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Scripta Materialia 51 (2004) 521–525

Grain boundary saturation and grain growth

Feng Liu *, Reiner Kirchheim

Institute fur Materialphysik, Universitat Gottingen, Tammannstr. 1, 37077 Gottingen, Germany

Received 2 August 2003; received in revised form 24 March 2004; accepted 25 May 2004

Available online 17 June 2004

Abstract

In nanoscale alloys with strong segregation tendency, the grain size at metastable thermodynamic equilibrium is determined by

the grain boundary energy, the enthalpy change of grain boundary segregation, and the solute excess of an equivalent grain bound-

ary monolayer at saturation. Good agreement between experiment and thermodynamic description has been found with respect to

the temperature-dependence of the metastable grain size.

� 2004 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

Keywords: Grain boundary energy; Grain boundary segregation; Grain growth

1. Introduction and theoretical background

A particularly interesting type of nanocrystalline

(NC) materials is characterised by a strong tendency

for grain boundary (GB) segregation, i.e. for an enrich-

ment of the solutes at the GBs [1–3,8–15]. On this basis,

two strategies have been devised to improve the thermalstability of NC material: a kinetic one, in which the GB

mobility m [1–4] is reduced, and a thermodynamical one,

in which the driving force, i.e. GB energy, rb, is sup-

pressed [6–14]. Standard metallurgical techniques for

realizing the kinetic approach typically involve the intro-

duction of impurity atoms or phases to pin the GBs, the

effectiveness of which has been demonstrated experi-

mentally for NC materials [1–3]. However, due to theArrhenius dependence of m on temperature, even a dras-

tic reduction in the GB mobility will eventually be over-

come at high temperature, and grain growth restarts

1359-6462/$ - see front matter � 2004 Acta Materialia Inc. Published by El

doi:10.1016/j.scriptamat.2004.05.042

* Corresponding author. Present address: Max-Planck-Institute for

Metals Research, Heisenbergstr. 3, 70569 Stuttgart, Germany. Tel.:

+49-711-6893325/551-395002; fax: +49-711-6893312/551-395012.

E-mail addresses: f.liu@mf.mpg.de, liufeng@ump.gwdg.de (F. Liu).

accordingly. Potentially more effective would be a reduc-

tion in rb, which exhibits only weak temperature

dependence [5].

According to the Gibbs adsorption theorem, rb can

be reduced by the addition of solutes that segregate to

the GBs [6,7]. Experiments have shown that the result-

ing effect on rb can be substantial [13,14], and theoreti-cal considerations predict that a reduction of rb to zero

is possible in alloy system with a high enthalpy of segre-

gation [8–12].

rb ¼ r0 � Cb0½RT lnX 0 þ DH seg� ð1Þ

where r0 is the GB energy for the pure solvent, X0 the

bulk concentration, DHseg the enthalpy change of GB

segregation per mole solute, R the gas constant, T theannealing temperature, and Cb0 the solute excess of an

equivalent saturated GB monolayer, available for segre-

gation. Normally, the number for the superficial atomic

density in the equivalent segregation monolayer (in units

of mol/m2) is adopted to describe the saturation cover-

age. If possible, it would be more informative to indi-

cate, in addition, the density in units of fraction of a

monolayer or of fraction of the saturation coverage.Unfortunately, data for these quantities are not availa-

ble.

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Fig. 1. Fitting of Eq. (4) to the experimental data points [15] at

different annealing temperatures, corresponding to GBs saturation

with half an equivalent monolayer of Ca. The values of the fitting

parameters: r0, Cb0 and DHseg, and the error of the fit are given in

Table 1.

522 F. Liu, R. Kirchheim / Scripta Materialia 51 (2004) 521–525

The Gibbs free energy of a polycrystal can be written

as [11]

G ¼X

lini þ rbA ð2Þ

where li represents the chemical potential of component

i, ni the amount of component i, and A the GB area. At

constant pressure, temperature and amount of material

the change in free energy can be written as [11],

dG ¼ rb dA ð3Þ

Consequently, whenever the GB energy is positive, grain

growth will decrease the free energy of the system, and

grain growth can only be suppressed is for rb=0. In

reality, precipitation of solutes or stable phases at GBs

results in substantial grain growth, a more stable equi-

librium. A stop of nanoscale grain growth in metastableNC system corresponds to saturated GBs with zero GB

energy. The grain diameter at GB saturation, D, is given

as [12],

D ¼ 3Cb0V M

X total � expr0�Cb0DH seg

Cb0RT

� � ð4Þ

with Xtotal as the summation of the bulk and the GB

concentrations, and VM the molar volume of the alloy.Apparently an increase of Xtotal will result in a reduction

of D, whereas an increase of T causes an increase in D.

For alloy systems with high DHseg and negligible solute

solubility, Eq. (4) can be approximated as,

D ¼ 3Cb0V M

X total

ð5Þ

Considering the temperature dependence of the metast-

able grain size, a simple relation was obtained [11],

dð1=DÞd ln T

¼ �X 0 lnX 0

3Cb0V M

ð6Þ

The right-hand side of Eq. (6) depends also on temper-

ature: changes in D result in changes in X0, i.e. the con-

figurational interfacial excess entropy. However, the

interfacial excess entropy plays a negligible role for thetemperature dependence of the metastable grain size

[11]. Some simple calculations (not shown here) prove

that the agreement with experimental data will not be af-

fected if the configurational (ideal) interfacial excess

entropy is taken into account. Therefore, Eq. (6),

neglecting the influence of a change of X0, turns out to

be useful for comparison with experimental results.

In the present paper, published data on grain growthin poly- andNC solid solutions [13–16] has been analyzed

in terms of a model for stabilization of the grain size

through GB segregation, i.e. Eqs. (1)–(6). A good agree-

ment between theory and experiment has been achieved.

2. Grain diameter at GB saturation

2.1. Grain boundary segregation in TiO2–Ca system

A systematic and quantitative study of size-depend-

ent segregation behaviour in the TiO2–Ca system wasconducted [15]. Highly dense, fine-grained TiO2 doped

with 0.34 mol.% Ca has been synthesized, and the Ca

coverage at GBs (which could be assumed to be equal

to the GB excess, Cb0) has been measured as a function

of grain size, in the range of 50–750 nm, using a STEM

micro-analytical technique. Ca segregates strongly to

GBs, and this segregation exhibits a clear size-depend-

ence below critical size 150–350 nm [15]. Above the crit-ical range the GBs become saturated at approximately

half of an equivalent monolayer of Ca [15]. Upon fur-

ther annealing (e.g. corresponding to 500 nm grain size),

a few CaTiO3 precipitates are observed, and then sub-

stantial grain coarsening takes place [15].

Applying Eq. (6), the critical grain size of TiO2, cor-

responding to GBs saturated with half of an equivalent

monolayer of Ca [15], could be approximately chosen as150, 200, and 350 nm, at 773, 1173, and 1323 K, respec-

tively. Fig. 1 presents the fits of Eq. (4) to these critical

values at GB saturation. Values for the fitting parame-

ters and the error of fit have been gathered in Table 1.

With reference to [17], the GB energy of TiO2 (e.g. 30–

200 nm) was found to be 0.5–1 J/m2, in the temperature

range of 873–1053 K, which corresponds to the value of

0.61 J/m2, obtained by fitting. The fitted values for Cb0and DHseg are almost equal to data given in [15]. The

GB coverage (@0.84·10�5 mol/m2) given in [15], corre-

sponding to half of an equivalent monolayer of Ca, is

Table 1

The values of the fitting parameters: r0, Cb0 and DHseg, and the error of the fitting of Eq. (4) to experimental data of the grain diameter at GB

saturation [15]

r0 (J/m2) DHseg (kJ/mol) Number of monolayer Cb0 (mol/m2) Error (%)

0.61 1.23 0.5 0.87·10�5 0.14

Table 2

GB excess of TiO2 at normal GBs at the beginning of annealing, for

three TiO2 contents and three annealing temperatures, respectively [16]

Solute content Cb0/1583 K Cb0/1603 K Cb0/1623 K

0.1 1.33·10�5 1.49·10�5

0.2 1.87·10�5 1.91·10�5 2.28·10�5

0.4 3.63·10�5 3.71·10�5 16.3·10�5

Fig. 2. Fitting of Eq. (4) to the grain sizes obtained in the single-phase

NC solid solutions of Pd100�xZrx upon annealing [13,14]. The values of

the fitting parameters: r0, Cb0 and DHseg, and the error of the fit are

given in Table 2.

F. Liu, R. Kirchheim / Scripta Materialia 51 (2004) 521–525 523

equal to Cb0; DHseg is close to a value of 1.1 eV, which

has been estimated for the elastic strain energy of the

cation size misfit of Ca in TiO2 [15].

2.2. Abnormal grain growth in BaTiO3 doped with excess

TiO2

Normal and abnormal grain growth are two differenttypes of grain growth. In normal growth the grain size

distribution remains uniform whereas in abnormal

growth a few grains grow at the expense of the fine ma-

trix grains which are often pinned. The abnormal grain

growth of BaTiO3 containing an excess of 0.1, 0.2, and

0.4 mol% of TiO2 has been studied [16]. It has been

found that the matrix grain size depends on TiO2 excess

but does not change with annealing temperature or time.The experimental data [16] strongly suggests that the

excess TiO2 in BaTiO3 causes the stop of normal grain

growth, but it is not obvious by which mechanism

TiO2 acts. Since the solubility of TiO2 in BaTiO3 is very

limited, one possibility could be that TiO2 forms a fine

precipitate dispersion within the matrix. In order to

check this possibility selected samples were thoroughly

examined in TEM [16], however, no evidence could befound of such a precipitate dispersion. The excess

TiO2 could be associated with the GBs. According to

the present thermodynamic treatment (see Eqs. (1)–

(6)), it is proposed that grain growth is suppressed due

to GB saturation.

Since BaTiO3 exhibits cubic crystal structure above

400 K [18], TiO2 excess could be assumed to be segre-

gated at (110) and (111) faces of the BaTiO3 cubic crys-tal (a=0.4018 nm), and the GB excess for one equivalent

monolayer is approximated as,

Cb0 ¼Cð110Þ þ Cð111Þ

2ð7Þ

with C(110) and C(111) as the GB excess of one equiv-

alent monolayer for different crystal faces. Cb0 could be

calculated from

Cð110Þ ¼ 2ffiffiffi2

pða2Þ

¼ 8:76 1018=m2;

Cð111Þ ¼ 2

ðsinð60Þ=2Þð2a2Þ ¼ 1:43 1019=m2 ð8Þ

as 1.153·1019/m2, or 1.91·10�5 mol/m2.

At the beginning of the annealing, the volume fraction

of the abnormal grain is relatively small [16], so all of the

TiO2 excess may be segregated to GBs of the normal

grain. The values for TiO2 excess at normal GBs have

been gathered in Table 2. Except for solute content of

0.1 mol% TiO2, the TiO2 excess is larger than the satura-

tion value for one equivalent monolayer (see below Eq.

(8)). The abnormal grains grow upon annealing, andconsume GBs. Therefore the TiO2 excess associated with

GBs has to be redistributed. Part of this TiO2 could be

collected by the advancing abnormal GBs, and the

remaining TiO2 could be shared by other unconsumed

normal GBs. Since the volume fraction of the abnormal

grains is substantially increased as abnormal grain

growth proceeds [16], the TiO2 excess at the normal

GBs will probably be increased. Therefore, the stop ofthe normal grain growth can be due to GB saturation.

Table 3

The values of the fitting parameters: r0, Cb0 and DHseg, and the error of the fitting of Eq. (4) to experimental data of the grain diameter at GB

saturation obtained in the single-phase NC solid solutions of Pd100�xZrx upon annealing [13,14]

Solute content r0 (J/m2) DHseg (kJ/mol) Cb0 (mol/m2) Error (%)

Pd90Zr10 0.7 55.9 2.07·10�5 2

Pd85Zr15 0.7 42.5 3.77·10�5 16

Pd80Zr20 (673–1400 K) 0.7 59.2 1.78·10�5 6

Pd80Zr20 (1073–1400 K) 0.7 59.2 1.78·10�5 3

524 F. Liu, R. Kirchheim / Scripta Materialia 51 (2004) 521–525

2.3. Grain growth in Pd–Zr solid solution

Using the ball milling technique, single-phase NC

solid solutions of Pd100�xZrx containing up to 20 at.%

Zr have been prepared and subsequently annealing in-duces segregation of Zr atoms to the GBs [13,14]. Fig.

2 shows the fit of Eq. (4) to grain sizes of the single-

phase NC solid solutions of Pd100�xZrx upon annealing

for 24 h at different temperatures. Values of the fitting

parameters and the error of fit have been gathered in

Table 3. Obviously, Eq. (4) provides much better fit for

Pd80Zr20 and Pd90Zr10 data than that for Pd85Zr15 data

(see Fig. 2). With reference to [13,14], values for the fit-ting parameters obtained for Pd80Zr20 and Pd90Zr10 data

are reasonable, but not for Pd85Zr15 data. The reason is

not clear. The slightly higher error obtained for Pd80Zr20data within 673–1400 K (cf. Table 3) implies that GBs

have not been saturated after 24 h annealing at temper-

atures within 673–1073 K. Significant grain growth in

Pd80Zr20 solid solution has not been initiated until

1173 K (see Fig. 2).GB excess was calculated and shown to increase from

0.2 to 2.1 mol/m2 with increasing annealing temperature

from 671 to 1400 K, and GB energy essentially vanishes

Fig. 3. Linear relation between 1/D and ln T according to Eq. (6) using

experimental data upon 24 h annealing within 1073–1400 K Pd80Zr20and Pd90Zr10 single-phase NC solid solution [13,14].

in the samples manifesting the greatest degree of Zr seg-

regation (T=1400 K) [14]. Good fits of Pd80Zr20 and

Pd90Zr10 data, however, indicate that grain growth stops

by reducing the GB energy to zero, independent of the

annealing temperature. Fig. 3 presents a linear relationof 1/D as a function of ln T according to Eq. (6) for data

with 10 and 20 at.% Zr (within 1173–1400 K). This fur-

ther provides evidence that rb has been reduced to zero

after 24 h annealing at the respective temperatures.

3. Conclusion

Applying a thermodynamic relation to published

experimental data, grain growth suppression is re-inter-

preted. The grain size at metastable thermodynamic

equilibrium is determined by the grain boundary energy,

the enthalpy change of grain boundary segregation, and

the solute excess of an equivalent grain boundary mono-

layer at saturation. It is confirmed that agreement be-

tween experiment and thermodynamic description wasfound with respect to the temperature-dependence of

the metastable grain size.

Acknowledgment

This research is supported by the Alexander von

Humboldt Foundation.

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