Homogenization kinetics of a cast Ti48A12W0.5Si alloy

MATERIALS SCIENCE & ENGINEERING

l

ELSEVIER

Materials Science and Engineering A220 (1996) 168-175

Homogenization kinetics of a cast Ti48A12W0.5Si alloy W.J. Zhang*, L. Francesconi, E. Evangelista Department of Mechanics, University of Ancona, Ancona 1-60t3I, Italy Received 14 March I996; revised 4 June 1996

Abstract The homogenization kinetics of a cast Ti48At2W0.5Si alloy with a duplex microstructure was studied in terms of 7-phase dissolution and a-grain growth. It was found that the measured volume fraction of remnant y grains can be well simulated by a model of interface-controlled dissolution in a dislocation mechanism, instead of a diffusion-controlled one. The activation energy for the ~/7 interface reaction was found to be Q~nt= 476 kJ m o l - 1, which is much higher than the interdiffusion activation energy in TiA1 alloy. The grain growth of a phase during homogenization can be categorized into three stages. During the first stage, where the volume fraction of remnant ~ grains is higher than about 10%, the growth of c~ grains follows the parabolic law D = kit °2, and the activation energy for grain growth was calculated to be Q1 = 442 kJ tool-~, very close to the Q~nt for a/y interface reaction. In the second stage, where few fine 7 grains (1-t0 vol.%) remained, a dramatic grain growth occurs. During the final stage, as the single a phase is obtained, the coarsening of a grains again satisfies the grain growth law D = Ic3t°'4, with the grain growth activation energy Q3 of 147 kJ mol - 1, lower than the reported interdiffusion energy of 7 phase. Keywords: Homogenization kinetics; Titanium aluminides; Two-phase alloys

1. Introduction N e a r - G a m m a titanium aluminides are currently considered a promising high-temperature candidate material by aerospace and automobile industries. Recent investigations have shown that four types of microstructure can be produced in these two-phase alloys by static heat treatment or thermomechanical treatment, namely, the equiaxed dual-phase, the duplex, the near lamellar, and the fully lamellar structure [1]. However, due to the double-cascading peritectic reactions in TiAl-based alloys, severe solidification segregation exists in the cast ingot, which even fails to be broken down by extensive hot working [2,3]. Therefore, homogenization in the high-temperature alpha field is essential to eliminate this compositional heterogeneity [4]. To date, there are few studies [5-9] focused on the homogenization process of TiAl-base alloys, and only Semiatin et al. have examined the responsible mechanism. They concluded that dissolution of 7 grains dur-

* Corresponding author. State Key Lab for Advanced Metals and Materials, University of Science and Technology Beijing, Beijing 100083, People's Republic of China.

ing homogenization is an interface controlled process. But, no detailed models were proposed to describe the kinetics of 7-phase dissolution and e-grain growth during homogenization. Since the properties of TiAl-base alloys are strongly dependent on the lamellar grain size [10], it is critical to establish a model which can predict the homogenization time and the resulting lamellar grain size of TiAl-base alloys. The objective of this investigation is to develop such a model on the basis of experimental data.

2. Experimental The alloy used in this study was provided by ABB Power Generation Ltd. in cast bars with a diameter of 16 nun under the US patent No. 5286443. The chemical composition and the heat treatment of the as-received material are given in Table 1. Cylindrical samples (8 x ~b16 ram) were sectioned from the bars, and protected with a high-temperature glass-alumina coating. Homogenization was conducted in a MoSi2 box furnace at 1350-1390°C (_+ 5°C) for 2 - 2 4 0 min. The samples were directly put into the furnace at the preset temperature, held for a set time, 0921-5093/96/$I5.00 © 1996 - - Elsevier Science S.A. All rights reserved PII 80921-5093(96)10451-2

169

IT'.]. Zhang et al. / Materials Science and Eng#~eering A220 (t996) t68-175

Table 1 Chemical composition and heat treatment of the as-receivedalloy Composition (wt?/o)

Heat Treatment

At

W

Si

O

32.3 HIP HT

8.98 0.2 0.0632 1185°C/172 MPa/3 h 1302°C/20 h(Ar)/GFCa+913°C 4 h(Ar)/GFC

N

H

Ti

61 ppm

12 ppm

bal.

GFC, gas fan cooling. and then controlled furnace-cooled (40 K min - t to ll00°C and then air-cooled). By this cooling rate, the high-temperature a grains completely transformed into lamellar colonies. The microstructure of the heat treated samples was examined by an optical microscope equipped with an image analyzer. To avoid the uncertainty of the cast core structure, the lamellar (a) grain size and the area fraction of 7 grains were quantified in the central region of 10 mm diameter on the longitudinal section of specimen. The data given in this paper is the average of at least two measurements.

3. Results

The as-received material shows a duplex microstructure, comprising of nearly equal amount of equiaxed 7 and lamellar grains, with a mean size of about 100 gm (Fig. l(a)). The lamellar (prior alpha) grain sizes developed during homogenization at different times and temperatures are shown in Fig. 2. The corresponding volume fractions of remnant 7 grains are given in Fig. 3. As can be seen, the homogenization process can be divided into three stages. In the stage I, the c~ grain growth follows the parabolic law, initially increasing very rapidly and then being relatively stable, while the volume fraction of )J grains changes inversely. The partially transformed microstructure is shown in Fig. l(b). During the stage II, a substantial grain growth occurs, although the 7 grains have not yet dissolved completely, with the fine remainings pinned on the grain boundaries (Fig. l(c)). These two stages are the so-called dissolution process of ~ phase during homogenization. In the stage III, a single c~ phase is obtained, and homogenization is just the coarsening process of single a grains, as the ~ phase completely dissolved (Fig. 1(d)); again, the grain growth follows the parabolic law.

4. Discussion

As shown in Figs. 2 and 3, the homogenization of TiA1 alloys with a duplex microstructure involves two processes, i.e. dissolution of ~ grains and growth of

grains. We will discuss each separately in the following sections. 4.1. Dissolution o f ~ grains (stage I and II) 4.1.1. Dissolution mechanism

The dissolution of a second phase during homogenization is the result of interface migration from the matrix into second phase. Depending on the interface mobility, this transformation can be categorized into three types, namely, diffusion-controlled, interface-controlled, and mixed controlled [11]. In the case that the interface has a very high mobility, e.g. an incoherent interface, dissolution takes place under diffusion control, because the interface will advance as fast as diffusion allows. As the interface mobility is very low, the rate controlling process will be interface reaction, and the dissolution process is said to be interface controlled. In this circumstance, a very small concentration gradient is sufficient to provide the necessary flux of atoms to and from the interface, therefore, no apparent composition gradient can be detected in the matrix and the dissolved phase. In the studied TiA1 alloys, the cz phase has a disordered hexagonal structure and the 7 phase an ordered L10 tetragonal structure. The ~(cz2)/7 interface is coherent with the orientation relationship: (111)~//(0001)~ and (]-10)~//(1i20)~ [12-14]. Therefore, the dissolution of 7 into the ~ matrix is just the case discussed in [11], that the two phases have a coherent interface with different crystal structures, which is generally dominated by interface-controlled growth. As shown in Fig. 4, if the growth of c~ phase is to occur by individual atomic jumps (i.e. so-called continuous growth), then an atom on a C site in the • phase must change into a B position. This results in a very high energy, unstable configuration with two atoms directly above each other on B sites. It can thus be seen that continuous growth at the ~/7 interface is very difficult, and a tow mobility is expected. To avoid the above situation, a 'ledge' or 'dislocation' mechanism was proposed [11,15]. In this model, interface migration is facilitated by the transverse motion of the ledges, across which atoms can transfer more easily. The ledges have been observed

170

W.J. Zhang et al.

/Materials Science and Engineering A220 (1996) 168-t75

(d) Fig. 1. Optical microstructure of Ti48A12W0.5Si altoy: (a) as-received, and after homogenization at 1360°C for (b) 30 min, (c) 1 h, and (d) 4 h (a small amount of fl-phase existing on the grain boundary due to W addition).

practically in some aluminium alloys with a height of hundreds of atom layers [16,t7]. Interestingly, the presence of ledges has also been observed by high-resolution TEM on the perfectly coherent c~2/yinterface in the 0 0 0

E

* 0 • 0

,4

1390°C 1375°C 1360'0 1350°0

N

4v"

:' ,,'

,_=

: .'

.: /

<'0

03

-Stage I ...... Stage II -Stage I11

0 0 yi

t

10

100

lamellar structure [15,18]. The ledges are two {111} planes high, indicating that growth of the e2 phase occurs by the migration of Shockley partial dislocation ledges on alternate {111 }-t plane parallel to the interface [15]. In this view, it is reasonable to consider that the inverse y--->c~ transformation during homogenization occurs in the same manner, i.e. an interface-controlled process by a dislocation mechanism, just as the fcc hcp transformation between the disordered c~ matrix and y' plates in A1-Ag alloy [17]. This argument is further supported by some experimental results [5,6,13]. Semiatin et al. found that no apparent concentration gradient (or very small gradient below the accuracy of the microprobe technique) of aluminium was detected by EDS within both the transformed alpha phase and the remnant gamma grains, and the measured dissolution time of gamma grains can not be interpreted by a diffusion-controlled model, rather than by an interfacecontrolled one [5,6].

1000

Time (min) Fig. 2. Lamellar (prior c0 grain size as a function of homogenization time in Ti48A12W0.5Si alloy at temperatures above the alpha transus.

4.1.2. Simulation of 7 dissolution To describe the interface-controlled dissolution of a second phase of R radius, Aaron and Kolter [19] extended the model proposed by Brice [20]: the net flux of

W.J. Zhang et al. /Materials Science and Engineering A220 (199i5) i68-175

171

0

o • 0 *

0

0

,,~ ~'i~,

1350'C 1360*C 1375°C 1390'C

1400

(experiment) (experiment) (experiment) {experiment}

1

I

I

I

l

/

Ca 0 L-

I~ ',3, ~\ ', o',, ~ ',, "-,

g

t

m L_

p, • ,,..>...

e..

/

I '~,%

1300'

E E

/i I/i t*

/

E

1200

Co

i:r.

7

1125 °C 0

U2,

1100 35 0

10

30~

2JO

40~

50 ,

iI

,

40

,

,

45

I 5'0

55

60

Atomic percent

AI

Time (rain) Fig. 3. The measured and calculated area fraction of remnant 7 grains as a function of dissolution time. The measured data is well fitted by the calculation (dotted line) using an interface-controlled model in a dislocation mechanism,

Fig, 5, Prediction of the phase equilibrium of the studied alloy and depiction of the parameters used in the interface-controlled dissolution model.

where atoms across the interface by the dislocation (ledge) mechanism is determined by the form

A~=A[(C;

-

c ,~ ) / c ~~ ~

(1)

[111]

y--TiAI A B C A B

[002] B C A B C

J,

B B

0 ( - - TiAI

? [0001 ]

O •

@

AI

Ti

AI/Ti

Fig. 4. The migration mechanism of the a/Y coherent interface in TiA1 alloy, by individual atom 1 jumping from the position C to B, or by Shokley partial dislocation.

A = A o [ T 2 exp( - Q~nt/RT)]

(2)

In the equation, Q~,t is the activation energy for interface reaction, R the universal gas constant, T the temperature in Kelvin and A 0 a constant; C~ is the maximum solute concentration allowed in the matrix under equilibrium condition at temperature T, and CT is the actual concentration at time t (see Fig. 5). Because the occurrence of an interface reaction during dissolution, the value C~ is always lower than C~. To use Eq. (1) simulate the experimental result, both C~ and C~ must be firstly specified. Fig. 5 shows the portion of binary Ti-A1 phase diagram by McCullough et al. [21]. Our previous metallographic study [22] revealed that the alpha transus of the studied alloy (exact A1 content: 48.4 at.%) is close to 1347°C. The A1 content of the lamellar (prior ~) and equiaxed 7 grains in the as-received state (treated at 1300°C) measured by EDS is 47.1 +0.4 and 49.8 ___0.4 at.%, respectively. From these data, modification of 2 at.% W and 0.5 at.% Si addition on the Ti-A1 phase equilibrium was predicted in Fig. 5 as dotted lines. The c~ transus is lowered by addition of W, a beta-phase stabilizer, as observed in [23]. The corresponding data for simulation are determined from Fig. 5, as summarized in Table 2. In this treatment, only the A1 concentration is considered, and the partition of W(Si) atoms in the ~ and 7 phases during dissolution was neglected based on the following evidences. The W content of the y grains in the as,received alloy was measured to be 1.2 + 0.4 at.%, lower than that of the ~ (lamellae) grains (2.3_ 0.4 at.%), tf we suppose that the 7-+ dissolution can only proceed after the equilibrium W concentration is reached, then, the dissolution speed of y grains would depend on the flux rate of W atoms into

W.J. Zhdmg et al. / Materials Science and Engineering A220 (1996) 168-i75

I72

the undissolved 7 grains, consequently, on the concentration gradient 3C w of W between ~ and ? phase according to the Fick's law. Because the 5C w only varies slightly during dissolution, from 2 . 3 - 1.2 = 1.1 at.% at the time t = 0 , to 1 . 9 - 1 . 2 = 0 . 7 at.% in the final stage of dissolution (1.9 at.% is the final W content in single c~ phase, i.e. the composition of the bulk sample), the dissolution rate of ?' grains should not change very rapidly, which is inconsistent with the experimental results shown in Fig. 3. The experimental result [4] has actually confirmed that the equilibrium of W concentration is unnecessary during the transformation from 7 to ~ phase, because the strong W segregation was observed to be still retained, as a single c~ phase structure is achieved in Ti45.6A15.2Nbl. 1W alloy after homogenization at the temperature above its c~ transus. Additionally, it seems difficult to explain the large difference of the calculated activation energies for c~ grain growth in the first and final stages (see Section 2), if the slow diffusion of W atoms would be the rate controlling process. Finally, the dissolution of ? phase in Ti47A12.5Nb0.3Ta alloy was found to mainly depend on the diffusion of A1 atoms, rather than the diffusion of Nb or Ta atoms [5]. Therefore, it is reasonable to only consider the diffusion of A1 atoms in the interfacecontrolled model. Here to note that the interface controlled dissolution of y phase does not refute the diffusion of A1 atoms as operative, but only that the latter process goes very rapidly compared with the interface movement. During homogenization, the A1 content of ~ (lamellar) and 7 grains will tend to reach the equilibrium composition CG and C~, or say, the A1 content will c~ y increase from C~/C~ at time t = 0 to C~/C, at time t, at the expense of ? grain partially dissolved. Since the process is interface controlled, diffusion of AI will be very fast, giving rise to no apparent concentration gradient in the 7 and cz grains [5]. As depicted in Fig. 6, the A1 concentration of ¢ grain (R~' radius) and ? grain (R~ radius) at time t can thus be written as cg • . R C g,_~ + ~R) c , = c,_~ + G / ~ c L ~oR/(

(3)

and

C [ = C [ _ 1 '}- (1 - .Pce/y)C~' - l t } - R / ( ~ [ - 1 - {}R)

(4)

Table 2 Data for simulation of the y-phase dissolution Temperature (°C) C~ CYm .P~/~ ~ A (fitted)

1350 48.48 50.6 0.835 165

1360 48.73 50.8 0.84 184

t= 0

C~ = 47.1 R~ = R~ = 50 ~m

1375 49.1 51.2 0.92 273 C~ = 49.8 V~ = 50%

C~ equal to CYm at the point of 0.1% remnant 7 phase.

1390 49.4 51.6 0.7 391

c O

tm

ct "r o

lg

O o

C(

....................

Ct c~

Ct_l

m

< e

grain

3' grain

I* . . . .

Rt-1 ...... "*i-- 4 ÷ - - 8R t I a t.......... Rt . . . . . . -,-I* . . . . . . . RtV --*q

Distance

(,um)

Fig. 6. Schematic illustration of the dissolution of ? grain into the c~ matrix.

Where P~/y is the partitioning parameter of A1 between and ? grains; the value of P~/~ was chosen such that the A1 content of remnant 7 grains just prior to the complete dissolution is close to the equilibrium value C~. The term 6R is the dissolved radius of 7 grain in the time interval (t - 1) --* (t); it can be given as

bR/bt = Addis

(5)

where A1 is a constant. The mean radius R~ and the volume fraction V~ of remnant 7 grains at time t is thus obtained R~ = Ry_ t - ~R

(6)

and

v~ = A2(R,~)2

(7)

where A 2 is a constant. Using the parameters in Table 2, the measured area fraction of ? grains in Fig. 3 can be fitted by Eqs. (1-7) using the numerical method. As can be seen in Fig. 3, the proposed model gives a reasonably good description of the experimental data, suggesting that the dissolution of 7 grains in the matrix is an interface-controlled process by a dislocation mechanism. The inaccuracy of the predicted phase equilibrium, and the existence of few fine fl-phase grains on the grain boundary may be the sources of the deviation of the calculated curves from tile experimental results. The corresponding variations of A1 concentration in c~ and 7 phases during dissolution are shown in Fig. 7. The fitted constant A in Eq. (2) at different temperatures are also given in Table 2. By taking logarithm of Eq. (2), the activation energy Qint can be obtained from the equation 2 log T - - log X = - log A o + Qi,t/2.3RT

(8)

W.J. Zhang et at. Materials Science and Engineering A220 (1996) t68-175 0

10

1 ,

,

,

20

~. I

I

I

I

30 f

/ /

/

I

I

[

I

I

I

50 l

r

r

I

60 r

r

.., ,.,'""

•

,"'"

C,

...,'*

..~

o.

................

Lq.

49.8 ---1350 '(3 . . . . . . 1360 °C

o

....

o

Table 3 The measured and calculated dissolution time of 7 grains (min)

m

//

/'

/ /

I

40

... ........

0%.-~'~

........... 1375,C .... 1390'0

...... 48.4

..

--" ~

r~

~'

E E O

10

20

1350 ~180 144 2.0

1360 ~60 58 1.44

1375 ~30 I8 0.82

1390 ~15 8 0.44

"The time of remained y grains less than 1 vol.%.

p = [(C~m- C~)/2rc(C~ -- C~)] 1/2 and

o

Daif(gm/s) = 1.5 x 104 exp( - Q r / R T )

(9C)

o

(9d)

is the interdiffusion coefficient in 7 phase, with the interdiffusion activation energy Qr = 180 kJ mol-~ [25]. The other terms have the same meaning mentioned above. As given in Table 3, the calculated tdif by Eq. (9) is much lower than the measured one, in agreement with the conclusion in [5], whereas the dissolution times tint calculated by the interface controlled model are quite close to the measured one (the fitted curves are basically determined by the first three points, so that the predicted tint data is convincing). These results indicate that the diffusion of A1 is very fast during homogenization, and the dissolution process is interface-controlled, rather than diffusion-controlled. W h e r e D d tf

30

40

50

60

Time [rain} Fig. 7. The calculated AI concentration variation during dissolution using the partitioning parameters shown in Table 2.

As shown in Fig. 8, the liner regression equation gives Qim = 476 kJ t o o l - 1. Compared with the interdiffusion activation energy of A1 in 7 phase Qy= 180-190 kJ mol-~ [24,25], the high value of Qi~t provides another strong evidence that the rate-limiting process during dissolution of 7 phase is interface reaction. To further support our conclusion, the time for complete dissolution td~ is also calculated by the diffusioncontrolled model of Whelan [26], which was discussed in detail by Semiatin et al. [5]. The time tail is given by the equations In r~ = ( - 2/~)arctan (/

(9)

where "cC = [2D diftdif/( R~)2]( C~n -- C~)/( CYm - C ~ )

(9a)

( / = (1 - p 2)~/2/p

(9b)

4,4

y=-ll+476.Tx

Temperature (°C) Measured a Interface-controlled~ Diffusion-controlled

o c

47.1 0

173

r^2=0.981

4.2. Grain growth o f ~ phase

The growth of ~ grain in Ti48A12W0.5Si alloy during homogenization can be divided into three stages as shown in Fig. 2. During the first and final stages, the grain growth follows the parabolic law. As known, the grain growth in single-phase polycrystal can be expressed as D = kt ~

(10)

where D denotes the mean diameter of grains, t the time and k a constant; the exponent n, is theoretically equal to 0.5 [27]. If the growth is a grain boundary diffusion process, the constant k in the above equation can be replaced by

4,2

k = ko exp( - Q / R T )

(11)

O

,-I I I-

4.0

O

,.l CM 3,8

Q=476

3.6

0.0305

kd/mol

I

I

1

0,0310

0.0315

0,0320

0.032S

1/2,3RT

Fig. 8. Liner regression of the activation energy for interface reaction.

where k0 is a constant, Q the activation energy for grain growth, R the gas constant and T the temperature. Fitting the experimental curves in the stage I by Eq. (10), an exponent n close to 0.2 was obtained (Table 4). Taking n = 0.2, the constant k at different temperatures can be obtained. Using the logarithmic form of Eq. (11), the activation energy for ~ grain growth in the stage I was calculated to be Q1--442 kJ tool-1, as shown in Fig. 9. Following the same procedure, the exponent n was determined to be 0.4 in the stage III (Table 4), and the corresponding growth activation energy Q3 of 147 kJ mol-1, which is lower than the

W.J. Zhang et al. / Materials Science and Engineering A220 (I996) I68-175

I74

Table 4 The fitted constants in grain growth law Tenlperature (°C)

1350

1360

1375

1390

Stage I

n k (n = 0.2)

0.179 73

0.209 93

0.178 131

0.229 160

Stage III

n k (n = 0.4)

0.398 89

0.397 100

0.412 119

previous reported interdiffusion energy of 180-190 kJ m o l - i in v-phase TiA1 alloy [24,25]. The values of n observed in most experimental cases for single phase alloy are in the range of 0.35-0.45, smaller than the theoretical value 0.5 [27]. But for the alloys containing second-phase particles, the exponent n can be as low as 0.05. During the first stage, the volume fraction of remnant 7 is higher than 10%, accordingly, a relatively low exponent n of 0.2 was resulted, as the c~ grain growth is hindered by these remaining 7 grains. In the final stage III, as the 7 grains are completely dissolved, a value of n = 0.4 is obtained. This value is quite acceptable, taking into account the effect of W and Si solutes. It is important to note that the calculated activation energy Q1 for grain growth in the first stage is very close to the activation energy Qi~, for c~/7 interface reaction. It likely implies that the a grain growth is limited by the dissolution of 7 phase, namely, the a/7 interface reaction, rather than the diffusion of A1 or W atoms. During the stage II, a very rapid grain growth was observed. Our current results fail to answer positively what occurred in this period. However, as the fraction of remnant 7 in this stage is lower than 10%, it might be the effect of grain coalescence; or it satisfies the Zener pinning grain growth model, i.e. the grain radius

2.4

• Stage Ill: y3=6.62-146.7x Q3=147 kJ/mol

r^2=0.987

2,2 b¢ O)

0 -J

2.0

1.8

0,03

13 Stage I:

yl=16-442.1x r'2-0.965 Q1-442 KJ/mot

0.031

0.032

R~ocR./fy, where R~ and f~ denote the radius and volume fraction of remnant 7 grains [6,28]. 5. Conclusions

The homogenization kinetics of a cast Ti48A12W0.5Si alloy was studied at temperatures above the alpha transus. The mean size of e grains and the volume fraction of remnant 7 grains were measured at the temperature range of 1350-1390°C for 2-240 min. The homogenization process was discussed in terms of the dissolution of 7 grains and the grain growth of c~ phase: (1) The measured volume fraction of remnant V grains can be well described by a model of interfacecontrolled dissolution in a dislocation mechanism, instead of a diffusion-controlled one. The activation energy for c~/? interface reaction was found to be Qint = 476 kJ tool-1, much higher than the interdiffusion activation energy in TiA1 alloy. The mechanism of V phase dissolving in the a matrix was discussed. (2) The cz grain growth during homogenization can be categorized into three stages. In the first stage, as relatively large amounts of undissolved V grains remained, the grain growth follows the parabolic law D = lqt °2, and the activation energy for grain growth was calculated to be Q1 = 442 kJ m o l - 1, which is very close to the calculated Qint for a/? interface reaction, indicating that the c/ grain growth is limited by the dissolution of V grains. During the stage II, an abnormal grain growth of a phase occurs, with few fine V grains remained. In the final stage III, a single a phase structure is obtained, and the coarsening of a grain satisfies the grain growth law D = ]¢3t°'4, with the grain growth activation energy of Q3 = 147 kJ tool- 1, lower than the reported interdiffusion energy of V phase.

Acknowledgements 0.033

1/2,3RT Fig. 9. Liner regression of the activation energy for c, grain growth during homogenization.

The authors appreciate the funds of C.N.R. We are pleased to thank Dr Nazmy (ABB Power Generation Ltd, Baden, Switzerland) for providing the material. One author (Zhang) acknowledges the financial support

W.J. Zhang et aL/ Materials Science and Engineer# N A220 (1996) 168-t75

of the European Union, grant No. 93.2009.1L. We are also in debt to the elegant and stimulating arguments of the referee.

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