351–369. E.C. Subbarao, J. Phys. Chem. Solids 23 (1962) 665–676. E.C. Subbarao, J. Am. Ceram. Soc. 45 (1962) 166–169. J.F. Ferna´ndez, A.C. Caballero, M. Villegas, Appl. Phys. Lett. 81 (2002) 4811–4813.
Contents lists available at SciVerse ScienceDirect
Materials Research Bulletin journal homepage: www.elsevier.com/locate/matresbu
Structural stability and cation disorder in Aurivillius phases M. Garcı´a-Guaderrama a,*, L. Fuentes b, A. Ma´rquez-Lucero b, O. Blanco a a b
Centro de Investigacio´n en Materiales DIP-CUCEI, Universidad de Guadalajara, Av. Revolucio´n 1500, Col. Olı´mpica, Guadalajara, Mexico Centro de Investigacio´n en Materiales Avanzados SC, Complejo Industrial Chihuahua, M. Cervantes 120, Chihuahua 31109, Mexico
A R T I C L E I N F O
A B S T R A C T
Article history: Received 23 April 2012 Received in revised form 11 July 2012 Accepted 6 August 2012 Available online 23 August 2012
A model to determine the structural stability and cation configuration of Aurivillius phases is presented. Structural stress caused by elastic coupling between the (M2O2) and (An1BnO3n+1) substructures that form the phase and the electrostatic attraction forces between them are considered. The model suggests that the cation exchanges observed in these substructures result from a balance between the cohesive electrostatic energy and the disruptive elastic coupling energy. The model explains and predicts the cation configuration of Aurivillius phases of type (Bi2xPbxO2)(PbyBi1y)n1BnO3n+1, where the cation exchange between M and A sites is of Bi3+ $ Pb2+ type. The PbBi2Nb2O9 phase is taken as a case study. This material has been studied quantifying this cation exchange and several theoretical models have been developed to explain the phenomenon. The cation equilibrium configuration determined by the present model is much more accurate than previous studies. Furthermore, a precise explanation of this phenomenon is provided. ß 2012 Elsevier Ltd. All rights reserved.
Keywords: A. Oxides D. Crystal structure D. Ferroelectricity
1. Introduction Aurivillius phases [1] present a M2An1BnO3n+3 type structure, which consist of (M2O2) fluorite-like substructures interleaved along the c-axis with (An1BnO3n+1) perovskites-like substructures. In this last expression, n is the number of BO6 octahedron, A is the cube-octahedral site occupied by 12-coordinated cations (Ca2+, Sr2+, Ba2+, Pb2+, La3+, Bi3+, etc.) and B is the octahedral site occupied by 6-coordinated cations of transition metals (Ti4+, Fe3+, Mn3+, Nb5+, Ta5+, W6+, etc.). The (M2O2) layer is described as a basal edgeshared MO4 group, where M occupies the apex of a square pyramid, with oxygen forming the basal plane. Classic studies on these materials [2,3] have considered this substructure invariant, consisting specifically of (Bi2O2)2+ layers, whereas more recent studies have quantified the partial substitution of Bi3+ in this layer by the same cations that are found in A [4–11]. Several lines of investigation have been focused on these materials, such as structural characterization [12–14], functional properties [15–17], and crystallochemical models [18–20]. One of the topics that recently has received attention and that could have incidence in all the previous studies, is the cation exchange between the M2O2 and (An1BnO3n+1) substructures. This phenomenon could be related with topics apparently disconnected such as the inexistence of phases with high n value [3,21–23], the variation of the structural parameters as a function of the thermal annealing
* Corresponding author. E-mail address: [email protected] (M. Garcı´a-Guaderrama). 0025-5408/$ – see front matter ß 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.materresbull.2012.08.038
conditions [9,10], changes of structural symmetry [24,25] or the narrow interval of cation substitutions [2,26]. In regard to last point, cations that are usually present in perovskites, such as Ho3+, Yb3+, Zr4+ or Ge4+, with ionic radii of 1.015, 0.985, 0.72 and 0.53 A˚, respectively [27], are absent in Aurivillius phases as single occupants of A or B positions. Analyzing the inclusion of these and other cations in Bi4Ti3O12 (n = 3), Armstrong and Newnham [2] proposed that the phase formation is set by the elastic coupling between (An1BnO3n+1) and (M2O2) substructures. According to these authors, in the uncoupled form, these two substructures have different ab plane dimension. The dimension of the (M2O2) layer is the feature that restricts the size of the cations that can be placed in the A and/or B site of the (An1BnO3n+1) substructure. When the size of these two substructures is modified through the coupling, elastic stresses appear between them and could be decisive on the Aurivillius phase formation. Kikuchi [3], using this idea, has developed a model that predicts the maximum energy that the Aurivillius phase can store due to elastic coupling between the (An1BnO3n+1) and (M2O2) substructures. According to Kikuchi, the volume changes generated by the coupling, generate elastic tensions, which are proportional to the disruptive energy. An important restriction to this model is that it assumes the (M2O2) layer as unalterable (M Bi3+). Experimental studies in those days did not suggest cation exchanges between this layer and the perovskite cube-octahedral sites. Only recently, this fundamental feature of Aurivillius phases has been reported in studies where the partial substitution of Bi3+ in the (M2O2) layer has been quantified for cations of alkaline earths [4–6], lanthanides [7,8] or with lone pair electrons [4,9,10].
M. Garcı´a-Guaderrama et al. / Materials Research Bulletin 47 (2012) 3850–3854
In addition to the Kikuchi model, several works have studied some features related to the stability of these materials: the absence of phases with a high number of perovskite layers [21], the reason why some cation substitutions are not allowed [26], and the origin of cation exchange between (M2O2) and (An1BnO3n+1) substructures [5,11,28,29]. The first part of our proposed model is based on the idea expressed by Armstrong and Newnham, and mathematically reformulated by Kikuchi. For this reason in order to understand our model, we present a brief review of these ideas. 2. Theory 2.1. Brief description of the Kikuchi model According to Kikuchi, aM0 , the lattice parameter of the (M2O2) layer, is significantly smaller than aP0 , the pseudo-tetragonal lattice parameter of (An1BnO3n+1)2–, when both substructures are in an uncoupled form. During the phase formation, the two substructures are coupled and their lattice parameters are equaled. This fact originates volume changes in both layers, DV M0 for (M2O2) and nDV P0 for (An1BnO3n+1), resulting in elastic tensions inside (M2O2), tM, and (An1BnO3n+1), ntP. These stresses can be expressed as:
t M ¼ K M
nt P ¼ nK P
DV M 0
(1)
V M0
DV P 0
(2)
V P0
where KM and KP are the (M2O2) and (An1BnO3n+1) bulk moduli. The elastic energies EM and EP that are stored in these substructures due to volume changes are: EM ¼
Z DV M0 0
EP ¼ n
1 2
t M dðDV M0 Þ ¼ K M
Z DV P0 0
DV M 0
2
V M0
V M0
1 DV P 0 2 t P dðDV P0 Þ ¼ nK P V P0 2 V P0
(3)
(4)
The sum of these two energies gives the disruptive energy EA, stored in the Aurivillius phase due to the coupling between the (M2O2) and (An1BnO3n+1) substructures EA ¼ EM þ EP
(5)
A relationship between KP and KM is obtained by equaling Eqs. (1) and (2)
a¼
KP 1 DV M0 =V M0 ¼ n DV P0 =V P0 KM
(6)
The disruptive energy EA can then be expressed as: EA ¼
1 K M f ðDV A Þ 2
f ðDV A Þ ¼ V M0
DV M 0 V M0
(7) 2
þ naV P0
DV P 0 V P0
2
(8)
The f(DVA) term in Eq. (8), is taken as a stability criterion of the Aurivillius phases, and can be calculated according to the following considerations. The coupling tensions appear only in the ab plane, while the lattice parameters in c plane are not modified. The relative volume changes can be expressed as a function of estimated parameters (V M0 , aM0 , V P0 and aP0 ) and the pseudo-tetragonal lattice parameter
3851
(aP) of the Aurivillius phase can be calculated according to:
DV M 0 V M0
DV P 0 V P0
¼1
¼1
aP aM 0
aP aP0
2
2
(9)
(10)
The parameter aM0 is estimated by analogy with isostructural
a-PbO material, which shows the same geometrical coordination of the (Bi2O2)2+ layers. According to Armstrong and Newnhan [2], the a-PbO lattice parameter, aPb0 , is 3.98 A˚, while the (Bi2O2)2+ lattice parameter, aM0 ¼ aBi0 , is 3.80 A˚. The volume of this last non-tensioned substructure, V M0 ¼ V Bi0 , was estimated by Kikuchi [3] as 65.4 A˚3. The volume of a perovskite unit in uncoupled state, V P0 , is V P0 ¼ ðaP0 Þ3
(11)
Also the mathematical formulation of the model, Kikuchi applies it to materials reported in the literature and some others synthesized by himself, determining as a general rule that Aurivillius phases with f(DVA) values smaller than 0.4 A˚3 are stable. Most of the materials analyzed by Kikuchi have a f(DVA) value lower than that mentioned in the paragraph above. Some others phases according to this rule should not exist, are commonly synthesized (BaBi2Nb2O9 and PbBi2Nb2O9, for example). Another material analyzed by Kikuchi (using the data reported by Ismailzade et al. [30]), is Bi9Ti3Fe5O27 (n = 8), the model predicts that the phase should be stable. However, several studies have shown that this material and other members with a high number of layers that belong to the series Bin+1Ti3Fen3O3n+3, present structural instability that results in the formation of superstructures [22,23]. 2.2. Development of a new model The model that we are proposing, based on the one developed by Kikuchi, quantifies the effect that the M $ A cation exchange type, between the (M2O2) and (An1BnO3n+1) substructures, have on structural stability. The model also incorporates two fundamentals concepts: (1) the idea expressed by Hervoches and Lightfoot [11], and Lightfoot et al. [29] that this phenomenon has the objective of reducing the coupling tensions between the substructures during the phase formation, and (2) the idea expressed by Rentchestler [31] that the cation exchanges modify the balance of electrostatic attractions between both substructures. We will only consider the case where the exchange between the sites M and A is of Bi3+ $ Pb2+ type, that is for (Bi2xPbxO2)(PbyBi1y)n1BnO3n+1 Aurivillius phases. The generalization to phases with other cation exchange types requires considerations that will not be treated quantitatively in this model. Then the (M2O2) layer is formed by (Bi2xPbxO2). The lattice parameter aM0 , of this substituted substructure, is given by x 2x (12) þ aPb0 aM0 ¼ aBi0 2 2 and its volume in coupled state V M0 , is V M0 ¼ V Bi0
aM0 aBi0
2
(13)
The aBi0 and aPb0 values considered in this model are the ones suggested by Armstrong and Newnham (3.80 and 3.98 A˚, respectively) [2], while the value of V Bi0 , is the one determined by Kikuchi (65.4 A˚3) [3].
3852
M. Garcı´a-Guaderrama et al. / Materials Research Bulletin 47 (2012) 3850–3854
In the Kikuchi model, the parameter a (Eq. (6)), is a proportionality constant that relates the changes of volume of (M2O2) and (An1BnO3n+1). This constant is calculated without taking into account cation exchange. In the model presented here, the volumes change in the substructures are a consequence of elastic coupling and cation exchange. To calculate this parameter, since a change in one substructure directly generates a change in the other one, only the part in which the change in volume that results by the coupling between substructures should be considered. The volume changes that are generated by cation exchange are not considered, because the volume increase or decrease in a substructure for the inclusion or migration of a cation, does not affect the volume of the other one. Therefore, to determine the parameter a we have 1 DV M0 =V M0 1 DV Bi0 =V Bi0 a¼ ¼ (14) n DV P0 =V P0 x¼0 n DV P0 =V P0 The lattice parameter of the (An1BnO3n+1) perovskite blocks in an uncoupled state, aP0 , is calculated with the empirical relationship estimated by Armstrong and Newnham [2]: aP0 ¼ 1:33RB þ 0:60RA þ 2:36
(15)
where RA represents the ionic radii of eight-coordinated A cation and RB is the ionic radii of six-coordinated B cation, both radii are taken from Shannon [27]. The error factor in using this equation is near 0.5% for the most reported perovskites. According to Armstrong and Newnham [2], the Bi3+ and Pb2+ ionic radii inside the perovskites, are 1.20 and 1.34 A˚ respectively, significantly different than the ones suggested by Shannon. These values will be used in later calculations. The exchange also has the effect of diminishing the cohesion energy between (M2O2) and (An1BnO3n+1) due to the difference of the Bi3+ and Pb2+ electric charge. This phenomenon was suggested by Rentchestler [31] and in our model it has been taken into account for the structural stability for the following reason. An Aurivillius phase with Bi3+ $ Pb2+ cation exchange, can be considered as an alternate stacking of substructures with electric charge Z according to Z ¼ ð2 xÞ e
(16)
where e is the electron electric charge and x is the exchange level in the layer (M2O2) = (Bi2xPbxO2). The cohesion energy in this type of system is a function of the electric charges and their distribution in the substructures. Due to the way in which the Bi3+ and Pb2+ cations are distributed inside the (Bi2xPbxO2) substructure, where half of them are located above the oxygen basal plane, while the other half are below it, two planes containing these cations exist with charge +Z/2 separated by the oxygen basal plane. Inside the (An1BnO3n+1) substructure, the A cation is absent in the cube-octahedral site of the octahedrons next to the (M2O2) layer; therefore, it may be considered that the negative charge is concentrated in the oxygen plane formed by the vertices of these octahedrons. Because the (An1BnO3n+1) substructure has two of these planes, the electric charge of each one is Z/2. The distance dMP, between these planes with charge +Z/2 and Z/2, will be considered a constant for any analyzed configuration. The electric charge surface density s, for each one of these planes is
s¼
Z=2 ðaP0 Þ2
Z=2 2e0
and the cohesion energy EC among all the (M2O2) and (An1BnO3n+1) substructures that conforms the phase can be written as " # 1 e2 ð2 xÞ2 (20) EC ¼ 2 ALR EC jMP ¼ ALR dMP 4 e0 ðaP Þ2 In this expression, ALR is a constant similar to that from Madelung and includes long range forces present among non adjacent charged planes. By integrating all the constants in a single one, KC, we have " # 1 ð2 xÞ2 (21) EC ¼ K C 2 ðaP Þ2 A structural stability criterion is proposed by taking into account the cohesive electrostatic energy EC, and the disruptive coupling energy EA. We assume that the cation equilibrium configuration is obtained when the relationship between EC and EA is maximized EC K C ð2 xÞ2 =ðaP Þ2 ¼ (22) E K f ðDV A Þ M A If the relationship KC/KM is considered as a constant for all the Aurivillius phases, we propose as a structural stability factor the expression: AA ¼
ð2 xÞ2 =ðaP Þ2 f ðDV A Þ
(23)
For any (Bi2xPbxO2)(PbyBi1y)n1BnO3n+1 Aurivillius phase type, this factor allows the calculation of relative stabilities and predicts its most stable cation configuration, starting from parameters that depend on the level of cation exchange (x). Some of these parameters (aM0 , and aP0 ) are calculated directly by knowing the value of x, while the value of the hypothetical lattice parameter in a coupled state as a function of x, aP = aP(x), can be calculated according to the following consideration: under coupling, the substructure (An1BnO3n+1) experiences a decrease in its lateral dimensions. A compression factor k, that will be assumed as a constant for any cation configurations which can take on an Aurivillius phase, is defined as: k¼
aP ðexpÞ aP0 ðxe Þ
(24)
where aP(exp) is the experimental lattice parameter, and aP0 ðxe Þ is the lattice parameter calculated at the cation exchange level xe, for the considered equilibrium configuration. This compression factor allows us to calculate aP for any value of x, knowing that aP0 ¼ aP0 ðxÞ: aP ¼ k aP 0
(25)
3. Calculation 3.1. Application of the model
(17)
and the electric field U inside the space between them is U¼
where e0 is the free space permittivity. The cohesion energy EC jMP between these planes is " # 1 e2 ð2 xÞ2 (19) dMP EC jMP ¼ U s ¼ 8 e0 ðaP Þ2
(18)
The Bi2PbNb2O9 (n = 2), was the first Aurivillius phase ferroelectrically characterized. Subbarao, in order to explain ferroelectric anomalies in this material [32,33], proposed a cation exchange of type Bi3+ $ Pb2+, but he did not show any structural evidence.
M. Garcı´a-Guaderrama et al. / Materials Research Bulletin 47 (2012) 3850–3854
The first experimental structural confirmation of cation exchange in this material (and the first one in any other Aurivillius phase) was carried out by Srikanth et al. [9] using neutron diffraction. However, the accuracy of the determinations were limited by the similar neutron coherent lengths of Pb2+ and Bi3+ cations. Ismunandar [10], using neutron diffraction and anomalous X-ray dispersion, reported a phase configuration as (Bi1.53Pb0.47O2)Pb0.53Bi0.47Nb2O7 when the material was cooled slowly after synthesis. For the same material, when cooled quickly, the phase configuration reported is (Bi1.33Pb0.67O2)Pb0.33Bi0.67Nb2O7. In the same work where the previous structures were presented, it was suggested that at high temperatures the cations Pb2+ and Bi3+ have a similar occupying preference for the A sites in (An1BnO3n+1) structure and for the M sites in (M2O2) structure; therefore a random statistical distribution is present between the two sites, which would explain, according to Ismunandar, that the slowly cooled material presents similar quantities of Pb2+ and Bi3+ in the A site. So far, in addition to this explanation, other work that tries to give a quantitative explanation of cation configurations for these phases was made by Pirovano et al. [28]. These authors, by applying computer simulation techniques, carry out energy minimization calculations for the possible PbBi2Nb2O9 cation configurations. Based on the lattice parameters values reported by Ismunandar [10], Pirovano concludes that the equilibrium for this material, (Bi2xPbxO2)Pb1xBixNb2O7, when it is cooled slowly after synthesis, is reached at a cation exchange level of x = 0.14, a value significantly far from the experimental result, x = 0.47. The same material will be analyzed according to the model that has been developed in this work. We will proceed to compare the relative stabilities of possible configurations that can adopt the phase, (Bi2xPbxO2)Pb1xBixNb2O7, for exchange level values from x = 0 until the point in which the aM0 value equals the aP value, with Dx = 0.02 intervals. The values of lattice parameters used in the following calculations are the ones reported by Ismunandar for the slowly cooled material. By considerations made in later works the parameters ALR, dMP, KM and KP are considered constants for any cation configuration and for any phase with (Bi2xPbxO2) (PbyBi1y)n1BnO3n+1 structural formula. The experimental cubic lattice parameter aP(exp) = 3.88442 A˚, at the experimental equilibrium configuration value of x(exp) = 0.47, is calculated based on the orthorhombic lattice parameters aort and bort (space group A21am), according to the equation: aP ðexpÞ ¼
aort þ bort pffiffiffi 2 2
3853
Fig. 1. Relative structural stability, expressed by the stability factor (AA), of the (Bi2xPbxO2)Pb1xBixNb2O7 Aurivillius phase as a function of the cation exchange (x). The first (a) and last (b) iteration resulting from the model are shown. xa is the cation exchange value where the iterative process started, and xe is the value where the structure achieved highest stability.
configuration to xe = 0.40. The following three iterations carry the equilibrium point to x = 0.44, 0.46 and 0.47. A sixth iteration does not move the equilibrium point (Fig. 2). Based on these calculations, we concluded that the cation configuration predicted by the model is x = 0.47, (Fig. 1b), in agreement with the experimental result.
(26)
With the values of aP(exp), ionic radii of A (Pb2+/Bi3+, 1.34/ 1.20 A˚) and B (Nb5+, 0.64 A˚), the model predicts an equilibrium configuration that comes closer to the experimental configuration by means of successive approaches. In the first step, the lattice parameter aP0 of (An1BnO3n+1) in an uncoupled state is calculated, (by Eq. (15)), for all configurations to be analyzed. These values will not be modified in the iterations. The values of aP(x) depend on the equilibrium cation exchange, x = xe, considered. In this point aP(exp) is assigned. For the first iteration, a configuration where the exchange is not present (xe = 0) is analyzed. With this value, the compression factor of (An1BnO3n+1) is calculated by Eq. (24) (k = 0.96743), which is used to calculate the hypothetical lattice parameter, aP, by Eq. (25), for any exchange value. By applying the model equations the maximum value of the stability factor AA, for this first iteration, is obtained at x = 0.29 (Fig. 1a). A second iteration, reassigning aP(exp) = 3.88442 A˚ at xe = 0.29, moves the equilibrium
Fig. 2. Equilibrium cation exchange value for each iteration. The value for the sixth iteration does not show change compared to the fifth iteration value; so, it is considered that the (Bi1.53Pb0.47O2)Pb0.53Bi0.47Nb2O7 cation configuration is the most stable, in agreement with the experimental configuration.
3854
M. Garcı´a-Guaderrama et al. / Materials Research Bulletin 47 (2012) 3850–3854
4. Discussion
5. Conclusions
Even though we consider the model proposed in this paper as correct, we also think that the remarkable concordance between the experimental result and the predicted one is to a certain extent coincidental. A great amount of physical simplifications and estimated parameters were applied. According to the main idea of the model, as the cation exchange level is increased, the (M2O2) substructure dimensions also increase, coupling with the (An1BnO3n+1) substructure, whose dimensions in turn decrease. This effect is due to the larger size of Pb2+ compared with the size of Bi3+. So, the cation exchange diminishes the disruptive energy EA, proportional to f(DVA), and arrives to a minimum value when the dimensions of (M2O2) equals (An1BnO3n+1), aP ¼ aM0 . If only this energy is considered, the cations jumping from M to A and vice versa, do not stop until the (M2O2) layer is free of elastic stresses. The mechanism that stops this cation exchange is the reduction of the electrostatic cohesion energy between the substructures, proportional to (2 x)2/(aP)2, due to the minor electric charge of Pb2+ compared with that of Bi3+. With these two mechanisms of opposed effect in the phase stability (both of them depend of the cation exchange level), a structural equilibrium is achieved when the energy relationship EC/EA is maximized. Applying the model to PbBi2Nb2O9 phase, with the lattice parameters reported by Ismunandar, it was verified that the phase acquires its maximum stability with the cation exchange level at x = 0.47. In addition to making quantitative predictions for cation exchanges in phases of the (Bi2xPbxO2)(PbyBi1y)n1BnO3n+1 type, the model can be used to explain qualitatively the inexistence of phases with a high value of n. In the isoestructural series as Pbn3Bi4TinO3n+3, for example, only phases with n 7 are reported [3,34]. In these materials, the cation exchange should contribute to the phase stability. However, considering a (Bi2O2)Pbn3Bi2TinO3n+1 structure without exchange, is evident that when the number of layers increase, the lattice parameter increase in an uncoupled state, aP0 of the (An1BnO3n+1) structure, according to Eq. (15). This increases at the same time as the disruptive energy. It is expected that for a certain value of aP0 (and therefore n), not even cation exchange is enough to achieve a stability factor AA that could allow the formation of the phase. The data reported in the literature indicates that for these particular phases the limit is n = 7. When the exchange is of type Bi3+ $ A, (A 6¼ Pb2+), it is possible to apply the model if cation A presents the same coordination environment (a lone-pair electron) as Bi3+, and the relationship between the lattice parameter of (Bi2O2)2+ and a hypothetical (A2O2) structure (both in an uncoupled state) is known. This estimation, however, has not been carried out in this investigation and is not in the literature either. In addition, if cation A does not present the lone-pair electrons, as it occurs in the most Aurivillius phases, the model cannot be used in these cases; however, it is possible that with appropriate modifications, it can be adapted to explain the cation disorder that has been reported in Aurivillius phases with this exchange type [4–8].
The parameter AA is a structural factor that results from certain logical considerations which allows us to calculate the cation configuration of Aurivillius phases of type (Bi2xPbxO2)(PbyBi1y)n1BnO3n+1. However, this parameter is not a universal indicator for the stability of Aurivillius phases since the model only considers the ionic radii and electrical charges of A cations and does not consider the electrical charges of the B cations, nor the octahedron deformation, nor Jahn-Teller effects, nor anion/cation vacancies, in the same way that the Goldschmidt equation only considers ionic radii to predict the stability of perovskites, and does not consider other factors. Experimental investigation focused on determining cation configurations, is required to confirm, modify, or discard what it is proposed here. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34]
B. Aurivillius, Ark. Kemi. 1 (1949) 463–499. R.A. Armstrong, R.E. Newnham, Mater. Res. Bull. 7 (1972) 1025–1034. T. Kikuchi, Mater. Res. Bull. 14 (1979) 1561–1569. B.J. Kennedy, Q. Zhou, Ismunandar, Y. Kubota, K. Kato, J. Solid State Chem. 181 (2008) 1377–1386. S.M. Blake, M.J. Falconer, M. McCreedy, P. Lightfoot, J. Mater. Chem. 7 (1997) 1609–1613. M.S. Haluska, S.T. Misture, J. Solid State Chem. 177 (2004) 1965–1975. S.N. Achary, S.J. Patwe, P.S.R. Krishna, A.B. Shinde, A.K. Tyagi, Pramana-J. Phys. 71 (2008) 935–940. N.C. Hyatt, J.A. Hriljac, T.P. Comyn, Mater. Res. Bull. 38 (2003) 837–846. V. Srikanth, H. Idink, W.B. White, E.C. Subbarao, H. Rajagopal, A. Sequeira, Acta Crystallogr. B 52 (1996) 432–439. Ismunandar, B.A. Hunter, B.J. Kennedy, Solid State Ionics 112 (1998) 281–289. C.H. Hervoches, P. Lightfoot, J. Solid State Chem. 153 (2000) 66–73. A.D. Rae, J.G. Thompson, R.L. Withers, A.C. Willis, Acta Crystallogr. B 46 (1990) 474–487. Ismunandar, T. Kamiyama, A. Hoshikawa, Q. Zhou, B.J. Kennedy, Y. Kubota, K. Kato, J. Solid State Chem. 177 (2004) 4188–4196. A. Snedden, D.O. Charkin, V.A. Dolgikh, P. Lightfoot, J. Solid State Chem. 178 (2005) 180–184. H. Irie, M. Miyayama, T. Kudo, J. Am. Ceram. Soc. 83 (2000) 2699–2704. K. Aizawa, Y. Ohtani, Jpn. J. Appl. Phys. 46 (2007) 6944–6947. D. Machura, J. Rymarczyk, J. Ilczuk, Eur. Phys. J. Special Topics 154 (2008) 131–134. Ph. Boullay, G. Trolliard, D. Mercurio, L. Elcoro, J.M. Perez-Mato, J. Solid State Chem. 164 (2002) 252–260. Ph. Boullay, G. Trolliard, D. Mercurio, L. Elcoro, J.M. Perez-Mato, J. Solid State Chem. 164 (2002) 261–271. R.L. Withers, J.G. Thompson, A.D. Rae, J. Solid State Chem. 94 (1991) 404–417. A. Sanson, R.W. Whatmore, J. Am. Ceram. Soc. 88 (2005) 3147–3153. G.N. Subbanna, L. Ganapathi, B. Mater. Sci. 9 (1987) 29–35. J.L. Hutchison, J.S. Anderson, N.R. Rao, Proc. R. Soc. London, Ser. A 335 (1977) 301–312. M.W. Chu, M.T. Caldes, L. Brohan, M. Ganne, A.M. Marie, O. Joubert, Y. Piffard, Chem. Mater. 16 (2003) 31–42. J. Tellier, Ph. Boullay, D.B. Jennet, D. Mercurio, Solid State Sci. 10 (2008) 177–185. A. Ramanan, J. Gopalakrishnan, C.N.R. Rao, J. Solid. State Chem. 60 (1985) 376–381. R.D. Shannon, Acta Cryst. A 32 (1976) 751–767. C. Pirovano, M.S. Islam, R.N. Vannier, G. Nowogrocki, G. Mairesse, Solid State Ionics 140 (2001) 115–123. P. Lightfoot, A. Snedden, S.M. Blake, K.S. Knight, Mat. Res. Soc. Symp. Proc. 755 (2003) 89–96. I.G. Ismailzade, V.I. Nesterenko, F.A. Mirishli, P.G. Rustamov, Sov. Phys. Cryst. 12 (1967) 400–404. T. Rentschler, Mater. Res. Bull. 32 (1997) 351–369. E.C. Subbarao, J. Phys. Chem. Solids 23 (1962) 665–676. E.C. Subbarao, J. Am. Ceram. Soc. 45 (1962) 166–169. J.F. Ferna´ndez, A.C. Caballero, M. Villegas, Appl. Phys. Lett. 81 (2002) 4811–4813.