Gottstein G., Shvindlerman L.S. Grain Boundary Migration in Metals: Thermodynamics, Kinetics, Applications

Подождите немного. Документ загружается.

234 3 Grain Boundary Motion

1.2 1.25 1.3 1.35 1.4 1.45

/T [K]

-10

-9

-8

-7

-6

[

]

1E-010

1E-009

1E-008

1E-007

1E-006

38.2°<111>; Al II

40.5°<111>; Al II

1.1 1.2 1.3 1.4 1.5 1.6

/T [K]

-10

-9

-8

-7

-6

[

]

1E-010

1E-009

1E-008

1E-007

1E-006

38.2°<111>; Al I

40.5°<111>; Al I

38.2°<111>; Al IV

40.5°<111>; Al IV

1.15 1.2 1.25 1.3 1.35 1.4

/T [K]

-10

-9

-8

-7

-6

[

]

1E-010

1E-009

1E-008

1E-007

1E-006

38.2°<111>; Al V

40.5°<111>; Al V

1.15 1.2 1.25 1.3 1.35 1.4

/T [K]

-10

-9

-8

-7

-6

[

]

1E-010

1E-009

1E-008

1E-007

1E-006

38.2°<111>; Al III

40.5°<111>; Al III

/s]

-7

-6

/s]

38.2° <111>; AI IV

40.5° <111>; AI IV

-9

-8

-7

-10

40.5° <111>; AI I

38.2° <111>; AI I

600 550 500 350400

38.2° <111>; AI II

40.5° <111>; AI II

450 550 500 450

T [°C]

-6

-8

-9

-10

1.20

1.25 1.30 1.35 1.40 1.451.1 1.2 1.3 1.4 1.5 1.6

T [°C]

(b)(a)

550 450500

-6

-7

-8

-9

-10

1.15 1.2

(c)

1.25 1.3 1.35 1.4

40.5° <111>; AI III

38.2° <111>; AI III

/s]

40.5° <111>; AI V

38.2° <111>; AI V

550 500 450

-6

-7

-8

-9

-10

1.15 1.20 1.25 1.30 1.35 1.40

1000/T [1/K]

T [°C]T [°C]

1000/T [1/K]

1000/T [1/K] 1000/T [1/K](d)

/s]

FIGURE 3.61

Temperature dependence of the reduced mobility A

for 38.2

◦

and 40.5

◦

111

tilt grain boundaries in diﬀerent types of pure aluminum.

by continuous tracking of the moving grain boundary (XICTD). The exper-

iments were carried out on bicrystals produced from high purity aluminum

of diﬀerent production. The total impurity content in each material was de-

ﬁned as the sum of the concentration of all elements found as determined by

glow discharge mass spectrometry (Table 3.4 [197, 255]). With regard to the

question which elemental concentration to consider as most relevant for the

overall migration behavior of the boundary, it was proposed to consider the

total impurity concentration as the relevant concentration for grain boundary

migration. There are several reasons to justify this assumption. All materials

used in the investigation were high purity aluminum, which were produced by

multiple pass zone reﬁning. During zone melting only those impurities which

strongly interact with lattice defects remain in the solid phase, the structure

of which is more disordered than the crystalline solid and, therefore, more

akin to a liquid.

The temperature dependencies of the reduced mobility (Sec. 3.5.2) of 38.2

◦

and 40.5

◦

111 tilt grain boundaries in aluminum of diﬀerent purity and the

concentration dependence of the activation enthalpy for these boundaries are

3.5 Experimental Results 235

represented in Figs. 3.61 and 3.8a, respectively. As evident from Fig. 3.8a,

even at the lowest impurity content, i.e. in the material of highest purity, the

activation enthalpy rises with increasing impurity concentration contrary to

predictions of impurity drag theory (Sec. 3.3.2).

There are several reasons for the discrepancy between the theories of impu-

rity drag and the experimental data. All theories are based on the assumption

of a small concentration both in the bulk and in the grain boundary. How-

ever, the boundary impurity concentration may be high despite a small bulk

impurity concentration. Furthermore, for high impurity concentrations in the

boundary it is necessary to take into account the mutual interaction of ad-

sorbed atoms in the boundary. Also, as shown experimentally [285, 286] and

theoretically [35], grain boundaries are inhomogeneous, i.e. not every site in

the grain boundary is equally favorable for impurity segregation. Thus the

interaction between the adsorbed atoms also should be taken into account.

The interaction of impurities with grain boundaries can be taken into ac-

TABLE 3.4

Impurity Content of Investigated Aluminum Samples

Material RRR Measured

impurity

content

(ppm)

Al I 21400 0.4

Al II 15000 1.0

Al III 5100 3.6

Al IV 6500 4.9

Al V 11500 7.7

(RRR — residual resistivity ratio = R (293K)/R(4.2K)).

count in terms of adsorption [197, 255]. The advantage of such an approach is

that it allows us to express the inﬂuence of adsorption on both the activation

enthalpy and the pre-exponential mobility factor. As shown above (Sec. 3.3.3),

impurity drag theory can be extended to take into account an interaction of

adsorbed impurities in the grain boundary and thus, provide a reasonable

agreement with experimental results for the concentration dependence of the

activation enthalpy and also for the pre-exponential mobility factor (Figs. 3.9

and 3.10). It should be noted that the chosen values of the parameters in

the equation of grain boundary mobility are physically reasonable. For exam-

ple, the enthalpy of adsorption in the grain boundary ranges from 0.82 eV

to 0.86 eV, the product of the coordination number and the heat of mixing

extends from 0.17 eV to 0.24 eV etc.. We emphasize once again that the best

agreement between theory and experiment can be achieved if the diﬀerence

in the partial activation areas ω

of the adsorbed atoms in the grain bound-

236 3 Grain Boundary Motion

ary is taken into account. Best agreement is obtained for a ratio β =0.05 of

the partial areas of impurity atoms and matrix atoms (see Sec. 3.3.3). The

boundary activity of the impurities can be described by the derivative of the

boundary surface tension γ with respect to the chemical potential μ

of the

impurity atoms

dγ

dμ

∼

−

(3.160)

This relation indicates that the impurity which most actively interacts with

the boundary should possess the smallest partial area. Despite the obvious

importance of the partial area in grain boundary thermodynamics, ignoring

it is the rule rather than the exception in current grain boundary physics.

The approach discussed in Sec. 3.3.3 principally allows us to explain the ori-

entation dependence of migration activation parameters in diﬀerent regimes of

impurity concentration (Fig. 3.58, Table 3.4). Fig. 3.62 shows the migration

activation enthalpy as a function of impurity concentration for special and

random grain boundaries using the Fowler-Guggenheim isotherm under the

natural conditions for the heat of mixing ε = ε

special

<ε

random

(the diﬀerence

between ε

special

and ε

random

is small), H

rand

. It is seen that the result

of the calculation complies with the scheme in Fig. 3.58. Actually, for a very

low impurity concentration (c

) the migration activation enthalpy for a ran-

dom grain boundary is lower than for a special one. The rate of increase of

the migration activation enthalpy with impurity content for a random grain

boundary is higher than for a special one, and at a deﬁned concentration c

the enthalpy for both boundaries becomes equal. There is a large gap be-

tween the values of enthalpy of activation at a concentration c

which belongs

to regime 2 in Fig. 3.58. With further increase in the impurity concentration

the migration activation enthalpies also become equal (c

, case 1 in Fig. 3.58).

3.5.6 Impurity Drag and Breakaway

According to the previous sections (Sec. 3.5.3–3.5.5) both pure metals and di-

lute alloys have to be treated as solid solutions with regard to grain boundary

migration because of the impact of segregation. In this section we shall con-

sider the results obtained on single-phase materials more quantitatively in the

framework of the impurity drag theories as put forward by L¨ucke and Detert,

Cahn, L¨ucke and St¨uwe, Westengen and Ryum [193]–[195], [287] (Sec. 3.3.2).

A comparison of theory and experiment can be carried out in two principally

diﬀerent directions. The ﬁrst way is the analysis of the experimental data in

terms of impurity drag, i.e. the analysis of the changes in grain boundary ve-

locity, mobility, migration, energy (enthalpy) of activation as a function of the

nature and content of impurity atoms. The theories of impurity drag include

the nature of the impurities as the energy of interaction U

between the im-

purity atoms and grain boundary and the volume diﬀusion coeﬃcient of the

3.5 Experimental Results 237

log c

random

special

log c

random

special

FIGURE 3.62

Computed dependence of activation enthalpy of grain boundary motion on

impurity concentration for special and random grain boundaries.

impurities. The second way consists of searching for eﬀects predicted by the

theories, in particular the eﬀect of detachment of the moving grain boundary

from adsorbed atoms. The requirements which are imposed by the theory on

experiments are tough: exactly known driving force of grain boundary migra-

tion; well-deﬁned grain boundary crystallography and geometry; physically

correct and reproducible experiments; and, ﬁnally, what is of critical impor-

tance, a pure metal doped with a deﬁned content of speciﬁc impurities. There

are no investigations capable of complying with all requirements mentioned,

and we shall only consider those closest to the ideal, i.e. an extremely small

number of investigations dedicated to the interaction of a moving single grain

boundary with impurities. We even will violate our principles and discuss

some experiments on polycrystals.

The concentration dependence of the grain boundary migration rate in lead

doped with Sn, Ag, Au is represented in Fig. 3.63 [277]. This dependence is

in qualitative agreement with the L¨ucke-Detert relationship (strictly speak-

ing, according to the L¨ucke-Detert theory the velocity should vary inversely

proportionally to the impurity content of the sample, but for a narrow range

of impurity content and a low concentration the predicted dependency can

be approximated by the observed exponential variation) and was observed

for all impurities studied by Aust and Rutter [277]. Evidently, the inﬂuence

of silver and gold is much stronger than the inﬂuence of tin. A reasonable

agreement between theory and experiment was also observed by Frois and

Dimitrov [288, 289] who studied grain growth in aluminum doped with Cu

and Mg (Fig. 3.64). Two branches of the concentration dependence were ob-

served: at low concentrations the velocity of grain growth did not depend on

238 3 Grain Boundary Motion

[

]

(Sn;Ag;Au)

[10

-4

at.%]

0.1

1.0

0 0.8 1.2 2.01.60.4

[

]

(Sn;Ag;Au)

[10

-4

at.%]

0.1

1.0

0 0.8 1.2 2.01.60.4

FIGURE 3.63

Grain boundary migration rate vs. impurity concentration in lead crystal for

special (low Σ) (top curve) and non-special (bottom curve) boundaries [277].

the impurity content, whereas at larger concentrations the velocity decreased

linearly with increasing concentration.

The concentration dependence of the activation enthalpy of grain boundary

migration comprised two diﬀerent regimes: at low content of Mg the enthalpy

of activation did not depend on concentration and equaled the value observed

for grain growth in pure Al (0.65eV). With rising concentration there was an

abrupt change of the activation enthalpy to 1.5eV, i.e. close to the activation

enthalpy of bulk self-diﬀusion [288, 289].

“Pure” Al doped with silver (Fig. 3.65) exhibited a slightly diﬀerent con-

centration dependency. The enthalpy of activation rose with increasing Ag

concentration, starting from a certain critical content of silver. A similar be-

havior was observed by Aust and Rutter for the concentration dependence

of the migration activation enthalpy of single grain boundaries in lead doped

with silver and gold (Fig. 3.66) [277]. Note the dramatic change in the migra-

tion activation enthalpy in Fig. 3.66.

The mobility of random grain boundaries and 100 non-special tilt grain

boundaries in aluminum bicrystals of diﬀerent purity was studied in a con-

centration range of solute atoms between 5 · 10

−5

and 2 · 10

−2

at% princi-

pal impurities in at%: Si (5 · 10

−6

− 1 · 10

−3

); P (5 · 10

−6

− 5 · 10

−3

), Zn

(1 · 10

−6

− 1 · 10

−3

), Ti (1 · 10

−6

− 5 · 10

−3

); Fe (5 · 10

−5

− 5 · 10

−4

); Cu

(1 · 10

−6

− 1 · 10

−2

) [227, 273]. The concentration dependence of the acti-

vation enthalpy for the motion of a single random grain boundary is shown

in Fig. 3.67a [227, 273]. The curve can be subdivided into three segments.

The ﬁrst segment is characterized by an activation enthalpy independent of

impurity content. The second segment stands out by a considerable increase

3.5 Experimental Results 239

-1

-3

-5

-7

-9

0.1 1 10010 1000

v [cm/s]

c [10

-6

at.%]

transition region (Mg)transition region (Cu)

growth-type 1

growth-type 2

annealing at 132°C

-1

-3

-5

-7

-9

0.1 1 10010 1000

v [cm/s]

c [10

-6

at.%]

transition region (Mg)transition region (Cu)

growth-type 1

growth-type 2

annealing at 132°C

FIGURE 3.64

Concentration dependence of grain growth rate in Al doped with Cu and

Mg [288, 289].

H [kJ/mol]

H [eV]

-6

-5

-4

-7

-3

[at.%]

(a)

0.5

1.0

0.5

100

H [eV]

-6

-5

-4

-3

H [kJ/mol]

[at.%]

(b)

150

1.5

1.0

100

H [kJ/mol]

H [eV]

-6

-5

-4

-7

-3

[at.%]

(a)

0.5

1.0

0.5

100

H [eV]

-6

-5

-4

-3

H [kJ/mol]

[at.%]

(b)

150

1.5

1.0

100

FIGURE 3.65

Enthalpy of activation of grain growth in Al doped with Mg and Ag [288, 289].

240 3 Grain Boundary Motion

0 0.2 0.4 0.6 0.8

0.9

0.6

0.3

1.2

1.5

c [ppm]

[eV]

1.0

[kJ/mol]

150

120

0 0.2 0.4 0.6 0.8

0.9

0.6

0.3

1.2

1.5

c [ppm]

[eV]

1.0

[kJ/mol]

150

120

FIGURE 3.66

Concentration dependence of migration activation enthalpy in lead doped with

Ag and Au [277].

in the activation enthalpy with rising solute content. Such variation in the

activation enthalpy can be described [227] on the basis of a concept of ad-

sorption of interacting atoms on a non-homogeneous boundary. In the third

segment of the H

gration does not only fail to increase with growing concentration, but even a

slight decrease in the activation enthalpy is observed. A maximum value of the

atoms contribute to the increase of the migration activation enthalpy, since

only a fraction

Γ < Γ

of impurity atoms moves jointly with the boundary.

Therefore, when the boundary adsorption Γ tends to saturate, the activation

enthalpy attains a maximum. A respective calculation predicts a reasonable

concentration for the maximum (∼ 10

−4

− 10

−3

at% [273]). The results ob-

tained from bicrystal experiments for random grain boundaries (Fig. 3.67a)

and 100 tilt boundaries in Al (Fig. 3.67b) agree with data of the activation

enthalpy of grain growth in aluminum doped with copper as measured by

Gordon and Vandermeer [290, 291] (Fig. 3.67c).

Thus far the impurity inﬂuence on migration of random grain boundaries

(or grain boundaries far from a coincidence misorientation) has been consid-

ered. That is the reason why the results of single grain boundary migration

were considered in context with grain growth, i.e. “polycrystal” experiments.

Apparently, in some investigations a reasonable agreement between experi-

ment and predictions of the impurity drag theory was observed. However,

the majority of experimental studies reveals fundamental and essential dis-

3.5 Experimental Results 241

c [at.%]

[kJ/mol]

[eV]

-5

-4

-3

-2

-1

100

200

(a)

c [at.%]

-5

-4

-3

-2

-1

100

200

[kJ/mol]

[eV]

(b)

H [eV]

(c)

100

200

H [kJ/mol]

c [at.%]

-5

-4

-3

-6

-7

(estimated)

E + Q

Cu in Al

c [at.%]

[kJ/mol]

[eV]

-5

-4

-3

-2

-1

100

200

(a)

c [at.%]

-5

-4

-3

-2

-1

100

200

[kJ/mol]

[eV]

(b)

H [eV]

(c)

100

200

H [kJ/mol]

c [at.%]

-5

-4

-3

-6

-7

(estimated)

E + Q

Cu in Al

FIGURE 3.67

Migration activation enthalpy for random (a) and non-special (b) 100 tilt

boundaries in Al of diﬀerent total impurity content. (c) Enthalpy of activation

for grain growth in zone reﬁned Al doped with Cu [290, 291].

242 3 Grain Boundary Motion

crepancies between theory and experiment, in particular a much too strong

dependence of the activation enthalpy on the impurity content. We shall in-

vestigate whether such behavior is only a consequence of the nature of random

grain boundaries or whether it also pertains to special boundaries.

Theory predicts that a grain boundary breaks away from an impurity cloud

at large velocities and high temperatures. A study of this breakaway phe-

nomenon allows us to extract additional information on the interaction of a

moving grain boundary with impurities. In the original L¨ucke-Detert theory

detachment occurs when the impurity atoms are not able to move with the ve-

locity prescribed by the boundary. The more reﬁned theories of L¨ucke-St¨uwe

and Cahn took into account that the concentration not only decreases with

increasing temperature, but also with increasing boundary migration rate (see

Sec. 3.3.2) which considerably speeds up detachment and results in a discon-

tinuous breakaway of the boundary from the adsorbed impurities.

The abrupt change in the velocity of a grain boundary during its detach-

ment from an impurity cloud can be considered as a kinetic phase transition

[292]. An experimental support of this viewpoint is the observed hysteresis of

the transition temperature. On heating, the detachment occurs at a higher

temperature than the re-attachment of the impurities to the boundary on

cooling [293, 294].

Also important is the dependence of the detachment parameters on the

ratio of grain boundary to bulk diﬀusion coeﬃcient: D

/D.Anincreaseof

/D shifts the detachment point toward higher driving forces. This has an

obvious physical explanation: the impurity atoms near the boundary become

more mobile as D

/D increases, and continue to move with the boundary at a

greater velocity, corresponding to a larger driving force. There is, however, an

interesting theoretical inconsistency. Having compared theoretical predictions

with experimental results, we see that for Al already at D

∼

the de-

tachment temperature exceeds the melting point. The calculated detachment

temperatures agree with those observed, if the diﬀusion coeﬃcient D

∼ 10D.

Evidently, D

cannot be identiﬁed with the coeﬃcient of impurity diﬀusion

along the boundary, since at the respective temperatures the boundary diﬀu-

sion coeﬃcient is usually 10

− 10

times the bulk value [295].

The eﬀect of grain boundary detachment from impurities was experimen-

tally observed and studied on single 111 tilt grain boundaries in gold

[296] and aluminum [293, 294, 297]; 10

10 and 11

20 tilt boundaries in

zinc [292, 293, 298, 299]; and 110 tilt boundaries in Fe-3%Si [300]. Figs. 3.68-

3.72 give the temperature and driving force dependencies of grain boundary

velocity (mobility) in gold, aluminum, zinc and Fe-3%Si. A characteristic fea-

ture of the dependencies considered is a dramatic change in the grain bound-

ary velocity (more correctly, mobility) in a narrow temperature range. Above

and below the breakaway regime the grain boundary mobility shows a usual

Arrhenius type temperature dependence.

A considerable body of experimental data has been accumulated to date.

The eﬀect was observed on special and close to special grain boundaries, but

3.5 Experimental Results 243

1.701.65 1.75 1.90 1.951.80 1.85 2.0

-2

-3

-4

-5

1000/T [1/K]

v [mm/min]

T [°C]

350

330 310

290 270 250 230

= 0.83eV

= 1.33eV

1.701.65 1.75 1.90 1.951.80 1.85 2.0

-2

-3

-4

-5

1000/T [1/K]

v [mm/min]

T [°C]

350

330 310

290 270 250 230

= 0.83eV

= 1.33eV

FIGURE 3.68

Temperature dependence of the migration rate of 111 tilt boundaries (ϕ =

◦

) in rolled gold [296].

1.4 1.5 1.6 1.7 1.8 1.9 2 2.1 2.2

1E-005

0.0001

0.001

0.01

0.1

1000/T [1/K]

1.6 1.8 2.0 2.2

/s]

-6

-7

-8

-9

1.4

440 350 280 230 180

T [°C]

1.4 1.5 1.6 1.7 1.8 1.9 2 2.1 2.2

1E-005

0.0001

0.001

0.01

0.1

1000/T [1/K]

1.6 1.8 2.0 2.2

/s]

-6

-7

-8

-9

-6

-7

-8

-9

1.4

440 350 280 230 180

T [°C]

FIGURE 3.69

Temperature dependence of the reduced mobility of 30

◦

10

10 tilt grain

boundaries in Zn.