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Adsorption-induced interface decohesion
ADSORPTION-INDUCED
INTERFACE
DECOHESION
M. P. SEAH Division of Chemical Standards, National Physical Laboratory, Teddington, Middlesex. England (Recefued
3 July 1979; in revisedform 26 Noucmbcr 1979)
Abaract-A new theory is presented to describe the segregation-induad changes in grain boundary cohesion. This approach enabks the changes to be calculated readily loom tabulated thermodynamic data of sublimitation enthalpiis and atom sizes. The current thermodynamic theory of the change in cohesion is shown to predict the same results. A general scheme is presented which shows the degree of embrittkmnt or of reduction in embrittkment in any &en matrix for the main boundary semsation of some 70 solutes. In agreement with experiment thd &ulations show thai in a ferrous &&r,-Bii S, Sb, Se, Sn and Te will be highly embrittling, followed in order of reducing &ct, by P, As, Ge, Si and Cu. In increasing order of their remedial e&t on embrittlemcnt in iron are N, B and C.
R&urn&--Nous ptcsentons une nounlk thtorie dCgivant ks variations de cohtsion intergranulaire induites par la &gr&gation. Ccttc mtthode permet de calcukr ccs variations dircctcment B partir des tables thermodynamiques de l‘enthalpie de sublimation et de la taille des atomes. Nous montrons que la thtorie thexmodynamiquc usuelle de la cohtsion conduit aux mSmes rcSultats. Nous p&tons la vakur de l’augmentation ou de la diminution de fragilitk par &&ation intergranulaire, dans une matrice donn& pour quelques 70 so1utCs.En accord~av&l’exp&iencc, ks cal&ls montrent que dans une matria fmeure Bi S Sb. Se. Sn et Te vont &re trQ fraailisants. ils sent suivis dans l’ordre d’une e&t dtcroisant, par P, As,.& Si et Cu. Inversement, N, B et-C, clas&s dans un ordre d’efficacitt croissante, diminuent la fragilit&du fer. ZusrmmenfaameEine nave Tbcork zur Bescbreibung der durch Segregation hervorgerufenen Anderung in &r KorngmnzkohPsion wird vorgekgt. Diem Theorie ermi)glicht, die #nderungen direkt aus tabellierten thermodynunicachen Daten der Sublimationsenthalpie und der Atomdurchmesser auszurechnen. Die gegenn&tip thermodynamischs Theo& der Kohiisioniinderung Nhrt -wie gezeigt wirdzum s&en Brgebnir Es wird ein al&m&es Schema entwkkelt, das den Verspr6dungsgrad oder die Verringerung der ~erspr&iung in &a gegebenen Matrix Nt die Korngrw tion von 70 Dotierstoffen angibt. In ~n~rnrn~g mit dem Experiment zeigen die Rechungen, Da# Bi, S, Sb, Se, Sn und Te in eina Eisenmatrix ~~~ wirken. gefolgt von (Reinhenfolge geringer werdenden Einflusses) P As, Ge, Si und Cu. Der Verspr&htng entgegenwirkend sind (mit ansteigcnder Wirksamkeit) N. B und C iNTRODUtXION During reoent yearq with the intensive application of Auger ekctron spectroscopy (AES), a considerabk volume of information has become available to characterise the segregation of solute atoms to grain boundaries [ 11. The precise location of the segregated solute atoms at the atom sites of the grain boundary has been demonstrated both by AES and by atomprobe field ion microscopy [l]. The types of species that segregate also have been well characterised. Generally solutes of a low solubility segregate strongly and vice versa [l, 23. In ternary or higher order systems certain solutes may interact and enhance the segregation of a given species. Solutes that enhance segregation tend to be those which have an intermediate free energy of interaction with the given species [3]. Those of a low free energy of interaction produce little effect whereas those of a high free energy cause tbc given solute to precipitate and again cause low segregation. This general model for segregation allows the detailed dependence of the solute segregation as a function of time and temperature to be calculated for low alloy steels. Such calculations for SAE 3 140 show that the segregation of phos955
phorus exactly parallels the development of the grain boundary brittleness, as measured by Charpy ductikb&k transition temperatures [4]. Many measurements show that the low temperature (< 100°C) intergranular embrittkment of steels is directly related to the degree of segregation of impurities&6]. In iron based systems, Bi {7j, S[2], Sb [5,8], Se [7,9], Sn [2,6] and Te [7,93 are ail very potent embrittlers, As [7,8], Ge [7], P [6, IO] and Si [?‘I are all moderate embrittlers and C[li] and B [ 123 improve the grain boundary cohesion. Thus the direct presence of atoms of the above species, at grain boundaries in iron or steel, causes a change in the cohesion’there in direct relation to the quantity and type of the atoms present. A number of theories have been proposed to explain the effect of segregation on grain boundary cohesion. The present paper puts forward a new approach which directly and simply provides a numerical estimate of the effects of different segregated chemicai species on the strength of grain boundariis. This approach, and most of the previous analyses, are concerned with predicting the effect of the segregant on the ideal strength of the grain boundary. During low temperature fracture it is visual&d
956
SEAH
ADSORPTION-INDUCED
INTERFACE
DECOHESION
that. under the applied stress, dislocations are emitted in the region of the crack tip and contribute to the plastic work term in the fracture process. As the applied stress rises, the plastic work increases, the region work-hardens and the local applied stress on the boundary increases until eventually failure occurs or the crack blunts [13]. If the ideal strength of the grain boundary is reduced, the failure mechanism, in
an otherwise ductile material, eventually changes from crack blunting to grain boundary separation. The present paper seeks to evaluate the relative shift in this ductile to brittle transition for the segregation of different chemical spe~ics to the grain boundaries. If this ideal grain boundary strength is reduced, failure occurs at a lower local stress and hence the overall plastic work involved also is reduced. Thus, although the plastic work term may be many orders of magnitude more than the ideal work of fracture of the grain boundary, the idea1term is important in that it governs the plastic term and the failure mode [14]. In this work, for simplicity,it is assumed that, of all the properties of the whole fracture process, the grain boundary segregation directly alters the properties only at the grain boundary. Thus, attention is focussed on the relative effects of different chemical species segregated at the grain boundary. In the next section the new theory is proposed which readily permits the calculation of the effects of different segregants on the change in cohesion of solid-solid interfaces. In section 3 the previous thermodynamic theories of fracture are examined in relation to this new theory. 2 l-HE NEW THXORY-PAIR BONDING In the case of low temperature fracture it is assumed that the grain boundary separates without redistribution of the solute atoms. The following pair-bonding or quasichemical approach gives an easy numerical evaluation of the behaviwr of the embrittlement, or otherwise, for the grain boundary segregation of some seventy solutes in the same number of solvent matrices. The fracture process is shown in Fig I. The actual energy required to break the bonds across the grain boundary may be simply determined by counting up the number of dangling bonds per unit area and summing their energies. These energies may be calculated from the sublimation enthalpies. A fair numeri-
cat estimate of this approach may be made using nearest neighbour bond terms For a binary system of solute B in solvent A the nearest neighbour bond
energies (negative values), e,,,,, r,. and cBB are assigned to AA, AB and BB neighbours respectively. The~fractureenergy per unit area of the pure solvent A, FE(O). is given by (1)
tbJ
Fig. 1. Schematic representation of the atoms (a), either aide of the unbroken grain boundary shown by the dotted iine, and (b) after facture of the grain boundary with the ruptured bonds bsh dangIing. The matrix A atoms arc denoted by open cir&a anznt solute B atoms by fibed
where 2, is the co-ordination of atoms in the layer on one side of the grain boundary to those in the adjacent layer on the other side. On average Z, can be shown to be related to the customary value for a perfect lattice Z, by the appropriate factor (I - y&y&. For a perfect lattice Z, would be 2 and 3 for b.c.e. and kc. lattices respectively where the total co-ordination, 2 , is 8 and I2 respectively. The ratio yJv, is typically i/3 [lSJ. In equation (1) ai is the area of an A atom in the surface. The fracture energy with a molar fractional monolayer, X, of B at the grain boundary is FE(Xb) * - 2 z* ((I - fXl)fC4,4 f X,(1 - *X‘)c,, + +Xbttbs)* If the AB system is a regular solution, the parameter o may be defined and WXb) - FE(O) = -
3 Wbb
- %4)
+ OX&(1- fX*)l.
(2)
A similar result is obtained if the segrcgant atoms in Fig. 1 are assumed to be coplanar. For an ideal solution, UII: 0. If also the two atoms, A and B have differ-
SEAH:
ADSORPTION-INDUCED
ing sizes, the segregation induced reduction in fracture energy may be written approximately FE(O) -
FE(X*) =
~~~_~)
where Xb is now the fractional area of the grain boundary covered by B atoms. The values of eAA and eDa may be determined from the molar sublimation enthalpies, HIvb [ 163 NT” p: -$ZNf,&4,
Hg” = -fZNeaa
where N is Avogadro’s number. Thus FE(O) - FE(Xb) = &(!$
- %)x1.
(3)
$X,[H:‘b. - J.&+-J
(4)
where H,$ represents the sublimation enthalpy of pure A or B per unit area derived from Hsub* = Hsub/Na2. Alternatively FE(o) -
FE(xb)
=
5/‘6 x,&f”
-
Yf”,
(3
where y:* and yf* are the surface energies of pure A and B, respectively Cl’& Of the above equations, equation (4) most readily leads to a unified numerical presentation. For all systems, the fracture energy reduction per monolayer of solute adsorbed on the grain boundary is simply (ZJZ) times the difference in ~blimatio~ enthalpy per unit area for the matrix and solute materials nspectively, and the degree of embrittlement is linearly dependetit on the kvel of segregation, as observed experimentally [S, 61. Before proceeding further, the previous thermodynamic treatment will be analysed to compare its prediction with equation (4).
Early theories to explain the effect of solute, segregation on low temperature grain boundary brittle fracture are summa&d by the author in a previous paper [lg]. This previous paper also gives references to many of the appropriate experimental observations which will be only briefly outlined in this work. In the previous paper the ideal work of fracture per unit area of the grain boundary, y. was calculated from the relation [ 193. Yb
boundary, respectively. If yI and yb are the equilibrium values for the fractured material, equation (6) is appropriate for slow crack growth at high temperatures where the solute diffusion is sufficiently rapid to establish equilibrium surface segregation at the crack tip. For fast, low temperature fracture it was assumed that y. would not be the equilibrium value and an attempt was made to estimate the value to use. The approach is briefly outlined below. For unit area of the grain boundary [20] i where V and S are the specific volumes and entropy of the interface zone, P is the pressure across the interface, T the temperature and r’ the quantity of species i with chemical potential pi per unit area of the interface. At constant temperature and pressure in a binary system, where i = A for the solvent and B for the solute, and applying the Gibbs-Duhem equation, the above relation reduces to dy, = [{X’/l - X”}ff - f-ad#
(6)
where 1: and yb are the energies per unit area of the freshly produced fracture surfaces and the prior grain
(7)
where X8 is the solute molar fractional content. If the solute obeys Henry’s law, since d$ = RT d In a where a is the solute activity, then d#’ = RT d In X’. For X3 6 1 and, for a dilute system in which ri a XB,it was possible to show by substitution in equation (7) and integrating, that ~6’= yf* - R7Tf
(8)
and similarly for the surface y: = y:” - RTf,B
(9)
where yf” and r;‘” are the grain boundary and surface energies of the pure matrix A, respectively. In the following all the bulk concentrations, surface or grain boundary excesses and chemical potentials referred to are those for B atoms and so that the atom denoting superscript will now be deleted from X, r and p respectively. Since, on fracture r, = +rb, it was found from equations (6) (8) and (9), that y=2$*-y;*
3. THE PREVIOUS THEORYTHERMOCHEMICAL
Y = 2Y,-
957
dYts = pdP--SdT-x:;d/,&,
Although equation (3) contains many approximations, the right hand side is in a form which permits a simple and general evaluation for many segregation systems. Two interesting alternatives of equation (3) may be written FE(O) - FE(X,) =
INTERFACE DECOHESION
(10) for a given solvent irrespective of the segregation level. Hirth and Rice in a series of publications, summarised recently [21], agree with this basic approach but present the argument in general terms and point out the omission of a term from equation (10). This term may be seen by considering the p---r diagram, as shown in Fig. 2. Figure 2, curve (a) shows the relation for the equilibrium excess of B atoms at the grain boundary, fb, as a function of the SOhte chemical potential, & Similarly curve (b) shows the surface excess for the two fracture surfaces, 2f,, as a function of p. For slow fracture at high temperature, where equilibrium of the solute is maintained throughout, the
958
SEAH:
ADSORPTION-INDUCED
I
r
Ibl
INTERFACE
DECOHESlON
and 2r,M = ru2Xx
2r:
exp
-
+$
(
>
where AGb and AG, are the free energies for the solute segregation to grain boundaries and surfaces, respectively and T’, is the value of r, at one monolayer coverage. Substitution of the above three equations in equation (12) gives y = 2y*o - yAo + T,(AG, - AG1 - RTln 2). (13)
0
Y”
Y
Fig. 2. The p dependence of r for solute atoms at sites on (a) the grain boundary and (b) the fracture surfaces.
system starts at K and, during fracture, moves to L. Here the total segregation level increases as a result of fracture. Application of equation (6) in the dilute approximation above gives
Equation (13) gives an indication of the magnitude of the difference between equations (11) and (12). Whether equation (11) or equation (12) is really correct would appear to depend on the state of the surface atoms immediately after fracture. If the surface atoms in their non-equilibrium state really have the low chemical potential that would be necessary for them to be in equilibrium with the bulk potential #‘, then equation (12) is correct and the work of fracture is reduced by segregation. If there is no reduction in chemical potential, equation (11) is valid and for intermediate cases the result must lie between equations (11) and (12).
Y = 2Y: - Yb” = 2r;” - y:” - RT(ZT,L - l-f) (constant I( process) Thus, a sizeable decrease in y occurs as a result of segregation in slow high temperature fracture. For fast fracture at low temperatures the system starts at K and moves to M with the total segregant level unchanged. Application of equation (6) in the dilute approximation gives v-2Y?-ti = 2~;’
- y$’ -
(11)
However, in moving from K to M Hirth and Rice [21] point out that the segregated solute atoms have been able to rekase a portion of energy
which, here, is simply (# - p)r,. fracture is reduced by this amount
Thus the work of
y = 2yf” - y,“” - QP - r”>r,.
The final results of the thermo&mical theory,, equations (12) and (13), may or may not be the best description of the effect of segregants on gram boundary cohesion. However, data for AG, and AG,, generally are not available and so the result is of limited use to the metallurgist. Equation (13) may be rewritten FE(O) - FE(X,)
RT(ZT: - rf)
= ato - yf” (constant p process)
4. COMPARISON OF TIiB PAIR BONDING AND THERMOCHEMICAL APPROACHES
(12)
measure of the new term may be obtained as follows. In the Henrian approximation for dilute systems
A
where p. is the chemical potential of a reference state. In the dilute approximation also [223
x’ AH, - AH, + T(AS, - AS,)}. = Nx'
(14)
The entropy termq AS, and AS,, to a large extent cancel each other [lSJ. Nearest neighbour bond analysis, using the approach described in section 2 and elsewhere [23], shows that in the dilute approximation for a regular solution AH,=
- H(fZ&,,
(15)
- e& + o&J.
This relation has been shown to be valid in predicting experimental values of surface segregation over a range of 100 kJ mole-’ with a standard deviation of 15 kJ mole-’ [23]. Thus, although the pair bonding approach may be inadequate if there is considerable relaxation of the surface bonds, as, for example, in silicon, experiments for many metallic systems show considerable agreement with the pair bonding approach. In a similar manner to equation (IS), AH* may be derived AH, = -H(f(Z,
- Z,)(E,, - (AA) +o(z,-
Z,)).
(16)
SEAH:
ADSORPTION-INDUCED
Thus equation (14) may be rewritten FE(X,)
- FE(O) “-7
2
I~Xb(Qll) - d
+ @Xb)
(17)
which is equivalent to equation (2) developed in the new theory. Thus, within the regular solution ap proximation the previous thermodynamic description given by equation (12) affords the same basic result as the bond breaking descriptions of equations (2) and (4). Since it is possible to perform general numerical calculations with equation (4) which are not directly possible with equation (l2), equation (4) is used in the next section to give the general predictions of the theory. I PREDI~ONS
OF THE THEORY
From equation (4) the loss in fracture energy of the grain boundary atoms is proportional to the difference in Hgub* of the matrix and segregant atoms calculated t?om the tabulated data for pb [16j and the atom sizes, u. a is derived from pNa3 = A
(18)
where p is the bulk density of the pure material and A
iNTERFACE DECOHESION
is its atomic weight. Values of Web* for 70 elements are plotted in Fig. 3 as a function of the atom size, a. Taking any given element as the matrix, elements lower in Fig 3 will, if segregated, cause embrittlement of the matrix grain boundaries whereas elements higher will be remedial to such embrittlement. The magnitude of these effects is, of course, in proportion to the distance vertically from the matrix element point. If iron is taken for the matrix the line (i) divides the embrittling and remedial elements. Because of the approximations used in deriving equation (4), points near the line are not so easily classified. Incorporation of the regular sofution parameter, W, ignored above, causes a term of approximately 1J/m’ to be added or subtracted from the Hsubr terms in equation (4) For ekments which form an intermetallic with iron (o negative) the grain boundaries become more cohesive than indicated in Fig 3 and vice versa. A second small error arises from the use of equation (18) For elements segregating in iron the atom size should best be determined from that of the compound which first appears as the solute content is raised [2]. For metals and metalloids equation (18) is sufficiently accurate but for P, N and 0 the value from the compound has been adopted. In alit with experiment, Fig. 3 shows that in iron Bi, S, Sb, Se, Sn and Te are all highly embrittling followed in order of reducing
CC t
Increasing grain boundary cohesion
,B oe .W
__*_____________(I)
\ Increaring grain bounday lmbrlttlement
I 0.2
Fig. 3. Ideal compilation for the iron if iron
I
w
I
Nam ,
0.3
959
lBO
I 0.b
oKIlabJcs 0.5
onm
solution general remedial/embrittkment plot for matrix and segrcgant elements based on the of Hab* values aa a function of the atom size calculated from equation (18). As an example matrix, line (i) represents the locus dividing those ekments which reduce the fracture energy from those which inacase it. Line (ii) is the similar locus for the critical fracture stress
960
SEAH:
ADSORPTION-INDUCED IOC
--
,_.._.
--‘-r’__‘_.
INTERFACE DECOHESION -----I
\ Incrsoimgg’al” boundarycohcsmn
us l sn lSb
I
a2
I
I
0.3
I
0.~
I
0.4
1
I
0.5
Fig. 4. Remedial/cmbrittkment plot for segregants in iron in the reguJu solution approximation. The position of ekments not shown may be estimated by recognking that tbc rektive poxitions of elements of similar chemistry remain approximately unaltered between Figs. 3 and 4. effect, by P, As, Ge, Si and Cu. In increasing order of their remedial e&ct on cmbrittkment are N, B and C. Of course, Fig. 3 does not take into account any bulk precipitation or changes in microstructure caused by these elements. The effect of the regular solution parameter, 0, can be included to provide a plot such as Fig, 3 for a particular matrix. For the matrix of iron this is shown in Fig. 4 for solutes for which o can be assessed Values of w are determined, as discussed by Swalin [24]. from tabulations [ 163 of the enthalpy of rnixing, the activity coefficients of the two species and details of the phase diagram. Thus, in iron, elements such as -Cu and Ti, which have similar marginal effects in the ideal solution approximation of Fig. 3, are shown to have clearly separated effects in the regular solution approximation of Fig. 4. Elements further removed from iron in Fig. 3 remain, however, relatively unaffected. The remedial ekments deserve a little further discussion. Equation (13) predicts a remedial etkct only if the free energy for grain boundary segregation is larger than that for surface segregation. There are two types of remedial atom lo consider; those occupying substitutional sites in the lattice and those occupying interstitial sites. Comparative surface and grain boundary segregation data for the substitutional ele-
ments are not availabk but analysis of surface segregation data [23] shows that as Hmw decreases, or as the atom size increases, the free energy of surface segregation increases in magnitude. Since the propensity to intergranular failure also increases in this way, substitutional ekments that are highly surface active tend to be the ones that are also highly embrittling. Likewise the high bond energy substitutional elements in the remedial sector are those which, to a large extent in binary systems, prefer the higher coordination sites of the lattice to those of the free surface and the grain boundary and, hence, do not segregate. In ternary systems, however, coupling of an embrittling atom to one of these remedial atoms may cause segregation of the latter so that its remedial property may be utilised In this way the reduction in the grain boundary embrittkment of a temper brittle steel for a given level of phosphorus (embrittling ekment) at the grain boundary by the synergistic segregation of molybdenum [25] (remedial element) may be explained. The behaviour of the interstitial remedial elements C, B and N dialers from the above behaviour in that the release of strain energy when the solute atom leaves the lattice dominates its intrinsic chemical wish to dcsegrcgate strongly from the surface and less strongly from the grain boundary. In this way strong
SEAH:
ADSORPTION-INDUCED
grain boundary segregation ooblrs and, for C B and N, since fewer of the highly energetic bonds are disrupted at the grain funds than at the free surface, the free energy of segregation to the grain boundary is larger than that to free surface. Data for the segregation of carbon in a-iron [26,27] supports this view. Thus, C, B and N interstitial atoms probably segregate to release strain energy, segregate more strongly to grain boundaries than free surf&es and at the grain boundaries improve their cohesion. Thus equation (131 in avant with quation (41 predicts that these elements will be remediil to the grain boundary embrittlement of iron as demonstrated experimentally [I 1, 12,281. It may be argued that the critical fracture stress of the atoms either side of the grain boundary should be considered instead of the work of fracture. As the applied stress on the fracturing sample increases, dislocations pile up in the crack tip region and contribute to the plastic work term mentioned earlier. At the point where the crack will just advance the atoms at the crack tip can either part, giving brittle fracture, or emit dislocations and so blunt the crack tip. The criterion for emitting dislocations depends on the shear stress across the slip plane where it intersects the crack front on the grain boundary [29,30], and this, in turn, is related to the stress acting across the grain boundary atoms at the crack tip. Thus the ductile to brittk transition may be goverened by the witicat fracture stress for the grain boundary and not the work of fracture. Of course, in perfectly brittle materials, where the choice is between cleavage and intergranular failure, other aiteria may be appropriate. Even allowing for this alternative criterion of critical fracture stress, instead of the fracture energy used previously, it will be shown below that Figs. 3 and 4 retain their full usefulness in assessing the embrittling and remedial properties of different segregants. The segregation of large atoms to the grain boundary displaces the stress-distance curves for the atoms across the boundary outwards. For the generally used potentials, e.g. Lennard Jones, Morse, Mie etc, it is easy to show that the fracture energy is proportional to the product of the critical fracture stress and the interplanar distance [18]. Thus, the ratio of the critical fracture stress of the segregated boundary, a(X,), to that of the clean boundary, u(O), is: 1
0 (xb) - 40)
=
1
+
i
+
+&(&,,a~
FE -
1)
6)
FE(o)
Hence a(Xb) PE 6Y
~xb(~,“b*/~~~~ 1
f
-
)xb(&,,ff,,
-
1)
t) *
the neutral line dividing the remedial and embrittling elements in Figs. 3 and 4 is now given by
Thus,
&ub’ =
aB -
(2A
HP”.
(19)
INTERFACE DECOHESION
961
This result is shown for elements in iron as line (ii) in Fig. 3 and its counterpart in Fig. 4. For iron the conclusions concerning the remedial and embrittling elements remain unchanged. Indeed, because of the way elements occur in Figs 3 and 4 it would be difficult to distinguish which of the two criteria of fracture energy and critical fracture stress is correct from a study of the remedial and embrittling elements in solvent matrices. Figure 3 may be used, as above, to define the ~brittling and remedial segregants in all matrices. Elements below the appropriate lines (i) and (ii) passing through the matrix ekment point are embrittling and vice versa. For instance bismuth will embrittle copper and potassium and phosphorus will embrittle tungsten, as have been shown experimentally [29,3 I, 321. CONCLUSIONS A simple theory, based on pair bonding, is proposed to describe the changes in grain boundary cohesion resulting from solute segregation. The theory is shown to be equivaknt to the detailed general thermodynamic treatment within the regular solution approximation. Numerical evaluation of the theory, through equation (4)1allows a general scheme of embrit~~g and remedial segregants to be defined for any solvent matrix. This scheme, shown in Fig. 3, predicts, in agreement with experiment, that Bi, S, Sb, Se, Sn and Te will be highly embrittling in iron followed, in order of reducing effect, by P, As, Ge, Si and Cu. In increasing order of their remedial effect are N, BandC. Ac~wf~ge~~~-T~e author would like to thank J. P. Hirth and J. R. Rice for material prior to publication and J. P. Hirth and E. D. Hondros for general discussions.
REFERENCES 1.
E. D. Hondros and M. P. Seah, Int. Metall. Rev. 22,
262 (1977). 2. M. P. Seah and E D. Hondros, PFOC.R. Sot. London
(A) 335, 191 (1973). 3. M. Guttmann and D. McLean, Proc. 1977 Materials Science Seminar on Intetjiucial Segregation, Chicago ill., 22-23 October 1977. 4. M. P. Seah, Acta Metall. 25, 345 (1977). 5. H. Ohtani, H. C. Feng, C. J. McMahon and R. A. Mulford, Mctall. Trans. IA, 87 (1976). 6. A. K. Cianel, H. C. Feng, A. H. Ucisik and C. J. McMahon, Metall. Trans. %A. 1059 (1977). 7. 8. J. Schulz and C. J. McMaho~ ASTM STP 499,104 (1972). 8. C. --..J. McMahon, -ASTM STP 407, I27 (3968). 9. C. Pichard, M. Guttmann, J. Rieu and C. Goux, L.-es Joints Intcrgranulaires dans les Mttaux, J. Physique 36, f 51 Colloque C4 Suppltment 10 (1975). 10. E. D. Hondros, M. P. Seah and C. Lea, Met. Mater January p. 26. (1976). 11. C. Pichard, J. Rieu and C. Goux, Mem. Sci. Rer. Merall. 70, 13 (1973).
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SEAH:
ADSORPTION-INDUCED
INTERFACE DECOHESION
12. H. Tana and A. Yoshikawa. Proc ICSTlS SIJDLI~. 22. E. D. Hondros and M. P. &ah, Melall. Trans. A&l363 ~. Trans.-&I Japan II, 1256 (1971). (1977). 23. M. P. &ah. J, Catal. 57,450 (1979). 13. R. Thompson, I. Mater. Sci. 13, 128 (1978). 24. R. A. SwaIin? T~er~~~ies MSolids, Wiley (I 962). 14. D. McLean, Grain Boundaries in Metals, Oxford Uni25. Ph. Dumouhn, M. Foucault, M. Palmier. M. Wayman, versity Press (1957). M. Biscondi and M. Guttmann. Metal. Sci. 14, I I 5. M. P. Seah and C. Lea, Phil. Mag. 31, 627 (1975). (19801 16. R. Hultgren, P. A. Desai, D. T. Hawkins, M. Glciser 26. H. J. Grabke, W. Paulitachkc, G. Tauber and H. Viefand IL K. Kelly, Selected values 01 the Thermodynamic haus, Surface Sci. 63,377 (1977). Properties of Elements and Selected Values oj de Ther27. J. M. Paoazian and D. N. Bcsbers, ~etaif. Trans. 2, uric Properties of Binary Allays, ASM Metals 497 (197lj Park (19733. 28. G. Taubcr and H. J. Grabke, Bcr. Bunsengca. Phys. 17. P. Wynblatt and R. C. Ku, Proceedings 1977 Materials Chem. 82.298 I19781 Science Seminar on lnre@cial Segregarlon, Chicago Ill, 29 E. D. H&droi and. D. McLean, Phil. Msg. 29, 771 22-23 October 1977. (1974). IS. M. P. Seah, Proc. R. Sot. London (A) 349,535 (1976) 30. A. Kelly, W. R. Tyson and A. H. Cottrelf, Phil. Ma@. 19. M. Polanyi, Z. Phys. 7, 323 (19233. 15,567 (1967). 20. R. H. Fowler and E. A. Guggenheim. S&&icaI Ther31. R. P. Simon, 0. J. Dooley and T. W. Haas,Meta~l. rn~ya~ic~ Cambridge University Press (1952). Tkwis. 5,585 (3974). 21. J. P. Hirth and J. R. Rice, Meralf. Trans. to be pub32. D. F. Stein, J. Vat. Sci. Technol. 12,268 (1975). lished.
INTERFACE
DECOHESION
M. P. SEAH Division of Chemical Standards, National Physical Laboratory, Teddington, Middlesex. England (Recefued
3 July 1979; in revisedform 26 Noucmbcr 1979)
Abaract-A new theory is presented to describe the segregation-induad changes in grain boundary cohesion. This approach enabks the changes to be calculated readily loom tabulated thermodynamic data of sublimitation enthalpiis and atom sizes. The current thermodynamic theory of the change in cohesion is shown to predict the same results. A general scheme is presented which shows the degree of embrittkmnt or of reduction in embrittkment in any &en matrix for the main boundary semsation of some 70 solutes. In agreement with experiment thd &ulations show thai in a ferrous &&r,-Bii S, Sb, Se, Sn and Te will be highly embrittling, followed in order of reducing &ct, by P, As, Ge, Si and Cu. In increasing order of their remedial e&t on embrittlemcnt in iron are N, B and C.
R&urn&--Nous ptcsentons une nounlk thtorie dCgivant ks variations de cohtsion intergranulaire induites par la &gr&gation. Ccttc mtthode permet de calcukr ccs variations dircctcment B partir des tables thermodynamiques de l‘enthalpie de sublimation et de la taille des atomes. Nous montrons que la thtorie thexmodynamiquc usuelle de la cohtsion conduit aux mSmes rcSultats. Nous p&tons la vakur de l’augmentation ou de la diminution de fragilitk par &&ation intergranulaire, dans une matrice donn& pour quelques 70 so1utCs.En accord~av&l’exp&iencc, ks cal&ls montrent que dans une matria fmeure Bi S Sb. Se. Sn et Te vont &re trQ fraailisants. ils sent suivis dans l’ordre d’une e&t dtcroisant, par P, As,.& Si et Cu. Inversement, N, B et-C, clas&s dans un ordre d’efficacitt croissante, diminuent la fragilit&du fer. ZusrmmenfaameEine nave Tbcork zur Bescbreibung der durch Segregation hervorgerufenen Anderung in &r KorngmnzkohPsion wird vorgekgt. Diem Theorie ermi)glicht, die #nderungen direkt aus tabellierten thermodynunicachen Daten der Sublimationsenthalpie und der Atomdurchmesser auszurechnen. Die gegenn&tip thermodynamischs Theo& der Kohiisioniinderung Nhrt -wie gezeigt wirdzum s&en Brgebnir Es wird ein al&m&es Schema entwkkelt, das den Verspr6dungsgrad oder die Verringerung der ~erspr&iung in &a gegebenen Matrix Nt die Korngrw tion von 70 Dotierstoffen angibt. In ~n~rnrn~g mit dem Experiment zeigen die Rechungen, Da# Bi, S, Sb, Se, Sn und Te in eina Eisenmatrix ~~~ wirken. gefolgt von (Reinhenfolge geringer werdenden Einflusses) P As, Ge, Si und Cu. Der Verspr&htng entgegenwirkend sind (mit ansteigcnder Wirksamkeit) N. B und C iNTRODUtXION During reoent yearq with the intensive application of Auger ekctron spectroscopy (AES), a considerabk volume of information has become available to characterise the segregation of solute atoms to grain boundaries [ 11. The precise location of the segregated solute atoms at the atom sites of the grain boundary has been demonstrated both by AES and by atomprobe field ion microscopy [l]. The types of species that segregate also have been well characterised. Generally solutes of a low solubility segregate strongly and vice versa [l, 23. In ternary or higher order systems certain solutes may interact and enhance the segregation of a given species. Solutes that enhance segregation tend to be those which have an intermediate free energy of interaction with the given species [3]. Those of a low free energy of interaction produce little effect whereas those of a high free energy cause tbc given solute to precipitate and again cause low segregation. This general model for segregation allows the detailed dependence of the solute segregation as a function of time and temperature to be calculated for low alloy steels. Such calculations for SAE 3 140 show that the segregation of phos955
phorus exactly parallels the development of the grain boundary brittleness, as measured by Charpy ductikb&k transition temperatures [4]. Many measurements show that the low temperature (< 100°C) intergranular embrittkment of steels is directly related to the degree of segregation of impurities&6]. In iron based systems, Bi {7j, S[2], Sb [5,8], Se [7,9], Sn [2,6] and Te [7,93 are ail very potent embrittlers, As [7,8], Ge [7], P [6, IO] and Si [?‘I are all moderate embrittlers and C[li] and B [ 123 improve the grain boundary cohesion. Thus the direct presence of atoms of the above species, at grain boundaries in iron or steel, causes a change in the cohesion’there in direct relation to the quantity and type of the atoms present. A number of theories have been proposed to explain the effect of segregation on grain boundary cohesion. The present paper puts forward a new approach which directly and simply provides a numerical estimate of the effects of different segregated chemicai species on the strength of grain boundariis. This approach, and most of the previous analyses, are concerned with predicting the effect of the segregant on the ideal strength of the grain boundary. During low temperature fracture it is visual&d
956
SEAH
ADSORPTION-INDUCED
INTERFACE
DECOHESION
that. under the applied stress, dislocations are emitted in the region of the crack tip and contribute to the plastic work term in the fracture process. As the applied stress rises, the plastic work increases, the region work-hardens and the local applied stress on the boundary increases until eventually failure occurs or the crack blunts [13]. If the ideal strength of the grain boundary is reduced, the failure mechanism, in
an otherwise ductile material, eventually changes from crack blunting to grain boundary separation. The present paper seeks to evaluate the relative shift in this ductile to brittle transition for the segregation of different chemical spe~ics to the grain boundaries. If this ideal grain boundary strength is reduced, failure occurs at a lower local stress and hence the overall plastic work involved also is reduced. Thus, although the plastic work term may be many orders of magnitude more than the ideal work of fracture of the grain boundary, the idea1term is important in that it governs the plastic term and the failure mode [14]. In this work, for simplicity,it is assumed that, of all the properties of the whole fracture process, the grain boundary segregation directly alters the properties only at the grain boundary. Thus, attention is focussed on the relative effects of different chemical species segregated at the grain boundary. In the next section the new theory is proposed which readily permits the calculation of the effects of different segregants on the change in cohesion of solid-solid interfaces. In section 3 the previous thermodynamic theories of fracture are examined in relation to this new theory. 2 l-HE NEW THXORY-PAIR BONDING In the case of low temperature fracture it is assumed that the grain boundary separates without redistribution of the solute atoms. The following pair-bonding or quasichemical approach gives an easy numerical evaluation of the behaviwr of the embrittlement, or otherwise, for the grain boundary segregation of some seventy solutes in the same number of solvent matrices. The fracture process is shown in Fig I. The actual energy required to break the bonds across the grain boundary may be simply determined by counting up the number of dangling bonds per unit area and summing their energies. These energies may be calculated from the sublimation enthalpies. A fair numeri-
cat estimate of this approach may be made using nearest neighbour bond terms For a binary system of solute B in solvent A the nearest neighbour bond
energies (negative values), e,,,,, r,. and cBB are assigned to AA, AB and BB neighbours respectively. The~fractureenergy per unit area of the pure solvent A, FE(O). is given by (1)
tbJ
Fig. 1. Schematic representation of the atoms (a), either aide of the unbroken grain boundary shown by the dotted iine, and (b) after facture of the grain boundary with the ruptured bonds bsh dangIing. The matrix A atoms arc denoted by open cir&a anznt solute B atoms by fibed
where 2, is the co-ordination of atoms in the layer on one side of the grain boundary to those in the adjacent layer on the other side. On average Z, can be shown to be related to the customary value for a perfect lattice Z, by the appropriate factor (I - y&y&. For a perfect lattice Z, would be 2 and 3 for b.c.e. and kc. lattices respectively where the total co-ordination, 2 , is 8 and I2 respectively. The ratio yJv, is typically i/3 [lSJ. In equation (1) ai is the area of an A atom in the surface. The fracture energy with a molar fractional monolayer, X, of B at the grain boundary is FE(Xb) * - 2 z* ((I - fXl)fC4,4 f X,(1 - *X‘)c,, + +Xbttbs)* If the AB system is a regular solution, the parameter o may be defined and WXb) - FE(O) = -
3 Wbb
- %4)
+ OX&(1- fX*)l.
(2)
A similar result is obtained if the segrcgant atoms in Fig. 1 are assumed to be coplanar. For an ideal solution, UII: 0. If also the two atoms, A and B have differ-
SEAH:
ADSORPTION-INDUCED
ing sizes, the segregation induced reduction in fracture energy may be written approximately FE(O) -
FE(X*) =
~~~_~)
where Xb is now the fractional area of the grain boundary covered by B atoms. The values of eAA and eDa may be determined from the molar sublimation enthalpies, HIvb [ 163 NT” p: -$ZNf,&4,
Hg” = -fZNeaa
where N is Avogadro’s number. Thus FE(O) - FE(Xb) = &(!$
- %)x1.
(3)
$X,[H:‘b. - J.&+-J
(4)
where H,$ represents the sublimation enthalpy of pure A or B per unit area derived from Hsub* = Hsub/Na2. Alternatively FE(o) -
FE(xb)
=
5/‘6 x,&f”
-
Yf”,
(3
where y:* and yf* are the surface energies of pure A and B, respectively Cl’& Of the above equations, equation (4) most readily leads to a unified numerical presentation. For all systems, the fracture energy reduction per monolayer of solute adsorbed on the grain boundary is simply (ZJZ) times the difference in ~blimatio~ enthalpy per unit area for the matrix and solute materials nspectively, and the degree of embrittlement is linearly dependetit on the kvel of segregation, as observed experimentally [S, 61. Before proceeding further, the previous thermodynamic treatment will be analysed to compare its prediction with equation (4).
Early theories to explain the effect of solute, segregation on low temperature grain boundary brittle fracture are summa&d by the author in a previous paper [lg]. This previous paper also gives references to many of the appropriate experimental observations which will be only briefly outlined in this work. In the previous paper the ideal work of fracture per unit area of the grain boundary, y. was calculated from the relation [ 193. Yb
boundary, respectively. If yI and yb are the equilibrium values for the fractured material, equation (6) is appropriate for slow crack growth at high temperatures where the solute diffusion is sufficiently rapid to establish equilibrium surface segregation at the crack tip. For fast, low temperature fracture it was assumed that y. would not be the equilibrium value and an attempt was made to estimate the value to use. The approach is briefly outlined below. For unit area of the grain boundary [20] i where V and S are the specific volumes and entropy of the interface zone, P is the pressure across the interface, T the temperature and r’ the quantity of species i with chemical potential pi per unit area of the interface. At constant temperature and pressure in a binary system, where i = A for the solvent and B for the solute, and applying the Gibbs-Duhem equation, the above relation reduces to dy, = [{X’/l - X”}ff - f-ad#
(6)
where 1: and yb are the energies per unit area of the freshly produced fracture surfaces and the prior grain
(7)
where X8 is the solute molar fractional content. If the solute obeys Henry’s law, since d$ = RT d In a where a is the solute activity, then d#’ = RT d In X’. For X3 6 1 and, for a dilute system in which ri a XB,it was possible to show by substitution in equation (7) and integrating, that ~6’= yf* - R7Tf
(8)
and similarly for the surface y: = y:” - RTf,B
(9)
where yf” and r;‘” are the grain boundary and surface energies of the pure matrix A, respectively. In the following all the bulk concentrations, surface or grain boundary excesses and chemical potentials referred to are those for B atoms and so that the atom denoting superscript will now be deleted from X, r and p respectively. Since, on fracture r, = +rb, it was found from equations (6) (8) and (9), that y=2$*-y;*
3. THE PREVIOUS THEORYTHERMOCHEMICAL
Y = 2Y,-
957
dYts = pdP--SdT-x:;d/,&,
Although equation (3) contains many approximations, the right hand side is in a form which permits a simple and general evaluation for many segregation systems. Two interesting alternatives of equation (3) may be written FE(O) - FE(X,) =
INTERFACE DECOHESION
(10) for a given solvent irrespective of the segregation level. Hirth and Rice in a series of publications, summarised recently [21], agree with this basic approach but present the argument in general terms and point out the omission of a term from equation (10). This term may be seen by considering the p---r diagram, as shown in Fig. 2. Figure 2, curve (a) shows the relation for the equilibrium excess of B atoms at the grain boundary, fb, as a function of the SOhte chemical potential, & Similarly curve (b) shows the surface excess for the two fracture surfaces, 2f,, as a function of p. For slow fracture at high temperature, where equilibrium of the solute is maintained throughout, the
958
SEAH:
ADSORPTION-INDUCED
I
r
Ibl
INTERFACE
DECOHESlON
and 2r,M = ru2Xx
2r:
exp
-
+$
(
>
where AGb and AG, are the free energies for the solute segregation to grain boundaries and surfaces, respectively and T’, is the value of r, at one monolayer coverage. Substitution of the above three equations in equation (12) gives y = 2y*o - yAo + T,(AG, - AG1 - RTln 2). (13)
0
Y”
Y
Fig. 2. The p dependence of r for solute atoms at sites on (a) the grain boundary and (b) the fracture surfaces.
system starts at K and, during fracture, moves to L. Here the total segregation level increases as a result of fracture. Application of equation (6) in the dilute approximation above gives
Equation (13) gives an indication of the magnitude of the difference between equations (11) and (12). Whether equation (11) or equation (12) is really correct would appear to depend on the state of the surface atoms immediately after fracture. If the surface atoms in their non-equilibrium state really have the low chemical potential that would be necessary for them to be in equilibrium with the bulk potential #‘, then equation (12) is correct and the work of fracture is reduced by segregation. If there is no reduction in chemical potential, equation (11) is valid and for intermediate cases the result must lie between equations (11) and (12).
Y = 2Y: - Yb” = 2r;” - y:” - RT(ZT,L - l-f) (constant I( process) Thus, a sizeable decrease in y occurs as a result of segregation in slow high temperature fracture. For fast fracture at low temperatures the system starts at K and moves to M with the total segregant level unchanged. Application of equation (6) in the dilute approximation gives v-2Y?-ti = 2~;’
- y$’ -
(11)
However, in moving from K to M Hirth and Rice [21] point out that the segregated solute atoms have been able to rekase a portion of energy
which, here, is simply (# - p)r,. fracture is reduced by this amount
Thus the work of
y = 2yf” - y,“” - QP - r”>r,.
The final results of the thermo&mical theory,, equations (12) and (13), may or may not be the best description of the effect of segregants on gram boundary cohesion. However, data for AG, and AG,, generally are not available and so the result is of limited use to the metallurgist. Equation (13) may be rewritten FE(O) - FE(X,)
RT(ZT: - rf)
= ato - yf” (constant p process)
4. COMPARISON OF TIiB PAIR BONDING AND THERMOCHEMICAL APPROACHES
(12)
measure of the new term may be obtained as follows. In the Henrian approximation for dilute systems
A
where p. is the chemical potential of a reference state. In the dilute approximation also [223
x’ AH, - AH, + T(AS, - AS,)}. = Nx'
(14)
The entropy termq AS, and AS,, to a large extent cancel each other [lSJ. Nearest neighbour bond analysis, using the approach described in section 2 and elsewhere [23], shows that in the dilute approximation for a regular solution AH,=
- H(fZ&,,
(15)
- e& + o&J.
This relation has been shown to be valid in predicting experimental values of surface segregation over a range of 100 kJ mole-’ with a standard deviation of 15 kJ mole-’ [23]. Thus, although the pair bonding approach may be inadequate if there is considerable relaxation of the surface bonds, as, for example, in silicon, experiments for many metallic systems show considerable agreement with the pair bonding approach. In a similar manner to equation (IS), AH* may be derived AH, = -H(f(Z,
- Z,)(E,, - (AA) +o(z,-
Z,)).
(16)
SEAH:
ADSORPTION-INDUCED
Thus equation (14) may be rewritten FE(X,)
- FE(O) “-7
2
I~Xb(Qll) - d
+ @Xb)
(17)
which is equivalent to equation (2) developed in the new theory. Thus, within the regular solution ap proximation the previous thermodynamic description given by equation (12) affords the same basic result as the bond breaking descriptions of equations (2) and (4). Since it is possible to perform general numerical calculations with equation (4) which are not directly possible with equation (l2), equation (4) is used in the next section to give the general predictions of the theory. I PREDI~ONS
OF THE THEORY
From equation (4) the loss in fracture energy of the grain boundary atoms is proportional to the difference in Hgub* of the matrix and segregant atoms calculated t?om the tabulated data for pb [16j and the atom sizes, u. a is derived from pNa3 = A
(18)
where p is the bulk density of the pure material and A
iNTERFACE DECOHESION
is its atomic weight. Values of Web* for 70 elements are plotted in Fig. 3 as a function of the atom size, a. Taking any given element as the matrix, elements lower in Fig 3 will, if segregated, cause embrittlement of the matrix grain boundaries whereas elements higher will be remedial to such embrittlement. The magnitude of these effects is, of course, in proportion to the distance vertically from the matrix element point. If iron is taken for the matrix the line (i) divides the embrittling and remedial elements. Because of the approximations used in deriving equation (4), points near the line are not so easily classified. Incorporation of the regular sofution parameter, W, ignored above, causes a term of approximately 1J/m’ to be added or subtracted from the Hsubr terms in equation (4) For ekments which form an intermetallic with iron (o negative) the grain boundaries become more cohesive than indicated in Fig 3 and vice versa. A second small error arises from the use of equation (18) For elements segregating in iron the atom size should best be determined from that of the compound which first appears as the solute content is raised [2]. For metals and metalloids equation (18) is sufficiently accurate but for P, N and 0 the value from the compound has been adopted. In alit with experiment, Fig. 3 shows that in iron Bi, S, Sb, Se, Sn and Te are all highly embrittling followed in order of reducing
CC t
Increasing grain boundary cohesion
,B oe .W
__*_____________(I)
\ Increaring grain bounday lmbrlttlement
I 0.2
Fig. 3. Ideal compilation for the iron if iron
I
w
I
Nam ,
0.3
959
lBO
I 0.b
oKIlabJcs 0.5
onm
solution general remedial/embrittkment plot for matrix and segrcgant elements based on the of Hab* values aa a function of the atom size calculated from equation (18). As an example matrix, line (i) represents the locus dividing those ekments which reduce the fracture energy from those which inacase it. Line (ii) is the similar locus for the critical fracture stress
960
SEAH:
ADSORPTION-INDUCED IOC
--
,_.._.
--‘-r’__‘_.
INTERFACE DECOHESION -----I
\ Incrsoimgg’al” boundarycohcsmn
us l sn lSb
I
a2
I
I
0.3
I
0.~
I
0.4
1
I
0.5
Fig. 4. Remedial/cmbrittkment plot for segregants in iron in the reguJu solution approximation. The position of ekments not shown may be estimated by recognking that tbc rektive poxitions of elements of similar chemistry remain approximately unaltered between Figs. 3 and 4. effect, by P, As, Ge, Si and Cu. In increasing order of their remedial e&ct on cmbrittkment are N, B and C. Of course, Fig. 3 does not take into account any bulk precipitation or changes in microstructure caused by these elements. The effect of the regular solution parameter, 0, can be included to provide a plot such as Fig, 3 for a particular matrix. For the matrix of iron this is shown in Fig. 4 for solutes for which o can be assessed Values of w are determined, as discussed by Swalin [24]. from tabulations [ 163 of the enthalpy of rnixing, the activity coefficients of the two species and details of the phase diagram. Thus, in iron, elements such as -Cu and Ti, which have similar marginal effects in the ideal solution approximation of Fig. 3, are shown to have clearly separated effects in the regular solution approximation of Fig. 4. Elements further removed from iron in Fig. 3 remain, however, relatively unaffected. The remedial ekments deserve a little further discussion. Equation (13) predicts a remedial etkct only if the free energy for grain boundary segregation is larger than that for surface segregation. There are two types of remedial atom lo consider; those occupying substitutional sites in the lattice and those occupying interstitial sites. Comparative surface and grain boundary segregation data for the substitutional ele-
ments are not availabk but analysis of surface segregation data [23] shows that as Hmw decreases, or as the atom size increases, the free energy of surface segregation increases in magnitude. Since the propensity to intergranular failure also increases in this way, substitutional ekments that are highly surface active tend to be the ones that are also highly embrittling. Likewise the high bond energy substitutional elements in the remedial sector are those which, to a large extent in binary systems, prefer the higher coordination sites of the lattice to those of the free surface and the grain boundary and, hence, do not segregate. In ternary systems, however, coupling of an embrittling atom to one of these remedial atoms may cause segregation of the latter so that its remedial property may be utilised In this way the reduction in the grain boundary embrittkment of a temper brittle steel for a given level of phosphorus (embrittling ekment) at the grain boundary by the synergistic segregation of molybdenum [25] (remedial element) may be explained. The behaviour of the interstitial remedial elements C, B and N dialers from the above behaviour in that the release of strain energy when the solute atom leaves the lattice dominates its intrinsic chemical wish to dcsegrcgate strongly from the surface and less strongly from the grain boundary. In this way strong
SEAH:
ADSORPTION-INDUCED
grain boundary segregation ooblrs and, for C B and N, since fewer of the highly energetic bonds are disrupted at the grain funds than at the free surface, the free energy of segregation to the grain boundary is larger than that to free surface. Data for the segregation of carbon in a-iron [26,27] supports this view. Thus, C, B and N interstitial atoms probably segregate to release strain energy, segregate more strongly to grain boundaries than free surf&es and at the grain boundaries improve their cohesion. Thus equation (131 in avant with quation (41 predicts that these elements will be remediil to the grain boundary embrittlement of iron as demonstrated experimentally [I 1, 12,281. It may be argued that the critical fracture stress of the atoms either side of the grain boundary should be considered instead of the work of fracture. As the applied stress on the fracturing sample increases, dislocations pile up in the crack tip region and contribute to the plastic work term mentioned earlier. At the point where the crack will just advance the atoms at the crack tip can either part, giving brittle fracture, or emit dislocations and so blunt the crack tip. The criterion for emitting dislocations depends on the shear stress across the slip plane where it intersects the crack front on the grain boundary [29,30], and this, in turn, is related to the stress acting across the grain boundary atoms at the crack tip. Thus the ductile to brittk transition may be goverened by the witicat fracture stress for the grain boundary and not the work of fracture. Of course, in perfectly brittle materials, where the choice is between cleavage and intergranular failure, other aiteria may be appropriate. Even allowing for this alternative criterion of critical fracture stress, instead of the fracture energy used previously, it will be shown below that Figs. 3 and 4 retain their full usefulness in assessing the embrittling and remedial properties of different segregants. The segregation of large atoms to the grain boundary displaces the stress-distance curves for the atoms across the boundary outwards. For the generally used potentials, e.g. Lennard Jones, Morse, Mie etc, it is easy to show that the fracture energy is proportional to the product of the critical fracture stress and the interplanar distance [18]. Thus, the ratio of the critical fracture stress of the segregated boundary, a(X,), to that of the clean boundary, u(O), is: 1
0 (xb) - 40)
=
1
+
i
+
+&(&,,a~
FE -
1)
6)
FE(o)
Hence a(Xb) PE 6Y
~xb(~,“b*/~~~~ 1
f
-
)xb(&,,ff,,
-
1)
t) *
the neutral line dividing the remedial and embrittling elements in Figs. 3 and 4 is now given by
Thus,
&ub’ =
aB -
(2A
HP”.
(19)
INTERFACE DECOHESION
961
This result is shown for elements in iron as line (ii) in Fig. 3 and its counterpart in Fig. 4. For iron the conclusions concerning the remedial and embrittling elements remain unchanged. Indeed, because of the way elements occur in Figs 3 and 4 it would be difficult to distinguish which of the two criteria of fracture energy and critical fracture stress is correct from a study of the remedial and embrittling elements in solvent matrices. Figure 3 may be used, as above, to define the ~brittling and remedial segregants in all matrices. Elements below the appropriate lines (i) and (ii) passing through the matrix ekment point are embrittling and vice versa. For instance bismuth will embrittle copper and potassium and phosphorus will embrittle tungsten, as have been shown experimentally [29,3 I, 321. CONCLUSIONS A simple theory, based on pair bonding, is proposed to describe the changes in grain boundary cohesion resulting from solute segregation. The theory is shown to be equivaknt to the detailed general thermodynamic treatment within the regular solution approximation. Numerical evaluation of the theory, through equation (4)1allows a general scheme of embrit~~g and remedial segregants to be defined for any solvent matrix. This scheme, shown in Fig. 3, predicts, in agreement with experiment, that Bi, S, Sb, Se, Sn and Te will be highly embrittling in iron followed, in order of reducing effect, by P, As, Ge, Si and Cu. In increasing order of their remedial effect are N, BandC. Ac~wf~ge~~~-T~e author would like to thank J. P. Hirth and J. R. Rice for material prior to publication and J. P. Hirth and E. D. Hondros for general discussions.
REFERENCES 1.
E. D. Hondros and M. P. Seah, Int. Metall. Rev. 22,
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