Hillert M. Phase Equilibria, Phase Diagrams and Phase Transformations: Their Thermodynamic Basis

Подождите немного. Документ загружается.

12.5 Classiﬁcation of sharp phase transformations 265

class I class II

class III

T >T

T <T

Figure 12.9 Different types of four-phase reactions in a ternary system, represented in a

compositional coordinate system.

α+β+γ

β+γ

α+γ

L+α+β

L+β+γ

L+β

L+α

L+β

L+α+β

β+γ

α+γ

L+β+γ

L+α+γ

234

(a) (b)

Figure 12.10 See Exercise 12.6.

four alloys, indicated by arrows, go through on cooling. Show projections of the four-

phase planes and draw lines representing the two sections.

Hint

In both diagrams all four three-phase ﬁelds connected to the invariant four-phase ﬁeld are

shown in the section. It is thus possible to know the type of transformation. In diagram

(a), three of the four fall above the invariant one and these three all contain liquid. It is

evident that this is a four-phase eutectic transformation. Both the alloys, 1 and 2, give

L → α + β + γ and the parts of these alloys, already solidiﬁed, remain unchanged. In

diagram (b) there are two three-phase ﬁelds on each side of the four-phase horizontal.

This must be a class II transformation and both alloys, 3 and 4, give L + α → β + γ.

Solution

The solution is shown in Fig. 12.11.

266 Sharp and gradual phase transformations

(a) (b)

Figure 12.11 Solution to Exercise 12.6.

12.6 Applications of Schreinemakers’ projection

Schreinemakers’ T, P diagram, introduced in Section 10.1, can be very useful in a dis-

cussion of phase transformations, in particular for higher-order systems where all other

methods of reducing the number of axes to two would yield much more complicated

pictures. As an introduction, consider the diagram in Fig. 10.1(b).Itshows that there are

three two-phase surfaces covering the angle between the (α) and (β) lines. By keeping

P and T constant at values within that angle and varying µ

we could expect the trans-

formations α ↔ δ, γ ↔ δ and β ↔ γ.Ofcourse, they should occur one after the other

and only two reaction sequences are possible, β → γ → δ → α or the reverse. By this

consideration we can immediately conclude that the phases are arranged in this order

along the composition axis, z

. This is conﬁrmed by Fig. 10.10. Similar considerations

based on the other angles will give less complete answers. Unfortunately, this very sim-

ple way of deciding the relative compositions of the phases taking part in an invariant

equilibrium gets much more complicated in higher-order systems. The method based on

coincidence, described in Section 10.4,may be more powerful.

Next, let us consider a transformation occurring by changing T or P and keeping

the other constant. If the composition is also constant, then the phase ﬁeld rule from

Eq. (10.6)would yield

d = c + 2 − p − n

+ n

= c + 2 − p − c + c − 1 = c + 1 − p

because n

= 1 + c − 1 = c and n

= c − 1. A sharp transformation should be obtained

for d = 0, i.e. p = c + 1, and should thus occur if the system would cross a uni-

variant line (for which p = c + 1, see Section 10.1). This can be accomplished by

a suitable choice of composition. For illustration, see the arrow in Fig. 10.9. Then

the question is, what type of sharp transformation will it be. From the projection

in Fig. 10.1(b) we would only know that the (δ) line should give a transformation

between α, β and γ.However, the following method can be used to give more detailed

information.

Since δ does not exist along the (δ) line, it can only exist on the other side of the

invariant point. It will thus exist on the upper sides of the other univariant lines but

not on their lower sides (see Fig. 12.12(a)). Using the same kind of information from

12.6 Applications of Schreinemakers’ projection 267

(δ)

(α)

(β)

(γ)

(δ)

(α)

(β)

(γ)

(a) (b)

Figure 12.12 Method to decide the type of transformation. For instance, the δ phase only exists

on the other side of the invariant point, counted from the δ-absent line, here denoted (δ). The

arrow in Fig. 10.9,which is an expansion of the lower part of the present ﬁgure in the z

direction, illustrates the γ → α + β transformation.

Figure 12.13 Detail of binary T, x phase diagram with a point of minimum for the β + L

equilibrium. There the transformation β → L will be congruent. Three sets of lines

representing equilibrium with α at different P are given.

the other lines we get the results shown in Fig. 12.12(b).Bycrossing the (δ) line from

left to right, i.e. by decreasing P, under a suitable constant value of z

,wethus get the

transformation α + β → γ.Itshould be emphasized that the transformations described

by the positions of the Greek letters in Fig. 12.12 only occur when a line is crossed in

the plane of the projected diagram. It gives no information on the transformations in any

other direction.

In Section 10.8 it was concluded that one can include in Schreinemakers’ projection

a curve showing where an equilibrium with p = c phases degenerates by the phases

falling on the same point for a binary system, on the same line for a ternary system,

etc. An example was shown in Fig. 10.27. Such a singular curve may originate from

a univariant line, as demonstrated in Figs 12.13 and 12.14, using a binary system for

illustration.

268 Sharp and gradual phase transformations

L +α L

α+β

Figure 12.14 Schreinemakers’ P, T diagram corresponding to Fig. 12.13. The univariant line

changes character at the black point. That is where the singular curve originates and there is a

compositional degeneracy in that point. The α phase does not take part in the reaction there.

In order to simplify the construction, it was here assumed that the β + L equilibrium is

not affected by P but an increased P will increase the stability of α. Lines for equilibrium

with α are presented for three P values. With the lowest P value, P

, the α + β + L

equilibrium is of the peritectic type. With the highest P value, P

,itisofthe eutectic

type and the intermediate P value, P

, shows the transition where α does not take part

in the transformation of L to β. That will give a singular point on the univariant line for

α + β + LinFig.12.14 and that is where the singular curve for β + L starts.

At lowP (to the left of the transition point in Fig.12.14)analloyof suitable composition

would transform by L +α → β on the univariant line if its composition is such that it

reaches the three-phase horizontal in Fig. 12.13 on cooling. Otherwise, it wouldtransform

by L → β at lower T. That would happen if the composition is to the right of the L point

for P

in Fig. 12.13.Inany case, the transformation L → β would be completed at or

before the point of minimum in Fig. 12.13, i.e. the singular curve in Fig. 12.14.Athigh

P (to the right of the transition point) an alloy of suitable composition would transform

by L → α + β on the univariant line. That would happen for compositions on both sides

of the L point for P

in Fig. 12.13,but usually after a proeutectic precipitation of α or

β.Ifthe liquid alloy can be undercooled by α not nucleating, it may solidify by L → β

according to the part of the L +β phase ﬁeld below the eutectic temperature. The lowest

temperature of solidiﬁcation by L → β according to the phase diagram is again the

minimum. However, this part of the L + β phase ﬁeld is only metastable at P

. That is

why the singular curve in Fig. 12.14 has been drawn with a dashed line to the right of

the transition point.

Figure 10.3 illustrated a congruent point in a ternary system and it was concluded

that the position of such points could also be illustrated by a line in Schreinemakers’

projection. The name ‘doubly singular’ was proposed. Such a curve can originate from a

transition point on a singular curve, much in the same way as the singular curve originates

from a univariant line. This is illustrated in Fig. 12.15 where an invariant equilibrium

is also included. The L +β + γ surface covers the area between the (α) and (δ) lines.

However, to the left of the singular point L/(β + γ) that surface, when coming from the

(α) line, will overshoot the (δ) line, reach the singular curve and then bend back and

end up on the (δ) line. If the composition is suitable, then the alloy will not transform by

12.6 Applications of Schreinemakers’ projection 269

β+γ

α +β+ γ

β+γ

L+γ

L+α

L+β+γ

(α)

(β)

(γ)

(δ)

(L)

β+γ

singular curve

univariant

equilibrium

invariant equilibrium

doubly singular curve

Figure 12.15 Schematic Schreinemakers’ projection of a ternary system, illustrating a possible

arrangement of a univariant line, a singular curve and a doubly singular curve.

L + α → β + γ on the univariant line because there will be no α present. Such an alloy

will solidify by L → β + γ below the univariant line but in any case not later than on

the singular curve. However, if there is a transition point on the singular curve, to the left

of which the solidiﬁcation reaction is L + γ → β, then the solidiﬁcation can only occur

by L → β if there is no γ present. The alloy may then pass the singular curve on cooling

but in any case it should have solidiﬁed before passing the doubly singular curve.

Exercise 12.7

What transformation would occur on crossing the (α) line in Fig. 10.14(b) by increasing

the value of Y

at constant values of Y

, X

and X

Hint

Use the method illustrated by Fig. 12.12.

Solution

β + γ → δ + ε.

Exercise 12.8

Use the three-phase reactions indicated by Fig. 12.12(b) to decide on the relative com-

positions of the four phases.

Hint

The reactions on decreasing T are δ → α + β, β + δ → γ, δ → α + γ and γ → α + β.

Solution

δ must fall between α and β, γ between β and δ, δ between α and γ and ﬁnally γ between

α and β. They must be arranged in the order β, γ, δ, α or in the opposite direction.

270 Sharp and gradual phase transformations

α +β+γ

δ+ε

Figure 12.16 See Exercise 12.9.

Exercise 12.9

Figure 12.16 gives a detail of Schreinemakers’ projection of a quinary system. It shows a

univariant line for the α + β + γ + δ + ε + φ equilibrium and a singular curve for the

α + β + γ + δ + ε singular equilibrium. What transformation can be expected when

the univariant line is crossed?

Hint

Use the fact that the singular curve is stable only on the indicated side of the singular

point.

Solution

To the left of the transition point one can avoid the univariant reaction and reach the

singular curve if φ is not present before the univariant line is reached. Thus, φ does not

form by the univariant reaction but would be consumed if it were present. The reaction

must be α + β + γ + φ → δ + ε.Tothe right of the transition point, the univariant

reaction cannot be suppressed, not even if φ is absent, and the reaction must be α + β +

γ → δ + ε + φ.

12.7 Scheil’s reaction diagram

In many types of systems, P has a negligible effect and without any loss of information

one can section at P = 1 bar. For a binary system one will thus get the usual T, x diagram.

Foraternary system there is one dimension more but one could project in the T direction

and use x

and x

as axes. Such diagrams are useful but tend to be overloaded with

phase boundaries if many phases are solutions because there will be lines showing the

compositional changes of all those phases. A simpler diagram would be obtained by

using µ

and µ

(or a

and a

)asaxes. However, much information would be missing.

Using the method illustrated in Fig. 12.12 one could easily ﬁnd what transformation

would occur on crossing a univariant line but that would be of limited use. In order to

hit the line one must now work with a constant heat content because the projected axis

is T. Furthermore, one would have to vary µ

or µ

which is rarely very practical.

A rather useful method was proposed by Scheil [25] for ternary systems. His reaction

diagram shows how the lines representing three-phase equilibria are connected to form

12.7 Scheil’s reaction diagram 271

L→β+γ

L→α+β+γ

L→α+β

L→α+γ

B−CB−AB−A −CA−C

Figure 12.17 Scheil’s reaction scheme for a simple ternary system with eutectic reactions.

L+λ→κ

L→κ+(Al)

L→θ+(Al)

L+θ+κ→τ

L+θ→τ+(Al)

L→κ+τ+(Al)

Fe−AI Fe−Ni Fe−Ni−AI Ni−AI

Figure 12.18 See Exercise 12.10.

four-phase equilibria as a function of T but with no regard for composition. His diagram

also shows what three-phase equilibria originate from the binary sides. In addition, the

reactions occurring on cooling through the four-phase equilibria are given explicitly in

boxes. The diagram for a simple eutectic system is presented in Fig. 12.17.Ofcourse,

similar diagrams can be constructed for quaternary systems, showing four- and ﬁve-

phase equilibria.

Exercise 12.10

Part of Scheil’s diagram for the Al–Fe–Ni system is shown in Fig. 12.18, reproduced

from a publication. A mistake was made by joining the binary (L + λ → κ) with (L →

κ + τ

+ (Al) ). Try to correct it.

Hint

What phases are common for the four-phase equilibria? What one-dimensional equilibria

should connect them?

Solution

The solution is shown in Fig. 12.19.

272 Sharp and gradual phase transformations

L+λ→κ

L→κ+(Al)

L→θ+(Al)

L+θ+κ→τ

L+θ→τ+(Al)

L→κ+τ+(Al)

Fe−AI Fe−Ni Fe−Ni−AI Ni – AI

Figure 12.19 Solution to Exercise 12.10.

α+β

α+γ

γ+β

α+β+γ

α+β

α+γ

γ+β

(a) (b)

Figure 12.20 A section at constant P and x

through a ternary phase diagram in (b), compared

with a binary diagram in (a).

12.8 Gradual phase transformations at ﬁxed composition

If a new component is added to a system where a sharp transformation with p = c + 1

has been found at constant P (see Fig. 12.20(a)), then the value of c increases by one

unit and for the same transformation one will now have p = c. The dimensionality of

the corresponding phase ﬁeld will thus increase by one unit. This case may be illustrated

by an x

section at a low value of x

(see Fig. 12.20(b)).

It is evident that the phase transformation between γ and α + β, occurring when T is

changed, can no longer be sharp but is somewhat gradual. However, if the addition of

the new component is small, its effect on the actual phase transformation should also be

small and one may still recognize its characteristic features, for instance in the resulting

eutectic microstructure illustrated in Fig. 12.5,inparticular if the temperature has been

changed enough to move the system from the γ phase ﬁeld to the α + β phase ﬁeld

before the transformation has started. The transformation may thus appear as sharp even

though it is classiﬁed as a gradual transformation on thermodynamic grounds. As an

example, we shall now examine a case involving three phases and three components.

Figure 12.21 shows the T projection of such a phase diagram under constant P.

12.8 Gradual phase transformations at ﬁxed composition 273

Figure 12.21 Temperature projection of the isobaric A–B–C phase diagram.

(a) (b)

Figure 12.22 Conditions for (a) a eutectic transformation and (b) a peritectic transformation in a

ternary system under constant pressure and a gradual decrease of the temperature.

In this particular case the same three-phase equilibrium occurs in two of the binary

systems but it has different character, being eutectic on one side and peritectic on the

other. Evidently, there must be a transition between the two types somewhere inside the

ternary system. In order to decide where the transition is situated we must ﬁrst examine

how we can recognize the two types when the compositions of the phases change during

the transformation. This is fairly easy if we consider a system which consists of an L

phase only and if it has the correct composition for equilibrium with the α and β phases.

As the temperature is lowered slightly, the three-phase triangle moves and covers the

composition of the system (see the cross in Fig. 12.22(a)). Evidently, we should expect

the reaction L → L +α + β. Here we have included the L phase on both sides because

it has different compositions and it would be impossible to satisfy the mass balance

condition if that is not taken into account.

The dashed line in Fig. 12.22(a) is the extrapolation of the direction in which the L

phase is moving. It goes through the L corner of the triangle and the average composition

of the system, and it intersects the opposite side, a and b being the intercepts. From

Fig. 4.5 it is easy to see that the α and β phases must form in the proportion b:a and this

will be the ratio between them in the microstructure. As far as α and β are concerned they

have formed from material corresponding to the circle. It may not be very important that

274 Sharp and gradual phase transformations

Figure 12.23 See Exercise 12.11.

this material has been drawn from a phase with a different composition. From this point

of view, the reaction is clearly of the eutectic type. The result will be quite different if the

extrapolation does not intersect the opposite side. An example is given in Fig. 12.22(b).

The composition of the system will then fall outside the new three-phase triangle and

inside a two-phase ﬁeld, L + β. The reaction will simply be L → L + β and L will not

move in the direction of the L line in the phase diagram (solid arrow) but straight away

from the β phase (dashed arrow). L will no longer be in equilibrium with α.

The limiting case is found when the extrapolation, i.e. the solid arrow, is directed

away from the β corner. Using that criterion one could ﬁnd the point of transition in the

phase diagram in Fig. 12.21 if it were equipped with a series of three-phase triangles for

slightly different temperatures. Even though the criterion was derived by considering an

alloy composed of an L phase only, it is more general because, in practice, it may often

be justiﬁed to neglect the diffusion inside the solidiﬁed material in comparison with the

rapid diffusion in the liquid phase. The progress of the reaction at each stage is thus

mainly determined by the momentary composition of the L phase and in which direction

it is moving. For a reaction, where three solid phases are involved, it may be necessary

to make a detailed analysis of the diffusion of all the elements in all the phases. In the

next section we shall consider a special case where one component diffuses much faster

than the others.

Exercise 12.11

The sketched detail of an isobarothermal section of a ternary phase diagram (Fig. 12.23)

shows how all the phases in a three-phase equilibrium change on cooling.

(a) Test how a melted specimen with the composition of the L corner will react on

cooling. Give a reaction formula.

(b) Test how a β phase specimen with the composition of the β corner will react on

cooling. Give a reaction formula.