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 445

The plastic strain rate associated with the grain shape variation is then given

˙ε

= A

jΩ

= A

Ωσ

c (4.170)

Here A is a numerical coeﬃcient; use was made of the relation N Ω=1

that holds for a pure material. In a similar way an expression for the plastic

strain rate can be obtained for the case when creep is controlled by diﬀusional

mass transfer via grain boundaries (Coble creep [560]) assuming a vacancy

mechanism of grain boundary diﬀusion

˙ε

Coble

= Aπ

Ωσ

(4.171)

Here D

is the vacancy diﬀusivity and c

the vacancy concentration

in grain boundaries. (An interesting modiﬁcation of the diﬀusion-controlled

creep, in which the 1/R

dependence is combined with the grain boundary

diﬀusion, was discussed in [562].)

The value V

, the excess of grain boundary free volume, is the vacancy

capacity of the grain boundary. However, when considering Coble creep [560],

the authors of [561] were forced to separate β and δ: the diﬀerence in the den-

sity between grain boundary and the bulk β and the grain boundary width

δ. The diﬀusion ﬂux during grain boundary diﬀusion is determined by the

grain boundary width δ. Nevertheless, we would like to stress that a physical

meaning has only the parameter V

. V

, as shown in [20, 564] (see also

Chapter 1) can be measured experimentally. For instance, for a 40

◦

111 tilt

grain boundary in Al V

∼

5 · 10

−11

In the case when no grain growth-induced vacancies are present, c in

Eq. (4.170) (and also in Eq. (4.171) where it enters implicitly, provided that

grain boundary diﬀusion is controlled by the vacancy mechanism) is equal to

. However, in the case when a vacancy supersaturation due to grain growth

prevails, c = λc

, i.e. the actual vacancy concentration in the grain interior

needs to be substituted. The modiﬁed creep rates by the Nabarro-Herring and

the Coble mechanism are then given by

˙ε

∗

= λ ˙ε

4NkTV

· ˙ε

(4.172)

Similar considerations can be applied to Coble creep. If thermodynamic equi-

librium between the grain interior and the grain boundaries is assumed to

be reached quasi-instantaneously (i.e. within a time much smaller than the

incubation time t

incubation

), the actual vacancy concentration in the grain

boundaries c

will be enhanced, as compared to the thermal equilibrium

value by the same factor of λ. This leads to an enhanced rate of Coble creep

˙ε

∗

coble

= λ ˙ε

coble

4NkTV

· ˙ε

coble

(4.173)

While an enhancement of the creep rate by a factor of λ is quite signiﬁcant,

the strain increment acquired during the incubation time, when the vacancy

446 4 Thermodynamics and Kinetics of Connected Grain Boundaries

concentration stays at a higher level, is even more impressive. It is given by

Δε



incubation

˙ε

∗

dt =6A

Ωσ

(4.174)

and

Δε

coble



incubation

˙ε

∗

coble

dt =

4π

Ωσ







Ωσ

(4.175)

for the Nabarro-Herring and the Coble type of creep, respectively. As shown

in [561], for creep accompanied by grain growth the “extra” strain Δε for

t>t

incubation

can be signiﬁcant. Thus, for the Nabarro-Herring mechanism

Δε

∼

2%. For the Coble mechanism the eﬀect is much stronger.

We would like to emphasize once more that the commonly accepted ap-

proach to diﬀusional creep based on a vacancy concentration gradient is in-

sensitive to the eﬀect discussed, as grain growth-induced vacancy generation

does not give rise to additional concentration gradients. The principal idea

of Herring that the diﬀusion creep mechanism is based on a gradient of the

chemical potential of atoms has to be utilized instead.

4.9 Problems 447

4.9 Problems

PROBLEM 4.1

Find the pinning force that particles exert on a boundary.

(a) The radius of the particles is r =10

−8

m; the number of the particles per

unit volume is N =10

−3

; the surface tension of the grain boundary is

equal to γ =0.5J/m

(b) r =5· 10

−9

m, N =10

−3

; γ =0.5J/m

PROBLEM 4.2

Find the pinning force that particles exert on triple junctions.

(a) The radius of the particles r =10

−8

m, the number of pareticles per

unit volume N =10

−3

, the surface tension of the grain boundary is

γ =0.5J/m

; the triple line tension γ



=10

−9

m, the mean grain size

<D>=10

−6

(b) r =10

−9

m; N =10

−3

; γ =0.5J/m

; γ



=10

−8

J/m; <D>=10

−7

Compare the result with the result of (a).

PROBLEM 4.3

Find the pinning force that particles exert on grain boundary quadruple junc-

tions.

(a) The radius of the particles is r =10

−8

, the number of the particles per

unit volume is N =10

−3

, the surface tension of the grain boundary is

0.5J/m

; the mean grain diameter is <D>=10

−6

(b) The radius of the particles is r =10

−7

m, the number of the particles

per unit volume is N =10

−3

, γ =0.5J(m

, the mean grain diameter

<D>=10

−6

PROBLEM 4.4

Show that the rate of change of the total area S in a polycrystal in the course

of grain growth is proportional to S

PROBLEM 4.5

Find the dependency of the grain area change rate

as a function of the

topological class n for a system with triple junction for diﬀerent parameters

Λ:

Λ=10.39(n<6); Λ = 10

(n<6)

Λ=1.11(n>6); Λ = 10.39(n>6)

Λ = 200(n>6); Λ = 10

(n>6)

γm

=10

−9

PROBLEM 4.6

Consider the steady-state motion of a grain boundary half-loop with facets

448 4 Thermodynamics and Kinetics of Connected Grain Boundaries

FIGURE 4.79

(a) Geometry of four triple lines (tl) meeting at the quadruple junction (qj).

(b) Forces acting on the quadruple junction.

and triple junction. Find relations between the facet length, the facet mobil-

ity, and the mobilities of grain boundary and junction.

PROBLEM 4.7

Use dimensional analysis to obtain

(a) a relation between the mean grain size and the annealing time for grain

growth.

(b) the dimensionless parameters which determine the motion of connected

grain boundaries, i.e. grain boundaries and grain boundary junctions.

with impurities.

PROBLEM 4.8

In Chapter 4 the steady-state motion of the boundary system with triple and

quadruple junctions is considered. It was shown that the behavior of such a

system is controlled by the dimensionless parameter Λ

= −

ln sin Θ

1+3cosΘ

(4.176)

where Θ is the tetrahedral dihedral angle Θ (Fig. 4.79).

In equilibrium Θ = Θ

∞

=arc cos(−1/3) ≈ 109 · 47

◦

, m

is the mobility

4.9 Problems 449

of the quadruple junction, x

the length which characterizes the grain size,

the mobility of the triple junction.

Use the results [m

γ =2· 10

−14

/s, m

γ =4· 10

−12

m/s, m

γ =

9 · 10

−5

−1

] to ﬁnd the criterion Λ and the angle Θ for 2D conﬁgurations

n<6andn>6inAl.Use<R>=5· 10

−8

m to ﬁnd the tetrahedral angle

at the tip of the quadruple point.

Computer Simulation of Grain Boundary

Motion

“The steady state

Is out of date

Unless my eyes deceive me ...”

— George Gamow

“If this be true, (and they aﬃrm it with great Conﬁdence)

it is much to be wished that their Observations were made public”

— Jonathan Swift

5.1 Introduction

With the advent of high performance computers new tools have become avail-

able to study grain boundary motion and related phenomena like recrystal-

lization and grain growth. Already in the 1970s the structure and energy of

low Σ grain boundaries had been computed and important information on

the relaxed structure of coincidence boundaries was obtained (see Chapter 2)

but the dynamics of grain boundaries could not be addressed with the com-

puter power available at that time. Molecular-dynamics (MD) simulations are

widely used to study the atomic-level behavior of GBs at ﬁnite temperatures.

The method itself is nowadays well established in the ﬁeld of atomistic simula-

tions. As early as the late 1950s [492] and early 1960s the ﬁrst MD simulations

were performed and the basis of this method was established [491]. During the

following decades, reﬁnements of the MD method by introducing ﬁnite tem-

perature thermostats [493, 494, 497, 498], progress in deriving sophisticated

interatomic potentials [495, 496, 499] and extending the MD method by in-

troducing the simulation box vectors as variables [500, 501] steadily increased

the potential of the method.

Wolf et al. [480] were the ﬁrst to address the motion of ﬂat grain boundaries

by molecular dynamics (MD) simulations. Srolovitz and coworkers also used

MD to study the curvature driven motion of a grain boundary half-loop, a

geometry akin to experimental approaches [481]–[488].

451

452 5 Computer Simulation of Grain Boundary Motion

In this chapter we will describe the fundamentals of the simulation method

and the essential results obtained so far. Overviews on MD simulations of

grain boundary energy and mobility were recently published [489, 490].

5.1.1 Classical Molecular Dynamics

The classical MD method utilizes concepts of classical mechanics. The aim

of this method is to compute the time evolution of a system of N particles,

usually atoms or molecules. Since in classical mechanics a particle system

can be successfully characterized by a Lagrangian or Hamiltonian function,

an equivalent approach can be utilized for an atomistic system. The simplest

case of an interatomic interaction only depends on the distance r

between

two atoms, i and j, and not on their orientation or the local environment. In

such case the Lagrangian, L,reads

L({r

}, {˙r

})=K({˙r

}) − V({r

})=



i=1

˙r

· ˙r

−



i=1



j=1

V (r

) ,(5.1)

where K({

}) represents the overall kinetic energy and V ({r

})theoverall

potential energy of the system

. The equations of motion of each atom, i,are

then derived by the Newtonian equation of motion

= −



j=1

dV (r

)

= −



j=i

dV (r

)

· r

, (5.2)

where r

= r

−r

and r

= |r

|. In this case the Hamiltonian, H({r

}, {p

· ˙r

}), reads

H({r

}, {p

})=E =



i=1

· p



i=1



j=1

V (r

) (5.3)

E represents the total energy of the system and is conserved during motion.

Time integration of Eq. (5.2) gives the trajectories of each particle, i, under

the constraints that E =const.,V = const., and N = const., i.e. the particle

system represents a microcanonical ensemble [491].

For most MD simulations it is desired to set the macroscopic temperature

T , constant. Then the constraints are T =const.,V = const., and N =const.,

and the system represents a canonical ensemble. For this, Eq. (5.2) as well as

the Lagrangian need to be altered (see Sec. 5.1.5).

denotes the transpose of A,whereA is a tensor of rank one or two.

5.1 Introduction 453

5.1.2 Parrinello-Rahman Method

A simulation based on the Parrinello-Rahman scheme [501] allows the simu-

lation box to vary in shape and size. This feature is important for GBs due

to the two-dimensional (2D) nature of the defect. For instance GBs have an

excess volume [503, 504] which can only be modelled correctly if the box di-

mensions are allowed to adjust to internal and/or external stresses. In this

particular case the dimension parallel to the GB normal is allowed to expand

or contract according to the internal stress. The in-plane dimensions within

the GB plane remain ﬁxed since it is assumed that the in-plane dimensions

are determined by the bulk properties far away from the GB region. In the

AA

A



B

B



C

C



AB

BC

CA

FIGURE 5.1

Simulation box notation according to the Parrinello-Rahman scheme: The

vectors a, b and c span the edges of the simulation box and its orientation is

described by the vectors b ×c, c ×a and a × b. It should be noted here that

these vectors do not necessarily need to be parallel to the (x, y, z)coordinate

system.

following the matrix convention introduced by Parrinello-Rahman is used, i.e.

the simulation box is described by a matrix h

the columns of which represent

the simulation box vectors a, b and c, h

=(a, b, c).

A schematic of a simulation box is shown in Fig. 5.1. The Parrinello-Rahman

scheme adds nine new degrees of freedom to the already 3N degrees of free-

dom of an atomistic system composed of N atoms. To derive a new set of

equations of motion for each atom and the simulation box matrix h

,thescal-

ing of the real space coordinates of each atom by the simulation box matrix,

, needs to be introduced into the Lagrangian which is expressed by r

= h·s

or s

= h

−1

·r

,wherer

and s

are the coordinates in real and reduced space,

454 5 Computer Simulation of Grain Boundary Motion

respectively. The new Lagrangian is then given by

L ({s

}, {˙s

}, {h}, {

h})

= K({

}, {h}) − V ({s

}, {h})+K

box

({

h}) − V

elastic

({h}) (5.4)



i=1

· G ·

−



i=1



j=1

V (r

W · TR(

h) − V

elastic

({h})

where W is constant and has the dimension of mass and TR stands for the

trace of a (3 ×3) matrix or second-rank tensor. Furthermore, G

= h

·h,and

in the derivation of Eq. (5.4) it was assumed that

can be approximated by

h · s

+ h · ˙s

≈ h · ˙s

For a general stress state the elastic energy V

elastic

({h})reads

elastic

{h} = p · (Ω − Ω

)+Ω

· TR



σ − p · 1



(5.5)

whereΩisthevolumeofthesimulationbox.

The equations of motion for each atom in reduced space do not depend on

the speciﬁc form of V

elastic

({h}) or the kinetic energy of the box K

box

({

h}),

because the reduced coordinates s

do not appear in these terms as obvious

from Eqs. (5.4) and (5.5). Utilizing the Lagrangian equation of motion for the

coordinate s

i,x

, s

i,y

and s

i,z

yields

= m

= −



j=i

dV (r

)

· s

− m

· G

−1

G · ˙s

(5.6)

To derive the equations of motion for the simulation box represented by the

matrix h

; the strain tensor ε in Eq. (5.5) needs to be expressed in terms of

the components of the simulation box vectors. In a linear approximation, the

strain tensor, ε

,isgivenby





−1

− 1





−1

− 1





(5.7)

and the rotation tensor, ω





−1

− 1



−



−1

− 1





. (5.8)

For the derivations of Eqs. (5.7) and (5.8) it is assumed that the atomistic

system deforms homogeneously, i.e. the displacement, u

,ofanatom,j,is

fully determined by the deformation of the simulation box h

when changing

from state h

to h. The displacement, u

,ofanatom,j,isgivenby

= r

− r

= h · s

− h

· s



h · h

−1

− 1



· r



h · h

−1

− 1



⎛

⎜

⎝

0,1

0,2

0,3

⎞

⎟

⎠

⎛

⎝

⎞

⎠

(5.9)

5.1 Introduction 455

Utilizing the displacement vector u

, the strain tensor components, ε

μ,ν

,and

the rotation tensor components, ω

μ,ν

, can be derived from

μ,ν



∂u

∂x

0,ν

∂u

∂x

0,μ



(5.10)

μ,ν



∂u

∂x

0,ν

−

∂u

∂x

0,μ



(5.11)

The equations of motion of the simulation box can be derived under the

assumption that the strain tensor, ε

, is given by the non-linear expression



−1

· G · h

−1

− 1



(5.12)

and reads



P − p · 1



· h

−1

Ω − h · Σ

= h · h

−1



P − p · 1



· h

−1

Ω − h · Σ = h DP

89:;

symmetric

(5.13)

During an MD simulation with an adjustable simulation box, Eq. (5.13) will

be numerically integrated and the evolution of h

and ε with time can be calcu-

lated. The incoporation of the simulation box vectors as dynamical quantities

of the system is far from being trivial. Following Ray and Rahman ﬁnally

leads to new equations of motion

for each atom, i.

= −



j=i

dV (r

)

·s

−m

·h

−1

h ·s

−2 ·m

·h

−1

h · ˙s

(5.14)

5.1.3 Time Integration Schemes

The main goal of MD simulations is to determine the temporal evolution

of a many particle system in phase space and to determine the structural

and thermodynamic properties of the system. This can be achieved by time

integration schemes based on ﬁnite diﬀerence approaches for the equations of

motion of the particle system (Eq. (5.2) or (5.6)). Several schemes to integrate

the equations of motion exist, which usually are derived from a Taylor series

expansion for position, velocity, acceleration and higher order time derivatives

of each atom at times t+δtandt−δt.Hereδt is the time step used to integrate

Eqs. (5.6) and (5.14), are slightly diﬀerent owing to the fact that in the second calculation

scheme, the term

· s

was not dropped from the equation

h · s

+ h · ˙s

in contrast to

the original calculation scheme of Parrinello and Rahman [505].