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

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556 6 Applications

mobility. Hutchinson has pointed out that the computed and measured angu-

lar misorientation distributions about 111 axes are not identical. However,

the computation certainly demonstrates that caution has to be exercised when

interpreting apparently non-random misorientation distributions.

6.2.4.4 Recrystallization in Single Phase Alloys

From innumerable experiments it is known that the addition of solute ele-

ments to a pure metal drastically aﬀects its recrystallization behavior. The

recrystallization kinetics slows down, the recrystallized grain size decreases

and the recrystallization texture may change completely even with small con-

tents of the alloying element.

Evidently, the eﬀect of alloying elements on recrystallization kinetics, grain

size and texture has to be attributed to a change in nucleation rate

N and/or

growth rate v. The deformation microstructure and deformation texture is

usually less dramatically inﬂuenced by minor additions of alloying elements;

thus the nucleation rate ought to be less aﬀected or at least not markedly

increased by alloying. For instance, the addition of only 0.1% P to pure Cu

hardly aﬀects the rolling texture but leads to a very diﬀerent recrystallization

texture upon annealing (Fig. 6.36) [599]. Other prominent examples are the

system Fe in Al or Nb in steel. Therefore, the delay of recrystallization kinet-

ics and the grain reﬁnement upon alloying have to be attributed to a decrease

in the growth rate rather than to an increase in nucleation rate. The eﬀect

can be very drastic, however, e.g. the addition of 0.01%Fe to Al increases the

recrystallization time by orders of magnitude (Fig. 6.37).

A qualitative explanation of these phenomena provides the impurity drag

theory, as presented in Chapter 3. Much more diﬃcult is a quantitative treat-

ment of the impurity drag eﬀect on recrystallization kinetics. This is due to the

fact that impurity drag is a consequence of segregation in grain boundaries,

and the understanding of segregation is still in its infancy. Only recently have

atomistic calculations on segregation in select systems been provided [601] for

a stationary grain boundary, but virtually nothing is known about dynamic

segregation in a moving grain boundary.

Another shortcoming of the impurity drag theory is its phenomenological

character, which does not make use of grain boundary structure. From the

experimental data presented in Chapter 3, in particular the diﬀerent behavior

of pre-exponential factor and activation enthalpy, respectively, it is very likely

that segregated elements not only exert an impurity drag on the boundary but

also aﬀect grain boundary structure and thus its mobility. An improvement

of the traditional impurity drag theory by also accounting for solute-solute

interaction in the boundary is a ﬁrst attempt to tackle this problem (see

Chapter 3). Also, it is not improbable that minute precipitates form in the

boundary and move along with the boundary, which would strongly reduce

the growth rate, as discussed in Chapter 3.

The eﬀect of segregation on grain boundary structure is particularly dras-

6.2 Recrystallization annd Grain Growth 557

rolling textures

recystallization textures

pure Cu Cu-0.1% P

(a) (b)

FIGURE 6.36

{111} Pole ﬁgures of rolling and recrystallization textures of pure Cu and

Cu-0.1%P (degree of rolling: 95%).

120 150 200 300 400 500

time [s]

temperature [°C]

pure Al

FIGURE 6.37

Temperature dependence of the recrystallization time of binary alloys of high

purity aluminum and 1/100 atomic percent of a second metal (after [600]).

558 6 Applications

5N Al

5N Al+10 ppm Ga

5N Al

5N Al + 10 ppm Ga

600 550 500 450 400

T [°C]

-6

-7

-8

-9

-10

1.1

A [m

/s]

1.15 1.2 1.25 1.3 1.35 1.4 1.45

1.5

1/T [10

/K]

FIGURE 6.38

Arrhenius plot of the reduced mobility of a 38.2

◦

111 tilt grain boundary in

pure Al and pure Al doped with 10 ppm Ga.

tic if the segregated solute promotes the tendency for a grain boundary phase

transformation, e.g. in terms of a prewetting or a regular wetting transition,

i.e. essentially the formation of a more or less continuous ﬁlm of solute in

the boundary. If this ﬁlm becomes viscous it may drastically change grain

boundary mobility. This is likely for immiscible elements as for the system

Al-Ga. Contrary to expectations, grain boundaries in Al doped with 10 ppm

Ga show a mobility far higher than grain boundaries in pure Al (Fig. 6.38).

This behavior cannot be understood in terms of the impurity drag theory;

rather, it provides unambiguous evidence for the eﬀect of segregated solute

atoms on grain boundary structure (see Chapter 3).

Of course, such behavior has a drastic inﬂuence on recrystallization kinet-

ics and microstructure evolution, i.e. small amounts of Ga added to pure Al

considerably speed up recrystallization kinetics and alter the recrystallized

grain size. In fact, most commercial Al alloys contain some amount of Ga.

6.2.5 Grain Growth

After primary recrystallization, when the grains impinge with a random mor-

phology, the grain boundary surface area can be reduced by arranging grain

boundaries in planar positions. Grain growth is basically caused by the fact

that it is impossible to reconcile a completely planar boundary arrangement

and an equilibrium of surface tensions at grain boundary junctions. This is

easy to visualize for a two-dimensional model. Equilibrium of surface tensions

6.2 Recrystallization annd Grain Growth 559

FIGURE 6.39

Equilibrium of grain boundary surface tensions at a triple junction. α

—

contact angles; Φ

— inclination; n

— boundary normal; b

—inplane

direction.

at a junction is obtained for the angles α

according to the Herring equation

[73] (Fig. 6.39)



i=1





∂γ

∂Θ





= 0 (6.41)

If the grain boundary energy does not depend on the spatial orientation of

the grain boundary plane, then the torque terms ∂γ/∂Θvanishand

sin α

(6.42)

In the absence of an orientation dependence of grain boundary energy α

= α

≡ α = 120

◦

. Both straight boundaries and α = 120

◦

can be achieved

only for an exclusively hexagonal grain structure. A single ﬁve-sided grain has

to bend at least one grain boundary to balance the force at all boundary junc-

tions (Fig. 6.40). The curvature of this boundary, however, results in a driving

force to straighten the boundary. The boundary will move toward the center of

curvature and thus mistune the equilibrium angles at the junctions and cause

repeated rearrangements. As a result, no equilibrium at all can be obtained. It

follows that grains with fewer than six sides have convexly curved boundaries

and thus shrink while grains with more than six sides have concavely curved

boundaries and thus expand. Since large grains are in contact with many small

grains, their boundaries are composed of a large number of concavely curved

segments, even though their macroscopic appearance may suggest a convex

shape (Fig. 6.41); consequently, they will expand. In three dimensions the

situation is similar, although more complex in detail. The balance of surface

560 6 Applications

FIGURE 6.40

Two-dimensional equilibrium structure that, except for one defect, consists

only of hexagons with 120

◦

contact angles.

tensions requires in this case an angle of 120

◦

at grain boundary junctions

and a tetrahedral angle where four grains are in contact. The impossibility of

reconciling these requirements with planar grain boundaries again causes an

expansion of large grains, while small grains shrink.

The kinetics of normal grain growth can be readily obtained by assuming

that the radius of curvature R is proportional to the average grain diameter

D (R = αD), and the grain diameter varies proportionally to the average

growth rate (v = βdD/dt). Eqs. (6.34b) and (3.6) combine to form

= m

2γ

αD

(6.43)

which integrates for D(t =0)=D

− D



(αβ)



t = K

t (6.44)

The kinetic constant K

has the temperature dependence of the grain bound-

ary mobility m

. For a small initial (primary) grain size D

, the grain diameter

is expected to grow proportionally to the square root of annealing time, which

is observed only for ultrapure materials. Normally, grain growth progresses

less rapidly (Fig. 6.42), so that the results can be better approximated by the

relation

D = K

(6.45)

where n<1/2andK

is a constant. Frequently, grain coarsening even ceases

completely, which may be due to a variety of reasons.

If the grain size becomes comparable to the smallest specimen dimen-

sions, grain growth is substantially reduced. This is immediately evident for a

bamboo structure in wires. In sheets, a two-dimensional growth ought to pro-

ceed after a columnar grain structure perpendicular to the sheet surface has

6.2 Recrystallization annd Grain Growth 561

FIGURE 6.41

Abnormal grain growth in Zn. Note that grain boundary is macroscopically

convex but microscopically concave [605].

1000 10000

1.0

0.1

average grain diameter [mm]

3 10 100

corrected annealing time [s]

10000 ppm

3000 ppm

1000 ppm

500 ppm

100 ppm

zone refined lead

tin added

n = 0.41

n = 0.40

n = 0.39

n = 0.42

n = 0.34

n = 0.103

n = 0,109

n = 0.140

n = 0.221

n = 0.228

n = 0.345

n = 0.40

FIGURE 6.42

Graingrowthinzonereﬁnedleadandleadwithdiﬀerenttincontents.The

deviation of the exponent from the ideal value of 0.5, and the increase of this

deviation with increasing grain size are apparent (after [606]).

562 6 Applications

been established. This is, however, prevented by thermal grooving of grain

boundaries at surfaces (see Chapter 3). The ﬁnal grain size is usually well

approximated by about twice the sheet thickness. For secondary recrystalliza-

tion, however, the driving forces are generally suﬃcient to overcome grooving,

and growth continues in two dimensions.

In the presence of inclusions of a second phase, grain boundary motion is

hindered (see Chapter 3). Grain coarsening will cease, once the driving force

(Eq. (6.34b)), which decreases with increasing grain size, is balanced by the

Zener drag P

=3fγ/2r

,wheref and r

represent volume fraction and

average size of the particles. Assuming R = αD, the ﬁnal grain size D

3αf

(6.46)

Even in solid solutions grain growth may tend to cease in the course of time.

Because of the decrease in driving force with increasing grain size, the growth

rate decreases and the boundary may change from a free to a loaded state (see

Chapter 3). This can reduce the growth rate by several orders of magnitude.

Grain coarsening can be substantially restricted in strongly textured mi-

crostructures. Since the grain boundary migration rate is very small for small

orientation diﬀerences, grain growth is absent, if the recrystallization texture

consists of only one component such as the cube texture in fcc metals, but

major texture changes can result if minor amounts of other orientations exist

in a sharply textured microstructure.

Thus far we have considered only the eﬀect of grain boundaries on the

evolution of microstructure during recrystallization and grain growth. As ev-

ident from Chapter 4, however, triple junctions may play an essential role in

the dynamics and kinetics of grain boundary assemblies. In this context the

criterion

Λ=

· λ

(6.47)

plays a role, where λ is the characteristic length, i.e. the grain size D in poly-

crystals or the subgrain size d in cellular deformed structures. Owing to the

almost complete neglect of triple junction properties on microstructure evo-

lution there is little quantitative data on the behavior of boundary-junction

systems. In Chapter 4 we already noted the inﬂuence of triple junctions on the

Von Neumann-Mullins relation and its drag eﬀect on grain boundary motion.

According to Eq. (6.47) a large impact of triple junction mobility may be ex-

pected if the characteristic length λ is very small, i.e. for ultraﬁne microstruc-

tures or even nanocrystalline material. Owing to the experimental diﬃculties

to properly characterize nanocrystalline microstructures and in particular to

account for impurities, second-phase particles produced during consolidation,

as well as a potentially altered grain boundary structure at large surface to

volume ratios, it is not possible to accurately compare the grain growth dy-

namics of nano- and microsized grain structures.

However, there are indications that triple junctions substantially slow down

6.3 On Precipitation-Controlled Grain Size 563

grain growth kinetics in ultraﬁne grain structures, and they may even give rise

to the suppression of recrystallization and set oﬀ abnormal grain growth. One

“suspicious” example is mechanically alloyed ODS superalloys, which are very

ﬁne grained (D ∼ 100nm) after hot extrusion and exhibit conspicuous resis-

tance to recrystallization and grain growth, which may be set oﬀ only at very

high homologous temperatures (T>0.8T

). Although not yet beyond the

state of speculation, this behavior may be due to triple junctions, since all

hypotheses that try to account for the observed eﬀect by second-phase parti-

cles, cannot yet be conﬁrmed. It is in particular noteworthy that intermediate

anneals at not so high temperature can set oﬀ abnormal grain growth at even

more elevated temperatures, i.e. small changes in grain size may render the

microstructure unstable [603, 604].

6.3 On Precipitation-Controlled Grain Size

It is textbook knowledge that second-phase particles slow down the process

of grain growth (see Chapter 3). However, in [601, 602, 607] the question

was risen whether the formation of a second phase always slows down grain

growth in polycrystals, in particular in nanocrystals. The authors of [607]

considered the grain boundary motion in an alloy with a miscibility gap. When

the sample (polycrystal) is single phase, grain boundary motion or the rate

of grain growth is described by the theories of impurity drag (see Chapter 3).

However, if we lower the temperature by ΔT or change the concentration

by Δc (Fig. 6.43) the systems changes to a two-phase ﬁeld. This causes two

changes, namely the formation of precipitates of a second phase and depletion

of solute atoms in the bulk of a grain. Decreasing the temperature by ΔT will

oﬀset the system to the two-phase ﬁeld and the mass W

of the precipitates

with respect to the solid solution W

sol.sol.

will be according to the lever rule

(Fig. 6.43)

sol.sol.

− c

(6.48)

V (c



= c

, precipitates) >V (c

) (6.49)

where V is the boundary migration rate.

Two situations have to be distinguished, namely, large immobile and small

mobile precipitates.

For large immobile particles the grain growth rate can be represented as

[607]

V = m



P − P

(V ) − P





P − P

(V )



(6.50)

where P =

αγ

is the driving force for grain growth, γ is the grain boundary

surface tension, P

and P

are dragging forces in the solid solution and

564 6 Applications

FIGURE 6.43

Binary phase diagram with miscibility gap (see text for details) [607].

the two-phase regime, respectively. P

is the dragging force by immobile par-

ticles, and α is a geometrical parameter that correlates grain size and grain

boundary curvature and will be taken to be α = 3 in the following. To assess

Eq. (6.50) the Zener- and the L¨ucke-Detert approximations for particle drag

and impurity drag, respectively

drag

3¯cγ

(6.51)

=Γ

kT (6.52)

¯c — the volume fraction of the particles, r — particle radius, Γ — adsorption

at the grain boundary, D

— bulk diﬀusion coeﬃcient of the solute atoms.

The next inequality expresses the acceleration of grain growth caused by

precipitation [607]

P −

3¯cγ

Γ(c



= c

)

Γ(c

)

(6.53)

Eq. (6.53) is valid under the assumption that the impurity diﬀusion coeﬃcient

is independent of concentration. With the driving force for grain growth P ≈

3 ·

, and the Henry adsorption isotherm Γ(c)=Bc, Eq. (6.53) reads

−

¯c

(6.54)

It is pointed out that, strictly speaking, for solids the isotherms should be used

which take into consideration the interaction between species (see Chapter 1).

6.3 On Precipitation-Controlled Grain Size 565

The Henry isotherm was used in [607] for a rough estimate to establish the

essential physical link between the parameters of the problem.

The next relation follows from Eqs. (6.48), (6.49) and (6.54) [607]

1 −

−c

(6.55)

or, using evident relationships between c

, c

(6.56)

Eqs. (6.55) and (6.56) determine the conditions under which precipitation ac-

celerates grain growth in the alloy. Since the relations (6.55) and (6.56) are

derived under the assumption that the second-phase particles are immobile,

the larger r (particle radius) and the smaller

R (mean grain size), the better

the approach. For c

∼ 1, c

∼ 10

−3

, c

∼ 10

−4

Eq. (6.56) yields

< 10

However, the phenomenon discussed holds as well for particles moving to-

gether with the grain boundary, in other words, for small, mobile particles.

The theory of grain boundary motion dragged by mobile particles was com-

prehensively considered in Chapter 3. In particular, it was shown that the

velocity of joint motion of a grain boundary and the particles with a contin-

uous size distribution ¯n(r) can be expressed as

V =

∞

¯n(r)m

(r)

(6.57)

where m

(r) is the mobility of a particle of size r. The total number of particles

per unit area n is as follows:

n =



∞

¯n(r)dr (6.58)

The border between free grain boundary motion and the joint motion of the

grain boundary with the particles is established by dimensionless criterion ρ:

ρ =



∞

¯n(r)m

(r)

dr (6.59)

If ρ  1thenV

∼

P , and the grain boundary velocity is determined by

the mobility of the boundary. If ρ  1 grain boundary motion is determined

by the mobility of the particles and their distribution function, which in the

simple case of a uniform size distribution r = r

can be represented as

¯n(r)=n

δ (r − r

)

V =

)

(6.60)

ρ =

)