Mechanism ofgrain grovvth inmetals

H. Fredriksson

A theoretical model has been developed to describe the grain growth process duringrecrystallisation of metals. The theory has been tested on various experimentalresults presented previously in the literature. It has beenfound that the effect of

-grain orientation and the effect of alloying elements on the process can be explainedby the new theory. MSTj 1120

© 1990 The Institute of Metals. Manuscript received 7 July 1989; in final form30 January 1990. The author is in the Department of Casting of Metals, TheRoyal Institute of Technology, Stockholm, Sweden.

Introduction

Theoretical and experimental investigations of recrystallis-ation and grain growth have been carried out for manyyears.1 The recrystallisation process has many similaritiesto other phase transformations.

The driving force for the recrystallisation process is adecrease of the stored strain energy in the material andthe driving force for grain growth is a decrease in grainboundary energy. In a recrystallisation process, the atomsleave the cold worked grain on one side of a grainboundary and are transferred across the boundary andincorporated into the recrystallised atomic lattice on theother side of the boundary. The process is often treated asa thermally activated process by means of the absolutereaction rate theory.2,3 However, it has been found thatthe pre-exponential term and the activation energy do notagree very well with the expected values for these twophysical parameters.

The recrystallisation process is influenced by the compos-ition of the alloy. 3,4 The rate of recrystallisation is decreasedwhen alloying elements are added to a pure metal. Thiseffect is explained by a segregation of alloying elements tothe grain boundary5 and has been discussed theoreticallyby Lucke and Detert6 and later by Cahn.7 These theoriesdescribe the growth process qualitatively and give a goodunderstanding of the growth mechanism.

More than 20 years ago, Hillert8 discussed the role ofinterfaces in phase transformations in the solid state. Inthat work, he assumed that the interface could be treatedas a separate phase. In the present paper, a modificationof Hillert's ideas on the effect of interphases will bepresented and used to analyse the recrystallisation process.

Recrystallisation and grain growth in puremetals

The structure of a grain boundary has been discussedextensively in the literature. Even 40 years ago, Smith9

presented an elegant illustration of grain boundaries usingbubble models and, since then, McLean,lO Aust andChalmers,l1 Blakely,12 and many others have done exten-sive work in this area. On the atomic scale, grain boundariesare considered as a thin layer by including differentamounts of disorder in the structure, compared with thestructure in the adjacent grains.

Grain boundaries are very difficult to describe analytic-ally and, therefore, thermodynamic expressions are oftenused to describe the state and properties of the boundary.

In the present discussion, this macroscopic method willbe used to describe the grain boundaries. It will be assumedthat the boundary layer can be treated as a thin layer of anew phase. This new phase will be termed boundary andwill be given a free energy in the same way as the otherphases. Following a treatment for interfaces made byHillert,t3 the difference in free energy L\Gm can be relatedto the free energy of the other phases by

L\G = (JVmm L1 . (1)

where

(J = boundary free energyVm = molar volume of interface

L1 = thickness of interface

The free energy for a system in equilibrium will vary asshown in Fig. la; where one phase has a lower free energythan the other, the variation of the free energy will be asshown in Fig. lb. The latter situation occurs during therecrystallisation process. The driving force for recrystallis-ation is denoted by L\G~w. It can now be seen that thesurface free energy in relation to the new (recrystallised)grain is greater than if it is related to the cold workedgrain. The free energy difference can be described by thefollowing relation

L\G~cw = L\G~ - L\G~w . (2)

where

L\G~ = grain boundary free energyL\G~w = cold work energy

The absolute reaction rate theory is often used to describethe rate of movement of a boundary.2,3 It will also beapplied in the present treatment; however, there will be amodification of the theory. Figure 1b shows the flow ofatoms into and out of the boundary. The transport ofatoms from the cold worked grain to the grain boundary,as well as from the grain boundary to the cold workedgrain, can be described (according to the absolute reactionrate theory) by the following expression

IB-CW = vexp( - RQT) {l-exp [ - (AG~;:G~W) J}. (3)

where

I = net number of atoms leaving grain boundaryv = vibration frequency of atoms

Q = activation energy for movement of atoms from grainto grain boundary and vice versa

R = universal gas constantT = temperature

Materials Science and Technology September 1990 Vol. 6 811

812 Fredriksson Mechanism of grain growth in metals

~Gm square bracket gives

GrainlBbUn1ary

By transformation, the net number of atoms jumping fromthe cold worked grain to the recrystallised grain can beequated to a rate of movement of the grain boundary vby

(7)

Now, it can be seen that both the pre-exponential termand the activation energy are influenced by the boundaryfree energy. Thus, this effect can describe previouslyreported discrepancies between theory and experimentalresults. To analyse this further, previously reported experi-mental results will be compared with the growth rate givenby equation (9).

In equation (9) it is difficult to estimate Q (the activationenergy for transfer of atoms into the boundary). It couldbe equal to the activation energy for grain boundarydiffusion along the boundary, but it could also be lessthan this value. In the present model, it will be assumedthat this energy is equal to the activation energy fordiffusion of vacancies. The Do value will be set equal to5 x 10-3 m S-1 in all the calculations given below. Thejump distance c5 from the grain into the boundary has beenset equal to one half of the interatomic distance.

Many years ago, Aust and Rutter4 and Gordon andVandermeer3 carried out very careful experiments on graingrowth in pure lead and in pure aluminium. Their resultsare shown in Figs. 2 and 3. Aust and Rutter analysed theeffect of grain orientation on grain boundary migration,as shown in Fig. 2. In this section of the paper calculationswill only be performed for a large angle (8 = 45°) betweenthe two grains. In the next section the effect of grainorientation will be discussed.

The results of Gordon and Vandermeer (Fig. 3) showalso the effect of small additions of copper on the graingrowth process. This will also be discussed below. In Fig. 3,the grain growth calculated using equation (9) is shown asa dotted line. The thermodynamic and physical parametersused in the calculations are given in Table 1. As can beseen, there is fairly good agreement between the calculatedand measured data. It should be pointed out that theentropy of the grain boundary has been selected to obtaingood agreement between experiment and theory. Thereason is that no value of this parameter was found in the

This equation is very similar to the typical expressionsderived from absolute reaction rate theory, with theaddition of the last term on the right hand side. This termcan be included in the diffusion coefficient and, thus, theexpression will be exactly the same as the typical relation-ship.2,3 However, the term cannot be neglected; it willinfluence both the activation energy and the pre-exponentialterm, as will be discussed below.

The boundary free energy per unit area O"B can bedivided into one energy term and one entropy term in thefollowing way

O"B = O"(u) - TO"(s) . (8)

whereO"(u) = internal grain boundary energyO"(s) = internal grain boundary entropy

According to Martin and Doherty,14 O"(u) will vary from0·1 to 1·0 J m -2 with typical values of about 0·5 J m -2.The entropy of the grain boundary O"(s) can vary from 0to 3 mJ K -1 m -2. Both O"(u) and O"(s) are dependent onthe crystallographic orientation between the grains on bothsides of the grain boundary.

Putting equation (8) into equation (6) gives

v = Do exp ((J~;m)AG~w exp [~ ( Q + (J(U~ Vm)]<5 RT RT (9)

. (4)

(0)

Distance

(b)

Recrystalli~edGrain

~G~

-----Blew

~~Gm%Cold'worked .grain

ICW-REX = ICW _ ]REX

Distance

Grain boundary free energy a related to free energyof surrounding grains and b in system with coldworked grain on one side and recrystallised grainon other side: see text for definition of symbols

v=; Doexp ~-tr ){exp ( _ ~G;) [exp C~~;W)-1J}. (5)

where Do is the diffusion coefficient and c5 is the diffusionor jump distance from the ordered grain structure to thedisordered structure of the grain boundary. In the presentmodel this distance will be set equal to one half of theinteratomic distance.

The free energy differences for transfer of atoms fromthe cold worked and recrystallised grains to the boundarycan be related to the boundary free energy by equation(1). Using this relation together with equation (5) gives

Do exp ( - RQr) { (. O"B Vm) [ (~G~W). J}v-----c5--- exp - LJRT exp RT -1

(6)

A series expansion of the exponential term inside the

Using the same assumption for the transport of atomsfrom the recrystallised grain to the grain boundary givesthe expression

/HEX=vexp( - R;')[l-exp( - ~G;)J . (3a)

The number of atoms inside the grain boundary must beconstant. This results in a net movement of the grainboundary, described by the difference between equations(3) and (3a)

Materials Science and Technology September 1990 Vol. 6

Fredriksson Mechanism of grain growth in metals 813

o Exp (4)'?o

i 10-7

eClc

'co 10-8

<5

2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8

1000/ToK

3 Comparison between theory and experimental data(From Ref. 3) for recrystallisation of aluminium

300'Co

oo

oo

o

Calculated

o 0

4

x 14

E 12

i P 10o 00, g 8c'§ 6Cl

2

o15 20 25 30 35 40 45

Orientation difference () (degs) (100) tilt in specimen

2 Comparison between theory and experimental data(From Ref. 4) for grain growth in lead

literature; on the other hand, the value selected does fallinto the range of 0-3 mJ K -1 m - 2 given by Martin andDoherty. 14

Effect of grain orientation

Many experimental investigations have been carried outto analyse the effect of orientation between the grains onthe growth process.4 The results of one such experimentare shown in Fig. 2. In the present report these results willbe analysed using the theory derived above. It will beassumed that the variation in growth rate arises from avariation in the surface free energy with the grainorientation.

In several investigations, it has been shown that thesurface tension varies with the grain orientation (see, forinstance, Ref. 9). To give a theoretical treatment ananalytical expression describing the surface free energyvariation must be found. In the present treatment thisexpression will be simplified as much as possible. It willbe assumed that the surface free energy is a sinusoidalfunction of the orientation. Both the internal energy of thesurface free energy and the internal entropy are assumedto be described by the following expressions

(jB(U) = (j~ (sin nO + A) . (10)(jB(S) = (j~ (sin nO + A) . (11)

where A is a factor varying from zero upwards to describethe size of the anisotropy and n is a factor describing theamplitude. By a careful selection of A and n it is found

that the experimental results for lead (Fig. 2) can bedescribed fairly well. In the present treatment, values ofA = 0 and n = 3 were selected. The result of this calculationis shown as a solid line in Fig. 2.

The correlation between the calculated and experimentalresults is very good. Even if only one comparison is madebetween theory and experiment, it -seems to be reasonableto suggest that an anisotropy in the grain boundary freeenergy can explain the observation that the grain growthis dependent on the grain orientation.

It is of further interest to note that the anisotropy inthe entropy term of the grain boundary free energy has astronger effect on the growth than the anisotropy in theinternal energy.

Grain growth in alloys

For alloys the situation will be somewhat different thanfor pure metals. It is well known that a grain boundaryoften has a different composition than the surroundinggrains. The alloy content is often higher in the grainboundary than inside the grain itself. An example is shownin Fig. 4. This figure is taken from a work by Guttman5

and shows the difference in concentration of seleniumbetween a high angle grain boundary and an iron matrix.It can be seen that the selenium content first increasesslowly and almost linearly with the composition in thematrix, then increases rapidly up to a high and almostconstant value. In the discussion below this type of

Table 1 Parameters used in calculations in this study~ 60

~-Te

c: 50o:;::as'- 40'EQ)og 30o>.~ 20"Cc::Jo 10.cc:.(U~ 0 20 40 60 80 100 120 140

Bulk concentration (wt.ppm)

4 Grain boundary segregation in iron alloyed withtellurium and selenium (From Ref. 5)

Lead

5 x 10-5

1·8 x 10-10

2·5 x 10-3

18·3 X 10-6

8·311·5 x 10-9

8 x 10-3

540·257310-4

1002·0oo105

103

Aluminium

5 x 10-5

1·5 x 10-10

2·5 X 10-3

10x10-6

8·311·1 x10-9

16580·5Various10-4

1162·0oo

Parameter

Do, m2 S-1

b,m0"(5), J K-1 m-2

V m3 mol-1R~J mol-1 K-1

.d,mI1GcW J mol-1

Q, kJ ~OI-1

O"(u), J m-2

T,KOgiff, m2 S-1

Qdiff, kJ mol-1kB/REX

A(u)A(5)Q~, J mol-1

OB, J mol-1

Materials Science and Technology September 1990 Vol. 6

814 Fredriksson Mechanism of grain growth in metals

Cold deformeda phase

phase

(0)

GrainBoundary

(0)

GrainBoundary

Composition, % BII

R"0::r~(J)

CD

GrainBoundary

X lew I X B/Rex Conc.

Cold deformeda grain

GrainBoundary

R

(b)

Recrystalliseda grain

o00·;-:J()

CD

6 Free energy curves for grain boundary, cold workedgrain, and recrystallised a phase and concentrationdistribution in grain boundary during recrystallis-ation process

(b)

"0::r~(J)

CD

oen';-:J()CD ,

a free energy curves; b concentration distribution

5 Grain boundary and solid rJ. phase at equilibrium

(13)

solubility curve will be used to explain the effect of alloyingelements on the grain growth process.

The effect of a higher concentration in grain boundarieson the grain growth will be a 'solute drag effect', discussedfirst by Lucke and Detert. 6 The same authors also gave atheoretical treatment of the effect of enrichment of solutesin the grain boundary on the grain growth. They assumedthat the movement of the boundary was determined bythe diffusion of solute elements behind the boundary.Cahn 7 and, later, Lucke and Stuwe15 have given a moresophisticated treatment of the solute drag effect on graingrowth. At present, it must be concluded that the solutedrag theory can provide an acceptable interpretation ofthe effects of impurities on the grain growth process.

The theoretical treatment presented in the last sectionwill now be used to analyse the effect of grain boundarysegregation on the grain growth. The discussion will belimited to binary alloys. Here it will be assumed that thesituation at equilibrium is as described in Fig. 4. The grainboundary is considered to be a thin layer of a separatephase termed boundary, The free energy diagram for analloy with a solid phase and a boundary phase will containtwo different curves, as shown in Fig. 5. This type ofdiagram can be used to determine the composition of eachphase. The equilibrium compositions of the solid and theboundary are given by drawing tangents to the free energycurve for the interphase parallel to each other. The twotangents are shown in Fig. Sa. The composition profile inthe two phases at equilibrium are shown in Fig. 5b.

The situation in which a recrystallised grain and a colddeformed grain are present at the same time will now beanalysed. The cold deformed grain has its own free energycurve, and the fre~ energy diagrams can be describedaccording to Fig. 6.

This figure shows that there is a driving force for themovement of the boundary. However, there is also asegregation of the alloying element to the grain boundary.For the boundary to move there must exist a driving forcefor the mass transport. This is shown as a concentrationgradient inside the boundary in Fig. 6b.

Figure 6 is drawn in such a way that the concentrationis higher in the boundary at the border with the

recrystallised grain than at the border with the colddeformed grain. Because of this, there will be a masstransport over the interface when the interface is moving.The concentration in the cold worked grain and in therecrystallised grain is the same and is equal to the originalcomposition.

The concentration at the interface between the boundaryand the recrystallised grain is given by the parallel tangentmethod. This is used to obtain the maximum driving forcefor atom transfer from the boundary to the recrystallisedgrain. This transition can be described by the absolutereaction rate theory, written in the following way

Do exp ( - RQT) [ (L\G~REX)JVREX = ------''-----'-- 1 - exp - --- (12)

c5 RTThe concentration at the interface between the boundaryand the cold worked grain cannot be given by the paralleltangent method, since there will be no driving force fordiffusion of alloying elements over the boundary. Thetransition of atoms from the cold worked grain to thegrain boundary is also given by the absolute reaction ratetheory and will be written as

vew = Do exp ~ -iT )[1-exp ( _ ~~~w)]Owirtg to the fact that the thickness of the grain boundarydoes not change, the movement of the boundary will begiven by the difference between equations (12) and (13),th us obtaining

VB = VREX _ vCw

. (14)

Materials Science and Technology September 1990 Vol. 6

This equation is similar to equation (9) used for themovement of grain boundaries in pure metals.

It was pointed out above that there must be a masstransport over the grain boundary. This mass transportcan be described by Fick's first law

D diff XBjREX - XBjCWV =A xBjREX _ Xo . (15)

whereDdiff = diffusion coefficient inside grain boundary

L1 =width of grain boundaryXBjREX = composition in boundary at side against

recrystallised grainXBjCW = composition in boundary at side against

cold worked grainXo = composition of a phase

From Fig. 6 it can be seen that ..1.G~cw can be dividedinto two parts, one related to the diffusion inside theboundary 1\Gdiff and one related to the cold work energyin relation to the boundary free energy.

To calculate XBjCW and XBjREX the boundary must betreated as a typical phase with well known thermodynamicproperties. This treatment is given in the Appendix.

Following the thermodynamic treatment of the boundary,the driving force for diffusion can be shown to be

1\Gdiff = RT(XBjREX - XBjCW) . (16)

This relation is found by using the chemical potential forcomponent A and assuming that the concentration differ-ences are low and that Raoult's law is valid. By puttingequation (16) into equation (15), the driving force can berelated to the rate of movement of the boundary as follows

..1.G~ff = RT~(XBjREX - Xo) . (17)Ddiff

The composition XBjREX is related to the original compos-ition by the partition coefficient kBjREX defined by equation(34) in the Appendix. Substituting in equation (17) gives

..1.G~ff = RT~Xo(kBjREX -1) . (18)Ddiff

In equation (14), ..1.G~cw can now be related to ..1.G~REXand ..1.G~ff by the following expression

1\G~cw = 1\G~REX - 1\G~w + ..1.G~ff . (19)

Putting equation (19) into equation (14) and performing aseries expansion gives

D (1\G~REX) (..1.G~w- ..1.G~ff)v=Jexp -~ RT . (20)

and relating ..1.Gdiffto the rate of movement of the boundary(equation (17)) gives

(Qdiff)

= Dgiff exp -lff Do x (..1.G~REX + Q) ..1.G~WIv 1\ XO(kBjREX -1) lJ e p RT RT

{Dg

iff

exp ( - ~) D [(..1.G~REX + Q)J}L1XO(kBjREX - 1) + J exp - RT

. (21)

The term ..1.GBjREXcan be related to the boundary freeenergy as follows

..1.GBjREX= "B _ "a. = o"B _ O"a. + RTln (1 XB

) (22)m rA rA rA rA 1_ xa.

where 11 is the chemical potential. Performing a seriesexpansion and using equation (34) gives

..1.G~REX= °lll- 01l~EX + RTXo(l - kBjREX) . (23)

Fredriksson Mechanism of grain growth in metals 815

10

Experimental - •

.000110-7

Tin Concentration7 Comparison between theory and experimental

results (From Ref.4) for grain growth in Pb-Snalloys

Relating this expression to the free energy of the grainboundary gives

l1G~REX = tTo;m + RTXW - kB/REX) • (24)

By combining equations (21) and (24) the movement ofthe grain boundary can be calculated as a function of theconcentration of the alloying element. It is of interest tonote that (J° is the surface free energy for the pure elements,as in equation (1).

It is of further interest to note, from equation (21), thatthe movement of the grain boundary is influenced greatlyby the values of the two terms in the chain bracket belowthe line. Either the first or second term dominates. If thefirst is much larger than the second, the result is anexpression similar to that given for pure metals (compareequation (9)). Thus, the entire movement is controlled bythe rate of movement of atoms into and out of the grainboundary. If the second term below the line is the dominantterm, the movement is determined by the diffusion ofelements inside the grain boundary.

Figures 3 and 7 show measurements of the rate ofmovement of the grain boundary in aluminium3 and lead,4respectively. To these elements certain amounts of copperand tin, respectively, have been added. The amounts addedare shown in the figures.

The experimentally measured data shown in Figs. 3 and7 have been compared with calculated values fromequations (21) and (24). In the calculations the physicalparameters given in Table 1 have been used. As a firstattempt, it was assumed that the partition coefficient forthe element between the grain and the grain boundary waslow and constant, i.e. that the region in question lies onthe lower part of the curve shown in Fig. 4a. It wasassumed that k was equal to 2 for both alloys shown inFigs. 3 and 7.

Good agreement was obtained between the calculatedand experimental results for aluminium alloyed with copper .

Materials Science and Technology September 1990 Vol. 6

816 Fredriksson Mechanism of grain growth in metals

For lead alloyed with tin, the agreementis very good fora tin content up to 10-5 wt-%. Above this value there is achange in the slope of the curve for the grain growth rate,for both the experimental and calculated results. However,the change in slope given by the theoryis not as great asthe experimental change. To attempt to explain thisdeviation it will be assumed that the partition coefficientfor tin in lead follows a similar curve as that derived fromFig. 4 for selenium in iron. By assuming that the thermo-dynamic properties according to the regular solutionmodel can be used for the matrix as well as for thegrain boundary, the k value is found by the expression

(niX XiX _ nB XB)k = ko exp R T ' . (25)

whereniX = regular solution interaction constant in matrixnB = regular solution interaction constant in grain

boundaryBy a careful selection of niX and nB (values given inTable 1), the experimental data shown in Fig. 7 can beexplained by the theory, using equations (21)-(25). Theresults of these calculations are shown by the dotted linein Fig. 7.

It should be noted that the activation energy selectedfor diffusion has a value much higher than the activationenergy for selfdiffusion in grain boundaries. The reason isthat the copper and lead atoms are much larger than thealuminium and tin atoms, respectively. If the atoms ofthe alloying element could have been the same size as thematrix atoms, then the activation energy probably wouldhave been suitable for grain boundary diffusion. Unfortun-ately, there is too little information about diffusion ingrain boundaries available, but it seems reasonable tomake this assumption.

Discussion

The absolute reaction rate theory is often used to describedifferent types of thermally activated phase transformations.It has also been used to describe grain growth in metals.A comparison between experimental results of grain growthin metals and the absolute reaction rate theory shows thatthe grain growth is described qualitatively by this theory.However, the comparison shows that the pre-exponentialterm and the activation energy derived from the experi-mental results do not agree with the theoretical values.The activation energy derived from the experiments inpure metals is often higher than that expected from thetheory and the pre-exponential term is often lower. Also,the absolute reaction theory cannot describe the effect ofgrain orientation on the growth rate.

By considering the grain boundary as a high energyregion over which the atoms should pass, and by consider-ing this region to be described by standard thermodynamiclaws, both the activation energy and the pre-exponentialterm will be changed. The activation energy will beincreased with the enthalpy value of the boundary freeenergy and the pre-exponential term will be multiplied bythe exponential value of the entropy term of the boundaryfree energy. By considering the grain boundary as a highenergy region over which the atoms should pass, it isshown that better agreement is obtained between theexperimental data and the theory for a thermally activatedprocess. It is also easy to describe the difference in graingrowth measured for different grain orientations.

Grain growth in alloys has also been described theoretic-ally 'as a thermally activated process combined with

Materials Science and Technology September 1990 Vol. 6

diffusion of alloying elements. Also, these theories6,7 candescribe the grain growth process, qualitatively. However,here the activation energy for the process also will deviatefrom the value derived Irom, the experimental data. Byreconsidering the grain boundary as a high energy regionwith its .,own composition, it will'.be possible'to describethe experimental data quantitatively. However, owing tothe lack of equilibrium values 'of the grain boundarycomposition, some assumptions must be made. Theseassumptions can be based on knowledge found in. theliterature. The diffusion coefficient of any alloying elementacross a grain boundary has not been measured yet. Inthe literature, only diffusion along the grain boundarieshas been measured. Therefore, it is difficult to select thecorrect value of activation energy for this diffusion process.The activation energy for diffusion across the boundary isprobably less than the activation energy along the grainboundaries. However, here also the activation energy musthave values between the value for selfdiffusion of the maincomponent and the value for diffusion of the alloyingelements along the grain boundary. Based on existingknowledge (from the literature) of grain boundary segreg-ation and of the activation energy for diffusion in theboundary, reasonable chemical and physical constants canbe selected. This has been carried out in the presenttheoretical treatment and fairly good agreement was foundbetween theory and experimental data. Not only has aqualitative understanding of the grain growth process beenfound, but a more quantitative description has beenpossible.

Concluding remarks

The grain growth process during the recrystallisation ofmetals is affected strongly by the grain boundaries. A newtheory has been, developed to describe the effect of thegrain ,boundaries. The theory is based on the assumptionthat the boundary is treated as a separate phase with itsown free energy. This ~ssumption, in combination with theabsolute reaction rate theory for transfer of atoms fromthe grains to the boundary, describes the grain growthprocess very well. The effect of grain orientation and theeffect of grain boundary segregation of alloying elementshave been described ·'by the theory. The theory could leadtonewdevelopments in the description of phase transforma-tions and could probably be used to describe otherphenomena, such as diffusionless transformation and den-dritic growth d4ringsolidification of metals.

Furthermore,' the theory can be used to describe thethermodynamic properties of grain boundaries and phaseboundaries, from grain growth as well as from phasetransformation studies.

Acknowledgment

Professor M. Hillert is gratefully acknowledged for manystimulating discussions through the years.

References

1. P. CETTERILL and P. R. MOULD: 'Recrystallisation and graingrowth in metals'; 1976, London, Surrey University Press.

2. J. W. CAHN, W. B. HILLING, and G. w. SEERS: Acta Metall.,1964, 12, 1421.

3. P. GORDON and R. A. VANDERMEER: Trans. A/ME, 1962, 224,917.

Fredriksson Mechanism of grain growth in metals 817

(29)

· (30)

· (31)

4. K. T. AUST and J. W. RUTTER: Trans. A/ME, 1956, 215, 119.5. M. GUTTMAN: Metall. Trans., 1977, SA, 1383.6. K. LUCKE and K. DETERT: Acta Metall., 1957,5, 628.7. J. w. CAHN: Acta Metall., 1962, 10, 789.8. M. HILLERT: 'Mechanism of phase transformation in crystalline

solids', 231; 1969, London, The Institute of Metals.9. c. S. SMITH: 'Metal interfaces'; 1952, Cleveland, OR, ASM.

10. D. MCLEAN: 'Grain boundaries in metals'; 1957, Oxford,Clarendon Press.

11. K. T. AUST and B. CHALMERS: Proc. R. Soc., 1950,A201, 210.12. J. M. BLAKELY: 'Introduction to the properties of crystal

surfaces'; 1973, Oxford, Pergamon Press.13. M. HILLERT: Spring Meeting, University of Pittsburgh, PA,

May 1974, The Metallurgical Society of AIME.14. J. W. MARTIN and R. D. DOHERTY: 'Stability of microstructure

in metallic systems'; 1976, Cambridge, Cambridge UniversityPress.

15. K. LUCKE and H. STUWE: 'Recovery and recrystallisation ofmetals', 131; 1963, New York, Interscience.

where Y is the activity coefficient. Using this relation andassuming that Henry's law and Raoult's law are valid ineach phase and that the concentration of element B is verylow gives

XB = xa Y~YB x [(OJ1~ - °J1A) + eJ1~ - °J1B)]B BYBYA e P RT

One such equation is obtained for the cold workedstructure aildone for the recrystallised structure.

The values of °J1A and °J1B in the cold worked structureare different from those in the recrystallised structure. Inthe present model, it will be assumed that the difference isequal to the cold work energy. It will be assumed that thechemical potential for both A and B increases with thesame amount of cold work energy. Thus

°J1Xw = °J1~EX + ~G~w

oJ1~w = °J1~EX + ~G~w

. (26)

At equilibrium, the concentrations in the two equationswill be equal.

(32)

· (34)

· (35)

. (33)

REXX~ = XBkBjREX

CWX~ = XBkBja

Furthermore, it will be assumed that neither the activitycoefficients nor the chemical potential of the boundarychange with the amount of cold work. Using thisassumption gives

(

~O +~o )REXXB = Xa A' exp J1A

RTJ1B

CWXB = x· AU exp (11°/lAR+TI1°/lB)

whereB (J.

A' = YAYBYBY~

A" Y~YB (~G~W)=--exp --YBY~ . RT

~oJ1A = change in chemical potential of element A

~ °J1B = change in chemical potential of element B

The two ,"terms A' exp [(~oJ1A + ~oJ1B)/RT] andA" exp [(~o/lA + ~o/lB)/RT] can be defined as partitioncoefficients. Thus, the following expressions are obtained

. (27)

J1m = J1c:n+ RTln am

where

J1m = chemical potential of element mJ1c:n= chemical potential of element m in its standard

stateam = activity of element m

Following the principle given by Hillert9 and, as shown inFig. Ib, the parallel tangent method gives

Appendix

(28)

It will be assumed that, according to Hillert,9 thefree energy of the boundary is described by typicalthermodynamic equations.

The chemical potential of each element in any phasecan .be written as follows

where the definitions. of the chemical potentials are shownin Fig. 5. Using the relations for the chemical potentials ofeach element and combining these with equation (27) gives

XBX(J. B aRTln ~ = 0J1B - 0J1a + 0J1(J._ 0J1B + RTln YAYB

X ~X B A A B B YA Y~

Materials Science and Technology September 1990 Vol. 6

First Announcement and Call for Papers

MATERIALS FOR COMBINED-CYCLE POWER PLANT

10-12 June 1991 Sheffield, UK

Sponsored and organised by the Materials Science Division of The Institute of Metals

Conference structureThe programme will consist of three days of plenary technical sessions. Each session will be headed byinternational keynote papers followed by contributions drawn from a call for papers. Posters may also bepresented.

ScopeA very effective method of improving the efficiency of electricity production, is the employment of a gasturbine, with the hot exhaust gases being implemented to raise steam for a steam turbine, thus producing a'combined cycle' generation unit. This practice permits the use of a variety of fuels, such as gas, liquid fuels,or coal (via gasification, pressurised combustion, or in the form of direct coal firing). The recent tendency bygovernments to remove barriers preventing the use of natural gas to generate electricity, coupled with theshort construction time of gas fired combined cycle plant, had made such plant a very attractive proposal.There is indeed an active building programme all over the world. The use of natural gas also allows acid gasemission limits to be readily met, and much lower carbon dioxide emissions per unit of electricity generatedthan coal or oil fired systems.

The technology of combined cycle plant is developing continually, with materials development and engineeringtaking the lead at each stage. This is particularly visible with the introduction of larger gas turbines withincreased firing temperatures and also with sophisticated blade cooling manufacturing techniques. Improvedsteels for rotors of single cylinder steam turbines and materials for coal fired systems in boilers, turbines, anddirect fired heat engines are also under active development.

Call for papersThis three day international conference will not only cover the materials development and engineering forthe components of a range of combined cycle plant with contributions from a call for papers, but will alsoinclude several invited papers to set the materials papers into context of plant efficiency, design, and operatingexperience.

Papers for oral or poster presentation are invited on the above aspects of materials development andengineering for combined cycle plant. Prospective authors should send a 200 word abstract as soon aspossible.

PublicationFull texts of all papers presented at the conference will be published in a conference preprint to be issued todelegates on arrival at the meeting.

Technical visits, social events, and accompanying persons programmeIt is planned to run technical visits relevant to the major themes of the conference at the end of the meeting.There will be two receptions during the conference. It is intended to hold a conference banquet at the CutlersHall in Sheffield. A social programme for accompanying persons will also be arranged.

SponsorshipCompanies interested in taking advantage of advertising and sponsorship at the conference will have theopportunity to do so.

Further informationProspective authors, delegates, and companies interested in advertising/sponsorship should contact:

Ms J. UptonConference Department (C112)The Institute of Metals1 Carlton House TerraceLondonSW1Y 5DBTel: 071-8394071; Fax: 071-839 3576; Telex: 8814813