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

Подождите немного. Документ загружается.

82 1 Thermodynamics of Grain Boundaries

50μm

sδD

0.1

1.0

[at.%]

48 80 119 281 332h

(a)

0.1

1.0

[at.%]

50μm

(b)

1.8

975

7.6

857

809

790

102h

745°C

-c

sδD

50μm

sδD

0.1

1.0

[at.%]

48 80 119 281 332h

(a)

0.1

1.0

[at.%]

50μm

(b)

1.8

975

7.6

857

809

790

102h

745°C

-c

sδD

FIGURE 1.29

The dependencies of GB zinc concentration c

on depth δ in a Fe-5at.%Si alloy.

(a)Atconcentrationc

the value of GB diﬀusivity sδD

changes abruptly at

a constant temperature of 735

◦

C and various annealing times t.Notethatc

does not depend on t. (b) At concentration c

the value of diﬀusivity sδD

changes abruptly. The concentration c

depends on temperature.

large slope at low Zn concentrations (Fig. 1.29). The transition from one type

of behavior to the other was found to occur at a deﬁnite Zn concentration c

at the GB. As shown, c

is independent of the annealing time. Consequently,

this concentration is an equilibrium characteristic of a grain boundary (Fig.

1.29). It is an equilibrium characteristic of a grain boundary and depends on

temperature. The ratio of the grain boundary diﬀusivities in the two regions

was approximately 10

, which is an indication of the quasi-liquid nature of the

layer present in the grain boundary at high Zn concentration. The dependence

of c

on temperature (grain boundary phase diagrams) has been determined

for alloys with diﬀerent Si contents. Let us summarize the main features of

the diagrams.

1. For all Si contents sharp peaks directed toward low Zn concentrations

were observed in the grain boundary phase diagrams at the peritectic

temperature of the Fe-Zn systems (782

◦

C) (Fig. 1.30a,b). The nature

of the peaks is connected with the fact that in the Fe-Zn system the

virtual point of solid solution decomposition lies only slightly above the

1.4 Applications of Grain Boundary Thermodynamics 83

peritectic temperature. It was shown [121] that the concentration c

which a grain boundary phase transition occurs depends on the grain

boundary surface tension γ and the surface tension γ

of the solid-

liquid interface according to the expression

= c

−

(γ − 2γ

)

n+1

b(Wn)

1/n

(1 + n

−1

)

n+1

(1.213)

where c

is the solubility limit of Zn in the alloy, W and n are the

constants describing the repulsive interaction between two solid-liquid

interfaces and b is a constant which may be determined from the thermo-

dynamic data describing the alloy. Actually, all the miscellany of wetting

phenomena may be obtained from an analysis of the free energy Ω

the wetting layer:

=2γ

+ λΔg + V (λ) (1.214)

where λ is the thickness of the wetting layer, Δg istheexcessfreeenergy

of the wetting phase. The latter term describes the interaction of the

“crystal-wetting phase” interfaces. The situation that the thickness of

the wetting layer gets inﬁnitely large when the line of phase coexistence

is approached (Δg → 0) is called complete wetting. Because the equi-

librium thickness of the wetting layer is determined by minimization of

Eq. (1.214) with respect to λ, complete wetting can be observed only in

the case when the two following conditions are satisﬁed:

(1) 2γ

<γ

(2) V (λ) must have a global minimum for  →∞, in other words, the

interfaces “crystal-wetting phase” must repel one another. In [100] it

was assumed that V (λ) depends on λ by a power law

V (λ)=W/λ

(1.215)

Expanding Δg into a power series in (c

− c

), where c

is the bulk

solubility limit (solvus or solidus) and restricting ourselves to the ﬁrst

term we obtain

Δg = b (b

− c

) (1.216)

The function Ω

(λ), determined by means of Eqs. (1.214)–(1.216), has a

minimum at λ



b (c

− c

/nW )

−1/n+1



,whereatΩ

(λ

)=Ω

= γ

the premelting transition occurs. Under this condition the relationship

(1.213) was obtained. In the vicinity of the critical point, γ

decreases

drastically, and, according to Eq. (1.213), c

also decreases.

2. In the alloy containing 5at%Si a peak in the grain boundary phase dia-

gram directed toward low Zn content was observed in the temperature

vicinity of the Curie point (Fig. 1.30). Such eﬀects are often observed at

the intersection of a line of a second order phase transition with a line

84 1 Thermodynamics of Grain Boundaries

0 5 10 15 20 25 30 35 40

[at.%]

T [°C]

950

900

850

800

750

700

650

(Fe-5at.%Si)

(pure Fe)

α+L

α+Γ

per

= 782°C

(a)

[at.%]

010200 10 100

700

900

800

T [°C]

Fe-5at.%Si Fe-10at.%Si Fe-12at.%Si

α+L

α+Γ

per

(b)

0 5 10 15 20 25 30 35 40

[at.%]

T [°C]

950

900

850

800

750

700

650

(Fe-5at.%Si)

(pure Fe)

α+L

α+Γ

per

= 782°C

(a)

[at.%]

010200 10 100

700

900

800

T [°C]

Fe-5at.%Si Fe-10at.%Si Fe-12at.%Si

α+L

α+Γ

per

(b)

FIGURE 1.30

(a) The temperature dependencies of c

and c

vs. Zn concentration. The

solubility limit of zinc in pure iron is also displaced (solidus and solvus). Γ is

an intermetallic compound and L is the liquid. (b) Section of Fe-Si-Zn phase

diagrams for alloys with diﬀerent silicon contents. The dotted lines show the

temperature dependencies of the concentration c

at which the GB diﬀusivity

changes drastically (premelting phase transition). c

is the bulk solubility limit

of Zn in the Fe-Si alloys. T

per

is the peritectic temperature of the binary Fe-Zn

system.

1.4 Applications of Grain Boundary Thermodynamics 85

Peritectic line

Curie line

Peritectic line

Curie line

FIGURE 1.31

Three-dimensional grain boundary premelting surface in “Zn concentration-Si

concentration-temperature” space.

of a ﬁrst-order phase transition [126, 127]. Here, a transition from com-

plete to incomplete wetting may occur. The transition of the alloy into a

ferromagnetic state creates a force of attraction between the grains. Ac-

curate calculations show that the “magnetic” part of the function V(λ)is

comparable with the “non-magnetic” part [128, 129]. All of the above

may be illustrated by a three-dimensional grain boundary phase dia-

gram with the coordinates “temperature-concentration-concentration”

(Fig. 1.31). Such a diagram looks like a two-dimensional surface below

which (at low Zn concentrations) grain boundaries with low Zn adsorp-

tion are stable and above which grain boundaries exist in a premelting

state.

3. Below the Curie point the premelting line is very close to the line of

the bulk solubility limit, and below the temperature of atomic ordering

A2-B2 in an alloy containing 12at%Si the complete wetting disappears

simultaneously with the grain boundary premelting phase transition. So

it can be concluded that atomic and spin ordering shorten the region

of stability of a grain boundary in a premelted state. In the case of a

ferromagnetic transition it is connected with the long-range attractive

contribution to the free energy of the premelting layer, the origin of

which was discussed above.

In the case of atomic ordering A2B2 in the bulk, an additional contribu-

tion to the free energy of both the grain boundary and the premelting

layer arises due to the disorder in the interface core. Since the premelt-

ing layer with a liquid-like structure is more disordered than the grain

boundary, such a contribution should be greater for a premelting layer,

and it loses its stability in the ordered state of the bulk.

4. It was shown [130] that when the hydrostatic pressure is increased the re-

86 1 Thermodynamics of Grain Boundaries

[

]

600

700

800

900

T [°C]

p [GPa]

per

[

]

600

700

800

900

T [°C]

p [GPa]

per

FIGURE 1.32

Three-dimensional phase diagram with the locations of the bulk solubility

limit of Zn (c

) and the concentration of the premelting transition on grain

boundaries, c

gion of stability of the premelting layer at the grain boundary decreases,

and at some critical pressure p

one can observe the transition from com-

plete to incomplete wetting of the grain boundary by the Zn-rich melt

with the simultaneous disappearance of the grain boundary premelting

transition. In Fig. 1.32a schematic three-dimensional grain boundary

phase diagram in coordinates “temperature - Zn concentration-pressure”

is shown in two sections, at p =0(isobaric)andatT = 905

◦

Cfordiﬀer-

ent pressures (isothermal). The disappearance of grain boundary wetting

and premelting is connected with a higher excess volume of the premelt-

ing layer as compared with that of an “ordinary” grain boundary.

As mentioned at the beginning of the discussion of this item, wetting of grain

boundaries by the melt was repeatedly observed [100, 113, 114]. Cahn’s paper

[116] gave a new impetus to research in this area [131]–[134]. Unfortunately,

the exploration of wetting phenomena at solid/solid interfaces is much more

poorly developed than, in part, that for planar solid substrates and ﬂuid mix-

tures. From our point of view this is due to the considerable experimental

diﬃculties connected with measurements of the boundary which is “hidden”

inside the sample.

There is a good reason to believe that the experimental technique developed

in [135]–[137] permits us to improve our understanding of this phenomenon.

What is a wetting transition? Let us consider a system in which a solid bicrys-

tal is in contact with a melt. The equilibrium of a bicrystal in contact with

a liquid metal can be characterized by the dihedral angle Θ at the site of

1.4 Applications of Grain Boundary Thermodynamics 87

(a)

(b)

(a)

(b)

FIGURE 1.33

Partial (a) and complete (b) wetting of a grain boundary by a melt (schemat-

ically); γ and γ

are the excess free energies of the grain boundary and

solid/liquid interface, respectively. L and S denote the solid and liquid phases,

respectively.

intersection of the grain boundary and the solid/liquid interface (Fig. 1.33).

The angle Θ is deﬁned by the condition of equilibrium

cos





2γ

(1.217)

The two situations Θ > 0 and Θ = 0 correspond to a partial and complete

wetting, respectively. If some intensive thermodynamic parameters like tem-

perature, composition, pressure or the grain boundary misorientation angle

are altered, a transition from partial to complete wetting can occur. This is

a wetting phase transition, which can be of ﬁrst or second order (Fig. 1.33b).

When the situation is reversed and the temperature is decreased or the pres-

sure is increased the angle Θ changes from zero to non-zero values. This is

called the dewetting phase transition (Fig. 1.34).

First of all we would like to discuss the eﬀect of grain boundary misorienta-

tion on the wetting parameters, then the inﬂuence of high pressure on the grain

boundary wetting. Finally, the important physical problem will be discussed

— to which order of phase transition the grain boundary wetting relates.

There are few experimental data concerning the eﬀect of the grain boundary

misorientation on the wetting parameters. In the Cu-In system the value of

the wetting temperature (T

) for a symmetrical 011 tilt grain boundary

close to the symmetrical coherent twin boundary was only about 30

◦

Chigher

than for the general 141

◦

011 grain boundary, though the excess free energy

diﬀerence for these grain boundaries was about 50% in pure Cu [135]. In the

Al-Sn system the value of T

for a symmetrical 38.5

◦

011 grain boundary

close to the Σ9 coincidence misorientation was about 13

◦

C higher than for

the general 32

◦

011 grain boundary [136].

One can see that there is an inﬂuence of the grain boundary misorientation

88 1 Thermodynamics of Grain Boundaries

50µm

0.01 GPa

0.30 GPa

0.85 GPa

1.17 GPa

1.4 GPa

Φ=35° Φ=38°

Fe-6at.% Si

50µm

0.01 GPa

0.30 GPa

0.85 GPa

1.17 GPa

1.4 GPa

Φ=35° Φ=38°

Fe-6at.% Si

FIGURE 1.34

Optical micrographs of contact areas between Fe-Si bicrystals (bottom, bright)

and Zn-rich melts (top, dark) after annealing at 950

◦

C under diﬀerent hydro-

static pressure.

1.4 Applications of Grain Boundary Thermodynamics 89

0 0.5 1.0 1.5

Pressure [GPa]

120

Contact angle Θ [deg]

Fe-12at.%Si, 38°

Fe-6at.%Si, 35°

Fe-6at.%Si, 38°

Fe-6at.%Si, 43°

0 0.5 1.0 1.5

Pressure [GPa]

120

Contact angle Θ [deg]

Fe-12at.%Si, 38°

Fe-6at.%Si, 35°

Fe-6at.%Si, 38°

Fe-6at.%Si, 43°

FIGURE 1.35

Pressure dependence of contact angle Θ for various grain boundaries. As the

pressure increases, the dewetting transition from complete to partial (incom-

plete) wetting occurs at p

and energy on the value of T

. However, the magnitude of the temperature

change cannot be predicted in a straightforward manner.

In [137] the inﬂuence of pressure on the wetting of grain boundaries in

bicrystals of Fe-Si by a Zn-rich melt was studied. The transition from com-

plete (Θ = 0) to partial (Θ > 0) wetting was found to occur as the pressure

increased (Fig. 1.35). The dewetting transition pressure is higher for special

boundaries than for general ones (Fig. 1.36). The investigation of the dewet-

ting under a high hydrostatic pressure enables us to estimate the resulting

excess volume of grain boundary and solid-liquid interfaces. Actually, from

Eq.(1.217) it follows that

Θ = 2 arc cos



+ pΔV

2(γ

)+pΔV



(1.218)

90 1 Thermodynamics of Grain Boundaries

0246

0.0

0.4

0.8

1.2

General GBsSpecial GBs

[GPa]

|ϕ-ϕ

Σ5

| [deg]

0246

0.0

0.4

0.8

1.2

General GBsSpecial GBs

[GPa]

|ϕ-ϕ

Σ5

| [deg]

FIGURE 1.36

Dependence of grain boundary phase transition pressure p

on deviation from

Σ5 coincidence misorientation. The region of existence of the special Σ5 grain

boundary is shaded.

where γ

, γ

are the excess energies of the grain boundary and solid-liquid

interfaces at ﬁxed constant Zn concentration; ΔV

and ΔV

are the excess

volumes of these interfaces, respectively. It was found that the excess volume

ΔV

for all four boundaries studied is ΔV

=2.5 ± 1.4

The problem of the order of the grain boundary wetting phase transition

was discussed in [136]. If the grain boundary wetting transition is of ﬁrst-

order a discontinuity in the grain boundary and surface entropies should be

observed, which in our case leads to a discontinuity of the temperature deriva-

tive of the grain boundary energy at T

:[∂γ/∂T − ∂ (2γ

) /∂T ]

T =T

=

0 [116, 134]. If the grain boundary wetting follows the second order,

[∂γ/∂T]

T =T

=[∂ (2γ

) /∂T ]

T =T

. The theory [131] predicts the law of

the temperature dependence Θ(T ) in the vicinity of the wetting point: for the

ﬁrst-order phase transition

Θ ∼



− T



1/2

;

for the second order

Θ ∼



− T



3/2

Grain boundary wetting in the system Al-Sn was studied with bicrystals with

symmetric tilt 011(001) grain boundaries with the misorientation angles

◦

(random grain boundary) and 38.5

◦

(near Σ9). The observed hysteresis is

consistent with the assumption that the wetting transition is of ﬁrst-order.

An additional reason for grain boundary wetting phase transition to be of

ﬁrst-order is the temperature dependence of the contact angle in the vicinity

of the wetting point (Fig. 1.37). As mentioned above, the theory predicts a

1.4 Applications of Grain Boundary Thermodynamics 91

0.001 0.01

0.1 1

[T-T

] / T

ϕ=32°

ϕ=38.5°

100

Θ [deg]

100

Θ [deg]

0.001 0.01

0.1 1

[T-T

] / T

ϕ=32°

ϕ=38.5°

100

Θ [deg]

100

Θ [deg]

100

Θ [deg]

100

Θ [deg]

FIGURE 1.37

The dependence of the contact angle Θ on (T − T

) /T

. Curves with slope

1/2 and 3/2 (dashed lines) correspond to wetting phase transition of ﬁrst and

second order, respectively [136].