Profiles of phosphorus concentration at grain boundaries in ferrous alloys

OOOL6160/85 53.00 + 0.00 Copyright Q 1985 Pergamon Press Ltd

~cro mrrafl. Vol. 33. No. 1, pp. 139-144. 1985 Printed in Great Britain. All rights rc~er~cd

PROFILES OF PHOSPf-IORUS CONCENTRATION AT GRAIN BOUNDARIES IN FERROUS ALLOYS T. SHINODAt Research Laboratory of Precision Machinery and Electronics, Tokyo Institute of Technology, Nagatsuda, Midori-ku, Yokohama 227, Japan (Received

7 June 1984)

Abstract-The CDPs of phosphorus across grain boundaries in the preceding alloys have been calculated on the assumption that v,,, the interaction parameter for Fe-P system, is negative in sign. The obtained P-CDPs are essentially the same in type as those in the previous study where vu was positive. When v,, < 0, the normalized P- concentration (with a variable range of O-l), xl’), at the first bulk layer adjacent to the grain boundary is able to reach a level as high as 0.8 or more, while the successive n,- values at the bulk layers of Znd, 3rd, etc. never exceed the order of magnitude of IO-’ however the parameters for calculation are chosen in each variable range. If another restriction that e,, < e,,, where e,, and e,, are the binding energies for the nearest neighbor pair of Fe-P and that of Fe-Fe respectively, is imposed besides the above condition of a,, < 0, the model used in the present study fails in the P-CDP calculation so far as the FcP binary system is concerned.

R&muG-Nous avons calcule les protils de concentration en profondeur (PCP) du phosphore normalement aux joints de grains dans Ies alliages FcP et Ni-P en supposant que le paramitre d’interaction v,s pour le systtme Fe-P Ctait nbgatif. Les PCP du phosphore obtenus sont essentiellement les memes que ceux qui avaient Ctt obtenus au cours d’ttudes anttrieures dans lesquelles v,, Ctait positif. Lorsque v,, est negatif, la concentration normali&e en phosphore (qui varie de 0 B I) n\‘) sur la premiere couche massive adjacente au joint de grains peut atteindre une valeur de 0,s ou plus, alors que les valeurs successives n, sur les secondres, troisiemes, etc couches ne dfpassent jamais un ordre de grandeur de IO-* quels que soient les parametres choisis pour le calcul dans chaquc domainc de variation dcs variables. MSme si l’on impose en plus de la condition ci-dessus vu c 0, la restriction e,, < e,,, oi e,, et e,, sont respectivement les energies de liaison pour les paires de premiers voisins Fe-P et Fe-Fe, le modtle utilis6 dans cette etude ne permet pas de calculer les PCP du phosphore en ce qui concerne le systeme binaire Fe-P.

Znsammenfassuog-Die Konxentrationsprofile von Phophor iiber die Komgrenxen in den friiher untersuchten Legierungen wurden berechnet. Hierzu wurde angenommen, dal3 der Wechselwirkungsparameter fiir das System Fe-P, v,, negatives Vorxeichen hat. Die erhaltenen P-Profile sind im wesentlichen dieselben wie die friiher fiir den Fall positiver v,, erhaltenen. Fib v,, < 0 ist die normalisierte P-Konzentration (mit einem Bereich O-l) n\” durchaus in der Lage, in der ersten Schicht an der Komgrenze ein Niveau bis zu 0,8 zu erreichen. Die nlchsten Lagen iiberschreiten niemals 10e2. Wird eine weitere Einschtinkung eingefiihrt, nlmlich e,, < e3)(Bindungsenergien fiir Nachstnachbarpaare

von Fe-P und Fe-Fe), dann IiiBt sich die& Model1 auf das biniire Fe-P-System nicht mehr anwenden.

1. INTRODUCTION

In a previous paper (11 the phosphorus CDP across grain boundaries in Fe-P and Fe-Ni-P systems under equilibrium was calculated on a discrete lattice model, which rests on the same theoretical bases as those for the models developed by One [21,Meijeting [3] and Williams and Nason [4]. The calculated results in the previous paper revealed that the Pconcentration at the first bulk layer adjacent to a grain boundary sharply decreases down to less than one tenth as large as that at the grain boundary. Such a “sharp interface*’ type of solute segregation at grain boundary or free surface has been evidenced by some experiments, e.g. with P-intergranular segregation in tPemranent address: Hitachi Research Laboratory, Hitachi Ltd. Hitachi 317, Japan.

a Fe-P alloy by means of a field ion microscopy [5] and with Pd-free surface segregation in a Pd-Ni alloy by using an Auger electron spectroscopic technique 161.

The previous calculations, however, were made on the assumption that the interaction parameter for a Fe-P binary system, vFep, is positive, say of the order of 20 kJ/mol. On the other hand, Miedema’s rule [ 171 indicates that the interaction parameter for a binary system which contains stable inter-metallic compounds generally takes negative sign. It is very likely that vrc-p is negative since the Fe-P system has stable compounds like Fe,P. This situation has necessitated us to perform supplemental CDP calculations based on the same model, but with the use of the negative sign of vrc-r.. The aim of the present note is to know how the phosphorus CDP’s for negative values of vre-r differ

139

140

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from those obtained in the previous study [1] and to know whether our model can be carried over even in this version of uFep being negative. 2. CALCULATIVE PROCEDURE

Basic dlrerence equations let us detine a normalized con~nt~tion of ith component at the rth bulk layer which is counted from the next one to the grain boundary layer, nj’), as follows

cp - c:

nf’=---.

Cf-Cf’

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

(1)

where Cf), C: and Cf are the atomic concentrations of ith component at the rth bulk layer, at a buIk layer which is far from the grain boundary and at the grain itself, respectively. The subscripts of 1, 2 and 3 adhering to some variable stand for P, Ni and Fe, respectively; e.g. uu refers to a,,. The superscripts of k),’ and “b” denote the grain ‘&g”, “t(=l,2,..., boundary layer (or phase), the rth bulk layer and a bulk layer far from the g-boundary, where k is a limited number of bulk layers. Thus, we can provide a set of the basic difference equations with respect to a set of unknown variables of n\‘), nj’), . . . , n{“) as follows

OF PHOSPHORUS

1 + R, .nj” 1 - S,*n\‘)-- S**n$”

(2)

ALLOYS

and that Cf, C’f, RT (s gas constant times absolute temperature) and a set of Ni- concentrations of Cj, cp! . . . , Cf) are all given from an experiment on Ni and P intergranular segregations in a low alloy steel [8]. The parameters necessary for calculations are after all reduced to m, z (or Z), K, u,,, w,* and a. Out of these, m is related with Z and z in the form m = (Z/22) - 1

(6)

where Z is the usual co-ordination number for a given crystal structure of the bulk phase and z the number of nn-pairs for one reference atom on a bulk layer to given atoms on its neighbouring layer (either of the two). The K is defined as K = (B’/2) (1 + W/N)

(7)

where B’ is a positive constant being a little less than unity [I], N the number of atoms on each bulk layer and NS is that on the g-layer. The vu is given by nil =

et3 -

(et l +

e331/2

(8)

where eii is the interaction energy per m-pair for the “i-j” bond (i,j = 1, 2 or 3). The w,* is thelSo-called Guttman’s Metal-Impurity interaction parameter [9] for the Ni-P coupling in a FeNi-P alloy, and is written in the form 42 The

=Q*ln

IN FERROUS

=

42 -

@I3

+

h3).

parameter a is a constant defined as a= (et, - eJ(2

*u13).

(9)

From equations (8) and (9) we can obtain the following relationship

and

(1 + a) = (er3 - e33)/a13.

yr

(10)

A*@ + 2(m + 1)n - H$‘) =Q*ln

1 + R,*nf) ( 1 - S, *nf’ - S2*n5”)

(r 12)

(3)

where R, = (Cf - Cf)/C:

(i=l,2)

Si = (cg - Cf)/( 1 - c: - c:, I A2nl’)=n~+‘)-2.n~)+np-‘f

(r 22)

Q = W-M2*v,-z -n)

(4)

B = (1 - K) (1 + a)/2*y, - [K +

(1- WI41

(5)

H~)=Q’[(n~*)+2*n~‘+Kf(l-K)/K] and

Values of rhe parameters For the sake of comparison of the present results with the previous ones, the present values of parameters, except for u,~, wlz, et, and K, were equated with the corresponding values in the previous calculation. The present parameters, together with the previous ones, are listed in Table 1. The value of e,, was this time estimated from the available data for the heat of sublimation of phosphorus [W-12]. Influences of uu, w12, K and m on the calculated results of nj” were investigated with changing the magnitude of any one of these parameters but fixing the magnitudes of the remainders to those values shown in Table 1. The dependences of the ratio of Cf/C: on n\‘) and np’ were also investigated in this study.

H$‘r= Q’[n$‘+‘) f 2.(m - I).# -I-@-‘l (rZ2)

3. RESULTS

with y, = (CI - C:) and Q’ = (w,JS2)/(2~u,,*S,).

Fe-P binary system

To solve the above difference equations, we will here assume that nf(l E 1)2n\1JTn{2)Z _ -***

I nf (s 0) Assumption

I

In A~umption I we have assumed that nf) 2 0. From this inequality and equation (3) in which both Hi2) and S, are to be zero due to the Fe-P system,

CONCENTRATION

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PROFILES

OF PHOSPHORUS

IN FERROUS

141

ALLOYS

Table 1. Parameters used in the present calculations and :hosc in the previous study

3.65 x IO-’ 3.65 x IO-’ 8.45 lo-*

0 np RT Z

u,,t wut c,,t

9.33 x 10-z See Table I in Ref. [I] 6.25 8

Previous

Present

Present/Previous

c: SE

-IS -64 -75

%t K m I

20 -169 -30 -104

0.5

SI 0.3 3.08

tUnit: kJ/mol.

we obtain

RHS of equation lationship 1 + R,.n$ Q-In , _S,.nl*‘L2m.nl*‘+nl”.

(5) since R, + 1, we find the re-

Bz-(l-K)(l+a)/2.y,-K(>O)

(13)

and then obtain a restriction with K It is readily shown that the above inequality never holds when u,, is negative [so that when Q is negative as can be seen from equation (4)], because the right hand side (RHS) of the above inequality is to be positive. In consequence, we necessarily conclude that all the value of n’;’ except for n\‘) must be zero in this case. Thus, we have only to solve the following equation with respect to unknown value of nj’) Q.ln *=(2m+l)X-B I’

(11)

where Xsignifies the unknown variable of nil). Equation (11) can be derived from equation (2) putting as n\‘) = 0, H$ = 0 and S, = 0. Figure 1 shows a plot of log .(I’) vs K on condition that z =8, z = 3.08 (i.e. m = 0.3) and u,, = - 15 kJ/mol. The value of n\‘) takes a maximum less than 0.1 at K = 0, but decreases with increasing K, followed by a sudden fall in n\‘) at around K = 0.65. This sudden fall in n{‘)is apparently related with the fact that there is a maximum K beyond which we can no longer find any solution to equation (11). This is directly because the constant B becomes negative when K is over the maximum, K,,. Thus, we know that the inequality BZO

KSK,,[=A/(lfA)]

(14)

where A ‘Y -(1 + a)/2*y,. This A is positive because -(1 - K) (1 + a)/2*y, 2 K from the preceding equation of (I 3) and further because 0 5 K 5 1 [I]. Consequently, the term (1 + a) must be negative. Hence we have the relationship

taking account of equations (8) and (10). Putting as e33= - 104 and e,, = -64 kJ/mol in the above relationship, we have a restriction with u,~, - uu 5 20 kJ/mol. Fe-Ni-P

(1%

ternary system

In this case all np’- values with 7” being two and over are not necessarily zero. This is because the positive term of H&‘)(r 2 2) remains on the left hand side (LHS) of equation (3). Figure 2 (a) demonstrates the plots of log P$) vs r for K = 0.2 and 1. It is interesting to note that the n\‘) becomes significantly greater than 0.1 when K is as small as 0.2. However, the values of nr) for r 2 2, without exception, fall down to the level less than 0.01. In Fig. 2 (a), the value of n12’is a little smaller than n(I’! This is of no matter in view of the main purpose of the present

(12)

must be warranted to solve equation (11) with a negative value of Q. Neglecting the third term on the

I

I

1

Fe-P

-1

-1

-‘I 0

0

0.2

0.4

0.6

0.8

Fig. I. Effect of K on the nj’)- value for FOP system. Values of the other kinds of parameters are shown in Table I.

2

L

6

8

10

12

1’4

(4 Fig. 2. (a) Variations in rr\‘) (or in n{“)) as a function of atomic layers from grain boundary, I, and (b) the nj’) (or nj2))vs K relationship for FeNi-P system. Values of the other kinds of parameters are shown in Table 1.

142

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IN FERROUS

ALLOYS

0.6

0.6

1.0

m Fig. 5. Variation in n{” (or in np)) as a function of m for Fe-N&P system. Values of the other kinds of oarameters are in Table 1.

Fig. 3. Comparison of the preceding (“unmodified”) log n\” vs r curve in Fig. 2 for K =0.2 with the corresponding (“modified”) curve based on a modified calculalion.

study, but is apparently inconsistent with Assumption I. To remove this shortcoming, we can modify the method for CDP calculations in such a way as nt’ will be accorded with n\‘+ ‘) (i 2 2) even when an event as ny+‘)Znp occurs during each computing performance. Figure 3 demonstrates a comparison between the preceding result given in Fig. 2 for K = 0.2 and the corresponding curve obtained from the calculation thus modified. As the difference between both the n\“- values obtained through the two methods is negligible as shown in Fig. 3, we will calculate the CDPs of interest by means of the method prior to modification, unless otherwise noted hereafter. Figure 2(b) is a plot of ,\I) vs K for this ternary alloy system, showing that the .(I’) value is continuously lowered as K is increased within the range of O-1. This is in contrast with the curve in Fig. 1 for the Fe-P binary system, where we experienced a drastic fall in the nj’- value at a certain plot of K. The n\*‘- value tends to increase as the corresponding n\‘) is decreased, although the former value does not exceed the level of IO-*. The relationship between nj” and ui3 is shown in Fig. 4. The n{‘) tends to increase as (-Q) is lowered,

0.8

S&R.

and this tendency becomes noticeable as K is lowered. The “solution-less range (SLR)” which often appears in Fig. 4 and the subsequent figures signifies a range where the calculated n{‘) exceeds unity. We may call this range the SLR in a sense that Assumption 1 fails there. It should be noted here that, even in this range of SLR, the nv)- values for r 12 never exceed the level of IO-*. The reverse V,,-dependence of n\*) to that of nj” is also found in this case. The variation in nl” as a function of m is illustrated in Fig. 5, where K is fixed to 0.5. It can-be,seen from this figure that the n\“- value is hardly affected by m, and that the SLR is found at a side of m close to zero. Such a moderate effect of m on the n{“- value has been aIso reported in the previous paper [I]. The np’ also here shows an inverse change to that of nl’) as a function of m. The magnitude of nj*’ is, in contrast w&h the case of n\“, fairly affected by the value of m as shown in Fig. 5; the n\*’ reaches the level of IO-* at around m = 0.5, but it never exceeds the order of magnitude of lo-* even when m = I. Figure 6 shows the dependences of (- ~‘3 on nj’) and ni*’ for K = 0.5. In contrast with the dependence of (-u,,) on n\“, the n\“- value increases with increasing (-w12). Although the n\‘) can reach a level more than 0.7 at around (-w,*) = 1IO kJ/mol, the value of nj*) is kept around a value as small as 4 x 10-3. Differently from the above cases, at this place the n$*’ as well as the nj’) increases with increasing (- w12).

0.6

_ 0.6 Y CT 04

0.6

c c

VIJ=-15kJhol K-=0.5

1

/

10 - V,,(

20 15 kJ/mol )

25

Fig. 4. Variations in n\‘) (or in n$) as a function of ( -I+$ with various values of K for I%-Ni-P system. The “S.L.R.” indicates a range where the calculated nj’- value exceeds unity.

Lo 20

1

60

I

16,

I

I

60 60 100 - Ml2 (kJ/mol 1

’

120

Joe

Fig. 6. Variation in nl’) (or in n\*)) as a function of (- w,& for Fe-Ni-P system. Values of the other kinds of parameters are in Table I.

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0.2

OF PHOSPHORUS

0

0.1

0.2

0.3

0.4

0.5

0.6

CPg Fig. 7. Variation in n\” (or in I*‘)as a function of Cf (or of the ratio of C; to C’:) for Fe-Ni-P system. The symbol A indicates the K-value for each calculation. The K,,,,, vs C; relationship here is obtained from equation (I 4).

ALLOYS

cl, < e3)

between n\‘) (or ni’)) and C; on condition that the value of Ci is changeable a littie within the range of 3.5 x lO-4-5 x 10m4. This means that the ratio of C;/Ci varies almost in proportion to C$ In Fig. 7, K is varied from 0.2 to 0.7, depending on different values of C;. These K, as shown in the figure, approximately line up on a curve parallel with the K_, vs C,: curve. This KmX vs C; relationship has been derived from equation (14) considering that the version of equation (14) is not only specific to the binary Fe-P system but also applicable to the ternary Fe-Ni-P. The nj”- value attains to a high level of 0.5 or more at a small ratio of C;/C:, but rapidly decreases down to less than 0.1 as the ratio of Ci/Ci is increased. The corresponding value of n\‘) shows the similar C;/Ci (or C;)- dependence to that of n\‘! but it never exceeds the level of 10m2even if the n\‘) reaches a level as high as 0.5. 4. DISCUSSION

Consideration of ui3 being negative, adopted in this study, seems to be evidenced by some experimental facts; e.g. by a measure of the activity of phosphorus in fused iron [13], from which a,, is worked out to be about - 23 kJ/mol. Furthermore, in conjunction with the evaluation of e,, ( =ep._&, the consideration of u,) < 0 seems to be more rational than the previous one that vi3 is to be positive [I]: the present value of e,i, -64 kJ/molt, has been estimated, as mentioned earlier, from the heat of sublimation of phosphorus

tThe value of e,, varies somewhat depending upon different data of the heat of sublimation of P (e.g. -43 kJ/mol from the data in Ref. [13]). but this does not cause significant difference in the CDP calculation results.

(16)

[lo, 141 leads us to trouble in conjunction with the Fe-P system such that we can no longer find the solution to equation (1 I). This is because the constant B, given in terms of equation (l3), becomes negative since the sign of the term (1 +a) turns positive if et3 < e33, as can be understood from equation (IO). Additionally, insiting the version in terms of equation (16), we must replace equations (14) and (15) by the following equations of (14’) and (IS), respectively, so far as the Fe-P system is concerned KL K,,,,(=

A/(1 +A))>0

(14’)

and - au 2 20 kJ/mol.

Figure 7 represents the relationship

143

while the e,, of - 169 kJ/mol in the previous study would not have any experimental support like this. The version of ui, < 0, however, brings some questions into our model for the CDP calculation. First, a widely accepted consideration that

ot 0.4 ; 0.6 ’

IN FERROUS

(15’)

Equations (14’) and (15’) can be derived from equation (13) with the positive sign of (I + a) in mind and from the similar procedure to that for obtaining equation (15). We have tacitly assumed the regime of equations of (14) and (15) to perform the present calculations of P-CDP for the Fe-P alloy, or else we could not carry out these calculations, in other words, could not solve equation (11) concerning this binary alloy system. If we insit the version of equation (16) under the condition of Y,, < 0, apparently we need to modify our model for the CDP calculation so that we may have the solution for the Fe-P system under those conditions. Such modifications will be. the subject of further studies as well as re-evaluation of the various approximations or assumptions for simplicity which were adopted in the previous study to derive equations (2) and (3) of the present work. With reference to the Fe-Ni-P, it is apparent that it is possible to have a solution to equation (2) even under either regime of equations (14) and (15) or that of equations (14’) and (15’). This is due to the existence of the positive term of Hi’) on the LHS of equation (2). This characteristic with regard to the Fe-Ni-P system does not provide any suggestion of which regime of equations (14) and (IS) or those of (14’) and (15’) is superior to each other. However, the following seems to be noteworthy: the regime of equations (14’) and (15’) recommends us to take K and (-vu) to be a higher value within the individual variable range, whereas vice uersa under the regime of equations (14) and (15). Hence, it follows that the calculated value of rzi’i becomes a lower one, say of the order of 0.1 under the former regime while it becomes a higher one, say of the order of 0.5 or more under the latter regime, as can be seen from Figs 2(b) and (4).

144

SHINODA:

CONCENTRATION PROFILES OF PHOSPHORUS IN FERROUS ALLOYS

The second question in connection with the version of vu c 0 is of how consistently we can coordinate the negative nature of the sign of u,, with the positive nature of the sign of the heat of intergranular segregation of phosphorus, AHp, in ferrous alloys. According to McLean’s classical theory [IS], the driving force of solute atoms to be enriched at grain boundaries is only the elastic interaction between the solute and solvent atoms. It does not seem unreasonable to think that a,), the Fe-P interaction parameter for the bulk phase of the alloy, can be so divided into the two parts of “elastic” and “chemical” interactions as a,, = u,, (elast) + uu (them). The similar division will be allowed with regard to of,, the Fe-P interaction parameter for the grain boundary phase, in a way as uf = of, (elast) + ufr (them). Roughly it is considered that vf, (elast)< 0.

(17)

Thus, it is shown that the term of (vu-ut) acts as the driving force for the solute intergranular segregation in McLean’s sense. However, this never means that the term of (uu - ut) covers the whole part of AH,,, because the AH, is usually estimated to be 40-50 kJ/mol [16] while the value of au (elast) only amounts to 12kJ/mol or so [fir]. A part of the discrepancy between (~1,~- 07,) and AHp may be filled up by the term of -(V/d) (a y g/K@ which has been proposed by the present author [18], where V is the specific volume per mole of the alloy interested, d the grain boundary thickness and y* is the grain boundary free energy per unit area of the grain boundary/bulk interface. Further arguments about this discrepancy is beyond the scope of the present study. Alternatively, equation (17) gives an interesting relationship such that 0 > au > ut if we accept the condition of au CO. These relationships signify that the chemical affinity between phosphorus and iron may be stronger in the grain boundary phase than in the bulk phase. Finally, we will have a word with the inter~anular CDP of Ni itself. The existence of Ni in the ternary alloy little changes the CDP of phosphorus in shape or type, compared with that for the simple Fe-P system. Namely, the “sharp interface” type of CDP of phosphorus also prevails in the case of Fe-Ni-P irrespective of the situation that the “diffused interface” type of CDP of nickel is formed across grain boundaries in this alloy [SJ. This independency of the CDP of P from that of Ni inversely suggests that the latter may be prescribed by quite a different way from what determines the former, that is, the way studied hitherto by us. Question of what prescribes the intergranular CDP of nickel remains to be answered by further studies.

5. SUMMARY The “con~ntration depth profile (CDP)” of phosphorus across grain boundaries in Fe-P and Fe-Ni-P alloys, calculated on condition that ub, the interaction parameter for Fe-P system, is negative in sign, is essentially the same in type as that obtained in the previous study where u,, was assumed to be positive. Namely, when au < 0, the normalized P- concentration (with a variable range of O-l), nil), at the first bulk layer adjacent to the grain boundary amounts to a level as high as 0.8 or more if the parameters needed for the CDP calculation are suitably chosen. However, the successive n,- values at the bulk layers of 2nd. 3rd, etc. never exceed the order of magnitude of IfJ;’ (In-the case of Fe-P, it is shown that n, ER, = , . . s 0) no matter how the magnitude of the corresponding n{‘)- value is close to unity. Thus, it can be said that the “sharp interface” type of the phosphorus CDP is also formed even when u,, < 0. If another restriction that e,, < e,, is imposed in addition to the condition of au < 0, the model used here fails in the P-CDP calculation so far as the Fe-P binary case is concerned. However, this is n& the case of the Fe-Ni-P alloy due to an additive term in the equations of interest, which is caused by the existence of Ni. At this place et3 and e,, are the binding energies for the nearest neighbor pair of Fe-P and that of Fe-Fe, respectively. Acknowledgernears-The author is grateful to Professor T. Suzuki, Research Laboratory of P.M. & E., Tokyo Institute of Technology, for his great encouragement in the present study.

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IS. T. Shinoda and T. Nakamura, Trans. Iron Steel Inst. Japan 19, 367 (1979).