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

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4.7 Grain Growth in 3D Systems 425

FIGURE 4.68 (SEE COLOR INSERT FOLLOWING PAGE 424)

Volume rate of change as a function of the topological class; the lines represent

the analytical approaches of Mullins, Rios, and Glicksman, and Hilgenfeldt et

al. The normalized volume rate of change was calculated directly from the

change of grain volume with time [554].

and computer simulations signiﬁes that there is no constant growth rate for

a given topological class. Hence, grains with the same number of triple junc-

tions may exhibit diﬀerent growth rates depending on the diﬀerence between

the mean width and the length of the triple junctions. Since this is the case,

none of the approaches that formulate the growth rate as a function of the

topological class (4.99, 4.102, 4.103) can be entirely valid for single crystals,

since they can yield a correct result only under very speciﬁc conditions, i.e. if,

at a certain time, there are grains fulﬁlling the relations. In order to test the

predictive power of these approaches, the volume rate of change as a function

of the topological class is plotted for these relations in Fig. 4.68 and compared

to simulation results ([554]). It is evident that each topological class can have

various growth rates in accordance with the Cahn-MacPherson-Srolovitz rela-

tions (4.104, 4.108). The mean normalized volume rate of change

−1/3

for each topological class is represented as a solid black circle, whereas open

circles represent the mean according to the Cahn-MacPherson-Srolovitz rela-

tion that was calculated from the mean L(D), e(D), and volume for each topo-

logical class, i.e.

−1/3

=2πm

γ (L(D)

− 1/6 a(D)

) · <V >

−1/3

If only the mean values for each topological class are compared, all approaches

show very good agreement with the simulation results. This has also been

pointed out by Wakai [555], Weygand [556] and in phase-ﬁeld simulations by

Kim [557]. The mean value determined by the Cahn-MacPherson-Srolovitz

426 4 Thermodynamics and Kinetics of Connected Grain Boundaries

relation renders also a good agreement; however, it must be stressed that

this relation is not meant to be used as a general descriptor of the average

−1/3

The main property which distinguishes the observed analytical geometrical

approaches is that all of them, except the Cahn-MacPherson-Srolovitz rela-

tion, do not include the metrics of the grain.

The theories of Mullins, Glicksman and Rios and Hilgenfeldt et al. are con-

structed by analogy with the Von Neumann-Mullins approach, they are purely

topological. That is why the major weakness of the approaches mentioned

are not quantitative discrepancies in computer simulations. All of these ap-

proaches give asymptotically a reasonable rate of the grain volume evolution,

but in the absence of metrics. This is the reason why it has been impossi-

ble in the frame work of these approaches to consider problems connected

to grain growth in systems with ﬁnite mobility of grain boundary junctions.

The inﬂuence of the metrics comes particularly into play when it is impossible

to assume that grain boundary junctions have an inﬁnite mobility. Such an

attempt was undertaken in [542].

When deriving Eq. (4.104) Cahn, MacPherson and Srolovitz assumed that

the dihedral angles at triple lines attain their equilibrium value of 120

◦

.Con-

sequently, the angle between triple lines at quadruple junctions must reach

the tetrahedral angle of ∼ 109.47

◦

[542]. Since a ﬁnite mobility of the junc-

tions inﬂuences the value of these angles, one might expect that the Cahn-

MacPherson-Srolovitz relation will not be able to predict adequately the vol-

ume rate of change of a grain in the case of a ﬁnite mobility of grain boundary

junctions. This principal problem was studied in [542, 554] for diﬀerent triple

junction and quadruple junction mobility. In all cases a tangible deviation

from the values predicted by Eq. (4.104) was observed. However, for a ﬁnite

quadruple junction mobility the eﬀect is much smaller than for a ﬁnite triple

junction mobility.

In Fig. 4.69 the results of computer simulation and diﬀerent analytical

approaches are presented. Under the action of a ﬁnite mobility of triple lines

(Λ

=0.2), the diﬀerent analytical approaches predict values of the volume

change rate which depart greatly from the simulation results both in a quan-

titative and qualitative sense. For a better agreement the diverse approaches

need to consider the change of curvature that comes with a boundary junction

of ﬁnite mobility. One major eﬀect of a ﬁnite quadruple junction mobility is

a change in the curvature of the adjoining triple lines, which implicitly alters

also their length. The total length of the triple lines d(D) for the conﬁguration

shown in Fig. 4.61 is given by the sum of all triple line lengths a and d.The

length of the triple lines for a hexagonal cross section can be easily calculated

as (Figs. 4.37 and 4.61) [427, 441]

a =2 =

ln sinΘ



1+cosΘ

sinΘ



(4.115)

4.7 Grain Growth in 3D Systems 427

Glicksman-Rios

simulation Λ

= 0.2

Hilgenfeldt

Mullins

Cahn-MacPherson-Srolovitz

Glicksman-Rios

simulation Λ

= 0.2

Hilgenfeldt

Mullins

Cahn-MacPherson-Srolovitz

FIGURE 4.69

A comparison of the diﬀerent approaches with simulation results for the case

of a ﬁnite triple junction mobility reveals a strong discrepancy [542, 554].

FIGURE 4.70

Dependency of the triple line length on the parameter Λ

[542].

428 4 Thermodynamics and Kinetics of Connected Grain Boundaries

FIGURE 4.71

A comparison of the diﬀerent approaches and with the simulation results for

the case of a ﬁnite triple mobility reveals a strong discrepancy [542].

where Θ can be expressed as a function of the criterion Λ

∼

2Λ

3Λ

(9Λ

+2)

(4.116)

On the other hand the length d can be represented as

d = d

− 2v

t (4.117)

where d

is the length of the longitudinal triple line at t =0,v

is the velocity

of the front and rear faces of the grain.

The simulation gives us the possibility of extracting the dependency of the

parameters a and L(D)onΛ

(Figs. 4.70 and 4.71). One can see that L(D)

increases rapidly for rather small Λ

to a value of L(D), which corresponds

to the curvature of the grain boundary for Λ

→∞. The dependence of the

parameters L(D)ande(D) on a ﬁnite quadruple junction mobility questions,

strictly speaking, the applicability of the Cahn-MacPherson-Srolovitz relation

to grain growth in real 3D systems. However, it can be expected that, similar

to the generalized Von Neumann-Mullins relation [427, 441, 442] there is a

generalized Cahn-MacPherson-Srolovitz relation

= f (L [Λ

, Λ

] e [Λ

, Λ

]) (4.118)

4.8 Kinetics of Grain Growth Inhibited by Vacancy Generation 429

4.8 Kinetics of Grain Growth Inhibited by Vacancy

Generation

Even for a crystal of a pure element it is thermodynamically proﬁtable to con-

tain “intrinsic impurities” — the vacancies. It it a rather elaborate problem

to account for the eﬀect of vacancies on the processes in solids. Even if we

restrict ourselves to processes where the vacancies are generated as a product

of a kinetics process the list is long enough. Vacancy production by moving

jogs on dislocations during plastic deformation, by a progressing solid/liquid

interface in solidiﬁcation, or, generally, in every ﬁrst-order phase transition,

which includes the crystalline phase, shrinking voids in sintering, or, ﬁnally,

during grain growth due to the diﬀerent density of grain boundary and the

ideal crystal [20], [455]–[457] (grain boundary excess free volume — see Chap-

ter 1). As for the latter example of vacancy production, which is the subject of

this paragraph, the point is that the excess of grain boundary free volume lib-

erated by the reduction of total grain boundary area can be considered a ﬂux

of vacancies into the bulk of a sample, a cause of elastic stresses or a source of

plastic deformation. However, in principle, the approach which considers the

elimination of volume excess by vacancies is most versatile. In fact, the way

to reduce the excess through elastic stresses is a deadlock. The relaxation of

elastic stresses is a long-term and slow process. That is why in a short time

the reduction of grain boundary area will result in a strong increase of free

energy. Moreover, as mentioned above, the relaxation of elastic stresses occurs

by a directed ﬂux of vacancies. The inhibiting eﬀect of vacancies on the very

process in which they are generated will be considered from a thermodynam-

ics viewpoint [11, 12], [458]–[460].

The excess free volume a system has to get rid of in such processes can be

released as vacancies which have to be accommodated by the crystal bulk. A

vacancy can be understood as an “impurity atom” which is dissolved in a crys-

tal, and the distinctive property of which (compared to ordinary impurities)

is that a vacancy is born from vacuum and disappears in vacuum, i.e. there is

no conservation rule. Other than that the properties of vacancies are similar

to the usual impurities: creating a solution even in absolutely pure crystals,

vacancies decrease the free energy; if the vacancy concentration exceeds the

solubility limit — the equilibrium concentration of vacancies c

— vacancies

try to leave the crystal or to precipitate as second-phase particles, i.e. voids.

The kinetics aspect of vacancy inﬂuence on grain growth was considered in

[198, 199]. In [11, 12], [458]–[460] it was shown that in the course of grain

growth a self-drag of grain boundaries can be observed, caused by the libra-

tion of excess free volume in terms of vacancy emission.

In the most general terms, the Gibbs free energy G of a system with vacan-

cies can be written as

G = G

non-vac

+ G

vac

(4.119)

430 4 Thermodynamics and Kinetics of Connected Grain Boundaries

where G

non-vac

is the non-vacancy part of the Gibbs free energy and G

vac

the contribution due to vacancies.

Obviously small deviations of the vacancy concentration c from its equilib-

rium value c

will increase the last term in Eq. (4.119), irrespective of the

sign of the diﬀerence c − c

. As shown in [459]



vac



NkT



1+exp



−H



∼

NkT

(4.120)

Here n and N are the number of vacancies and the number of atomic sites per

unit volume, respectively, and H

is the vacancy formation enthalpy. Then,

taking into account that at equilibrium

vac

= 0, we arrive at

G = G

non-vac

+ G

vac

NkT

(c − c

)

(4.121)

Higher order terms in c−c

have been neglected. It is noted that this compact

formula for the vacancy part of the free energy embraces both the vacancy

formation enthalpy term and the entropy contribution,

The formulation given in [459] makes it possible to look at the kinetic

processes involving vacancies from a viewpoint of their thermodynamic feasi-

bility. Obviously, a kinetic process will be promoted thermodynamically if the

derivative of G with respect to time t, G, is negative, i.e.

G =

non-vac

NkT

(c − c

)˙c<0 (4.122)

The time derivative of the vacancy concentration ˙c can be written as

˙c =˙c

− ˙c

−

(4.123)

representing a competition between the vacancy production (˙c

)andtheva-

cancy removal (˙c

−

) rates. While the latter quantity can be expressed in a

generic form

˙c

−

(c − c

sink

) (4.124)

where D

is the vacancy diﬀusivity, d is the sink spacing and c

sink

the vacancy

concentration at a sink, the vacancy production rate depends on the particular

mechanism of vacancy generation and is related to the rate of variation of

the ﬁrst, “non-vacancy,” term of the free energy

G. In what follows, kinetic

processes which are signiﬁcantly aﬀected by this coupling with their “by-

product,” i.e. the vacancies produced, will be investigated.

As a “by-product” of this process, vacancies are released into the crystal

bulk. Indeed, the density of a grain boundary is generally lower than that of

the bulk (see Chapter 1).

The excess free volume released during the reduction of the grain boundary

area has to be accommodated by the bulk. This assumption is supported by

4.8 Kinetics of Grain Growth Inhibited by Vacancy Generation 431

recent computer simulations of grain boundary motion [457]. The supply of

vacancies by moving grain boundaries may produce a vacancy supersaturation

in the bulk, raising the Gibbs free energy and producing a thermodynamic

force on the boundary. As can be expected intuitively, particularly by analogy

with the Le Chatelier principle, this thermodynamic force will oppose grain

bounday migration. Under certain conditions considered in [11, 458, 459], this

eﬀect may be strong enough to temporarily suppress grain growth altogether.

In the approach proposed uninhibited grain growth was considered to occur

only during a limited time t

∗

(Eq. (4.125)). After that time, on reaching the

condition when the time derivative of the free energy of the system becomes

positive, “locking” of grain growth occurs

t ≥ t

∗

γc

NkT(V

)

(4.125)

Under the assumption that t

∗

 τ = d

the expression for the eﬀective

velocity of grain growth V

eﬀ

can be written as

eﬀ

∗

V =

γD

NkTZ(V

)





(4.126)

Here d and D

are the average spacings between vacancy sinks and the va-

cancy diﬀusivity, correspondingly. V is the “unperturbed” rate of grain growth

driven by the boundary energy; γ, δ, β and m are grain boundary charac-

teristics: free energy per unit area, thickness, the relative excess free volume,

and mobility, respectively;

R is the average grain size (radius), D

is the

bulk self-diﬀusion coeﬃcient, N is the number of atoms per unit volume and

Z is the coodination number; kT has its usual meaning. Clearly it is more

correct to express the product βδ as a grain boundary excess free volume





.Theratio

determines the characteristic vacancy annihila-

tion time. Smoothing the discontinuous solution by replacing V

eﬀ

with the

time derivative of the average grain size d

R/dt and solving Eq. (4.125) yields

the time law for the grain growth model of [11, 459]

−

γD

NkTZ(V

)

(4.127)

where

is the initial grain size.

The model outlined above accounts for the inhibiting eﬀect of vacancies on

grain growth, but, of course, it is not more than an approximation of the real

continuous process of grain growth. Guided by the principle that “natura non

facit saltus,” the authors of [12, 460] suggested a description of grain growth

as a continuous process.

An equation expressing the balance of energy associated with an increment

of grain size dR within a time increment dt can be written as

−



3γ



R =



R/dt



dt +

NkT

(c − c

)

R (4.128)

432 4 Thermodynamics and Kinetics of Connected Grain Boundaries

Eq. (4.128) expressed the energy balance: the energy release associated with

the reduction of the GB area (the left-hand side of Eq. (4.128)) is distributed

between the dissipation due to the drag forces (ﬁrst term on the right-hand

side of (4.128) and the vacancy sub-system — second term on the right-hand

side of Eq. (4.128)).

˙c =

−

(c − c

) (4.129)

Eqs. (4.128) and (4.129) render a full description of the evolution of the grain

system in terms of the average grain radius and the vacancy concentration.

In a dimensionless form they can be rewritten as

dξ

+ AΨC −A = 0 (4.130)

dξ

− C (4.131)

Here ξ =

R/d, C = c − c

, A = m

γ/D

,Ψ=

4NkT(V

)

, p =6V

/d;the

time t is now non-dimensional and is measured in units of d

Numerical solutions of the above set of equations for a broad range of pa-

rameters have shown that for suﬃciently small initial grain size ξ

= R

the grain growth uninhibited by vacancies is preceded by an incubation time

during which the growth rate is substantially reduced, the time dependence of

the grain size exhibiting a plateau-like behavior (Fig. 4.72). For large values

of ξ

no incubation time is observed. The incubation time is deﬁned as the

time at which the grain growth rate is a maximum. This time corresponds to

the termination of the plateau-like behavior and a transition to uninhibited,

parabolic grain growth (Fig. 4.72). During the incubation time the vacancy

concentration stays approximately constant (Fig. 4.72). Numerical results also

show that the non-dimensional incubation time is inversely proportional to ξ

and can be represented by the following formula

incubation

P Ψ

(4.132)

or in dimensional form

incubation

=24

NkT(V

(4.133)

where D

is the bulk self-diﬀusion coeﬃcient.

For times larger than the incubation time a common parabolic grain growth

law follows, while within the incubation time an approximate analytical solu-

tion of the set of Eqs. (4.132) and (4.133) reads

C =1/Ψ (4.134)

4.8 Kinetics of Grain Growth Inhibited by Vacancy Generation 433

Eq. (4.139)Eq. (4.139)

FIGURE 4.72

Dependence of non-dimensional grain size (a) and of excess vacancy concen-

tration (b) on the non-dimensional time calculated from Eqs. (4.130), (4.131)

for the following values of parameters: A =10, Ψ = 100, and p = 100 [12].

−

Pξ

(4.135)

or rewritten in dimensional form

−

γD

NkT(V

)

(4.136)

This demonstrates that the approximation [11, 458] based on the intermittent

locking-unlocking scheme with smoothing, provides a quantitatively adequate

representation of continuous grain growth in the regime where the inhibiting

eﬀect of grain growth-induced vacancies is operative. Indeed, Fig. 4.72 demon-

strates an excellent match between the ξ(t) dependencies over the incubation

period given by Eqs. (4.135) and (4.136).

The condition for the occurrence of the vacancy-induced stabilization

against grain growth can be extracted from a comparison of the inhibited grain

growth with the common, parabolic grain growth kinetics. In fact, parabolic

growth occurs if the vacancy eﬀect can be neglected, i.e. if the vacancy con-

centration is suﬃciently close to its equilibrium value. From Eq. (4.130) it

then follows

− ξ

=2At (4.137)

Obviously, the solution of the set of Eqs. (4.130) and (4.131) coincides with the

parabolic law given by Eq. (4.137) for a time tending to zero and also for times

much larger than the incubation time, when C tends to zero, cf. Fig. 4.72. A

real inhibition of grain growth means that the rate of growth in the plateau

region, described by Eq. (4.135) is much smaller than that corresponding to

“free” uninhibited growth described by Eq. (4.137). This condition reads

ξ  (ApΨ)

1/3

(4.138)

434 4 Thermodynamics and Kinetics of Connected Grain Boundaries

or in the dimensional form



∗



24NkTZ(V

)



1/3

(4.139)

One can see that this condition is not very material sensitive, particularly due

to the fact that the ratio of the grain boundary mobility and the coeﬃcient

of bulk self-diﬀusion is not strongly material dependent, and also due to the

power of 1/3. One possible source of structural dependence is the sink spacing

d, for example if it is related to the inverse of the square root of the dislocation

density that may bring about some variability of the quantity on the right-

hand side of inequality (Eq. (4.139)).

One particular case needs to be considered separately, though. It is the case

when the initial grain size is so small that no other vacancy sinks but the grain

boundaries themselves are available, so that d is to be identiﬁed with the grain

size

R. Eq. (4.136) then changes to



dξ

− C



(4.140)

Eq. (4.135) remains unchanged. However, the meaning of the non-dimensional

grain size and the non-dimensional time are diﬀerent now: ξ =

R(V

), and

time is measured in units of (V

)

Solving the set of Eqs. (4.135) and (4.140) numerically, one can see that the

temporal behavior of the grain size in this case is diﬀerent from the behavior in

the case of constant sink spacing d. After a short initial period of time t

trans

the case of vacancy-inhibited growth slows down considerably (Fig. 4.73).

After that grain growth eﬀectively stays inhibited for very long times. The

growth rate is thus always lower than the respective rate of usual parabolic

growth (Fig. 4.73). Again, using the fact that after the transient period t

trans

the vacancy concentration is sustained at an approximately constant level and

that the grain size does not change signiﬁcantly, we arrive at an approximate

solution of the set of Eqs. (4.135) and (4.140) which reads

R −

γD

NkTZ(V

)

(4.141)

From Fig. 4.73 one can see that Eq. (4.140) indeed provides a good analytical

approximation for suﬃciently short times, which nevertheless should be longer

than t

trans

The stability condition of nanocrystalline material against grain growth can

be expressed by the inequality



=24NkTZ(V

)

(4.142)