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Introduction to Alloy Phase Diagrams

Introduction to Alloy Phase Diagrams

ASM Handbook, Volume 3: Diagrams Copyright © 1992 ASM International® Hugh Baker, editor, p 1.1-1.29 All rights reserved. DOI: 10.1361/asmhba0001123 www.asminternational.org

Section 1 Introduction to Alloy Phase Diagrams

Hugh Baker, Editor

ALLOY PHASE DIAGRAMS are useful to exhaust system). Phase diagrams also are con- terms "phase" and "phase field" is seldom made, metallurgists, engineers, and materials sulted when attacking service problems such as and all materials having the same phase name are scientists in four major areas: (1) development of pitting and intergranular , hydrogen referred to as the same phase. new alloys for specific applications, (2) fabrica- damage, and hot corrosion. Equilibrium. There are three types of equili- tion of these alloys into useful configurations, (3) In a majority of the more widely used commer- bria: stable, metastable, and unstable. These three design and control of heat treatment procedures cial alloys, the allowable composition range en- conditions are illustrated in a mechanical sense in for specific alloys that will produce the required compasses only a small portion of the relevant Fig. l. Stable equilibrium exists when the object mechanical, physical, and chemical properties, . The nonequilibrium conditions is in its lowest energy condition; metastable equi- and (4) solving problems that arise with specific that are usually encountered inpractice, however, librium exists when additional energy must be alloys in their performance in commercial appli- necessitate the knowledge of a much greater por- introduced before the object can reach true stabil- cations, thus improving product predictability. In tion of the diagram. Therefore, a thorough under- ity; unstable equilibrium exists when no addi- all these areas, the use of phase diagrams allows standing of alloy phase diagrams in general and tional energy is needed before reaching meta- research, development, and production to be done their practical use will prove to be of great help stability or stability. Although true stable equilib- more efficiently and cost effectively. to a metallurgist expected to solve problems in rium conditions seldom exist in objects, the In the area of alloy development, phase dia- any of the areas mentioned above. study of equilibrium systems is extremely valu- grams have proved invaluable for tailoring exist- able, because it constitutes a limiting condition ing alloys to avoid overdesign in current applica- from which actual conditions can be estimated. tions, designing improved alloys for existing and Common Terms Polymorphism.The structure of elements new applications, designing special alloys for and compounds under stable equilibrium condi- special applications, and developing alternative Before the subject of alloy phase diagrams is tions is crystalline, and the structure of alloys or alloys with substitute alloying elements discussed in detail, several of the commonly used each is unique. Some elements and compounds, to replace those containing scarce, expensive, terms will be discussed. however, are polymorphic (multishaped); that is, hazardous, or "critical" alloying elements. Appli- Phases. All materials exist in gaseous, , or their structure transforms from one crystal struc- cation of alloy phase diagrams in processing in- solid form (usually referred to as a phase), de- ture to another with changes in and cludes their use to select proper parameters for pending on the conditions of state. State variables , each unique structure constituting a dis- working ingots, blooms, and billets, fmding include composition, temperature, pressure, mag- tinctively separate phase. The term (ex- causes and cures for microporosity and cracks in netic field, electrostatic field, gravitational field, isting in another form) is usually used to describe castings and welds, controlling heat and so on. The term "phase" refers to that region polymorphic changes in chemical elements. treating to prevent damage caused by incipient of space occupied by a physically homogeneous of and alloys is discussed , and developing new processing technol- . However, there are two uses of the term: in a later section of this Introduction; the allo- ogy. the strict sense normally used by physical scien- tropic transformations of the elements are listed In the area of performance, phase diagrams give tists and the somewhat looser sense normally used in the Appendix to this Volume. an indication of which phases are thermodynami- by materials engineers. Metastable Phases. Under some conditions, cally stable in an alloy and can be expected to be In the strictest sense, homogeneous means that metastable crystal structures can form instead of present over a long time when the part is subjected the physical properties throughout the region of stable structures. Rapid is a common to a particular temperature (e.g., in an automotive space occupied by the phase are absolutely iden- method of producing metastable structures, but tical, and any change in condition of state, no some (such as Fe3C, or"") are produced how small, will result in a different phase. at moderately slow cooling rates. With extremely For example, a sample of solid metal with an rapid freezing, even thermodynamically unstable apparently homogeneous appearance is not truly structures (such as "glasses") a single-phase material, because the pressure con- can be produced. dition varies in the sample due to its own weight Systems. A physical system consists of a sub- in the gravitational field. stance (or a group of substances) that is isolated In a phase diagram, however, each single-phase from its surroundings, a concept used to facilitate field (phase fields are discussed in a following study of the effects of conditions of state. "Iso- Ill section) is usually given a single label, and engi- lated" means that there is no interchange of mass neers often find it convenient to use this label to between the substance and its surroundings: The (a) (b) (c) refer to all the materials lying within the field, substances in alloy systems, for example, might regardless of how much the physical properties of be two metals, such as and ; a metal the materials continuously change from one part and a nonmetal, such as and ; a metal Fig. I Mechanical equilibria: (a) Stable. (b) Metas- of the field to another. This means that in en- and an compound, such as iron and table. (c) Unstable gineering practice, the distinction between the cementite; or several metals, such as aluminum, 1*2/Introduction to Alloy Phase Diagrams , and . These substances constitute the components comprising the system and should not be confused with the various quid) phases found within the system. A system, how- 4 ever, also can consist of a single component, such Solid 2 as an element or compound. Liquid / Phase Diagrams. In order to record and visual- ize the results of studying the effects of state variables on a system, diagrams were devised to show the relationships between the various I Ot phases that appear within the system under equi- librium conditions. As such, the diagrams are variously called constitutional diagrams, equilib- f rium diagrams, or phase diagrams. A single- Temperature component phase diagram can be simply a one- or two-dimensional plot showing the phase changes in the substance as temperature and/or Fig. 2 Schematic pressure-temperature phase diagram pressure change. Most diagrams, however, are two- or three-dimensional plots describing the phase relationships in systems made up of two or more components, and these usually contain beled fields. Stable equilibrium between any two phases occurs along their mutual boundary, and fields (areas) consisting of mixed-phase fields, as Composition B well as single-phase fields. The plotting schemes invariant equilibrium among all three phases oc- A in common use are described in greater detail in curs at the so-called , O, where the three boundaries intersect. This point also is Fig. 3 Schematic binary phase diagram showing mis- subsequent sections of this Introduction. cibility in both the liquid and solid states System Components. Phase diagrams and the called an invariant point because, at that location systems they describe are often classified and on the diagram, all externally controllable factors named for the number (in Latin) of components are fixed (no degrees of freedom). At this point, in the system: all three states (phases) are in equilibrium, but any The Gibbs applies to all states of changes in pressure and/or temperature will cause matter (solid, liquid, and gaseous), but when the Number of Name of one or two of the states (phases) to disappear. effect of pressure is constant, the rule reduces to: components system or diagrum Univariant Equilibrium. The phase rule says One Unary that stable equilibrium between two phases in a f=c-p+ 1 Two Binary unary system allows one degree of freedom (f= Three Temary 1 - 2 + 2). This condition, called univariant The stable equilibria for binary systems are sum- Four Quatemary equilibrium or monovariant equilibrium, is illus- Five Quinary marized as follows: Six Sexinary trated as lines 1, 2, and 3 separating the single- phase fields in Fig. 2. Either pressure or tempera- Seven Septenary Number of Number of Degrees of Eight Octanary ture may be freely selected, but not both. Once a components ph~es freedom Equilibrium Nine Nonary pressure is selected, there is only one temperature Ten Decinary that will satisfy equilibrium conditions, and con- 2 3 0 Invariant 2 1 Univariant versely. The three curves that issue from the triple 2 l 2 Bivariant point are called triple curves: line 1, representing Phase Rule. Thephase rule, first announced by the reaction between the solid and the gas phases, J. Willard Gibbs in 1876, relates the physical state is the sublimation curve; line 2 is the melting of a to the number of constituents in the curve; and line 3 is the vaporization curve. The system and to its conditions. It was also Gibbs vaporization curve ends at point 4, called a criti- who first called each homogeneous region in a cal point, where the physical distinction between Miscible , Many systems are comprised of components having the same crystal structure, system by the term "phase." When pressure and the liquid and gas phases disappears. temperature are the state variables, the rule can be Bivariant Equilibrium. If both the pressure and the components of some of these systems are written as follows: and temperature in a unary system are freely and completely miscible (completely soluble in each arbitrarily selected, the situation corresponds to other) in the solid form, thus forming a continu- f=c-p+2 having two degrees of freedom, and the phase rule ous . When this occurs in a binary says that only one phase can exit in stable equi- system, the phase diagram usually has the general appearance of that shown in Fig. 3. The diagram where f is the number of independent variables librium (p = 1 - 2 + 2). This situation is called (called degrees of freedom), c is the number of bivariant equilibrium. consists of two single-phase fields separated by a two-phase field. The boundary between the liquid components, and p is the number of stable phases field and the two-phase field in Fig. 3 is called the in the system. ; that between the two-phase field and the Binary Diagrams solid field is the solidus. In general, a liquidus is Unary Diagrams the locus of points in a phase diagram repre- senting the at which alloys of the Invariant Equilibrium. According to the phase If the system being considered comprises two various compositions of the system begin to rule, three phases can exist in stable equilibrium components, a composition axis must be added to freeze on cooling or finish melting on heating; a only at a single point on a unary diagram (f= 1 - the PT plot, requiring construction of a three- solidus is the locus of points representing the 3 + 2 = 0). This limitation is illustrated as point O dimensional graph. Most metallurgical problems, temperatures at which the various alloys finish in the hypothetical unary pressure-temperature however, are concerned only with a fixed pressure freezing on cooling or begin melting on heating. (PT) diagram shown in Fig. 2. In this diagram, the of one atmosphere, and the graph reduces to a The phases in equilibrium across the two-phase three states (or phases)--solid, liquid, and gas--- two-dimensional plot of temperature and compo- field in Fig. 3 (the liquid and solid ) are are represented by the three correspondingly la- sition (TX diagram). called conjugate phases. Introduction to Alloy Phase Diagrams/I-3

point P, an invariant point that occurred by - L L cidence. (Three-phase equilibrium is discussed in the following section.) Then, if this two-phase field in the solid region is even further widened so that the solvus lines no longer touch at the invariant point, the diagram passes through a series of configurations, finally taking on the more familiar shape shown in Fig. 6(b). The three-phase reaction that takes place at the invari- ro ant point E, where a liquid phase freezes into a 0E mixture of two solid phases, is called a eutectic I-- I reaction (from the Greek word for "easily melted"). The alloy that corresponds to the eutec- tic composition is called a eutectic alloy. An alloy having a composition to the left of the eutectic a b B point is called a hypoeutectic alloy (from the A Composition Greek word for "less than"); an alloy to the right (a) Composition is a hypereutectic alloy (meaning "greater than"). Schematic binary phase diagram with a mini- In the described above, the two Fig. 5 mum in the liquidus and a gap in the components of the system have the same crystal L solid state structure. This, and other factors, allows complete miscibilitybetween them. Eutectic systems, how- ever, also can be formed by two components If the solidus and liquidus meet tangentially at having different crystal structures. When this oc- some point, a maximum or minimum is produced curs, the liquidus and solidus curves (and their in the two-phase field, splitting it into two por- extensions into the two-phase field) for each of

:3 tions as shown in Fig. 4. It also is possible to have the terminal phases (see Fig. 6c) resemble those a gap in miscibility in a single-phase field; this is for the situation of complete miscibility between shown in Fig. 5. Point Tc, above which phases tXl system components shown in Fig. 3. a)E I.-- and ~2 become indistinguishable, is a critical Three-Phase Equilibrium. Reactions involv- point similar to point 4 in Fig. 2. Lines a-Tc and ing three conjugate phases are not limited to the b-Tc, called solvus lines, indicate the limits of eutectic reaction. For example, upon cooling, a solubilityof component B in A and Ain B, respec- single solid phase can change into a mixture of tively. The configurations of these and all other two new solid phases or, conversely, two solid phase diagrams depend on the thermodynamics phases can react to form a single new phase. A B Composition of the system, as discussed later in this Introduc- These and the other various types of invariant (b) tion. reactions observed in binary systems are listed in Eutectic Reactions. If the two-phase field in the Table 1 and illustrated in Fig. 7 and 8. solid region of Fig. 5 is expanded so that it touches Intermediate Phases. In addition to the three Fig. 4 Schematic binary phase diagrams with solid- the solidus at some point, as shown in Fig. 6(a), solid terminal-phase fields, (~, [~, and e, the dia- state miscibility where the liquidus shows a complete miscibility of the components is lost. gram in Fig. 7 displays five other solid-phase maximum (a) and a minimum (b) Instead of a single solid phase, the diagram now fields, 7, 5, fi', ~q, and ~, at intermediate composi- shows two separate solid terminal phases, which tions. Such phases are called intermediate are in three-phase equilibrium with the liquid at phases. Many intermediate phases, such as those

L

L+~ L+I3 / %SS,,, ~ .~ ~ % I

%• E % •p "t'... _~ i r" .s -S ~ ~+~ c~+B / A Composition B A Composition B A Composition (a) (b) (c)

Fig° 6 Schematic binary phase diagrams with invariant points. (a) Hypothetical diagram of the type shown in Fig. 5, except that the in the solid touches the solidus curve at invariant point P; an actual diagram of this type probably does not exist. (b) and (c) Typical eutectic diagrams for components having the same crystal structure (b) and components having different crystal structures (c); the eutectic (invariant) points are labeled E. The dashed lines in (b) and (c) are metastable extensions of the stable-equilibria lines. 1-4/Introduction to Alloy Phase Diagrams

Critical L Allotropl¢ ¢onsruent ~ruent I \ = +L \ / L, + ,/ --\ .o~=.o~o - \ "/; I \ V / \ 7 + Ls \/L + L2 t ::7 /" ......

7/Co~ngr~7 + 71 71 Monoteetold

Eutectoid ~ + 6'

...... Y_o_lLmo_r p_h_z c_ ......

~+~

A Composition B A Composition B

Hypothetical binary phase diagram showing intermediate phases formed by Fig, 8 Hypothetical binary phase diagram showing three intermetallic line com- Fig. 7 various invariant reactions and a polymorphic transformation pounds and four melting reactions illustrated in Fig. 7, have fairly wide ranges of dimensions becomes more compticated. One op- diagram, reading values from them is difficult. homogeneity. However, many others have very tion is to add a third composition dimension to the Therefore, ternary systems are often represented limited or no significant homogeneity range. base, forming a solid diagram having binary dia- by views of the binary diagrams that comprise the When an intermediate phase of limited (or no) grams as its vertical sides. This can be represented faces and two-dimensional projections of the homogeneity range is located at or near a specific as a modified isometric projection, such as shown liquidus and solidus surfaces, along with a series ratio of component elements that reflects the nor- in Fig. 9. Here, boundaries of single-phase fields of two-dimensional horizontal sections (iso- mal positioning of the component in the (liquidus, solidus, and solvus lines in the binary therms) and vertical sections (isopleths) through crystal structure of the phase, it is often called a diagrams) become surfaces; single- and two- the solid diagram. compound (or line compound). When the compo- phase areas become volumes; three-phase lines Vertical sections are often taken through one nents of the system are metallic, such an interme- become volumes; and four-phase points, while comer (one component) and a congruently melt- diate phase is often called an intermetallic com- not shown in Fig. 9, can exist as an invariant ing binary compound that appears on the opposite pound. (Intermetallic compounds should not be plane. The composition of a binary eutectic liq- face; when such a plot can be read like any other confused with chemical compounds, where the uid, which is a point in a two-component system, true binary diagram, it is called a quasibinary type of bonding is different from that in becomes a line in a ternary diagram, as shown in section. One possibility is illustrated by line 1-2 and where the ratio has chemical significance.) Fig. 9. in the isothermal section shown in Fig. 10. A Three intermetallic compounds (with four types Although three-dimensional projections can be vertical section between a congruently melting of melting reactions) are shown in Fig. 8. helpful in understanding the relationships in a binary compound on one face and one on a dif- In the hypothetical diagram shown in Fig. 8, an alloy of composition AB will freeze and melt isothermally, without the liquid or solid phases undergoing changes in composition; such a phase change is ailed congruent. All other reactions are Liquidus surfaces incongruent;, that is, two phases are formed from L+/~ one phase on melting. Congruent and incongruent \\\ L+a Solidus phase changes, however, are not limited to line Solidus compounds: the terminal component B (pure surface phase e) and the highest-melting composition of intermediate phase 8' in Fig. 7, for example, ~,e::,...... -~ freeze and melt congruently, while 8' and e freeze \ Solvus / and melt incongruently at other compositions. Solvus surface Metastable Equilibrium. In Fig. 6(c), dashed surface lines indicate the portions of the liquidus and solidus lines that disappear into the two-phase solid region. These dashed lines represent valu- able information, as they indicate conditions that would exist under metastable equilibrium, such as might theoretically occur during extremely rapid cooling. Metastable extensions of some sta- ble-equilibria lines also appear in Fig. 2 and 6(b).

Ternary Diagrams A When a third component is added to a binary 9 Ternary phase diagram showing three-phase equilibrium. Source: 56Rhi system, illustrating equilibrium conditions in two Fig, Introduction to Alloy Phase Diagrams/I,5

C C C a:Z r, r, r, ,. A/VV\ r, r~ r, AA/kAAA/k r, r2

A B A xa ~ B A

Fig. 10 Isothermal section of a ternary diagram with Fig. 11 Triangular composition grid for isothermal sec- phase boundaries deleted for simplification tions; x is the composition of each constituent Fig. 12 Liquidus projection of a ternary phase diagram in or percent showing isothermal contour lines. Source: Adapted from 56Rhi ferent face might also form a quasibinary section tion, component A is placed at the bottom left, B (see line 2-3). formed at the intersections of two surfaces. Ar- at the bottom right, and C at the top. The amount All other vertical sections are not true binary rowheads are often added to these lines to indicate of component A is normally indicated from point diagrams, and the term pseudobinary is applied the direction of decreasing temperature in the to them. A common pseudobinary section is one C to point A, the amount of component B from trough. where the percentage of one of the components is point A to point B, and the amount of component held constant (the section is parallel to one of the C from point B to point C. This scale arrangement faces), as shown by line 4-5 in Fig. 10. Another is often modified when only a comer area of the ThermodynamicPrinciples is one where the ratio of two constituents is held diagram is shown. Projected Views. Liquidus, solidus, and solvus constant and the amount of the third is varied from The reactions between components, the phases surfaces by their nature are not isothermal. There- 0 to 100% (line 1-5). formed in a system, and the shape of the resulting fore, equal-temperature (isothermal) contour Isothermal Sections. Composition values in phase diagram can be explained and understood lines are often added to the projected views of the triangular isothermal sections are read from a through knowledge of the principles, laws, and these surfaces to indicate their shape (see Fig. 12). triangular grid consisting of three sets of lines terms of thermodynamics, and how they apply to In addition to (or instead of) contour lines, views parallel to the faces and placed at regular compo- the system. often show lines indicating the temperature sition intervals (see Fig. 11). Normally, the point Internal Energy. The sum of the kinetic energy troughs (also called "valleys" or "grooves") of the triangle is placed at the top of the illustra- (energy of motion) and potential energy (stored energy) of a system is called its internal energy, Table I Invariant reactions E. Internal energy is characterized solely by the Type Reaction state of the system. Closed System. A thermodynamic system that Eutectic Lt undergoes no interchange of mass (material) with (involvesliquid and solid) I.,2> V < S Monoteetic its surroundings is called a closed system. A L closed system, however, can interchange energy S~ > \/ < $2 Eutectic with its surroundings. First Law. The First Law of Thermodynamics, S, as stated by Julius yon Mayer, James Joule, and L > V < S~ Catatectic (Metatectic) Hermann von Helmholtz in the 1840s, states that energy can be neither created nor destroyed. Eutectoid St Therefore, it is called the Law of Conservation of (involvessolid S~ > V < S~ Monotectoid Energy. This law means that the total energy of only) St an isolated system remains constant throughout ,%> \/ < S~ Eutectoid any operations that are carded out on it; that is, for any quantity of energy in one form that disap- pears from the system, an equal quantity of an- Peritectic other form (or other forms) will appear. (involvesliquid Lt > A < I.~ Syntectic and solid) S For example, consider a closed gaseous system to which a quantity of heat energy, ~Q, is added St Peritectic L> A < and a quantity of work, 5W, is extracted. The First ,% Law describes the change in intemal energy, dE, of the system as follows: Peritectoid (involvessolid St > A < ~ Peritectoid only) S~ dE = ~2-aW

In the vast majority of and material applications, the only work done by or on a system is limited to pressure/volume terms. 1-6/Introduction to Alloy Phase Diagrams

Any energy contributions from electric, mag- C = 8Q Second Law. While the First Law establishes netic, or gravitational fields are neglected, except ST the relationship between the heat absorbed and for electrowinning and electrorefining processes the work performed by a system, it places no such as those used in the production of copper, However, if the substance is kept at constant restriction on the source of the heat or its flow aluminum, magnesium, the alkaline metals, and volume (dV = 0): direction. This restriction, however, is set by the the alkaline earths. With the neglect of field ef- Second Law of Thermodynamics, which was ad- fects, the work done by a system can be measured &2 = dE vanced by Rudolf Clausius and WilliamThomson by summing the changes in volume, dV, times (Lord Kelvin). The Second Law states that the each pressure causing a change. Therefore, when and spontaneous flow of heat always is from the field effects are neglected, the First Law can be higher temperature body to the lower tempera- written: ture body. In other words, all naturally occurring processes tend to take place spontaneously in the dE = ~3Q - PdV direction that will to equilibrium. . The Second Law is most conveniently . Thermal energy changes under con- If, instead, the substance is kept at constant pres- stated in terms of entropy, S, another property of stant pressure (again neglecting any field effects) sure (as in many metallurgical systems), state possessed by all systems. Entropy represents are most conveniently expressed in terms of the the energy (per degree of absolute temperature, enthalpy, H, of a system. Enthalpy, also called T) in a system that is not available for work. In heat content, is defined by: terms of entropy, the Second Law states that all natural processes tend to occur only with an H=E+PV increase in entropy, and the direction of the proc- CP= L dT Jp ess always is such as to lead to an increase in Enthalpy, like internal energy, is a function of the entropy. For processes taking place in a system in state of the system, as is the product PV. equilibrium with its surroundings, the change in Heat Capacity. The heat capacity, C, of a sub- and entropy is defined as follows: stance is the amount of heat required to raise its temperature one degree; that is: ~Q dE + PdV dS =_ T- T

r3 i i 2

A Composition A Composition C~ q

(a) (b) (c)

T~ ..... 4 ...... £ r, I I r2

r3

r, (X i i i rs

A Composition B A Composition B A Composition

(d) (e) (f)

Fig. 13 Use of Gibbs energy curves to construct a binary phase diagram that shows miscibility in both the liquid and solid states. Source: Adapted from 66Pri Introduction to Alloy Phase Diagrams/lo7

r~ r2

t t == (D o~ .g_ L9 (.9 I T66 i 7 I I J I I I I I I I I I I I i I 1 I I A Composition B A Composition B A Composition B

(a) (b) (c)

r, rs 1 T 1 t t r~ r3 == L L /3 r,

1~9 ~ 10 i r5 I I I I [ ,' A Composition A Composition B A Composition

(d) (e) (f)

Fig. 14 Use of Gibbs energy curves to construct a binary phase diagram of the eutectic type. Source: Adapted from 68Gor

Third Law. A principle advanced by Theodore independent variables, pressure and absolute tem- equilibrium is altered, a reaction occurs that Richards, Walter Nemst, Max Planck, and others, perature, which are readily controlled experimen- opposes the constraint, i.e., a reaction that par- often called the Third Law of Thermodynamics, tally. If the process is carded out under conditions tially nullifies the alteration. The effect of this states that the entropy of all chemically homoge- of constant pressure and temperature, the change theorem on lines in a phase diagram can be seen neous materials can be taken as zero at absolute in Gibbs energy of a system at equilibrium with in Fig. 2. The slopes of the sublimation line (1) zero temperature (0 K). This principle allows its surroundings (a reversible process) is zero. For and the vaporization line (3) show that the system calculation of the absolute values of entropy of a spontaneous (irreversible) process, the change reacts to increasing pressure by making the denser pure substances solely from heat capacity. in Gibbs energy is less than zero (negative); that phases (solid and liquid) more stable at higher Gibbs Energy. Because both S and V are diffi- is, the Gibbs energy decreases during the process, pressure. The slope of the melting line (2) indi- cult to control experimentally, an additional term, and it reaches a minimum at equilibrium. cates that this hypothetical substance contracts on Gibbs energy, G, is introduced, whereby: freezing. (Note that the boundary between liquid and ordinary , which expands on freez- G =_ E + PV- TS =_ H- TS Features of Phase Diagrams ing, slopes toward the pressure axis.) Clausius-Clapeyron Equation. The theorem and of Le Ch~telier was quantified by Benoit Clapey- The areas (fields) in a phase diagram, and the ron and Rudolf Clausius to give the following position and shapes of the points, lines, surfaces, dG = dE + PdV + VdP - TdS - SdT equation: and intersections in it, are controlled by thermo- dynamic principles and the thermodynamic prop- However, erties of all of the phases that constitute the sys- dP AH tem. dT TAV dE=TdS-PdV Phase-field Rule. The phase-fieM rule speci- fies that at constant temperature and pressure, the Therefore, number of phases in adjacent fields in a multi- where dP/dT is the slope of the univariant lines in component diagram must differ by one. a PT diagram such as those shown in Fig. 2, AV dG = VdP-SdT Theorem of Le Chfitelier. The theorem of is the difference in molar volume of the two Henri Le Ch~telier, which is based on thermody- phases in the reaction, and AH is the difference in Here, the change in Gibbs energy of a system namic principles, states that if a system in equi- molar enthalpy of the two phases (the heat of the undergoing a process is expressed in terms of two librium is subjected to a constraint by which the reaction). 1,8/IntroductionL to Alloy Phase Diagrams L

~ L+(x~ ~ i J"

" %~ Incorrect /// L+ccJ /

@ % I a+13 o¢+15 I G

(a) (b) A B Composition

Fig. 1 ~; Examplesof acceptable intersection angles for Fig. 16 An exampleof a binaryphase diagramwith a minimum in the liquidusthat violatesthe Gibbs-KonovalovRule. ~boundaries of two-phasefields. Source: 56Rhi Source: 81Goo

Solutions. The shapes of liquidus, solidus, and points 1 and 2, where these compositions intersect as diagrams with multiple three-phase reactions, solvus curves (or surfaces) in a phase diagram are temperature T3, is called a tie line. Similar tie lines also can be constructed from appropriate Gibbs determined by the Gibbs energies of the relevant connect the coexisting phases throughout all two- energy curves. Likewise, Gibbs energy surfaces phases. In this instance, the Gibbs energy must phase fields (areas) in binary and (volumes) in and tangential planes can be used to construct include not only the energy of the constituent ternary systems, while tie triangles connect the ternary phase diagrams. components, but also the energy of mixing of coexisting phases throughout all three-phase re- Curves and Intersections. Thermodynamic these components in the phase. gions (volumes) in temary systems. principles also limit the shape of the various Consider, for example, the situation of complete Eutectic phase diagrams, a feature of which is a boundary curves (or surfaces) and their intersec- miscibility shown in Fig. 3. The two phases, field where there is a mixture of two solid phases, tions. For example, see the PT diagram shown in liquid and solid tz, are in stable equilibrium in the also can be constructed from Gibbs energy Fig. 2. The Clausius-Clapeyron equation requires two-phase field between the liquidus and solidus curves. Consider the temperatures indicated on that at the intersection of the triple curves in such lines. The Gibbs energies at various temperatures the phase diagram in Fig. 14(f) and the Gibbs a diagram, the angle between adjacent curves are calculated as a function of composition for energy curves for these temperatures (Fig. 14a-e). should never exceed 180 ° or, alternatively, the ideal liquid solutions and for ideal solid solutions When the points of tangency on the energy curves extension of each triple curve between two phases of the two components, A and B. The result is a are transferred to the diagram, the typical shape must lie within the field of third phase. series of plots similar to those shown in Fig. 13(a) of a eutectic system results. The mixture of solid The angle at which the boundaries of two-phase to (e). c~ and g that forms upon cooling through the fields meet also is limited by thermodynamics. At temperature Th the liquid solution has the eutectic point k has a special microstructure, as That is, the angle must be such that the extension lower Gibbs energy and, therefore, is the more discussed later. of each beyond the point of intersection projects stable phase. At T2, the melting temperature of A, Binary phase diagrams that have three-phase into a two-phase field, rather than a one-phase the liquid and solid are equally stable only at a reactions other than the eutectic reaction, as well field. An example of correct intersections can be composition of pure A. At temperature T3, be- tween the melting temperatures of A and B, the Gibbs energy curves cross. Temperature T4 is the Correct Incorrect melting temperature of B, while T5 is below it. Construction of the two-phase liquid-plus-solid L L field of the phase diagram in Fig. 13(f) is as follows. According to thermodynamic principles, the compositions of the two phases in equilibrium with each other at temperature T3 can be deter- mined by constructing a straight line that is tan- gential to both curves in Fig. 13(c). The points of tangency, 1 and 2, are then transferred to the phase diagram as points on the solidus and liquidus, respectively. This is repeated at sufficient tem- Line peratures to determine the curves accurately. compound If, at some temperature, the Gibbs energy curves I Line for the liquid and the solid tangentially touch at compound some point, the resulting phase diagram will be similar to those shown in Fig. 4(a) and (b), where a maximum or minimum appears in the liquidus and solidus curves. A B A . The two-phase field in Fig. 13(0 ComposlUon Composition consists of a mixture of liquid and solid phases. (a) (b) As stated above, the compositions of the two phases in equilibrium at temperature T3 are C1 glo. 1 7 Schematicdiagrams of binary systemscontaining congruent-meltingcompounds but having no associationof -'b the component atoms in the melt common. The diagram in (a) is consistent with the Gibbs-Konovalov Rule, and C2. The horizontal isothermal line connecting whereasthat in (b) violates the rule. Source: 81Goo Introduction to Alloy Phase Diagrams/11,9

Typical Phase-Rule Violations 10. When two phase boundaries touch at a point, Problems Connected With Phase-Boundary they should touch at an extremity of tempera- Curvatures (See Fig. 18) ture. 11. A touching liquidus and solidus (or any two Although phase rules are not violated, three addi- 1. A two-phase field cannot be extended to become touching boundaries) must have a horizontal tional unusual situations (21, 22, and 23) have also part of a pure-element side of a phase diagram common tangent at the congruent point. In this been included in Fig. 18. In each instance, a more at zero solute. In example 1, the liquidus and the instance, the solidus at the is too subtle thermodynamic problem may exist related to solidus must meet at the melting point of the pure "sharp" and appears to be discontinuous. these situations. Examples are discussed below where element. 12. A local minimum point in the lower part of a several thermodynamically unlikely diagrams are 2. Two liquidus curves must meet at one composi- single-phase field (in this instance, the liquid) considered. The problems with each of these situ- tion at a eutectic temperature. cannot be drawn without an additional boundary ations involve an indicated rapid change of slope of 3. A tie line must terminate at a phase boundary. in contact with it. (In this instance, a horizontal a phase boundary. If such situations are to be associ- 4. Two solvus boundaries (or two liquidus, or two monotectic line is most likely missing.) ated with realistic thermodynamics, the temperature solidus, or a solidus and a solvus) of the same 13. A local maximum point in the lower part of a (or the composition) dependence of the thermody- phase must meet (i.e., intersect) at one compo- single-phase field cannot be drawn without a namic functions of the phase (or phases) involved sition at an invariant temperature. (There should monotectic, monotectoid, syntectic, and sintec- would be expected to show corresponding abrupt and not be two values for a phase boundary toid reaction occurring below it at a lower tem- unrealistic variations in the phase diagram regions at one temperature.) perature. Alternatively, a solidus curve must be where such abrupt phase boundary changes are pro- 5. A phase boundary must extrapolate into a two- drawn to touch the liquidus at point 13. posed, without any clear reason for them. Even the phase field after crossing an invariant point. The 14. A local maximum point in the upper part of a onset of in a phase does not normally validity of this feature, and similar features re- single-phase field cannot be drawn without the cause an abrupt change of slope of the related phase lated to invariant temperatures, is easily demon- phase boundary touching a reversed monotectic, boundaries. The unusual changes of slope considered strated by constructing hypothetical free-energy or a monotectoid, horizontal reaction line coin- here are: diagrams slightly below and slightly above the ciding with the temperature of the maximum. invariant temperature and by observing the rela- When a 14 type of error is introduced, a mini- 21. Two inflection points are located too closely to tive positions of the relevant tangent points to mum may be created on either side (or on one each other. 22. An abrupt reversal of the boundary direction the free energy curves. After intersection, such side) of 14. This introduces an additional error, (more abrupt than a typical smooth "retro- boundaries can also be extrapolated into metas- which is the opposite of 13, but equivalent to 13 grade"). This particular change can occur only table regions of the phase diagram. Such ex- in kind. if there is an accompanying abrupt change in the trapolations are sometimes indicated by dashed 15. A phase boundary cannot terminate within a temperature dependence of the thermodynamic or dotted lines. phase field. (Termination due to lack of data is, 6. Two single-phase fields (Ix and 6) should not be of course, often shown in phase diagrams, but properties of either of the two phases involved in contact along a horizontal line. (An invariant- this is recognized to be artificial.) (in this instance, ~ or ~, in relation to the bound- temperature line separates two-phase fields in 16. The temperature of an invariant reaction in a ary). The boundary turn at 22 is very unlikely to contact.) binary system must he constant. (The reaction be explained by any realistic change in the com- 7. A single-phase field (Ix in this instance) should line must he horizontal.) position dependence of the Gibbs energy func- not be apportioned into subdivisions by a single 17. The liquidus should not have a discontinuous tions. line. Having created a horizontal (invariant) line sharp peak at the melting point of a compound. 23. An abrupt change in the slope of a single-phase at 6 (which is an error), there may be a tempta- (This rule is not applicable if the liquid retains boundary. This particular change can occur only tion to extend this line into a single-phase field, the molecular state of the compound, i.e., in the by an abrupt change in the composition depend- Ix, creating an additional error. situation of an ideal association.) ence of the thermodynamic properties of the 8. In a binary system, an invariant-temperature line 18. The compositions of all three phases at an invari- single phase involved (in this instance, the should involve equilibrium among three phases. ant reaction must be different. phase). It cannot be explained by any possible 9. There should be a two-phase field between two 19. A four-phase equilibrium is not allowed in a abrupt change in the temperature dependence of single-phase fields (Two single phases cannot binary system. the Gibbs energy function of the phase. (If the touch except at a point. However, second-order 20. Two separate phase boundaries that create a temperature dependence were involved, there and higher-order transformations may be excep- two-phase field between two phases in equilib- would also be a change in the boundary of the e tions to this rule.) rium should not cross each other. phase.)

11 17 11 17 L

12 18 13

16 16 2 16

a 4

5 a+7 i 1D 2O

a i , ,

100 Composition B Composition B

:l:|~,,,b 1 ,.,R Hypothetical binary phase diagram showing many typical errors of construc- Fig. 19 Error-free version of the phase diagram shown in Fig. 18. Source: 910kal tion. See the accompanying text for discussion of the errors at points 1 to 23. Source: 910kal lol0/Introduction to Alloy Phase Diagrams seen in Fig. 6(b), where both the solidus and solvus lines are concave. However, the curvature of both boundaries need not be concave; Fig. 15 shows two equally acceptable (but unlikely) in- tersections where convex and concave lines are mixed. Congruent Transformations. The congruent point on a phase diagram is where different C f phases of same composition are in equilibrium. The Gibbs-Konovalov Rule for congruent points, which was developed by Dmitry Konovalov from a thermodynamic expression given by J. Willard Gibbs, states that the slope of phase boundaries at congruent transformationsmust be zero (horizon- tal). Examples of correct slope at the maximum cell / and minimum points on liquidus and solidus curves can be seen in Fig. 4. Often, the inner curve A "/ on a diagram such as that shown in Fig. 4 is erroneously drawn with a sharp inflection (see Fig. 20 a space lattice Fig. 16). Fig. 21 Crystal axes and unit-cell edge lengths. Unit- cell faces are shown, but to avoid confusion A similar common construction error is found they are not labeled. in the diagrams of systems containing con- gruently melting compounds (such as the line compounds shown in Fig. 17) but having little or or molecules in the interior of a crystal is called no association of the component atoms in the melt its crystal structure. The unit cell of a crystal is all three directions, all unequal lengths must be (as with most metallic systems). This type of error the smallest pattern of arrangement that can be stated to completely define the crystal. The same is especially common in partial diagrams, where contained in a parallelepiped, the edges of which is true if all interaxial angles are not equal. When one or more system components is a compound form the a, b, and c axes of the crystal. The defining the unit-cell size of an alloy phase, the instead of an element. (The slope of liquidus and three-dimensionalaggregation of unit cells in the possibility of crystal ordering occurring over sev- solidus curves, however, must not be zero when crystal forms a space lattice, or Bravais lattice eral unit cells should be considered. For example, they terminate at an element, or at a compound (see Fig. 20). in the copper- system, a supedattice forms having complete association in the melt.) Crystal Systems. Seven different crystal sys- that is made up of 10 cells of the disordered lattice, Common Construction Errors. Hiroaki tems are recognized in crystallography, each hav- creating what is called long-period ordering. Okamoto and Thaddeus Massalski have prepared ing a different set of axes, unit-cell edge lengths, Lattice Points. As shown in Fig. 20, a space the hypothetical binary phase shown in Fig. 18, and interaxial angles (see Table 2). Unit-cell edge lattice can be viewed as a three-dimensional net- which exhibits many typical errors of construc- lengths a, b, and c are measured along the corre- work of straight lines. The intersections of the tion (marked as points 1 to 23). The explanation sponding a, b, and c axes (see Fig. 21). Unit-cell lines (called lattice points) represent locations in for each error is given in the accompanying text; faces are identified by capital letters: face A con- space for the same kind of or group of atoms one possible error-free version of the same dia- tains axes b and c, face B contains c and a, and of identical composition, arrangement, and orien- gram is shown in Fig. 19. face C contains a and b. (Faces are not labeled in tation. There are five basic arrangements for lat- Higher-Order Transitions. Fig. 21.) Interaxial angle tx occurs in face A, tice points within a unit cell. The first four are: The transitions at considered in this Introduction up to this point angle [3 in face B, and angle y in face C (see Fig. primitive (simple), having lattice points solely have been limited to the common thermodynamic 21). cell comers; base-face centered (end-centered), types called first-order transitions---that is, Lattice Dimensions. It should be noted that the having lattice points centered on the C faces, or changes involving distinct phases having differ- unit-cell edge lengths and interaxial angles are ends of the cell; all-face centered, having lattice ent lattice parameters, , , den- unique for each crystalline substance. The unique points centered on all faces; and innercentered sities, and so on. Transitionsnot involvingdiscon- edge lengths are called lattice parameters. The (body-centered), having lattice points at the cen- tinuities in composition, enthalpy, entropy, or term lattice constant also has been used for the ter of the volume of the unit cell. The fifth ar- molar volume are called higher-order transitions length of an edge, but the values of edge length rangement, the primitive rhombohedral unit cell, and occur less frequently. The change in the mag- are not constant, varying with composition within is considered a separate basic arrangement, as netic quality of iron from ferromagnetic to param- a phase field and also with temperature due to shown in the following section on crystal struc- agnetic as the temperature is raised above 771 °C thermal expansion and contraction. (Reported lat- ture nomenclature.These five basic arrangements (1420 °F) is an example of a second-order transi- tice parameter values are assumed to be room- are identified by capital letters as follows: P for tion: no phase change is involved and the Gibbs temperature values unless otherwise specified.) the primitive cubic, C for the cubic cell with phase rule does not come into play in the transi- Interaxial angles other than 90 ° or 120° also can lattice points on the two C faces, F for all-face- tion. Another example of a higher-order transition change slightly with changes in composition. centered cubic, I for innercentered (body-cen- is the continuous change from a random arrange- When the edges of the unit cell are not equal in tered) cubic, and R for primitive rhombohedral. ment of the various kinds of atoms in a multicom- ponent crystal structure (a disordered structure) to an arrangement where there is some degree of Table 2 Relationshipsof edge lengths and of interaxial angles for the seven crystal systems crystal ordering of the atoms (an ordered struc- Crystal system Edge lengths lntera~dalangles Examples ture, or superlattice), or the reverse reaction. Tficlinic (anorthic) a ¢ b # c Ix # ~ # 7 ¢ 90° HgK Monoclinic a ¢ b ¢ c ~ = y = 90 ° # [3 13-S; CoSb2 Orthorhombic a # b # c 0t = 13 = 7 = 90° or-S; Ga; Fe3C (cementite) Crystal Structure Tetragonal a = b # c ot = 13 = 7 = 90° 13-Sn (white); TiO2 Hexagonal a = b ~ c o~ = 13 = 90°; y = 120 ° Zn; Cd; NiAs Rhombohedral(a) a = b = c ct = [5 = Y # 90° As; Sb; Bi; calcite Acrystal is a solid consisting of atoms or mole- Cubic a = b = c ot = 13 = T = 90° Cu; Ag; Au; Fe; NaCI cules arranged in a pattern that is repetitive in three dimensions. The arrangement of the atoms (a) Rhombohedral crystals (sometimes called trigonal) also can be described by using hexagonal axes (rhombohedral-hexagonal).