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The theorem of corresponding relation between neighboring phase regions and their boundaries in phase diagrams (I)
CALPHAD Vo1.7, No.3. Printed in the USA.
NEIGHBORING
pp. 185-199,
0364-5916/83 $3.00 + .oo (c) 1983 Pergamon Press Ltd.
1983
THE THEOREM OF CORRESPONDING RELATION BETWEEN PHASE REGIONS AND THEIR BOUNDARIES IN PHASE
DIAGRAMS
(I)
Muyu Zhao (Department of Chemistry, Jilin University, Changchun, People's Republic of China) (This paper was presented at CALPHAD XI, Argonne Illinois, May 1982) ABSTRACT.
Several fundamental concepts in phase diagrams have been discussed. The theorem of corresponding relation between the number of different phases in the neighboring phase regions Qt and the dimension R, of their phase boundary in the phase diagrams is derived from the phase rule. The five corollaries of this theorem for isobaric phase diagrams are derived also. By applying this theorem and its corollaries, the following results can be obtained. Both the boundary rule and contact rule of phase regions in phase diagrams may be deduced. The ten empirical rules for constructing the complex ternary isobaric phase diagrams from phase diagram units may be put on a theoretical basis. The relation between neighboring phase regions and their boundary in phase diagrams of one-, two--, three- and four-components of all types and in the different horizontal and vertical sections of ternary phase diagrams may be explained without introducing any supplementary concept. 1.
Introduction
A phase diagram is composed of several phase regions, it is very important to find out the implicit relation between them. For this purpose, we must study: (1) what is the kind of another phase region which can be bounded to a given phase region? (2) what is the characteristic of their boundary? That is, we must study the relation between neighboring phase regions (abbreviated to NPRs) and their boundaries, which is important for interpreting phase diagrams and in constructing of phase diagrams,since the phase rule usually cannot accomplish this task directly. Therefore there have existed many empirical rules and two general rules dealing with NPRs and their boundary. The general rules are as follows: 1.1 The boundarv rule (I_). It points out that any phase region containing Qll phases can be bounded only by the phase region containing $2 phases: “2-J,
i
(R -
R;,
(1)
where R is the dimension of the phase diagram, i.e. the number of independent variables among temperature (I'),total pressure (P) and concentration (x.), which constructthe phase diagram or a section of it; R; is the dimensioh of the boundary between two NPRs. This boundary is, in fact, a set of system points, which is called alloy points in other references. 1.2 The contact rule or law of adjoining phase regions derived by Palatnik and Landau(Z) is , =R-D+-D-3 0 Rl and R have been already defined; D+ or I)- is the number of phases which r disappear in crossing the boundary from one phase region to another Received 18 May 1982 185
respectively. The boundary discussed by PaLatnik and Landau is, also, a set of system points. Although these two rules have been well accepted and used extensively, there exist some shortcomings. In fact, these two rules are used to treat two different aspects of the same problem: the relation between NPRs and their boundary, but they fail to uniteand to be deduced from each other. There are many exceptions to them, therefore some supplementary concepts such as degenerate regions should be introduced, or the invariant regions in isobaric phase diagrams of N components must be considered as (N+l) phase regions, etc. According to phase ruler we have derived the theorem of corresponding relation between NPRs and their phase boundary, and its five corollaries by which the following results may be achieved: (1) Both the boundary rule and contact rule are derivable. According to our derivation, it is easy to point out when the rules would be invalid. (2) The ten empirical rules, derived by F. N. Rhines (3) for constructing complex ternary phase diagrams from phase diagram units may be interpreted theoretically. (3) The relation between NPRs and their boundaries in phase diagrams of one-, two-, three- and four-components of all types, and in different horizontal and vertical sections of ternary phase diagrams may be explained without introducing any supplementary concept. 2.
Some Fundamental Concepts for Phase Diagrams
2.1 The dimension of an isobaric phase diagram, R is equal to the number of components N in the system. 2.2 The phase region and the number of phases in it can readily be defined in such a way that its dimension equals the dimension of the phase diagram given. Let the number of phases in any phase region be I$, evidently, the minimum number of phases in any phase region is 1, so that a)l. In an isobaric phase diagram, since the dimension of phase region should equal that of the phase diagram, the temperature of the system must be an independent variable at least. Otherwise, if the temperature maintains constant, the phase region can exist in an isothermal section of the phase diagram only. Since temperature is an independent variable at least, then the degrees of freedom of the system fbl. According to the phase rule, 4=N+l-f (pressureis constant), as f>.f, the number of phases existing in any phase region $$N, so N3@)7. 2.3 The neighboring phase regions When discussing the two NPRs on both sides of common boundary, let $1 and $2 be the numbers of phases in the first and second NPRs respectively, and let the total number of different phases in both NPRs be @. If many phase regions meet one another at this boundary, then Q, is also the total number of different phases in all these phase regions. On calculating of Q, if the same phase appears in several phase regions, it should be counted once only. A+ is the number of common phases existing in both NPRs, evidently, @=i$r+$2-A+t because the common phases have been counted twice in 41 and a2, so that A@=Q1+@2-Q, 2.4 The maximum number of phases existing in any one phase region of two NPks,Q,ax. It has been proven that the maximum number of phases existing in any phase
NEIGHBORTNG PHASE REGIONS AND THEIR
BOUNDARIES
187
region is i$ \cN. And the maximal number of phases existing in any one phase region of t!%xtwo NPRs is rjmax
The boundary and phase boundary of the NPRs.
The common boundary of two or more NPRs may be understood in two different points of view: The boundary is set of system points, by which the NPRs are separated from each other. This is the case in the boundary rule and contact rule. Let the dimension of boundary be R'l.
On the other hand, the phase boundary (represented by PB) is regarded as the set of phase equilibrium points of the system existing at the common boundary. Phase points are more important than system points for discussion on equilibrium problems. Let the dimension of PB be RI. In phase diagrams, R1 may equal R1 or not, The parameters which characterize the relation between the NPRs and their boundary may be explained with Fig. 1. Example 1: sI+L/sl+sp are the two NPRs. $1=2, (SI and L), 1$2=2, (sland SZ) @=3, (L, s1 and ~2) A$=1 (the common phase is ~1).
b
The boundary is line cE, Ri =l. The PB are three points: C(Sl) I d(sp) and E(L), Rl=O so R; # R1 in this example. Example 2, the two NPRs are L/L+s,. Their boundary is line aE. The PB are lines aE and aC. The common phase is I J L. Most part of the system exists at Ae fB line aE, only an infinitesmal of the system exists at line aC. So PB is mainly the line aE, i.e. the PB is mainly the set of equilibrium points of the Figure 1. Binary isobaric phase common phase in equilibrium with the diagram with eutectics L-liquid phase, sir s2 other phases in the two NPRs. Then, in this case, boundary is identical with -solid solutions the PB, Rl=Rj . If the regularities and ranges of variations of @, A@, @max, RI and R; and the relation between them may be determinedin all different phase diagrams, the relation between the NPRs and their boundary in these phase diagrams can be explained thoroughly. 3.
3.1
The Theorem of Correspondina Relation and its Corollaries tar Isobaric Diasrams
The theorem of corresponding relation.
Since the PB of NPRs is a set of equilibrium phase points of the system, therefore the system existing at the PB must obey phase rule. The general form of phase rule is known as: R,= f=(N-r-Z) + 2-@+K where, R,, f and Nwere defined previously, r is the number of independent
(3)
chemical reactions in the system, Z means the other independent conditions which constrain the concentrations (except the conditions Xx$=1, where x? denotes the mole fraction of the i-th component containing in the l$-th phase,ii-I, 2,...., N), K is the number of independent variables other than T, P and Xi; in general, R=d, The dimension of PB, RI, equals degrees of freedom f. The PB is shared by two or more NPRs , so the different phases, which exist at PB, otherwise in these NPRs, must be the same existing in the system these NPRs cannot meet each other at this PB, so @=O
(41
# and $ have the same value, but they are distinct in physical meaning e Sub-
stituting f4) into (31, we have: RI=(N-z-Zf+2-WR
(51
@=(N-r-2)+2-RI+K
(5a)
l?qus(5) and (5a) show us the definite corresponding relation between QI and RI. This relation is valid practically for relationships between Q, and RI for all cases, so it is called the theorem of corresponding relation between 0 and R1 in phase diagrams, and expressed by TCR. It is able to solve all the problems treated by the contact rule. This theorem is derived from the p&e rule accompanied by the relation at=4, therefore TCR is not a variant of the phase rule. 3-2 The corollaries of the theorem of corresponding relation (TCR) for isobaric phase diagrams. The first corollary: Since RI@, we have:
The range of variation,of ct,in any phase diagram. (N-Z-r)+2+K>@.
There must be two or more phases in two NPRs, i.e. @>2; therefore: (N-Z-r)+2+K)@$?
(6)
The secondcorollary: The range of variation of RI in any phase diagram. @32, (N-r-Z)+X3RL, therefore:
Since
(N-r-z)+K)RI)O
(71
When Z=r=K=O* P= constant, (6) and (7) become: (Nil))@>,2 (N-l)>R1>0
(6a1 (7a)
respectively: the ranges of variations of @ and R 1 in usual isobaric phase diagrams of N components are thus defined.
The third corollary. In isobaric phase diagrams of N&2 components and with Z=r=K=O, there are two cases in which A$=0 in the two NPRs. First in phase diagrams mentioned above , one-phase region may meet each other at single points only. In this case, A+=O, RI--O, Proof: If two one-phase regions i and j meet each other at a particular PB, then all the common phase points in this PB must have:
x:=x{ ,
i-
x2-x
ej,
..*
. ..*
i
‘
xN=xN
j
(8)
NETGHBORTNG PHASE REGIONS AND THETR BOUNDAR‘LES
189
According to the TCR, when P=const-, then: RI=(N-r-2)+1-@ There are (N-l) independent equations in (8), so Z=N-1, (In a phase diagram with Z=r=K=O in general; there may exist some PB in which Z#O), and r=O I @=2, therefore: R1=N-(N-1)+1-2=0 Since degree of freedom vanishes, only single points may satisfy th is condit ion. So in an isobaric phase diagram of N components and Z=rk=O, one-pnase regions can meet each other at single points only. Secondly,ininvariant regionsof the phase diagram mentioned Rl=O, there is only one case in which A$=0 in the two NPRs. that, if @>N, then Qmax\N, so $max in the equal to N, the minimum number of phases in any phase region
above, @=N+l, It has been proved two NPRs may be 4min'l. Therefore:
In all other cases, A$>l. The fourth corollary.
In the phase diagram mentioned above, if Rl+l, then: (Q-l),A@bl
(IO)
Let us consider (A@)min in this case. According to the definition of A@, A$ is an inteqer (x0). If we can prove A� in the above condition, then AQlgf. Let us apply the reduction to absurdity. Assume that there is no common phase in the two NPRs and the dimension of their PB, R1)l. Since R,al, the PB must be an equilibrium phase line at least. According to the characteristics of phase diagrams, if Rl>l and there is no common phases in the two NPRs, there must be one phase in each of these NPRs, which meet each other at this PB, otherwise the two NPRs cannot meet each other atthis PB. But according to the third corollary, one-phase regions can meet each other at single points only. Thus, one phase regions cannot meet each other ata PB with Rl>l. So the assumption is wrong. Therefore, if there is a PB with R1)l between two NPRs, these two NPRs must have one or more common phases, i.e. A$>l. The PB with Rl>l between the two NPRs is the phase boundary of this or these common phases. Then consider (A"i~ax,:,:~ta~~o~~~~~d~~ &: ;r,z;'_;:; sinOIif Pk;z;e;jA;l; gram mentioned above, A4 must be less than $max by one at least, otherwise, the two NPRs Then A@ is less than @ by one at least also, i.e. (Qk??&'be identical ?)>/A@. Combining ;he conditions of (A+)max and (AQ)minr (10) may be obtained. The fifth corollary.
In the phase diagram mentioned above, if Rl=O, then: (@-2)>A+>,O
(11)
Since Rl=O, according to TCR, @=N+l, so O>N, then emax
MuwZRAo
190
4.
Other Rules Governing the NPRs and Their Boundaries in Phase Diagrams
4.1The relation between R:
and R1.
4.1.1A characteristic of phase boundary. Let us consider the PB of isobaric phase diagrams of N()2) components and with Z=r=K=O, RI&I, A1+22. According to the 3rd corollary of TCR, one-phase regions may meet each other at single points only: they can neither cross each other nor meet along a PB with Rr>,l. A PB with RI>? is a portion of a common phase in the two NPRs, therefore, the two or more PBS with Rx)1 must possess the same characteristic as the two or more one-phase regions possess, i.e. they can meet each other at single points only. The condition is rather simple. Consequently, the boundary, which is enclosed by two or more PBS, cannot be very complicated. 4.1.2 The relation between R; P=const.
and R1 in phase diagrams with Z=r=K=O, N>2 and
4.1.3 Rl>l, then: Ri =Rl+A4-l
(12)
According to the 4th corollary$f R1)1,A.$31.WhenA$=I, there is one PB with RI>?, and the boundary (set of system points) is identical with the PB, then, R'l = RI. When A$=2, there are two PBS with dimension of R1 between the two NPRs. These two PBS enclose a boundary with dimension of Ri which is one more than Rl, Rf =Rl+l. ff A4=3, there are three PBS with dimension of R1 between the two NPRs. These three PBS enclose a boundary with dimension R; =Rt+2, and so on. In short,
Ri =Rl+A$-1
This relation is shown in Table 1. Table 1
Rl=l A@
1
2
3
The number of common phase lines
1
2
3
Ri =Rl+A@-1
1
2
3
The character of boundary
line
surface*
space**
* The two corresponding equilibrium phase points of the two equilibrium phase lines join to a tie-line, the locus of the tie-lines forms a curved surface. ** The three corresponding equilibrium phase points of the three phase boundary lines form a triangle, the locus of these triangles forms a space of three dimensions.
The conditions of R1)2 are similar, i.e. Ri =Rl+A$-1 also, discussion is to be omitted. 4.1.4 Rl=O and there is no invariant region with (4max+l) phases between the two
NPRs, the transition from one NPR to another occurs only when the total composition of the system is changed. In this case, A$)l, Ri =Rl+A+-1. (12a) Take sl+L/L+sl in Pig. 1 for example. The boundary and PB of sl+L/L+sa are points, Rl=R, A+=?, Rf =R,+A+-?=0+1-7=0. This condition is not importantin a phase transition of closed system, it needs no further discussion. 5 Rr=O and there is an invariant region with (d;max +?) phases between the two s: the two different cases may be: There exists an eutectic or peritectic reaction. There exists maximum (or minimum) melting (or boiling) point. both cases, Ri =R,+A+
a. b.
In
Take the boundary line cE between SI+L/S~+S~ in Fig. 1 for example. this case, Rl=O, Ai$=t8 Ri ==Rt+h~$=O+l=t.
(13) In
It is shown in table 2, how the boundary varies with 54, when RI=O.
Table 2.
Rl=O
I
Ri
=R1
1
0
AQ,
I
+ag
The character of boundary
5.
The
5
point
I
I
I
2
I
7
tie-line
a region of triangle
Rule Governinq~ the Process of Cooling
As the temperature decreases, the system changes from the phase region stable at higher temperature to the stable one at lower temperature, RI decreases simultaneously or remains unchanging, until Ri decreases to zero. If the temperature decreases further, R1 may increase once: then decreases again or remains unchanged , until RI decreases to zero* and so on. This rule is obtained by inductive method, and is valid for most cases. This rule reflects such a physical background: when the temperature is high enough, the entropy effect is significant, the mutual solubilities of components are large. There is only one phase in the gaseous state under ordinary pressure or only one phase in liquid state frequently. That is to say: the number of phases in a particular phase region at high temperature is small, the value of Ipof the two NPRs is small too, then Rt of the PB is large at high temperature. When the temperature decreases, the solubilities betweendifferent components decrease, hence separation and precitation occur: the number of phases in a given phase region increases, and the value of cb increases too, then the value of RI decreases. 6.
Deduction of the Boundary Rule and the Contact Rule of Phase Regions
Both rules are not universally valid. From TCR and its corollaries, both of them may be deduced in those cases in which they are valid.
MYU z?IAo
192
6.1 The deduction of the contact rule Ri
=R-D+-D-
(14)
By the definition of D+ and D-, (D++D-) is the number of different phases belonging to either one only of the two NPRs, i.e. it does not include the number of common phases existing in both NPRs. Since 0 includes the number of the common phases, AO, then: (P=D++D-+A@
(15)
In generalfor isobaric phase diagrams with r=Z=K=O and P=const, according to TCR: R1=N+l+
(16)
substituting (15) into (16), (17) is obtained: R1=N+l-(D++D-+A$) In an isobaric phase diagram, the dimension R is equal to N. this relation onto (17), and by rearranging, we obtain:
(17) Substituting
R1+A@-l=R-D+-D-
(18)
According to (12), when R1)1, AQ>l, R; =R1+AQ-1; from (18), we have: =R-D+-DR; 6.2 The deduction of the boundary rule: $z=Ql*(R-R; 1
(19)
Since the order of $1 and a2 may be arbitrary, we may appoint the phase region containing more phases in the two NPRs as the first phase region, and R>R; , according to (19), so: (20)
+I’+2 then (19) may be rewritten as:
$2=$1-(R_Ri 1 (21) similarly, if Z=r=K=O, P=const, then R=N, substituting it into (21), we have: $z=+(N-R; (22)
1
(22)
can be proven as follows.
The boundary rule is not valid for all cases, it is correct only when $1=0, and invalid for a>$,_ For instance, let us consider the PB of the NPRs L+s1/L+s:! in a ternary isobaric phase diagram, where @=3, 411=2, 0>$1, Ri =I. But according to boundary rule, then: $2=$1+(R-R; )=2+(3-I)=0 or +4 It does not tally with the real phase diagram. Many other similar examples may be cited. In a word, the boundary rule is valid only for @=$1. Then, in the effective scope of boundary rule, we may let $1=@, therefore: @2=(2$1-2@)+$2=$1-0+($1+$2-@)=@1-@+A$ (23) According to TCR, Q=N+l-R1, substituting this relation into (23), and rearranging we get: @2=$1- N-(R1+AQ-1) According to (12), when R131, Ri =R1+A@-1, then (22) may be obtained and (21) may be obtained from (22) directly. As already mentioned, we choose Q1>@2 just for convenience, however, the order of phase regions is practically arbitrary, if +2>$1, then: $2=$1+(R-Rf )
(24)
193
NEIGHBORINGPHASE REGIONSAND THEIR BOUNDARIES
Combining
(21) with (24), (19) is obtained too.
4.3 The limitations of the two rules. In the course of derivation of both rules, the limitations are obvious. Since our theorem is valid for all cases and in the course of deducing these two rules, several supplementary equations have been employed; because these supplementary equations are valid only in certain cases, the validity of these two rules is thus limited. The limitations are as follows. The contact rule is invalid: (a) When Rt=O and there is an invariant region with (Qmax+7) phases between the two NPRs. (b) When Z or r#O. The boundary rule is invalid in these two cases also. is invalid further when @>+I. 7.
The boundary rule
The Establishmentof Two NPRs and Their Boundary
7.1 If the combination of phases in the first phase region and the characteristics of common boundary are known, the combination of phases in the second phase region may be fixed. According to the conditions given, +1, RI and R; are known, Since @=N+1-R,, Q may be evaluated. From (12) and (13), A$ may also be determined from R1 and Ri . Since A@=@ltQ2-0 and Qr 01 and A@ are known, $2 may be calculated by: QZ=@+AQ-+i
(25) The number of common phases existing in both phase regions is A+, the number of phases belonging to the first phase region only is (@l-A+). The number of phases belonging to the second phase region only is: ~2-A~=(~+A~-~,)-A~=~-~l. 7.2 If the combination of phases in the two NPRs are known, the characteristics a? the common boundary may be determined. Since the combinations of phases in the two NPRs are known, all the values of ($1, +?, Ad, and @ may be established. According to TCR, the value of RI may be calculated from @. Furthermore, the value of Ri may be calculated from Rr and 5Q also, so that the characteristics of the common boundary are well determined. 8.
The Phase Diagram of Unary Systems
Phase diagrams of unary systems are very simple. But both the contact rule and the boundary rule fail to interpret them directly. However, they can be easily interpreted by TCR without any additional concept. For the sake of saving space, the discussion is omitted. 9.
Phase Diagrams of Binary and Ternary Systems
Applying TCR and its corollaries to isobaric phase diagrams of binary and ternary systems, the results obtained are listed in Table 3 and 4.
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194
Table 3.
of binary systems
RI, 0, AQ and Ri
r=Z=K=O, P=const, R1=3-@
I
RI> (l>R,aO)
I
1
0
@(3>@>2)
2
3
Q, max
2
2
A4
A+=1
laA@>O
Ri
R1 = R1 +A@-1=1
R1 =Ri+AQ=AQ,
The combinations of the phases in the two NPRs
L/L+si
A@=O, Ri =O
L+Si/Si
L/Sl+SZr
A@=l, RI =7
si/sl+ S2
L+Si/L+Sj,
Table 4.
RI, QI, AQ and Ri
r=Z=K=O
P=const
Ri
(2bR1>0)
I
L+Si/Sj
L+Sj/S,+
of ternary systems
R1=4-0
-
I
2
1
0 -
@(4)@&2)
2
3
4
@ma,
2
3
3
A4
Ri
I
S2
-
A@=1
2sAQl
R; =Rl+A$-I=2
The combinations of the phases in the
L/L+Si
two NPRs
Si/Si+S
2>A@bO
Ri =R1+A$-l=A~ A~$=11 Ri ~1
L+Si/Si j
Ri =R~+Ac$=A+ A$=O, Ri =O
L/L+Si+Sj
L/sl+sz+sB
L+Si/L+S.
L+Si/Sj+Sk
3 L+Si/Si+S*
3
-
L+Si+Sj/Sk
L+Si+Sj/Si
AQ=l,
Si/Sl+S2+S3
L+Si/L+Sj+Sk
SifSj/Sj+Sk
L+Sj,/Sl+S2+S3
Ri =I
L+Si+Sj/Sj+Sk L+Si/L+Si+S.
3
AQ=2,
R1' =2
L+Si+Sj/Si+S j
L+Si+Sj/S'+S2+S3
Si+Sj/Sl+S2+S$
L+Si+Sj/L+Sj+Sk
Notes for Table 3 and 4: L denotes liquid phase: sl, s,and s3denote_soJid solution phases: i, j, k=l, 2 3; i#j#k. In a few cases, there are fi/fJ, Q=2, Z=l, or 2, Rx=O. Since the case, where R&=0 and Rf =Rl+A@-I is not important in usual phase transition of closed system, it is not discussed here.
NEIGHBORINGPEASE REGTONSAND THEIR BOUNBARIES
195
These tables show all the relations between the NPRs and their boundaries in these binary or ternary phase diagrams, There are three kinds of systems which obey the relation shown in the above tables: (a)The components are completely miscible both in liquid and solid state. (b)The components are miscible in the liquid state, but there occurs an eutectic reaction between one or more pairs of the components. The system has a Simple Peritectic (i.e. there is no compound formed) If there exists a maximum (or minimum) melting (or boiling) point or compound formation in the phase diagram, we must use (5) to treat it, the relation between the NPRs and their boundaries in this phase diagram may also be obtained. If there are two or more invariant regions in the phase diagram, then the phase diagram may be divided into two or more units, each of which contains one invariant region. These units may be analyzed separately. (c)
The possible forms of transition from one phase diagram unit (abbreviated to PDU) to another may be discussed also. Take a binary system for instance, three phases form an invariant region and a PDU. Let these phases be f', f* and f3. There are phases, f4, f5, . . . . f3+p in the phase diagram also. Let i, j, k, =I, 2, 3 and i#j#k; 1, m=4, 5, . . . 3+p, Pfm, f' and fm denote the phases that appear when transferring from the PDU to another. The possible forms of transition from one PDU (composed of f', f2and f3 in this case) to another PDU are:
invariant transition Rl=O, @=3, A$=0
univariant transition RI=?, @=2, A$==1 fi/fi,fl
fi/fl+fm fi+fj/fl Rl=O, G-3, A@=1 fi+fj/fj+fl Rl=O, 0=2, Z=l, A$=0 fi/fl
The possible forms of transition from one PDU to another in a complicated ternary phase diagram are similar. Discussion of it may be omitted. The relation between the NPRs and their boundaries in phase diagrams of four-component systems could be explained also by applying TCR and its corollaries too. An article about quaternary phase diagrams is to be published later. 10.
Explanation of Rhines'Rules
(3)
We can now turn to F. N. Rhines' ten empirical rules which must be obeyed when constructing a complicated ternary phase diagram from phase diagram units. These rules are as follows:(J) 1)
One-phase regions may meet each other only at single points, which are also temperature maxima or minima.
MUYU ZHAO
196
One-phase regions are elsewhere separated from each other by two-phase regions representing the two phases concerned; thus the bounding surfaces of one-phase regions are always boundaries of two-phase regions. 3) One-phase fields touch three-phase regions only along lines which are generally nonisothermal. 4) One-phase regions touch four-phase reaction planes only at single points. 5) Two-phase regions touch each other along lines which, in general, are nonisothermal. 6) Two-phase regions are elsewhere separated by one- and three-phase regions by bounding surfaces they are enclosed. 7) Two-phase regions meet three-phase regions upon "ruled" bounding surfaces generated by the limiting tie-lines. 8) Two-phase regions touch four-phase reaction planes along single isothermal lines, which are limiting tie-lines. 9) Three-phase regions meet each other nowhere except at the four-phase reaction isotherms. 10) Three-phase regions are elsewhere separated and bounded by two-phase regions involving those phases which are held in common by neighboring three-phase regions. All these ten rules are included in the Table 4 and its notes. They are expressed in more general forms as follows. Let i, j, k, 1 = 1, 2, 3, 4 and they do not equal one another. 2)
1)
fi/fj, @=2, Z=2, A$=O, Rl=O. There are only single points satisfying this condition.
2) 3)
fi/fi+fj/fj @=2, R1=2, A$=1 , R'l=R1=2.
There are phase boundary surfaces.
fi/fi+fj+fk, 0=3, Rl=l, A$=l, RV1=R1=l, phase boundary lines. fi/fj+fk+fl , 9=4, R1=O, A@=O, RUI=Rl+A@=O single phase points.
4) 5) fi+fj/fj+fk , 0=3, Rl=l, A@=l, R'l=Rl+A$-191, nonisothermal phase lines. 6)7) fi+fj,'fj/fj+fk 0=2, R1=2, A$=l, R',=R1=2, phase boundary surfaces. fi+fj/fi+fj+fk/fj+&3, RI=1 I A@=2, Rtl=Rl+A$-l= I 1+2-I=2 bounding surface generated by the limiting tie-lines. 8)
fi,fj/fi,fk+fl , @=4, R1=O, A@=l, R'l=R,+A$=l, tie-lines.
9) fi+fj+fk/fi+fk+fl , 0=4, Rl=O, A+=2, if there is an invariant phase region between the two NPRs,Ri =Rl+A$=O+2=2. If there is no invariant phase region between the two NPRs i.e. the two NPRs are located at the same side of the invariant phase plane, Ri =Rl+A$-1=0+2-l=l, the boundary is a tie-line. Except the four-phase reaction plane, the two three-phase regions cannot touch each other elsewhere. Since 2)R190 in ternary isobaric phase diagrams; R1 equals zero only in a four-phase plane, it equals one or two elsewhere. If Rl=l, 84~1, let the first phasekrepion contain fi+fj+fk and A$=l; then the second phase region contains f +f +fm, a=5 accordingly. But this is impossible, because 0~4 in ternary phase diagrams. If AQ=2, then CJ=QJ~+@~-A$I= 3+3-2~4, SO, Rl=O; it is impossible, because it has been assumed R1=l. If AQ=3, then the two three-phase regions are identical. A+ cannot be larger than 3 in a ternary system, therefore, there is no other possibility. If rq=2, @=4-R1=4-2=2; it is impossible because there are three phases in the first three-phase region yet, 0 cannot be less than bl. Therefore, the two three-phase regions cannot touch each other elsewhere except on the fourphase reaction planes. (10) fi+fj+fk,fj+fk,fj+fk+f1,@=3, R,=l, A$=2, R;=R.,+A$-1=1+2-1=2, bounding surfaces generated by the'limiting tie-lines. Rhines' consideration was not complete. A supplement may be made: three-phase regions may be separated by one-phase regions i.e.
197
NEIGHBORING PHASE REGIONS AND THEIR BOUNDARIES
fi+fj+fk/fk/fk+fl+fm, in this case, @=3, Rl=l, A$=l, 11.
Ri
=I.
The Horizontal Sectionsof Ternary Phase Diagrams.
The difference between line ac (ab, bc also) and line fg (ad, ce, cj, hi, . . . . also) cannot be distinguished by contact rule; because according to this rule, for both lines ac and fg, Ri =R-D+-D-=2-1=1, therefore there is no difference. But according to TCR, for line fg; R,=N-@=3-2=1 (for T, P are kept constant)there is one degree of freedom. This line is the phase boundary line. For line ac, the PB of Sl+L/L+S1+S2, R1=3-3=0, there is no degree of freedom, A@=2, thereA fore, line ac is a tie-line. Since the dimension of the PB in the horizontal section is, in general, less than the dimension of the PB by one in the space diaqram,then if the dimension of the PB in the horizontal section has been fixed, so the character of the corresponding PB inthe space diagram may be known. Figure 2.
12.
Horizontal section of a ternary phase diagram
The Vertical Sectionsof the Ternary Phase Diagrams
In the vertical section of the ternary phase diagram, there are three indepedent variables, i.e. T, xl, and x2, R=3, though the vertical section is drawn on a plane. R1=4-@. A 1
x
Figure 3.
m
n
Y
C B A vertical section, (x, A 40% B 6O%.y, A-40% C 60%)
12.1The character of the boundary lines and boundary points on the vertical section. There are three different kinds of boundary lines and two different kinds of boundary points on the vertical section. If the different equilibrium phases in each phase region are known, the different character of the boundary lines and boundary points may be distinguished.
198
mYuzw
The first kind of bounaary lines: de, ef and fg, R1=2. Take line de for example. There are phases L and s2 in the two NPRs L/L+sz around line de, @=2, Rl4-2=2, Ae=l, Ri =2, so both the corresponding boundary and PB are phase boundary surfaces. de is the section of the planar P& in the space diagram on the vertical section, de is a phase boundary line also. There are phase points on this line. The character of ef and fg is similar. The second kind of boundary lines: mjt jh, hk, ki, II, In, Ri=l. Take hj for example. There are three phases L , SI and sz in the two NPRs, L+sI+s2/sk+ There are two corresponding Zfnear PBS in the space diaszr #=3, Rl=l, &I=2. gram which are not shown in the vertical section usually. Ri =Rl+Ae-1=1+2-t*2 therefore the boundary of the two NPRs is a boundary surface generated by the limiting tie-Iines. The boundary line hj in the vertical section is the section of the bounding surface in the space diagram on the vertical section. This line generally is not a true phase boundary line, there axe only system points on the line. The third kind of boundary Lines jki, Rt=O. There are four phases: L, sir phase s2 and sg in the NPRs, @=4, R1=0. There are four equilibrium isothermal. points in the space diagram, but none of them is shown in this vertical section: j, k and 1 are all system points. The first kind of boundary points: e and f, R1=l. There are three phases in the NPRs around point e (or f), $=3, Ri=l, A+=?, Rj =Rl+&$-l=l+l-I=1so the corresponding boundary and PB in the space diagram are the same linear PP. Point e (or f) is the section of this phase line on the vertical section, this point is a true phase point. The second kind of boundary points: j, k and I, Rl=O. These have been discussed in the third kind of boundary lines. 13. The Variation of Phase Regions and their Boundaries with Temperature. How does the phase region vary when it passes the boundary, or what is the combination of the phases in the phase region below the boundary during the process of cooling? At higher temperature L (liquid phase) is stable. Let the temperature decrease because 2kRtbOinaternary diagram, then the dimension RI of PB between L and the phase regions at lower temperature is probably equal to two (since if the system points lie just on line elE, e2E or eaE, WL+si+s., Rx=l; and when the system point Lies just at point E, L.-*sI+ s2+s3, RI-O. E&ept these three lines and one point, R1 must be equal to 2.) According to table 4, R1=2, #=2, the liquid phase L must transformtoI,+si (irl, 2, 3) in these cases. The phase region near 60%B contains Phases L+s, the-region near 60%C in the case under Present discussion contains phases L+saI and the middle region contains phases, Lisl&s thetemperaturedecreases further, then below the phase regions L+si, RX=1 (for R1 decreases or does not change when temperature decreases not too therefore there must appear a new phase in the second much), #=3, and 41=2, phase region, and when temperature is not too low the liquid phase does not disappear completely, so the combination of the phases in the phase regions below lines hk and hi must be phases L+si+s,. Below hk, it is L+s~+s~, because the system contains more B: below ki, it isjL+sl+ss, because the system contains more C.When thetemperaturedecreases further; there are two possibilities; RI=1 and Rr=O. If R1=l, (P=3,but @,=3,the disappearance of some phases (oracertain phase phase) in the second phase region must happen. In this case, the liquid L (stable at high temperature}will disappear, while the phases SI+S~ or .sl+sg would remain. The latter ones are the cases below the lines hj or il. If Rk=O, a=4 there will appear a new phase in the second phase region (for $1=3). And one original phase existing in the first phase region must disappear in the second phase region, because 42 cannot be larger than three. L disappears in this case. Therefore the phases s1+s2+89 would remain. This is the case under Iine j k 1.
~~G~OR~NG PHASE REGIONSANB TRRTR BOUNTIES
199
Thus, it can be seen that the vertical section can be explained well by applying our method. Based upon all the previous discussions, we might say that our theorem and its corollaries probably have the following advantages: (a) (b) Cc)
Our method might be helpful in teaching multicomponent phase diagrams. Our theorem might be useful for the determination and calculation of phase diagrams. If some necessary equilibrium information are known, one might predict some features of the phase diagram by our method.
14.
References
1.
Gordon, P., "Principles of Phase Diagrams in Materials Systems." McGraw-Hill, New York, (1968).
2.
Palatnik, A. S. and Landau, A. I., "Phase Equilibria in Multicomponent Systems", Holt, Rinehart and Winston Inc., New York (1964).
3.
Rhines, F. N,, "Phase Diagrams in Metallurgy" McGraw-Hill, New York. (1956)
NEIGHBORING
pp. 185-199,
0364-5916/83 $3.00 + .oo (c) 1983 Pergamon Press Ltd.
1983
THE THEOREM OF CORRESPONDING RELATION BETWEEN PHASE REGIONS AND THEIR BOUNDARIES IN PHASE
DIAGRAMS
(I)
Muyu Zhao (Department of Chemistry, Jilin University, Changchun, People's Republic of China) (This paper was presented at CALPHAD XI, Argonne Illinois, May 1982) ABSTRACT.
Several fundamental concepts in phase diagrams have been discussed. The theorem of corresponding relation between the number of different phases in the neighboring phase regions Qt and the dimension R, of their phase boundary in the phase diagrams is derived from the phase rule. The five corollaries of this theorem for isobaric phase diagrams are derived also. By applying this theorem and its corollaries, the following results can be obtained. Both the boundary rule and contact rule of phase regions in phase diagrams may be deduced. The ten empirical rules for constructing the complex ternary isobaric phase diagrams from phase diagram units may be put on a theoretical basis. The relation between neighboring phase regions and their boundary in phase diagrams of one-, two--, three- and four-components of all types and in the different horizontal and vertical sections of ternary phase diagrams may be explained without introducing any supplementary concept. 1.
Introduction
A phase diagram is composed of several phase regions, it is very important to find out the implicit relation between them. For this purpose, we must study: (1) what is the kind of another phase region which can be bounded to a given phase region? (2) what is the characteristic of their boundary? That is, we must study the relation between neighboring phase regions (abbreviated to NPRs) and their boundaries, which is important for interpreting phase diagrams and in constructing of phase diagrams,since the phase rule usually cannot accomplish this task directly. Therefore there have existed many empirical rules and two general rules dealing with NPRs and their boundary. The general rules are as follows: 1.1 The boundarv rule (I_). It points out that any phase region containing Qll phases can be bounded only by the phase region containing $2 phases: “2-J,
i
(R -
R;,
(1)
where R is the dimension of the phase diagram, i.e. the number of independent variables among temperature (I'),total pressure (P) and concentration (x.), which constructthe phase diagram or a section of it; R; is the dimensioh of the boundary between two NPRs. This boundary is, in fact, a set of system points, which is called alloy points in other references. 1.2 The contact rule or law of adjoining phase regions derived by Palatnik and Landau(Z) is , =R-D+-D-3 0 Rl and R have been already defined; D+ or I)- is the number of phases which r disappear in crossing the boundary from one phase region to another Received 18 May 1982 185
respectively. The boundary discussed by PaLatnik and Landau is, also, a set of system points. Although these two rules have been well accepted and used extensively, there exist some shortcomings. In fact, these two rules are used to treat two different aspects of the same problem: the relation between NPRs and their boundary, but they fail to uniteand to be deduced from each other. There are many exceptions to them, therefore some supplementary concepts such as degenerate regions should be introduced, or the invariant regions in isobaric phase diagrams of N components must be considered as (N+l) phase regions, etc. According to phase ruler we have derived the theorem of corresponding relation between NPRs and their phase boundary, and its five corollaries by which the following results may be achieved: (1) Both the boundary rule and contact rule are derivable. According to our derivation, it is easy to point out when the rules would be invalid. (2) The ten empirical rules, derived by F. N. Rhines (3) for constructing complex ternary phase diagrams from phase diagram units may be interpreted theoretically. (3) The relation between NPRs and their boundaries in phase diagrams of one-, two-, three- and four-components of all types, and in different horizontal and vertical sections of ternary phase diagrams may be explained without introducing any supplementary concept. 2.
Some Fundamental Concepts for Phase Diagrams
2.1 The dimension of an isobaric phase diagram, R is equal to the number of components N in the system. 2.2 The phase region and the number of phases in it can readily be defined in such a way that its dimension equals the dimension of the phase diagram given. Let the number of phases in any phase region be I$, evidently, the minimum number of phases in any phase region is 1, so that a)l. In an isobaric phase diagram, since the dimension of phase region should equal that of the phase diagram, the temperature of the system must be an independent variable at least. Otherwise, if the temperature maintains constant, the phase region can exist in an isothermal section of the phase diagram only. Since temperature is an independent variable at least, then the degrees of freedom of the system fbl. According to the phase rule, 4=N+l-f (pressureis constant), as f>.f, the number of phases existing in any phase region $$N, so N3@)7. 2.3 The neighboring phase regions When discussing the two NPRs on both sides of common boundary, let $1 and $2 be the numbers of phases in the first and second NPRs respectively, and let the total number of different phases in both NPRs be @. If many phase regions meet one another at this boundary, then Q, is also the total number of different phases in all these phase regions. On calculating of Q, if the same phase appears in several phase regions, it should be counted once only. A+ is the number of common phases existing in both NPRs, evidently, @=i$r+$2-A+t because the common phases have been counted twice in 41 and a2, so that A@=Q1+@2-Q, 2.4 The maximum number of phases existing in any one phase region of two NPks,Q,ax. It has been proven that the maximum number of phases existing in any phase
NEIGHBORTNG PHASE REGIONS AND THEIR
BOUNDARIES
187
region is i$ \cN. And the maximal number of phases existing in any one phase region of t!%xtwo NPRs is rjmax
The boundary and phase boundary of the NPRs.
The common boundary of two or more NPRs may be understood in two different points of view: The boundary is set of system points, by which the NPRs are separated from each other. This is the case in the boundary rule and contact rule. Let the dimension of boundary be R'l.
On the other hand, the phase boundary (represented by PB) is regarded as the set of phase equilibrium points of the system existing at the common boundary. Phase points are more important than system points for discussion on equilibrium problems. Let the dimension of PB be RI. In phase diagrams, R1 may equal R1 or not, The parameters which characterize the relation between the NPRs and their boundary may be explained with Fig. 1. Example 1: sI+L/sl+sp are the two NPRs. $1=2, (SI and L), 1$2=2, (sland SZ) @=3, (L, s1 and ~2) A$=1 (the common phase is ~1).
b
The boundary is line cE, Ri =l. The PB are three points: C(Sl) I d(sp) and E(L), Rl=O so R; # R1 in this example. Example 2, the two NPRs are L/L+s,. Their boundary is line aE. The PB are lines aE and aC. The common phase is I J L. Most part of the system exists at Ae fB line aE, only an infinitesmal of the system exists at line aC. So PB is mainly the line aE, i.e. the PB is mainly the set of equilibrium points of the Figure 1. Binary isobaric phase common phase in equilibrium with the diagram with eutectics L-liquid phase, sir s2 other phases in the two NPRs. Then, in this case, boundary is identical with -solid solutions the PB, Rl=Rj . If the regularities and ranges of variations of @, A@, @max, RI and R; and the relation between them may be determinedin all different phase diagrams, the relation between the NPRs and their boundary in these phase diagrams can be explained thoroughly. 3.
3.1
The Theorem of Correspondina Relation and its Corollaries tar Isobaric Diasrams
The theorem of corresponding relation.
Since the PB of NPRs is a set of equilibrium phase points of the system, therefore the system existing at the PB must obey phase rule. The general form of phase rule is known as: R,= f=(N-r-Z) + 2-@+K where, R,, f and Nwere defined previously, r is the number of independent
(3)
chemical reactions in the system, Z means the other independent conditions which constrain the concentrations (except the conditions Xx$=1, where x? denotes the mole fraction of the i-th component containing in the l$-th phase,ii-I, 2,...., N), K is the number of independent variables other than T, P and Xi; in general, R=d, The dimension of PB, RI, equals degrees of freedom f. The PB is shared by two or more NPRs , so the different phases, which exist at PB, otherwise in these NPRs, must be the same existing in the system these NPRs cannot meet each other at this PB, so @=O
(41
# and $ have the same value, but they are distinct in physical meaning e Sub-
stituting f4) into (31, we have: RI=(N-z-Zf+2-WR
(51
@=(N-r-2)+2-RI+K
(5a)
l?qus(5) and (5a) show us the definite corresponding relation between QI and RI. This relation is valid practically for relationships between Q, and RI for all cases, so it is called the theorem of corresponding relation between 0 and R1 in phase diagrams, and expressed by TCR. It is able to solve all the problems treated by the contact rule. This theorem is derived from the p&e rule accompanied by the relation at=4, therefore TCR is not a variant of the phase rule. 3-2 The corollaries of the theorem of corresponding relation (TCR) for isobaric phase diagrams. The first corollary: Since RI@, we have:
The range of variation,of ct,in any phase diagram. (N-Z-r)+2+K>@.
There must be two or more phases in two NPRs, i.e. @>2; therefore: (N-Z-r)+2+K)@$?
(6)
The secondcorollary: The range of variation of RI in any phase diagram. @32, (N-r-Z)+X3RL, therefore:
Since
(N-r-z)+K)RI)O
(71
When Z=r=K=O* P= constant, (6) and (7) become: (Nil))@>,2 (N-l)>R1>0
(6a1 (7a)
respectively: the ranges of variations of @ and R 1 in usual isobaric phase diagrams of N components are thus defined.
The third corollary. In isobaric phase diagrams of N&2 components and with Z=r=K=O, there are two cases in which A$=0 in the two NPRs. First in phase diagrams mentioned above , one-phase region may meet each other at single points only. In this case, A+=O, RI--O, Proof: If two one-phase regions i and j meet each other at a particular PB, then all the common phase points in this PB must have:
x:=x{ ,
i-
x2-x
ej,
..*
. ..*
i
‘
xN=xN
j
(8)
NETGHBORTNG PHASE REGIONS AND THETR BOUNDAR‘LES
189
According to the TCR, when P=const-, then: RI=(N-r-2)+1-@ There are (N-l) independent equations in (8), so Z=N-1, (In a phase diagram with Z=r=K=O in general; there may exist some PB in which Z#O), and r=O I @=2, therefore: R1=N-(N-1)+1-2=0 Since degree of freedom vanishes, only single points may satisfy th is condit ion. So in an isobaric phase diagram of N components and Z=rk=O, one-pnase regions can meet each other at single points only. Secondly,ininvariant regionsof the phase diagram mentioned Rl=O, there is only one case in which A$=0 in the two NPRs. that, if @>N, then Qmax\
above, @=N+l, It has been proved two NPRs may be 4min'l. Therefore:
In all other cases, A$>l. The fourth corollary.
In the phase diagram mentioned above, if Rl+l, then: (Q-l),A@bl
(IO)
Let us consider (A@)min in this case. According to the definition of A@, A$ is an inteqer (x0). If we can prove A� in the above condition, then AQlgf. Let us apply the reduction to absurdity. Assume that there is no common phase in the two NPRs and the dimension of their PB, R1)l. Since R,al, the PB must be an equilibrium phase line at least. According to the characteristics of phase diagrams, if Rl>l and there is no common phases in the two NPRs, there must be one phase in each of these NPRs, which meet each other at this PB, otherwise the two NPRs cannot meet each other atthis PB. But according to the third corollary, one-phase regions can meet each other at single points only. Thus, one phase regions cannot meet each other ata PB with Rl>l. So the assumption is wrong. Therefore, if there is a PB with R1)l between two NPRs, these two NPRs must have one or more common phases, i.e. A$>l. The PB with Rl>l between the two NPRs is the phase boundary of this or these common phases. Then consider (A"i~ax,:,:~ta~~o~~~~~d~~ &: ;r,z;'_;:; sinOIif Pk;z;e;jA;l; gram mentioned above, A4 must be less than $max by one at least, otherwise, the two NPRs Then A@ is less than @ by one at least also, i.e. (Qk??&'be identical ?)>/A@. Combining ;he conditions of (A+)max and (AQ)minr (10) may be obtained. The fifth corollary.
In the phase diagram mentioned above, if Rl=O, then: (@-2)>A+>,O
(11)
Since Rl=O, according to TCR, @=N+l, so O>N, then emax
MuwZRAo
190
4.
Other Rules Governing the NPRs and Their Boundaries in Phase Diagrams
4.1The relation between R:
and R1.
4.1.1A characteristic of phase boundary. Let us consider the PB of isobaric phase diagrams of N()2) components and with Z=r=K=O, RI&I, A1+22. According to the 3rd corollary of TCR, one-phase regions may meet each other at single points only: they can neither cross each other nor meet along a PB with Rr>,l. A PB with RI>? is a portion of a common phase in the two NPRs, therefore, the two or more PBS with Rx)1 must possess the same characteristic as the two or more one-phase regions possess, i.e. they can meet each other at single points only. The condition is rather simple. Consequently, the boundary, which is enclosed by two or more PBS, cannot be very complicated. 4.1.2 The relation between R; P=const.
and R1 in phase diagrams with Z=r=K=O, N>2 and
4.1.3 Rl>l, then: Ri =Rl+A4-l
(12)
According to the 4th corollary$f R1)1,A.$31.WhenA$=I, there is one PB with RI>?, and the boundary (set of system points) is identical with the PB, then, R'l = RI. When A$=2, there are two PBS with dimension of R1 between the two NPRs. These two PBS enclose a boundary with dimension of Ri which is one more than Rl, Rf =Rl+l. ff A4=3, there are three PBS with dimension of R1 between the two NPRs. These three PBS enclose a boundary with dimension R; =Rt+2, and so on. In short,
Ri =Rl+A$-1
This relation is shown in Table 1. Table 1
Rl=l A@
1
2
3
The number of common phase lines
1
2
3
Ri =Rl+A@-1
1
2
3
The character of boundary
line
surface*
space**
* The two corresponding equilibrium phase points of the two equilibrium phase lines join to a tie-line, the locus of the tie-lines forms a curved surface. ** The three corresponding equilibrium phase points of the three phase boundary lines form a triangle, the locus of these triangles forms a space of three dimensions.
The conditions of R1)2 are similar, i.e. Ri =Rl+A$-1 also, discussion is to be omitted. 4.1.4 Rl=O and there is no invariant region with (4max+l) phases between the two
NPRs, the transition from one NPR to another occurs only when the total composition of the system is changed. In this case, A$)l, Ri =Rl+A+-1. (12a) Take sl+L/L+sl in Pig. 1 for example. The boundary and PB of sl+L/L+sa are points, Rl=R, A+=?, Rf =R,+A+-?=0+1-7=0. This condition is not importantin a phase transition of closed system, it needs no further discussion. 5 Rr=O and there is an invariant region with (d;max +?) phases between the two s: the two different cases may be: There exists an eutectic or peritectic reaction. There exists maximum (or minimum) melting (or boiling) point. both cases, Ri =R,+A+
a. b.
In
Take the boundary line cE between SI+L/S~+S~ in Fig. 1 for example. this case, Rl=O, Ai$=t8 Ri ==Rt+h~$=O+l=t.
(13) In
It is shown in table 2, how the boundary varies with 54, when RI=O.
Table 2.
Rl=O
I
Ri
=R1
1
0
AQ,
I
+ag
The character of boundary
5.
The
5
point
I
I
I
2
I
7
tie-line
a region of triangle
Rule Governinq~ the Process of Cooling
As the temperature decreases, the system changes from the phase region stable at higher temperature to the stable one at lower temperature, RI decreases simultaneously or remains unchanging, until Ri decreases to zero. If the temperature decreases further, R1 may increase once: then decreases again or remains unchanged , until RI decreases to zero* and so on. This rule is obtained by inductive method, and is valid for most cases. This rule reflects such a physical background: when the temperature is high enough, the entropy effect is significant, the mutual solubilities of components are large. There is only one phase in the gaseous state under ordinary pressure or only one phase in liquid state frequently. That is to say: the number of phases in a particular phase region at high temperature is small, the value of Ipof the two NPRs is small too, then Rt of the PB is large at high temperature. When the temperature decreases, the solubilities betweendifferent components decrease, hence separation and precitation occur: the number of phases in a given phase region increases, and the value of cb increases too, then the value of RI decreases. 6.
Deduction of the Boundary Rule and the Contact Rule of Phase Regions
Both rules are not universally valid. From TCR and its corollaries, both of them may be deduced in those cases in which they are valid.
MYU z?IAo
192
6.1 The deduction of the contact rule Ri
=R-D+-D-
(14)
By the definition of D+ and D-, (D++D-) is the number of different phases belonging to either one only of the two NPRs, i.e. it does not include the number of common phases existing in both NPRs. Since 0 includes the number of the common phases, AO, then: (P=D++D-+A@
(15)
In generalfor isobaric phase diagrams with r=Z=K=O and P=const, according to TCR: R1=N+l+
(16)
substituting (15) into (16), (17) is obtained: R1=N+l-(D++D-+A$) In an isobaric phase diagram, the dimension R is equal to N. this relation onto (17), and by rearranging, we obtain:
(17) Substituting
R1+A@-l=R-D+-D-
(18)
According to (12), when R1)1, AQ>l, R; =R1+AQ-1; from (18), we have: =R-D+-DR; 6.2 The deduction of the boundary rule: $z=Ql*(R-R; 1
(19)
Since the order of $1 and a2 may be arbitrary, we may appoint the phase region containing more phases in the two NPRs as the first phase region, and R>R; , according to (19), so: (20)
+I’+2 then (19) may be rewritten as:
$2=$1-(R_Ri 1 (21) similarly, if Z=r=K=O, P=const, then R=N, substituting it into (21), we have: $z=+(N-R; (22)
1
(22)
can be proven as follows.
The boundary rule is not valid for all cases, it is correct only when $1=0, and invalid for a>$,_ For instance, let us consider the PB of the NPRs L+s1/L+s:! in a ternary isobaric phase diagram, where @=3, 411=2, 0>$1, Ri =I. But according to boundary rule, then: $2=$1+(R-R; )=2+(3-I)=0 or +4 It does not tally with the real phase diagram. Many other similar examples may be cited. In a word, the boundary rule is valid only for @=$1. Then, in the effective scope of boundary rule, we may let $1=@, therefore: @2=(2$1-2@)+$2=$1-0+($1+$2-@)=@1-@+A$ (23) According to TCR, Q=N+l-R1, substituting this relation into (23), and rearranging we get: @2=$1- N-(R1+AQ-1) According to (12), when R131, Ri =R1+A@-1, then (22) may be obtained and (21) may be obtained from (22) directly. As already mentioned, we choose Q1>@2 just for convenience, however, the order of phase regions is practically arbitrary, if +2>$1, then: $2=$1+(R-Rf )
(24)
193
NEIGHBORINGPHASE REGIONSAND THEIR BOUNDARIES
Combining
(21) with (24), (19) is obtained too.
4.3 The limitations of the two rules. In the course of derivation of both rules, the limitations are obvious. Since our theorem is valid for all cases and in the course of deducing these two rules, several supplementary equations have been employed; because these supplementary equations are valid only in certain cases, the validity of these two rules is thus limited. The limitations are as follows. The contact rule is invalid: (a) When Rt=O and there is an invariant region with (Qmax+7) phases between the two NPRs. (b) When Z or r#O. The boundary rule is invalid in these two cases also. is invalid further when @>+I. 7.
The boundary rule
The Establishmentof Two NPRs and Their Boundary
7.1 If the combination of phases in the first phase region and the characteristics of common boundary are known, the combination of phases in the second phase region may be fixed. According to the conditions given, +1, RI and R; are known, Since @=N+1-R,, Q may be evaluated. From (12) and (13), A$ may also be determined from R1 and Ri . Since A@=@ltQ2-0 and Qr 01 and A@ are known, $2 may be calculated by: QZ=@+AQ-+i
(25) The number of common phases existing in both phase regions is A+, the number of phases belonging to the first phase region only is (@l-A+). The number of phases belonging to the second phase region only is: ~2-A~=(~+A~-~,)-A~=~-~l. 7.2 If the combination of phases in the two NPRs are known, the characteristics a? the common boundary may be determined. Since the combinations of phases in the two NPRs are known, all the values of ($1, +?, Ad, and @ may be established. According to TCR, the value of RI may be calculated from @. Furthermore, the value of Ri may be calculated from Rr and 5Q also, so that the characteristics of the common boundary are well determined. 8.
The Phase Diagram of Unary Systems
Phase diagrams of unary systems are very simple. But both the contact rule and the boundary rule fail to interpret them directly. However, they can be easily interpreted by TCR without any additional concept. For the sake of saving space, the discussion is omitted. 9.
Phase Diagrams of Binary and Ternary Systems
Applying TCR and its corollaries to isobaric phase diagrams of binary and ternary systems, the results obtained are listed in Table 3 and 4.
MunJ7sWo
194
Table 3.
of binary systems
RI, 0, AQ and Ri
r=Z=K=O, P=const, R1=3-@
I
RI> (l>R,aO)
I
1
0
@(3>@>2)
2
3
Q, max
2
2
A4
A+=1
laA@>O
Ri
R1 = R1 +A@-1=1
R1 =Ri+AQ=AQ,
The combinations of the phases in the two NPRs
L/L+si
A@=O, Ri =O
L+Si/Si
L/Sl+SZr
A@=l, RI =7
si/sl+ S2
L+Si/L+Sj,
Table 4.
RI, QI, AQ and Ri
r=Z=K=O
P=const
Ri
(2bR1>0)
I
L+Si/Sj
L+Sj/S,+
of ternary systems
R1=4-0
-
I
2
1
0 -
@(4)@&2)
2
3
4
@ma,
2
3
3
A4
Ri
I
S2
-
A@=1
2sAQl
R; =Rl+A$-I=2
The combinations of the phases in the
L/L+Si
two NPRs
Si/Si+S
2>A@bO
Ri =R1+A$-l=A~ A~$=11 Ri ~1
L+Si/Si j
Ri =R~+Ac$=A+ A$=O, Ri =O
L/L+Si+Sj
L/sl+sz+sB
L+Si/L+S.
L+Si/Sj+Sk
3 L+Si/Si+S*
3
-
L+Si+Sj/Sk
L+Si+Sj/Si
AQ=l,
Si/Sl+S2+S3
L+Si/L+Sj+Sk
SifSj/Sj+Sk
L+Sj,/Sl+S2+S3
Ri =I
L+Si+Sj/Sj+Sk L+Si/L+Si+S.
3
AQ=2,
R1' =2
L+Si+Sj/Si+S j
L+Si+Sj/S'+S2+S3
Si+Sj/Sl+S2+S$
L+Si+Sj/L+Sj+Sk
Notes for Table 3 and 4: L denotes liquid phase: sl, s,and s3denote_soJid solution phases: i, j, k=l, 2 3; i#j#k. In a few cases, there are fi/fJ, Q=2, Z=l, or 2, Rx=O. Since the case, where R&=0 and Rf =Rl+A@-I is not important in usual phase transition of closed system, it is not discussed here.
NEIGHBORINGPEASE REGTONSAND THEIR BOUNBARIES
195
These tables show all the relations between the NPRs and their boundaries in these binary or ternary phase diagrams, There are three kinds of systems which obey the relation shown in the above tables: (a)The components are completely miscible both in liquid and solid state. (b)The components are miscible in the liquid state, but there occurs an eutectic reaction between one or more pairs of the components. The system has a Simple Peritectic (i.e. there is no compound formed) If there exists a maximum (or minimum) melting (or boiling) point or compound formation in the phase diagram, we must use (5) to treat it, the relation between the NPRs and their boundaries in this phase diagram may also be obtained. If there are two or more invariant regions in the phase diagram, then the phase diagram may be divided into two or more units, each of which contains one invariant region. These units may be analyzed separately. (c)
The possible forms of transition from one phase diagram unit (abbreviated to PDU) to another may be discussed also. Take a binary system for instance, three phases form an invariant region and a PDU. Let these phases be f', f* and f3. There are phases, f4, f5, . . . . f3+p in the phase diagram also. Let i, j, k, =I, 2, 3 and i#j#k; 1, m=4, 5, . . . 3+p, Pfm, f' and fm denote the phases that appear when transferring from the PDU to another. The possible forms of transition from one PDU (composed of f', f2and f3 in this case) to another PDU are:
invariant transition Rl=O, @=3, A$=0
univariant transition RI=?, @=2, A$==1 fi/fi,fl
fi/fl+fm fi+fj/fl Rl=O, G-3, A@=1 fi+fj/fj+fl Rl=O, 0=2, Z=l, A$=0 fi/fl
The possible forms of transition from one PDU to another in a complicated ternary phase diagram are similar. Discussion of it may be omitted. The relation between the NPRs and their boundaries in phase diagrams of four-component systems could be explained also by applying TCR and its corollaries too. An article about quaternary phase diagrams is to be published later. 10.
Explanation of Rhines'Rules
(3)
We can now turn to F. N. Rhines' ten empirical rules which must be obeyed when constructing a complicated ternary phase diagram from phase diagram units. These rules are as follows:(J) 1)
One-phase regions may meet each other only at single points, which are also temperature maxima or minima.
MUYU ZHAO
196
One-phase regions are elsewhere separated from each other by two-phase regions representing the two phases concerned; thus the bounding surfaces of one-phase regions are always boundaries of two-phase regions. 3) One-phase fields touch three-phase regions only along lines which are generally nonisothermal. 4) One-phase regions touch four-phase reaction planes only at single points. 5) Two-phase regions touch each other along lines which, in general, are nonisothermal. 6) Two-phase regions are elsewhere separated by one- and three-phase regions by bounding surfaces they are enclosed. 7) Two-phase regions meet three-phase regions upon "ruled" bounding surfaces generated by the limiting tie-lines. 8) Two-phase regions touch four-phase reaction planes along single isothermal lines, which are limiting tie-lines. 9) Three-phase regions meet each other nowhere except at the four-phase reaction isotherms. 10) Three-phase regions are elsewhere separated and bounded by two-phase regions involving those phases which are held in common by neighboring three-phase regions. All these ten rules are included in the Table 4 and its notes. They are expressed in more general forms as follows. Let i, j, k, 1 = 1, 2, 3, 4 and they do not equal one another. 2)
1)
fi/fj, @=2, Z=2, A$=O, Rl=O. There are only single points satisfying this condition.
2) 3)
fi/fi+fj/fj @=2, R1=2, A$=1 , R'l=R1=2.
There are phase boundary surfaces.
fi/fi+fj+fk, 0=3, Rl=l, A$=l, RV1=R1=l, phase boundary lines. fi/fj+fk+fl , 9=4, R1=O, A@=O, RUI=Rl+A@=O single phase points.
4) 5) fi+fj/fj+fk , 0=3, Rl=l, A@=l, R'l=Rl+A$-191, nonisothermal phase lines. 6)7) fi+fj,'fj/fj+fk 0=2, R1=2, A$=l, R',=R1=2, phase boundary surfaces. fi+fj/fi+fj+fk/fj+&3, RI=1 I A@=2, Rtl=Rl+A$-l= I 1+2-I=2 bounding surface generated by the limiting tie-lines. 8)
fi,fj/fi,fk+fl , @=4, R1=O, A@=l, R'l=R,+A$=l, tie-lines.
9) fi+fj+fk/fi+fk+fl , 0=4, Rl=O, A+=2, if there is an invariant phase region between the two NPRs,Ri =Rl+A$=O+2=2. If there is no invariant phase region between the two NPRs i.e. the two NPRs are located at the same side of the invariant phase plane, Ri =Rl+A$-1=0+2-l=l, the boundary is a tie-line. Except the four-phase reaction plane, the two three-phase regions cannot touch each other elsewhere. Since 2)R190 in ternary isobaric phase diagrams; R1 equals zero only in a four-phase plane, it equals one or two elsewhere. If Rl=l, 84~1, let the first phasekrepion contain fi+fj+fk and A$=l; then the second phase region contains f +f +fm, a=5 accordingly. But this is impossible, because 0~4 in ternary phase diagrams. If AQ=2, then CJ=QJ~+@~-A$I= 3+3-2~4, SO, Rl=O; it is impossible, because it has been assumed R1=l. If AQ=3, then the two three-phase regions are identical. A+ cannot be larger than 3 in a ternary system, therefore, there is no other possibility. If rq=2, @=4-R1=4-2=2; it is impossible because there are three phases in the first three-phase region yet, 0 cannot be less than bl. Therefore, the two three-phase regions cannot touch each other elsewhere except on the fourphase reaction planes. (10) fi+fj+fk,fj+fk,fj+fk+f1,@=3, R,=l, A$=2, R;=R.,+A$-1=1+2-1=2, bounding surfaces generated by the'limiting tie-lines. Rhines' consideration was not complete. A supplement may be made: three-phase regions may be separated by one-phase regions i.e.
197
NEIGHBORING PHASE REGIONS AND THEIR BOUNDARIES
fi+fj+fk/fk/fk+fl+fm, in this case, @=3, Rl=l, A$=l, 11.
Ri
=I.
The Horizontal Sectionsof Ternary Phase Diagrams.
The difference between line ac (ab, bc also) and line fg (ad, ce, cj, hi, . . . . also) cannot be distinguished by contact rule; because according to this rule, for both lines ac and fg, Ri =R-D+-D-=2-1=1, therefore there is no difference. But according to TCR, for line fg; R,=N-@=3-2=1 (for T, P are kept constant)there is one degree of freedom. This line is the phase boundary line. For line ac, the PB of Sl+L/L+S1+S2, R1=3-3=0, there is no degree of freedom, A@=2, thereA fore, line ac is a tie-line. Since the dimension of the PB in the horizontal section is, in general, less than the dimension of the PB by one in the space diaqram,then if the dimension of the PB in the horizontal section has been fixed, so the character of the corresponding PB inthe space diagram may be known. Figure 2.
12.
Horizontal section of a ternary phase diagram
The Vertical Sectionsof the Ternary Phase Diagrams
In the vertical section of the ternary phase diagram, there are three indepedent variables, i.e. T, xl, and x2, R=3, though the vertical section is drawn on a plane. R1=4-@. A 1
x
Figure 3.
m
n
Y
C B A vertical section, (x, A 40% B 6O%.y, A-40% C 60%)
12.1The character of the boundary lines and boundary points on the vertical section. There are three different kinds of boundary lines and two different kinds of boundary points on the vertical section. If the different equilibrium phases in each phase region are known, the different character of the boundary lines and boundary points may be distinguished.
198
mYuzw
The first kind of bounaary lines: de, ef and fg, R1=2. Take line de for example. There are phases L and s2 in the two NPRs L/L+sz around line de, @=2, Rl4-2=2, Ae=l, Ri =2, so both the corresponding boundary and PB are phase boundary surfaces. de is the section of the planar P& in the space diagram on the vertical section, de is a phase boundary line also. There are phase points on this line. The character of ef and fg is similar. The second kind of boundary lines: mjt jh, hk, ki, II, In, Ri=l. Take hj for example. There are three phases L , SI and sz in the two NPRs, L+sI+s2/sk+ There are two corresponding Zfnear PBS in the space diaszr #=3, Rl=l, &I=2. gram which are not shown in the vertical section usually. Ri =Rl+Ae-1=1+2-t*2 therefore the boundary of the two NPRs is a boundary surface generated by the limiting tie-Iines. The boundary line hj in the vertical section is the section of the bounding surface in the space diagram on the vertical section. This line generally is not a true phase boundary line, there axe only system points on the line. The third kind of boundary Lines jki, Rt=O. There are four phases: L, sir phase s2 and sg in the NPRs, @=4, R1=0. There are four equilibrium isothermal. points in the space diagram, but none of them is shown in this vertical section: j, k and 1 are all system points. The first kind of boundary points: e and f, R1=l. There are three phases in the NPRs around point e (or f), $=3, Ri=l, A+=?, Rj =Rl+&$-l=l+l-I=1so the corresponding boundary and PB in the space diagram are the same linear PP. Point e (or f) is the section of this phase line on the vertical section, this point is a true phase point. The second kind of boundary points: j, k and I, Rl=O. These have been discussed in the third kind of boundary lines. 13. The Variation of Phase Regions and their Boundaries with Temperature. How does the phase region vary when it passes the boundary, or what is the combination of the phases in the phase region below the boundary during the process of cooling? At higher temperature L (liquid phase) is stable. Let the temperature decrease because 2kRtbOinaternary diagram, then the dimension RI of PB between L and the phase regions at lower temperature is probably equal to two (since if the system points lie just on line elE, e2E or eaE, WL+si+s., Rx=l; and when the system point Lies just at point E, L.-*sI+ s2+s3, RI-O. E&ept these three lines and one point, R1 must be equal to 2.) According to table 4, R1=2, #=2, the liquid phase L must transformtoI,+si (irl, 2, 3) in these cases. The phase region near 60%B contains Phases L+s, the-region near 60%C in the case under Present discussion contains phases L+saI and the middle region contains phases, Lisl&s thetemperaturedecreases further, then below the phase regions L+si, RX=1 (for R1 decreases or does not change when temperature decreases not too therefore there must appear a new phase in the second much), #=3, and 41=2, phase region, and when temperature is not too low the liquid phase does not disappear completely, so the combination of the phases in the phase regions below lines hk and hi must be phases L+si+s,. Below hk, it is L+s~+s~, because the system contains more B: below ki, it isjL+sl+ss, because the system contains more C.When thetemperaturedecreases further; there are two possibilities; RI=1 and Rr=O. If R1=l, (P=3,but @,=3,the disappearance of some phases (oracertain phase phase) in the second phase region must happen. In this case, the liquid L (stable at high temperature}will disappear, while the phases SI+S~ or .sl+sg would remain. The latter ones are the cases below the lines hj or il. If Rk=O, a=4 there will appear a new phase in the second phase region (for $1=3). And one original phase existing in the first phase region must disappear in the second phase region, because 42 cannot be larger than three. L disappears in this case. Therefore the phases s1+s2+89 would remain. This is the case under Iine j k 1.
~~G~OR~NG PHASE REGIONSANB TRRTR BOUNTIES
199
Thus, it can be seen that the vertical section can be explained well by applying our method. Based upon all the previous discussions, we might say that our theorem and its corollaries probably have the following advantages: (a) (b) Cc)
Our method might be helpful in teaching multicomponent phase diagrams. Our theorem might be useful for the determination and calculation of phase diagrams. If some necessary equilibrium information are known, one might predict some features of the phase diagram by our method.
14.
References
1.
Gordon, P., "Principles of Phase Diagrams in Materials Systems." McGraw-Hill, New York, (1968).
2.
Palatnik, A. S. and Landau, A. I., "Phase Equilibria in Multicomponent Systems", Holt, Rinehart and Winston Inc., New York (1964).
3.
Rhines, F. N,, "Phase Diagrams in Metallurgy" McGraw-Hill, New York. (1956)