Theoretical estimates of phosphorus concentration profiles across grain boundaries in Fe-P and Fe Ni-P systems

THEORETICAL ESTIMATES OF PHOSPHORUS CONCENTRATION PROFILES ACROSS GRAIN BOUNDARIES 1N Fe--P AND Fe---Ni P SYSTEMS -I: SHINODA Kescarch Laboratory of Precision Machinery and Electronics. Tokyo lnstitu~c ol”I’cchnology. Nagatsuda,

Midori.

Yokohama

227. Japant

Abstract---Theequilibrium “intergranular segregation extent (ISE)“of P in the above two alloy systems has been calculated on the basis ofa model presented here. The ISE of P is defined as the monolayer thicknessof a P-enriched bulk region adjacent to the grain boundary, where a normalized concentration of P is greater than one-tenth ofthat ofthegrain boundary.Themodel isin principleanalogous toOno’sdiscretela~~iceapproach based on a simple regular solution model under the assumption of the nearest neighbor interaction ofatoms. The ISE of P thus calculated is. irrespective of the Ni-concentration of the alloys, 5 one monolayer thick at most whatever magnitudes are chosen for the various parameters adopted within their limited ranges in the present model. This suggeststhat the equilibrium intergranular segregation of P occurs exclusively at grain boundaries per se in ferrous alloys. R&sum&Nous avons calcult I’ “&endue de la dgrbgation inter granulaire” (ESI) d’bquilibre du phosphore dans les deux alliages Fe-P et Fe-Ni-P B partir d’un modile que nous prksentons. Nous d&fin&sons 1’ ES1 du P comme l’ipaisseur d’une rkgion massive enrichie en P adjacente au joint de grains, oti la concentration normalide en P est suptrieure au dixitme de celle du joint de grains. Ce modile est analogue en principe z+l’approche du rkseau discret d’Ono reposant sur un modele simple de solution r&uIi&e, dans le cadre de l’approximation des interactions entre plus proches voisins. L’ epaisseur de 1’ ES1 du P ainsi calculize est de l’ordre de grandeur de la monocouche, quelle que soit la concentration en Ni des alliages et quelles que soient les grandeurs choisies pour les les divers paramttres du pr&ent modtle. Ceci donne g penser que la dgr6gation intergranulaire d%quilibre du P se produit exclusivement aux jounts de grains per se dans les alliages ferreux. Zusammenfassung-Die Ausdehnung der intergranularw Gleichgewichtssegregation (BE) von phosphor wird mit einem hier beschriebenen Model1 fiir die beiden Systeme Fe-P und Fe-Ni-P berechnet. Diese Ausdehnung ist definiert als die Monolagendicke einer mit Phosphor angereicherten Zone nahe der Komgrenzz, in der normalisierte Konzentration des phosphors ein Zehntel des Wertes in der Komgrenze iiberschreitet. Das Model1 ist im Prinzip analog zu dem disireten Gittermodell von ono, welches auf einem

einfachen Model1 einer regelmBL3igenLGsung unter Annahme einer Wechselwirkung zwischen niichsten Nachbam beruht. Das so berechnete ISE von Phosphor ist, unabhingig von der Nickelkonzentration der kgierungen, hiichstens eine Monolage dick, egal welche Werte Wr die verschiedenen Parameter innerhalb ihns Geltungsvereiches im vorgelegten Model1 angesetzt werden. Dieses Ergebnis legt nahe, dal3 die intergranulare Gleichgewichtssegregation von Phosphor in Eisenlegienmgen ausschlieDlich an den Komgrenzen auftritt.

NOMENCLATURE In this text, subscripts of g, b. and (r) (r = I,2 ,...) denote, respectively, thegrain boundary1ayer.a bulklayerfarfrom the g-layer and the rth number of layers away from the layer next to the grain boundary into the bulk. Superscripts of i and j (i.j = 1, 2 or 3) stand for the alloy components in a ternary alloy, where I and 2 indicate the solute speciesand 3 the sol-

vent. q’,

C: and Cf = atomic concentrations of ith component at the rth, “b” and “g” layers, respectively It’;’ and qf = normalized concentrations defined as r$’ = (c’-c)/(C: -CF). and hence tl; = 1 s,iand e,,(or&etc.) = binding energy for the nearest neighbor (nn-)pairofa like&i) bond and that ofunlike(i-j)one, respectively.TheEf,,etc.arethesimilarquantitiesofthe g-phase vi, or yt, = interaction parameter between i and jth t Permanent address: Hitachi Research Laboratory, Hitachi Ltd., Saiwaicho, Hitachi, I baraki 3 16. Japan.

components in an “i-j” binary system; vij = E,~(s,, +~,~)/j)lz.The 18 is the similar quantity as for the g-phase o,,orwf, = interaction parameter between i andjth solutes in a ternary system; (uij = vi,-(~*,3 + )I~,): the wfl, is the similar quantity of the g-phase Y,~ and o,~ = with the Fe-Psystem, v,, shows rpmFc and ,I,,~ means w,._~, for Fe-Ni-P system 2 = the so-called “co-ordination number” for the crystal structure of the b-phase z(z*andz’) = thenumberof,ln-pairsforareferenceatomon a layer IO given atoms on its neighboring layer in the bnh.,m 4. dnd ,? are similar q&tities?or-the case the y,.‘G.u.G reference atom lies on the Ist-layer or on the g-layer, respectively m = non-dimensional quantity with a range of 0 to 1; VI = (Z/22) - I y - “---A:----;onal positive quantity less than unity. ” - “““-“““c”J’ defined by equation(29)inthetext.denotingthedegree of a relaxed state of the g-phase X,(or X:) = an approximated solution oT$;‘. X,’ is the best approximation of I/‘;’

2051

2052

y,

s

SHINODA:

PHOSPHORUS

CONCENTRATION

(“;_c:

R; = ?‘, ct. ti = t or 2) s, = !‘, i I - c: - c;t, (i = 1 or 2) fi,, = k7-jqZ*r,,._V,) ? = li:, , -‘:,,),‘(2.v,,) II, = [h’+fI-K)IR,]+(l-K)(1+*)/(2.y,) 1. INTRODUCTION

confusion with reference to the problem of how a solute species segregates in equilibrium at a grain boundary and its adjacent bulk region in a solid alloy. On the one hand it is believed that the solute atoms are arrayed exclusively at the grain boundary, say like a monolayer type of segregation (this might be called the “sharp interfaced hypothesis). On the other hand such a thought coexists that a solute enriched interfacial region extends over a few atomic layers or more adjacent to the grain boundary (the “diffuse interface” hypothesis). In this latter view, people would take a solute “concentration depth profile (CDP)“, just observed by means of Auger electron spectroscopy (AES), with the aid of ionsputtering technique, to be a true picture of the solute CDP or very similar to the true one in its shape. Some investigators are in favor of the former hypothesis through the AESexperimentson thesurface segregation of solutes in alloys, with its analogy to the intergranular segregation in mind [l-33. However, there seems to be little observation to prove this hypothesis also true with regard to the intergranular segregation, especially of P in ferrous alloys. In this situation theoretical trial to clarify which hypothesis is more probable is considered to be very useful. The aim of the present study is first to derive a method for calculating the CDP of a solute species across a grain boundary in a ternary alloy system. The solute CDP to be studied here is not what concerned with a grain boundary which has a specific crystallographic relation with its adjacent bulk layers, but such one as obtained by averaging over many grain boundary facets in a polycrystalline alloy. The second aim lies upon the respect that, by using the derived method, we compute the CDP of P in a ferrous alloy with or without containing Ni as a ternary alloy element, and compare those results with the available AESdata forpracticalsteelscontaining PandNi [4-63.

There seems to be some

The

calculation

approach. model

which

[7],

pnirwise

is based is

on

a discrete

in principle analogous

using

the simple

interaction

and

nearest

regular

lattice

to Ono’s

neighbor

solution

(nn)-

approxi-

mations.

2. DERIV,\TION Consider for simplicity

a ternary

OF BASIC

EQUATIONS

alloy crystalline material

assumed

to be consisted

which is

of the grain

___._

~__-

GRAIN

BOUNDARIES

boundary (g)-phase of one (or mono) atomic layer, locatingatthecenterofthematerial,andofthebulk(b)phase regions on both the sides of the g-layer. Let each bulk region on both sides of the grain boundary be made of I-atomic layers arraying in parallel with the glayer, and this ternary alloy be made of two kinds of dilute solutes designated 1 and 2 and of a solvent 3. In view of the symmetrical nature of this material, we will hereafter compute some needed thermodynamical variables only with this halfpart of the material, i.e. the bulk phase crystal of /-atomic layers on one side plus an “imaginatively” halved region of the monolayer of g-phase.t For convenience’ sake, let us number all the atomic layers on each side of the bulk phase as Ist, 2nd, . . . , Ith in sequence from the layer next to the grain boundary towards the bulk interior. Here, we have taken 1to be a great value. To compute the total enthalpy of the system of interest, U, it is convenient to derive U into three parts : (1) Ub for a bulk region consisting of layers with the ordinal number more than 2nd, (2) U,, for an interphase region made ofthe g, 1st and 2nd layers, and (3) U,for halfa region madeof lst-g-1st layers.Then, U is defined as U = Ub + Urb + U,. Next, we will define the concentration of ith component (i = I,2 or 3) at the rth layer

cl”=nl”/nn”‘,

(r-

42 ,“.( fj

(11

where nf’ is the number of atoms of ith component on the layer r and

Likewise, the concentration g-layer is given by Cf = nf/ne,

of ith component (i = I,2

and 3)

for the (2)

where nf and n* have the same meanings as nf” and n”) defined above, respectively. The subscripts of i andj for each variable will be used hereafter as the symbol to denote the species of the alloy components, and the superscripts of g, b and (r) as the symbols to refer to the g-layer (or phase), a bulk layer far from the g, and the rth bulk layer, respectively. Start with thecomputationofU~:todothis,wemust first count the number of all nn-pairs of i-i bond, (i.e. between two “like” atoms) and i-j bond (between two “unlike” atoms) on the assumption that all atoms in each layer are distributed quite at random. Choosing any set of three successive layers in the b-phase,r- 1, r and r+ 1 (but r 2 2) and then counting the number of nn-pairs of the i-i bond,JE’, around one ith atom on the r-layer, we know that Jf;‘= (1/2),,,.Ct”.(:.CI’-‘)+2nl.z.Cj”+r.C~+”). Hcrc it is assumed that the number of atoms on each layer is the same as one another, that is, nr” = 11”’ = I .. I? E n.Thefactor 1/2isso;~stocount~~ch pair of interactions only once. The value : siands for the number of fm-pairs for one reference atom on a layer to

SI-IINOIM:

~ON~~NT~ATION

PHOSPHORUS

GKAIN BOUND,AIZIES

2053

neighboring layer (either ofthe two) and nr is delined as

given ;Itoms on its in the b-phase

Of = (Z/22) - I

(3)

with 2 being the usual co-ordination number for the crystal structure of the bulk phase. The total number of n+pairs of the i-i bonds, Jii, for the bulk phase is worked out to be where z’{” is dclined as -

(I{” = 1’1.C’j”-t C~“/Z

c=2

r-2

~2~.~‘)*~)+~*‘.~+1’ I

3

*

a

f‘

(4)

Using a newly defined variable

CY’= m.CI”+{CI’+”

f @- “)/2,

(r 2 2)

(5)

and with the use of the relationship C~‘fif$‘+~~‘=m+l

(rr2)

(6)

Inequation(1 l},~~and~~ja~sorepresentthejnteraction energy for a /m-pair ofthe i-i bond and that for a “a-pair of the i-j bond, respectively, but, in this case, one atom of the pair lies on the g-layer while the other one on the next layer 1. Similarly, z* means the number ofr’n-pairs for one atom on the layer 1 to given atoms on the glayer. Let us here assume, for simplicity, that

fii can be reduced into a simple form

z”.g =“‘;.3 = const. ( = p’), (Assumption I)

{(cp * Q’)}.

Jii = z * n i

(-0

r=2

Likewise, we can sum up the number of nn-pairs of the i-j bond in the bufk phase, Jil+.llb as follows Jij+Jjj

=

Z’fl

i

(i = 1, 2 and 3).

2 Eii

2 Eij

where i,j = 1,2 and 3 except i = j. Thus, equation (11) can be reduced into the form

(8)

(@*Lljr’+Cjj”*fl’f.

t=2

If the interaction energy per nn-pair of the i-i bond and that of the i-j bond are referred to as .Q and a!j, respectively, the value of U, is written as U,=E11~~,,+E220~22+E33*533

In this equation, Z?f’ (k = 1, 2 and 3) and /? are expressed in the following expression q”

+~,2(~,2+~2,)+&23(~23+~32)+~3,(~31

(%i

+

(13)

and

&JjY2

/I = (1+ 8’)/2.

and

(9) cl;‘=

m~1’+(C~2~+~.C?}/2 (i= 1,2and3)

With the use of variables such as ‘u = so -

= Q”+D**CJ/2=

+J13)*

cf”.c)“+C)“.cI”

It seems to be noteworthy that the variable i”$” obeys the relationship as being analogous to equation (6)

We can rewrite U,, in terms of the form

C\“+C$“+C\”

s,,{(m+ t)C${‘+O;‘}/Z

(14)

= (m+P).

(15)

We can calculate U, in a similar way to the earlier cases U, =(1/2)zs.ng(C1.~~.&f1+Cfi.~gr.E’i2 +cg.Cg.&g,+(c(t.e4+csi.C9)sf,+(C8,.

After similar calculative performance to the above, we can get the expression for IJ,,.

ef

+Cg*C1)eHJ+(C~.17aliC~,Cg)~e3,)

(16)

where cf (i = 1,2 and 3) is defined by using CB 1 = 111”Cr+ t C!” ,

‘skk

(17)

with m’ as m’ = (Zg/2z*)-

1

(18)

similarton’intermsofequation(3).Inequation(l6),the factor l/2 comes from the definition of U,, mentioned earlier. In equations (16) and (18). 9 stands for the number ofnrr-pairs for one atom on the g-layer to given atoms on layer

I, which

is intentionally

discriminated

SHINODA:

XSJ

PHOSPHORUS

from :* bccausc the quantities are not always identical

the

for an imaginary

structure

of the grain

three-dimensional

boundary

phase.

BOUNDARIES

expression

with each other. The value Zg is the co-ordination number

GRAIN

CONCENTRATION

z*vtJ[(i -2~c!‘+“)+2m(l-2’cy,“)

crystal

introducing

+(i-2*~Cf’-“)]+z(m+

tf(zit-eJJ) f k T In (C[‘/C$‘) + z - cq,[ c) r+ 1)

here such variables as \~yj = E~j-(Ek + &~j)/2

+2m.q’+Cj’-‘bJ

and

(19) Cfj = Cf++C,r*Cf

and usingequations( 16)-(18) theexpression for U,can be simpIi~ed into the form

1

the form

(24)

vij-(yi3+vj3)

where i,j = 1 and 2 except i = j. Considering that cf” and 0;’ converge successively constant values Ct and C”,.when t = I, namely, at a layer far from the g-layer, we can derive the following relationship from equation (23) 2*Z~V&l+

total free energy of the system, F, is computed by merely adding the configurational entropy terms to the total enthalpy, U. On the basis of the usual regular solution approximation, we can express”F” in terms of The

(23)

where i,j = I and 2 except i = j. In this equation, Wtj is given by aij =

*vf* . (20)

+cq,.i~f2fC43’vS~+CQ,

= rlt

l)(l-2*Cf)+Z(m+

1)(&l,-&as)

+kT In (C~/Cb,)f2*z*w,,.(mfl).C~

= Al (25)

where i,j = 1 and 2 except i - j, Eliminating II from equations (23) and (25), we find out the relationship kT In ((~)/C$‘)(C~/Ch) = 2.z*v,,(A2~~f2(m+I)(CI:‘-C~)}

F = Ub+Ug_b+UO+n*kT

-z.w,j{A2C”;‘+2m(Cj”-C~)1, (i,j = 1 and 2 except i -1) kT has the usual meaning. As the system under consideration is thought of being a closed one, the total number of atoms of each component in the system has to remain constant. Hence, the following constraint condition holds where

N,+nf/2

= n 2 ,=

Cf’i-rF*

Q/2

(26)

where AzqB means the second dilTirencc with respect to c, AzcI*’ =. ($+“_2*Cf”+cI’-” (i = 1 or 2, and 2 s r s I). (27) In a similar way, from dF*/aQ6*’ = 0, we can get another reIationship

= const.

1

kT In {(C~‘)/c(:))(C’fC~)} (i = 1,2 and 3) (22)

= $Z*ViJf2(2l?t+

l)(~“-C~)+2(~2’-cj1)

where

i-(1 -2*C$-K(I

c

Ni =

n\“‘.

-2*Cf))-zW,J

x (2(m+l)(c’,“-Cb,)+(c;2)-2.c(I”)

k=I

+K’~~+z(Eil-E~j)(I-K)/2 ofsoluteconcentrations which we want toknow,thatis,(Cgl,C’~‘,..., t?!‘)and(C$Cy’,..., Cy’)

Two sequences

are consequently minimize, Following

F, under

obtained

from

the

the constraint

convention,

condition

of equation

let us introduce

where i,j = 1 and 2 except i = j, and K is given by

to

K+(*+f). n

(22).

a function

Finally, from i?F*/aCf

F*(Cy, C’;‘. C!‘) such that

relationship

t -i., z (

cp'+IP*cq/2

II

,=t

where

Y = 1.2 . . . . . I. and

multipliers.

L, and i,

ITron Ihccondilion

>

are Langrange

that (:F*/?Cr’

2 I I’ 5 I) and after some rc:lrrangemcnts.

= O(but wc obtain

(2g)

= 0, we can derive

(29) the following

SHINODA:

PHOSPHORIJS

CONCIINTRAl‘ION

where i,j = I and 2 except i =,j, and K’ is pul

;IS

In principle, to

solving the nonlinear type simultaneous of (264, (28) and (30). we would he able

know

values

the

of

(C:,C’”

IK)l:NI>ARIES

2055

AI/‘,’’ = II’,” - q’,“. In equations (33) and (35), the \~;lriahlcs such ;IS r&l’. I/>“. , and C’s,C!, Cy and C: arc taktli for Lllowll V:llueS in SOlllL‘ manner, for instance, through AEScxperiments.Consequently, wehaveonjy lo solve the sin~ultancous equations (33) and (35) with rcspcct IO the unknown variables of I+,“. I/‘,~). , tf. wIwrc

(311

equations

GRAIN

C-y’,

cc;, cc,’‘. . . . , C$‘)

and hence (CJ, C$”,..., $). ‘;f thi required parameters such as /I’, rig/n and zg/?/= were all known. However, our subsequent eNort in this study will be restricted to calculate a sequence of normalized concentrations of component I, i.e. q\“, $‘.. . , I#‘. Here, s’[’ is defined as follows

cy - c: s’;‘=-c--c’;’

(32)

(r= 1,2 ,..., f).

(Assumption II)

This is because (1) the main subject ofthe present study is to estimate the CDP or the “intergranular segregation extent (BE)” of solute “1” rather than to know the sequence of its absolute concentrations, i.e. haveverylittlequantitative cr 0;‘,cc: ,...,and(2)we kibwledge of the grain boundary parameters such as K’, 9, m’ and nl/n, which must be known for calculating those absolute concentrations. Now, from equation (26), we can obtain a set of basic difference equations with respect to #‘, #,“, . . . , $,‘I

A’&‘+2(m+

r%2*(Cl--Ct) 2.v,,(c:-c’i)

l).$,“-

x {A2’1’2”+2m.&‘} = 2.v

In

kT 13.z*(Cf-C!)

1 +R, *s:” l-S,*&‘-S2.#

(r 2 2) (33)

where ~2’ is normalized concentration of solute 2 at layer r, which can be defined in terms of quite a similar equation to equation (32) for VP).A2t$ means that A2#

= #+I’-2*#‘+#-‘1,

(i = 1 or 2)

where R1 _ --

q-cp

(i = 1 or 2) (34)

s,= 1!Lii+,

(i = 1 or 2)

The last basic equation equation (28) as follows A$,“+(2m+

I)*$,“-

of interest is induced from

W2KI-a 2v,,tc:

x

{#+2m.?$‘+(K+

Our problem formalized in theprecedingsection was solved numerically (see Appendix I) using the assumplion that

The calculative procedure obeys the following steps: Step 1: determination ofXf, the best approximation Of’Il ‘I’ 3 from equation (35) with the use of the inequalities of #’ I q(,” and $t’ 2 0. Step 2 : determination of Xz, the best approximation of r/‘,2’,from equation (33) with the use of the inequalities of g\“’ < #’ and $;’ 2 0. ofX,*(r 2 31,the best Step 3 : iterativedetermination approximation of rf,” (r 2 3), in the same way as in Step 2. Among the parameters used for the present calculation, thesetsofvaluesof(Cq and Ci)and(C! and C:) and the value of T were basically chosen so as to accord with the corresponding experimental data due to Viswanathan [S] on a low alloy steel containing Ni and P. This choice of a reference data is merely for convenience’sake. For thesame reason, the set ofvalues of $,‘I (r r I) were read from Viswanathan’s data [S], taking the monolayer distance to be 1.43A, say, one half of the lattice constant for the b.c.c. iron at room temperature. These sets of known parameters used for calculation are listed in Table 1. Some other sets of Cq and Ct, besides the reference sets, were additionally adopted. These additional sets of Cq/Ct were so chosen as to have considerably greater ratios of Cj to Ci than that for the reference set. Hereafter, the subscripts of 1,2 and 3 used for such symbols as Ci”, vi, \tij, (oij and ci, stand for P, Ni and Fe, respectively. Needless to say, since we will use the “kJ/moi” unit in the actual calculations, every kT term in equations (33), (35) and (A3) must be replaced by RT, where R is the gas constant. Table

I. Inter-

-a

y)l+(K+

for

y)

and

intra-granularconcentrationsof

the present

calculation

c:

= 3.68 x 10-d

C; = 3.68 x IO-*

c:

= 8.45

cy = 9.33 x 10-z

x IO-’

v’i’ = 0.920.

0.860.

0.795,

0.750.

0.710.

0.570. 0.557. 0.540. successively I.2 ..,..

+2(;_Kc:)(‘+7$9 NOW:

kT = 2v,,(C:-c91n

I -S,.4I’-S2’&’

(35)

and(r)m

0.675.

0.645.

according

used

data

0.615.

0.590.

as I increases as

12.

C = atomic

Subrcripls

l+R,*q\”

P and Ni

as a set of reference

fraction

and

,l = normalized

of I and 2 show P and Ni. rcspcctively. thcgrain

bulk layer away

boundary.a

from the grain

bulk layerhrfrom boundary.

concentralion. Superscripts

it and

respectively.

g. b

thcrth

2056

SHINODA:

r’:lramelcrs

Typical veluc and variuble range in parentheses

._-..-_. - -----

s’13 (kJ.‘mol)

PHOSPHORUS

R&S

GRAIN

BOUNDARIES

Fe-P

9 8

20 (IO-30) 30 13%70) -169 -104 6.25 (at T = 753 K) 0.3 (0. I-0.9) 8or 14 (0.4-0.998)

-w,~ IkJ/mol) c,, (kJ,mol) taJ (kl. mol) RT (kJ.‘mol) m 2 K

CONCENTRATlON

10 5 -

Note: subscriptsof 1,2 and 3 stand successivelyfor

P, Ni and Fe.

oiz ( = cop_& set to 20 and - 30 kJ/mol, respectively, referred to the published data [8,9]. The effects of some other values of v,, and w, 2 around these typical values were also studied in view of the absence from certain information of these values. The value of sSJ was taken to be 104kJ/mol, from the data of heat of sublimation for iron [lo]. On the other hand, the E, , (= .+_+fvalue for the nn-pair of P was regarded as - 169 kJ/mol, which is quite different from that estimated for the data of heat of sublimation for P, that is, -64 kJ/mol [ IO,11-j. The value of - 169 kJ/mol for st, was calculated using a relation ofs,, = -es3 -~(v,~-E,~)~ derived from equation (9) and using such values of .sSa = -104,v,,= 20andst3 = -116kJ/molwhichwas taken from Ref. [ 11J. A third group of the parameters are Z, t and M, among which there are only two independent variables because they are related to one another by equation (3). In the following calculation, m was set to a certain value within therangeofO-&say, typicallyO.3.Thisrangefor m was dete~ined by trial and error from various combinations of the magnitude of Z(typically 8 but 14 as a subsidiary value) and that of z, which depend upon crystallographic planes of b.c.c. structure of interest. Finally, the value of K, defined in terms of equation (29), wasalso taken for a parameter unknown, but within the range of O-l. The suitability of these choices of magnitude for the parameters vt3, sir, ct t, M and K, will be discussed later. Typical values for these unknown parameters vtS, o,~, Z, m and K, together The typical values of vt 3( = v&and

were

with thevaluesofe,

2

Number of atomic

layers fmm groin bo~dofy

tc)

Fig 1. (a) Log &r vs r relationship with variation in K, and (b) the q\‘r vs K relationship for Fe-P system, whore m = 0.3, Z = 8 and vtJ = 20 kJ/mol. The broken lines in (a) and (b) show the ease e,, = -64 kJ/mol.

variation in g\” as a function of K has quite an inverse tendency compared with the curve for the case e, , = - 169kJ/mol, The value of $,‘r for the case sl, = - 64 kJ/mol comes up to a maximum of the order of 0.1 at around I< = 0.925, In this case, no solution for the difference equations of equations (33) and (35) longer existsin the lower range ofK lessthan 0.925.In Fig. 2(b), the two curves, irrespective of the difference in their magnitudesof2, show a roughly similar trend of $,‘r in that it increases with a decrease in m within the range of C&l,although a small hump is found on each curve at around m = 0,775; with the curve for 2 = 8, $‘shows a tendency to get to the maximum of the order of 2 x 10e2 at mclose too, whileit attains themaximum of 5.6 x 10e2 at m p: 0.2 in the case 2 = 14. In this latter case, no solution of the di&rence equations longer exists ifthe m-valueis lowered to less than 0.2,As shown in Fig. 3(b), r#r increases with an increase in v,s, but never exceeds the level of about 5 x 10e2 even though

,laJJand RT,areshowninTable2.

3.2. Results (A) Fe-P system. The erects ol K, m and vxJ on the of $,r’, the normalized concentration of P at the 1st atomic layer, lor a Fe-P binary system with a fixed set of Ct (= 8.45 x 10er) and Ct (= 3.68 x IO-‘) are indicated in Figs f(b), 2(b) and 3(b). The effects of I<, m and I’~, on the relationship between logarithm of &’ (r 2 I) and the designated number of atomic layer, r, (simply the log q(,% relationship) are demonstrated, respectively,in Figs l(a). Z(a)and 3(a),in which theother kinds of parameters were kept constant. As can be seen from Fig. I, the value oTr)‘,‘r monotonously increases up value

to the level of 2 x IO- z as K is incrqtsed this figure, -64

corresponding

results

for

up to unity. In the cast

c, , =

kJ/mol are exhibited in passing, where the

Fig. 2. (a) Log $’ YS t r~l~ti~n~hip with variation in II?and %. and(h) the~~‘,“‘vsmrrlationshipsettingZto heXand 14,forthe Fe-P system, where K = 0.95 and Y, 3 = 20 kJ/moi.

SHINODA:

PHOSPHORUS

CONCENTRATION

here were used for the t/\“s

(r)

Fig. 3. (a} Log q’;’ vs r relationship with various sets of Y, 3 and relationship for Fe-P system. where E,,,and(b)thert’,“vsv,~ K = 0.95, nz = 0.3 and Z = 8. The values in parentheses in(b) denotes,, (kJ/mol).

v,,comesup to -30 kJ/mol. in thesecalcuIations,each value of&,, was changed for different values of V,3 ; the s,,-values for diRerent values of u13, shown in parentheses in Fig. 3, were all computed in the same manner as in the earlier case where v13 = 20 kJ/mol, using the same values ofEas = - 104 kJ/mol and sl 3 = - 116 kJ/mol. Figure 3(b) indicates that no solution of the difference equations longer exists if vrs is over 32.5 kJ/mol under the parameter condition that K = 0.95, m = 0.3 and 2 = 8. Figure4demonstrates the variations in q\l’and in the log q’;‘-r relation when the set of values of ef and Ct were variously changed by putting the other parameters into constants such as m = 0.3,Z = 8 and v,~ = 20 kJ/mol. Only the value of K was varied together with the difference in the C:/Ci-value-s. This

(a) Fe-P

,.,2x,o-9

--A----0

I 2

‘057

1301:NDAKlliS

Fig. 5. The allowable ranges of A’ for the solu~i~ts of I/(,‘~ ~tll various sets ofCy/C’; (in parrnthese0. The upper Ilmits of h’

-IO/--’

-10

GXAlN

I 6

I 4

I 8

I IO

ff)

Fig. 4. With Fe-P system, (a) log &’ vs r relationship with various sets of q/C: and (0 : 0.5/7.0 x 10m4and 0.49, A : 0.X/5.0 x lo-* and 0.66, 0: 8.45 x IO-‘/3.65 x 10q4 and 0.995,V:O.f3/5.96 x IO-‘and0.97,and 0: 1.0 x IO-“t2.44 x 10-z and 0.98),and (b) the $,” vs 0: relationship using the upper limit of K for the same value of Q:, shown in Fig. 5. In Fig.4.thevaluesofm = 0.32 = 8,v,, = ZOkJ/molwereused. The values in parentheses in (b) are the ratios of @/Cy.

calculation

in Fig. 4(b).

in K was necessirated from satisfying a mathematical condition so that the diference equations might have their solutions. The allowable range for K where the solutions can exist is shown by a vertical bar in Fig. 5 for every set of C:/C’;. The ratio of Cf/Cl;, the grain boundary enrichment factor, for each set of C~/C~ is shown in parentheses in Figs 4(b) and 5. Figureqb)exhibits the variation in themaximum value of$/)for each set ofCf/CF.This maximum valueofrl:” was built up when K was set to its upper limit within each allowable range, as can be seen from the results shown in Fig. 1. Among these sets of CT/C’; adopted in Fig. 4(b), the Ct value for the sets with Cf-values being lower than -0.15 was determined by using the McLean eqilitibrium segregation theorem of the form [ 121 variation

Cf/(l -Cl)

2 C; exp(AH,/RT)

(36)

where AH, is the heat ofsegregation, being worked out to be about 35 kJ/mol from the reference set of C~/C~, shown in Table 1 setting as T = 753 K. The C’;-value for the other sets with Cl-values being greater than -0.15 was chosen within the range of 4 x 10m4 to 7 x 10e4 so that the ratio of Cl to Ci may change at relevant intervals. It is seen from Fig. 4(b) that, with increasing C8, the maximum value of $,I’ for each set of Cl/C’; first rapidly increases, passes a maximum oftheorder of5 x lo-‘at around Cq = 0.13 and then decreases gradually. It is interesting that the ratio of Ct to Ct,on thecontrary, becomes a minimum just around this peak of the curve. A noteworthy thing in conjunction with Fig. 5 is that the allowable range for K is lowered as a whole as the magnitude of Cq is increased ; for instance, the upper limit ofK keeps a high level more than 0.9 so far as Ct is set to the value less than 0.2, while it decreases down to the order of 0.5 when Cf is increased up to the level of 0.5. The horizontal broken line depicted in connection with each log q(,“-r curve for the Fe-P system implies that the value of $,‘I (or X:) never decreases however increases the number of atomic layer, r, beyond a certain number of it. The existence of this minimum leveiofq([‘probablyhasadeeprelationwith method for calculation,

although

the present

the reason is not yet

2058

StIINODA:

PHOSPHORUS

CONCENTRATION

0

-1

-2

-3

-41 0

I 2

I 4

I 8

I 6

I IO

I 12

I 14

(II

Fig. 6. (a) Log r$’ vs r relationship with various sets of Cy/C’( and K Ic): 8.45 x IO-‘/3.68 x IO-“ and 0.89..- A: 2.44 x IO-‘/‘I3 x 10-0and0.~85,~:0.28/4,56 x 10-qand0.60, and 0: 0.5/7.0 x IO-* and 0.38), and (b) the ‘1:‘) vs c( relationship (0-O) and the corresponding curve of the Kupper limit vs C; relationship (O-O), for Fe-Ni-P system, where 1~ = 0.3, Z = 8. 1*t5= 20 kJ/mol and o),~ = -30 kJ/mol. The values in parenthesis in (b) are the ratio Cj/Cj.

clear. The minimum level ofr]r’ is, however, apparently dependent on the ratio of Cf/Ci as can be seen from Fig. 4. (B) Fe-ASP system. Figure 6 provides an illustration for the Fe-Ni-P system which is just comparable to that for the Fe-P in Fig. 4, Both the figures have thesame parametriccondition with respect to the magmtudes ofm, Z and vt S. The value ofw,, in Fig. 6 has been set to - 30 kJ/mol. The ratio of Cf to C: for each set of Cq/Q in Fig. 6(b) is also depicted in each parenthesis. Comparing Fig. 6(b) with Fig. 4(b), we are able to see that both the $,I’ vs cSprelationships are in qualitative agreement with each other except that the maximum of q’,” for the former, 9 x lo-‘. is approximately twice as great as that for the latter, 5 x lo-*, and that the peak of the Q\‘)--C; curve is not built up at the point where the ratio of Cl/C! would become a minimum. The relationship be-

tween the upper limit of K and CF in Figs 6(b) and 5

--w12fkJ

-4

!-

0

.-I---. 2

: [ i

-wi21 kJ/mol)

ImoC)

“,8

1

I

I

I

I

I

4

6

8

IO

12

14

GRAIN

BOUNDARIES

aresimilar toeachotherinaq~litat~ve~nse;~eup~r limit of K for the Fe-Ni-P system is somewhat smaller than that for the Fe-P system as a whole when compared at the same point of C: or C$K$ratio. Different to the curves for the Fe-P, the log #-r curve for the Fe-Ni-Psystem hasalong tail, but itslevel of$,” being of the order of 1O-2-lO-3 at most. Figure 7 demonstrates the effects oF~>,,,~on the value of $,“, on the K upper limit, and on the log t&“-r relationship for the Fe-Ni-P system, with setting the other parameters as Cf = 8.45 x lo-‘, C: = 3.68 x 10e4,Z = 8,andm = 0.3.Figure7(b)showsthatthe maximum value of &’ corresponding to each upper limit of K reaches 0.10-0.12 so long as{-o,,)is set to within a range of40-70 kJ/mol, while the upper limit of K itself has to be lowered to a value less than 0.4 if the value of ( - w , z) is raised up to more than 70 kJ/mol. 4 DISCUSSION

Validity of the present results would be rather dependent on the propriety of the assumptions and magnitudes of parameter adopted in this study. With regard to Assumption 11,there would be no problem to

discuss further, at least so far as the P~quj~ibr~urn intergranular segregation in ferrous alloys is concerned. Assumption I signifiesthat the binding energy for a nn-pair of one atomic bond in an alloy crystal depends upon the relaxation state of the crystal structure rather than the type of atomic bond itself, i.e. “like” or “unlike” one. This assumption would be correct at least to the first approximation. Once we recognize Assumption I, such a condition that lv$i < fv,,j, which is one of the necessary requirements for solute i to he enriched at grain boundaries in an i-j (solvent) binary alloy system in equilibrium 19%121, is explicable consistently in terms of the present model : the first necessary presumption such that lefjl < levl has been recently supported by a theoretical study based on the electronic theory [ 133. The second necessary presumption that {z* 1< izl would be acceptable in view of the crystal structure of the g-phase being possibly more relaxed in a geometrical sense than that of the b-phase. These two presumptions would vindicate that 0 c p’ < I in the context of A~umption I. Thus we can successfulfy interpret that rvtl < Ivu[, using equations (9) and (19) with the aid of Assumption 1. Let us turn to the discussion of the propriety of adopted magnitudesorpa~meterssuchas~~ t.~,3, Km andca,,. Withtheuseofthemagnitudeof-64kJ/moi for I:, , which is estimated from the data of the heat of sublimation of P, the value or r. which is defined in terms o~~quiItion (A3). is made to be a positive. lfz is positive. the term U,, also deiined in terms of equation (A3). will become the minimum of unity just when K is identical with unity, as can be easily understood from the ~~e~ii~~ti~~n of R,. This situ;\tion brings a serious problem into the present C;llCUlittiOil, Sitcil thitl our din‘crcnce equations ilFt! no longer solved in case

J’,

(E

o&r

c;

-r:)

exceeds

iL

lhat the dilrcrcncc

033;tirt

tll;igl~illidC

cqu;ltioris

nay

01’

if ;

h;l\,e

ii)

Ihcir

in 1aws l’u~tction

ofcqtf:fiictrt ot’

cithcr

(31. %

or

:.

IIW

v:~lrrc

With

of fir

rcg:lrd

cll;ifjgCs

;fs ;j

to tile

b.c.c.

slructurc. % is idcnlical lo S ir WConly cake the first ,],Ipairwisc inlcr;lclion into account For instance, the ,,IAppmdin ill. It rOllows that this rcyuifemcnt will never values for pktncs of ( It)O),( I I I ) :tlld (I IO) in tbc b.c.c. I~s~I~ist~c~jify, 2 O.tScvrn when H., look its ~~~i~~i~i~t~~~ structure arc $1. l/Z antf 1. rcspmivcly. As nlCntj(~ned of unity on the condition of ii positive value of x. carlicr, howcvcr. fff slhmtti hC l00hCti upon 8s ;t const;+nt Alternatively, if I:, , is such ;I v:hrc Ihal makes (1 i-Y) ;ivcragctl over the various types of phncs in the b.c.c. nep:iGve, the trouble menGoned above can be rcsolvcd. structure rztther than the value relalcti (0 ;i specific This is readily understood in view of the dclinition UT crysr:lIlogr;lphic plane. More ztltcntion is pxid to tlic R, ; in that case. S,can bea small positive value. as close present rcsuit 11~~1111hs ljltic cfliic~ on Lhc value off/(,” tozemuspossihle.ifh’issct t~~CiCCl%li~ vatt~e\~i~l~in lhc itsdf,so long:~s% isequA ~o~.itss~l(~~v~i in Fig. Z(b). The range of O- f . The value of - 169kJ/mol for E$,. adoptcxi curve for 2 = 14 in lhis figure indicates ;I possible case in this study, apparently makes (1 fr) negative. when we exgcnd the concept of “~1” 10 the second WHowever, the value of - 169 kJ/mol for I:~, nearly pairwise inleraclion approxim:i~ion. Even in that cast. causesa problem ofwhether the binding force for a P-P I)Jfalls’within the rangeoTO-I ; for instance. 111 becomes couple could really be stronger than that for a pair of 0.4for thecaseofthe( 10O)parallelism sinceZ = 14and Fe-Fe & = - IO4 kJ/mol), for instance, under some z = 5 at that time. It will bc pointless to discuss furlhet condi~ionsof~heai?oysysfem.Thisqu~s~ionrema~ns to thiscasebecausethepresentn~odeliscons~i~u~edon the be answered in the future. On the other hand, we can basis of the first nrl-pairwise interaction approxiderivesuch a relation that (1 +I) = (E,~-E&*,~ from mation. equations (9) and (A3). Hence, (1 $ z) is to be negative Just as in the case of I*,3, we have been very short of only if vlJ > 0 and E,~ < +p Some investigators credible information about ( -wi2). The present [11,14, IS] have presented some cafculafions that calculation, however, has shown the possibility of supporfthe~ssibifitytha~s,~ < ~~~.Asregardsv,,,no ( -wi2) being as large a level as 7.5kJ/mol, as estimated credible value has been found as yet, at least to the by Seah [8), if the K for one material could be lowered present author’s knowledge. According to acalculation down to the level of -0.5 [see Fig. 7(b)]. The present results obtained thus far has shown that based on continuum mechanics [9], the v,~ has been estimated to be about 12 kJ/mol (positive). A higher the calculative ISE of P is approximately “one estimation of this quantity may be possible from the monolayer” thick at most and never exceeds “two view that vi 3 should be smaller than the value of AH, monolayers’” no matter how we choose the magnitude cited in equation (36), say, + 35 kJ/moI. Thus, if turns ofeach parameter needed for the calcufation within its out that our estimations for v13 and E,, adopted in the range referred to ( this is true even if-we would adopt the present study are never too unreasonable. value of E, , = -64 kJ/mol), and that the calculative Through the definition of K in terms ofequafion(29), ISEs’are fairly small compared with the observed ISEs’ if can be regarded as a positive value less than unity. by means of the AES depth analytical technique, which This is proved from the following. First, p’ 5 I, as has havea~~ained“six monolayers or such” as shown in Fig. already been verified in the previous section. Next, the 8. Here, the ISE of P has been defined as the monolayer possibility that n’ 5 n would be reasonably acceptable thickness of a P-enriched bulk region adjacent to the sc~lutions.

Such a Wily

the nlagnilude CIS II,,

X ?‘r hS

crl’ U.4 nlust

tx

10 he! IcSS Ihilll

rcstriclcd _ 0. I5

ill

[WC

since the crystal structure of the g-phase would be more or less relaxed geometrically relative to that of the b-phase. In view of the bold assumptions and simpIifi~tions adopted in the present model, K in this model would not be a material constant as its literal sense; this is suggested, for instance, from the fact that the upper limits of K for Fe-Ni-P system shown in Fig. 6(b) are not in accord with those for the Fe-P system shown in Fig. 5. However, it should be noted here that K for an alloy, as wet1as its values of v, s and W%2rplays an important role in determining a possible value of Cz under a given condition of the other parameters. The lower limits of K shown in Fig. 5 are prescribed from a mathematical requirement that the term B, in equation (A I) or (A2) is to be positive. This is formulated by K > A,‘(t +A)

where A = --_(I +0)/(2*y,)-y,/R, E -(t +r)f(;!*~~) sincey,/R, cc 1 in the cases investigated here. In Fig. 5, the lower limit of K for each set of C;/C’; was obtained in this means.

-.-

Coiculoted

P-COP

for Fe-NI-P

-+-

ColculoteG

P-CDP

for Fe-P

Qf AES

-CDP

of P for steel

(Refs C4 -61

0

2

4

6

8

10

J

12

from groin boundary in the monoloyef unst of 1.43&

Dfstonce

Fig. 8. Comparison between the calcuk~ted log I/‘;’ VJ Y relationship (showing the highest values of #,” for the Fe- I’ and Fe-Ni-P) and the observed one by means nT Al3 technique for some alloying sleek

2060

SHINODA:

PHOSPHORUS

CONCENTRATION

grain boundary, where ,I’{’is greater than one-tenth of TV’; ( E I 1.After all, the present calculation has supported the “sharp interface” hypothesis, at least so lar as the P intergranular

equilibrium

is concerned.

In fact. such a monolayer

segregation in ferrousalloys type of solute

segregation has been evidenced by Stoddart et al. [3] through a skilful AES experiment on the Pd surface segregation in Pd-Ni system. In view of the good analogy between both the equilibrium intergranular and surface segregations, it seems to be quite probable that the same thing occurs with regard to the P intergranular segregation in ferrous alloys. If the “sharp interface (or monolayer type)” hypothesis will be correct, we have to answer the question of why one necessarily gets an exponential type of CDP when he observes one monolayer type of solute atom arrays built up at grain boundaries, by using the AES depth analytical technique. A successful answer to this question may be given by the idea, due to Palmberg and Marcus [ 11,on the basis ofthe statictical nature of ion-sputtering process. Alternatively, if the true picture of a solute CDP is unable to be seized correctly by means of the AES analysis, there will be inevitably a question of whether the direct usage of the AES data for Ni are allowable, without any correction

as was done in the present calculation. Concerning this respect, we are able to say that the AES-Ni data are usable without any correction, because the discrepancy between the true CDP of a solute species and its AES

image may be reduced rapidly if the ISE of the solute extended over more than a certain thickness of monolayers. A more detailed account will be presented by the present author in another paper. 5. SUMMARY The equilibrium “concentration depth profile (CDP)” of phosphorus across grain boundaries in ferrous alloys with or without Ni has been calculated, based on a model, which in principle is analogous to Ono’s discrete lattice approach, using the simple nearest neighbor (nrt)-pairwise interaction and regular solution approximations. The calculation has been performed on the assumption that the P-concentration at the grain boundary (g)-layer, Cf, and that at a bulk (b)-layer far from the g-layer, CF. and the Ni-concentrations at the g and its successive bulk layers are all given. The calculated CDP of P are demonstrated in terms of the relationship between a normalized P-concentration, II’,”[I (<“;I- Cr)/(Cy -C:)], and r, the number ofbulk layers ;~way from the Ist-layer next to the g-layer towards (he bulk interior. Here, C’,” is the P concen)r;ttion at the rth bulk layer. With the Fe-P system, the “intergranular segregation c\tt‘n~ (ISE)” or P. which is defined as iI bulk region adjacent IO 111~g-ph;rsc where ‘I’,” is more than one-tenth err/; ( E I ).is - one monolayer thick at most ;rrrd ne\cr e.sccrtls two monolayers choose the

no matter how we

uurgnitudss or parameters needed for the

GRAIN

BOUNDARIES

The same is true with the Fe-Ni-P system ; these suggest that the equilibrium intergranular segregation of P in ferrous alloys occurs exclusively at grain boundaries per se. This suggestion may be supported by the experimental evidence of the monolayer type of equilibrium surface segregation of a solute in some alloy system. The parameter, K, newly defined in this model as a constant whose magnitude shows a degree of the relaxed state of grain boundary structure plays an important role in determining the possible value of q, togetherwith thevariousinteractionparametersforthe couple of atoms. calculation.

Acknowledgements-The author is grateful to Professor S. Umekawa and Professor T. Suzuki. Research Laboratory of P.M. & E., Tokyo Institute of Technology, for their great encouragement in the present work. REFERENCES 1. P. W. Palmbergand H. L. Marcus, Trans. AmSoc. Mefals

62,1016(1969). 2. E. P. Hondrosand M. P. Seah.Segregarion fo Interfaces,p. 262. Int. Metal Rev. 13977L

3. C.T. H. Stoddart, R. i. M&s and D. Potx.SurfSci. 53.241 4. H. L. Marcus, L. H. Hackett Jr and P. W. Palmberg, ASTM Data Ser.. STP 499.90 11972). 5. R. Viswanathan. Metal. T&s. i, 80s (1970). 6. R. Viswanathan and T. P. Sherlok, Mete/. Trans. 3,459 (1971). 7. S. Ono. Mem. Fat. Engrs, Kyushu Univ., Japan 10, 195 (1947). 8. M. P. Seah. Acta metal/. 25,345 (1977). 9. T. Shinoda and T. Nakamura, Trans. Japan Inst. Metals 21,781(1980). 10. R.Hultgren,P.A.Desai,D.T.Hawkins,M.GleiserahdK. K. Kelly, Selected Values ojThermodynamic Properties of Binary Alloy. Am. Sot. Metals, Metals Park, OH (1973). It. W. G. Hartweck. Scripta merals. 15,453 (1981). 12. D. McLean, Grain Boundaries in Metals. Clarendon Press, Oxford (I 967). 13. K. Masuda. M. Hashimoto. Y. Ishida, R. Yamamoto and M. Doyama, J. Phys. Sot. Japan S&3990 (1982). 14. A. Yoshikawa. Trans. Iron Srerl Inst. Japan Swpl. . . 11, 1263 (1971). 15. M. Hashimoto. Y. Ishida, R. Y amamoto, M. Doyama and T. Fujiwara, Scripta metafl. 16,267 (1982). APPENDIX

The best approximations oftf,“.

#‘,

I

...

For simplicity, consider the case when r&q\“,. . . , ~“2are all zero. Since Ar#r = B\sr-r#r 5 0 and v\*r 2 0 from equation (35)and Assumption 11,weobtainacoupleoHnequalities with respect to X, and X,, which stand successively for approximations of B’,” and r$’ [X,, henceforth, denotes an approximation of 4’;’ for any integer of r] AX’,“z

X,-X,

= B,,

In

l+R,*X,

I-s,*x, -B,-2(m+

1).X,

s 0

(Al)

-B,-(2v1+

I)*‘X,

t 0

tA2)

and

X2 = F(X,) = B,, In

I+R,*X,

I-s,-x,

SH[NODA:

PHOSPIiORUS

CONCENTRATION

GRAIN

ROUNDARIES

206 I

where

B,

I

[K+(I

-K)/R,]+(l

-K)(I

+T)/(~.v,).

Taking accoum of the fact that the parameters of R,. S,. y,, B, ,. z, B, and rn are all positive as shown in the (ext. we can easily determine a range within which Xr falls, Xf 5 X, $ X7, as schematically shown in Fig A I(a). First, pm as x, = x:1 s (Xi+ X:)/2. One can readily determine the value of X, from equation (A2) as X\‘r = F(X’,“). Let us here remark that the magnitude of X2 increases with an increase in X,, as can be seen from Fig. Al(a). The range for X, is similarly by given using the following inequalities which are introduced from equation (33) and

(b)

Assumption 11 AX\”

= X,-X,

= B,, In

l+R,-X2 I-S(.X, -X’,“-(2m+

1)*X,

5 0

(A4)

r 0.

(A5)

and

X2 =G(X,)=B,,ln

1+&*X2 1 -s,

‘Xs -X’,“-2m*X,

IfthevaluesofX\“andX$“given abovestand successively for the “good” approximations of r#’ and #‘, the X$‘r must fall within a range, X$5 Xyr < X2, which is prescribed from the inequalities of(A4)and (AS), as illustrated in Fig. A l(b). To the contrary. if Xs < 0 because X\“is a “wrong” approximation, theSx”isexpected to takeavaluelower thanXf. In that case, X, must be rearranged to a smaller starting value such as Xl” = (X\r’+Xt)/2. In the case AX’:) > 0, since Xi” is “wrong”, the Xi“ will fall on the higher side of Xy so that we may rearrange Xt as Xrtzr= (Xl’) +X:)/2. Generally, the range for X, (r z 3) can be conditioned in terms of the following inequalities [see Fig. Al(c)) AXj*),X,+,-X,=8,,

ins -

1.

I

-X!~*-(2m+l)X,50

(A6)

(d)

and X r+l=G(X,)=B,,ln-

I+R,*X, l--&.X,

-X!‘It -2m*X,

2 0 (A7)

where the values of Xr),Xtt’J,....X$., are all “good” approximations for a certain starting value ofXyr, with I being the iterative number of rearrangements of Xt. At this place, if the X, for the iteration number of I, X?[as G(X$$L ,)I, fails to satisfy either the inequality of (A6) or (A7), we must accordingly rearrange thevalueofX, oncemoreso that X\‘+ I) maybeequaftoeither(X$u + Xa/20r(qrr + X:)/2,whereXy designates a greater one of either Xy-tt or X$‘-2r and Xsf’ is vice versa. These procedures are iterated until )X~+‘r-Xt,‘)j becomes smaller than a given calculation error, b. When IX:‘+tr-Xttrr] s b, we regard X:[E (X:(‘*r+Xt,r’)/2] as the best approximation of r#t. Once XT is given, X:, the best approximation of qrt2’,can be determined in the same way as XT was computed. This time, we start with the inequalities of (A4) and (AS), but where Xtt” on the left hand side of the equations must be substituted by XT. In a similar way, we can easily obtain X:, X2, etc.

Fig. Al. Schematic illustrations for the numerical solution of the nonlinear difference equations treated in this study.

APPENDIX II A necessary requirement /or B, to yield the solution, XT In general, the digerence equations under consideration have not the solution with respect to X,, unless equation 5, and equation t2, which are defined below, intersect with each

StIINODA:

2062

i’l4OSPt-IC)RliS CONCENTRATfON

GRAIN

BOUNDARIES

sineeRr w I. if24, *S,/M CCI asin thecaseofinierest,X~is further approximated to the form that X(: 3 B, JM. In order that 5, mayintersectwith~,,thetermB, x y, mustbesmaller than A~,,,ascanbeseenfrom Fig. AI(d). Here.A<,, is given in the form that

and r2 = M~.v, ‘X,

(A9)

where M is either of 2(m + 1) or (2~ + I). Let us here define a function ol X, such as

A< = 5, (bur B, = 0)-r,

Putting as RT = 6.25 kJ/mol. vrs = 20 kJ/mol. o = 3.08, 1O.t with y, and M being 0.3 and 1.6, respectively, we know that A<, -w 0.16. Provided r&r&r , . . . arenot zero, the same is trueonly if we replace 8, in each equation with S; which is defined as R, = 600 and Xt ok B,,/M

As can be readily understood, this function AC takes a maximum, A<,,,, at X, = Xy, which is nearly equal to

X+&/ii),

Note added in proof-According

(a I)

(All)

where B,, = %(cf4%(2%,.Y,I4

to Miedima’s rule [J. less-common Met& 32, 117 (1973)], the interaction parameter for Fe-P system, vu, is expected to be negative because this binary system containa stable intermctdlic compounds. conducting similar calculations for the Fe-Ni-P system based on the present model with the use of, e.g. such parametric vahma a~ vu = - 15 kJ/mol, q1 = - 75 kJ/moi, Z = 8. z = 3.08, e3,= - 104kJ/mol, e,, = - 64 kJ/mol and with the Table 1-v&o for C:, Cf, t#, etc. we obtain more interesting results such that )I,1’)= 0.44 and 0.12 according as K = 0.2 and 0.98 but qjo(i 2 2) are leas than 6 x 10eJ in both the cases. this implies that the conclusions mentioned above hold csscntially without correction at least in the mining that the “sharp interface” type of in~r~n~ar segregation of P is also deduced from the present mode1 even in the case of taking vlf to be negative.