Empirical Analysis of Stability of A_n+1B_nO_3n+1 Ruddlesden–Popper Phases Using Reciprocal n-Values

Abstract

Layered A_n+1B_nO_3n+1 (n = 1…∞) Ruddlesden–Popper (RP) phases are a promising system for a variety of applications. Within the RP family, the thermodynamic properties of the phases are essentially additive with variation in the n value, but at present, there are no general approaches that would allow one to evaluate the individual stability of the RP phases and the possibility of their interconversion. The aim of this paper is to present a novel concept for performing a thermodynamic analysis of RP phases using the reciprocal values of the index n. We present an empirical equation ΔG_1/n = ΔG_P + B₁/n + B₂/n², where ΔG_1/n and ΔG_P are the molar Gibbs energies of formation of the Ruddlesden–Popper (RP) phase (AO)_1/nABO₃ and the parent ABO₃ perovskite, respectively, and n is a stoichiometry index of A_n+1B_nO_3n+1 RP phase. The correlation was validated using available thermodynamic data for the systems Sr-Ti-O, Ca-Ti-O, Sr-Zr-O, La-Ni-O, and La-Co-O. For all A-B combinations, the equation quantitatively describes the Gibbs energy of RP phase formation. Predicted values for the non-linear approximation lie within the experimental uncertainty in determining ΔG_1/n. The proposed correlation was used to analyze the relative stability of the RP phases and to determine the feasibility of synthesizing new compounds.

1. Introduction

Compounds with the ABO₃ composition and perovskite structure, as well as layered perovskite-like A_n+1B_nO_3n+1 Ruddlesden–Popper (RP) phases, are among the most attractive and interesting mixed oxides due to their outstanding functional properties [1]. In the last decade, RP phases based on rare earth oxides and transition metals Mn, Fe, and Co have attracted particular interest as functional materials for solid fuel elements [2,3], oxygen-permeable materials [4] and electrocatalysts as a single-phase [5,6] or perovskite-based composites [7].

A_n+1B_nO_3n+1 RP phases are a homologous series of layered substances which originate from perovskite ABO₃. The structure of the n-th member of the RP family can be visualized as alternating AO layers with a rock salt structure, separated by n layers of ABO₃ with a perovskite structure. The stoichiometry of such phases is usually written as A_n+1B_nO_3n+1 or AO(ABO₃)_n (n = 1…∞). Thus, the first member of the series is the compound A₂BO₄ (n = 1), and the last is perovskite ABO₃ (n = ∞).

A theoretical study of the electronic structure of Sr_n+1Ti_nO_3n+1 (n = 1–3, ∞) shows that with increasing index n, the A_n+1B_nO_3n+1 phases become unstable with respect to dissociation, the predicted limit of phase stability being n ≈ 3 [8,9]. This is in line with experimental published data for Sr-Ti-O [10], Sr-Ru-O [11,12], Ca-Ti-O [13], La-Ni-O [14,15] series; compounds with n = 1–3 were fabricated using standard high-temperature methods of ceramic synthesis. Nevertheless, the high n members of the RP Sr_n+1Ti_nO_3n+1 family were prepared using molecular beam epitaxy and pulsed laser deposition methods for n = 4–6 [16], n  = 1–5 and 10 [17], or an unprecedented n = 20 RP phase, AO·(ATiO₃)₂₀ (A = Ba_0.6Sr_0.4) [18].

An analysis of the literature has shown that, at present, very few studies have been published in which the thermodynamic properties of RP phases are systematized, making it possible to quantitatively predict their stability when the value of n changes. The thermodynamic properties of A_n+1B_nO_3n+1 (n = 1–4, ∞) compounds for Ca-Ti-O, Sr-Ti-O, Sr-Zr-O, Sr-Mo-O, and Sr-W-O systems have been analyzed to examine the stabilities of perovskites and RP phases [19]. The chemical potential diagrams (a log[a(A)/a(B)] vs. log P_O2) were constructed using a computer program and the available estimated or experimental thermodynamic data. A thermodynamic analysis was performed on the dissociation reaction of A₂MO₄ compounds into perovskites AMO₃ and alkaline earth oxides (AOs) using empirical correlations between stabilization energy and the tolerance factor. It was shown that the correlation between the tolerance factor and enthalpy of formation among perovskites was found to be very useful to predict an absence of some perovskite phase in certain systems. Thermodynamics of layered perovskites was systematically considered in [20]. Based on thermodynamic data for perovskites and RP phases of various chemical compositions, the paper demonstrates that thermodynamic layer values are substantially additive, thus permitting prediction of the properties for materials with unknown values. Similar conclusions were drawn in paper [21]. Based on dissolution enthalpies of La_n+1Ni_nO_3n+1 complex oxides, the thermochemical characteristics of these oxides from binary oxides were calculated. It was found that there is a linear dependence (1) between the enthalpy of formation of any complex oxide in the RP series and the value of n:

ΔH = 30.9 − 971n (kJ/mol La)

(1)

These approaches, based on the additivity of properties and their linear correlation with the n value, have two significant drawbacks. Firstly, strict additivity implies full reversibility of a transformation. For example, the reversibility of PR phase dissociation, Equation (2), results in a failure to synthesize a pure AO(ABO₃)_n phase. Therefore, it is necessary to take into account deviations from linearity (additivity), the magnitude of which, despite their small values, will most likely determine the stability of the products.

2 AO(ABO₃)_n = AO(ABO₃)_n−1 + AO(ABO₃)_n+1

(2)

Secondly, when analyzing the dependence F = f(n), where F is any property and n = 1…∞, the ABO₃ perovskite (n = ∞) falls out of consideration.

In this paper, we consider a new approximation to describe the stability of RP phases using reciprocal values of index 1/n = 1, 1/2, 1/3 …0. The aim of the study is to formulate general relationships covering a wide range of RP phases and test these relations using published thermodynamic experimental data for different sets of A and B cations that can form layered perovskite-like phases.

2. Materials and Methods

Here, and below, the abbreviation RPn is used for an n-th member of the Ruddlesden–Popper family (A_n+1B_nO_3n+1, AO·(ABO₃)_n or (AO)_1/n·ABO₃, n = 1…∞).

Gibbs energies of formation (ΔG_n) were used as a measure of RPn phase stability. The thermodynamic quantities of the RPn-forming systems (n = 1–3,∞) were collected from the appropriate publications (Sr-Ti-O [19,22]; Ca-Ti-O [23]; Sr-Zr-O [19]; La-Ni-O [15]; La-Co-O [24,25]). Values of ΔG_n at 298 K were used without any transformation as they appear in the parent papers. High temperature values of ΔG_n were calculated for a reference temperature 1200 K using the equations given in the original sources. Values of ΔG_n for stoichiometry A_n+1B_nO_3n+1 were converted to the reciprocal scale 1/n (stoichiometry (AO)_1/n·ABO₃, 1/n = 1, 1/2, 1/3, 0) using the obvious expression (3):

ΔG_n(A_n+1B_nO_3n+1) = n·ΔG_1/n((AO)_1/nABO₃)

(3)

3. Results and Discussion

We will now construct the n-th member of the RP family by stacking n layers of perovskite on an AO layer. The work to create such a structure can be written as (4), and dividing both sides of the equation by n, we obtain (4a).

ΔG_n = nΔG_P + ΔG_AO      for AO(ABO₃)_n phase 

(4)

ΔG_1/n = ΔG_n/n = ΔG_P + ΔG_AO/n for (AO)_1/nABO₃ phase

(4a)

where ΔG_n, ΔG_1/n—molar Gibbs energies of AO(ABO₃)_n and (AO)_1/nABO₃ formation, accordingly;

ΔG_P, ΔG_AO—increments corresponding to the addition of the perovskite or AO layer.

As a first approximation, let us assume that the values of ΔG_AO and ΔG_P do not depend on n. This means that the expression (4a) is valid for any n and that the value of ΔG_1/n (molar Gibbs energy of (AO)_1/nABO₃ formation) is a linear function of 1/n ΔG_1/n = A′ + B₁′/n. In this equation, A′ (y-intercept) is equal to ΔG_P (the Gibbs energy of ABO₃ perovskite formation), since ΔG_1/n = ΔG_P at n = ∞ and B₁′ is a slope (empirical fitting parameter).

Using available thermodynamic data from [15,19,22,23,24,25], the values of ΔG_1/n were calculated for different A-B combinations, and the results were plotted against 1/n (Figure 1a). It can be seen that the experimental points for all systems satisfactorily correspond to the linear dependence; for all cases, Pearson product–moment correlation coefficients were found to be not less than 0.998. Figure 1b shows the approximation of the points for (SrO)_1/nSrTiO₃ by a linear function (4a) in more detail; the residual plot for the regression of ΔG_1/n vs. 1/n is shown in Figure 1b, inset (red line). A pronounced non-random U-shaped pattern suggests a better fit for a non-linear model (5), where B₁’ and B₂′ are empirical fitting parameters. To assess correspondence of the linear (4a)/non-linear quadratic model (5) to the data, a successive F-test was employed to check whether the 1/n² term results in a significant improvement. It shows that at a 0.05 significance level, model (5) is more likely to be correct (F_exp = 1035.9 > F_crit(0.95,1,1) = 18.5). It should be pointed out that in the case of (SrO)_1/nSrTiO₃, predicted values for the non-linear approximation lie within the experimental uncertainty in determining ΔG_1/n and the same regularities are valid for the rest of the A-B cations under consideration (Figure 1a).

ΔG_1/n = ΔG_P + B₁′/n + B₂′/n²

(5)

The existence of dependence (5) gives rise to a number of significant and unanticipated consequences.

• As follows from Figure 1b (inset), the accuracy of the approximation of ΔG_1/n by Equation (4) is higher than the uncertainty of its experimental determination. Thus, Equation (5) can be used to estimate the Gibbs energy of formation of any member of the RP family, which cannot be determined experimentally, and to verify the values obtained experimentally.
• Let us consider the possibility of the disproportionation reaction of an RPn phase into RP(n − 1) and RP(n + 1) phases by Equations (2) and (6):

(AO)_1/nABO₃ = (n − 1)/(2n) (AO)_1/(n−1)ABO₃ + (n + 1)/(2n) (AO)_1/(n+1)ABO₃

(6)

It is easy to show that the change in the Gibbs energy ΔG_r,n of reaction (6) is reduced to expression (7) if correlation (5) is valid. Given that B′₂ > 0 (Figure 1b), this means that for any n > 1, ΔG_r,n is positive. Therefore, there is no inherent instability of PRn phases which was predicted in theoretical studies for n ≥ 3 ([8,9]).

$$∆{\mathrm{G}}_{\mathrm{r},\mathrm{n}}={\displaystyle \frac{{\mathrm{B}}_2^{\prime }}{n^2(n^2-1)}}$$

(7)

3.

The absolute value of the change in the Gibbs free energy of reaction (6) of interconversion of the RP phases is small. In particular, in the case of strontium titanates, it does not exceed 281 J/mol at 1200 K. This is exactly what causes the difficulty of synthesizing the RP phases A_n+1B_nO_3n+1 with a high n value. On the other hand, low values of ΔG_r,n make it possible to influence the relative stability of the RP phases and, accordingly, to shift the equilibrium of their interconversion, using a factor that affects the thermodynamics of compounds with different n to varying extents. Oxygen non-stoichiometry, for instance, can act as such a factor for systems based on transition metals (Co, Ni, Fe). It is known that double perovskites (AA′)MO₃ and the RP phases (A,A′ = rare and alkaline earth metals, M = Co, Ni, Fe), are capable of reversibly adding/eliminating oxygen (8) to form non-stoichiometric compounds depending on the temperature and partial pressure of oxygen.

A_n+1M_nO_3n+1 = A_n+1M_nO_3n+1–δ + δ/2 O₂

(8)

To date, it has not been possible to find systematic experimental data in the literature on the influence of oxygen non-stoichiometry δ on the thermodynamic quantities (enthalpy, entropy, Gibbs energy) of A_n+1M_nO_3n+1 compounds with different n values. Therefore, it is not possible to quantitatively confirm the assumption made. However, a qualitative illustration of the relationship between non-stoichiometry and stability of the RP phase with a certain value of the index n is the observation of the sequential disproportionation of perovskite LaCoO₃ to La₄Co₃O₁₀, La₂CoO₄ and CoO at 1100–1400 K with a decrease in the partial pressure of oxygen from 10⁵ to 10⁻¹⁰ Pa [25]. It has also been suggested that direct decomposition of La₃Ni₂O_7−δ leads to the formation of La₂NiO_4−δ and La₄Ni₃O_10−δ under the process of electrochemical oxygen removal at 1400 K [26]. The formation and development of oxygen and ruthenium vacancies by adjusting the oxygen partial pressure was shown to facilitate a phase transition from SrRuO₃ to Sr₃Ru₂O₇ [27]. An additional indication of the validity of this assumption is the reversible interconversion (9) of the type ABO₃+A₂BO₄↔A₃B₂O₇ for the system 0.3 Gd₂O₃ – 1.2 SrO – CoO, occurring at 1243–1473 K with oxygen pressure variations in the range from 2∙10⁴ to 1 Pa [28].

(Gd_xSr_1−x)CoO_3−δp + (Gd_ySr_1−y)₂CoO_4−δ1 ↔ (Gd_0.2Sr_0.8)₃Co₂O_7−δ2 + (δ₂ − δ₁ − δ_p)/2 O₂

(9)

4.

A further generalization of the approximation (5) can be obtained by reducing the equation to a dimensionless form (10), where B₁ = B₁′/ΔG_P and B₂ = B₂′/ΔG_P.

G_n/P = (ΔG_1/n/ΔG_P − 1) = (ΔG_1/n − ΔG_P)/ΔG_P = B₁/n + B₂/n²

(10)

This transformation allows us to describe the energetics of the RP phases in all A-B-O series based on a minimum number of empirical parameters. Figure 2 shows the experimental data transformed according to Equation (10); the lines are fitted experimental values of (ΔG_1/n/ΔG_P − 1) to the reciprocal index n.

Table 1. Number of (AO)_1/nABO₃ phases (N) included in calculation (fitting), their indexes (n) and estimates of parameter B₁ and B₂, Equation (10).

System	N	df	n	B1	B2
Srn+1TinO3n+1 (1200K)	4	2	1–3,∞	0.34985 ± 0.00046	−0.02873 ± 0.00051
Can+1TinO3n+1 (1200K)	3	1	2,3,∞	0.17996 ± 1 × 10−16 *	−0.15427 ± 2 × 10−16 *
Lan+1ConO3n+1 (1200K)	3	1	1,3,∞	0.90318 ± 6 × 10−16 *	−0.13991 ± 7 × 10−16 *
Srn+1TinO3n+1 (298K)	4	2	1–3,∞	0.37907 ± 0.00085	−0.01143 ± 0.00097
Srn+1ZrnO3n+1 (298K)	4	2	1–3,∞	0.34264 ± 0.00028	−0.0056 ± 0.0003
Lan+1NinO3n+1 (298K)	4	2	1–3,∞	0.69749 ± 0.0015	−0.01505 ± 0.00169

* extremely small value of standard error results from insufficient number of experimental points (N = 3) and degrees of freedom (df = 1).

shows the estimates of parameter B₁ and B₂ which were obtained using a non-linear least squares estimation procedure. The dimensionless form (10) of Equation (5) represents the most complete generalization of the implicit dependence of the Gibbs energy of formation of a A_n+1B_nO_3n+1 member of the RP family on the index n. Transformation of the experimental points using Equation (10) results in a set of closely spaced curves passing through the origin, individual differences associated with the nature of the A and B cations being much less pronounced (Figure 1a and Figure 2). Moreover, a comparison of the curves at 298 K and 1200 K for the Sr_n+1Ti_nO_3n+1 system (Figure 2; ■ and Δ points) demonstrates that the influence of temperature is effectively neutralized.

It is noteworthy that no significant correlation was found between B₁ and B₂ coefficients at p < 0.05 (Pearson’s correlation coefficient 0.0736, Spearman rank order correlation ρ = 0.2619). We can therefore treat the terms B₁/n and B₂/n² as independent and ascribe to them a reasonable, simple physical interpretation. It seems that there are at least three main factors that could potentially influence the stability of the AO(ABO₃)_n phase structure. These factors could be (i) the nature of the parent perovskite ABO₃, (ii) the dimensionality of the perovskite block n, and (iii) the modulation of the crystal structure by AO layers. Thus, the term G_n/P = (ΔG_1/n/ΔG_P–1) in Equation (10) is related to the thermodynamics of the parent perovskite ABO₃ and corresponds to the relative excess of the molar Gibbs energy of the RP phase (AO)_1/nABO₃ compared to that of ABO₃. The term B₁/n reflects the additive properties of the structure and accounts for the progressive decrease in the contribution of the insertion of the AO layer to the Gibbs energy with increasing dimension n of the perovskite block. The negative non–linear term B₂/n² corresponds to the contribution to the stabilization of the structure. This can be attributed to the interaction of the layers of rock salt with each other.

The present paper does not aim to define the exact physical meaning of B coefficients. Instead, it introduces the conception of performing thermodynamic analysis using the reciprocal value of n. At this stage, the coefficients B₁ and B₂ are empirical fitting parameters and are determined from experimental data. Further development of Equation (10) may involve a theoretical or semi–empirical evaluation of these coefficients, which will allow the calculation of the Gibbs free energy values. Figure 3 clearly shows the obvious correlation between the coefficient B₁ and Goldschmidt’s tolerance factor (TFper), which provides evidence that such assessments are possible. This dependence is quite logical, as TFper is an indicator for the stability and distortion of perovskite structures, and, as mentioned above, the coefficient B₁ can be related to the dimensionality of the perovskite block. The relationship of the B₂ coefficient and specific physical and chemical characteristics is more problematic. However, the theoretical calculation of these parameters is beyond the scope of this paper and is the subject of future research.

4. Conclusions

The proposed dependence (5) permits for the calculation of the molar Gibbs energies of the formation of the Ruddlesden–Popper phases (AO)_1/nABO₃, wherein the reciprocal value of the index n is employed as a variable. The calculation is based on the molar Gibbs energy of formation of the parent perovskite ABO₃ and incorporates two supplementary empirical parameters.

This dependence (5) can be transformed into a dimensionless Equation (10), which, when employed, translates the experimental points into a set of analogous curves that pass through the origin. In this case, the first parameter (B₁) is responsible for the slope of the curve, and the second (B₂) determines its curvature. While the exact physical interpretation of B₁ and B₂ parameters is currently unclear, it is reasonable to assume that the first one is associated with the description of the additive changes in Gibbs energy with an increase in the perovskite block dimension, while the second (non–linear) relates to stabilization of the layered structure.

We believe that the approach associated with the description of the thermodynamic properties of RP phases in the space of reciprocal values of the index n is promising for the search of the most general regularities, which will yield an extensive set of possibilities for thermodynamic prediction within and across layered perovskite–like RP phases, and possibly can be extended to other layered perovskites (Aurivillius and Dion–Jacobson phases).