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

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576 6 Applications

of the metals and the temperature of motion. Let us estimate the dimen-

sionless criterion λ

junctions,b

using the particular value of quadruple junction

mobility



≈ 2 · 10

−4

−1



and grain boundary and triple junction mobil-

ity for Al (Table 4.4, 110 tilt grain boundary) at 200

◦

C.Onecanseethat

for

R =10

−8

mandγ =0.5J/m

the criteria are equal to

=10

−13

;

=3· 10

−6

and

junctions,b

=10

−13

It is obvious that the main contribution to λ

junctions,b

comes from triple junc-

tions drag. It is stressed that the eﬀects of the diﬀerent grain boundary junc-

tions are closely connected; indeed,

(6.86)

At high temperatures where the mobility of the triple junctions due to their

high activation enthalpy is large, the major drag of grain growth in nanocrys-

talline materials is exerted by quadruple junctions whereas at rather low tem-

perature the retardation is aﬀected by triple junctions

Owing to the loss of excess volume during grain growth, vacancies are gener-

ated to temporarily compensate the volume change. This leads to a drag eﬀect

during grain growth [12], [458]–[460] (see Chapter 4). The criterion λ

vac,b

for

this type of drag can be expressed as

vac,b

beﬀ

γπ(n − 6)/3

Rγ

NkTZ(V

)

(6.87)

where A

= m

γ is the reduced grain boundary mobility, N is the number of

the atoms per unit volume, Z is the coordination number, D

is the bulk

self-diﬀusion coeﬃcient, V

is the grain boundary excess free volume. For

pure Al at 300

◦

C λ

vac,b

≈ 10

R,andfor

R ≈ 10

−8

m this yields λ

vac,b

≈ 10

−5

In general two types of particle drag eﬀects on grain growth should be

considered: the drag by small mobile particles and by large immobile particles.

However, the consideration in [612] is conﬁned to nanocrystalline materials.

The grain size in nanostructures is so small that it is diﬃcult to imagine the

existence of large immobile particles which are comparable in size with the

grains. That is why only the drag by particles moving jointly with the grain

boundaries was analyzed in [612]. The velocity of a grain boundary moving

together with particles is given in Chapter 3 Eq. (3.71). The eﬀective mobility

of such a grain boundary can be expressed as

eﬀ

)

(6.88)

6.5 Grain Boundary Junction Engineering 577

where m

) is the mobility of particles of radius r

,andn

is the number

of particles per unit area of the boundary, respectively. For the second-phase

particles considered in [612], the criterion λ

part,b

can be represented as [607]

part,b

part

beﬀ

(r)

(6.89)

where the number of second-phase particles n is determined by the phase

diagram and the cooling interval. As shown in [612] for a system with rather

small solubility the criterion λ

part,b

canbeexpressedasλ

part,b

∼

−15

.In

other words, only “suﬃciently large” particles will exert an eﬃcient drag on

grain growth in nanocrystalline systems.

It is of interest to consider the relative eﬃciency of particles and triple

junctions on retardation of grain growth. The relation which describes the

relative eﬃciency of triple junction drag compared to mobile particle drag

was derived in [612]:

tj,p

beﬀ

πm

) r

(6.90)

With the same values of the parameters used in the evaluations discussed

above, relation (6.90) can be transformed to

tj,part

≈ 2 · 10

−3

· r

(6.91)

Obviously, triple junction drag becomes comparable with particle drag only

for r ≥ 10

−5

m, in other words for very large particles, r>10 μm.

From this analysis it can be concluded that grain boundary junctions are

most eﬀective for microstructural stabilization of very ﬁne-grained microstruc-

tures in the range of the used parameters.

6.5 Grain Boundary Junction Engineering

The concept of grain boundary engineering (GBE) has been embodied in

the papers of the groups of Watanabe and Palumbo, respectively [614]–[618].

The practical realization of this idea was reduced to thermomechanical treat-

ment, most often to the thermocycling. Such treatment results in an increase

in the number of special grain boundaries, predominantly twin boundaries.

While not questioning the prospects of such an approach the authors would

like to call the reader’s attention to the fact that a deeper understanding

of well-known processes or physical phenomena which permit us to create

materials with given properties and (or) distribution of grain boundaries is

578 6 Applications

subsumed under GBE as well. Strictly speaking, all processes which trans-

form the properties of materials due to a change in grain boundary properties

and distribution, in other words, in grain microstructure — recovery, recrys-

tallization and especially grain growth — fall in this category. This is our

understanding of GBE. In particular, the essential diﬀerence between grain

microstructure obtained during grain growth under diﬀerent regimes — triple

junction regime and grain boundary kinetics — indicates that there is a way

to utilize this behavior to inﬂuence microstructural evolution, which will be

referred to as grain boundary junction engineering, that utilizes junction prop-

erties for microstructure control [447]. Grain boundary junction engineering

is a new branch of GBE. The consideration given in [619] is conﬁned to grain

growth. Grain growth aﬀects the grain microstructure but its development

depends on the internal parameters of the sample — chemical nature of the

matrix, i.e. impurities, composition — and the characteristics of the process

— temperature, pressure, time of annealing. To generate the desired grain

microstructure is only half of the problem. It is equally important to stabilize

the formed microstructure. This issue is of special importance for ultraﬁne-

grained and nanocrystalline materials. To stabilize a small grain size usually

drag forces by impurities or particles are introduced. However, both methods

require a special phase and chemical composition of the material on the one

hand, and change, as a consequence, its physical and mechanical properties

(see Chapter 6). In [619] another approach is proposed, based on the essential

diﬀerence in grain microstructures formed by junction kinetics and by bound-

ary kinetics. The theoretical foundations of grain boundary junctions kinetics

and the special features of grain growth aﬀected by boundary junctions are

considered comprehensively in Chapter 4. Here we would like to dwell on some

aspects of the problem.

As shown in Chapter 4, in the course of grain growth inﬂuenced by triple

junctions the grains with a number of sides n with n

∗

<n<n

∗

become

locked and can neither grow nor shrink. The inﬂuence of such a “layer” of

locked grains on the stability of grain microstructure will be more severe for

ﬁne-grained and nanocrystalline materials.

It is emphasized repeatedly that a limited triple junction mobility always

slows down the evolution of the grain microstructure of polycrystals, irrespec-

tive whether the topological class of the considered grain is smaller or larger

than 6. However, the ﬁnite mobility of triple junctions not only slows down

grain microstructure evolution, it changes the ﬁnal distribution of the grains

of diﬀerent topological classes, as well. Actually, relation (4.74) provides a way

of estimating the time dependence of the criterion Λ [447, 619]

S =

dΛ

= β



· Λ ·

dΛ

(6.92)

6.5 Grain Boundary Junction Engineering 579

where β



, α

and α

are geometrical coeﬃcients of the order of

one. From (6.92) and (4.74) we arrive at

dΛ

= −β ·

2π − n (π − 2θ)

(6.93)

where β =



The rate of change of the parameter Λ depends on the topological class n

of a grain. A more speciﬁc relation for

dΛ

can be obtained in the vicinity of

θ = π/3 (see Chapter 4).

The temporal change of Λ during grain growth is given in Fig. 6.48. It

can be seen from Fig. 6.48 and Eq. (6.93) that triple junction drag aﬀects the

growth of grains with diﬀerent topological class n markedly diﬀerent. Further-

more, Fig. 6.48 illustrates that the junction drag eliminates the unique border

between growing and shrinking grains: for rather small Λ the derivative

dΛ

is positive for n = 5 and negative for n = 7. This issue is the key for under-

standing the inﬂuence of triple junctions on grain microstructure evolution.

Moreover, this eﬀect provides another explanation of abnormal grain growth

in ﬁne grained materials. Namely, a grain of rather large topological class will

break free of triple junction drag and will start to grow at boundary kinetics

much faster than the grains with small or not so large n.Evidently,sucha

strong acceleration of the growth of an individual grain is characteristic of

abnormal grain growth. A possible way of junction kinetics treatment (JKT)

is schematically depicted in Fig. 6.49, which illustrates a sequence of anneal-

ings. As shown in Chapter 4, the annealing at a relatively low temperature

initiates grain growth at triple junction kinetics, and the grain microstructure

resulting from this treatment is a typical “junction” microstructure, a system

of polygonal grains which tends to assume an equilateral shape, etc. Fig. 6.49

presents the results of grain growth studied by computer simulations. Evi-

dently, a subsequent annealing at boundary kinetics conditions (subsequent

to junction kinetics) requires a much larger time to reach the same grain

size. An increase in the eﬀect of JKT as expressed by the grain area ratio

after junction-controlled growth, S

, conspicuously delays regular grain

growth under boundary kinetics. This phenomenon can be seen more clearly

in Fig. 6.50, where the ratio of the growth rates with and without JKT is

presented for diﬀerent W =

d(S

)/dt

afterJkT

d(S

)/dt

.ThenumeratorofW is the

slope of the time dependency of the mean grain area change after JKT, while

the denominator is the rate of mean grain area change at boundary kinetics.

The value W

−1

is a measure for the stabilization of the grain microstructure

in the course of such a treatment.

580 6 Applications

n = 5

100

120

140

n = 3

n = 4

(a)

(b)

n = 12

n = 7

100

-4

-2

-8

-6

6040

n = 5

100

120

140

n = 3

n = 4

n = 5

100

120

140

n = 3

n = 4

(a)

(b)

n = 12

n = 7

100

-4

-2

-8

-6

6040

n = 12

n = 7

100

-4

-2

-8

-6

6040

FIGURE 6.48

The dependence of

dΛ

on Λ for grains with (a) n =3,4,5and(b)n = 7, 12,

respectively [447].

6.5 Grain Boundary Junction Engineering 581

grain boundary kinetics

time

grain boundary junction kinetics

temperature

grain boundary kinetics

time

grain boundary junction kinetics

temperature

FIGURE 6.49

Procedure of junction kinetics heat treatment (JKT): two consecutive anneal-

ings: at junction and grain boundary kinetics, respectively [447].

582 6 Applications

0 3000880000 885000 890000

100

120

140

160

Grain boundary

junction kinetics

Grain boundary

kinetics

0 3000 1385000 1390000

0 3000 1914000 1920000

100

(a)

(c)

(b)

0 3000880000 885000 890000

100

120

140

160

Grain boundary

junction kinetics

Grain boundary

kinetics

0 3000 1385000 1390000

0 3000 1914000 1920000

100

(a)

(c)

(b)

FIGURE 6.50

The kinetics of mean grain area change for JKT: boundary kinetics (dashed

line) and junction kinetics (solid line),  — annealing at boundary kinetics

after heat treatment at junction kinetics (Λ = 10

−4

) for diﬀerent starting

ratio

: 10 (a); 15 (b); 20 (c) correspondingly; the time of annealing is given

in arbitrary units [447].

6.5 Grain Boundary Junction Engineering 583

FIGURE 6.51

The dependence of W =

d(S

)/dt

afterJkT

d(S

)/dt

[447].

Appendices

7.1 Appendix A

7.1.1 Mass Transport in Solids in Terms of Osmotic Pressure

If the given mass, enclosed as before, is divided into two parts, each of which

is homogeneous and ﬂuid, by a diaphragm which is capable of supporting an

excess of pressure on either side, and is permeable to some of the components

and impermeable to others, we shall have the equations of conditions

δη



+ δη



=0,δν



=0,δν



and for the components which cannot pass by the diaphragm

δm



=0,δm



=0,δm



=0,δm



=0, etc.

and for those which can

δm



+ δm



=0,δm



+ δm



=0, etc.

With these equations of condition, the general condition of equilibrium will

give the following particular conditions:



= t



and for the components which can pass the diaphragm, if actual components

of both masses



= μ



,μ



= μ



, etc.

but not



= p



nor



= μ



,μ



= μ



, etc.

([1]), where the signs



and



denote the values in two parts of the system

accordingly; η is entropy, v and m are the volume of the parts of the system

and the masses of the components, respectively, t and p are the absolute tem-

perature and pressure, respectively.

585

586 7 Appendices

The osmotic pressure is a thermodynamic eﬀect in the sense that it arises

as a result of tendency of the system to change in a manner such that the

chemical potentials of the components become equal at all points of the sys-

tem. Since the magnitude of the osmotic pressure depends on the diﬀerence

between the chemical potentials (for the dilute solutions — on the diﬀerence

between the concentrations) and drops down to zero when they are equal.

However, the reason of the osmotic pressure is kinetic — the diﬀerence be-

tween the mobilities of atoms of diﬀerent species. Traditionally the osmotic

pressure was considered in the limiting case when the mobility of one of the

component is equal to zero. The theory of osmotic pressure in solutions was

developed for liquids, but the arguments can be transferred to solids. The sit-

uation when the atoms of components move with diﬀerent velocities is most

typical for the diﬀusion in metals. The Kirkendall eﬀect in self-diﬀusion re-

sults from the diﬀerence between mobilities, so that the original interface is

displaced (a marker placed at this interface is displaced).

In an analysis of self-diﬀusion, the driving force for the diﬀusive mixing is

usually assumed to be gradients of the chemical potentials of the components,

∇μ

, in which only the entropy term RT∇lnn

,(n

is the density of particles in

species i) was incorporated. This approach leads to the well-known Darken’s

results [620]. However, in the general case ∇μ

also contains a term propor-

tional to the pressure gradient (v

∇p,wherev

is a partial volume). This term

is usually neglected, reasoning that p = const. The situation changes where

there is osmotic pressure [621]. To show the diﬀerence between the classi-

cal approach [620] and this one, which takes into consideration the osmotic

pressure [621], let us brieﬂy review Darken’s analysis of self-diﬀusion. The

component ﬂuxes in the laboratory coordinate system can be expressed as

= −D

∇n

+ Vn

= −D

∇n

+ Vn

(7.1)

where D

and D

are the partial diﬀusion coeﬃcients of the components, and

V is the interface velocity.

Assuming n

+ n

= n =const.andj

+ j

= 0, we ﬁnd

V =(D

− D

) ∇N

(7.2)

where N

= n

/n.

Since j

= −

D∇n

,wehave

D = D

+ D

(7.3)

which determines the rate of mixing of the components and is obtained from

the Boltzman-Mattano solution.

D is often called the chemical diﬀusion coef-

ﬁcient.

Let us consider now the process taking into account in the expression for

the driving force X

the pressure gradient

= −kT ln n

− v

∇p

(7.4)