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EQUILIBRIUM I N HETEROGENEOUS SYSTEMS OF T W O OR MORE COMPONENTS LOUIS A. DAHL National Bureau of Standards, Washington, D. C.
REQUIREMENTS OF PHASE RULE THEORY
The phase rule, derived by J. Willard Gibbs (1) from the laws of thermodynamics, provides the investigator of phase equilibria with an indispensable guide in testing conditions of equilibrium in heterogeneous systems. In deducing the phrase rule it is assumed that the variables determining an equilibrium state are temperature, pressure, and composition. The phase rule as commonly stated therefore applies only when no other variables are involved. For example, if any phase is so h e l y divided that surface forces exert an appreciable influence on the state of equilibrium, the usual statement of the phase rule must be modified before it can be applied. The phase rule expresses the relation between the number of components, the number of phases, and the number of degrees of freedom of a system in a state of equilibrium. These terms must be defined before the phase rule can be presented and its applications discussed. A heterogeneous system is composed of different parts, each homogeneous but separated from the others by boundmg surfaces. Each of these parts is a, phase. In considering the number of phases in a system, all of those parts which are identical in physical and chemical character constitute a single phase. The components of a system may be chosen somewhat arbitrarily, but the number of components is fixed. As the components of a system there must be chosen the smallest number of con~tituents~by which the composition of each phase present in any state of equilibrium may be expressed. This smallest number is the number of components of the system. The number of degrees of freedom is defined by MacDougall (2) as "the number of intensive variables which can be altered independently and arbitrarily without bringing about the disappearance of a phase or the formation of a new phase." The number of degrees of freedom is sometimes referred to as the variance of a system. Systems possessing zero, one, two, or three degrees of freedom are invariant, univariant, bivariant, or trivariant, respectively. The number of degrees of freedom possessed by a system in a state of equilibrium may be determined by the phase rule, which is expressed by the equation,
Systems in which the vapor phase is absent require large changes in pressure to produce small changes in the temperature of equilibrium. Such systems are termed condensed systems. Except when the effects of large pressure changes are under considerat,ion, the pressure variable is ignored, and the phase rule is expressed by the equation, F = C + l - P
(2)
Systems in which the vapor pressures of liquid and solid phases are extremely low, and in which the conditions of equilibrium a t atmospheric pressure consequently differ only slightly from those existing.when the liquid and solid phases are under the pressure of their own vapor, are treated as condensed systems. Investigations to determine equilibrium conditions in systems of this type are usually carried out with charges in open containers. Equation (2) may be applied to such systems if the vapor phase is ignored. Applying the phase rule to a one-component system, it is found that F = 3 - P. When three phases are present the system is invariant, when two are present it is univariant, and when one is present it is bivariant. This is shown in Figure. 1;'which shows the phases
Tigun, 1. Equilibrium Between Solid. Liquid. e n d Vss.. in a On. c o m p o n e n t system
present in a solid-liquid-vapor system at various temperatures and pressures. Three phases, solid, liquid, and vapor, can exist at equilibrium only at the temperature and pressure represented by point 0. Since changes cannot be made in any direction from F=C+2-P (1) point 0 without causing the disappearance of at least in which F is the number of degrees of freedom, C the one of the phases, point 0 is an invariant point. number of components, and P the number of phases. At any point on the curve OC liquid and vapor may 411
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coexist in a state of equilibrium. Changes in temperature and pressure following the curve OC may be made without causing the disappearance of either phase. Successive increments of such changes are in small segments of the curve, and these small segments may he regarded as straight lines. Since a straight line has only one dimension, the combined change in temperature and pressure in each successive segment occurs in only one dimension. The curve therefore represents one degree of freedom and is referred to as a univariant curve. Within the areas designated as solid, liquid, and vapor in Figure 1, changes may be made in temperature or pressure, or both, that is, in two dimensions. A system in any one of these areas possesses two degrees of freedom and is said to be bivariant. From the foregoing considerations it may be seen that an invariant condition is represented by a point, a univariant condition by a curve, a bivariant condition by a surface, etc. The number of degrees of freedom is the same as the number of dimensions in which small changes in the variables can be made without causing any phase to disappear or a new phase appear. This holds true for systems of any number of componepts and furnishes a 'guide in determining m~hetherthe relations indicated in a phase diagram or space model are in harmony with the phase rule. It should be noted that examinations of the diagram or model with reference to the phase' rule are made as it is constructed in the course of an investigation, as a check on the reliability of observations, and are seldom needed in considering the completed work. Binary Systems. The phase diagram of a binary system differs from that of a one-component system in the fact that there are regions in the diagram in which no equilibrium state is indicated directly. That is, a point in such a region indicates a composition incapable of existing as a single phase a t the corresponding temperature. The hypothetical system in Figure 1 may now be reexamined for the purpose of comparison. Any point in the area8 designated as liquid, solid, and
Per Cent Phenol by Weight Pisum 2.
Tha S ~ t e mPhenol-Wote~ at 160 Mm.
vapor represents the temperature and pressure at which the single phase designated may exist in a state of equilibrium. Any point on the curves AO, OB, or OC represents the temperature and pressure a t which two phases, indicated in the areas on each side of the curve, may coexist, while the triple point 0 common to the three areas, represents the conditions under which all three phases may coexist. The diagram in Figure 2, representing the binary system phenolwater a t atmospheric pressure, illustrates the different situation encountered in the phase diagram of a binary system. Since the pressure is constant, equation (2) may be applied in determining the number of degrees of freedom in systems in which one or two liquids are present in equilibrium. For a single liquid phase, F = 2, and for two liquid phases, F = 1. The conditions under which one liquid phase may exist in a state of equilibrium or two phases may coexist are therefore represented by a bivariant surface or a univariant curve, respectively. The area outside the curve in the figure is a bivariant surface indicating at each point within it the temperature at which a liquid of corresponding composition may exist in a state of equihbrium. The area inside the curve has an entirely different significance, since it indicates at each point within it a temperature at which a liquid of corresponding composition may not exist in a state of eqnilihrium as a single phase. For example, a mixture of the composition 25 per cent phenol, 75 per cent water, a t 35"C., represented by the point P, does not exist as a single liquid phase, hut separates into two liquid phases, L1 and Lz, on the univariant curve. The area inside the curve is not a bivariant surface but a solubzlzty gap. Each point on bhe univariant burve in Figure 2 represents the temperature a t which a liquid of corresponding composition may exist in equilibrium with another liquid phase. This other phase is found at another point on the curve, at the same temperature. For example, at 35°C. liquid L, (9.9 per'cent phenol) may coexist with liquid L2 (67.6 per cent phenol). The line L1L2is a tie-lime, or conode, indicating a t its extremities the phases which may coexist at 35% Although equilibrium conditions are not indicated directly in a solubility gap, it is convenient to designate in the gap the phases which are present when a state of equilibrium represented by the extremities of any tie-line has been attained. Thus, there are two liquid phases indicated in the solubility gap in Figure 2, one more than in a bivariant surface. The solubility gap is a special case in which the phases indicated a t the extremities of tie-limes through the gap are in the same state of aggregation, that is, both are liquids or both solids. There are other gaps in which the phases are not in the same state of aggregation. An example is shown in Figure 3, which represents a hypothetical system in which the components A and B are miscible (intersoluble) in the solid state in all proportions. he univarikt curves A1L3B1 and A1S3B1are the liquidus and solidus curves, re-
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spectively. The areas above the liquidus and below the solidus are bivariant surfaces representing conditions in which only a single phase is present at equilibrium. The area between the curves is a gap in which there is no equilibrium state indicated directly. Any horizontal tie-lme through the gap connects points on the liquidus and solidus curves representing the compositions of a liquid solution and solid solution capable of coexisting a t equilibrium at the temperature indicated by the tie-line. For example, liquid L2 and solid X , may coexist a t temperature T3. AS in Figure 2, the number of phases indicated in the gap is the number associated with a univariant curve, since for any point in the gap representing the coinposition and temperature of a mixture the equilibrium state is shown by reference to the boundary curves. Solidus and liquidus curves are located experimentally by determining the temperatures at which a series of solid solutions begin to liquefy and the temperatures a t which complete melting occurs. I n locating the liquidus it is not necessary to start with solid solutions. A mechanical mixture of components A and B, Figure 3, in the proportions corresponding to composition X will be completely melted a t the same temperature as a solid solution of that composition. On the other hahd, for determination of temperatures to locate the solidus each charge must be a homogeneous solid solution. This may be seen by considering the course of crystallization in the system shown in Figure 3. . Point X o in Figure 3 represents composition X at temperature To, when it is entirely liquid. When cooled to temperature T I the liquid may coexist with solid S,, which appears when the temperature is reduced slightly. As cooling proceeds the solid phase changes in composition along the solidus curve from S, to X3, while the liquid follows the liquidus curve from X I to Lz, at which point all of the liquid disappears, leaving a solid solution of composition X. This is the ideal course of crystauization, described on the assumption that equilibrium is continuously attained during the process of cooling. That is, it is assumed that after each infinitesimal drop in temperature,*constant temperature is maintained to permit all of'the previously crystallized solid phase to transform to the composition of the solid phase in equilibrium at that temperature. Because of the time required for such transfonnations in the solid state the ideal course of crystallization is seldom followed. Instead, the solid phase separating out at any moment is deposited on the surface of crystals already present. This results in composition gradients in the crystals. For example, as composition X , Figure 3, is cooled from temperature T I to T 2 , solid S , separates out first, and solid solutions from S, to Sa are deposited upon the original crystals. Liquid L, is then in equilibrium with solid Sa on the surfaces of the cryst,als, although the system as a whole is not in a state of equilibrium. The composition of the crystals as a whole is between that of S1 and Sz, say S,. Since the line SEX,is longer than SzX2the quantity of liquid L, is greater than in the ideal course of
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Figure 3. Hypothetical Isobe~ic System-Components Intersolubl~in Solid State in A l l Proportions
crystallization. As this process continues in cooling to temperature T3, liquid LZ fails to disappear at that temperature and further crystallization takes place with additional cooling. The last liquid present may be on the liquidus between L and B,, its composition depending upon the rate of cooling. If crystallization is complete a t temperature Ta, for example, the surface of the crystals and the last liquid present will have the compositions S3 and L1, respectively. The effect of the composition gradients in the crystals, from S1 in the interior to S3 on the surface, would be to lead one to locate .the solidus point for composition X at X p instead of X3. Upon heating, the process is reversed, with liquid L3 appearing a t temperature Ta,again leading one to locate the solidus poirit'at X4 instead of X3. On the other hind, if homogeneous crystals of composition X are prepared, point X s may be located by observing the temperature at which liquid first appears. The areas bounded by univariant curves in the diagram of a binary system are termed state regions by Masing (5). Similarly, the spaces ;between bivariant surfaces in a space model are termed state spaces. State regions and spaces are of two kinds: homogeneous, illustrated by the bivariant surfaces in Figures 2 and 3, in which states of equilibrium of single phases are indicated; and heterogeneous, in which there is no equilibrium state, illustrated by the gaps in the figures. In determining whether the number of phases indicated for points, curves, surfaces, etc., are in harmony with the phase rule, it should be borne in mind that gapsthat is, heterogeneous state spaces-are excluded from consideration. A hypothetical system in which the solid phases are only partially intersoluble is shown in Figure 4. In this system the components A and B form a compound designated as G. The regions in which only a single phase, ar, 8, and y , or liquid L, is indicated are bivariant surfaces. Those in xvhich tmo phases are indicated are gaps, or heterogeneous state regions, in which t,here is no equilibrium state indicated directly.
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example, the univariant curves mAl and AIP represent the compositions of phases capable of participating in two-phase equilibria. Similarly, the invariant points m, GI, and P represent compositions of phases capable of participating in three-phase equilibria. Invariant points P and E represent compositions of liquids, each capable of coexisting with two solid phases. Point E is between the points representing the solid phases with which it may coexist, and liquid E may therefore be formed from these solid phases in proper proportions. Conversely, it may separate into the two solid phases as it crystallizes. I t is classed as a eutectic point. On the other hand, point P is not between m and GI, which represent the compositions of solid phases with which liquid P may coexist. Point Composition P is classed as a peritectic point. This distinction Pigum 4. Hypothetical Isobaric Smtem4omponents Partially Inturoluble in Sdid State between eutectic and peritectic points applies to systems of any number of components. As a result of this Horizontal tie-lines through these gaps connect points difference in location of liquids P and E with respect on univariant curves representing compositions of to the solid phases with which they may coexist, there phases capable of coexisting at equilibrium a t the tem- is one gap above mP and two below it, while there are perature indicated by the position of the tie-line. The two gaps above rs and one below it. Marsh (4) refers tie-line ab, for example, connects a solid solution a to this as "an important means of distinguishing beof composition a with a liquid phase L of composition tween eutectic and peritectic types." Components A and B, Figure 4, do not decompose b. These phases coexist a t temperature TI. A mixture of composition X is composed of these phases at below their melting points, and are said to melt contemperature Ti. The quantities of the phases L and gruently. Compound G decomposes into liquid and a are proportional to the lengths of the line segments solid phase a,and is said to melt incongruently. The fact that the regions for solid phases a, 8, and ax, and x,b, respectively. The tie-line ab may be considered to drop as the tem- y in Figure 4 are narrow indicates that A, B, and G are perature is lowered, until it coincides with the line mP intersoluble in the solid state to only a limited extent. at temperature Tz. This line touches thzee homo- If A, B, and G are mutually insoluble in the solid state, geneous state regions a t m, GI, and P. These points of these regions are reduced to straight lines, as in Figure contact represent the compositions of three phases 5 . Phase diagrams of the type shewn in Figure 5 may .. capable of coexisting a t temperature Tz, and they are therefore invariant points. Three phases may coexist at temperature T2, but their proportions change as heat is added or withdrawn, without change of temperature. This results in the presence of only two coexisting phases a t maximum or minimum heat content a t that temperature. For example, mixture X is composed of the phases of a and L a t maximum heat content at temperature T z , in quantitites p~portional to the line segments X 9 and mX2, respectively, while at minimum heat content at temperature T2 it is composed of phases a and y in quantities proportional to line segments XzGl and mXz, respectively. The transformation occurs without change of temperature. Tieline mP is not a phase boundary, but is inserted as a Composition part of the diagram to connect compositions of phases capable of participating in three-phase equilibria and to separate gaps differing with respect to the pairs of Figun 5. Hypothetical Isobaric System4omponents Inaolubla in Solid State phases indicated. The line rs has similar significance. Comparison of Figures 1 and 4 shows an important difference between phase diagrams in which composition result from evidence that the components are immiscible is a variable and those in which it is not. In Figure 1 or practically immiscible in the solid state, but it is a univariant condition is represented by a single curve probable that in many instances the degree of inter' and an invariant condition by asingle point. In Figure 4 solubility in the solid state has not been investigated. When the boundaries of the homogeneous state region a univariant condition is represented by a single curve, and an invariant condition by a single point. For in a binary isobaric system have been located the phase
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diagram is completed by the insertion of horizontal tielines connecting invariant points. That is, the phase diagram is fully established when the boundaries of homogeneous state spaces have been determined. Ternary Systems. A ternary isobaric system may be completely represented geometrically by a space model in the form of a triangular prism, in which the base is a triangle to represent composition, and the axis perpendicular to the base represents temperature. A simple space model of this type is .shown in Figure 6.
rig6. space Model of Hypothetical Isobaris TomSystem. Solid Component. Immiscihlo
The sides of the model are binary systems with eutectics a t El, E2, and Ex. From these points the curves EIE, and E2E and E,E slope downward to E, the eutectic for the ternary system ABC. These curves are boundaries of the bivariant (liquidus) surfaces EE1T.E8, EE2T,E3,and EElToE2,which intersect at E. Any point, curve, or surface in Figuqe 6, which as drawn is only schematic, may be projected perpendicularly to the base of the space model, to obtain a triangular diagram, as in Figure 7. Isotherms are introduced to show the slopes of curves and surfaces with reference to the temperature axis. The isotherms are projections of horizontal contours on the curved bivariant surfaces. The spacing between isotherms indicates the steepness of the surfaces, and it is therefore possible to visualize the surfaces in the space model by considering the isotherms as contour lines. In Figure 6 it is assumed that the components A, B, and C are completely insoluble in the solid state. This simplifies projection of the space model on the triangular base, as there are no. curves or surfaces above one another to be projected. Systems in which the components are insoluble in the solid state, or in which the solid state is negligible, are therefore peculiarly suited to representation in a plane diagram. An
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example of this type, in which most of the solid phases are practically immiscible, is the system Ca0-AlzOr SiOz, Figure 8. This system was investigated by Rankin and Wright (5) and modified by Bowen and Greig (6, 7). Rankin and Wright reported that only the compound Ca0.Si02 takes up other compounds in solid solution to an appreciable extent. Greig (7) discovered the immiscibility of two liquids in the SiOz region. More recently it has been found by Lagerqvist, Wallmark, and Westgren (8) and by Tavasci (9) that the comvound identified as 3Ca0.5AbO. - - bv " ~ & m and Wright actually has the composition Ca0.2AlzOa. This is confirmed in experiments reported by Goldsmith (10). With the exception of regions involving CaOSiOz and those involving immiscible liquids the diagram is generally interpreted with reference to equilibria between liquid and pure solid phases. Assuming the absence of solid solutions, the diagram supplies all of the information concerning equilibria between liquid and solid phases which is obtainable from a space model of the system. It will now be assumed that the components of the system in Figure 6 are partially intersoluble. The rcgions for solid soh~tionsill the linary .y.rems on rhe fnct.3 of the wncr m d e l haw been inrrodurcd in I.'ieurc 9. The solid solutions in which A, B, and C predominate are designated a, P, and y, respectively. The regions for these solid solutions are boundaries of homogeneous state spaces in the space model. The boundaries in the interior are not shown in this figure, but are introduced in Figure 10. Interior surfaces bounding the spaces for the 4 and r. -phases are shaded, while the.space for the or phase is shown in outline only. An exception is the interior surface bnT,r, which is left open to give a better view of th-e horizontal triangle abc. This triangle, which touches one-phase spaces at four invariant points, a, b, e, and E, is similar in signifi-
F b m 7. Projection of Space Model in F i w m 6 on Triangular B-. (T.mpe~.tu.es on IsotherIntroduced Arbitrarily.)
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cance to the line rs in Figure 4. At the temperah-e of this plane (the eutectic temperature) transformations occur with change of heat content. For example, mixtures in the triangle which may be formed by joinmg a, b, and E with straight lines, are composed of liquid E and solid phases a and P at maximum heat content at the eutectic temperature, and of solid phases a, 8, and y a t minimum heat content. At intermediate heat contents the four phases, a, 6, 7, and liquid E coexist a t equilibrium. In this case there are three heterogeneous state spaces (gaps) above the triangle abc and one below, corresponding to the two gaps above and one below the line rs in Figure 4.
and is represented more clearly and completely in a series of horizontal (isothermal) seotions. Certain principles applying to the space model of an isobaric ternary system are not only useful in drawing isothermal sect,ions, but provide checks on the accuracy of the data. I t is because of their application to testing data that they are considered here. With the exception of horizonti! planes separating gaps in the space model, surfaces between gaps are the loci of horizontal tie-lines connecting points on univariant curves. For example, tie-lines between the curves at and cu in Figure 10 generate the ruled surface* atuc, which separates the space for the phases a and 7 from that for the phases or,-?, and liquid L. Similarly, the ruled surface* a'acc' separates the space for Surfaces in the interior of the space model in Figure or and y from the space for a, P, and 7 . Considering 10 are below the liquidus surfaces, EE1T.,E3, EEZTCEd, and EE,T,E,. I n addition, there are interior surfaces corresponding surfaces bounding three-phase spaces, above and below one another. For example, the it may be seen that horizontal planes passing through surface c's'sc is below a portion of the surface CUT& these spaces intersect the boundaries of the spaces in and this in turn is below a portion of the liquidus sur- straight lines, forming triangles. Referring to isoface EE3T,E2. Thus, in a limited area projections of thermal sections, Marsh (4), says that "if the threethree surfaces are superimposed. The projection is space regions are not triangular, then something is simplified if interest is confined t o equilibria between wrong either with the interpretation of the data or with solid phases and liquid, since there is then no need of the data themselves." *A "ruled surface" is a suriaee generated by the motion of s projecting surfaces below the plane in which the tri- straight line. The ruled surfaces in Figure 10 are each generated angle abc lies. However, a system of the type shown in by a. horiaontd line moving in such a. manner that it passes canFigure 10 does not lend itself to satisfactory projection, tinuously through two relnkd curves in space.
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Pi9. Space Model of Hypothetical Isobaric Ternary System. Solid Components Pertially Intemolubla. (Only Exterior S u f s c m Shoam in Thi. Figum.)
Triangular diagrams of ternary systems are generally incomplete, except in the simple case in which there is only one phase of variable composition. Because of the fact that it is usually not feasible to attempt a complete investigation of a more complex ternary system, estahlishing a space model or its equivalent, various devices are employed to make up part of the lack. In some instances the diagram is a projection of the liquidus surfaces only, with symbols or shaded areas indicating the regions in which solid solutions were encountered in the course of the investigation. Another device is to connect by tie-lines points on univariant curves bounding liquidus surfaces with points representing the compositions of solid phases with which these liquids may coexist a t equilibrium. The latter method should be particularly useful when the phase diagram is to be applied to the study of a process in which only partial fusion occurs. Multi-Component Systems. The composition variables in a system of N components may be represented geometrically in a figure of N-l dimensions. The composition variables in a ternary system are therefore represented in two dimensions (the triangular diagram). As has been shown, the introduction of another variable, temperature, leads to the necessity of using a space model for the complete representation of an isobaric ternary system. A quaternary system requires three dimensions for the composition variables, and these variables are usually represented by a space model in
Schreinemakers (11) has pointed out that in a section through the space model of a ternary system extensions of boundaries of one-phase regions meeting at a point should either both be in a three-phase region or they should be in different two-phase regions. This may he seen in the base of the space model in Figure 10. If the curves t'a' and m'a' are extended, the extensions will be in the three-phase region, m p y. Extensions of r'b' and n'b' are in the two-phase regions, or p and p y, respectively. , The hypothetical system in Figure 104s one in which the components are mutually intersoluble in the solid state to a limited extent, and in which transformations are of the eutectic type. I t is only one of a large number of possible systems involving intersolubility in the solid state. For example, two pairs of components may be partially intersoluble in the solid state, while the third pair may be completely intersoluble. In other instances there may be transformations of the peritectic type, or there may be immiscible liquids in the system. In any case, however, equilibria between phases of variable composition lead to the presence of bivariant surfaces above and below one another in the space model. This results in difficulty in projecting the curves and surfaces on the trianmlar base in such a way as to represent the system c%mpletely in a plane. Types of space models of ternary systems are treated Piwn 10. space of xypothotical ~aobaric unary systmm. from standpoint in various works On with Sdid Components Partially Int~reoluble. (Interior Lines and equilibria in heterogeneous systems (3,4,11,12'). cvlsar ~ ~ t t . d .~ ~surfaces t shaded.) ~ ~ i ~ ~
+ +
+
+
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the form of a tetrahedron. A fourth dimension would the form of tetrahedra. In each of these space models be required to represent the temperature variable in a the percentage of one of the components is constant. manner equivalent to the representation of tempera- Each model may be designated by a single equation. ture in the space model of an isobaric ternary system. For example, the equation of a space model in which Since a four-dimensional space model cannot be con- component A is constant a t 15 per cent is A = 15. structed, complete representation of a quaternary Another plan, used by Eubauk (16) is to develop triisobaric system in a space model is impossible. It is angular diagrams in which the percentages of two theoretically possible to introduce isothermal surfaces components are constant. The equations of these in the tetrahedron representing the composition vari- diagrams are in pairs, such as A = 0, B = 5 ; A = 15, ables in such a system, but the difficulties from a prac- B = 25, etc. To understand the significance of points tical standpoint are so great that it is not feasible to do curves, etc., appearing in space models or triangular so. The intersections of isothermal surfaces with diagrams used in this manner it is necessary to know phase boundary surfaces are curves, and these may be the types of intersection which are made with invariant shown on the latter surfaces. While this method fails points, univariant curves, etc., in four-dimensional to show the isothermal surfaces as such, it will give a space. These are indicated in the following table. fair idea of their location. Plane sections through the The table refers to condensed systems of five compospace model may show isotherms, that is, curves which nents, and the vapor phase is therefore ignored. are intersections of the plane section with isothermal surfaces in the model. Investigation of a quaternary INVESTIGATION OF HETEROGENEOUS SYSTEMS system then involves the separate investigation of a A heterogeneous system is completely represented by series of plane sections. This plan was adopted by Lea and Parker (IS) in their investigation of the system a geometrical figure in which the axes are the inCa0-2Ca0~Si02-5Ca0~3A12034CaO~A1~~Fez0~.dependent variables influencingthe conditions of equilibThe occurrence of solid solution in an isobaric quater- rium. The investigation of a heterogeneous system nary system introduces greater difficulties than are may be regarded as an exploration of this geometrical encountered in the investigation of a similar ternary figure to determine the location of significant points, system, since complete representation requires four curves, etc. It is not proposed to present here an exdimensions. In some instances it may be sufficient haustive study of methods which may be employed in to introduce tie-lines from points on univariant liquidus the investigation of heterogeneous systems. A few curves to the compositions of solid phases with which examples may serve to demonstrate that the investigathe liquids may coexist a t equilibrium. In general, tion of a system is guided by a knowledge of the prinhowever, analytical methods are needed for adequate ciples involved in the construction and interpretation of the phase diagram or space model of a system. The examples cited pertain to methods and to types of Tetre variant TriDiUniInsystems with which the author is familiar, and are inhyper- variant variant variant variant tended only as illustrations of the manner in which the volume volume surface curve point investigation of a system may proceed systematically. Number of solid phases in equilibrium with liquid To obtain a view of some of the methods which may be 1 2 3 4 5 employed, let us consider the hypothetical ternary Intersections with soace model and trirtnrmlsr diamam system in Figure 7. In this system the solid phases * Space model Volume Surface Curve Point are assumed to be immiscible, and interest is confined Triangular Area Cube t Point t to the liquidus surfaces projected on this figure from the disgram a * May appear as point in space model, but only when the space model, and to the boundaries of those surfaces. Liquids in the region designated as A are similar in one equation of the space model is satisfied by the invariant point. t May appear as a point in the triangular diagram, but only respect, that solid A is the first crystalline phase to when both of the equatmns of the triangulm diagram are satisfied by some point on the univarisnt curve. The point in the d i e appear in the normal course of crystallization. The aram then represents that particular point, not the univariant reeion is known as the A ~ r i m a r vnhase rezion. Sirni,, .. curve. Inrly, the regions d c s i p & l as R snd Care the prirnnry :May appear ar a point in the tri~n~ular diagram, hut only w h m holh oi the emaiiuw of the trianrmlnr d i n m m arc satisfied phase regions for and (:, rrspertirrl?.. It is app3rent. by the invariant phint. then, that the boundaries of the pr&rtry phase regions may be located by melting a large number of mixtures, consideration of such a system. Methods of analytical and observing in each case the crystalline phase which treatment of multicomponent systems have been given first appears on cooling, that is, the primary phase. little attention in the literature, but reference may be For example, a series of mixtures from X to P will have C as the primary phase, while at P both A and C made to papers by Morey (14) and Dahl(16). Systems of more than four components require more will appear as primary phases. From P to Y the than three dimensions for the composition variables primary phase will be found to be A. Thus, by alone. When only the liquidus in an isobaric quinary "bracketing," point P may be located as a point on the system is under consideration it is possible to represent univariant curve EJE. Another method of locating point P is to determine equilibrium conditions by a series of space models in
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melting points on the line X Y , and to plot these temperatures as ordinates against distances on the line XY, as in Figure 11. Let us suppose that the points designated by crosses in this figure represent the compositions and corresponding observed melting points. Curves drawn through these points intersect at a point, locating P with reference to the line X Y . In addition, the points designated by circles are entered on the curve at 10' intervals, and serve to locate intersections of the line XY with isotherms in the triangular diagram. The line X Y in Figure 7 has been drawn arbitrarily to emphasize the fact that exploration along any straight line will yield data of value. It is also apparent that it is an advantage at any stage to choose mixtures which lie in a straight line. In actual practice the lines along which to explore are selected systematically, starting with lines on the boundaries of the system, in this case the lines AB, BC, and AC. All of the lmes may not he selected a t the outset, but they may be chosen as the investigation proceeds. For example, if it is planned to develop an incomplete phase diagram in which no isotherms appear, short straight l i e s crossing phase boundaries may be sufficient. In that case, portions of phase boundaries established a t any given stage may give an indication of the short straight lines to be selected in the work which follows. Condensed Systems of the Refractory Oxides. In the investigation of systems of the refractory oxides the primary phase for any given composition is determined by the quenchmg method. A charge of that composition is held for a time at a temperature slightly beloni the melting point, and then quenched by dropping suddenly into water or mercury. The liquid in the charge is cooled too rapidly to permit crystallization, and the primary phase is then found imbedded in the supercooled melt, or glass. The primary phase is identified by microscopic examination of the cooled charge. It may appear that only one determination of the primary phase is needed in each field, since melting point determinations by means of heating or cooling curves may serve to locate the phase boundaries. However, difficulties are frequently enwuntered in applying the heating and cooling curve methods, due to such causes as sluggish energy changes and supercooling. It may then be necessary to employ the quenching method in the determination of melting points. Charges of identical composition quenched at successively higher temperature will consist of crystalline phases and glass a t temperatures below the melting point, and glass alone a t temperatures above the melting point. Thus the melting point and the primary phase for any given composition may be determined simultaneously by this procedure. The quenching method is applicable when sudden cooling results in "freezing" the charge to a state identical with the state at the temperature in the furnace but with the liquid converted to glass. In some instances crystallization occurs so rapidly that quenching fails to freeze the charge to a state correspondmg to its stat,e at the furnace temperature. For example, in an
investigation of a portion of the system Ca0-Aldk Fe203 it was found by Hansen, Brownmiller, and Bogue (17) that crystallization was so rapid that glasses could not he obtained by quenching melts, and it was therefore necessary to use heating curves to determine melting points. In a later investigation of the system by Swayze (18) it was found that reduction of the size of charge to 2 to 15 mg. led to satisfactory quenching in most cases, hut that in a portion of the field even the use of 2 mg. charges failed to give satisfactory quench'mg. The choice of a method of determining melting points is dependent upon the characteristics of the system under investigation, such as the viscosity of liquids, rate of crystallization, etc. These characteristics may vary in different parts of a system to such an extent as to necessitate a change in method i~ passing from one part of the system to another. m e n both methods can be used they may serve as a check on one another. Systems with Intersoluble Solid Phases (Solid Solutions). The presence of solid solutions in a system introduces homogeneous (one-phase) state regions or spaces for the solid solutions, and a k heterogeneous ~ state regions or spaces (gaps), in the diagram or space model completely representing the system. Actual diagrams or space models may fail to include such regions or spaces, but their absence leads to difficulties in interpretation. It will be assumed here that complete representation of a system is sought, within the range of conditions considered in an investigation. In the investigation of an isobaric system the range of temperatures considered may be that involved in the process of crystallization or fusion. That is, the investigation may be concerned only with equilibria between solid phases and liquid. In that case, the occurrence of solid solution leads to the necessity of locating solidus curves in a binary system, or solidus surfaces in a ternary system. In the system shown in Figure 4 the solidus curves are the curves Aim, Glr, and B,s, which may be located by determining the temperatures a t which liquid first appears when solid solutions a,y, and 8, respectively, are heated.
JOURNAL OF CHEMICAL EDUCATION
Firmre 14. laothrrmal %&ion of Figure 10 (604. Below Binwg Eut.rtiu and Above Ternary Eutectic
It was mentioned earlier, in the discussion of Figure 10, that the space model of an isobaric ternary system in which solid solution occurs may be represented by a series of isothermal sections. An understanding of the manner in which homogeneous and heterogeneous state spaces appear in isothermal sections is helpful in planning procedures for the development of data equivalent to that required in the construction of a space model. Figures 12 to 14 are designed to illustrate condition? which are encountered. It is assumed in each figure that the liquidus surfaces are those shown in Figure 7. Figure 12 is an isothermal section of Figure 6,-which represents a ternary system in which solid solution does not occur. The curved houndaries of the liquid 'phase region, which is designated as L, are the 100' isotherms in Figure 7. Tie-lines to these houndaries radiate from the vertices representing the pure components. The proportions of liquid and solid a t 10O0, for any composition in a two-phase region, may be estimated from the tie-line passing through the composition. For example, a mixture of composition P is composed of solid C and liquid L at 100" in proportion to the lengths of the lines VPand PC, respectively. The composit.ion of the liquid phase is indicated a t point v. C
Figurn 13. Isothermal Section of Figure 10 (lW"). A~OW B ~ W ~ ~ t~~~~~~~t~~~~ ~ ~ t i ~
In Figure 13 the boundaries of the liquid phase region are the same as in the preceding figure. However, this figure is an isothermal section of the space model in Figure 10, and includes sections through the spaces for solid phases a, p, and y. The curve fg is the intersection of the surface cuT,s, Figure 10, with the 100" plane. The boundaries of the ar and 0 regions have a similar significance. Tie-lines across the twophase regions connect liquid and solid phases capable of coexisting at equilibrium at 100". Extensions of the tie-lines do not necessarily pass through the vertices of the triangle. The compositions of liquid and solid phases a t 10O0,and their proportions, may he estimated for any composition in a two-phase region from the tie-line passing through the composition. For example, a mixture of composition P is composed of the y phase and liquid in quantities proportional to the lengths of the lines v P and Pu. In this system there would he little error in drawing tie-lines to pass through the vertices when making such estimates, hut in many systems it is necessary to locate tie-lines experimentally in sufficient number to permit accurate interpretation of the diagram. At temperatures below the binary eutectics but above the ternary eutectics an isothermal section of the space model in Figure 10 is of the type shown in Figure 14. In this figure the curved boundaries of the liquid phase region are the 60' isotherms in Figure 7. The relation of the regions for solid solutions a,8, and y to the space model may be seen by considering the boundaries of the y region. The curve f'g' is a section of the solidus surface cuT,s in Figure 10, at a lower level than the corresponding curve fg in Figure 13. The curves af' and g'b are sections of the surfaces c'u'uc and c'css', respectively, in Figure 10. The regions in which two solid phases appear are solubility gaps or, since they refer to solid phases, they may he referred to as miscibility gaps. In addition to these regions there are regions in which three phases, one liquid and two solid, are indicated. It will he noted that heterogeneous state regions-that is,. regions in which more than one phase isindicated-are separated from one another by
AUGUST, 1949
421
straight lines. As a result, the three-phase spaces are methods of attack. Other types of ternary systems always triangular. Advantage may be taken of this have not been discussed, but these are treated in detail fact when exploring a system, since only two points in various works on the subject (3, 4, 11, 12). In are required to establish the location of a straight line. dealing with more simple systems it is sometimes The heterogeneous state spaces (two or three phases) possible to show in a single triangular diagram the isoare established when the boundaries of homogeneous therms of liquidus surfaces and the boundaries of threestate spaces have been determined. It may therefore phase regions a t successive temperatures. This form appear to be unnecessary to locate heterogeneous state of treatment was adopted by Bowen (19) in his investispaces experimentally. However, the fact that three- gation of the system diopside-anorthite-albite. In this phase spaces are triangular provides a.means of locating system the albite and anorthite form a complete series significant points in the boundaries of the spaces for of solid solutions (plagioclase), while diopside is not solid solutions, such as points f' and g' in Figure 14. miscible with either of these components. To illustrate, let us consider the triangle f'Pf"in this LITERATURE CITED figure. Point P has been located by melting point (1) GIBBB,J. W., Trans. Conn. Acad. Arts Sei.; "Collected determinations establishing the isotherms on the liquiWorks," Longmans, Green and Co., New York, 1928, Vnl. dus (Figure 7). Explorations may be conducted with - - ~1-, n. c~ Q -f-i.F. H., "Thermodynamics and Chemistry," compositions along the line RS. In this exploration (2) MACDO~GALL, John Wiley & Sons, New York, 1939. the quenching method or the heating curve method (3) MAsmG, G., "Ternary Systems," translated by B. A. may be employed to determine temperatures a t which ROGERS, Reinhold Publishing Corp., New York, 1944. changes occur in the number of phases present. A (4) MARSH,J. S., "Principles of Phase Diagrams," McGrawHill Book Co., New York, 1935. vertical section of the space model is obtained by plot(5) RANKIN, G. A., AND F. E. WRIGHT,Am. J. Sei., 39, 1 ting these temperatures-against composition on the line 11915). ~----, RS. The curves in the vertical section, in the range of (6) BOWEN,N. L.,AND J. W. GREIG,J . Am. Ceram. Soc., 7 , temperatures between the ternary eutectic and the 238, 410 (1924). (7) GREIG,J. W., Am. J.S&.,13,1 (1927). lowest binary eutectic, will separate regions correV., S. WALLMARK, AND A. WESTGREN, 2. (8) LAGERQVIST, sponding to those crossed by the line RS in Figure 14. anorg.ehem., 234, l(1937). The 60' point on the curve between the regions a! y B., Tonid-Ztg.,61,717,729 (1937). (9) TAVASCI, and L rr y locates point m in Figure 14, while the (10) GOLDSMITH. J. R.. J. Geol..56.1(1948). 60' point on the curve between the regions L a! +,.y and L y locates point n. A similar exploration along the line R'S' locates points m' and n'. Upon drawing IYII. the straight lines ~ n Pn', , and mm' to their interse; (12)VOGEL, R,, HeterOgene Gleichgewichte~~; uHandbuch tions, points f' and f" are located. Similar treatment dsr Metallphysik," edited by G. Masmo, Akademische of other isothermal sections will locate successive Verlaeseesellschaft. Lei~sie.1937. , Trans. Roy. Sac., Doints on the univariant curves c u and cs (Figure 10). (13) LEA, F.-M., AND T . W . ' P ~ K E RPhil. . 234, 1 (1934).which are edges of the space, and will also locate (14) MoREE, ,, ,. J , Phys,'Chem,, 34, (1930), corresponding edges of the rr and p spaces. (15) D a m , L. A,, J. Phys. & Coll. Ch~m., 52,698(1948). A ternary isobaric system in which the components (16) EUBANK, W. R.,Dissertstion, Johns Hopkins University, 1947. are mutually soluble to a limited extent in the solid AND R.H. BOGWE, model are (17) HANSEN,W. C., L. T. BROWNMILLER, state, and in which the sides of the J. Am. Chem. Soc., 50, 396 (19287;-Portland Cement binary systems of the eutectic type,'has been selected Association Fellowship Paper 13. to illustrate the problems involved in hvestigating (1s) SWAYZE. M. A,. ~ mJ..Sei..244. I. 65 (1946). systems in which solid solution occurs, and to suggest (ioj BOWEN,'N. ~.,kbid.,40,236,16i ('1915)
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