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Phase Relations in Ternary Systems in the Subsolidus Region: Methods To Formulate Solid Solution Equations and To Find Particular Compositions Victor E. Alvarez-Montaño,*,† Mario H. Farías,† Francisco Brown,‡ Iliana C. Muñoz-Palma,§ Fernando Cubillas,‡ and Felipe F. Castillón-Barraza*,† †

Universidad Nacional Autónoma de México - Centro de Nanociencias y Nanotecnología, km. 107 Carretera Tijuana-Ensenada, Ensenada, B.C. 22860, México ‡ Departamento de Investigación en Polímeros y Materiales, Universidad de Sonora, Rosales y Luis Encinas s/n col. Centro, Hermosillo, Sonora 83000, México § Departamento de Ciencias Químico-Biológicas, Universidad de Sonora, Rosales y Luis Encinas s/n col. Centro, Hermosillo, Sonora 83000, México S Supporting Information *

ABSTRACT: A good understanding of ternary phase diagrams is required to advance and/or to reproduce experimental research in solid-state and materials chemistry. The aim of this paper is to describe the solutions to problems that appear when studying or determining ternary phase diagrams. A brief description of the principal features shown in phase diagrams of ternary systems in the subsolidus region is included. We present a systematic procedure to obtain specific compositions of particular interest as well as to calculate binary and ternary ratios of compounds inside the Gibbs triangle and a step-bystep methodology to formulate solid solution equations in binary, ternary, or highercomposition compounds. Specific problems and their solutions are presented as a practical guide, linking the learned concepts with their applications in ceramics and solid-state chemistry research. In addition, ready-to-go exercises with worked-out solutions are included for practice. The procedures described herein may be helpful to those interested in phase diagram interpretation, preparation, and analysis. KEYWORDS: Upper-Division Undergraduate, Graduate Education/Research, Physical Chemistry, Problem Solving/Decision Making, Phases/Phase Transitions/Diagrams, Solid State Chemistry, Inorganic Chemistry, Stoichiometry

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INTRODUCTION

On the other hand, the study of different ternary systems (including pure metals, alloys, minerals, and ceramic oxides) has contributed to new fundamental and applied knowledge. Phase diagrams are used to determine the parameters to be employed in the fabrication of a material of interest, to design a material with desired properties (mechanical, physical, and/or chemical), to solve problems related to materials performance, and to develop new materials.14 One of the best-known examples of phase diagram application is in the production of Portland cement, where specific proportions of components like CaCO3, SiO2, Al2O3, Fe2O3, etc. are carefully manipulated, mixed, and heated to get the desired properties of their products.15 Another example is in the steel industry, which is currently very advanced thanks to the existence of hundreds of alloys, and part of its success is due to knowledge of the Fe−C equilibrium diagram, which was established many years ago (1895−1899).11

Research on phase diagrams of pure and combined substances has been very successful in science, technology, and engineering for many years. One- and two-component systems are the most common in teaching literature.1−5 Liquid ternary systems have been described in this Journal.6−9 Solid ternary systems have been previously tackled; however, only partial attention has been given to the subsolidus region.10−12 The subsolidus region can be understood as any set of thermodynamic equilibrium states in which no gas or liquid phases exist. Great emphasis has been placed on the interpretation and the general use of ternary phase diagrams;10−14 however, no detailed attention has been focused on the specific calculations involved, despite their growing importance in materials research in the past decades. Generally, this knowledge is transmitted from mentors to students in few research groups. For this reason, it is very common to see students in trouble when they are facing ternary system descriptions of advanced solid-state phase relations, and it is even more difficult for them to get involved in experimental studies in this field. © XXXX American Chemical Society and Division of Chemical Education, Inc.

Received: March 30, 2017 Revised: May 29, 2017

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DOI: 10.1021/acs.jchemed.7b00237 J. Chem. Educ. XXXX, XXX, XXX−XXX

Journal of Chemical Education

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two compositions). If we fix the temperature and pressure in a subsolidus region, the phase rule becomes P + F = C. According to this, the maximum number of phases in mutual equilibrium that can be present in a subsolidus three-component system is three.13 Procedures to interpret ternary phase diagrams are welldescribed in the literature.10,18,40,41 If all of the components have been specified, it is sufficient to establish the proportions of two of them, and the third component can be obtained by subtraction. For example, if a system has the components A, B, and C and consists of 30% A and 50% B, the proportion of C is clearly 20%. In order to optimize an experimental procedure, the diagram with planned composition coordinates should be proposed previous to the experiment. Also, to avoid mistakes, it is important to define the order of the components in the ternary diagram scheme and in compositions. In research publications, each author decides this order. Throughout this paper we will be using the order of compositions A−B−C, starting from the superior corner of the phase diagram with triangular shape and following a clockwise direction. In all of these representations, we are using compositions in a mole ratio (or mole percent), as is preferred in practice for inorganic systems.18 All of the systems in the present article represent isothermal and isobaric sections in the subsolidus region. Lines joining stoichiometric binary/ternary phases from different faces of the triangle are called pseudobinary sections.14 The procedures here described, although focused on ternary phase diagrams of the subsolidus region, can be useful for other ternary systems under isobaric or isothermal conditions, such as finding coordinates in liquid systems. The present work should not be considered a complete guide to solve phase diagram determination problems. We are interested in communicating procedures and personal methods used by the authors in phase diagram interpretation and determination. These procedures may be useful for undergraduate and graduate students working in ceramics, solid-state chemistry, and materials science.

Considerable time and effort is devoted during ternary phase diagram determination. During this process, a series of calculations, mainly related to chemical compositions, are required. An example is finding the specific ratio of three components inside their ternary diagram, which corresponds to a predetermined molar ratio of the final species in a pseudobinary section. Another practical problem is to locate the ternary ratio corresponding to the exact intersection of two pseudobinary sections. Among the most common issues faced by students in this research area is the formulation of solid solution equations for ternary, quaternary, or higher-composition chemical compounds. These and other problems will be addressed systematically in the succeeding sections of this article, which describe step-by-step procedures that are useful to interpret and determine ternary phase diagrams.

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IMPORTANCE OF SOLID SOLUTIONS For many years, the study of phase relations in ternary systems at high temperatures has been a convenient procedure to find and synthesize new materials for technological applications. Of the many possible ternary systems that can be investigated, only a small fraction have been studied. Thus, it is expected that several thousand new compounds may be found.16 From the solid-state physics point of view, it is highly important to study ternary and higher-composition systems, since many physical properties and phenomena can be discovered. A solid solution is a perfect example of how much the physical properties can be modified with a subtle change in composition while retaining the same crystal structure.17,18 Thus, solid solutions exhibit remarkable uses in the design of new materials with specific properties. An excellent description of solid solutions is given in the classic book by Professor A. R. West.18 Solid-solution-based materials show interesting mechanical,19,20 chemical,21,22 magnetic,23,24 electrical,25−28 optical,29,30 and thermal properties.31,32 Very commonly, students have problems in establishing equations for solid solutions in ternary, quaternary, or highercomposition systems. One reason for this is the lack of systematic procedures to be followed. Complications arise since diverse parameters must be considered, such as the oxidation states of components, crystal chemistry coordination, ionic radii, etc.33 Nevertheless, in a general case, it is only necessary to consider how the chemical species change in the different compositions. In this work, we describe a basic general method to formulate solid solution equations based on the compositional changes of binary, ternary, and quaternary hypothetical compounds.

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DESCRIPTION OF BASIC FEATURES EXPRESSED IN TERNARY PHASE DIAGRAMS IN A SUBSOLIDUS REGION Figure 1 depicts a hypothetical isothermal and isobaric section of a ternary phase diagram for the A−B−C system in a subsolidus region. The most common features in a ternary phase diagram are illustrated. Often, scientific reports present ternary phase diagrams without gridlines and numerical scales in order to gain clarity in graph figures. All of the figures in the present article are presented on triangular graph paper in the Supporting Information, including some figures with numerical scales. Different symbols are used to express the presence of one-, two-, or three-phase zones in ternary phase diagrams. In this paper, solid circles represent single-phase compounds. These compositions usually refer to stoichiometric compounds. In Figure 1, the pure components A, B, and C are located at the corners, and the formation of the compounds AB, AB2, CB, and AC is shown in their respective binary systems. One example found in the literature is the work by Grivel and Thyden,42 who describe the ternary system SrO−In2O3−CuO at 900 °C in air, with analogous binary oxides such as Sr2CuO3, In2Cu2O5, SrIn2O4, SrCuO2, etc. at the three sides of the ternary system. Following with the description of Figure 1, now let us focus on the Z3 coordinate, where a single phase is shown. An initial question is the following: What is the formula of the compound

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THE PHASE RULE AND PHASE DIAGRAM INTERPRETATION The phase rule, derived by J. W. Gibbs, should be initially explained, since a phase diagram is a graphic description of this rule.13 Its basic mathematical expression is P + F = C + 2, where C is the number of components, P is the number of stable phases in the system, and F is the number of independent variables (also called degrees of freedom). The deduction of the phase rule and its applications can be found in many textbooks.13,14,34−39 In this equation, C + 2 represents the maximum number of phases that can coexist in equilibrium, since F can never be less than zero.13 Moreover, to define the state of a phase in a three-component system, no more than four variables need to be specified (temperature, pressure, and B



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Continuing with our description, three zones where two phases coexist in equilibrium are shown (see Figure 1). In the A−Z1−AC area, tie lines radiate from AC toward the A(ss) line. In the A−AB−Z1 area, tie lines radiate from AB to A(ss), and compound Z1 coexists in equilibrium with different solid solutions from zone Z2. Each tie line represents a different twophase equilibrium. The compounds at the two ends of each tie line are stable along the line (e.g., the AC−A(ss) zone, the line AC−Z1, and also CB−B and AB−AB2), and along the line only the relative amounts of the resulting phases change. The lever rule can be applied to get compound ratios, as is usually described in binary isobaric phase diagrams. Finally, the seven dark solid triangles (AC−Z1−CAx, Z1−CB−CBx, Z1−Z3−CB, AB−Z3−Z1, AB−AB2−Z3, AB2−B−Z3, and Z3−B−CB) indicate different areas where three phases coexist in equilibrium. The stable phases are always those in the corners of the triangle. Many three-phase areas can be found in the phase diagrams reported by Xue and co-workers,50,51 Shi et al.,52 and Zhan et al.53 An excellent example of applied phase diagrams research, in which almost all of the features described here are present, is the ternary system studied by Harvey et al.,54 and its importance related to physical properties is cited later in the review article by Hoel et al.55

Figure 1. Hypothetical isothermal and isobaric section of a ternary phase diagram for the A−B−C system. Schematized are the most common results, as presented in experimental studies in the subsolidus region. Bold solid lines (e.g., A−Z1), dark areas (e.g., Z2), and solid circles represent one phase; lightface solid lines (e.g., AC−A and AC− Z1) represent tie lines where two phases coexist in equilibrium; and solid triangles (e.g., CB−Z3−Z1 and AB2−B−Z3) indicate areas where three phases coexist in equilibrium.

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DETERMINATION OF BINARY AND TERNARY RATIOS IN PSEUDOBINARY SECTIONS Sometimes it is necessary to study a line of compositions between two existing or nonexisting phases in a ternary system. A nonexisting phase is usually represented in phase diagrams. This means that at the given temperature and pressure, this particular phase, although it is schematized, does not exist. In these studies it is important to know the ratio between the chemical ending species. Figure 2 shows a scheme of a ternary phase diagram, a hypothetical A−B−C system. In the A−C and C−B binary systems, there exist analogous phases (AC and CB); in the A−

formed at point Z3? This point is at the coordinate where the C−AB2 section crosses the pseudobinary CB−AB. In this particular case, it can be read from a scaled triangular coordinate diagram,43 and it is exactly AB2C. Since more complicated cases can occur, we provide a systematic guide for the solution of related questions. Similar compounds inside the ternary diagram are the new ternary phases Sr2CuWO6 and Sr8CuW3O18, which exist in the SrO−WO3−CuO system at 800 °C in air.44 The bold solid line (A−Z1) represents an extended solid solution of the crystal structure of A (A(ss)) in the ternary system. In this representation, A is able to dissolve both B and C along the line A−CB until composition Z1. Some important questions arise now, such as the following: What is the formula of the compound formed at point Z1? A solid solution exists along A−Z1; what equation does it have? The ability to answer these and related questions should be acquired after reading this paper. The dark area (Z2) represents a single-phase zone. Experimental articles in the literature describe ternary areas of solubility, for example, solid solutions in ternary metallic alloys45 and in mineral systems.46,47 The limits in such solid solution areas are determined by systematic preparation of samples along specific composition lines using X-ray diffraction techniques and the parametric method.48 A classic textbook where the basic principles of the parametric method are described is that by Professor B. D. Cullity.49 In the case of area Z2 (Figure 1), the crystal structure of C dissolves both A and B. Some questions arise, such as the following: In the pseudobinaries C−AC and C−CB, what are the formulas for the solid solutions? These are CAx and CBx, respectively. We will discuss a few more complicated systems and elaborate the questions that might arise in a systematic way.

Figure 2. Coordinate scheme used during ternary phase diagram determination for the hypothetical A−B−C system. P1, P2, and P3 are predefined compositions that can be obtained by the methods described in this paper. T1 is in the ratio AB:C = 1:1, and T2 is in the ratio AB:C = 3:1. C



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A PROCEDURE TO FORMULATE SOLID SOLUTION EQUATIONS FOR BINARY OR PSEUDOBINARY SYSTEMS Figure 4 represents only a limited section of the ternary system A−B−C, also called a subsystem. If a solid solution occurs between species A and AB in a subsystem, like the one in Figure 4a, the process to write the equation representing the solid solution can be described as follows. First, the point where the solid solution starts is selected. This point is usually where a stoichiometric species is formed; in this example, it is point S1 (composition and crystal structure of AB). Then it is assumed that this solution is extended to the other end (A, pure composition). From where the solid solution begins, the subscript of species A changes from A1 to A1, or A1→1 (i.e., no change), whereas chemical species B changes from B1 to B0, or B1→0, which means that there is no more B at the theoretical end of this solid solution. Finally, the difference between the final and initial states is calculated for each species. Three different situations can result, as follows: (1) If the difference is negative, these species will have a subscript n−mx, where n is the initial subscript, m is the difference between the final and initial subscripts, and x is the composition variable. (2) If the difference is positive, these species will have a subscript in the form n+mx, with the same meaning for all parameters. (3) Finally, if the difference is zero, these species will keep the same subscript (n) and, as m = 0, will have no subscript involving the variable x; such is the case of A in this example. In summary, for each element, the final form of the subscript will be n±mx, depending on the value of m. Applying this procedure to the solid solution in Figure 4a would give the formula AB1−x. It can be noticed that the value of x in the limit of this solid solution is not 1. What is the limit value for x? In general, the value of x is an experimental result obtained by applying the parametric method.49 However, we can follow the next procedure to solve this problem. If it is assumed that the solid solution reaches the point S2 (which is in the ratio A:B = 7:3), then first we would write an equation with the stoichiometry A:B = 7:3, i.e., 7A + 3B → A7B3; since the A7B3:AB1−x ratio should be fulfilled, we can write the equation 3/7 = 1 − x. Solving, we get x = 4/7. AB3/7 is the limit compound, and the complete equation for the solid solution can be written as AB1−x (0 ≤ x ≤ 4/7). In Figure 4b, an intermediate compound between A and AB in the same solid solution occurs, for instance, a compound with the composition A:B = 4:1, whose formula is A4B. Then in order to get the solid solution equation, it should be considered that the solid solution occurs from AB up to A4B, rather than to A. Applying the above-described procedure, now A changes from A1 to A4, or A1→4, and B goes from B1 to B1, or B1→1, which means no changes for B. The resulting n and m values for A would be 1 and 3, respectively. Then the equation representing this solution would be A1+3xB. If this solid solution, like the last problem, comes up to point S2, following the above procedure we can obtain the value x = 4/9, and the complete solid solution equation can be written as A1+3xB (0 ≤ x ≤ 4/9).

B binary system, there are two phases, named AB and AB2. Finding the correspondence between the binary and ternary ratios in the pseudobinary section AB−AC can be easy. For example, the composition for the ratio AB:AC = 1:1 is in the middle of this line, i.e. starting from A, B, and C in the ratio A:B:C = 2:1:1, or 50% A, 25% B, and 25% C (point P1 in Figure 2), which can be observed directly from the gridlines of a scaled triangular graph paper. If the ending species of this line are others, a different result could be obtained. Therefore, it is necessary to use a procedure that can be applied to a wide range of situations. Let us begin with the study of the AC−AB2 section (Figure 2) to find the ratio AC:AB2 = 1:1. First, we select the mole ratio of interest and write a stoichiometric reaction between the ending compounds: AC + AB2 → A2CB2. Then the product of the assumed reaction is selected, and a new equation is proposed to obtain this product from the three components of the ternary system: A + B + C → A2CB2. Finally, the equation is balanced: 2A + 2B + C → A2CB2. The ternary molar ratio is equivalent to taking the stoichiometric coefficients of the last balanced equation, i.e., A:B:C = 2:2:1, or 40% A, 40% B, and 20% C (point P2 in Figure 2). Following this procedure, the ratio AC:AB2 = 1:2 will be found at point P3 (A:B:C = 3:4:1), the ratio AB:C = 1:1 is in the ternary ratio A:B:C = 1:1:1 (point T1), and the ratio AB:C = 3:1 has A:B:C = 3:3:1, or 42.85% A, 42.85% B, and 14.28% C (point T2). Now we consider Figure 3, in which the hypothetical ternary system AO−BO2−C2O3 is represented. A(II), B(IV), and

Figure 3. Coordinate scheme used during ternary phase diagram determination for the hypothetical AO−BO2−C2O3 system, with elements A, B, and C combined with oxygen (O). Q1, Q2, Q3, are predefined compositions that can be obtained by the methods described in this paper. U1 is in the ratio ABO3:C2O3 = 1:1, and U2 is in the ratio ABO3:C2O3 = 3:1.

C(III) are hypothetical cations combined with oxygen (O). If we consider the same questions applied to the system in Figure 2 and follow the described procedure, then the ratio AC2O4:ABO3 = 1:1 is at point Q1 (AO:BO2:C2O3 = 2:1:1), the ratio AC2O4:AB2O5 = 1:1 at point Q2 (AO:BO2:C2O3 = 2:2:1), and the ratio AC2O4:AB2O5 = 1:2 at point Q3 (AO:BO2:C2O3 = 3:4:1). Finally, the ratio ABO3:C2O3 = 1:1 corresponds to the coordinate U1 and the ratio ABO3:C2O3 = 3:1 to U2.

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SOLID SOLUTION EQUATIONS INSIDE THE TERNARY SYSTEM One important thing to remember is that in experimental practice, when a total solid solution occurs between two compounds, they must have the same crystal structure; however, if two compounds exhibit the same crystal structure, D



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Figure 4. (a) Hypothetical isothermal and isobaric section of a ternary phase diagram for the A−AB−AC subsystem, with a solid solution from S1 (composition AB) up to S2 (composition A:B = 7:3), without any intermediate compound in the AB−A section. The formula for this solid solution is AB1−x (0 ≤ x ≤ 4/7). (b) Hypothetical isothermal and isobaric section of a ternary phase diagram for the A−AB−AC subsystem with a solid solution from S1 up to S2 with an intermediate compound A4B in the A−AB section. The formula for this solid solution is A1+3xB (0 ≤ x ≤ 4/9).

Figure 5. (a) Hypothetical isothermal and isobaric section of a ternary phase diagram for the A−B−C system, with a total solid solution between AB and CB. The formula for this solid solution is AxBC1−x (0 ≤ x ≤ 1). A solid solution exists from coordinate X up to Y, with formula A2BC2−x (0 ≤ x ≤ 1). If this solid solution is extended from X until Z, the solid solution equation will change to A2+2xB1+xC2−x (0 ≤ x ≤ 1). (b) Hypothetical isothermal and isobaric section of a ternary phase diagram for the AO−BO2−C2O3 system, with a total solid solution between ABO3 and C2BO5. The formula for this solid solution is A1−xBC2xO3+2x (0 ≤ x ≤ 1).

they will not necessarily form a total solid solution.18 Suppose we have the chemical species AB and CB (Figure 5a) with the same crystal structure and there exists a total solid solution between them. The composition along the solid solution varies in A and C, but B remains constant. Applying the same procedure described in the last section, and starting at AB, we have that A changes from A1 to A0, or A1→0, C changes from C0 to C1, or C0→1, and for B there is no change in the subscript (B1→1). From this, n = 1 and m = −1 for A, n = 1 and m = 0 for B, and n = 0 and m = 1 for C. Thus, the formula will be A1−xBCx (0 ≤ x ≤ 1). It should be noted that the equation proposed inversely, AxBC1−x (0 ≤ x ≤ 1) would be correct too, with its use depending only on the selection where the solid solution starts. An example of a total solid solution between two compounds similar to that described in this paper is the SnO2−TiO2−Y2O3 system studied by Sun et al.56 In that report, a total solid solution exists between Y2Sn2O7 and Y2Ti2O7 with the formula Y2(Ti1−xSnx)2O7 (or Y2Ti2−2xSn2xO7) for 0 ≤ x ≤ 1 and the pyrochlore crystal structure. In the same experimental system there are partial solid solutions Ti1−xSnxO2 (0 ≤ x ≤ 0.22) and Ti1−xSnxO2 (0.68 ≤ x ≤ 1), corresponding to the TiO2-rich and SnO2-rich zones, respectively. Let us consider the solid solution that exists in Figure 5a from point X to point Y. The coordinates of point X are A:B:C = 2:1:2, hence, the compound A2BC2 is formed. Coordinates of point Y are A:B:C = 2:1:1, so A2BC is formed. To obtain the

solid solution equation, we proceed as described before. The subscripts of A and B do not change between points X and Y, while C starts with 2 and ends with 1 as the subscript. Thus, the equation can be written as A2BC2−x (0 ≤ x ≤ 1). If this solid solution would end at point Z, with coordinates A:B:C = 4:2:1, then the final compound would be A4B2C, and the solid solution equation would change to A2+2xB1+xC2−x (0 ≤ x ≤ 1). Another experimental example is in the Ga2O3−In2O3−SnO2 system, and its formula is Ga3−xIn5+xSn2O16 (0.2 ≤ x ≤ 1.6).57 Many species forming solid solutions are ternary or quaternary compounds. Let us consider a ternary system of hypothetical oxides, as in Figure 5b, where a solid solution exists between species ABO3 and C2BO5. Following the procedure described above and starting from ABO3, we see that A1 changes to A0 (or A1→0), C goes from C0 to C2 (or C0→2), B remains constant (B1→1), and O changes from O3 to O 5 (or O3→5 ). The corresponding formula would be A1−xC2xBO3+2x (0 ≤ x ≤ 1). The formula for the same solid solution but starting with C2BO5 should be C2−2xAxBO5−2x. Whenever an equation is obtained as above, it is useful to perform a series of proofs to verify that the proposed equation is true for all cases. Now let us focus on Figure 6. Suppose the chemical species ABO3 and C2BO5 have different crystal structures and both form partial solid solutions in their pseudobinary section. Application of the described procedures shows that E



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Figure 6. Hypothetical isothermal and isobaric section of a ternary phase diagram for the AO−BO2−C2O3 system, with two partial solid solutions between ABO3 and C2BO5. The formula for the solid solution starting at ABO3 is A1−xBC2xO3+2x (0 ≤ x ≤ 0.3), and the solid solution starting at C2BO5 is AxBC2−2xO5−2x (0 ≤ x ≤ 0.5).

Figure 7. Coordinate scheme used during ternary phase diagram determination for the hypothetical A−B−C system with a point of interest, R, at the intersection of the pseudobinaries C−A2B and AC−“C2B”. N1 and N2 are specific compositions that can be obtained by the methods described in this paper.

A1−xBC2xO3+2x (0 ≤ x ≤ 0.3) is the equation for the solid solution starting at ABO3 and with limit in composition A:B:C = 7:10:3. Also, AxBC2−2xO5−2x (0 ≤ x ≤ 0.5) is the formula for the solid solution starting at C2BO5 and finishing with composition A:B:C = 1:2:1. Cai et al.58 reported a solid solution between Co3BO5 and Co2InBO5 with formula Co3−xInxBO5 (0 < x ≤ 1), which is similar to the hypothetical cases described here. A good exercise could be to find the equations for the solid solutions in Figure 6 from point X up to point Y (A2BC4−2xO10−3x) and from point X up to point Z (A2+2xB1+xC4−2xO10+x). Other exercises for practice can be found in the Supporting Information.

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ratio, or 12.5% A, 25.0% B, and 62.5% C) and N2 (A:B:C = 1:2:1 ratio, or 25% A, 50% B, and 25% C) can be obtained following the same procedure.

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FINAL QUESTIONS RELATED TO COORDINATE P3 FROM FIGURE 2 Finally, three interesting questions about point P3 in Figure 2 can be exposed: (i) What is the A:B:C ratio at this coordinate? (ii) What is the AC:AB2 molar ratio? (iii) What is the AB:CB ratio? At this moment, we have the tools to answer this kind of question. To answer question (i), we write a solid solution equation in the pseudobinary system AC−AB2. This solid solution is AB2xC1−x. The point of interest (P3) is also on the line AB−CB with a different hypothetical solid solution equation, A1−xBCx. On this line, the ratio A1−xCx:B = 1:1 must be satisfied. Thus, we can say that the ratio AC1−x:B2x should also be 1:1; therefore, 1 + 1 − x = 2x, and x must be 2/ 3. Substituting this value into AB2xC1−x, we get AB4/3C1/3, and applying the same procedures used in this paper, we find that the A:B:C coordinates of point P3 are 37.5:50.0:12.5. Questions (ii) and (iii) are easy to answer. Since point P3 corresponds to AB4/3C1/3, starting from AC and AB2 the AC:AB2 ratio must be 1:2, and starting from AB and CB the ratio must be AB:CB = 3:1.

DETERMINATION OF THE COMPOSITION WHERE TWO LINES INTERSECT IN A TERNARY SYSTEM

Solid Solutions To Find Coordinates

Suppose it is necessary to know the composition of each starting compound to prepare a mixture corresponding to point R in Figure 7. This point is the only one where the lines C− A2B and AC−“C2B” cross (the quotation marks represent a nonexistent phase, or an unstable compound). How is this precise composition obtained as a function of the three components? The following procedure can be applied for the same or similar problems. First, we write the equation for a hypothetical solid solution whose line passes through point R. For this example, we select the pseudobinary AC−“C2B”. The equation will be A1−xBxC1+x (0 ≤ x ≤ 1). It should be noticed that the composition at point R is on the line C−A2B. This line has the characteristic that the relationship between A and B is always 2 to 1. This serves to generate a simple set of equations that establishes 1 − x = A (quantity in mole of A) and x = B (quantity in mole of B). Since we know that A = 2B should be satisfied (the molar amount of A must be twice that of B), we must have 1 − x = 2x. Solving this equation, we end up with x = 1/3. Substituting x = 1/3 into the equation for the solid solution results in the formula A2/3B1/3C4/3, which can be obtained from the molar ratio A:B:C = 2:1:4, or 28.57% A, 14.28% B, and 57.14% C. The coordinates N1 (A:B:C = 1:2:5

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CONCLUSIONS In the present article we have discussed some problems faced during the determination, preparation, and description of ternary phase diagrams in the subsolidus region. We have introduced procedures to facilitate the necessary calculations carried out previous to any experimental phase diagram determination research. A systematic technique to establish solid solution equations in binary, ternary, and highercomposition compounds has been presented. Also, a step-bystep method for obtaining specific composition coordinates of interest in a ternary system has been proposed. Many exercises F



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have been solved from the graphs in this article, and other exercises to find ratios and coordinates and to formulate solid solution equations are included in the Supporting Information and would be convenient for the reader to put into practice. Ternary phase diagram determination is an exciting field, although most of the times it is a challenge. The authors of this paper want to make available to the next generation of ceramists and solid-state chemists the tools for research on the synthesis of materials from literature phase diagrams as well as for obtaining a correct interpretation of them. This paper is helpful for anyone interested in interpreting and analyzing phase diagrams.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available on the ACS Publications website at DOI: 10.1021/acs.jchemed.7b00237. Figures in this article shown on triangular graph paper (PDF, DOCX) Exercises to find ratios and coordinates (PDF, DOCX) Figures for exercises to find ratios and coordinates (PDF) Exercises to formulate solid solution equations (PDF, DOCX) Figures for exercises to formulate solid solution equations (PDF)

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AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected], [email protected]. *E-mail: ff[email protected]. ORCID

Victor E. Alvarez-Montaño: 0000-0003-4159-8558 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors express their gratitude to Mariá Isabel Pérez Montfort and Jonathan Guerrero for their useful comments on the English redaction. One of us (V.E.A.M.) gratefully acknowledges DGAPA-UNAM for a postdoctoral fellowship at Centro de Nanociencias y Nanotecnologiá (CNyN-UNAM).

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REFERENCES

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