A qualitative look at the thermodynamics of ternary phase diagrams. Elyse Waham

Transcription

1 Technical Report 2 National Science Foundation Grant DMR Metals and Metallic Nanostructures Program A qualitative look at the thermodynamics of ternary phase diagrams By Elyse Waham May 2016 Hyperfine Interactions Group Department of Physics and Astronomy Washington State University Pullman, Washington USA

2 This report is the Senior Thesis of Elyse Waham, submitted in partial fulfillment of requirements for the degree of BS in Physics, Washington State University, Pullman. May 2016

3 A qualitative look at the thermodynamics of ternary phase diagrams Elyse Waham Senior Thesis for Submission to Professor Dickinson Of Washington State University In partial fulfillment of the requirements for the degree of Bachelor of Science Professor Gary S. Collins, Advisor Department of Physics and Astronomy Washington State University May 2016

4 ABSTRACT A qualitative look at the the rmodynamics of ternary phase diagrams Elyse Waham Department of Physics and Astronomy Washington State University Bachelor of Science In 2014 Collins research group embarked on experiments using Perturbed Angular Correlation (PAC) spectroscopy to measure interactions between solute atoms in ternary alloys. PAC requires a probe atom, 111 In, making the systems strictly quaternary, but the probe is present only at the extremely low mole fraction of 10 11, and has dissolved in almost every phase studied in Collins s lab in the past. Ideal for measurements of interactions between solute atoms are ternary phases A m B n (X) containing about 1 at.% of X, dissolved on either or both sublattices of the A B phase. However, many attempts to make such samples did not yield the desired result. Instead, two alternative outcomes were commonly observed: the solute atom did not dissolve in the host alloy and, instead, retained its elemental crystal structure, or the solute atom reacted with one or other of the two elements, forming an A X or B X phase, and thereby leaving a depleted A B phase. In this thesis, Gibbs free energy diagrams for binary and ternary systems are examined to give insight into the reasons for these outcomes. By looking at the geometry of inferior tangent lines and planes below Gibbs free energy curves and surfaces, one can understand the observed outcomes.

5 CONTENTS 1. Introduction 1 2. Thermodynamics of alloys 4 i. Binary Phase Diagrams ii. Termina l and intermediate phases a. Dilute solute of 1 at.% solute in a terminal phase iii. Ternar y Phase Diagrams a. Complete solubility 13 b. Terminal phases and no intermediate phase 14 c. Terminal phases and a unique intermediate phase 16 d. Binary A 2 B intermetallic with or without solute incorporation 18 e. Competing reactions Conclusion Acknowledgements References 23

6 1 Introduction Starting in fall 2014, the Collins research group began studies of solute atoms in ordered intermetallic compounds. The goal of this research is to measure interaction energies between pairs of solute atoms. Using the method of Perturbed Angular Correlation (PAC) of gamma rays, one can detect quadrupole interactions at nuclei of probe atoms when a solute atom is within the first atomic shell surrounding the probe. The probe that was used in these studies is the radioisotope 111 In, which decays into 111 Cd with a halflife of 2.8 days. By measuring the temperature dependence of the site fraction of probesolute close pairs, one can determine the interaction energy. If the interaction is attractive, then the site fraction will be high at low temperature and decrease with increasing temperature. If the interaction is repulsive, then the site fraction will be low at low temperature and increase with increasing temperature. The mole fraction of 111 In required for a PAC measurement is extremely dilute, of the order of At such dilution 111 In has been found to be soluble in every host (out of hundreds) studied so far by Collins s group and therefore acts as a ``neutral observer`` that does not disturb the host alloy. A solute mole fraction of ½ to 1 at.% is ideal for measurements of either attractive or repulsive interactions. Consider a binary alloy such as A 2 B having the C15, or cubic Laves, structure, shown in the diagram. The system A 2 B(+ 111 In) is strictly ternary, but in practice is effectively binary, with the C15 structure preserved and with a site preference for the 111 In probe on either sublattices A or B. When ~1 at.% of solute element X is added, to make A 2 B(X)( 111 In) the system strictly is quaternary, but effectively ternary.

7 2 The desired alloy for the measurements is one that retains the A 2 B crystal structure, with both solute X and probe 111 In dissolved on one or other, of the sublattices. To date, about 30 samples have been made by arc melting under argon, with three differen t outcomes: C15 crystal structure A 2 B: A atoms dark, B atoms light. 1. The A 2 B structure was maintained, with X dissolved on either the A or B sublattice according to its site preference. This is the desired outcome. 2. Element X did not get incorporated in the A 2 B structure at all, but retained the crystal structure of pure X. There was no mixing of X with A or B. 3. Element X reacted with one of the two host constituents, making an A X or B X phase and thereby depleting one of the host components so that the desired A 2 B phase could not form. Another outcome, not observed, has the three elements A B X forming a true ternary phase having a unique structure not existing in any of the A B, A X or B X systems. Nearly all measurements made in Collins group conducted over a one year period starting October 2014 led to undesired outcomes (2) or (3). Professor Collins had naively expected that outcome (1) would be routinely observed, based on the ease with which 111 In is incorporated into any system. However, this was not the case. To try to identify better ternary systems, a subscription was purchased for access to the alloy phase diagram database by American Society for Metals (ASM) International in August Ternary

8 3 phase diagrams were examined and suggested that solute could be dissolved by up to a few percent in various binary compounds. Since then, a number of successful measurements have been made or are in progress. These include systems based on binary GdAl 2, which has the C15 structure and on which an extensive study was made earlier of site preferences for the 111 In probe [1]. In that study, it was shown that the 111 In probe tends to occupy the Gd sublattice in Gdpoor alloys and the Al sublattice in the Al poor alloys. The width of the GdAl 2 phase field has not been measured, however the phase appears as a line compound on the binary phase diagrams and probably therefore, has a width of the order 0.3 at.% close to the 33.3 at.% stoichiometric composition. Also, independent of composition the site fraction of the 111 In probes on the Al sites was found to increase with temperature [2]. Studies on GdAl 2 (A g), GdAl 2 (Au) and GdAl 2 (Cu) have worked well, with outcome (1) being observed. As an instance with outcome (2), an attempt to dissolve Sn in NiAl failed. A signal was only observed for the 111 In probe in the Sn metal. This means not only that there was a lack of solution of Sn in the NiAl phase, but that there was strong segregation of the 111 In probe to the Sn phase during solidification of the sample after arc melting. Outcome (3) was observed during an attempt to make a Ni 2 In 3 (Sn) system. No evidence of incorporation of Sn was observed in any phase, but the measured spectrum showed signals for Ni 2 In 3 as well as for the more In rich Ni 3 In 7 phase. It was therefore concluded that Sn had alloyed with Ni to produce a Ni Sn phase leaving behind a Ni In alloy depleted of Ni that formed a mixture of the two indium phases. These undesired, but common, outcomes led to a search for explanations. In this thesis I explore the thermodynamics relating to these systems. First I review the basic

9 4 thermodynamics of binary systems and the origin of binary phase diagrams. I then extend concepts for binary systems to ternary systems and their phase diagrams. Thermodynamics of alloys To examine the free energies of ternary systems, one must understanding the fundamental concepts related to their construction. General information about the Gibbs free energies of phases, or for short free energies, and the theory of solutions as it affects the free energies is required to understand phase diagrams. How phase diagrams are constructed will be discussed in turn for binary and ternary diagrams. Phase diagrams are based on underlying free energy curves and surfaces of the various equilibrium phases. The Gibbs free energy is the useful work done in a process at constant temperature and pressure [3]. It is a minimum in thermal equilibrium. It is defined as G = H TS, (1) in which G is Gibbs free energy, H is the enthalpy, T is the Temperature and S is the entropy [4]. For a particular phase, G, H and S are all functions of the composition of the phase.. Experiments were carried out in vacuum (a constant pressure) and at various constant temperatures. The entropy term gets larger in magnitude as the temperature increases. Therefore, for our purposes, one can consider the free energy of each phase as functions of T, P and the mole number N. G (T,P,N) = U + PV TS, (2)

10 5 in which U is the internal energy, P is the pressure, V is the volume, T is the temperature and S is the entropy. The Gibbs free energy of a phase varies as a function of its composition. This is related to the introduction of point defects for nonstoichiometric alloys and bond energies between the atoms. Next is considered why and how separation into multiple phases occurs in a sample. Phase Mixture The reason for separation into multiple phases is related to minimization of the free energy. A phase is a physically homogeneous state of matter with a specific composition and bonding [5]. All phase transformations occur to reduce the total free energy of the sample. The total free energy of a sample is (3) where X i is the mole fraction of each element in a phase and G i is its corresponding free energy per mole. This equation shows how the separate free energies in a mixture of phases combine. Binary Phase Diagrams In order to derive a phase diagram from free energy diagrams we examine different important aspects of free energy curves. Shown in figure 1 are free energy curves for two terminal phases, and, of a binary A B alloy, at a specific temperature, plotted as a function of the mole fraction of element B. For a given sample composition, the equilibrium free energy will be the one on the lowest curve. If the average mole fraction is

11 6 in the domain on the left, then there will be only α phase. If the mole fraction is in the domain to the right, there will be only β phase. However, if the average mole fraction is in the middle domain, then the free energy will be lowered below the free energies of either the α or β phases if a mixture of the two phases is made. In that case, the average free energy will fall on the inferior tangent line connecting the two curves, as shown by the dashed line. The two phases will have compositions given by the vertical dashed lines, with volume fractions determined by a lever rule to be consistent with the average composition. Fig 1: Free energy curves of the terminal and phases of an A B alloy at a particular temperature. The equilibrium free energy is given by the family of tangent lines that are inferior to the two curves. A special inferior tangent curve is shown by the dashed line, joining minima of the and phase free energy curves. Within the range of composition between the tangent points, the free energy is lowered by having a mixture of and phases. From reference [4]. To construct a phase diagram one needs to consider how free energy curves change as a function of temperature. Figure 2 shows how free energy diagrams at different temperatures determine a phase diagram. On these graphs there are three pure phases, two solid phases α and β and a liquid phase l and also three phase mixtures l+α, l+β and α+β. It should also be noted that the energy curves move differently as a function of temperature. The higher the temperature the lower the liquid phase, l will be relative to the solid phase curves, α and β, owing to the higher entropy of a liquid. To reduce the free

12 7 energy of a system, the liquid phase is favored at high temperature because of its higher entropy, which makes intuitive sense. Fig 2: Free energy curves for a two component system at different temperatures produce different tangent lines therefore, various phases. This information is used to construct a phase diagram with temperature at the y axis. The lines on the phase diagram that represent one temperature are called isothermal lines From reference [4]. The phase rule was developed by Gibbs to see how phases were affected by variables of the system. The degrees of freedom, f, of a system to include temperature, pressure and concentration. Gibbs proved a phase rule given by the relation f = c p +1, (4) in which p is the number of coexisting phases and c is the number of component elements. This equation shows, for example, that only a limited number of phases are available to form. This version of the phase rule is commonly used by experimentalists when the pressure is held constant; the full version, in which the pressure can also vary, has

13 8 numerical constant 2. The following table show how this relationship plays out for unitary, binary and ternary systems with varying degrees of freedom. C P F Free variables Example T Liquid water (T fixed) Unary eutectic, all variables fixed such as Boiling point of liquid water T, X A NiCu solid solution T (X A, X B fixed) Binary alloy, two phase field (T fixed, X s fixed) B inary eutectic point, all variables are fixed T, X A, X B ( X C = 1 X A X B ) Mixture of water, alcohol, acetic acid T, X A α + β, two phase mixture with X dissolved such as Ni 2 Al 3 + Sn T Ni 2 In 3 + Ni 3 In 7 + NiSn (fixed components) (T and all X s fixed) Ternary eutectic, all variables fixed Fig 3: Shows all possible number of phases that can be present with given variables fixed or determined. Notice that pressure, is always fixed. Examples of each kind of system are given. As is seen in figure 3 there are many ways to make a system. With this foundation on phase diagrams and free energy curves we shall examine some examples of systems.

14 9 Terminal phases and intermediate phases First binary systems will be examine. It is easier to understand the how free energy curves affect the phase diagrams and will help to develop an intuitive perspective for the following complexity of ternary systems. A few examples will be presented as they relate to ternary systems that were tested by Collins s group. Fig 4: Labeling of a phase diagram to demonstrate terminology. The diagram shows two terminal phases, and, and one intermediate phase. At high temperature is a liquid phase. The six other fields are two phase mixtures of and liquid phases. Figure 4 labels phases that can be present in a binary A B system. The x axis shows the mole fraction of B; the y axis shows the variation in temperature. The far right and left phases α and γ are terminal phases because they have the crystal structure of their respective elements. The β phase is an intermediate phase having a crystal structure A 2 B 3. This figure will be a useful reference as we look at the different possibilities of phase formation.

15 10 a. A dilute solute, X of 1 at.% in a terminal phase A binary system with 1 at.% of a third element can result in either full incorporation or none in a single phase. The phase diagrams of these systems will show how this concept relates to their free energies and resulting geometry. Fig 5: A binary phase diagram with two terminal phases α and β. Here it shows that solubility of B in A is roughly 10 at.% below temperature T1.. From reference [4]. The phase diagram in figure 5 displays a terminal solubility of B in single phase A of roughly 10 at.%. This then can be related back to the free energy curves in a diagram such as figure 1. This is a consequence of the inferior tangent lines of phase curves below temperature T 1 contacting the α phase curve at compositions of about 10 at.% of B. For other A B diagrams, the terminal phase fields may have much smaller widths, even much less than 1 at.%. Representation of a phase diagram with an intermediate phase was shown in figure 4. It shows that there is an intermediate phase over a range of composition for this binary

16 11 system. If one considers free energy curves, then this means that phase β has inferior tangent lines joining the α and γ terminal phase curves. This geometry results in two terminal phases, one intermediate phase, and mixed phases present at a range of temperatures and different compositions in each phase. Compared with the simpler Fig. 5, Fig. 4 illustrates the main features of a binary system. Ternary phases The Gibbs free energy curves are now surfaces in three dimensions, as a function of the compositions of elements A, B, C. An artist`s impression of these surfaces is shown in figure 6. The analogs of inferior tangent lines for free energy curves in a binary system are inferior tangent planes. These surfaces and planes can be hard to visualize but an attempt to make the geometry understandable will be made and is the main goal of this thesis. In figure 6, an inferior tangent plane is drawn that touches the bottom of three free energy surfaces. These planes and their relative geometry are essential to understand phase formation in ternary systems.

17 12 Fig 6: 3 D perspective representation of free energy surfaces of a ternary alloy at a particular temperature. An inferior tangent plane is drawn that touches the three surfaces. From free energy surfaces and inferior tangent planes such as in figure 6, a phase diagram of the ternary system can be determined. Figure 7 shows the layout of a ternay phase diagram, with the tree pure elements located in the corners and varying in at.%, usually. The diagonal lines are tie lines used to determine composition of all three components inside the diagram. Consider in turn four different scenarios.

18 13 Fig 7: Layout of a ternary phase diagram. Pure elements are in the corners and tie lines are used to determine composition in the diagram. a. Complete solubility Fig 8: A free energy curve that demonstrates complete solubility of components A, B and C at a temperature. The free energy surface in figure 8 shows a system that allows complete solubility of all three components over all compositions. This free energy surface would result in a uniform ternary phase diagram such as figure 7, with no phase boundaries. This means there is only one phase possible for all compositions. In practice, alloy systems rarely or

19 14 never exhibit this behavior, however an example would include liquid mixtures of water (H 2 O), acetic acid (CH 3 COOH) and alcohol (C 2 H 6 O). b. Terminal phases with no intermediate phase Fig 9: Gibbs free energy surfaces for a system in which only the crystal structures of elements A, B and C are observed. Demonstrates what the geometry of the free energy curves could be in order to construct the necessary tangent planes from tangent lines between curves to get the resulting phases in the phase diagram of figure 11. The free energy diagram in figure 9, shows three surfaces for three terminal phases α, β and γ. This would result in one special inferior tangent plane that determines the composition range of the α+β+γ mixture. Other planes determine ranges of the three terminal phases: α, β and γ and the mixed phases: α+γ, α+β, β+γ. The complete set of inferior planes results in the ternary phase diagram shown in figure 10. Looking at figure 10 the resulting phase regions are where the tangent lines intersect surfaces and reflected in the phase boundaries as in binary systems. This geometry then creates all possible mixed phase regions present in figure 10.

20 Fig 10: Phase diagram for the system shown in figure 9. There are three terminal phases shown in the corners: α, β and γ. Then there are three two phase regions along the edges and one three phase region in the middle of the diagram. These phases can be seen to show up geometrically when examining the tangent surfaces of the free energy curves. From reference [4]. 15

21 16 c. Terminal phases with a unique intermediate phase Fig 11: A free energy diagram with three terminal phases: α, β and γ and one intermediate phase, δ. The minimum point of the surface for phase lies below an inferior plane touching the and surfaces. However, only tangent planes inferior to all free energy surfaces determine which equilibrium phases are present. The free energy surface diagram in figure 11 is gaining complexity. There are now three terminal phases, three different mixed phases and an intermediate phase. Looking at the geometry of the surfaces it is noticed that the middle, δ phase surface is particularly stable and therefore, many inferior tangent planes will touch this surface. However, only the most inferior tangent planes govern phase formation. The resulting phase diagram at the temperature of Fig. 11 may look as shown in figure 12. The inferior planes create the

22 17 terminal, mixed and intermediate phases in this diagram 12. While figure 12 was drawn for a liquid intermediate phase, the same diagram can apply for solid phase. Fig 12: A phase diagram showing three terminal phases: α, β and γ, a unique liquid phase l, and mixtures of phases: α+l, β+l, γ+l. From reference [4]. The geometry of the phase surfaces at a given temperature and the resulting tangent planes determine the phase regions that will be present in a ternary phase diagram.

23 18 d. Binary A 2 B intermetallic with or without solute incorporation Fig 13: This free energy surface diagram has three terminal phases: α, β, γ and an intermediate binary A2B phase: δ. The geometry of these curves may or may not result in either incorporation of C up to 1 at.% into the A2B phase or not, depending on the detailed shapes of the A2B and phase free energy surfaces. The energy surfaces in figure 13 result in a more complex phase diagram than the examples examined previously. X marks the desired composition of the sample. Here there are two possible outcomes: solubility exceeded or solubility not exceeded based on the geometry of the A 2 B and γ phase surfaces. Depending on the detailed shape of the surfaces, solute C may or may not have solubility in the A 2 B phase. The drawn lines on the bottom and side of the ternary phase diagram outline a pseudo binary phase diagram between A 2 B and C (γ) phases. Figure 14 demonstrate two possible geometric scenarios for figure 13. In figure 14a, the geometry results in solubility less than 1 at.% of C due to

24 19 the close intersection of the tangent line with the pure A 2 B axis resulting only in phase A 2 B and terminal phase of C. Instead, in Figure 14b the inferior tangent line touches the A 2 B curve such that a composition of several atomic percent of solute C could dissolve in the A 2 B phase. Fig 14: Shows two pseudo binary curve diagrams that demonstrate possible geometric outcomes that affect the solubility of C in the system A2B. (Left): The inferior tangent line intersects the A2B curve very close to the axis,resulting in little solubility of C. (Right): the tangent line intersects the free energy curve at a distance that would allow solubility of several at.% of C in the phase. Fig 15: Ternary phase diagram of nickel, tin and aluminum at 298 K. An attempt was made to make a sample of Ni 2 Al 3 containing 1 at.% Sn. According to the diagram, it should be possible to dissolve several percent, but no solution of Sn was observed. Instead, a signal was observed for pure Sn metal, with all the 111 In probe having segregated to that phase. From reference [6].

25 20 Figure 15 show the phase diagram for Ni Al Sn at 25 o C. An attempt was made to prepare Ni 2 Al 3 phase with ~1 at.% Sn dissolved. The phase diagram indicates that one should be able to dissolve up to about 2 at.% Sn in Ni 2 Al 3 (see small blue area close to Ni Al axis near 60 at.% Al), but none dissolved at temperatures in the range C. Instead, the only signal observed was for 111 In in Sn metal, from which it is concluded that the Ni 2 Al 3 phase was produced, but that all 111 In segregated very strongly to the Sn metal phase. This is consistent with the white color sliver region, indicating presence of twophases, extending from Ni 2 Al 3 phase to Sn metal, except that the maximum solubility of Sn in Ni 2 Al 3 appears to be much less than 1 at.%. This is an example of outcome (2). Fig. 16: Phase diagram for Gd Al Ag. The diagram appears to show that one can alloy up to ~10 at.% Ag in intermetallic GdAl2 (see slanting blue line extending from GdAl2 parallel to the Ag Al axis). Measurements showed solution of 1 2 at.% Ag, consistent with the diagram. From reference [7]. Fig. 16 shows the phase diagram for Gd Al Ag at 497 o C. The diagram appears to show that one can alloy up to ~10 at.% Ag in intermetallic GdAl 2 (see slanting blue line

26 21 extending from GdAl 2 parallel to the Ag Al axis). Measurements accordingly showed solution of 1 2 at.% Ag. This was desired outcome (1). e. Competing reactions Competing reactions may prevent the desired phase from forming. Figure 17 shows schematically the existence of a number of intermediate phases, and the equilibrium phases for a particular composition will be governed as usual by the inferior tangent plane construction procedure. Fig 17: A free energy diagram that has three terminal phases: α, β, γ and phases at A3B, A2B and AC. More phase surfaces results in more tangent planes and therefore a more complex ternary phase diagram.

27 22 Fig 18: A ternary phase diagram of Tin, Nickel and Indium that demonstrates the ideas represented in figure 17. An attempt was made to dissolve 1 at.% Sn in Ni 2 In 3. While the diagram shows that this should be possible, measurements showed only signals from the phases Ni 2 In 3 and the more In deficient Ni 3 In 7, with no indication that any Sn had dissolved in those phases. The implication is that Sn had reacted with Ni, perhaps forming the phase Ni 3 Sn 4, thereby depleting the amount of Ni needed to constitute the Ni 2 In 3 phase. From reference [8]. An attempt was made to form Ni 2 In 3 with 1 at.% Sn dissolved. Figure 18 shows the phase diagram for Ni In Sn, which appears to indicate that one can replace 10% of In in Ni 2 In 3 with Sn (see horizontal blue line extending from Ni 2 In 3 parallel to the In Sn axis. ). However, what was observed were signals only from phases Ni 2 In 3 and Ni 3 In 7, indicating depletion of Ni, and without obvious solution of Sn. The interpretation is that Sn reacted with Ni to form a phase such as Ni 3 Sn 4, leaving an insufficient amount of Ni to combine with In to make the desired Ni 2 In 3 phase. This is an example of outcome (3).

28 23 Conclusion This study demonstrates how the principles for generating ternary phase diagrams follow from the simpler principles for binary phase diagrams and can help to interpret the results of experiment. In principle, one can calculate free energy surfaces. This would require an understanding of bond energies between atoms in all the relevant phases, including crystal structures and local atomic environments, which is a formidable task. Acknowledgements I would like to thank Professor Gary Collins for his constant encouragement, advice, guidance and patience throughout the research and writing process. This work was supported in part by the National Science Foundation under grant DMR (Metals Program). References [1] Zacate, Matthew O., Collins, Gary S., Temperature and composition driven changes in site occupation of solutes in Gd1+3xAl2 3x, Physical Review B69, , [2] Kasal, Krystal. Investigations of Ternary Alloys using Perturbed Angular Correlations in Technical Report. Pullman: Washington State University, Cf. [3] Kittel, Charles and Kroemer, Herbert. Thermal Physics. New York: W.H. Freeman and Company, [4] Prince, A.. Alloy Phase Equilibria. Netherlands: Elsevier Publishing Company, [5] Campbell, F.C.. Phase Diagrams : Understanding the Basics. Materials Park, OH: A S M International, ProQuest ebrary.

29 24 [6] Prince, A. Aluminum Nickel Tin ternary, isothermal section. ASM International Alloy Phase Diagram Database [7]Stelmakhovych B.M., Zhak O.V., Bilas N.R., and Kuz ma. Silver Aluminum Gadolinium ternary, isothermal section, ASM International Alloy Phase Diagram Database, [8] Bhargava, M. K., and Schubert, K.. Indium Nickel Tin ternary, isothermal section. ASM International Alloy Phase Diagram Database