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

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4.8 Kinetics of Grain Growth Inhibited by Vacancy Generation 435

FIGURE 4.73

Dependence of non-dimensional grain size on the non-dimensional time cal-

culated from Eqs. (4.130) and (4.140) for the following values of parameters:

A =10, Ψ = 100 for two diﬀerent time scales [12].

These results have an interesting interpretation: a nanocrystalline mate-

rial cannot be stable if condition (4.142) or (4.139) (whichever is relevant)

is violated. In that sense, the critical radius

∗

can be regarded as a

limiting stable grain size above which grain growth uninhibited by vacancies

is possible. Obviously, as seen from (4.139) and (4.142) the critical grain size

is temperature dependent. As the activation energies for grain boundary mo-

bility are generally diﬀerent, the sense of the temperature variation of the

critical grain size will depend on the interplay of the temperature dependen-

cies of the mobility and self-diﬀusion coeﬃcient. It is emphasized that the

grain boundary mobility may depend very strongly on the type of the bound-

ary, the texture in the material and its purity (see Chapter 3), which makes

an estimation of the critical radius to a fairly diﬃcult task. In [12] such an

estimation was attempted for Al, the only material for which m

is known

with some conﬁdence. At 300

◦

C one obtains both

and

∗





1/3

the order of 100 nm. These estimates can, from our viewpoint, explain the

relative stability of nanocrystalline materials against grain growth.

The results of the described approach were compared with grain growth ex-

periments in nanocrystalline Fe [452]. As intimated by the authors of [452] the

theory developed is able to account for the growth rate discrepancy between

nanocrystalline and microcrystalline Fe without recourse to impurity eﬀects.

We would like to remind the reader that the results of [452] were considered

in the previous paragraph as an experiment, which supports the grain bound-

ary junction drag approach, predicting under deﬁnite conditions a linear de-

pendence between average grain size

R and annealing time. The vacancy drag

concept suggests also another explanation. A quantitative experiment only

can say which approach gives more adequate description of reality. The set of

Eqs. (4.135) and (4.136), or (4.132) and (4.133) yields solutions that predict a

436 4 Thermodynamics and Kinetics of Connected Grain Boundaries

FIGURE 4.74

Dependence of non-dimensional grain size on the non-dimensional time cal-

culated from Eqs. (4.130) and (4.131) for various levels of initial vacancy

supersaturation.

decrease of the average grain size for the case of an initial vacancy supersatura-

tion. In other words, it can be speculated that grain reﬁnement can be induced

by a high vacancy concentration, which can be created in diﬀerent ways, e.g.

by quenching from a high temperature, irradiation with energetic particles

or plastic deformation. An example of the computed variation of the average

grain size with time for three diﬀerent initial excess vacancy concentrations is

shown in Fig. 4.74. Although the magnitude of grain size decrease is small the

very possibility of vacancy-induced grain reﬁnement appears interesting. The

physical reason for this grain reﬁnement is similar to the well-known eﬀect of

grain reﬁnement during discontinuous precipitation [462], diﬀusion-induced

grain boundary migration [463] and discontinuous ordering reactions [464]. In

all cases the loss of energy associated with an increase of grain boundary area

is compensated for by the energy gain in the bulk due to decomposition of a

supersaturated solution, formation of a solid solution, or bulk ordering of the

quenched disordered alloy behind moving grain boundaries.

In [460] the discussed approach was applied to grain growth in thin ﬁlms,

one of the most important subjects of modern information technology. Grain

growth in thin ﬁlms has some distinctive properties, which are mainly con-

nected with the stresses developing in thin ﬁlms and with a strong inﬂuence

of the free surface on grain boundary motion [465, 466]. The consideration

of the ﬁrst feature dates back to Chaudhari [13, 14] [467]–[469]. First, the

cohesion between a ﬁlm and a substrate should lead to internal stresses in

4.8 Kinetics of Grain Growth Inhibited by Vacancy Generation 437

the ﬁlm. Second, under certain conditions an equilibrium grain size can exist

beyond which no grain growth occurs. The latter results from consideration of

the interplay between the elastic strain energy and the total grain boundary

energy during grain growth. The model proposed by Chaudhari appears to

be commonly accepted, see, e.g., a review [14]. In that approach it was tac-

itly assumed that the excess free volume of grain boundaries released during

grain growth disappears instantly, giving rise to a tensile stress. This strong

assumption limits the applicability of the model, as the free volume release

would normally occur via generation of lattice defects, notably the injection

of vacancies into the bulk of the material [457].

Let us consider brieﬂy the situation analyzed in [13, 14]. A polycrystalline

thin ﬁlm is deposited on a substrate, and we assume the ﬁlm is initially in

a stress-free state. In the course of grain growth the reduction of the excess

grain bounday free volume would lead to a decrease in the free volume, and,

as a result (at constant ﬁlm thickness), to a decrease in the area of the ﬁlm.

Then the stress in a given direction of the isotropic ﬁlm can be estimated as

σ =

2(1 − ν)



−



(4.143)

Here E/(1 − ν) is the biaxial elastic modulus expressed in term of Young’s

modulus E and Poisson’s ratio ν.

However, Eq. (4.143) going back to [13] does not take into account the elon-

gation of the specimen due to the generation of vacancies. One possibility is to

convert it into individual vacancies. As long as the vacancies thus generated

stay in the ﬁlm, no net volume change occurs. (This implies that the excess

free volume is considered to be equal to the total volume of vacancies gen-

erated, i.e. possible relaxation eﬀects are neglected.) However, when vacancy

removal by diﬀusion (e.g. to the ﬁlm surface) occurs, ﬁlm shrinkage due to

loss of grain boundary area will outweigh its extension due to vacancies. If an

individual vacancy leads to the specimen elongation αb,whereb is the atomic

spacing in the direction of interest and α is the numerical coeﬃcient of the

order of unity, the resulting stress reads [460]:

σ =

(1 − ν)





−



− α (c − c

)



(4.144)

The coeﬃcient α can be extracted from a simple consideration. In the case

when no vacancy removal occurs the stress should vanish:



−



= α (c − c

) (4.145)

On the other hand one has under such “adiabatic” condition

c − c

= V



−



(4.146)

438 4 Thermodynamics and Kinetics of Connected Grain Boundaries

From (4.145) and (4.146) we get α =0.5.

As mentioned above, Chaudhari’s approximation presumes that the excess

volume disappears instantaneously, while in our approach part of it is con-

tained in the vacancies. In this case the resulting stress can be expressed by

σ =

2(1 − ν)





−



− (c −c

)



(4.147)

The Gibbs free energy of the ﬁlm (per unit volume) can be written as a sum

of three contributions:

The elastic energy:

(1 − ν) (4.148)

the grain boundary free energy (per unit volume):

(4.149)

where γ is the boundary surface tension; the relation (4.149) is derived under

the assumption that the grains are cylindrical with height equal to the ﬁlm

thickness; and, ﬁnally the “vacancy” free energy:

vac

NkT

(c − c

)

(4.150)

Consider the derivative of the full Gibbs free energy G = G

+ G

vac

with respect to the average grain size R



NkT

(c − c

) −





(1 − ν)





−



− (c − c

)



− γ



(4.151)

The behavior of the system can be diagnosed by an analysis of the sign of this

derivative. The thermodynamic viability of the grain growth process requires

that the derivative be negative.

The evolution of the vacancy concentration with time t is given by

−

(c − c

) (4.152)

The ﬁrst term on the right-hand side represents the rate of vacancy gener-

ation accompanying grain growth (V being the grain growth rate, which is

proportional to the velocity of grain boundary migration). The second term

describes the vacancy removal by diﬀusion to the ﬁlm surface, with D

being

the vacancy diﬀusivity and d the ﬁlm thickness.

Two essentially diﬀerent situations of grain growth in thin ﬁlms were consid-

ered in [460]. When vacancy removal can be neglected (which can be referred

4.8 Kinetics of Grain Growth Inhibited by Vacancy Generation 439

to as the “adiabatic” case), σ = 0, and the last term in (4.152) can be dropped

leading to

(4.153)

Equation (4.151) then assumes the form



−γ +(V

)

NkT



−



(4.154)

It is seen that in the early stages of grain growth, when R is close to R

,the

derivative

is negative, meaning that the process is favorable thermody-

namically. However, at a certain critical value of R given by

1 −

γc

)

NkT

(4.155)

the derivative

vanishes. This critical value of R can be seen as an “equilib-

rium” grain size beyond which no growth will occur. Obviously, such a critical

grain size is only possible if the denominator of Eq. (4.155) is positive, i.e. if

)

NkT

γc

(4.156)

A simple estimate using representative values of the parameters

=2· 10

−11

,γ=1J/m

,T= 500K,c

=10

−8

shows that this condition is fulﬁlled for R

< 1.5 · 10

−5

m. The estimation is

given for V

=2· 10

−11

[460]. However, the direct measurements of

grain boundary excess free volume (Chapter 1) show that V

might be in the

range of 5 ·10

−11

, which gives us the value of R

equal to R

≈ 10

−4

Of course, this value is strongly temperature dependent, primarily through

the thermal equilibrium vacancy concentration. If inequality (4.156) is not

fulﬁlled, grain growth will be unrestricted.

The second situation, which can be called the “Chaudhari” approximation

[467], also follows from Eqs. (4.152) and (4.153). Indeed, assuming the exis-

tence of such a critical radius and thus setting V to zero and further assuming

that the vacancy concentration is equal to its thermal equilibrium value (a

tacit assumption in Chaudhari’s paper [467]) we ﬁnd that

∼

1 −

4γR

(1−ν)

E(V

)

(4.157)

This formula produces Chaudhari’s result [13] which was revisited by Thomp-

son and Carel [14].

The modiﬁed energy balance equation reads

=(1− ν) ·

dσ

NkT

(c − c

)

βδ

(4.158)

440 4 Thermodynamics and Kinetics of Connected Grain Boundaries

The set of Eqs. (4.158) and (4.153) with σ given by Eq. (4.144) describes this

problem completely.

A convenient non-dimensional form of this set of equations reads

dχ

− c (4.159)



A +(1−B)C −



−



dχ

= −



−

− C



+ Q



dχ



(4.160)

where C = c

− c, χ =

, χ

, A =

4γ

(1 − ν),

B =

4NkT

(1 − ν)andQ =

4(1 − ν)V

(4.161)

is the vacancy diﬀusivity, d is the sink spacing.

The set of equations (4.159) and (4.161) describing the evolution of the

grain size and the vacancy concentration provide a full description of the sys-

tem [460]. Some results of a detailed numerical study are given below.

The maximum growth rate is observed at the very beginning of the process.

An almost uninhibited, nearly parabolic growth regime is observed at large

times (Fig. 4.75).

The length of the plateau-like incubation stage depends on the value of

model parameters and the initial grain size χ

. An interesting feature should

be noted — during the incubation time the vacancy concentration remains

constant (Fig. 4.76). The end of the incubation period is signiﬁed by a pre-

cipitous drop in C and a peak in the time dependence of the grain growth

rate

dχ

(Fig. 4.77) [460]. The dependence of the incubation time τ on 1/χ

is shown in Fig. 4.78; this dependence can be represented by a simple formula

τ =

(4.162)

It should be stressed that τ is not sensitive to the dissipation parameter Q.

The formula (4.162) can be rewritten as a dimensional relation

inc.

)

NkT

γR

(4.163)

expressing the incubation time τ

inc.

in terms of the physical parameters of the

system, temperature and sink spacing [460].

The thermodynamic-kinetic approach developed in [11], [460] makes it pos-

sible to give a full description of the evolution of the average grain size for

many systems. Similar consideration can be applied to grain growth in rein-

forcement particles in a composite to grain growth in a multiphase system, or

the kinetics of void dissolution [459].

4.8 Kinetics of Grain Growth Inhibited by Vacancy Generation 441

012345

5x10

=2.5x10

Non-dimensional grain size/10

Non-dimensional time/10

FIGURE 4.75

Variation in the grain size time for various values of the initial grain size. (All

quantities are given in dimensionless units [460].)

012345

2x10

-8

4x10

-8

6x10

-8

8x10

-8

1x10

-7

2.5x10

=5x10

Excess vacancy concentration

Non-dimensional time/10

FIGURE 4.76

Evolution of the excess vacancy concentration with the non-dimensional time

for three diﬀerent values of the initial grain size [460].

442 4 Thermodynamics and Kinetics of Connected Grain Boundaries

012345

5x10

=2.5x10

/dt)

Non-dimensional time/10

FIGURE 4.77

The grain growth rate as a function of time for three diﬀerent values of the

initial grain size. (All quantities are given in dimensionless units [460].)

0 5 10 15 20

A=50

A=5

Incubation time/10

/Initial grain size

FIGURE 4.78

The initial grain size dependence of the incubation time. (Note the diﬀerent

scales for the two diﬀerent values of the parameter A used [460].)

4.8 Kinetics of Grain Growth Inhibited by Vacancy Generation 443

4.8.1 Diﬀusion-Controlled Creep in Nanocrystalline

Materials under Grain Growth

The developed approach to the processes in materials where vacancy gener-

ation is considered as an integral part of grain growth was applied to creep

in nanocrystalline materials governed by diﬀusion [558]–[560], [563]. Let us

consider a frequent situation in polycrystals when creep deformation occurs

in parallel with grain growth.

The amount of vacancy supersaturation and the corresponding grain size

variation, as mentioned before, can be described by a set of coupled diﬀer-

ential equations (Eqs. 4.130, 4.131, and 4.158). One expresses the balance of

energy associated with an increment dR of the mean grain size R,andthe

other describes the change of vacancy concentration in the bulk.

As shown in [561] for nanocrystalline materials it is natural to assume that

no vacancy sink of dislocations type are available in the bulk of a grain, in

other words, the sink spacing d can be identiﬁed with average grain size R.

The vacancies generated at grain boundaries will be ﬁrstly adsorbed by the

boundaries and then directed to the free surface of a sample by boundary

diﬀusion. Nevertheless, the main concept of the approach developed in [11],

[458]–[460] is valid in this case as well. Although the concept discussed above

related to non-equilibrium processes, like grain growth in polycrystals and in

thin ﬁlms on a substrate, void dissolution, and sintering, it considered the

phenomena in terms of equilibrium thermodynamics, i.e. the change of va-

cancy concentration of the system was felt as a change of the vacancy part of

the free energy. So, in our case the increase in the local vacancy concentra-

tion in a boundary results, in accordance with equilibrium between the grain

boundary and the adjacent bulk, in an increase in the vacancy concentration

in the grain.

We would like to remind the reader that qualitatively the physics of the

phenomenon can be described as follows. Grain growth is suppressed over a

characteristic incubation time t

incubation

, followed by “normal” grain growth

characterized by a parabolic time law. During this time, the average vacancy

concentration in the bulk is maintained at a nearly constant level above the

equilibrium one. After the incubation time the vacancy concentration drops

down to thermal equilibrium.

The vacancy concentration during the incubation time can be estimated by

setting the rate of the concentration variation ˙c to zero and substituting for

dR/dt an expression following from [460]

dR/dt =

γD

NkT(V

)

(4.164)

The resulting vacancy concentration reads

c =



4NkT(V

)



(4.165)

444 4 Thermodynamics and Kinetics of Connected Grain Boundaries

Estimates show that the second term stemming from the vacancy generation

eﬀect is at least an order of magnitude larger than the ﬁrst one so that

c =

4NkT(V

)

≡ λc

(4.166)

holds. At ﬁrst glance, it may appear strange that the magnitude of the eﬀect

is inversely proportional to the “vacancy eﬃciency” of grain boundaries rep-

resented by the grain boundary excess free volume V

. However, the level of

vacancy supersaturation is the result of an interplay between the number of

vacancies injected in the bulk due to the shrinkage of the total volume of the

boundaries and the rate of this shrinkage. While the ﬁrst quantity is propor-

tional to V

, the latter is inversely proportional to the square of V

, as seen

from Eq. (4.164). It should be noted that the time

incubation

=24

NkT(V

)

γD

(4.167)

during which the increased vacancy concentration is maintained is propor-

tional to (V

)

.HereR

is the initial grain size.

We would like to discuss here one long-standing misconception in the sci-

entiﬁc literature. If we take into account that the vacancy concentration is

increased during grain growth the expressions for Nabarro-Herring or Coble

[558]-[560] creep need to be modiﬁed. However, according to a standard text-

book treatment of diﬀusional creep, no eﬀect of this enhancement of the va-

cancy concentration would follow. It is customary to attribute diﬀusional creep

to the non-uniform vacancy concentration in an inhomogeneously stressed

solid. In this case the diﬀusive ﬂux is determined by the gradient of the va-

cancy concentration and an overall increase in the vacancy concentration, as

with grain growth-induced vacancies, would not aﬀect the creep rate. This

concept is physically wrong, though, because the vacancies in an inhomoge-

neously stressed solid are in thermodynamic equilibrium and, therefore, their

chemical potential is zero everywhere, even though their concentration is non

uniform. As Herring [559] pointed out correctly, it is the inhomogeneous chem-

ical potential distribution of the atoms that gives rise to a diﬀusion ﬂux in

diﬀusion creep. In the presence of a stress σ the chemical potential of an atom

is changed by an amount σΩ. A uniaxial stress causes a chemical potential

gradient of the order σΩ/R on a grain scale, which gives rise to a diﬀusion

ﬂux, and hence, diﬀusion creep. The overall vacancy concentration does af-

fect the creep rate since the atomic mobility is proportional to the total local

vacancy concentration. The atom ﬂux across a grain is then given by

j =

N ·

Ωσ

(4.168)

where D denotes the bulk diﬀusivity of atoms. For the vacancy mechanism of

bulk diﬀusion it can be written as

D = D

c (4.169)