MATERIALS SCIENCE AND ENGINEERING Solubility Limit PDF

9.2 Solubility Limit • 299 which are also common terms, were defined in Section 4.3. Another term used in this context is system, which has two meanings. System may refer to a specific body of material under consideration (e.g., a ladle of molten steel); or it may relate to the series of possible al...

9.2 Solubility Limit • 299 which are also common terms, were defined in Section 4.3. Another term used in this context is system, which has two meanings. System may refer to a specific body of material under consideration (e.g., a ladle of molten steel); or it may relate to the series of possible alloys consisting of the same components but without regard to alloy composition (e.g., the iron–carbon system). The concept of a solid solution was introduced in Section 4.3. To review, a solid solution consists of atoms of at least two different types; the solute atoms occupy either substitutional or interstitial positions in the solvent lattice, and the crystal structure of the solvent is maintained. system 9.2 SOLUBILITY LIMIT solubility limit Tutorial Video: Phases and Solubility Limits What is a Solubility Limit? For many alloy systems and at some specific temperature, there is a maximum concentration of solute atoms that may dissolve in the solvent to form a solid solution; this is called a solubility limit. The addition of solute in excess of this solubility limit results in the formation of another solid solution or compound that has a distinctly different composition. To illustrate this concept, consider the sugar–water (C12H22O11–H2O) system. Initially, as sugar is added to water, a sugar–water solution or syrup forms. As more sugar is introduced, the solution becomes more concentrated, until the solubility limit is reached or the solution becomes saturated with sugar. At this time, the solution is not capable of dissolving any more sugar, and further additions simply settle to the bottom of the container. Thus, the system now consists of two separate substances: a sugar–water syrup liquid solution and solid crystals of undissolved sugar. This solubility limit of sugar in water depends on the temperature of the water and may be represented in graphical form on a plot of temperature along the ordinate and composition (in weight percent sugar) along the abscissa, as shown in Figure 9.1. Along the composition axis, increasing sugar concentration is from left to right, and percentage of water is read from right to left. Because only two components are involved (sugar and water), the sum of the concentrations at any composition will equal 100 wt%. The solubility limit is represented as the nearly vertical line in the figure. For compositions and temperatures to the left of the solubility line, only the syrup liquid solution exists; to the right of the line, syrup and solid sugar coexist. The solubility limit at some temperature is the composition that corresponds to the intersection of the given temperature coordinate and the solubility limit line. For example, at 20C, the maximum solubility of sugar in water is 65 wt%. As Figure 9.1 indicates, the solubility limit increases slightly with rising temperature. Figure 9.1 The solubility of sugar 100 200 Temperature (°C) 150 60 Liquid solution + solid sugar Liquid solution (syrup) 40 100 20 50 Water 0 0 20 40 60 80 100 100 80 60 40 20 0 Composition (wt%) Temperature (°F) Solubility limit 80 Sugar (C12H22O11) in a sugar–water syrup. 300 • Chapter 9 / Phase Diagrams 9.3 PHASES phase Tutorial Video: Phases and Solubility Limits What is a Phase? 9.4 Also critical to the understanding of phase diagrams is the concept of a phase. A phase may be defined as a homogeneous portion of a system that has uniform physical and chemical characteristics. Every pure material is considered to be a phase; so also is every solid, liquid, and gaseous solution. For example, the sugar–water syrup solution just discussed is one phase, and solid sugar is another. Each has different physical properties (one is a liquid, the other is a solid); furthermore, each is different chemically (i.e., has a different chemical composition); one is virtually pure sugar, the other is a solution of H2O and C12H22O11. If more than one phase is present in a given system, each will have its own distinct properties, and a boundary separating the phases will exist, across which there will be a discontinuous and abrupt change in physical and/or chemical characteristics. When two phases are present in a system, it is not necessary that there be a difference in both physical and chemical properties; a disparity in one or the other set of properties is sufficient. When water and ice are present in a container, two separate phases exist; they are physically dissimilar (one is a solid, the other is a liquid) but identical in chemical makeup. Also, when a substance can exist in two or more polymorphic forms (e.g., having both FCC and BCC structures), each of these structures is a separate phase because their respective physical characteristics differ. Sometimes, a single-phase system is termed homogeneous. Systems composed of two or more phases are termed mixtures or heterogeneous systems. Most metallic alloys and, for that matter, ceramic, polymeric, and composite systems are heterogeneous. Typically, the phases interact in such a way that the property combination of the multiphase system is different from, and more desirable than, either of the individual phases. MICROSTRUCTURE The physical properties and, in particular, the mechanical behavior of a material often depend on the microstructure. Microstructure is subject to direct microscopic observation using optical or electron microscopes; this is touched on in Sections 4.9 and 4.10. In metal alloys, microstructure is characterized by the number of phases present, their proportions, and the manner in which they are distributed or arranged. The microstructure of an alloy depends on such variables as the alloying elements present, their concentrations, and the heat treatment of the alloy (i.e., the temperature, the heating time at temperature, and the rate of cooling to room temperature). The procedure of specimen preparation for microscopic examination is briefly outlined in Section 4.10. After appropriate polishing and etching, the different phases may be distinguished by their appearance. For example, for a two-phase alloy, one phase may appear light and the other phase dark. When only a single phase or solid solution is present, the texture is uniform, except for grain boundaries that may be revealed (Figure 4.15b). 9.5 PHASE EQUILIBRIA equilibrium free energy phase equilibrium Equilibrium is another essential concept; it is best described in terms of a thermodynamic quantity called the free energy. In brief, free energy is a function of the internal energy of a system and also the randomness or disorder of the atoms or molecules (or entropy). A system is at equilibrium if its free energy is at a minimum under some specified combination of temperature, pressure, and composition. In a macroscopic sense, this means that the characteristics of the system do not change with time, but persist indefinitely—that is, the system is stable. A change in temperature, pressure, and/or composition for a system in equilibrium results in an increase in the free energy and in a possible spontaneous change to another state by which the free energy is lowered. The term phase equilibrium, often used in the context of this discussion, refers to equilibrium as it applies to systems in which more than one phase may exist. Phase 9.6 One-Component (or Unary) Phase Diagrams • 301 metastable equilibrium is reflected by a constancy with time in the phase characteristics of a system. Perhaps an example best illustrates this concept. Suppose that a sugar–water syrup is contained in a closed vessel and the solution is in contact with solid sugar at 20C. If the system is at equilibrium, the composition of the syrup is 65 wt% C12H22O11–35 wt% H2O (Figure 9.1), and the amounts and compositions of the syrup and solid sugar will remain constant with time. If the temperature of the system is suddenly raised—say, to 100C— this equilibrium or balance is temporarily upset and the solubility limit is increased to 80 wt% C12H22O11 (Figure 9.1). Thus, some of the solid sugar will go into solution in the syrup. This will continue until the new equilibrium syrup concentration is established at the higher temperature. This sugar–syrup example illustrates the principle of phase equilibrium using a liquid–solid system. In many metallurgical and materials systems of interest, phase equilibrium involves just solid phases. In this regard the state of the system is reflected in the characteristics of the microstructure, which necessarily include not only the phases present and their compositions, but, in addition, the relative phase amounts and their spatial arrangement or distribution. Free energy considerations and diagrams similar to Figure 9.1 provide information about the equilibrium characteristics of a particular system, which is important, but they do not indicate the time period necessary for the attainment of a new equilibrium state. It is often the case, especially in solid systems, that a state of equilibrium is never completely achieved because the rate of approach to equilibrium is extremely slow; such a system is said to be in a nonequilibrium or metastable state. A metastable state or microstructure may persist indefinitely, experiencing only extremely slight and almost imperceptible changes as time progresses. Often, metastable structures are of more practical significance than equilibrium ones. For example, some steel and aluminum alloys rely for their strength on the development of metastable microstructures during carefully designed heat treatments (Sections 10.5 and 11.9). Thus it is important to understand not only equilibrium states and structures, but also the speed or rate at which they are established and the factors that affect that rate. This chapter is devoted almost exclusively to equilibrium structures; the treatment of reaction rates and nonequilibrium structures is deferred to Chapter 10 and Section 11.9. Concept Check 9.1 What is the difference between the states of phase equilibrium and metastability? [The answer may be found at www.wiley.com/college/callister (Student Companion Site).] 9.6 ONE-COMPONENT (OR UNARY) PHASE DIAGRAMS phase diagram Much of the information about the control of the phase structure of a particular system is conveniently and concisely displayed in what is called a phase diagram, also often termed an equilibrium diagram. Three externally controllable parameters that affect phase structure— temperature, pressure, and composition—and phase diagrams are constructed when various combinations of these parameters are plotted against one another. Perhaps the simplest and easiest type of phase diagram to understand is that for a one-component system, in which composition is held constant (i.e., the phase diagram is for a pure substance); this means that pressure and temperature are the variables. This one-component phase diagram (or unary phase diagram, sometimes also called a pressure–temperature [or P–T] diagram) is represented as a two-dimensional plot of 302 • Chapter 9 / Phase Diagrams Figure 9.2 Pressure–temperature phase 1,000 b 100 Pressure (atm) diagram for H2O. Intersection of the dashed horizontal line at 1 atm pressure with the solid– liquid phase boundary (point 2) corresponds to the melting point at this pressure (T = 0C). Similarly, point 3, the intersection with the liquid–vapor boundary, represents the boiling point (T = 100C). Liquid (Water) Solid (Ice) 10 c 3 2 1.0 0.1 Vapor (Steam) O 0.01 a 0.001 ⫺20 0 20 40 60 80 100 120 Temperature (°C) pressure (ordinate, or vertical axis) versus temperature (abscissa, or horizontal axis). Most often, the pressure axis is scaled logarithmically. We illustrate this type of phase diagram and demonstrate its interpretation using as an example the one for H2O, which is shown in Figure 9.2. Regions for three different phases—solid, liquid, and vapor—are delineated on the plot. Each of the phases exist under equilibrium conditions over the temperature–pressure ranges of its corresponding area. The three curves shown on the plot (labeled aO, bO, and cO) are phase boundaries; at any point on one of these curves, the two phases on either side of the curve are in equilibrium (or coexist) with one another. Equilibrium between solid and vapor phases is along curve aO—likewise for the solid–liquid boundary, curve bO, and the liquid–vapor boundary, curve cO. Upon crossing a boundary (as temperature and/or pressure is altered), one phase transforms into another. For example, at 1 atm pressure, during heating the solid phase transforms to the liquid phase (i.e., melting occurs) at the point labeled 2 on Figure 9.2 (i.e., the intersection of the dashed horizontal line with the solid–liquid phase boundary); this point corresponds to a temperature of 0C. The reverse transformation (liquid-to-solid, or solidification) takes place at the same point upon cooling. Similarly, at the intersection of the dashed line with the liquid–vapor phase boundary (point 3 in Figure 9.2, at 100C) the liquid transforms into the vapor phase (or vaporizes) upon heating; condensation occurs for cooling. Finally, solid ice sublimes or vaporizes upon crossing the curve labeled aO. As may also be noted from Figure 9.2, all three of the phase boundary curves intersect at a common point, which is labeled O (for this H2O system, at a temperature of 273.16 K and a pressure of 6.04 * 10-3 atm). This means that at this point only, all of the solid, liquid, and vapor phases are simultaneously in equilibrium with one another. Appropriately, this, and any other point on a P–T phase diagram where three phases are in equilibrium, is called a triple point; sometimes it is also termed an invariant point inasmuch as its position is distinct, or fixed by definite values of pressure and temperature. Any deviation from this point by a change of temperature and/or pressure will cause at least one of the phases to disappear. Pressure–temperature phase diagrams for a number of substances have been determined experimentally, which also have solid-, liquid-, and vapor-phase regions. In those instances when multiple solid phases (i.e., allotropes, Section 3.6) exist, there appears a region on the diagram for each solid phase and also other triple points. Binary Phase Diagrams Another type of extremely common phase diagram is one in which temperature and composition are variable parameters and pressure is held constant—normally 1 atm. There are several different varieties; in the present discussion, we will concern ourselves 9.7 Binary Isomorphous Systems • 303 with binary alloys—those that contain two components. If more than two components are present, phase diagrams become extremely complicated and difficult to represent. An explanation of the principles governing, and the interpretation of phase diagrams can be demonstrated using binary alloys even though most alloys contain more than two components. Binary phase diagrams are maps that represent the relationships between temperature and the compositions and quantities of phases at equilibrium, which influence the microstructure of an alloy. Many microstructures develop from phase transformations, the changes that occur when the temperature is altered (typically upon cooling). This may involve the transition from one phase to another or the appearance or disappearance of a phase. Binary phase diagrams are helpful in predicting phase transformations and the resulting microstructures, which may have equilibrium or nonequilibrium character. 9.7 BINARY ISOMORPHOUS SYSTEMS isomorphous Possibly the easiest type of binary phase diagram to understand and interpret is the type that is characterized by the copper–nickel system (Figure 9.3a). Temperature is plotted along the ordinate, and the abscissa represents the composition of the alloy, in weight percent (bottom) and atom percent (top) of nickel. The composition ranges from 0 wt% Ni (100 wt% Cu) on the far left horizontal extreme to 100 wt% Ni (0 wt% Cu) on the right. Three different phase regions, or fields, appear on the diagram: an alpha (a) field, a liquid (L) field, and a two-phase a + L field. Each region is defined by the phase or phases that exist over the range of temperatures and compositions delineated by the phase boundary lines. The liquid L is a homogeneous liquid solution composed of both copper and nickel. The a phase is a substitutional solid solution consisting of both Cu and Ni atoms and has an FCC crystal structure. At temperatures below about 1080C, copper and nickel are mutually soluble in each other in the solid state for all compositions. This complete solubility is explained by the fact that both Cu and Ni have the same crystal structure (FCC), nearly identical atomic radii and electronegativities, and similar valences, as discussed in Section 4.3. The copper–nickel system is termed isomorphous because of this complete liquid and solid solubility of the two components. Some comments are in order regarding nomenclature: First, for metallic alloys, solid solutions are commonly designated by lowercase Greek letters (a, b, g, etc.). With regard to phase boundaries, the line separating the L and a + L phase fields is termed the liquidus line, as indicated in Figure 9.3a; the liquid phase is present at all temperatures and compositions above this line. The solidus line is located between the a and a + L regions, below which only the solid a phase exists. For Figure 9.3a, the solidus and liquidus lines intersect at the two composition extremes; these correspond to the melting temperatures of the pure components. For example, the melting temperatures of pure copper and nickel are 1085C and 1453C, respectively. Heating pure copper corresponds to moving vertically up the left-hand temperature axis. Copper remains solid until its melting temperature is reached. The solid-to-liquid transformation takes place at the melting temperature, and no further heating is possible until this transformation has been completed. For any composition other than pure components, this melting phenomenon occurs over the range of temperatures between the solidus and liquidus lines; both solid a and liquid phases are in equilibrium within this temperature range. For example, upon heating of an alloy of composition 50 wt% Ni–50 wt% Cu (Figure 9.3a), melting begins at approximately 1280C (2340F); the amount of liquid phase continuously increases with temperature until about 1320C (2410F), at which point the alloy is completely liquid. 304 • Chapter 9 / Phase Diagrams Figure 9.3 (a) The copper–nickel phase Composition (at% Ni) diagram. (b) A portion of the copper–nickel phase diagram for which compositions and phase amounts are determined at point B. 1600 (Adapted from Phase Diagrams of Binary Nickel Alloys, P. Nash, Editor, 1991. Reprinted by permission of ASM International, Materials Park, OH.) 0 20 40 60 80 100 2800 1500 Liquid 1453°C 2600 Solidus line Liquidus line 1300 2400 ␣ +L B 1200 2200 ␣ A 1100 Temperature (°F) Temperature (°C) 1400 2000 1085C 1000 0 40 20 80 100 Composition (wt% Ni) (Cu) 1300 60 (Ni) (a) Liquid Temperature (°C) Tie line B ␣ + Liquid ␣ + Liquid ␣ 1200 R S ␣ 20 40 30 CL C0 Composition (wt% Ni) 50 C␣ (b) Concept Check 9.2 The phase diagram for the cobalt–nickel system is an isomorphous one. On the basis of melting temperatures for these two metals, describe and/or draw a schematic sketch of the phase diagram for the Co–Ni system. [The answer may be found at www.wiley.com/college/callister (Student Companion Site).] 9.8 Interpretation of Phase Diagrams • 305 9.8 INTERPRETATION OF PHASE DIAGRAMS For a binary system of known composition and temperature at equilibrium, at least three kinds of information are available: (1) the phases that are present, (2) the compositions of these phases, and (3) the percentages or fractions of the phases. The procedures for making these determinations will be demonstrated using the copper–nickel system. Phases Present : VMSE Isomorphous (Sb-Bi) The establishment of what phases are present is relatively simple. One just locates the temperature–composition point on the diagram and notes the phase(s) with which the corresponding phase field is labeled. For example, an alloy of composition 60 wt% Ni–40 wt% Cu at 1100C would be located at point A in Figure 9.3a; because this is within the a region, only the single a phase will be present. However, a 35 wt% Ni–65 wt% Cu alloy at 1250C (point B) consists of both a and liquid phases at equilibrium. Determination of Phase Compositions : VMSE Isomorphous (Sb-Bi) tie line The first step in the determination of phase compositions (in terms of the concentrations of the components) is to locate the temperature–composition point on the phase diagram. Different methods are used for single- and two-phase regions. If only one phase is present, the procedure is trivial: the composition of this phase is simply the same as the overall composition of the alloy. For example, consider the 60 wt% Ni–40 wt% Cu alloy at 1100C (point A, Figure 9.3a). At this composition and temperature, only the a phase is present, having a composition of 60 wt% Ni–40 wt% Cu. For an alloy having composition and temperature located in a two-phase region, the situation is more complicated. In all two-phase regions (and in two-phase regions only), one may imagine a series of horizontal lines, one at every temperature; each of these is known as a tie line, or sometimes as an isotherm. These tie lines extend across the two-phase region and terminate at the phase boundary lines on either side. To compute the equilibrium concentrations of the two phases, the following procedure is used: 1. A tie line is constructed across the two-phase region at the temperature of the alloy. 2. The intersections of the tie line and the phase boundaries on either side are noted. 3. Perpendiculars are dropped from these intersections to the horizontal composition axis, from which the composition of each of the respective phases is read. For example, consider again the 35 wt% Ni–65 wt% Cu alloy at 1250C, located at point B in Figure 9.3b and lying within the a + L region. Thus, the problem is to determine the composition (in wt% Ni and Cu) for both the a and liquid phases. The tie line is constructed across the a + L phase region, as shown in Figure 9.3b. The perpendicular from the intersection of the tie line with the liquidus boundary meets the composition axis at 31.5 wt% Ni–68.5 wt% Cu, which is the composition of the liquid phase, CL. Likewise, for the solidus–tie line intersection, we find a composition for the a solidsolution phase, Ca, of 42.5 wt% Ni–57.5 wt% Cu. Determination of Phase Amounts : VMSE Isomorphous (Sb-Bi) The relative amounts (as fraction or as percentage) of the phases present at equilibrium may also be computed with the aid of phase diagrams. Again, the single- and two-phase situations must be treated separately. The solution is obvious in the single-phase region. Because only one phase is present, the alloy is composed entirely of that phase—that is, the phase fraction is 1.0, or, alternatively, the percentage is 100%. From the previous example for the 60 wt% Ni–40 wt% Cu alloy at 1100C (point A in Figure 9.3a), only the a phase is present; hence, the alloy is completely, or 100%, a. 306 • Chapter 9 lever rule / Phase Diagrams If the composition and temperature position is located within a two-phase region, things are more complex. The tie line must be used in conjunction with a procedure that is often called the lever rule (or the inverse lever rule), which is applied as follows: 1. The tie line is constructed across the two-phase region at the temperature of the alloy. 2. The overall alloy composition is located on the tie line. Tutorial Video: Phase Diagram Calculations and Lever Rule The Lever Rule 3. The fraction of one phase is computed by taking the length of tie line from the overall alloy composition to the phase boundary for the other phase and dividing by the total tie line length. 4. The fraction of the other phase is determined in the same manner. 5. If phase percentages are desired, each phase fraction is multiplied by 100. When the composition axis is scaled in weight percent, the phase fractions computed using the lever rule are mass fractions—the mass (or weight) of a specific phase divided by the total alloy mass (or weight). The mass of each phase is computed from the product of each phase fraction and the total alloy mass. In the use of the lever rule, tie line segment lengths may be determined either by direct measurement from the phase diagram using a linear scale, preferably graduated in millimeters, or by subtracting compositions as taken from the composition axis. Consider again the example shown in Figure 9.3b, in which at 1250⬚C both a and liquid phases are present for a 35 wt% Ni–65 wt% Cu alloy. The problem is to compute the fraction of each of the a and liquid phases. The tie line is constructed that was used for the determination of a and L phase compositions. Let the overall alloy composition be located along the tie line and denoted as C0, and let the mass fractions be represented by WL and Wa for the respective phases. From the lever rule, WL may be computed according to S R + S (9.1a) Ca - C0 Ca - CL (9.1b) WL = or, by subtracting compositions, Lever rule expression for computation of liquid mass fraction (per Figure 9.3b) WL = Composition need be specified in terms of only one of the constituents for a binary alloy; for the preceding computation, weight percent nickel is used (i.e., C0 = 35 wt% Ni, Ca = 42.5 wt% Ni, and CL = 31.5 wt% Ni), and WL = 42.5 - 35 = 0.68 42.5 - 31.5 Similarly, for the a phase, Lever rule expression for computation of a-phase mass fraction (per Figure 9.3b) R R + S (9.2a) = C0 - CL Ca - CL (9.2b) = 35 - 31.5 = 0.32 42.5 - 31.5 Wa = 9.8 Interpretation of Phase Diagrams • 307 Of course, identical answers are obtained if compositions are expressed in weight percent copper instead of nickel. Thus, the lever rule may be employed to determine the relative amounts or fractions of phases in any two-phase region for a binary alloy if the temperature and composition are known and if equilibrium has been established. Its derivation is presented as an example problem. It is easy to confuse the foregoing procedures for the determination of phase compositions and fractional phase amounts; thus, a brief summary is warranted. Compositions of phases are expressed in terms of weight percents of the components (e.g., wt% Cu, wt% Ni). For any alloy consisting of a single phase, the composition of that phase is the same as the total alloy composition. If two phases are present, the tie line must be employed, the extremes of which determine the compositions of the respective phases. With regard to fractional phase amounts (e.g., mass fraction of the a or liquid phase), when a single phase exists, the alloy is completely that phase. For a two-phase alloy, the lever rule is used, in which a ratio of tie line segment lengths is taken. Concept Check 9.3 A copper–nickel alloy of composition 70 wt% Ni–30 wt% Cu is slowly heated from a temperature of 1300⬚C (2370⬚F). (a) At what temperature does the first liquid phase form? (b) What is the composition of this liquid phase? (c) At what temperature does complete melting of the alloy occur? (d) What is the composition of the last solid remaining prior to complete melting? Concept Check 9.4 Is it possible to have a copper–nickel alloy that, at equilibrium, consists of an a phase of composition 37 wt% Ni–63 wt% Cu and also a liquid phase of composition 20 wt% Ni–80 wt% Cu? If so, what will be the approximate temperature of the alloy? If this is not possible, explain why. [The answers may be found at www.wiley.com/college/callister (Student Companion Site).] EXAMPLE PROBLEM 9.1 Lever Rule Derivation Derive the lever rule. Solution Consider the phase diagram for copper and nickel (Figure 9.3b) and alloy of composition C0 at 1250⬚C, and let Ca, CL, Wa, and WL represent the same parameters as given earlier. This derivation is accomplished through two conservation-of-mass expressions. With the first, because only two phases are present, the sum of their mass fractions must be equal to unity; that is, Wa + WL = 1 (9.3) For the second, the mass of one of the components (either Cu or Ni) that is present in both of the phases must be equal to the mass of that component in the total alloy, or WaCa + WLCL = C0 (9.4) 308 • Chapter 9 / Phase Diagrams Simultaneous solution of these two equations leads to the lever rule expressions for this particular situation, WL = Ca - C0 Ca - CL (9.1b) Wa = C0 - CL Ca - CL (9.2b) For multiphase alloys, it is often more convenient to specify relative phase amount in terms of volume fraction rather than mass fraction. Phase volume fractions are preferred because they (rather than mass fractions) may be determined from examination of the microstructure; furthermore, the properties of a multiphase alloy may be estimated on the basis of volume fractions. For an alloy consisting of a and b phases, the volume fraction of the a phase, Va, is defined as a phase volume fraction—dependence on volumes of a and b phases Va = va va + vb (9.5) where va and vb denote the volumes of the respective phases in the alloy. An analogous expression exists for Vb, and, for an alloy consisting of just two phases, it is the case that Va + Vb = 1. On occasion conversion from mass fraction to volume fraction (or vice versa) is desired. Equations that facilitate these conversions are as follows: Va = Conversion of mass fractions of a and b phases to volume fractions Wa ra Wb Wa + ra rb (9.6a) Wb Vb = rb Wb Wa + ra rb (9.6b) and Wa = Conversion of volume fractions of a and b phases to mass fractions Wb = Vara Vara + Vbrb Vbrb Vara + Vbrb (9.7a) (9.7b) 9.9 Development of Microstructure in Isomorphous Alloys • 309 In these expressions, ra and rb are the densities of the respective phases; these may be determined approximately using Equations 4.10a and 4.10b. When the densities of the phases in a two-phase alloy differ significantly, there will be quite a disparity between mass and volume fractions; conversely, if the phase densities are the same, mass and volume fractions are identical. 9.9 DEVELOPMENT OF MICROSTRUCTURE IN ISOMORPHOUS ALLOYS Equilibrium Cooling At this point it is instructive to examine the development of microstructure that occurs for isomorphous alloys during solidification. We first treat the situation in which the cooling occurs very slowly, in that phase equilibrium is continuously maintained. Let us consider the copper–nickel system (Figure 9.3a), specifically an alloy of composition 35 wt% Ni–65 wt% Cu as it is cooled from 1300C. The region of the Cu–Ni phase diagram in the vicinity of this composition is shown in Figure 9.4. Cooling of an alloy of this composition corresponds to moving down the vertical dashed line. At 1300C, point a, the alloy is completely liquid (of composition 35 wt% Ni–65 wt% Cu) and has the microstructure represented by the circle inset in the figure. As cooling begins, no microstructural or compositional changes will be realized until we reach the : VMSE Isomorphous (Sb-Bi) Figure 9.4 Schematic representation of the development of microstructure during the equilibrium solidification of a 35 wt% Ni–65 wt% Cu alloy. L L (35 Ni) L (35 Ni) ␣ (46 Ni) a 1300 L (32 Ni) ␣ b L ␣ (46 Ni) c Temperature (°C) + ␣ (43 Ni) ␣ (43 Ni) L (24 Ni) d ␣ L (32 Ni) ␣ ␣ ␣ 1200 L (24 Ni) e ␣ (35 Ni) ␣ ␣ ␣ ␣ ␣ ␣ ␣ ␣ ␣ ␣ ␣ ␣ ␣ ␣ ␣ (35 Ni) 1100 20 ␣ 30 40 Composition (wt% Ni) ␣ ␣ 50 310 • Chapter 9 / Phase Diagrams liquidus line (point b, ~1260C). At this point, the first solid a begins to form, which has a composition dictated by the tie line drawn at this temperature [i.e., 46 wt% Ni–54 wt% Cu, noted as a(46 Ni)]; the composition of liquid is still approximately 35 wt% Ni–65 wt% Cu [L(35 Ni)], which is different from that of the solid a. With continued cooling, both compositions and relative amounts of each of the phases will change. The compositions of the liquid and a phases will follow the liquidus and solidus lines, respectively. Furthermore, the fraction of the a phase will increase with continued cooling. Note that the overall alloy composition (35 wt% Ni–65 wt% Cu) remains unchanged during cooling even though there is a redistribution of copper and nickel between the phases. At 1250C, point c in Figure 9.4, the compositions of the liquid and a phases are 32 wt% Ni–68 wt% Cu [L(32 Ni)] and 43 wt% Ni–57 wt% Cu [a(43 Ni)], respectively. The solidification process is virtually complete at about 1220C, point d; the composition of the solid a is approximately 35 wt% Ni–65 wt% Cu (the overall alloy composition), whereas that of the last remaining liquid is 24 wt% Ni–76 wt% Cu. Upon crossing the solidus line, this remaining liquid solidifies; the final product then is a polycrystalline a-phase solid solution that has a uniform 35 wt% Ni–65 wt% Cu composition (point e, Figure 9.4). Subsequent cooling produces no microstructural or compositional alterations. Nonequilibrium Cooling Conditions of equilibrium solidification and the development of microstructures, as described in the previous section, are realized only for extremely slow cooling rates. The reason for this is that with changes in temperature, there must be readjustments in the compositions of the liquid and solid phases in accordance with the phase diagram (i.e., with the liquidus and solidus lines), as discussed. These readjustments are accomplished by diffusional processes—that is, diffusion in both solid and liquid phases and also across the solid–liquid interface. Because diffusion is a time-dependent phenomenon (Section 5.3), to maintain equilibrium during cooling, sufficient time must be allowed at each temperature for the appropriate compositional readjustments. Diffusion rates (i.e., the magnitudes of the diffusion coefficients) are especially low for the solid phase and, for both phases, decrease with diminishing temperature. In virtually all practical solidification situations, cooling rates are much too rapid to allow these compositional readjustments and maintenance of equilibrium; consequently, microstructures other than those previously described develop. Some of the consequences of nonequilibrium solidification for isomorphous alloys will now be discussed by considering a 35 wt% Ni–65 wt% Cu alloy, the same composition that was used for equilibrium cooling in the previous section. The portion of the phase diagram near this composition is shown in Figure 9.5; in addition, microstructures and associated phase compositions at various temperatures upon cooling are noted in the circular insets. To simplify this discussion, it will be assumed that diffusion rates in the liquid phase are sufficiently rapid such that equilibrium is maintained in the liquid. Let us begin cooling from a temperature of about 1300C; this is indicated by point a¿ in the liquid region. This liquid has a composition of 35 wt% Ni–65 wt% Cu [noted as L(35 Ni) in the figure], and no changes occur while cooling through the liquid phase region (moving down vertically from point a¿). At point b¿ (approximately 1260C), a-phase particles begin to form, which, from the tie line constructed, have a composition of 46 wt% Ni–54 wt% Cu [a(46 Ni)]. Upon further cooling to point c¿ (about 1240C), the liquid composition has shifted to 29 wt% Ni–71 wt% Cu; furthermore, at this temperature the composition of the a 9.9 Development of Microstructure in Isomorphous Alloys • 311 Figure 9.5 Schematic representation L of the development of microstructure during the nonequilibrium solidification of a 35 wt% Ni–65 wt% Cu alloy. L (35 Ni) L (35 Ni) 1300 ␣ +L a⬘ ␣ (46 Ni) ␣ b⬘ L(29 Ni) ␣ (40 Ni) ␣ (46 Ni) Temperature (°C) c⬘ L(24 Ni) ␣ (42 Ni) d⬘ L(21 Ni) ␣ (35 Ni) e⬘ 1200 L(29 Ni) ␣ (46 Ni) ␣ (40 Ni) ␣ (38 Ni) ␣ (31 Ni) L (24 Ni) ␣ (46 Ni) ␣ (40 Ni) ␣ (35 Ni) f⬘ ␣ (46 Ni) ␣ (40 Ni) ␣ (35 Ni) ␣ (31 Ni) L (21 Ni) ␣ (46 Ni) ␣ (40 Ni) ␣ (35 Ni) ␣ (31 Ni) 1100 20 30 40 Composition (wt% Ni) 50 60 phase that solidified is 40 wt% Ni–60 wt% Cu [a(40 Ni)]. However, because diffusion in the solid a phase is relatively slow, the a phase that formed at point b¿ has not changed composition appreciably—that is, it is still about 46 wt% Ni—and the composition of the a grains has continuously changed with radial position, from 46 wt% Ni at grain centers to 40 wt% Ni at the outer grain perimeters. Thus, at point c¿, the average composition of the solid a grains that have formed would be some volume-weighted average composition lying between 46 and 40 wt% Ni. For the sake of argument, let us take this average composition to be 42 wt% Ni–58 wt% Cu [a(42 Ni)]. Furthermore, we would also find that, on the basis of lever-rule computations, a greater proportion of liquid is present for these nonequilibrium conditions than for equilibrium cooling. The implication of this nonequilibrium solidification phenomenon is that the solidus line on the phase diagram has been shifted to higher Ni contents—to the average compositions of the a phase (e.g., 42 wt% Ni at 1240C)—and is represented by the dashed line in Figure 9.5. There is no comparable alteration of the liquidus line inasmuch as it is assumed that equilibrium is maintained in the liquid phase during cooling because of sufficiently rapid diffusion rates. At point d¿ (~1220C) and for equilibrium cooling rates, solidification should be completed. However, for this nonequilibrium situation, there is still an appreciable 312 • Chapter 9 Photomicrograph showing the microstructure of an as-cast bronze alloy that was found in Syria, and which has been dated to the 19th century BC. The etching procedure has revealed coring as variations in color hue across the grains. 30. (Courtesy of George F. Vander Voort, Struers Inc.) / Phase Diagrams proportion of liquid remaining, and the a phase that is forming has a composition of 35 wt% Ni [a(35 Ni)]; also the average a-phase composition at this point is 38 wt% Ni [a(38 Ni)]. Nonequilibrium solidification finally reaches completion at point e¿ (~1205C). The composition of the last a phase to solidify at this point is about 31 wt% Ni; the average composition of the a phase at complete solidification is 35 wt% Ni. The inset at point f ¿ shows the microstructure of the totally solid material. The degree of displacement of the nonequilibrium solidus curve from the equilibrium one depends on the rate of cooling; the slower the cooling rate, the smaller this displacement—that is, the difference between the equilibrium solidus and average solid composition is lower. Furthermore, if the diffusion rate in the solid phase increases, this displacement decreases. There are some important consequences for isomorphous alloys that have solidified under nonequilibrium conditions. As discussed earlier, the distribution of the two elements within the grains is nonuniform, a phenomenon termed segregation—that is, concentration gradients are established across the grains that are represented by the insets of Figure 9.5. The center of each grain, which is the first part to freeze, is rich in the high-melting element (e.g., nickel for this Cu–Ni system), whereas the concentration of the low-melting element increases with position from this region to the grain boundary. This is termed a cored structure, which gives rise to less than the optimal properties. As a casting having a cored structure is reheated, grain boundary regions melt first because they are richer in the low-melting component. This produces a sudden loss in mechanical integrity due to the thin liquid film that separates the grains. Furthermore, this melting may begin at a temperature below the equilibrium solidus temperature of the alloy. Coring may be eliminated by a homogenization heat treatment carried out at a temperature below the solidus point for the particular alloy composition. During this process, atomic diffusion occurs, which produces compositionally homogeneous grains. 9.10 MECHANICAL PROPERTIES OF ISOMORPHOUS ALLOYS We now briefly explore how the mechanical properties of solid isomorphous alloys are affected by composition as other structural variables (e.g., grain size) are held constant. For all temperatures and compositions below the melting temperature of the lowestmelting component, only a single solid phase exists. Therefore, each component experiences solid-solution strengthening (Section 7.9) or an increase in strength and hardness by additions of the other component. This effect is demonstrated in Figure 9.6a as tensile strength versus composition for the copper–nickel system at room temperature; at some intermediate composition, the curve necessarily passes through a maximum. Plotted in Figure 9.6b is the ductility (%EL)–composition behavior, which is just the opposite of tensile strength—that is, ductility decreases with additions of the second component, and the curve exhibits a minimum. 9.11 BINARY EUTECTIC SYSTEMS Another type of common and relatively simple phase diagram found for binary alloys is shown in Figure 9.7 for the copper–silver system; this is known as a binary eutectic phase diagram. A number of features of this phase diagram are important and worth noting. First, three single-phase regions are found on the diagram: a, b, and liquid. The a phase is a solid solution rich in copper; it has silver as the solute component 9.11 Binary Eutectic Systems • 313 50 300 40 30 200 0 (Cu) 20 40 60 80 50 40 30 20 0 (Cu) 100 (Ni) Composition (wt% Ni) Elongation (% in 50 mm [2 in.]) 60 400 Tensile strength (ksi) Tensile strength (MPa) 60 20 40 60 80 Composition (wt% Ni) (a) 100 (Ni) (b) Figure 9.6 For the copper–nickel system, (a) tensile strength versus composition and (b) ductility (%EL) versus composition at room temperature. A solid solution exists over all compositions for this system. and an FCC crystal structure. The b-phase solid solution also has an FCC structure, but copper is the solute. Pure copper and pure silver are also considered to be a and b phases, respectively. Thus, the solubility in each of these solid phases is limited, in that at any temperature below line BEG only a limited concentration of silver dissolves in copper Composition (at% Ag) 0 20 40 60 80 100 2200 1200 A 2000 Liquidus 1000 Liquid 1800 F ␣ +L ␣ 800 1600 779°C (TE) B ␤+L E 8.0 (C␣ E) 71.9 (CE) 91.2 (C␤ E) G 1400 ␤ 1200 600 Temperature (°F) Temperature (°C) Solidus 1000 Solvus ␣ +␤ 800 400 C 600 H 200 0 (Cu) 20 40 60 Composition (wt% Ag) 80 400 100 (Ag) Figure 9.7 The copper–silver phase diagram. [Adapted from Binary Alloy Phase Diagrams, 2nd edition, Vol. 1, T. B. Massalski (Editor-in-Chief), 1990. Reprinted by permission of ASM International, Materials Park, OH.] 314 • Chapter 9 solvus line solidus line liquidus line The eutectic reaction (per Figure 9.7) eutectic reaction / Phase Diagrams (for the a phase), and similarly for copper in silver (for the b phase). The solubility limit for the a phase corresponds to the boundary line, labeled CBA, between the a/(a + b) and a/(a + L) phase regions; it increases with temperature to a maximum [8.0 wt% Ag at 779C (1434F)] at point B, and decreases back to zero at the melting temperature of pure copper, point A [1085C (1985F)]. At temperatures below 779C (1434F), the solid solubility limit line separating the a and a  b phase regions is termed a solvus line; the boundary AB between the a and a  L fields is the solidus line, as indicated in Figure 9.7. For the b phase, both solvus and solidus lines also exist, HG and GF, respectively, as shown. The maximum solubility of copper in the b phase, point G (8.8 wt% Cu), also occurs at 779C (1434F). This horizontal line BEG, which is parallel to the composition axis and extends between these maximum solubility positions, may also be considered a solidus line; it represents the lowest temperature at which a liquid phase may exist for any copper–silver alloy that is at equilibrium. There are also three two-phase regions found for the copper–silver system (Figure 9.7): a + L, b + L, and a + b. The a- and b-phase solid solutions coexist for all compositions and temperatures within the a + b phase field; the a + liquid and b + liquid phases also coexist in their respective phase regions. Furthermore, compositions and relative amounts for the phases may be determined using tie lines and the lever rule as outlined previously. As silver is added to copper, the temperature at which the alloys become totally liquid decreases along the liquidus line, line AE; thus, the melting temperature of copper is lowered by silver additions. The same may be said for silver: the introduction of copper reduces the temperature of complete melting along the other liquidus line, FE. These liquidus lines meet at the point E on the phase diagram, which point is designated by composition C E and temperature T E; for the copper–silver system, the values for these two parameters are 71.9 wt% Ag and 779C (1434F), respectively. It should also be noted there is a horizontal isotherm at 779C and represented by the line labeled BEG that also passes through point E. An important reaction occurs for an alloy of composition CE as it changes temperature in passing through TE; this reaction may be written as follows: cooling L(CE) m a(CaE) + b(CbE) heating (9.8) In other words, upon cooling, a liquid phase is transformed into the two solid a and b phases at the temperature TE; the opposite reaction occurs upon heating. This is called a eutectic reaction (eutectic means “easily melted”), and CE and TE represent the eutectic composition and temperature, respectively; CaE and CbE are the respective compositions of the a and b phases at TE. Thus, for the copper–silver system, the eutectic reaction, Equation 9.8, may be written as follows: cooling L(71.9 wt% Ag) m a(8.0 wt% Ag) + b(91.2 wt% Ag) heating Tutorial Video: Eutectic Reaction Vocabulary and Microstructures Eutectic Reaction Terms Often, the horizontal solidus line at TE is called the eutectic isotherm. The eutectic reaction, upon cooling, is similar to solidification for pure components in that the reaction proceeds to completion at a constant temperature, or isothermally, at TE. However, the solid product of eutectic solidification is always two solid phases, whereas for a pure component only a single phase forms. Because of this eutectic reaction, phase diagrams similar to that in Figure 9.7 are termed eutectic phase diagrams; components exhibiting this behavior make up a eutectic system. 9.11 Binary Eutectic Systems • 315 Composition (at% Sn) 0 20 40 60 80 100 327°C 600 300 Liquid 500 Temperature (°C) ␣ 200 ␤ +L 183°C 400 ␤ 18.3 61.9 97.8 300 100 ␣ + ␤ Temperature (°F) 232°C ␣ +L 200 100 0 0 (Pb) 20 40 60 Composition (wt% Sn) 80 100 (Sn) Figure 9.8 The lead–tin phase diagram. [Adapted from Binary Alloy Phase Diagrams, 2nd edition, Vol. 3, T. B. Massalski (Editor-in-Chief), 1990. Reprinted by permission of ASM International, Materials Park, OH.] Tutorial Video: Reading a Phase Diagram How do I Read a Phase Diagram? In the construction of binary phase diagrams, it is important to understand that one or at most two phases may be in equilibrium within a phase field. This holds true for the phase diagrams in Figures 9.3a and 9.7. For a eutectic system, three phases (a, b, and L) may be in equilibrium, but only at points along the eutectic isotherm. Another general rule is that single-phase regions are always separated from each other by a two-phase region that consists of the two single phases that it separates. For example, the a + b field is situated between the a and b single-phase regions in Figure 9.7. Another common eutectic system is that for lead and tin; the phase diagram (Figure 9.8) has a general shape similar to that for copper–silver. For the lead–tin system, the solid-solution phases are also designated by a and b; in this case, a represents a solid solution of tin in lead; for b, tin is the solvent and lead is the solute. The eutectic invariant point is located at 61.9 wt% Sn and 183C (361F). Of course, maximum solid solubility compositions as well as component melting temperatures are different for the copper–silver and lead–tin systems, as may be observed by comparing their phase diagrams. On occasion, low-melting-temperature alloys are prepared having near-eutectic compositions. A familiar example is 60–40 solder, which contains 60 wt% Sn and 40 wt% Pb. Figure 9.8 indicates that an alloy of this composition is completely molten at about 185C (365F), which makes this material especially attractive as a low-temperature solder because it is easily melted. 316 • Chapter 9 / Phase Diagrams Concept Check 9.5 At 700⬚C (1290⬚F), what is the maximum solubility (a) of Cu in Ag? (b) Of Ag in Cu? Concept Check 9.6 The following is a portion of the H2O–NaCl phase diagram: 10 50 Liquid (brine) 40 ⫺10 30 Salt ⫹ Liquid (brine) Ice ⫹ Liquid (brine) 20 10 Temperature (°F) Temperature (°C) 0 0 ⫺20 ⫺10 Ice ⫹ Salt NaCl H2O ⫺30 0 100 ⫺20 10 90 20 80 30 70 Composition (wt%) (a) Using this diagram, briefly explain how spreading salt on ice that is at a temperature below 0⬚C (32⬚F) can cause the ice to melt. (b) At what temperature is salt no longer useful in causing ice to melt? [The answers may be found at www.wiley.com/college/callister (Student Companion Site).] EXAMPLE PROBLEM 9.2 Determination of Phases Present and Computation of Phase Compositions For a 40 wt% Sn–60 wt% Pb alloy at 150⬚C (300⬚F), (a) what phase(s) is (are) present? (b) What is (are) the composition(s) of the phase(s)? Solution Tutorial Video: Phase Diagram Calculations and Lever Rule Calculations for a Binary Eutectic Phase Diagram (a) Locate this temperature–composition point on the phase diagram (point B in Figure 9.9). Inasmuch as it is within the a + b region, both a and b phases will coexist. (b) Because two phases are present, it becomes necessary to construct a tie line across the a + b phase field at 150⬚C, as indicated in Figure 9.9. The composition of the a phase corresponds to the tie line intersection with the a/(a + b) solvus phase boundary—about 11 wt% Sn–89 wt% Pb, denoted as Ca. This is similar for the b phase, which has a composition of approximately 98 wt% Sn–2 wt% Pb (Cb). 9.11 Binary Eutectic Systems • 317 600 300 Liquid ␣+L 200 ␤+L ␣ B 400 ␤ 300 ␣+␤ 100 Temperature (°F) Temperature (°C) 500 200 C␤ 100 0 0 (Pb) 20 C␣ 60 80 C1 100 (Sn) Composition (wt% Sn) Figure 9.9 The lead–tin phase diagram. For a 40 wt% Sn–60 wt% Pb alloy at 150⬚C (point B), phase compositions and relative amounts are computed in Example Problems 9.2 and 9.3. EXAMPLE PROBLEM 9.3 Relative Phase Amount Determinations—Mass and Volume Fractions For the lead–tin alloy in Example Problem 9.2, calculate the relative amount of each phase present in terms of (a) mass fraction and (b) volume fraction. At 150⬚C, take the densities of Pb and Sn to be 11.23 and 7.24 g/cm3, respectively. Solution Tutorial Video: Phase Diagram Calculations and Lever Rule How do I Determine the Volume Fraction of Each Phase? (a) Because the alloy consists of two phases, it is necessary to employ the lever rule. If C1 denotes the overall alloy composition, mass fractions may be computed by subtracting compositions, in terms of weight percent tin, as follows: Wa = Wb = Cb - C1 Cb - Ca = 98 - 40 = 0.67 98 - 11 C1 - Ca 40 - 11 = = 0.33 Cb - Ca 98 - 11 (b) To compute volume fractions it is first necessary to determine the density of each phase using Equation 4.10a. Thus ra = 100 CSn(a) rSn + CPb(a) rPb 318 • Chapter 9 / Phase Diagrams where CSn(a) and CPb(a) denote the concentrations in weight percent of tin and lead, respectively, in the a phase. From Example Problem 9.2, these values are 11 wt% and 89 wt%. Incorporation of these values along with the densities of the two components leads to ra = 100 11 89 + 7.24 g/cm3 11.23 g/cm3 = 10.59 g/cm3 Similarly for the b phase: rb = 100 rSn = CPb(b) CSn(b) + rPb 100 98 2 + 7.24 g/cm3 11.23 g/cm3 = 7.29 g/cm3 Now it becomes necessary to employ Equations 9.6a and 9.6b to determine Va and Vb as Va = Wa ra Wb Wa + ra rb 0.67 10.59 g/cm3 = = 0.58 0.67 0.33 + 10.59 g/cm3 7.29 g/cm3 Wb Vb = rb Wb Wa + ra rb 0.33 7.29 g/cm3 = = 0.42 0.67 0.33 + 10.59 g/cm3 7.29 g/cm3 9.12 DEVELOPMENT OF MICROSTRUCTURE IN EUTECTIC ALLOYS Depending on composition, several different types of microstructures are possible for the slow cooling of alloys belonging to binary eutectic systems. These possibilities will be considered in terms of the lead–tin phase diagram, Figure 9.8. The first case is for compositions ranging between a pure component and the maximum solid solubility for that component at room temperature [20C (70F)]. For the lead–tin system, this includes lead-rich alloys containing between 0 and about 2 wt% Sn (for the a-phase solid solution) and also between approximately 99 wt% Sn and pure 9.12 Development of Microstructure in Eutectic Alloys • 319 M A T E R I A L S O F I M P O R T A N C E Lead-Free Solders S Table 9.1 Compositions, Solidus Temperatures, and Liquidus Temperatures for Two Lead-Containing Solders and Five Lead-Free Solders Solidus Temperature (ⴗC) Composition (wt%) Liquidus Temperature (ⴗC) Solders Containing Lead 63 Sn–37 Pb a 50 Sn–50 Pb 183 183 183 214 Lead-Free Solders 99.3 Sn–0.7 Cua 227 227 a 96.5 Sn–3.5 Ag 221 221 95.5 Sn–3.8 Ag–0.7 Cu 217 220 91.8 Sn–3.4 Ag–4.8 Bi 211 213 219 235 97.0 Sn–2.0 Cu–0.85 Sb–0.2 Ag a The compositions of these alloys are eutectic compositions; therefore, their solidus and liquidus temperatures are identical. 400 L 300 Temperature (°C) olders are metal alloys that are used to bond or join two or more components (usually other metal alloys). They are used extensively in the electronics industry to physically hold assemblies together; they must allow expansion and contraction of the various components, transmit electrical signals, and dissipate any heat that is generated. The bonding action is accomplished by melting the solder material and allowing it to flow among and make contact with the components to be joined (which do not melt); finally, upon solidification, it forms a physical bond with all of these components. In the past, the vast majority of solders have been lead–tin alloys. These materials are reliable and inexpensive and have relatively low melting temperatures. The most common lead–tin solder has a composition of 63 wt% Sn–37 wt% Pb. According to the lead–tin phase diagram, Figure 9.8, this composition is near the eutectic and has a melting temperature of about 183C, the lowest temperature possible with the existence of a liquid phase (at equilibrium) for the lead–tin system. This alloy is often called a eutectic lead–tin solder. Unfortunately, lead is a mildly toxic metal, and there is serious concern about the environmental impact of discarded lead-containing products that can leach into groundwater from landfills or pollute the air if incinerated. Consequently, in some countries legislation has been enacted that bans the use of lead-containing solders. This has forced the development of lead-free solders that, among other things, must have relatively low melting temperatures (or temperature ranges). Many of these are tin alloys that contain relatively low concentrations of copper, silver, bismuth, and/or antimony. Compositions as well as liquidus and solidus temperatures for several lead-free solders are listed in Table 9.1. Two leadcontaining solders are also included in this table. Melting temperatures (or temperature ranges) are important in the development and selection of these new solder alloys, information available from phase diagrams. For example, a portion of the tin-rich side of the silver–tin phase diagram is presented in Figure 9.10. Here, it may be noted that a eutectic exists at 96.5 wt% Sn and 221C; these are indeed the composition and melting temperature, respectively, of the 96.5 Sn–3.5 Ag solder in Table 9.1. +L 221°C 232°C  Sn + L 96.5 200  Sn  +  Sn 100 13°C 0 80  Sn 90 100 Composition (wt% Sn) Figure 9.10 The tin-rich side of the silver–tin phase diagram. [Adapted from ASM Handbook, Vol. 3, Alloy Phase Diagrams, H. Baker (Editor), ASM International, 1992. Reprinted by permission of ASM International, Materials Park, OH.] / Phase Diagrams Figure 9.11 Schematic 400 representations of the equilibrium microstructures for a lead–tin alloy of composition C1 as it is cooled from the liquid-phase region. L w (C1 wt% Sn)  L a b L 300 Liquidus c  Temperature (°C) 320 • Chapter 9    +L  Solidus 200 (C1 wt% Sn)  100  + w⬘ 0 10 C1 20 30 Composition (wt% Sn) tin (for the b phase). For example, consider an alloy of composition C1 (Figure 9.11) as it is slowly cooled from a temperature within the liquid-phase region, say, 350C; this corresponds to moving down the dashed vertical line ww¿ in the figure. The alloy remains totally liquid and of composition C1 until we cross the liquidus line at approximately 330C, at which time the solid a phase begins to form. While passing through this narrow a + L phase region, solidification proceeds in the same manner as was described for the copper–nickel alloy in the preceding section—that is, with continued cooling, more of the solid a forms. Furthermore, liquid- and solid-phase compositions are different, which follow along the liquidus and solidus phase boundaries, respectively. Solidification reaches completion at the point where ww¿ crosses the solidus line. The resulting alloy is polycrystalline with a uniform composition of C1, and no subsequent changes occur upon cooling to room temperature. This microstructure is represented schematically by the inset at point c in Figure 9.11. The second case considered is for compositions that range between the room temperature solubility limit and the maximum solid solubility at the eutectic temperature. For the lead–tin system (Figure 9.8), these compositions extend from about 2 to 18.3 wt% Sn (for lead-rich alloys) and from 97.8 to approximately 99 wt% Sn (for tin-rich alloys). Let us examine an alloy of composition C2 as it is cooled along the vertical line xx¿ in Figure 9.12. Down to the intersection of xx¿ and the solvus line, changes that occur are similar to the previous case as we pass through the corresponding phase regions (as demonstrated by the insets at points d, e, and f). Just above the solvus intersection, point f, the microstructure consists of a grains of composition C2. Upon crossing the solvus line, the a solid solubility is exceeded, which results in the formation of small b-phase pa