Gottstein G., Shvindlerman L.S. Grain Boundary Migration in Metals: Thermodynamics, Kinetics, Applications
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638 8Solutions
PROBLEM 4.7
The major ideas of a dimensional analysis were put forward by Newton,
Fourier, Buckingham, Rayleigh, Tolman, Bridgman, and Huntley. In partic-
ular, Fourier’s contribution should be distinguished because he introduced
important concepts. The first is the description of the so-called “dimension
formula,” while the second is a clear understanding of the dimensional homo-
geneity of physical expressions [624].
The “dimensional formula” reflects the requirement that physical relations
have to be independent of the units used. The second Fourier concept ex-
presses the dimensional homogeneity of the physical relations; the relation
must be “dimensionally homogeneous.” “For each physical quantity there is
one, and only one, dimension formuled. Conversely, corresponding to each di-
mensional formula there is one, and only one, physical quantity” [625]. There
are some exceptions, which we will not discuss here. Another presentation of
this principle is given in the so-called “π-theorem” (Buckingham theorem):
The physical equations can be expressed as a relationship between dimension-
less complexes. In accordance with the π-theorem the number of independent
dimensionless complexes is equal to the number of quantities considered minus
the number of elementary dimensions used . The meaning of the approaches
mentioned will be discussed in the following for specific examples (“funda-
mental quantities”).
(a) The first step is to define the quantities which determine the motion of a
grain boundary, the time t which is necessary to obtain the given mean
grain size <D>. Obviously, t is a function of grain boundary mobility
m
b
, grain boundary surface tension γ andmeangrainsize<D>;in
other words,
t = f (m
b
,γ,<D>) (8.222)
In accordance with the principle of dimensional homogeneity we arrive
at
T = m
a
b
γ
b
<D>
c
(8.223)
Since the dimension of m
b
is
L
4
E·T
=
L
2
M
−1
T
, γ is
E
L
2
=
MT
−2
,
<D>[L], where E is the dimension of the energy, L of the length, T
of the time, M of the mass, we have:
—exponentofT :1=a − 2b
—exponentofL:2a + c =0
—exponentofM: −a + b =0
The solution is a = −1; b = −1; c =2.
Then the expression for the desired dependence can be presented as
<D>
2
= Cm
b
γt (8.224)
where C is a numerical coefficient.
It can be seen that presenting the surface tension as γ
E
L
2
and the
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8Solutions 639
grain boundary mobility as m
b
L
4
ET
we can arrive at the result (8.224)
even faster. We accept as an axiom that, in the framework of a system
of fundamental quantities, the dimension formula of a physical variable
can be expressed in a unique way. However, the choice of the “funda-
mental” quantities to a certain extent is a question of convenience [626].
Moreover, this can be achieved by practice only, and the shortest way
at the first stage is to use a standard set of fundamental quantities.
In the framework of the concept of the π-theorem, the result (8.224)
manifests that there is only one dimensionless complex which can be
formed from the quantities m
b
, γ, <D>,andtimet.
(b) The primary aim of the second problem is to find the dimensionless
complexes which determine the motion of connected grain boundaries.
The first step is to decide what physical and structural quantities are
involved in this process. We chose the following physical parameters of
the problem:
— grain boundary velocity V
L
T
— distance between the junctions a[L]
— grain boundary mobility m
b
L
2
T
M
— triple junction mobility m
tj
LT
M
— quadruple junction mobility m
qp
T
M
— driving force of grain boundary motion P
M
T
2
L
As can be seen the number of independent dimensionless complexes
is equal to the number of quantities considered minus the number of
fundamental quantities: 6 − 3=3.
Then
V
a
a
b
m
c
b
m
d
tj
m
e
qp
P
f
= K (8.225)
—exponentofL: a + b +2c + d − f =0
—exponentofT : −a + c + c + e − 2f =0
—exponentofM: −c − d − e + f =0
Finally
— c = −a −
b
2
−
d
2
— e =
b
2
−
d
2
— f = −a
The complexes obtained look like this:
V
Pm
b
− I;
am
1/2
qp
m
1/2
b
− II;
m
tj
m
1/2
b
· m
1/2
qp
− III (8.226)
It is easy to see that we get the main dimensionless criteria which were
derived analytically from microscopic considerations. Indeed:
Λ
tj
=
m
tj
a
m
b
=(II) · (III); Λ
gp
=
m
gp
a
2
m
b
=(II)
2
(8.227)
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640 8Solutions
GB
x
GB
x
FIGURE 8.12
Diagram of a grain boundary (GB) which moves in the direction x.
(c) The problem “3” is closely connected with the approach put forward by
W. Williams (1892), Noon and Spencer (1949) and analyzed in Huntley
[624].
As can be seen from the given examples, a solution to a problem can be
found by dimensional analysis when the number of independent quan-
tities exceeds the number of the fundamental quantities L, M, T...)by
one. The principal idea of the approach described is that, if it is nec-
essary to increase the number of fundamental quantities, considering
components of the length in three mutually perpendicular directions. In
this case the fundamental quantities are [L
x
], [L
y
], [L
z
]. This simple
change, at first glance, makes many dimensional expressions more infor-
mative. This is particularly true for the velocity; the difference between
LT
−1
and L
x
T
−1
is evident.
Let us apply this approach to a problem: to find a relation for the veloc-
ity of a flat grain boundary which moves into the direction x under the
action of a constant driving force in the system with the concentration
c of dissolved impurities (Fig. 8.12).
The major physical quantities which define the problem are the velocity
of grain boundary motion, driving force for motion, grain boundary mo-
bility, concentration of the impurities. In accordance with the approach
described and Fig. 8.12 we get
V
L
x
T
,P
ML
x
T
2
L
y
L
z
,m
b
L
2
x
T
M
,c
1
L
x
L
y
L
z
(8.228)
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8Solutions 641
Then
L
x
T
= K
ML
x
T
2
L
y
L
x
a
L
2
x
T
M
b
·
1
L
x
L
y
L
z
e
(8.229)
L
x
:1=a +2b − e
L
y
:0=−a − e
L
z
:0=−a − e
T : −1=−2a + b
M : a − b =0
This yields:
b =1; a =1; c = −1
Finally we obtain:
V = K
Pm
b
c
(8.230)
where K is a numerical coefficient.
Expression (8.230) was derived by L¨ucke and Detert in 1957 from macro-
scopic considerations. The model put forward by L¨ucke and Detert is
convincing and physically clear.
Finally, we would like to cite Rayleigh; this equation is relevant to all
problems considered in this paragraph.
“I have often been impressed by the scanty attention paid even by orig-
inal workers in physica to the great principle of similitude. It happens
not infrequently that results in the form of ‘laws’ are put forward as nov-
elties on the basis of elaborate experiments which might have predicted
a priori after a few minutes’ consideration.”
PROBLEM 4.8
(1) The criterion Λ
gp
=
m
gp
x
0
m
tj
=
9·10
−5
·10
−8
4·10
−12
∼
=
1.1.
The tetrahedral angle Θ is equal to ∼ 108.5
◦
.
(2) The criterion Λ for a 2D configuration n<6Λ=
m
tj
<R>
m
b
=
4·10
−12
5·10
−8
2·10
−14
=10
−5
.
The angle Θ can be found from the relation
Λ=
2Θ
2cosΘ− 1
Θ=10
−5
.
For the configuration n>6
Λ=−
ln sin Θ
1 − 2cosΘ
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642 8Solutions
The angle Θ for Λ = 10
−5
is equal to
∼
=
89.8
◦
.
PROBLEM 4.9
(a) The rate of change of
˙
S for pure grain boundary kinetics (triple junc-
tion mobility, or, more correctly, the criterion Λ →∞) can be found using
Eq. (4.71)
d
˙
S
dn
=
m
b
γπ
3
(8.231)
It can be seen from Eq. (8.231) that the rate of change of
˙
S with n is constant;
when n →∞,
˙
S tends to infinity as well.
(b) In the intermediate case, when the effect of the triple junctions is large, and
grain growth kinetics can be viewed as grain boundary motion, the generalized
Von Neumann-Mullins relation can be expressed by relation (4.82) for n<6
and relation (4.85) for n>6.
Therateofchangeof
˙
S with topological class can be represented as
d
˙
S
dn
=
m
b
γπ
3
·
6+
√
3Λ
2+
√
3Λ
(8.232)
d
˙
S
dn
=
m
b
γπ
3
·
1 −
6
πΛB
(8.233)
Onecanseethattherate
d
˙
S
dn
is positive; however, it depends on the criterion Λ.
For Λ →∞the rate
d
˙
S
dn
tends to the constant value
m
b
γπ
3
, characteristic of pure
grain boundary kinetics. The effect of Λ on the parameter
d
˙
S
dn
is qualitatively
different for n<6andn>6. Namely, for n<6
d
dΛ
d
˙
S
dn
= −
4
√
3m
b
γπ
3
2+
√
3Λ
2
< 0 (8.234)
and for n>6
d
dΛ
d
˙
S
dn
=
2m
b
γ
Λ
2
B
> 0 (8.235)
For both branches the rate of change of
d
˙
S
dn
with Λ tends to zero as Λ increases.
(c) As shown in Chapter 4, under triple junction kinetics, the rate of grain
area change
˙
S is expressed by Eq. (4.89).
Contrary to the relations for
˙
S for grain boundary kinetics and for inter-
mediate kinetics, for pure triple junction kinetics at n →∞the function
˙
S
tends to a product ∞·0.
To remove this uncertainty the L’Hˆopital’s rule can be utilized: if we are
looking for the limit of the fraction
f(x)
g(x)
and the lim
x→∞
f
(x)
g
(x)
= A the limit of
the fraction
f(x)
g(x)
, lim
x→∞
f(x)
g(x)
= A as well.
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8Solutions 643
Let us represent our junction
˙
S as
2
˙
S =
−m
tj
γn
¯
Rsin
2π
n
2sin
π
n
− 1
1/n
(8.236)
As easy to see the expression
d
˙
S
dn
n→∞
is equal to
d
˙
S
dn
n→∞
= lim
n→∞
−m
tj
γ
˜
R
2π
n
2
cos
(
2π
n
)[
2sin
(
π
n
)
−1
]
+
2π
n
2
sin
(
2π
n
)
cos
(
π
n
)
1/n
2
=
=2πm
tj
γ
˜
R (8.237)
Computer simulations of 2D grain growth under triple junction kinetics [441,
442] Chapter 4, Fig. 4.44, confirm the results found in (8.237).
2
In our calculations we restrict ourselves to the expression with exterior radius
¯
R.Weleave
to the reader the possibility of finding the limit for the expression with the interior radius
¯r.
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9
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