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A kinetic model of non-equilibrium grain-boundary segregation
0001-6160~89 $3.00 + 0.00 Copyright 0 1989 Pergamon Press plc
Acta metali. Vol. 37, No. 9, pp. 2499-2506, 1989 Printed in Great Britain. All rights reserved
A KINETIC MODEL OF NON-EQUILIBRIUM GRAIN-BOUNDARY SEGREGATION XU TINGDONG and SONG SHENIUA Department of Mate~als Science and Engin~~ng, Wuhan Iron and Steel University, Wuhan, Hubei, China (Received 28 November 1988)
Abstract-An isothermal kinetics of non-equilibrium grain-boundary segregation was developed both for segregation processes and for desegregation processes within a phenomenological theory. An effective time concept of a cooling process was discussed. On these bases, a simple and accurate method for evaluation of the levels of non-equilibrium grain-boundary segregation during cooling was proposed. According to the method, we have calculated the levels of non-equilibrium segregation to austenite or prior austenite grain boundaries for boron in Fe-3O%Ni alloy, aluminium in Inconel600, chromium in 2.25%Crl%Mo steel and tin in 2.25%Crl %MoO.OB%Sn steel in different experimental conditions respectively. Results calculated from the kinetic model in this paper are in satisfactory agreement with the observed data of experimental measurements for all the above samples. R&au&---Dans le cadre dune theorie ph~nom~nolo~que, nous d~veloppons une cinetique isotherme de la segregation inter~anulaire hors d’equilibre, $ la fois pour les mecanismes de la s&egation et pour ceux de la desegregation. Nous discutons un concept de temps efficace pour le m&&me de refroidissement. Sur ces bases, nous proposons une methode simple et precise pour &valuer les taux de segregation intergranulaire hors d’equilibre au tours du refroidissement. Selon cette methode, nous avons calcule les taux de segregation hors d’equilibre vers les joints de grains d’austenite actuels ou anttrieurs dans les cas suivants: bore dans un alliage de fer $ 30% de nickel, aluminium dans l’Inconel600, chrome dans un acier ri 2,25% de chrome at 1% de molybdbne, et Ctain dans un acier B 2,25% ce chrome, 1% de molybd6ne et 0,08% d’ktain, respectivement dans des conditions expkrimentales diffkentes. Les rbsultats calculkes A partir du modele cinetique propose dans cet article sont en accord satisfaisant avec les resultats exp&imentaux obtenus pour tous les echantillons que nous venons de titer. Z~~e~a~u~-Innerhalb einer ph~nomenologi~hen Theorie wird eine Nichtgleich~ich~ theorie der isothermen Kinetik der Komgren~e~egation sowohl fiir An-wie such fur Abreicherung entwickelt. Hierzu wird ein Konzcpt der effektiven Zeit eines Abk~prozes~s diskutiert. Auf dieser Basis wird eine einfache und genaue Methode fur die Auswertung der Nichtgleichgewichtssegregation an Komgrezen wahrend des Abkiihlens vorgeschlagen. Mit dieser Methode werden die entsprechenden Niveaus der Segregation fur Bor an austenitischen oder vorher austenitischen Komgrenzen in der Legierung Fe-30%Ni, fur Aiuminium in Inconel 600, fur Chrom im Stahl 2,25%Crl%Mo und fiir Zinn im Stahl 2,25%Crl %MoO,OB%Sn unter verschiedenen experimentellen Bedingungen berechnet. Die erhaltenen Ergebnisse stimmen mit den Beobachtungen an allen oben angegebenen Materialien zufriedenstellend iiberein. 1. INTRODUCTION of the grain-boundary
of
quantity of recombined complexes of solute atoms and vacancies exists. These three parts: solute atom,
solute has attracted considerable attention in recent years, since it is widely recognized that such solute segregations not only control many of the mechanical, chemical and electronic properties of technologitally important materials but also induce grain boundary structural transformations [l]. The solute segregation to grain boundaries is classified into equilibrium and non-equilibrium segregation. The works of McLean [2] on the thermodynamics and isothermal kinetics of equilibrium segregation founded an important basis for understanding the properties of equilibrium segregation. In the late 1960s a non-equilibrium grain-boundary segregation theory with respect to the solute atoms was proposed by Aust ef al. [3] and Anthony 14). It is suggested that the mechanism of segregation be based on an equilibrium in which a sufficient
vacancy and their recombined complex, are in equilibrium with each other. A sample properly maintained at a solid dissolution treatment temperature will, when cooled to a certain lower tem~ratu~, exhibit a loss of vacancies along the grain boundaries, whereby it attains the equilibrium vacancy concentrations at low temperatures. The decrease in the vacancy concentration causes the dissociation of the recombination complexes into vacancies and solute atoms. This in turn gives rise to the decrease in the recombined complex concentration near the grain boundary. Meanwhile, in regions remote from the grain boundary, where no other vacancy traps are present, vacancies would recombine with solute atoms and so the vacancy concentration is reduced. This makes the recombination complex concentration increase in the regions remote from the grain
The subject
segregation
2499
2500
XU TINGDONG and SONG SHENHUA:
boundary. Consequently, a concentration gradient appears between the grain boundary and regions beyond it. The gradient of complexes drives the recombination complexes to diffuse from regions remote from the grain boundary to the grain boundary. This diffusion causes excessive solute atoms to concentrate in the vicinity of grain boundary and results in non-equilib~um grainboundary segregation. Faulkner [5] evaluated the maximum non-equilibrium segregation level for the samples cooled from a higher temperature to a lower temperature, and took for granted that this maximum level was equal to the segregation level found in the process of segregation between the two temperatures at any cooling rate. Such a postulate is obviously a coarse one. Doig and Flewitt [6] gave a kinetic analysis for the non-equilibrium grain-boundary segregation of solute atoms. In their analysis, no an equation of diffusion suitable for the diffusion of recombined complexes was found and so a number of assumptions, which differ from the actual diffusion processes in non-equilibrium segregation and their experimental conditions were made. Xu Tingdong [;1 proposed an isothermal kinetic model only for the segregation processes of non-equilibrium segregation and suggested a critical time concept of non-equilibrium segregation and an effective time concept of a cooling processes [S, 91. In the present work, we further above works to propose a isothermal kinetic model of non-equilibirum segregation both for segregation processes and for desegregation processes. Using the results of the above concepts we suggest a simple and accurate method evaluating the levels of non-equilib~um segregation during cooling. Finally data calculated by the author’s model show a reasonable agreement with the observed experimental data for boron in Fe-30%Ni alloy, aluminium in Inconel 600, chromium in 2.25%Crl%Mo steel and tin in 2.25%Crl%MoO.O8%Sn steel. 2. MODEL 2.1. Segregation and desegregation The results of experiments [8,9] on non-equilibrium grain-boundary segregations have shown that for a sample being cooled quickly enough from a higher solution temperature to a certain lower temperature and then held at this lower temperature, there is a critical time t,, to which when the holding time of the sample at this lower temperature is equal, the level of segregation will be maximum. If the holding time of the sample at this lower temperature is shorter than the critical time, the levels of segregation at grain boundaries will increase with increasing holding time. If the holding time is longer than the critical time, the levels of segregation will decrease with increasing holding time. It is clear from the above experimental results that when holding time is shorter than the critical time, the segregation of the complexes from the center to the grain boundary will be dominant,
GRAIN-BOUNDARY SEGREGATION called segregation processes, and when holding time is longer than the critical time, the diffusion of solute atoms from grain boundaries to the center will be dominant, called desegregation processes. The critical time formula at a certain temperature T was given in Ref [6,7] as t,(T) = RZ In [(D,/Q)/4S (D, - Q)]
(1)
where 6 is the critical time constant, according to the method suggested in [9], S = 11.5 can be calculated and will be adopted in the following calculation of the present work. Di and D, are respectively the diffusion coefficients of solute atom and complex of solute atom and vacancy at temperature T, and R is the grain size. The fact that both segregation process and desegregation process exist is the most considerable characteristic for non-equilibrium grain-boundary segregations. It is obvious that different physical processes and essentials are holding for these two processes of segregation so it is necessary that different isothermal kinetic models should be founded for these two processes. 2.2. The isothermal kinetics for segregation processes For solid solution, it is postulated that there is an equilibrium between solute atom I, vacancy Y and recombined complex C formed by the former two components z+v=c where it is assumed that one recombined complex is made up of one solute atom and one vacancy. As discussed in [ii, 141,the following equation is held for this equilbrium Cc/C, = & expK& - Q)lkTl
(2)
where K, is a geometric constant, i$, is the energy of formation of the recombined complex, Ef is the energy of formation of vacancy and E,, is, in generai, smaller than Ef, k is Boltzmann’s constant C, is the concentration of recombined complexes, C, is the concentration of solute atoms. It was assumed here that equation (2) is only heId in any of the localized regions even though it is not necessary for the whole system to be in a system of equilib~um. This assumption is justified because this reaction is only concerned with short range diffusion in small regions [lo]. As has been discussed in [S, 71, when a sample is cooled from solution-treatment temperature, TO, to any lower temperature, T, the maximum concentration of non~quilibrium segregation induced during the cooling at grain boundary, C:(T), is given by
expW& - WWJ
- [& - W/W1
(3)
where C, is the concentration of solute atoms within the grain. It is obvious that the maximum concentration, C;(T), in equation (2) is only dependent on the
XU TINGDUNG
and SONG SHENHUA:
difference between solution-treatment temperature T, and the lower temperature T and is independent of the cooling rate from T, to T. If the solid solutiontreatment temperature of samples is thought to be a constant, Cr(T) will only be a function of temperature T. Equation (2) is therefore an ~mpo~ant thermodynamic equation describing the non-equilibrium grain-boundary segregation level. It can be seen in the following discussion, that on the basis of it, one can establish the isothermal kinetic relationship of non-equilibrium segregation. Suppose that the sample is cooled so quickly from temperature Tj down to 7;, 1(T, > Tj+ 1) that no mass transfer occurs in the specimen during cooling. Then the temperature is maintained at Ti+ 1for a period of time, during which time the recombined complexes will diffuse from within the grain toward the grain boundary, and the solute atoms are co~~ntrated at the grain boundary. Let r, I< Ti < To and denote C:(Tj)/C, by xi, and C:(T,, ,),lC, by ai+ I, obvious& a,, 1 > ai. As the width of the concentrated layer of solute atoms d, is relatively small as compared with the grain size, and ai + f x d is very small as compared with the grain diameter, the solute atoms concentrated at the grain boundary will be almost entirely furnished from the very narrow region in the vicinity of the grain boundary and the concentration of solute atoms within the grain remains a constant one C,. With the above conditions being met, the diffusion of recombined complexes towards the grain boundary can be simpli~ed into a steady linear flow of recombined complexes into the grain boundary in a semi-infinite medium. The corresponding di~usion equation for combined complexes is, therefore
where D, is the coefhcient of diffusion of the recombined complex in the matrix and Cc is the concentration of recombined complexes. In order to get a diffusion equation which is relative to the concentration of solute atoms, C,, substituting equation (2) into equation (4) gives (5) L),a2c/axr = t?C~at where C = C, is the concentration of solute atoms and r>, still is the coefficient of diffusion of the recombined complexes. Evidently equation (5) is an equation of diffusion of a special form, in which the coefficient of diffusion is concerned with the recombined complexes while the concentration is concerned with the solute atoms. It properly depicts the diffusion of recombined complexes dragging the solute atoms toward the grain boundary, causing the difference in concentration of solute atoms in different parts of the crystal. In view of the extremely small thickness of the iayer at the grain boundary where the concentration gradient can be neglected, one can imagine an interface between the grain interior and boundary concen-
GRAIN-BCKJNDARY
SEGREGATION
trated layer where the concentration
2501
is
C = C,(t)/%+ I where Cb(t) is the concentration of the concentrated layer when holding time of sample at temperature T,,i is equal to t, and will change with the holding time at the constant temperature. For simphcity in computation, the interface between the grain interior and the concentrated layer at x = 0 is chosen. According to the First Law of Diffusion and the Principle of Mass Conservation, the interface should suffice the following condition (C),=0= ~,(ac/a.~),,,
C,(l)l%,, = (dP)*(X,(t)/dr)
=(1/2)~i.+]~d~(iiCjat),=,
(6)
where d is the width of the concentrated layer and the factor l/2 is related to the fact that at both sides of grain bounda~es the solute atoms diffuse towards the grain boundaries. Assuming equation (6) to be the boundary condition, the solution of equation (5) is given (see Appendix 1) &(t) = C?(T,, !) - C;(a,,, x exp(4Dtjaf+, Rearranging
--a,)
. d2)erfc(2(Dt)i~2a,+ ,+d].
we obtain
KG(f) - GX37)l/[GX?;+
I) - GVi)3
= 1 - exp(4Dt/af+ 1~d2)“erfc[2{DI)“2iaj, ,-d]
(7)
where
- Jo where D = D,. Equation (7) is an isothermal kinetic relationship for non-equilibrium segregation for the segregation processes. It describes the non-equilibrium segregation concentration Cbft) of the solute atoms at the grain boundary as a function of the holding time f at the constant temperature when I < t, i.e. the segregation process of the complexes to the grain bounda~es form the center is dominant. 23. 7% isofkmaf &esseS
kinetics for desegregation pru-
When the holding time at temperature Ti+ i is equal to the critical time at this temperature, t,( T,, , ), the solute concentration in the concentrated layers along the grain boundaries, Cb[t, (T, + , )] can be obtained by substituting t,(T,+ ,> into equation (7). So allowing for the concentration gradient in the concentrated layers can be neglected as above, we can assume that the concentration profile of solute atoms near the grain boundaries is the same as those shown in Fig. 1, for the sampies held for the period being equal
XU TINGDGNG
2502
-d/2
0
and SONG SHENHUA:
+d/Z
X
GB Fig. I. The concentration profiles near the grain boundaries as the holding times are equal to the critical time I,. t,, fz (f, < t, < t2) at T,, , .
to the critical time t, at the temperature q+, {i.e. when (-d/2)(+d/2),C=C,f.Herex=O is put on the center of concentrated layer and the orientation of x is normal to the grain boundary. It is clear that the concentration distribution of solute atoms is a jump discontinuous function of x at x = -d/2 and x = +-d/2 when held for the time being equal to critical time at temperature Ti+, . As assumed in Section 2.2, the width of the concentrated layers along grain boundary, d, is relatively so small as compared with the grain size that interior region of the grain could be thought of as a semi-infinite medium. When the holding time at temperature Ti+ 1 is longer than the critical time of this temperature, solute atoms will diffuse from the concentrated layer near the grain boundary to the center of grain to make the concentration of concentrated layer decreased and desegregation will be dominant. An indication of the imaged concentration profiles, quantified for different time, r, , f, , f2, during desegregation process is shown in Fig. 1. Using the error solution of diffusion equation to describe the diffusion process, the concentration of solute atoms as a function of x and time t can be obtained as following (see Appendix 2)
CL&- 4) =
cg+ (1/2)(C~(#~) - cg,
x (erf[(x + d/2)/[4D,(t - t,)]‘“] -erf[(x
- d/2)/[4D,(r - t,)]“2)
(8)
where t is the isothermal holding time at T,, ,, t, = tc(I;:, ,), Di is the diffusion coefficient of solute atoms in the matrix. Evidently, equation (8) is only concerned with desegregation and not segregation Thus the condition t > t, is necessitated for equation (8). For any sample held isothermally for the time longer than the critical time r,(Tj+ r), the level of non-eq~lib~um segregation at grain ~unda~es can be obtained by making x = 0 in equation (8)
C,(r)=C*+f[C,(t,)-C,l x {erf[d/2/[4D,(t - r,)J’/2] -erf[-d/2/[@/2/[4D,(t
- tJin]).
(9)
Equation (9) is an isothermal kinetic ~Iations~p for desegregation processes of non-equilibrium segrega-
GRAIN-BOUNDARY
SEGREGATION
tion. It describes the non-equilibrium segregation concentration, C&(t), of the solute atoms at grain boundary as a function of the holding time t at constant temperature when t > r,, i.e. the diffusion process of solute atoms from grain boundaries to center is dominant. 2.4. A method for c&dating the eon-equilibria ~egreg~~io~ level for ~on~~~o~ cooling For most of the experiments and the practical problems about non-equilibrium grain-boundary segregation, segregations will always occur during the continuous cooling. It is considerable importance how to calculate the non-equilibrium segregation levels for continuous cooling. In this section, a simple and accurate method to calculate the non-equilibrium segregation levels during cooling will be suggested. As discussed in [7,8], any continuous cooling curve for a sample can be replaced by a correspondent stepped curve, each step of which was formed by horizontal and vertical segments so as to calculate an effective time at some temperature for this cooling sample. The effective time formula of the stepped curve consisting of n steps at temperature T was given by t,(T)=
f: tjeXp[-EA(Ti=i
Tj)/RTT,]
(10)
where EA is the activation energy for diffusion of complexes in the matrix and it is assumed here that E,isgivenbytheaverageofthevaluesofthevacancy movement energy and the solute atom diffusion activation energy. t, and T, are respectively isothermal holding time and temperature at the ith step of the stepped curve. When the chosen steps are small enough, the effective time of the stepped curve at some temperature will accurately enough be equal to the effective time of the continuous cooling at same temperature. It is expected that the effect of diffusion of solute atoms (or complexes) during the continuous cooling is the same as that of the sample held at a constant temperature for the same time as the effective time of the continuous cooling at the same temperature. This means it is possible that calculating segregation levels of solute atoms (or complexes) during continuous cooling can be changed into calculating segregation levels during isothermal holding processes by using the effective time concept of the continuous cooling. For any sample which was cooled continuously from a higher temperature to a lower temperature, we can get the effective time corresponding to its continuous cooling curve at some temperature as above then compares it with the critical time of that sampfe at the same temperature, which can be calculated from equation (I), and the segregation levels of the sample can be calculated by substituting the effective time into the equation (7), which describes the segregation processes, if it is shorter than the cortical time of the same temperature and by substituting the effective time into equation (9), which describes the desegre-
XU TINGDONG
and SONG SHENHUA:
GRAIN-BOUNDARY
SEGREGATION
2503
gation processes, if it is longer than the critical time. It should be noted that this method is suitable for both non-equilibirum and equilibrium grain boundary segregation. In the situation of equilibrium segregation we only substitute the effective time of continuous cooling into the equation of the isothermal kinetics of equilibrium segregation suggested by McLean [2].
minium, chromium and tin segregations in their respective experimental materials were measured experimentally by the newly developed Scanning Transmission Electron Microscope (STEM). This technique has a spatial resolution of about from 1000 to 1500 A [23,22].
3. A COMPARISON OF CALCULATED RESULTS AND OBSERVED DATA
A number of authors have recently reported their findings on non-equilibrium grain-boundary segregations on the basis of the mechanism of the formation of recombined complexes of vacancies and solute atoms, and the diffusion toward the grain-boundary during cooling. Also, many different analyses have been introduced [5-71. The defects in the kinetic analysis described in [5,6] have been discussed in [7]. Here a new kinetic model both for the segregation processes and desegregation processes of non-equilibrium segregation is proposed. Comparing the present works with the results obtained by us in [7j, this new kinetic model has evidently furthered the kinetic model in [7] to make it not only suitable for the segregation processes but suitable for the desegregation processes as well. It provides a generalized relation for the behavior of non-equilibrium grainboundary segregation in a kinetic manner, just as Mclean’s model does in the case of equilibrium grain-boundary segregation, and especially in this model, the segregation level for the continuously cooled samples can also been calculated by the method of using the effective time concept and isothermal kinetic equations for segregation processes and desegregation processes. This method is more simple and more accurate than the method suggested by us in [7], in which some mathematical approximations were adopted during the derivation of correction factor. It is obvious and considerably important that the model presented here correlates the non-equilibrium segregation levels not only to the solid dissolution treatment temperature but also to the cooling rate of samples both in the case of segregation processes and in the case of desegregation processes of non-equilibrium segregation. As a matter of fact, it has been shown by many experiments that non-equilibrium grin-bounda~ segregation level is closeIy related to both the solid dissolution temperature and the cooling rate the specimen undergoes. Calculations based on the present model for the specimens of experimental alloys in Table 3 indicate that the condition f,< t, is met and at a constant solid dissolution temperature, the lower the cooling rate, the greater the segregation of solute atoms to the grain-boundaries; whereas at the same cooling rate, the higher the solid dissolution treatment temperature, the greater the segregation of solute atoms. Such a conclusion has been verified in the observed data listed in Table 3, adopted from (5, 11, 121. According to this model, it can be calculated that when the effective time for a
Using the above model, we have calculated the level of non-equilibrium segregation at the grain boundaries for boron segregation in Fe-30%Ni alloy [I I], aluminium segregation in Inconel 600 [5], chromium segregation in 2a%Crl%Mo steel [12] and tin segregation in 2p!Crl%Mo O.O8%Sn steel [12] respectively for their different experimental conditions offered by their authors. Fe-30%Ni alloy and Inconel 600 are austenitic alloys. And the martensite transfo~ations for Zi%Crl %Mo steel and 2f%Crl %MoO.O8%Sn steel commence at about 700 k, i.e. -0.4 T, which can be calculated [13]. The detailed compositions for the four alloys are given in Table 1. The data used for theoretical calculation of non-equihbrum segregations for these four systems are given in Table 2. The energy of formation of the recombined complexes, Ebr for boron in Fe-30%Ni alloy is taken from the result of Williams et al. [4] and for aluminium in Inconel600 is taken from the result of Faulkner [5] and for chromium in 2$Krl%Mo steel and tin in Z~~Cri%Mo0.08°~Sn steel are taken from the results of Doig and Flewitt [12]. The diffusion activation energy for vacancies, E, = 2.6 eV [6]. EA can be caIculated by averaging the values of the vacancy diffusion activation energy and solute atom diffusion activation energy. The concentration of solutes in the grain centres, C,, are taken as the analysed concentration for the segregation solute atoms, taken from Table 1 and converted to atomic percent. The grain boundary segregation thickness d = 0.15 pm is chosen in the present work. For all the experimental alloys, experimental conditions, calcufated results and observed data quoted from their respective reference are listed in Table 3. It should be noted that for the two kind of steei, the functional relation of temperature and cooling time was described from [6] by T,= T,.exp(-@)
(11)
where Z’, is solution-treatment temperatures, t is the cooling time, d, = 1 for the chromium and tin in their respective alloys. The fact that the calculated results agree reasonably with the experimental data is obvious. For boron segregation in Fe30%Ni alloy, the observed non-equilibrium segregation levels, C.,, , was measured by the assessment of the Particle Tracking Autoradiography (PTA) [I I]. The PTA has a spatial resolution of about 2.0 pm [20]. The alu-
4.
DISCUSSION
N.O. = Not observed.
Tin in 2$rl%Mo O.O8%Sn
l%MO
980 1423 1323 1223 1423 1323 1223
11SO 1220 lOS@ 1220 llS0 IOSO
Ft3O%Ni Boron in alloy
0.005
0.02
P 0.009
27 100 85 60 100
E
z
:=;
z 290 290
2.2
15.7 2.1
Cr -
-
0.46
0.38 0.53
Mn -
Comt&tion
0.006 -
N -
0.15
14.4 0.1
Ni 29.1
(wt%)
0.61 2.23 2.23 2.2s 0.037 0.037 0.037
0.0052 o.OOs2 o.oos2 o.OOs2 0.0052 0.0052
(a&
[16] [lq
Alloys
15.72 0.53 0.10 0.01s
0.72 z 2Oi 9.46 2.80
l,(Z) (5)
1200.75 3174.87 2377.52 I337.35 610.69 390.85 219.85
87.83 87.83 87.83 87.63 87.83 87.83
r,Wi) (5)
calculated results
2.7 2.6 Lq 720 0.37 (q 1000
1.87 x lo-‘exp(-2.8/H’) 3.36 x lo-‘ yf----/kT)
Al in Inconcl 600
-
0.24 -
Ti 0.033
1.0
0.08 1.0
MO -
0.08
-
Sn -
2.86 2.47 1.81 0.91 0.43
:::s
X:?? 0.13 0.088
0.073 0.098
2.5 3.23 2.54 2.17 3.81 1.30 0
E:: N:O: High Medilun Low
Maguitude (at%) Cl&,) (h < I,) CbU.-~e)(rc~G) C&I
2:55 2.6 [6] 720 OS [12] 1000
3.21 x lo-‘exp(-2.5/H’) 118) S x 10-‘f?x4~~-;.SS/kT) [16]
Cr in 2$rl %Mo
Table 2. Data used for theoretical calculation
0.46
0.11 0.38
Si -
Table 1. Alloy compositions
Table 3. Exlwrima~td ad
[IS] [lq
0.01
0.02
S 0.006
1400 1z SO
(d-l,
1.7s 2.6 [q 720 0.5 (141 IWO
(2)
Ahuninium in hone1 600 Chromium in 2$%Cr
-
2 x lo-‘exp(-0.91/W) S x 10v5 exf;l:$/kT)
SYat@m
sial
0.08 -
-
CO
B in Fe-30%Ni
0.09
2~hCrl%Mo0.08%So Steel
2$Xrl
600
0.04 0.07
C 0.08
%Mo
Inc&e~
Fe-30%Ni AIIov
Alloy
-
-
B 0.001
::: [q 720 0.75 [12] loo0
1.6 x 10-5exp(-2.0/kT) S x lo-’ e;$$ii3/kT)
[19] [16]
Balaoaz
8.6 Balance
Fe Balance
Sn in 2~Crl%MoOO.8%Sa
-
0.28 -
Al -
XU TINGDONG
and SONG SHENHUA:
continuously cooled sample at a temperature is longer than the critical time of this sample at the same temperature, te> r,, the lower the cooling rate of the sample, the less the segregations, at a constant solid dissolution temperature; whereas the higher the solid dissolution temperature, the greater the segregations, at a same cooling rate. Karlsson and Norden [24] gave the experimental results to verify indirectly the above calculated results for the situation of te> t,, that for boron segregation to grain boundaries the amount of segregation increased with increasing starting temperature and was largest at intermediate cooling rates (about 13 c/s after cooling from 1250°C). Contrasting the calculated result of boron segregation of samples solution-treated at 1200°C and cooled at the rate of 1400°C s-’ with that of samples solution-treated at 1050°C and cooled at 50°C s-‘, it will be found that segregation of boron to grain boundaries should not have been observed experimentally for the later. But the experimental result is negative. The author of the present article maintains that when cooled at a lower rate, because precipitation of extremely fine grain of borides is probably induced by grain-boundary segregation, the transition process of which in turn, prompts the concentration of boron toward the grain boundary. When cooled at a faster rate, however, the process of precipitation as mentioned above can hardly occur due to the negligible amount segregated and in the short time. It follows that the quantitative differences in segregation induced by the different starting temperature and different cooling rate appears larger than would be expected from the calculation. PTA method can not be used to identify the segregations of minute boride particles and boron atoms at the grain boundary. For aluminium in Inconel, chromium in 2$‘Xrl%Mo steel and tin in 2~%Crl%MoO.O8%Sn steel, just as shown in Table 3, the calculated results are mostly in agreement with the observed data in order of magnitude. The observed data for aluminium, chromium and tin segregations in Table 3 were measured experimentally by the newly developed Scanning Transmission Electron Microscope (STEM). A spatial resolution of about 1~~1500 w was held 122,231. The width of the concentrated layer along grain boundaries, d, is chosen as 0.15 pm in the calculation of the present work and this is because that (a) the non-equilibrium segregation is a process of kinetics and appears mainly during cooling within the region of high temperature. Consequently the ~gregation extent is generally larger and is from about hundreds of millimicrons to one micron [S, 14,221, (b) the spatial resolution of STEM is normally from 1000 to 1so0 A. 5. SUMMARY A new kinetic model of non-equilib~um grain boundary segregation has been developed for both
GRAIN-BOUNDARY SEGREGATION
2505
segregation processes and desegregation processes, which relates the non~quilib~um segregation levels to the solute dissolution temperature and the cooling rate of the discussed samples. Calculated results obtained from the new kinetic model agree satisfactorily with the observed data in [S, 11,121. Both the results predicted by the kinetic model of the present work and the observed data in experiments show that, when effective time t, is not greater than the critical time r, for specimens cooled at the same cooling rate, the higher the solid dissolution temperature, the greater the segregation; whereas for specimens treated at the same temperature, the lower the cooling rate, the greater the segregation. When effective time te is greater than the critical time tc for specimens treated at the same temperature, the lower the cooling rate, the less the segregation whereas for specimens cooled at the same cooling rate, the higher the solid dissolution temperature, the greater the segregation. Acknowledgements-Thus
work was supported by the Science Foundation Grant in the Education Division of the Metallurgical Ministry of China.
REFERENCES 1. K. E. Sickafus and S. L. Sass, Acfa met&. 35,69 (1987). 2. D. Mclean, Grafn-Roundarjes in Metals, p. 116. Oxford Univ. Press (1957). 3. K. T. Aust, S. J. Armijo, E. F. Koch and J. A. Westbrook, Trans. Am. Sot. Metals 60, 360 (1976). 4. T. R. Anthony, Acta mefall. 17, 603 (1969). 5 R. G. Faulkn&, J. Mater. Sci. 16, 37j (1981). 6. P. Doig and P. E. J. Flewitt, Actu mefall. 29, 1831 (1981). 7. Xu Tingdong, J. Mater. Sci. 22, 337 (1987). 8. Xu Tingdong, .i. Mater. Sci. Left. 7, 241 (1988). 9. Song Shenhua, Xu Tingdong and Yuan Zhexi, Acta metall. 37, 319 (1989). 10. T. R. Anthony, in LI@sion in Sol& (edited by A. S. Norwich and J. E. Burton). Academic Press, New York. 11. Song Shenhua, Xu Tingdong and Yuan Zhexi, unpublished works. 12. P. Doig and P. E. J. Flewitt, MetaN. Trans. ISA, 399 (1987). 13. W. Stevens and A. G. Haynes, .I. Iron Steel Insf. 183, 349 (1956). 14. T. M. Williams, A. M. Stoneham and D. R. Harries, Netall Sci. 10, 14 (1976). 15. M. E. Warga and C. Wells, f. Mefals 5, (1953). 16. F. S. Buffincton, K. Hirano and M. Cohen, Acta metall. 9, 434 (1961).
17. R. A. Swalin and A. Martin, Metall. Trans. A.I.M.E. 206, 576 (1956).
18. J. R. Macewan, J. U. Macewan and L. Yaffe, Can. J. Chem. 37, 1623 (1959). 19. A. W. Bowen and G. M. Leak, Metall. Trans. 1,2769 (1970). 20. C. 0. Bannister and W. D. J. Jones, J. Iron S&eelInsf. 124, 71 (1931). 21. He X. L. and Chu Y. Y., J. Phys. D Appl. Phys. 16, 1145 (1983). 22. R. G. Faulkner and T. C. Hopkins, X-Ray Specrromerry 6, 73 (1977). 23. R. G. Faulkner and K. Norrgard, ibid. 7, I84 (1978). 24. Leif Karlsson and Hans Norden. Acta mefail. 36, 13 (1988).
XU TINGLWNG and SONG SH~HU~
2506
APPENDIX 1 Solution of Equation (5) Equation (5) is simplified by using the Laplace transform; putting C=
* e -P’ c dt. s0
Equation (5) becomes
w/ax2 - q2 c = - C$o
641)
here q* =p/D and Cs is the initial uniform concentration in the crystal. This has solutions of the form e-gx and e [email protected] keep C bounded as x + cst we take the former, and young for the constant term in equation (Al) obtain C = M *e -qx f CJp.
(A2)
Now the Laplace transform of (6) is D@C/ax),=, = a,+ r&2 [PC - (e&xi+I ).C,l (A3) (ui/ai+ , ). C, being the crystal concentration at x = 0 immediately after quenching. Inserting (A2) into (A3) gives M=(u,-a,+,).Cg.d/Dq(ai+,qd+2).
(A4)
Replacing M in (A2) C =C&Wi+,)-
l]e-q’/Dg
(q +2/ei+t+)+C&
GUN-BOU~ARY
where S is the amount of diffu~onal component, which is held constant from the beginning of diffusion process to the end. The diffusion process described by Gauss solution are those at the beginning of which the diffusion component is all concentrated in a infinitestimal region and the concentration amplitude of diffusion component in this infinitestimal region will be infinite. For the situation that the diffusional component is all concentrated in a finite region at the beginning of the diffusion process, such as the desegregation model mentioned above, we can divide the finite region into intinitestimal parts and use Gauss solution in every infinitestimal part. Then, according to the principle of superposition, the solution for the finite region is the sum of all the Gauss solutions for the infinit~timal parts. Substituting integral for sum, we can get the error solution of diffusion equation C(x, t) = .4 + B erf[x/(4Dt)“*] where A and R are constant and 6(401)1/Z erf[x/(4Dt)‘j2] = -!exp(-y*) Lf n s 0
+ C.erq(x
x expj(2x/a,+ ,+d) + (4Dt/cr!+, d2)] x erfc([x/(Dt)“2] + [2(Dt)i’2/ai+ld]f. Rutting x = 0 gives the crystal concentration in contact with the region of the non-equilibirum segregation and on multiplying by a,+, we then obtain C,(t)=C~(T,+,)-C,(ai+,-ui)
(A@
x < -d/2, C(x, t) = A + B erf( - co) + C erf( - co) =A-B-C=C,, -d/2
=A+B-C=Cb[t,(Ti+,)] x>d/2,
(All)
C(x,t)=A+Berf(+co)+Cerf(+co) =A+B+C=C,.
(A12)
From e-quation (AlO), (Al 1) and (A12), it is obtained that
A
=C,,-B=f[C,(t,(I;:+,)-C,l,C=-[C&T,+,)-CJ.
Conseouentlv. substituting the values of A. B and C into equation (A!$ gives -
2
The Derivationof Equation (8) The Gauss solution of Diffusion equation is as following C = [S/(f14Dt)1/2].exp(-x2/4Dt)
W0)
[C~ft)-c~(~)l/[c~(Ti+,)- cZ(T,)l = I- exp(4in/crf+, .d2)e~c[2(Dt)‘~~ai+,*d$
APPENDIX
- d/2)/[4D& - t,)]‘@} (A9)
where A, B and C are constant to be determined, t is isothermal holding time at Ti+, , t = t,(Ti+ t ). At tire beginning of desegregation process, t = t,, the interfaces between the grain interiors and concentrated layers should s&lice the folfowing conditions
Equation (A6) can be transformed into
which is the equation for the kinetics of non~~ib~urn grain-boundary segregation at constant temperature only for segregation processes.
dy.
C(x, t) = A + B .erf{(x + d/2)/[4D,(t - t,)]‘“)
(As)
C=C*-C*V--W&+,)1
(A@
For the desegregation model mentioned above, two error solutions are used and we can get
and so, from Tables of Laplace tr~sfo~
x exp(4Dt/af+, d2) x erk(2(Dt)‘~2/a,+I4.
SEGREGA~ON
(A7)
C(x,t)-C,+f[C,(t,T,+,)-C,l x {erf[(x + d/2)/[4D, (t - tc)]“2 - erf[(x - d/2)/[4D,(t - tc)]“z]].
Acta metali. Vol. 37, No. 9, pp. 2499-2506, 1989 Printed in Great Britain. All rights reserved
A KINETIC MODEL OF NON-EQUILIBRIUM GRAIN-BOUNDARY SEGREGATION XU TINGDONG and SONG SHENIUA Department of Mate~als Science and Engin~~ng, Wuhan Iron and Steel University, Wuhan, Hubei, China (Received 28 November 1988)
Abstract-An isothermal kinetics of non-equilibrium grain-boundary segregation was developed both for segregation processes and for desegregation processes within a phenomenological theory. An effective time concept of a cooling process was discussed. On these bases, a simple and accurate method for evaluation of the levels of non-equilibrium grain-boundary segregation during cooling was proposed. According to the method, we have calculated the levels of non-equilibrium segregation to austenite or prior austenite grain boundaries for boron in Fe-3O%Ni alloy, aluminium in Inconel600, chromium in 2.25%Crl%Mo steel and tin in 2.25%Crl %MoO.OB%Sn steel in different experimental conditions respectively. Results calculated from the kinetic model in this paper are in satisfactory agreement with the observed data of experimental measurements for all the above samples. R&au&---Dans le cadre dune theorie ph~nom~nolo~que, nous d~veloppons une cinetique isotherme de la segregation inter~anulaire hors d’equilibre, $ la fois pour les mecanismes de la s&egation et pour ceux de la desegregation. Nous discutons un concept de temps efficace pour le m&&me de refroidissement. Sur ces bases, nous proposons une methode simple et precise pour &valuer les taux de segregation intergranulaire hors d’equilibre au tours du refroidissement. Selon cette methode, nous avons calcule les taux de segregation hors d’equilibre vers les joints de grains d’austenite actuels ou anttrieurs dans les cas suivants: bore dans un alliage de fer $ 30% de nickel, aluminium dans l’Inconel600, chrome dans un acier ri 2,25% de chrome at 1% de molybdbne, et Ctain dans un acier B 2,25% ce chrome, 1% de molybd6ne et 0,08% d’ktain, respectivement dans des conditions expkrimentales diffkentes. Les rbsultats calculkes A partir du modele cinetique propose dans cet article sont en accord satisfaisant avec les resultats exp&imentaux obtenus pour tous les echantillons que nous venons de titer. Z~~e~a~u~-Innerhalb einer ph~nomenologi~hen Theorie wird eine Nichtgleich~ich~ theorie der isothermen Kinetik der Komgren~e~egation sowohl fiir An-wie such fur Abreicherung entwickelt. Hierzu wird ein Konzcpt der effektiven Zeit eines Abk~prozes~s diskutiert. Auf dieser Basis wird eine einfache und genaue Methode fur die Auswertung der Nichtgleichgewichtssegregation an Komgrezen wahrend des Abkiihlens vorgeschlagen. Mit dieser Methode werden die entsprechenden Niveaus der Segregation fur Bor an austenitischen oder vorher austenitischen Komgrenzen in der Legierung Fe-30%Ni, fur Aiuminium in Inconel 600, fur Chrom im Stahl 2,25%Crl%Mo und fiir Zinn im Stahl 2,25%Crl %MoO,OB%Sn unter verschiedenen experimentellen Bedingungen berechnet. Die erhaltenen Ergebnisse stimmen mit den Beobachtungen an allen oben angegebenen Materialien zufriedenstellend iiberein. 1. INTRODUCTION of the grain-boundary
of
quantity of recombined complexes of solute atoms and vacancies exists. These three parts: solute atom,
solute has attracted considerable attention in recent years, since it is widely recognized that such solute segregations not only control many of the mechanical, chemical and electronic properties of technologitally important materials but also induce grain boundary structural transformations [l]. The solute segregation to grain boundaries is classified into equilibrium and non-equilibrium segregation. The works of McLean [2] on the thermodynamics and isothermal kinetics of equilibrium segregation founded an important basis for understanding the properties of equilibrium segregation. In the late 1960s a non-equilibrium grain-boundary segregation theory with respect to the solute atoms was proposed by Aust ef al. [3] and Anthony 14). It is suggested that the mechanism of segregation be based on an equilibrium in which a sufficient
vacancy and their recombined complex, are in equilibrium with each other. A sample properly maintained at a solid dissolution treatment temperature will, when cooled to a certain lower tem~ratu~, exhibit a loss of vacancies along the grain boundaries, whereby it attains the equilibrium vacancy concentrations at low temperatures. The decrease in the vacancy concentration causes the dissociation of the recombination complexes into vacancies and solute atoms. This in turn gives rise to the decrease in the recombined complex concentration near the grain boundary. Meanwhile, in regions remote from the grain boundary, where no other vacancy traps are present, vacancies would recombine with solute atoms and so the vacancy concentration is reduced. This makes the recombination complex concentration increase in the regions remote from the grain
The subject
segregation
2499
2500
XU TINGDONG and SONG SHENHUA:
boundary. Consequently, a concentration gradient appears between the grain boundary and regions beyond it. The gradient of complexes drives the recombination complexes to diffuse from regions remote from the grain boundary to the grain boundary. This diffusion causes excessive solute atoms to concentrate in the vicinity of grain boundary and results in non-equilib~um grainboundary segregation. Faulkner [5] evaluated the maximum non-equilibrium segregation level for the samples cooled from a higher temperature to a lower temperature, and took for granted that this maximum level was equal to the segregation level found in the process of segregation between the two temperatures at any cooling rate. Such a postulate is obviously a coarse one. Doig and Flewitt [6] gave a kinetic analysis for the non-equilibrium grain-boundary segregation of solute atoms. In their analysis, no an equation of diffusion suitable for the diffusion of recombined complexes was found and so a number of assumptions, which differ from the actual diffusion processes in non-equilibrium segregation and their experimental conditions were made. Xu Tingdong [;1 proposed an isothermal kinetic model only for the segregation processes of non-equilibrium segregation and suggested a critical time concept of non-equilibrium segregation and an effective time concept of a cooling processes [S, 91. In the present work, we further above works to propose a isothermal kinetic model of non-equilibirum segregation both for segregation processes and for desegregation processes. Using the results of the above concepts we suggest a simple and accurate method evaluating the levels of non-equilib~um segregation during cooling. Finally data calculated by the author’s model show a reasonable agreement with the observed experimental data for boron in Fe-30%Ni alloy, aluminium in Inconel 600, chromium in 2.25%Crl%Mo steel and tin in 2.25%Crl%MoO.O8%Sn steel. 2. MODEL 2.1. Segregation and desegregation The results of experiments [8,9] on non-equilibrium grain-boundary segregations have shown that for a sample being cooled quickly enough from a higher solution temperature to a certain lower temperature and then held at this lower temperature, there is a critical time t,, to which when the holding time of the sample at this lower temperature is equal, the level of segregation will be maximum. If the holding time of the sample at this lower temperature is shorter than the critical time, the levels of segregation at grain boundaries will increase with increasing holding time. If the holding time is longer than the critical time, the levels of segregation will decrease with increasing holding time. It is clear from the above experimental results that when holding time is shorter than the critical time, the segregation of the complexes from the center to the grain boundary will be dominant,
GRAIN-BOUNDARY SEGREGATION called segregation processes, and when holding time is longer than the critical time, the diffusion of solute atoms from grain boundaries to the center will be dominant, called desegregation processes. The critical time formula at a certain temperature T was given in Ref [6,7] as t,(T) = RZ In [(D,/Q)/4S (D, - Q)]
(1)
where 6 is the critical time constant, according to the method suggested in [9], S = 11.5 can be calculated and will be adopted in the following calculation of the present work. Di and D, are respectively the diffusion coefficients of solute atom and complex of solute atom and vacancy at temperature T, and R is the grain size. The fact that both segregation process and desegregation process exist is the most considerable characteristic for non-equilibrium grain-boundary segregations. It is obvious that different physical processes and essentials are holding for these two processes of segregation so it is necessary that different isothermal kinetic models should be founded for these two processes. 2.2. The isothermal kinetics for segregation processes For solid solution, it is postulated that there is an equilibrium between solute atom I, vacancy Y and recombined complex C formed by the former two components z+v=c where it is assumed that one recombined complex is made up of one solute atom and one vacancy. As discussed in [ii, 141,the following equation is held for this equilbrium Cc/C, = & expK& - Q)lkTl
(2)
where K, is a geometric constant, i$, is the energy of formation of the recombined complex, Ef is the energy of formation of vacancy and E,, is, in generai, smaller than Ef, k is Boltzmann’s constant C, is the concentration of recombined complexes, C, is the concentration of solute atoms. It was assumed here that equation (2) is only heId in any of the localized regions even though it is not necessary for the whole system to be in a system of equilib~um. This assumption is justified because this reaction is only concerned with short range diffusion in small regions [lo]. As has been discussed in [S, 71, when a sample is cooled from solution-treatment temperature, TO, to any lower temperature, T, the maximum concentration of non~quilibrium segregation induced during the cooling at grain boundary, C:(T), is given by
expW& - WWJ
- [& - W/W1
(3)
where C, is the concentration of solute atoms within the grain. It is obvious that the maximum concentration, C;(T), in equation (2) is only dependent on the
XU TINGDUNG
and SONG SHENHUA:
difference between solution-treatment temperature T, and the lower temperature T and is independent of the cooling rate from T, to T. If the solid solutiontreatment temperature of samples is thought to be a constant, Cr(T) will only be a function of temperature T. Equation (2) is therefore an ~mpo~ant thermodynamic equation describing the non-equilibrium grain-boundary segregation level. It can be seen in the following discussion, that on the basis of it, one can establish the isothermal kinetic relationship of non-equilibrium segregation. Suppose that the sample is cooled so quickly from temperature Tj down to 7;, 1(T, > Tj+ 1) that no mass transfer occurs in the specimen during cooling. Then the temperature is maintained at Ti+ 1for a period of time, during which time the recombined complexes will diffuse from within the grain toward the grain boundary, and the solute atoms are co~~ntrated at the grain boundary. Let r, I< Ti < To and denote C:(Tj)/C, by xi, and C:(T,, ,),lC, by ai+ I, obvious& a,, 1 > ai. As the width of the concentrated layer of solute atoms d, is relatively small as compared with the grain size, and ai + f x d is very small as compared with the grain diameter, the solute atoms concentrated at the grain boundary will be almost entirely furnished from the very narrow region in the vicinity of the grain boundary and the concentration of solute atoms within the grain remains a constant one C,. With the above conditions being met, the diffusion of recombined complexes towards the grain boundary can be simpli~ed into a steady linear flow of recombined complexes into the grain boundary in a semi-infinite medium. The corresponding di~usion equation for combined complexes is, therefore
where D, is the coefhcient of diffusion of the recombined complex in the matrix and Cc is the concentration of recombined complexes. In order to get a diffusion equation which is relative to the concentration of solute atoms, C,, substituting equation (2) into equation (4) gives (5) L),a2c/axr = t?C~at where C = C, is the concentration of solute atoms and r>, still is the coefficient of diffusion of the recombined complexes. Evidently equation (5) is an equation of diffusion of a special form, in which the coefficient of diffusion is concerned with the recombined complexes while the concentration is concerned with the solute atoms. It properly depicts the diffusion of recombined complexes dragging the solute atoms toward the grain boundary, causing the difference in concentration of solute atoms in different parts of the crystal. In view of the extremely small thickness of the iayer at the grain boundary where the concentration gradient can be neglected, one can imagine an interface between the grain interior and boundary concen-
GRAIN-BCKJNDARY
SEGREGATION
trated layer where the concentration
2501
is
C = C,(t)/%+ I where Cb(t) is the concentration of the concentrated layer when holding time of sample at temperature T,,i is equal to t, and will change with the holding time at the constant temperature. For simphcity in computation, the interface between the grain interior and the concentrated layer at x = 0 is chosen. According to the First Law of Diffusion and the Principle of Mass Conservation, the interface should suffice the following condition (C),=0= ~,(ac/a.~),,,
C,(l)l%,, = (dP)*(X,(t)/dr)
=(1/2)~i.+]~d~(iiCjat),=,
(6)
where d is the width of the concentrated layer and the factor l/2 is related to the fact that at both sides of grain bounda~es the solute atoms diffuse towards the grain boundaries. Assuming equation (6) to be the boundary condition, the solution of equation (5) is given (see Appendix 1) &(t) = C?(T,, !) - C;(a,,, x exp(4Dtjaf+, Rearranging
--a,)
. d2)erfc(2(Dt)i~2a,+ ,+d].
we obtain
KG(f) - GX37)l/[GX?;+
I) - GVi)3
= 1 - exp(4Dt/af+ 1~d2)“erfc[2{DI)“2iaj, ,-d]
(7)
where
- Jo where D = D,. Equation (7) is an isothermal kinetic relationship for non-equilibrium segregation for the segregation processes. It describes the non-equilibrium segregation concentration Cbft) of the solute atoms at the grain boundary as a function of the holding time f at the constant temperature when I < t, i.e. the segregation process of the complexes to the grain bounda~es form the center is dominant. 23. 7% isofkmaf &esseS
kinetics for desegregation pru-
When the holding time at temperature Ti+ i is equal to the critical time at this temperature, t,( T,, , ), the solute concentration in the concentrated layers along the grain boundaries, Cb[t, (T, + , )] can be obtained by substituting t,(T,+ ,> into equation (7). So allowing for the concentration gradient in the concentrated layers can be neglected as above, we can assume that the concentration profile of solute atoms near the grain boundaries is the same as those shown in Fig. 1, for the sampies held for the period being equal
XU TINGDGNG
2502
-d/2
0
and SONG SHENHUA:
+d/Z
X
GB Fig. I. The concentration profiles near the grain boundaries as the holding times are equal to the critical time I,. t,, fz (f, < t, < t2) at T,, , .
to the critical time t, at the temperature q+, {i.e. when (-d/2)
CL&- 4) =
cg+ (1/2)(C~(#~) - cg,
x (erf[(x + d/2)/[4D,(t - t,)]‘“] -erf[(x
- d/2)/[4D,(r - t,)]“2)
(8)
where t is the isothermal holding time at T,, ,, t, = tc(I;:, ,), Di is the diffusion coefficient of solute atoms in the matrix. Evidently, equation (8) is only concerned with desegregation and not segregation Thus the condition t > t, is necessitated for equation (8). For any sample held isothermally for the time longer than the critical time r,(Tj+ r), the level of non-eq~lib~um segregation at grain ~unda~es can be obtained by making x = 0 in equation (8)
C,(r)=C*+f[C,(t,)-C,l x {erf[d/2/[4D,(t - r,)J’/2] -erf[-d/2/[@/2/[4D,(t
- tJin]).
(9)
Equation (9) is an isothermal kinetic ~Iations~p for desegregation processes of non-equilibrium segrega-
GRAIN-BOUNDARY
SEGREGATION
tion. It describes the non-equilibrium segregation concentration, C&(t), of the solute atoms at grain boundary as a function of the holding time t at constant temperature when t > r,, i.e. the diffusion process of solute atoms from grain boundaries to center is dominant. 2.4. A method for c&dating the eon-equilibria ~egreg~~io~ level for ~on~~~o~ cooling For most of the experiments and the practical problems about non-equilibrium grain-boundary segregation, segregations will always occur during the continuous cooling. It is considerable importance how to calculate the non-equilibrium segregation levels for continuous cooling. In this section, a simple and accurate method to calculate the non-equilibrium segregation levels during cooling will be suggested. As discussed in [7,8], any continuous cooling curve for a sample can be replaced by a correspondent stepped curve, each step of which was formed by horizontal and vertical segments so as to calculate an effective time at some temperature for this cooling sample. The effective time formula of the stepped curve consisting of n steps at temperature T was given by t,(T)=
f: tjeXp[-EA(Ti=i
Tj)/RTT,]
(10)
where EA is the activation energy for diffusion of complexes in the matrix and it is assumed here that E,isgivenbytheaverageofthevaluesofthevacancy movement energy and the solute atom diffusion activation energy. t, and T, are respectively isothermal holding time and temperature at the ith step of the stepped curve. When the chosen steps are small enough, the effective time of the stepped curve at some temperature will accurately enough be equal to the effective time of the continuous cooling at same temperature. It is expected that the effect of diffusion of solute atoms (or complexes) during the continuous cooling is the same as that of the sample held at a constant temperature for the same time as the effective time of the continuous cooling at the same temperature. This means it is possible that calculating segregation levels of solute atoms (or complexes) during continuous cooling can be changed into calculating segregation levels during isothermal holding processes by using the effective time concept of the continuous cooling. For any sample which was cooled continuously from a higher temperature to a lower temperature, we can get the effective time corresponding to its continuous cooling curve at some temperature as above then compares it with the critical time of that sampfe at the same temperature, which can be calculated from equation (I), and the segregation levels of the sample can be calculated by substituting the effective time into the equation (7), which describes the segregation processes, if it is shorter than the cortical time of the same temperature and by substituting the effective time into equation (9), which describes the desegre-
XU TINGDONG
and SONG SHENHUA:
GRAIN-BOUNDARY
SEGREGATION
2503
gation processes, if it is longer than the critical time. It should be noted that this method is suitable for both non-equilibirum and equilibrium grain boundary segregation. In the situation of equilibrium segregation we only substitute the effective time of continuous cooling into the equation of the isothermal kinetics of equilibrium segregation suggested by McLean [2].
minium, chromium and tin segregations in their respective experimental materials were measured experimentally by the newly developed Scanning Transmission Electron Microscope (STEM). This technique has a spatial resolution of about from 1000 to 1500 A [23,22].
3. A COMPARISON OF CALCULATED RESULTS AND OBSERVED DATA
A number of authors have recently reported their findings on non-equilibrium grain-boundary segregations on the basis of the mechanism of the formation of recombined complexes of vacancies and solute atoms, and the diffusion toward the grain-boundary during cooling. Also, many different analyses have been introduced [5-71. The defects in the kinetic analysis described in [5,6] have been discussed in [7]. Here a new kinetic model both for the segregation processes and desegregation processes of non-equilibrium segregation is proposed. Comparing the present works with the results obtained by us in [7j, this new kinetic model has evidently furthered the kinetic model in [7] to make it not only suitable for the segregation processes but suitable for the desegregation processes as well. It provides a generalized relation for the behavior of non-equilibrium grainboundary segregation in a kinetic manner, just as Mclean’s model does in the case of equilibrium grain-boundary segregation, and especially in this model, the segregation level for the continuously cooled samples can also been calculated by the method of using the effective time concept and isothermal kinetic equations for segregation processes and desegregation processes. This method is more simple and more accurate than the method suggested by us in [7], in which some mathematical approximations were adopted during the derivation of correction factor. It is obvious and considerably important that the model presented here correlates the non-equilibrium segregation levels not only to the solid dissolution treatment temperature but also to the cooling rate of samples both in the case of segregation processes and in the case of desegregation processes of non-equilibrium segregation. As a matter of fact, it has been shown by many experiments that non-equilibrium grin-bounda~ segregation level is closeIy related to both the solid dissolution temperature and the cooling rate the specimen undergoes. Calculations based on the present model for the specimens of experimental alloys in Table 3 indicate that the condition f,< t, is met and at a constant solid dissolution temperature, the lower the cooling rate, the greater the segregation of solute atoms to the grain-boundaries; whereas at the same cooling rate, the higher the solid dissolution treatment temperature, the greater the segregation of solute atoms. Such a conclusion has been verified in the observed data listed in Table 3, adopted from (5, 11, 121. According to this model, it can be calculated that when the effective time for a
Using the above model, we have calculated the level of non-equilibrium segregation at the grain boundaries for boron segregation in Fe-30%Ni alloy [I I], aluminium segregation in Inconel 600 [5], chromium segregation in 2a%Crl%Mo steel [12] and tin segregation in 2p!Crl%Mo O.O8%Sn steel [12] respectively for their different experimental conditions offered by their authors. Fe-30%Ni alloy and Inconel 600 are austenitic alloys. And the martensite transfo~ations for Zi%Crl %Mo steel and 2f%Crl %MoO.O8%Sn steel commence at about 700 k, i.e. -0.4 T, which can be calculated [13]. The detailed compositions for the four alloys are given in Table 1. The data used for theoretical calculation of non-equihbrum segregations for these four systems are given in Table 2. The energy of formation of the recombined complexes, Ebr for boron in Fe-30%Ni alloy is taken from the result of Williams et al. [4] and for aluminium in Inconel600 is taken from the result of Faulkner [5] and for chromium in 2$Krl%Mo steel and tin in Z~~Cri%Mo0.08°~Sn steel are taken from the results of Doig and Flewitt [12]. The diffusion activation energy for vacancies, E, = 2.6 eV [6]. EA can be caIculated by averaging the values of the vacancy diffusion activation energy and solute atom diffusion activation energy. The concentration of solutes in the grain centres, C,, are taken as the analysed concentration for the segregation solute atoms, taken from Table 1 and converted to atomic percent. The grain boundary segregation thickness d = 0.15 pm is chosen in the present work. For all the experimental alloys, experimental conditions, calcufated results and observed data quoted from their respective reference are listed in Table 3. It should be noted that for the two kind of steei, the functional relation of temperature and cooling time was described from [6] by T,= T,.exp(-@)
(11)
where Z’, is solution-treatment temperatures, t is the cooling time, d, = 1 for the chromium and tin in their respective alloys. The fact that the calculated results agree reasonably with the experimental data is obvious. For boron segregation in Fe30%Ni alloy, the observed non-equilibrium segregation levels, C.,, , was measured by the assessment of the Particle Tracking Autoradiography (PTA) [I I]. The PTA has a spatial resolution of about 2.0 pm [20]. The alu-
4.
DISCUSSION
N.O. = Not observed.
Tin in 2$rl%Mo O.O8%Sn
l%MO
980 1423 1323 1223 1423 1323 1223
11SO 1220 lOS@ 1220 llS0 IOSO
Ft3O%Ni Boron in alloy
0.005
0.02
P 0.009
27 100 85 60 100
E
z
:=;
z 290 290
2.2
15.7 2.1
Cr -
-
0.46
0.38 0.53
Mn -
Comt&tion
0.006 -
N -
0.15
14.4 0.1
Ni 29.1
(wt%)
0.61 2.23 2.23 2.2s 0.037 0.037 0.037
0.0052 o.OOs2 o.oos2 o.OOs2 0.0052 0.0052
(a&
[16] [lq
Alloys
15.72 0.53 0.10 0.01s
0.72 z 2Oi 9.46 2.80
l,(Z) (5)
1200.75 3174.87 2377.52 I337.35 610.69 390.85 219.85
87.83 87.83 87.83 87.63 87.83 87.83
r,Wi) (5)
calculated results
2.7 2.6 Lq 720 0.37 (q 1000
1.87 x lo-‘exp(-2.8/H’) 3.36 x lo-‘ yf----/kT)
Al in Inconcl 600
-
0.24 -
Ti 0.033
1.0
0.08 1.0
MO -
0.08
-
Sn -
2.86 2.47 1.81 0.91 0.43
:::s
X:?? 0.13 0.088
0.073 0.098
2.5 3.23 2.54 2.17 3.81 1.30 0
E:: N:O: High Medilun Low
Maguitude (at%) Cl&,) (h < I,) CbU.-~e)(rc~G) C&I
2:55 2.6 [6] 720 OS [12] 1000
3.21 x lo-‘exp(-2.5/H’) 118) S x 10-‘f?x4~~-;.SS/kT) [16]
Cr in 2$rl %Mo
Table 2. Data used for theoretical calculation
0.46
0.11 0.38
Si -
Table 1. Alloy compositions
Table 3. Exlwrima~td ad
[IS] [lq
0.01
0.02
S 0.006
1400 1z SO
(d-l,
1.7s 2.6 [q 720 0.5 (141 IWO
(2)
Ahuninium in hone1 600 Chromium in 2$%Cr
-
2 x lo-‘exp(-0.91/W) S x 10v5 exf;l:$/kT)
SYat@m
sial
0.08 -
-
CO
B in Fe-30%Ni
0.09
2~hCrl%Mo0.08%So Steel
2$Xrl
600
0.04 0.07
C 0.08
%Mo
Inc&e~
Fe-30%Ni AIIov
Alloy
-
-
B 0.001
::: [q 720 0.75 [12] loo0
1.6 x 10-5exp(-2.0/kT) S x lo-’ e;$$ii3/kT)
[19] [16]
Balaoaz
8.6 Balance
Fe Balance
Sn in 2~Crl%MoOO.8%Sa
-
0.28 -
Al -
XU TINGDONG
and SONG SHENHUA:
continuously cooled sample at a temperature is longer than the critical time of this sample at the same temperature, te> r,, the lower the cooling rate of the sample, the less the segregations, at a constant solid dissolution temperature; whereas the higher the solid dissolution temperature, the greater the segregations, at a same cooling rate. Karlsson and Norden [24] gave the experimental results to verify indirectly the above calculated results for the situation of te> t,, that for boron segregation to grain boundaries the amount of segregation increased with increasing starting temperature and was largest at intermediate cooling rates (about 13 c/s after cooling from 1250°C). Contrasting the calculated result of boron segregation of samples solution-treated at 1200°C and cooled at the rate of 1400°C s-’ with that of samples solution-treated at 1050°C and cooled at 50°C s-‘, it will be found that segregation of boron to grain boundaries should not have been observed experimentally for the later. But the experimental result is negative. The author of the present article maintains that when cooled at a lower rate, because precipitation of extremely fine grain of borides is probably induced by grain-boundary segregation, the transition process of which in turn, prompts the concentration of boron toward the grain boundary. When cooled at a faster rate, however, the process of precipitation as mentioned above can hardly occur due to the negligible amount segregated and in the short time. It follows that the quantitative differences in segregation induced by the different starting temperature and different cooling rate appears larger than would be expected from the calculation. PTA method can not be used to identify the segregations of minute boride particles and boron atoms at the grain boundary. For aluminium in Inconel, chromium in 2$‘Xrl%Mo steel and tin in 2~%Crl%MoO.O8%Sn steel, just as shown in Table 3, the calculated results are mostly in agreement with the observed data in order of magnitude. The observed data for aluminium, chromium and tin segregations in Table 3 were measured experimentally by the newly developed Scanning Transmission Electron Microscope (STEM). A spatial resolution of about 1~~1500 w was held 122,231. The width of the concentrated layer along grain boundaries, d, is chosen as 0.15 pm in the calculation of the present work and this is because that (a) the non-equilibrium segregation is a process of kinetics and appears mainly during cooling within the region of high temperature. Consequently the ~gregation extent is generally larger and is from about hundreds of millimicrons to one micron [S, 14,221, (b) the spatial resolution of STEM is normally from 1000 to 1so0 A. 5. SUMMARY A new kinetic model of non-equilib~um grain boundary segregation has been developed for both
GRAIN-BOUNDARY SEGREGATION
2505
segregation processes and desegregation processes, which relates the non~quilib~um segregation levels to the solute dissolution temperature and the cooling rate of the discussed samples. Calculated results obtained from the new kinetic model agree satisfactorily with the observed data in [S, 11,121. Both the results predicted by the kinetic model of the present work and the observed data in experiments show that, when effective time t, is not greater than the critical time r, for specimens cooled at the same cooling rate, the higher the solid dissolution temperature, the greater the segregation; whereas for specimens treated at the same temperature, the lower the cooling rate, the greater the segregation. When effective time te is greater than the critical time tc for specimens treated at the same temperature, the lower the cooling rate, the less the segregation whereas for specimens cooled at the same cooling rate, the higher the solid dissolution temperature, the greater the segregation. Acknowledgements-Thus
work was supported by the Science Foundation Grant in the Education Division of the Metallurgical Ministry of China.
REFERENCES 1. K. E. Sickafus and S. L. Sass, Acfa met&. 35,69 (1987). 2. D. Mclean, Grafn-Roundarjes in Metals, p. 116. Oxford Univ. Press (1957). 3. K. T. Aust, S. J. Armijo, E. F. Koch and J. A. Westbrook, Trans. Am. Sot. Metals 60, 360 (1976). 4. T. R. Anthony, Acta mefall. 17, 603 (1969). 5 R. G. Faulkn&, J. Mater. Sci. 16, 37j (1981). 6. P. Doig and P. E. J. Flewitt, Actu mefall. 29, 1831 (1981). 7. Xu Tingdong, J. Mater. Sci. 22, 337 (1987). 8. Xu Tingdong, .i. Mater. Sci. Left. 7, 241 (1988). 9. Song Shenhua, Xu Tingdong and Yuan Zhexi, Acta metall. 37, 319 (1989). 10. T. R. Anthony, in LI@sion in Sol& (edited by A. S. Norwich and J. E. Burton). Academic Press, New York. 11. Song Shenhua, Xu Tingdong and Yuan Zhexi, unpublished works. 12. P. Doig and P. E. J. Flewitt, MetaN. Trans. ISA, 399 (1987). 13. W. Stevens and A. G. Haynes, .I. Iron Steel Insf. 183, 349 (1956). 14. T. M. Williams, A. M. Stoneham and D. R. Harries, Netall Sci. 10, 14 (1976). 15. M. E. Warga and C. Wells, f. Mefals 5, (1953). 16. F. S. Buffincton, K. Hirano and M. Cohen, Acta metall. 9, 434 (1961).
17. R. A. Swalin and A. Martin, Metall. Trans. A.I.M.E. 206, 576 (1956).
18. J. R. Macewan, J. U. Macewan and L. Yaffe, Can. J. Chem. 37, 1623 (1959). 19. A. W. Bowen and G. M. Leak, Metall. Trans. 1,2769 (1970). 20. C. 0. Bannister and W. D. J. Jones, J. Iron S&eelInsf. 124, 71 (1931). 21. He X. L. and Chu Y. Y., J. Phys. D Appl. Phys. 16, 1145 (1983). 22. R. G. Faulkner and T. C. Hopkins, X-Ray Specrromerry 6, 73 (1977). 23. R. G. Faulkner and K. Norrgard, ibid. 7, I84 (1978). 24. Leif Karlsson and Hans Norden. Acta mefail. 36, 13 (1988).
XU TINGLWNG and SONG SH~HU~
2506
APPENDIX 1 Solution of Equation (5) Equation (5) is simplified by using the Laplace transform; putting C=
* e -P’ c dt. s0
Equation (5) becomes
w/ax2 - q2 c = - C$o
641)
here q* =p/D and Cs is the initial uniform concentration in the crystal. This has solutions of the form e-gx and e [email protected] keep C bounded as x + cst we take the former, and young for the constant term in equation (Al) obtain C = M *e -qx f CJp.
(A2)
Now the Laplace transform of (6) is D@C/ax),=, = a,+ r&2 [PC - (e&xi+I ).C,l (A3) (ui/ai+ , ). C, being the crystal concentration at x = 0 immediately after quenching. Inserting (A2) into (A3) gives M=(u,-a,+,).Cg.d/Dq(ai+,qd+2).
(A4)
Replacing M in (A2) C =C&Wi+,)-
l]e-q’/Dg
(q +2/ei+t+)+C&
GUN-BOU~ARY
where S is the amount of diffu~onal component, which is held constant from the beginning of diffusion process to the end. The diffusion process described by Gauss solution are those at the beginning of which the diffusion component is all concentrated in a infinitestimal region and the concentration amplitude of diffusion component in this infinitestimal region will be infinite. For the situation that the diffusional component is all concentrated in a finite region at the beginning of the diffusion process, such as the desegregation model mentioned above, we can divide the finite region into intinitestimal parts and use Gauss solution in every infinitestimal part. Then, according to the principle of superposition, the solution for the finite region is the sum of all the Gauss solutions for the infinit~timal parts. Substituting integral for sum, we can get the error solution of diffusion equation C(x, t) = .4 + B erf[x/(4Dt)“*] where A and R are constant and 6(401)1/Z erf[x/(4Dt)‘j2] = -!exp(-y*) Lf n s 0
+ C.erq(x
x expj(2x/a,+ ,+d) + (4Dt/cr!+, d2)] x erfc([x/(Dt)“2] + [2(Dt)i’2/ai+ld]f. Rutting x = 0 gives the crystal concentration in contact with the region of the non-equilibirum segregation and on multiplying by a,+, we then obtain C,(t)=C~(T,+,)-C,(ai+,-ui)
(A@
x < -d/2, C(x, t) = A + B erf( - co) + C erf( - co) =A-B-C=C,, -d/2
=A+B-C=Cb[t,(Ti+,)] x>d/2,
(All)
C(x,t)=A+Berf(+co)+Cerf(+co) =A+B+C=C,.
(A12)
From e-quation (AlO), (Al 1) and (A12), it is obtained that
A
=C,,-B=f[C,(t,(I;:+,)-C,l,C=-[C&T,+,)-CJ.
Conseouentlv. substituting the values of A. B and C into equation (A!$ gives -
2
The Derivationof Equation (8) The Gauss solution of Diffusion equation is as following C = [S/(f14Dt)1/2].exp(-x2/4Dt)
W0)
[C~ft)-c~(~)l/[c~(Ti+,)- cZ(T,)l = I- exp(4in/crf+, .d2)e~c[2(Dt)‘~~ai+,*d$
APPENDIX
- d/2)/[4D& - t,)]‘@} (A9)
where A, B and C are constant to be determined, t is isothermal holding time at Ti+, , t = t,(Ti+ t ). At tire beginning of desegregation process, t = t,, the interfaces between the grain interiors and concentrated layers should s&lice the folfowing conditions
Equation (A6) can be transformed into
which is the equation for the kinetics of non~~ib~urn grain-boundary segregation at constant temperature only for segregation processes.
dy.
C(x, t) = A + B .erf{(x + d/2)/[4D,(t - t,)]‘“)
(As)
C=C*-C*V--W&+,)1
(A@
For the desegregation model mentioned above, two error solutions are used and we can get
and so, from Tables of Laplace tr~sfo~
x exp(4Dt/af+, d2) x erk(2(Dt)‘~2/a,+I4.
SEGREGA~ON
(A7)
C(x,t)-C,+f[C,(t,T,+,)-C,l x {erf[(x + d/2)/[4D, (t - tc)]“2 - erf[(x - d/2)/[4D,(t - tc)]“z]].