role of entropy in anistropy of grain boundary segregation

SURFACE AND INTERFACE ANALYSISSurf. Interface Anal. 26, 800È805 (1998)

Role of Entropy in Amistropy of Grain BoundarySegregation¤

Pavel Lejc— ek*Institute of Physics, Academy of Sciences of the Czech Republic, Na Slovance 2, 180 40 Praha 8, Czech Republic

From the viewpoint of existence of linear dependence between entropy and enthalpy of grain boundary segregation,the meaning of the entropy term is discussed. It is suggested that negative values of segregation entropy for siliconsegregation at individual grain boundaries in a-iron reÑect an “orderingÏ tendency of the interfaces due to thepresence of silicon. On the other hand, positive values of segregation entropy suggest a “disorderingÏ of the structureof these grain boundaries by segregation of carbon and mainly of phosphorus. The e†ect of “disorderingÏ is espe-cially high for special grain boundaries with compact intrinsic structure, resulting in high values of segregationentropy. The slope of the dependence of enthalpy on entropy of solute segregation at di†erent grain boundaries canbe considered as a compensation temperature. Its existence implies that an anisotropy of grain boundary chemicalcomposition will possess a qualitatively di†erent character above and under this temperature. This anisotropymeasured above the compensation temperature will be characterized by high values of grain boundary concentra-tion for special boundaries as compared to general boundaries. 1998 John Wiley & Sons, Ltd.(

KEYWORDS: anisotropy ; entropy ; grain boundaries ; solute segregation

INTRODUCTION

In the past decade, a systematic study of orientationdependence of grain boundary segregation was per-formed using well-characterized grain boundaries inbicrystals of an FeÈ3.55 at.% Si alloy. This studyresulted in the determination of sets of thermodynamicparameters of segregation at individual interfaces.1h9Despite this intensive work resulting in the constructionof the Ðrst experimental grain boundary segregationdiagrams,10 as well as in the speciÐcation of all special[100] tilt grain boundaries in a-Fe,6 many questionsremain unanswered until now. It is generally supposedthat so-called special grain boundaries represent anextreme on a structure/property dependence and thattheir properties are closer to the bulk behaviour thanthose of the general grain boundaries.11 From the segre-gation point of view it should mean that special grainboundaries exhibit lower segregation e†ects thangeneral grain boundaries. Besides the numerous proofsof this rule, there are studies of anisotropy of grainboundary segregation in the literature that show di†er-ent tendencies.7 For a classic example of such a result,Stolarz and LeCoze12 found maxima of silicon segrega-tion in austenitic stainless steel at the M013N, M012N and

* Correspondence to : P. Lejc— ek, Institute of Physics, Academy ofSciences of the Czech Republic, Na Slovance 2, 180 40 Praha 8, CzechRepublic.

¤ Dedicated to Professor Siegfried Hofmann on the occasion of his60th birthday.

Contract grant sponsor : Grant Agency of the Academy of Sciencesof the Czech Republic ; Contract grant no. : A1010708.

Contract grant sponsor : COST Action 517 ; Contract grant no. :OC517.40.

Contract grant sponsor : VEGA Grant Agency, Slovak Republic ;Contract grant no. 2/5161/98.

M023N symmetrical tilt boundaries that are widelyaccepted as special. A similar result was obtained forphosphorus segregation at the M012N, M013N and M015Nsymmetrical tilt grain boundaries in FeÈ3.55 at.% Sialloy at higher temperatures.7h9 Is this result really aconsequence of complex segregation behaviour of indi-vidual grain boundaries in multicomponent systems7 ordoes it reÑect the real segregation behaviour of individ-ual types of grain boundaries? Besides this heretic ques-tion, there is another problem connected to the valuesof segregation entropy. It was theoretically estimated bySeah13 that the absolute value of segregation entropyshould not exceed 3.3R. However, in the same paper, tinsegregation in polycrystalline iron is described by avalue of segregation entropy of 5.4R. Similarly, theentropy was found to reach the value of 5.2R for carbonsegregation in polycrystalline bcc iron14 as well as forphosphorus segregation at the M013N symmetrical tiltgrain boundary.15 Do these high values have a physicalmeaning? What is the reason for the fact that highvalues of segregation entropy correlate systematicallywith the segregation behaviour of special grain bound-aries? In the following, these questions will be dis-cussed.

EXPERIMENTAL DATA

The thermodynamic parameters of segregation(enthalpy and entropy of silicon, phos-*H

I0 *S

I0)

phorus and carbon in pure a-iron were determined fromAuger electron spectroscopy (AES) measurements of thetemperature dependence of the chemical composition ofwell-characterized grain boundaries in bicrystals of amulticomponent FeÈ3.55 at.% Si base alloy containingtraces of phosphorus and carbon. All experimental

CCC 0142È2421/98/110800È06 $17.50 Received 9 March 1998( 1998 John Wiley & Sons, Ltd. Accepted 23 May 1998

ENTROPY IN ANISOTROPY OF GRAIN BOUNDARY SEGREGATION 801

details on chemical composition of the alloy, prep-aration of bicrystals and AES measurements, as well ason the results of these measurements, have been sum-marized previously.4,6,7 To make the following treat-ment clearer, the values of and are listed in*H

I0 *S

I0

Table 1. Let us note that these values were obtainedfrom the correlation of measured data on the basis ofthe Guttmann quasichemical model of grain boundarysegregation in a multicomponent alloy.16 In fact, thevalues of and (although determined from the*H

I0 *S

I0

data measured in a multicomponent alloy) representthermodynamic parameters describing grain boundarysegregation of solute I in a dilute binary a-FeÈI system.

RELATIONSHIP BETWEEN SEGREGATIONENTHALPY AND ENTROPY

Recently, Rittner and Seidman17 revealed the existenceof a linear relationship between enthalpy and entropy ofpalladium segregation in nickel from the data calculatedfor individual grain boundary sites at numerous inter-faces. They also proved that this relationship is valid forgold segregation in platinum as well as for our experi-mental data on solute segregation in a-iron. In Fig. 1, aplot of our data on the segregation of silicon, phos-phorus and carbon at [100] tilt interfaces of a-iron isredrawn for all data given in Table 1. As seen from Fig.1, the linear relationship is fulÐlled for all the data. Inaddition, thermodynamic data found in the literaturefor di†erent segregants (cf. Table 2) in a-iron are plottedin Fig. 1.

The relationship between both thermodynamicparameters for the di†erent grain boundaries apparentfrom Fig. 1 can be expressed by

*HI0\ q

I*S

I0] u

I(1)

The values of parameters and Ðtted by the simpleqI

uI,

least-squares-method, are given in Table 3.

Figure 1. The dependence of on for Si, P and C for*HI0 *S

I0

different grain boundaries in a-iron (empty symbols). The data aretaken from Table 1. The dashed line represents the summary fit ofboth C and P data. For comparison, the literature data on grainboundary segregation of Si, P, C, Sb, Sn and S in other bicrystalsand polycrystals of a-iron base materials displayed in Table 2 arealso plotted (solid symbols).

It follows from Eqn (1) that is the segregationuIenthalpy of I for the boundary with The other*S

I0\ 0.

parameter

qI\Ad*H

I0

d'BNAd*S

I0

d'B

(2)

represents the “compensationÏ temperature at whichis constant for all grain boundaries, i.e. at which*G

I0

the segregation of I is the same at all grain boundaries.The derivation d/d' represents the change of the vari-able with the change of the boundary structure.

Existence of the compensation temperature was orig-inally observed for grain boundary di†usion31 andmigration.32,33 In all these cases, however, the ther-mally activated kinetic processes are considered. It wasdeduced that the compensation temperature is identicalto a temperature of Ðrst-order transition, such as the

Table 1. Segregation enthalpy (in J mol—1) and entropy (in JDHI0 DS

I0

mol—1 K—1) of silicon, phosphorus and carbon for individual [100 ] tiltgrain boundaries4,6,7

Grain boundary *HSi0 *S

Si0 *H

P0 *S

P0 *H

C0 *S

C0

Ê016Ì É16000 É15 É31000 17 É49000 1

Ê015Ì É12000 É9 É16000 38 É43000 7

Ê014Ì É14000 É9 É35000 19 É50000 2

Ê013Ì É8500 É3 É13000 45.2 É40000 12

Ê0klÌ É17000 É13 É37000 18 É51000 6

Ê0 7 15Ì É12000 É3 É31000 25 É45000 6

Ê012Ì É4100 0.2 É10900 42.5 É35000 12

Ê059Ì É12000 É5 É34000 20 É48000 4

Ê058Ì É16000 É11 É37000 16 É53000 É1

(018)/(047) É10000 É8 É32000 19 É50000 3

(001)/(034) É9000 É9 É25000 29 É44000 6

(017)/(011) É6100 É2.2 É14500 39.3 É36000 14

(0 3 11)/(097) É11000 É5 É32000 21 É48000 3

(001)/(011) É6000 2 É19000 38 É39000 11

(001)/(012) É10200 É5.3 É26000 28 É47000 4

(001)/(013) É7000 É2 É19000 35 É38000 13

(001)/(015) É8000 É3 É22000 32 É41000 10

(011)/(015) É7000 É2 É18000 37 É40000 9

(011)/(013) É6000 É1 É17000 37 É36000 14

(011)/(012) É6000 É2 É16800 36.1 É34000 16

( 1998 John Wiley & Sons, Ltd. Surf. Interface Anal. 26, 800È805 (1998)

802 P. LEJC‹ EK

Table 2. Compilation of values of (in J mol—1) andDHI0

entropy (in J mol—1 K—1) found in the literatureDSI0

(cf. Ref. 7)

Matrix Segregant *HI0 *S

I0 Ref.

Fe–12.9 at.% Si Ê013Ì Si É8100 É6.9 18

Polycrystals a-Fe–Si Si É17000 0a 19

Polycrystals a-Fe P É34300 21.5 20

É38000 0 21

É50000 0a 22

É21200 37.3 23

É32000 22 24

C É80000 0 22

É37700 43.2 23

É57000 21.5 25

É79000 É13 26

S É51000 0a 27

Sn É13000 45 13

É22500 26.1 28

Sb É13000 0a 29

É19000 28 30

Polycrystalline Sn É13100 39.1 13

low-alloy steel

a Segregation entropy was not specified in some papers but it issupposed that it was considered to possess the value DS

I0 ¼0.

melting point and point of solid-state transformation.33In the present case of segregation, however, the samelinear dependence was found between standard molarenthalpy and standard molar entropy of an equilibriumstate. This fact suggests that there is a more generalthermodynamic relationship between enthalpy andentropy that is not limited to thermally activated pro-cesses. In di†erence to those Ðndings from kineticexperiments, the compensation temperatures for segre-gation do not seem to be related to transformation tem-peratures in a-iron or in its alloys with silicon,phosphorus and carbon (cf. Table 3 and Ref. 34).

Looking at Fig. 1, several questions arise. The Ðrstquestion is why the data for silicon are separated,although it seems that the data for the other segregants,carbon and phosphorus, can be Ðtted by a commonline, as shown in Fig. 1 by the dashed line (cf. Table 3).We can also ask for the reason why there are generallynegative values of segregation entropy of silicon butpositive values for carbon and rather high positivevalues for phosphorus.

To explain these questions, let us start at the deÐni-tion of segregation entropy. The segregation of animpurity I at a grain boundary in a binary system MÈIshould follow a “chemical reactionÏ

I] M' % I' ] M (3)

This means that the matrix element M at the boundary' is replaced by the impurity I from the bulk. The stan-dard molar free energy of equilibrium segregation is

then deÐned as

*GI0\ (G

I0' ] G

M0) [ (G

I0] G

M0') (4)

where and are the standard molar free ener-GI(M)0' G

I(M)0gies of I(M) at the grain boundary and in the bulk,respectively. The standard molar segregation entropy isdeÐned in a similar way

*SI0\ (S

I0' ] S

M0) [ (S

I0] S

M0') (5)

In Eqn (5), and are the standard molarSI(M)0' S

I(M)0entropies of I(M) at the grain boundary and in the bulk,respectively.

It is known that entropy represents a measure of “dis-orderÏ of the system (in the most general sense of theword). This means that a positive value of the change inentropy suggests the Ðnal state to be less “orderedÏ thanthe starting state. From the viewpoint of segregation, apositive value of segregation entropy suggests that thepresence of an impurity atom at the grain boundary canbe considered as a “disorderingÏ of the boundary. This isprobably the case of phosphorus and carbon segrega-tion at grain boundaries of a-iron. On the other hand, adecrease of entropy, i.e. a negative segregation entropy,should be connected to some “orderingÏ tendency at theinterface. The negative values of were obtained in*S

I0

our measurements for silicon, suggesting that its segre-gation in an a-FeÈSi system leads to an “orderingÏ of theboundaries. It is known that the a-FeÈSi system exhibitsalloy ordering at concentrations of D10È13 at.% Si(depending on temperature).34 Because silicon segrega-tion provides a silicon boundary concentration close tothese values, it can be expected that an “orderingÏ occursat the boundary, as was indicated by the measurementsof silicon segregation at single grain boundaries ina-FeÈSi bicrystals.18 On the other hand, phosphorusand carbon tend to form stable phosphide and(Fe3P)carbide respectively, and thus both of these ele-(Fe3C),ments segregate at the boundaries in the form of over-saturated solid solutions. This is in agreement withpositive values of segregation entropy.

It is also interesting to consider the behaviour ofother segregants in a-iron from the viewpoint of therelationship between enthalpy and entropy of segrega-tion. The published data on segregation enthalpy andentropy obtained from measurements on polycrystals orbicrystals are also plotted in Fig. 1. As seen from Fig. 1,the majority of the points Ðt well to the (P ] C) branchof the dependence between segregation enthalpy andentropy. This is true for nearly all the data for phos-phorus, tin, antimony and sulphur. In the case ofcarbon, the data seem to be shifted somehow. Unfor-tunately, some data found in the literature werepublished incomplete. In these cases, only the value ofsegregation enthalpy is given but no value of segrega-tion entropy is mentioned. Usually, segregation entropyis neglected in such cases, and thus it is supposed topossess the value This is shown in Table 2*S

I0\ 0.

Table 3. Values of parameters (in K) and (in J mol—1) (I = Si, P, C)sI

xIfor Eqn (1)

qSi

uSi

qP

uP

qC

uC

qP½C

uP½C

750 É6000 920 É51000 1100 É52000 900 É51000

Surf. Interface Anal. 26, 800È805 (1998) ( 1998 John Wiley & Sons, Ltd.


concerning the data on sulphur as well as partially onphosphorus, antimony and carbon (cf. Fig. 1). Based ona comparison of Fig. 1 with the literature data,34 we canconclude that the solutes forming chemical compoundswith iron but segregating at the boundaries in the formof oversaturated solid solutions (e.g. P, S, Sn, Sb, C)should possess, in general, positive values of segregationentropy and fulÐl the dependence found in our case forphosphorus and carbon. On the other hand, the depen-dence suggesting the tendency of the boundary of “ord-eringÏ is somehow shifted to negative values ofsegregation entropy (Fig. 1). Similar behaviour might beexpected for Al, Co, Ga and Ge for example (cf. Ref. 34).

Because the special (singular) grain boundaries aremostly formed by simple structural units, it is clear thatthe presence of a foreign atom will represent a muchhigher “disorderÏ of their structures than in the case ofgeneral boundaries with much larger and complexstructures. Therefore, the special boundaries shouldpossess higher values of segregation entropy thangeneral boundaries. This is valid for the segregation ofall three elements in iron as measured in our experi-ments.4,6

IMPACT OF THE ENTHALPY/ENTROPYRELATIONSHIP ON MUTUAL BEHAVIOUROF INDIVIDUAL TYPES OF GRAINBOUNDARIES

The close relationship between enthalpy and entropy ofsegregation has another very important consequence.As discussed above, the special grain boundaries possesshigher values of segregation entropy than generalboundaries because the presence of the foreign atom inthe boundary core represents much higher “disorderingÏat the former interfaces as compared to the latter. Onthe other hand, the special grain boundaries exhibit alower tendency to grain boundary segregation thangeneral boundaries, as documented by the lower absol-ute values of segregation enthalpy. Let us model theconcentrations of individual solutes, in pure binaryX

I',

a-FeÈI systems at individual symmetrical tilt grainboundaries on the basis of the thermodynamic datagiven in Table 1. According to the simple McLeanmodel that can be applied to non-interacting dilutebinary systems, the grain boundary concentration canbe determined as

XI' \

XI

expC[ *H

I0 [ T *S

I0

RTD

1 ] XIexpC[ *H

I0[ T *S

I0

RTD (6)

For the modelling, we consider three binary systems :FeÈSi at.%), FeÈP at.%) and FeÈC(XSi\ 3 (XP\ 0.01

at.%). The results of this simple modelling(XC\ 0.01are shown in Fig. 2. It is clearly apparent from Fig. 2that there is a pronounced anisotropy of grain bound-ary segregation, characterized by minima of grainboundary concentration at special M012N, M013N andM015N symmetrical tilt grain boundaries for all threesolutes at low temperatures. This is in agreement withthe conclusion about the character of individual inter-

Figure 2. The plot of calculated interfacial concentrations ofsilicon (a), phosphorus (b) and carbon (c) in respective a-Fe–Ibinary systems on misorientation angle for individual symmetricaltilt grain boundaries listed in Table 1. Bulk composition of solutesin individual binary systems: Fe–Si, at.% ; Fe–P,X

Si¼3 X

P¼0.01

at.% ; Fe–C, at.%.XC

¼0.01

faces.7 At high temperatures, however, the orientationdependence of grain boundary enrichment qualitativelychanges and the maxima of grain boundary concentra-tion are observed at all these boundaries in the case ofphosphorus. Maxima of silicon segregation areobserved at the M015N and M013N boundaries, while alocal minimum still appears at M012N. As for carbon,there are practically no di†erences in the chemical com-position of any interface at higher temperatures. Noticethat the di†erences in chemical composition of specialgrain boundaries are not so dramatic for di†erent tem-peratures as compared to general boundaries. This isthe consequence of low absolute values of segregationenthalpy for special interfaces. We must bear in mindthat in the present modelling, pure binary dilute systemswere considered where no interaction comes intoaccount. Therefore, these orientation dependencesreÑect the intrinsic behaviour of individual grain bound-


804 P. LEJC‹ EK

aries. Due to the low values of segregation enthalpy,low interfacial concentrations of solutes are alsodetected at the special grain boundaries at low tem-peratures where the entropy term is low. However, thecontribution of the entropy term (high for specialboundaries and small for general boundaries) at hightemperatures qualitatively changes the character of theorientation dependence of the grain boundary concen-tration.

The above, apparently surprising, result is a conse-quence of the existence of the compensation temperature

At the compensation temperature the concentrationqI.

of a solute is the same (statistically) at all grain bound-aries, as discussed above. The values of are listed inq

ITable 3 for all three elements. Due to di†erent pairs ofthermodynamic values of segregation for di†erent grainboundaries, the ratio of the interfacial concentrationwill di†er qualitatively under and above Forq

I.

example, the fact that phosphorus at low temperaturesexhibits low enrichment of special grain boundaries ascompared to general boundaries has a serious conse-quence in the di†erent abilities of these boundaries totemper embrittlement : general boundaries with segre-gated phosphorus exhibit brittle fracture more easilythan special boundaries. At temperatures above qP \920 K, however, segregation at special grain boundarieswill be higher than that at general boundaries, but theenrichment is already too low to cause substantial dif-ferences in temper embrittlement of these two types ofgrain boundaries. This is probably the reason why the“oppositeÏ anisotropy of grain boundary segregation hasnot been mentioned in the literature. The only excep-tion showing this kind of anisotropy is silicon segrega-tion at grain boundaries of austenitic stainless steel,12where maxima of its enrichment were found for specialM013N, M012N and M023N symmetrical tilt grain bound-aries measured at 923 K. The value of is probablyqSilow enough also in this fcc base material for this kind ofanisotropy to be revealed. For comparison, in bcca-iron the value of the compensation temperature israther low K, Table 3). From the viewpoint of(qSi\ 750the compensation temperature we can ask what thevalue is for oxygen segregation in molybdenum, whereBiscondi35 reported no orientation dependence for the[100] symmetrical tilt grain boundaries (although theobjection7,9 that he made this conclusion on the basisof only two points remains true). In any case, however,all examples presented here clearly document that theanisotropy of grain boundary segregation can hardly be

represented by a simple orientation dependence of grainboundary concentration because it can bring qualit-atively di†erent results even for the same system.

CONCLUSIONS

The detection of the linear relationship between segre-gation entropy and enthalpy brought up a new pointfor understanding the behaviour of grain boundariesand their structure/property relationship. It is now clearthat the values of the segregation entropy are higher forspecial grain boundaries as compared to general grainboundaries due to higher “disorderingÏ of the formerinterfaces by the presence of impurity atoms as com-pared to the latter interfaces. The negative value of seg-regation entropy found for silicon segregation in a-ironsuggests a tendency to “orderingÏ of the grain boundaryduring solute segregation. Similar behaviour might beexpected for aluminium, cobalt, gallium and germa-nium, for example. On the other hand, solutes such asphosphorus, carbon, sulphur, antimony and tin, whichsegregate in the form of a solid solution, represent a“disorderÏ for the grain boundary structure that is reÑec-ted in the positive values of segregation entropy. Thelinear dependence between segregation entropy andenthalpy of a given solute at di†erent grain boundariesnecessarily implies that the so-called compensation tem-perature at that segregation is the same at all grainboundaries. As a practical consequence, anisotropy ofgrain boundary segregation under this compensationtemperature will di†er qualitatively from that observedat temperatures above it. This means that the orienta-tion dependence of the chemical composition of grainboundaries may lead to confusing conclusions. The exis-tence of the compensation temperature for an equi-librium state as well as for thermally activated kineticprocesses suggests a more general thermodynamicrelationship between enthalpy and entropy.

Acknowledgements

This work originated partially in the framework of projects of theGrant Agency of the Academy of Sciences of the Czech Republic(grant no. A1010708), of the international collaboration COST Action517 (grant no. OC517.40) and of the VEGA Grant Agency (SlovakRepublic, grant no. 2/5161/98). This work could not originate withoutmuch fruitful discussion of these problems with Siegfried Hofmann.

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role of entropy in anistropy of grain boundary segregation

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