KINETICS OF THE GRAIN SIZE CHANGE
IN POLYCRYSTALS AND HALL - PETCH
EFFECT
A. Ya. Temkin
Department of Interdisciplinary Studies
Faculty of Engineering
Tel-Aviv University
Ramat-Aviv
Tel-Aviv 69978
Israel
E-mail: temkin@eng.tau.ac.il
April 29, 2001
1
ABSTRACT
The kinetics of grain coalescence and disintegration, as well as the grain growth
(and decrease in size) in polycrystals is developed. A system of kinetic equations for
grains of different radii describing these processes is obtained. The average grain
radius approximation is introduced and applied to the effect of the electric current and
of ultra-short laser pulses upon sample properties, as well as to the ageing of metals.
A possible microscopic mechanism for the creation and rupture of bonds between
grains in a conducting polycrystal (e. g., metal) is considered and (in combination
with the average grain radius approximation) is used for the treatment of the last two
of these effects.
The expression for time dependence of the average grain radius is obtained.
Substituting this expression into Hall-Petch equation, one obtains the time
dependence of the yield strength. In the case when this is the electric current that
produces the coalescence and disintegration of grains, the approximated formulae for
small and large values of the current allow, in combination with experimental data,
the determination of the coefficients in the expression for the grain average radius.
According to these formulae the grain average size decreases (therefore, according to
the Hall-Petch equation, the yield strength rises) at small values of the current and
increases (therefore, according to the Hall-Petch equation, the yield strength
decreases) at large values of the current.
The change of the grain average radius and metal properties by ultra-short laser
pulses (of moderate or low intensity) is considered and the analytical solution
obtained. The spontaneous change of the grain average radius without action of any
external factor is considered. This process can be interpreted as the metal ageing.
2
1.
INTRODUCTION
In real polycrystal materials, elementary monocrystals are joined into clusters
that are usually called grains. Grains are of different sizes and shapes. Neighbor
grains often come into contact with each other. We suppose that there is a
microscopic mechanism of bond creation between neighbor grains coming into
contact, as well as the one of inter-grain bond rupture, but here do not specify their
nature. In the framework of this model the existence itself of grains having different
radii proves that such a mechanism exists. It may cause also the change of grain
distribution with respect to their size and shapes by their coalescence and
disintegration in a polycrystal under the influence of various external factors and even
spontaneous, without any external factor action upon the sample. The last case can be
identified with the material ageing.
We shall use the following model. Consider a grain surrounded by
grains. Denote by
 other
 e the average value of  defined on the set of all grains in the
sample. In our model we shall assume for simplicity that the number of
surrounding grains is the same for all grains and is equal to  e , thus neglecting the
fluctuations of
 from grain to grain. Denote by p the probability in unit of time of
one bond creation between two of neighbor contacting grains. Assume that the
creation of each bond occurs independently of acts of other bonds' creation. We
accept also that two of neighbor grains come into contact with each other are merged
(coalesced) into one grain, iff the number of bonds created between them
where
  0 ,
 0  0 is an integral number.
We model the grains as balls of the radius R, which, in general, differs between
grains. The ball model is introduced to avoid considering the relative orientations of
3
neighboring grains and, as a consequence, anisotropy of their coalescence processes.
The ball model represents a certain averaging of these processes with respect to all
possible orientations. We shall see below that a certain "deviation" from this model is
necessary to introduce the surface of contact between two neighboring grains.
We model the grains created as a result of coalescence also as balls and denote
the radius of such balls by Rc ,  , where
 is the number of primary balls merged into
the considered one. This radius is determined from the equality

4
4
  R3  Rc3,  ,
3  1
3
where
(1)
 is the density of each grain, assumed to be the same for all grains. From Eqn.
(1) we find
Rc ,   3

 R3
(2)
  1


If   R   R0  const , then
Rc ,   3  R0
(3)
For example, for the coalescence of two equal grains we obtain
Rc, 2  3 2R0
(4)
In our model for the coalescence of primary grains it must be
   c . However, the
analogous number for coalescence of grains obtained themselves as result of
coalescence(s), must be determined. It might not satisfy this inequality.
4
According to another model of grain creation and change, monocrystallites in a
grain are not connected by any kind of bonds, but, simply, remain together during
noticeable time-interval because the trajectories of their movement are close with
each other during this time-interval [1]. It refers us to the real gas theory of Mayer and
Montroll [2]. According this theory, gas (many-particle system) consists of virtual
(with very short time-of-life) clusters of different numbers of particles. In the case of
liquids the time-of-life of such clusters becomes long enough so that liquids possess
temporary order of their microstructures, and this order was observed experimentally.
Thus, a polycrystal, e. g., a metal, is represented as a "liquid" of elementary
monocrystallites forming Mayer-Montroll clusters that in this case are called grains.
The viscosity of such "liquids" is much larger than that of genuine liquids, so the
grain time-of-life is also essentially larger. According to this model the change of the
grain size occurs as a consequence of the monocrystallite movement itself, but not of
the grain coalescence and disintegration [1]. We shall include in kinetic equation the
both mechanisms considered in this Section.
2. KINETIC EQUATION OF GRAIN SIZE CHANGE BY
COALESCENCE, DISINTEGRATION AND GROWTH
In a real metal only a certain percentage of all grains genuinely come into
contact with their neighbor ones. The majority of them are separated from their
neighbor grains by a certain narrow split, e. g., of a few of atomic radii. Such factors
as the electric current, magnetic and electric fields may temporary, for a time-interval
 , draw them together, up to they come into contact. If    0 bonds created
5
between them during this time-interval  , these two grains will be joined into one
grain.
Let us introduce a grain distribution function N(R,t)with respect to the grain
radii. This distribution function depends on time t during the time when the current is
on. N(R,t) is the number of grains (balls) of radius R in the unit of volume. Note that R
is discrete variable.
The number of grains of a given radius R decreases because their coalescence
with neighbor grains of any radius that come in contact with grains of the radius R,
and increases (if R  R 0 ) because the coalescence of grains of radii smaller than R.
In this case the following condition must be satisfied:

 R3
3
R ,
(5)

which follows from Eqn. (5)
Now the mass conservation law must be taken into account in the kinetic
equation ( R0  const , N 0  const ):
Rmax
3
3
 R N R, t dR N 0 R0
(6)
R0
Note that the use of the integral in Eqn. (6) is an acceptable approximation only
under special conditions. The case when the process of the grain coalescence and
disintegration can be approximately considered as a continuum process will be
considered below. But, in general, condition (6) should be written as follows:

R  R0  R  Rmax 
R 3 N R, t   N 0 R03 ,
(7)
where R is a discrete variable.
6
It must be demanded that condition (1), or, equivalently (2) be satisfied for each
elementary act of grain coalescence. We shall replace in Eqn. (2) Rc ,  by R to make
it compatible with notations convenient for kinetic equations. It is evident the
condition (7) is automatically satisfied, if mass conservation is taken into account for
each elementary act of the coalescence, and, therefore, (7) need not be used.
In the case when the change of the grain size occurs because of grain growth and
decrease [1], the matter can be accepted by the considered grain from its neighbors
and can be transferred from the considered grain also to its neighbors. Then, in the
framework of the model of balls, the mass conservation law leads to the following

relations: R   R  i e1 Ri for the growth and decrease, correspondingly,
3
3
3
when the goal is to obtain a grain of radius R. Here R' and R''3I are the radii of the
primary grain (which is to be transformed into the one of the radius R) and its
neighbors, correspondingly. We shall use the approximation when this sum will be
replaced by its average value S with respect to the whole ensemble:
e
i 1 Ri 3  S .
Now we can write the kinetic equation determining the distribution function
change in time under a certain influence, e. g, of electric current or ultra-short laser
pulses, or because the state of the system is a nonequilibrium one. This system of
equations expresses the balance of grains of a certain radius R (in a unit of volume)
with respect to 4 following processes:
1)
Grain of radius R coalescence with those having the same and other
radii. 2) Grain of radius R disintegration into grains having radii smaller than
R. 3) Grain of radius R creation by the coalescence of grains of radii smaller
than R. 4) Grain of radius R creation by the disintegration of grains having
radii larger than R. The grain growth and decrease produced by the
7
mechanism described in [1] cannot be represented, if R is considered as a
discreet variable. By this reason these processes are not included in the first
of following kinetic equations, but only in the second one where R is
considered as a continuous variable.
Thus, we can write:
N R, t 

t
Rmax
Rmax



W R, R N R, t N R , t  R,3 R 3  R3 
R   R0 R   3 R 3  R 3
0
R
R3  R3
0
  W R, R , R N R , t N R , t  R , R  R  
3
3
3
R   R0 R   R0
3
R 3  R03
3

R   R0
3
R 3  R03
 P R, R , R N R, t  R ,
3
R   R0
3

R   R0
1  
R 3  R03
R 3  R03
 P R, R , R N R, t [ R,
R   R0
R  , R 
R  3  R  3
  2
R , 3 2R
3
R 3  R 3
(8)


 R, R]
with the initial condition
N R,0  N 0  R , R0 ,
where
(9)
 ik  1 when i  k ,  ik  0 when
i  k , is the Kroneker symbol, t is the
time, W  is the probability (in unit of time) of a grain having the radius R merging
with another grain having radius R', while W  is the probability (in unit of time) that
an act of coalescence of two grains having radii R" and R' , correspondingly, creates a
grain having the radius R, P+ and P- are the disintegration probabilities (in unit of
time). The probabilities W+ and W- include the probabilities of drawing of two
neighbor grains into contact during the time-interval  by, for example, electric
8
current. The Kroneker symbols in the right hand side of Eqn. (8) express the mass
conservation in each elementary act of the grain coalescence (see above) and
disintegration. Note that only elementary acts of the simultaneous two grains
coalescence are included in the system of equations (8). This is an approximation that
is valid when processes of the simultaneous coalescence of more than two grains can
be neglected.
Let us rewrite Eqns. (8) and (9) for the case when continuous variables can
approximately replace discrete variables. The processes of grain growth and decrease
[1] are included into the following kinetic equation. Then we obtain:
N R, t 

t

Rmax
Rmax

R0
3
R


3
R
3


W R, R N R, t N R , t  R   3 R 3  R  3 dR dR  
 R03
R 3  R03
 W R, R , R N R , t N R , t  
R0
R0


3
 R  R   R  dR dR  
3

3
3
R 3  R03 3 R 3  R03
3
3



R   R0
R 3  R03 3 R 3  R03
3
  R  R   R  dR dR  
3
 P R, R , R N R, t 

R   R0

R   R0
 P R, R , R N R , t 
R   R0
 N R , t { R   3 R 3  R  3 
1   R   R   2 R  
3

 dR w R, R N R , t  R  
R
R   R0
Rmax

R R

2 R  R   R  
3

R3  S 


dR w R, R N R , t  R   3 R 3  S  AN R, t 
with the initial condition
9
(10)
N  R0 , t   N 0   R  R0  ,
where
(11)
  x  x  is Dirac  -function, w and w are the probabilities of grain of
radius R creations by the growth from grains of radii R'<R and R'>R,
correspondingly. Notice that in this case R is a continuous variable, W+ and W- are
densities of probabilities (in unit of time), but not the probabilities themselves.
The probability of
 bond creation between two neighbor grains contacting
each other is
p  p ,
(12)
if bond creations are statistically independent events. Then, taking into account that
for coalescence the condition
   0 must be satisfied and denoting wR,R' the
probability in unit of time to draw two neighbor grains into contact (if they already
came into contact, wR,R' =1), we find


ln    ln 
  
W  w R , R R , R p      0   
  0  ,
ln p   ln p


(13)


ln    ln 
  
W R, R , R   w R, R R, R p      0   
  0  , (14)
ln p   ln p


where   x  x   1, when x  x , and   x  x   0, when x  x  ,
 R , R is the
surface of the contact of two neighbor grains. Note that the ball model is used here
with the purpose to characterize grain by only one parameter and to avoid the
consideration of the dependence of grain coalescence on their reciprocal orientation.
At the same time it must be kept in mind that real grains may be of any shape. Then it
is possible that the contacting area
contacting balls, for which
 R, R  0 in distinct from the case of two
 R, R  0 always. The second and third Heavyside's  -
functions in the right-hand sides of Eqns. (13) and (14) express that the time of
10
 bond creation 1 p must be less than  and that the maximum of bonds created
during the time-interval  would be still more than
0.
3. AVERAGE GRAIN RADIUS APPROXIMATION
Let us consider now the same problem, using the approximation that we shall
call the "average grain radius approximation". Probably this approximation can be
obtained from Eqn. (8) or, if the grain grows is taken into account, from Eqn. (10) as
the one for the zero-order moment of the distribution function. But here we shall
obtain it independently of Eqn. (8) or Eqn. (10), but simply from the balance of the
number of grains and the condition of their total mass conservation. The last condition
will allow us to transform this balance equation into an equation for the average
radius R t  change in time. This condition can be written as follows:
R 3 t N R , t   R03 N 0 ,
(15)
 
where N R , t is the total number of grains in a unit of volume, when their average
radius is equal to R ,
R0  R 0 and N 0  N R0 ,0 . From Eqn. (15) one obtains
N R , t   R03 N 0 R 3 t   N R t  ,
(16)
dN R t  dR  3R03 N 0 R 4 t 
(17)
and
dN R t  dt  3R03 N 0 R 4 dR t  dt 
(18)
Now one can write the above mentioned balance equation:
dN R t  dt  dN R , t  dR dR t  dt  
~
 aN 2 R , t   b N R , t ,
11
(19)
~
where a is the average coefficient of grains coalescence, b  b  c  f , b is the
average coefficient of the grain disintegration, c and f are the ones of the grain noncoalescent growth and non-disintegrative size decrease, respectively. The initial
condition is as follows:
R 0  R0
(20)
When R t  is so large that each grain remains isolated from other ones during


the noticeable time because N R t  is small according Eqn. (16), c is not constant.
In the limiting case, when contacts between grains necessary for the matter exchange,
one can write, using Eqn. (16), c  c R
it must replace
3
t , where
c  const . Then in Eqn. (19)
1 3
1
~
aR0 by aR03  c  and b by b   b  f  const . We do not
3
3
~
consider here this case and limit ourselves with the case when b  const . This
means, we consider the concentration of grains still large enough so that each grain
has neighbors all the time for the matter exchange. If the coalescence and
disintegration of grains are absent, the average radius of the grain increases in time
when c>f, which corresponds to results reported in [1].
Notice that R t   0, but can be R t   R0 . If Rmin is the average radius of
an elementary monocrystallite in the sample, then the condition R t   Rmin must be
satisfied.
Using Eqns. (16)-(18), we bring Eqn. (19) to the equation
dR t  1 3
1
1~
 aR0 N 0 2  b R t 
dt
6
R t  6
(21)
that can be rewritten as follows:
12
dyt  a 3
1~
 R0 N 0  b yt  ,
dt
2
2
(22)
where
yt   R 3 t 
(23)
From Eqns. (22) and (23), and the initial condition (20) one finds
a
a


 1~ 
R t   R0 3 ~ N 0  1  ~ N 0  exp  b t 
b
 b

 2 

Notice that lim R t   R0 exp 
a 0
~
b
6

(24)
t and lim R t   R0 3 1  a2 t . If
b 0
~
a  0 and b  0, then lim R t   R0 3  a ~  N 0 .
t 
 b
Substituting Eqn. (24) to Hall-Petch equation [3]
s  s 0  k 2 R t 
1
2
,
(25)
where s is the yield strength and k  const for a given material, one obtains the
following dependence of the yield strength on time:
st   s 0  k 2 R0 
4.
1
a
a


 1 ~ 
2
N

1

N
exp


  b t 
0
0
~
~

 b

 2 
b
1
6
(26)
GRAIN COALLESCENCE AND DISINTEGRATION UNDER
THE INFLUENCE OF THE ELECTRIC CURRENT I
Beginning from this place we consider only the processes of coalescence and
disintegration of grains.
13
In this consideration we shall accept that p  I and w  I . This hypothesis
must be checked by the comparison of obtained theoretical results with experimental
data or by other ways.
It is clear that R  I , t  depends on the electric current I because coefficients a
and b depend on I (and only by this way). Accordingly, the coalescence demands the
creation of an average number of bonds
breaking
   0
   0 , while disintegration requires
 1 bonds. Thus, one can suppose that
a  a 0 I  1
(27)
and
b  b0 I   0 1 ,
(28)
where a 0  const and b0  const depend on the material chemical and physical
properties. From Eqns. (25) and (26) we obtain
a a0  0

I
b b0
(29)
TWO LIMITS.
a) At small values of current I it is expected that b>a according to Eqns. (25)
and (26). Then the approximate solution should be
R t   R0 3
2
 1 
exp  bt  at R t   Rmin
b
 6 
(30)
Substituting the Eqn.(30) to Eqn. (25), one obtains
st   s 0  2
2
3
kb
1
6
 R0  
1
2
1 
exp bt 
 12 
14
(31)
This means, at small values of the electric current the average size of a grain
decreases and, as a consequence, apparently, the yield strength and the hardness
rise.
b) At large values of current I it is expected b<a according to Eqns. (25) and
(26). Then the approximate solution should be
R t   R0 3 1 
a
N 0t
2
(32)
Substituting the Eqn. (32) to Eqn. (25), one obtains
a


st   s 0  2 2 kR0 2 1  N 0 t 
2


1
1
1
6
(33)
This means, at large values of the electric current the average size of a grain
increases and, as a consequence, apparently, the strength and the hardness
decrease.
Eqn. (30) can be rewritten as follows:
R I , t   R0 3 2 b0 I   0 1 
 13
at R I , t   Rmin
 b

exp  0 I   0 1t 
 6

(34)
Eqn. (32) can be rewritten as follows:
R  I , t   R0 3 1 
a0 
I N 0t
2
(35)
Using Eqns. (34) and (35), one can obtain instead Eqns. (31) and (33) the
explicit dependence of s  I , t  on the electric current.
POSSIBLE EXPERIMENTS.
15
Small values of I. Measure experimentally R I 1 , t1  and R I 2 , t 2  .
Thereupon one will calculate bI 1  and bI 2  . Indeed, one obtain from Eqn. (30)
 R  I 1 , t1  
2
 1


exp  bI 1 t1 


 2

 R0  exp erim bI 1 
3
(36)
and the similar equation for I 2 and t 2 . The left-hand side of Eqn. (36) is determined
by the experiment, and so Eqn. (36) is a transcendent equation for the calculation of
bI 1  . The same way is to be used to calculate b I 2  . Then one obtains from Eqn.
(30)
bI 1  I 1  0 1

,
b I 2  I 2  0 1
(37)
where the value of the right-hand side is already known. One finds from Eqn. (37)
   0  ln bI 1  bI 2  ln I 1 I 2 
(38)
Large values of I. Determine experimentally R I 3 , t 3  and R I 4 , t 4  .
Thereupon one will calculate a I 3  and a  I 4  . Indeed, one obtains from Eqn. (32)
2  R3  I 3 , t 3  
aI 3  
1


N 0 t 3  R0
 exp erim
3
(39)
and the same formula for I 4 and t 4 . Then we shall obtain
aI 3  I 3 1
,

aI 4  I 4 1
(40)
whence

aI  
3
  ln

 aI 4 
 I3 
ln   1
 I4 
(41)
Then
16

aI  
3
 0  ln

 aI 4 
 I 3  ln bI 1  bI 2 
1
ln  
ln I 1 I 2 
 I4 
(42)
Now we can find a 0 and b0 from Eqns. (27) and (28), correspondingly, using a
and b determined by Eqns. (39) and (36), as well as values of
 and  0 determined
by Eqns. (41) and (42).
Thus, we know all constants and can use them to calculate R  I , t  and s t 
using the Eqn. (24) and (26), correspondingly, for all values of I and t.
5.
ON A POSSIBLE MECHANISM OF BOND CREATION
AND RUPTURE.
In this work we do not consider such pure macroscopic mechanisms as, for
example, the diffusive penetration of atoms from one grain to its neighbor and vice
versa, like it occurs at the contact surface of two metallic pieces. Only macroscopic
movements in the polycrystal can separate grains joined by this mechanism. Such
kinds of the coalescence and especially of the separation processes occur too slow to
be responsible for effects studied in few-of-minutes-duration experiments. Besides of
it, this way of the grains' binding may lead to the practically irreversible increase of
the grain size up to the collapse of the whole grain system into only one big grain, just
because the separation process is hampered and extremely slow.
Usually, only part of the grain surface comes into contact with the one of its
neighbor. The rest remains free, more precisely, almost free, because in contrast with
the free surface of a monocrystal in vacuum, an interaction exists with the nearest part
of the surface of the contacting neighbor grain. Thus, for a grain of a polycrystal the
17
physical conditions agree (though not exactly, but only approximately) with those
considered by Tamm [4] and in the following works (see, for example, [5, 6, 7, 8]).
Consequently, bound surface electronic states could be formed on free parts of the
grain surface, similar to the bound electronic states formed according to the theory of
Tamm on the free surface of a monocrystal. The principle point of the Tamm
consideration [2] is the break of the periodicity, e. g., of the Kronig-Penny potential,
as the cause of the rise a special kind of bound states. The contacting parts of
surfaces of two neighbor grains despite they are not free surfaces, are also able to
break the periodicity of the potential, which may cause the appearance of bound states
of conductance electrons on the boundary between these two grains. This is a
hypothesis that is to be checked by the corresponding Tamm-like theoretical
consideration. This is not a purpose of the present work. We shell limit ourselves with
the following qualitative consideration. Let be two Tamm surface electron states on
the both of these surfaces, exactly face to face. Let the distance between them is being
adiabatically decreased. If it does not lead to the repair of the periodicity at the
distance zero, these two Tamm states can be merged into one bound state such that the
electron wave function would belong to atoms of these two neighbor grains. In other
words, such a state cannot be identified with one of these grains, but belongs to the
both of them binding them, if this state is not empty. If a conductance electron is
captured by this quantum "catch", it will realize a one-electron bond between these
two neighbor grains coming into contact. When a hot electron knocks out the electron
realizing this bond, it will be ruptured. Macroscopic motions of grains also may break
such bonds, if the breaking stress surpass the binding energy of all bonds binding two
considered grains. The mechanism considered here seems to be one of possible
mechanisms of the grain size change by the electric current. We accept this
18
hypothesis to find out its consequences for the studied macroscopic kinetic processes.
Possibly, it will allow one to check this hypothesis by the comparison of obtained
theoretical results with experimental kinetic dependencies.
According to the theory of Regge poles [9] a resonance electron scattering
maximum is expected at such a positive value of the energy which is approximately
symmetric (relative the zero on the energy axis) to an electron bound state having
negative energy. This resonance maximum corresponds to a resonance state of the
system. In contrast to bound state, it possesses the complex eigenvalue of the energy
and so it decays in some time after its creation to one of a number of possible ways of
this decay (resonance state final channels) "chosen" according its quantum
probability.
If there are electrons with energy in the vicinity of such a resonance, their
resonance scattering can lead to important consequences that will be considered
below. Indeed, among the final channels of the considered resonance state decay are
those corresponding to the energy transfer to the lattice. They are accompanied with
the scattered electron transition from the temporary resonance state having a positive
real part of the energy complex eigenvalue to the bound state having a real negative
energy eigenvalue. It may be a bound state mentioned above between two atoms
belonging to two different grains coming into contact with each other. Then a oneelectron bond between these two grains will be created. But in this case it occurs as a
result of the electron - boundary surface resonance scattering. However, another issue
(with the corresponding channel of the resonance state decay) is possible when the
energy transferred to the lattice breaks an existing bond between two grains. This
means, the resonance scattering of conductance electrons may produce reciprocally
inverse effects. The result of the competition between these processes on the
19
microscopic and macroscopic scale depends on the relative probabilities of the
corresponding channels and kinetics of these processes. But it must be taken into
account that in this competition on the kinetic scale participate also the processes of
bond rupture mentioned above.
In the equilibrium state conductance electron are distributed according to FermiDirac distribution [10]:
dnE  
gVm
3
2
2 
2
3
E dE
e
E   / T
1
,
(43)
where the degeneracy g=2 for an electron, E is the electron energy,
 is the chemical
potential, T is the absolute temperature in energy units (which means that Boltzmann's
constant k is equal to 1), and V is the system volume.
In distribution (43) there are always a certain part of electrons possessing the
resonance energies. Therefore, even in equilibrium, they can produce changes of the
polycrystal microstructure including those that are irreversible, which leads to slow,
irreversible, adiabatic change of the metal with time, provoking the ageing of the
material (see Sec. 8).
Note that at high temperatures of free electron gas, the equilibrium distribution
(43) may be approximated by the classic Maxwell distribution, and the kinetic
Boltzmann equation for nonequilibrium distribution of electron also need not take
quantum effects into account.
The population of high-energy (hot) electrons can be increased by laser light
irradiation (see, for example, [11], [12], [13], [14]), by an electric field or by other
factors which disturb the equilibrium distribution. In these cases, the number of
electrons having energy in the vicinity of the resonances mentioned above may
20
increase in comparison with the one in the equilibrium. Consequently, the numbers of
created and broken bonds between grains increase. But their increase depends
differently on, for example, laser light intensity.
6. ELECTRON RESONANCE INELASTIC COLLISIONS
We consider here the simplest case when the cross-section of the electron resonance
collision can be expressed by the one-resonance-level Breit-Wigner formula [15]:
 fi E   g i 2
i  f
E  E r 
2
1 
  
2 
2
,
(44)
where i and f denote correspondingly the initial and final states, E is the electron
energy, Er is the resonance energy, i and  f are correspondingly the widths of the
initial and final states,     f is the total width,
 is the DeBroglie wavelength,
f
and gI is the statistical weight of the initial state. Then the number of i  f
transitions in the system per unit of time in a unit of volume (the reaction rate) is as
follows:

Y fi  n c  dEvnE  fi E  ,
(45)
0
where nc is the number density of scattering centers and v is the electron velocity.
Substituting Eqn. (44) into Eqn. (45), one obtains:
21
n E 

Y fi  n c g i i  f  dEv 2
0
E  E r 2   1  
(46)
2
2 
Notice that v  h ( 2 E m 2 ) .
3
2
Eqn. (46) can be rewritten as follows:
Y fi  nc i  f Q ,
(47)
where

Q  g i  dE
0
n E 
1
2E m
3
2
E  E r 2
1 
  
2 
2
 const
(48)
Note that more than one resonance maximum may contribute to resonance collision
cross-section, but in the present work we shall limit ourselves to a one-resonance
model.
7. GRAIN COALESCENCE AND DISINTEGRATION INDUCED
BY FEMTO-SECOND LASER PULSES
In this section we consider only those neighbor grains that come into contact, i.
e., the distance between them is of the order of atom dimension.
Femto-second laser pulses disturb the equilibrium distribution (43) of
conductance electrons in metals (see, for example, [9, 10, 11, 12]) increasing the
number density of hot electrons. Among hot electrons are those that have the energy
within the resonance maximum. According Ref. [12], their number density varies
approximately as  J , where J is the laser pulse intensity. Eqn. (21) can be
2
rewritten for one laser pulse of duration
 as follows:
22
def
R t , t     R t     R t  
a J  3
1
b J 
R0 N 0 2  
R t 
6
6
R t 
(49)
Instead Eqns. (27) and (28) one can write:
a J   a 0 J 2
(50)
b J   b0 J 2  0 1
(51)
and
If is possible to accept approximately that a0 and b0 remain unaffected by the
previous pulses' action, i. e., do not depend on the considered pulse number in the
sequence of pulses, one can write the change of R after l identical pulses as follows:
R t  l  
l
b J 
 a J  3

R t     
R0 N 0 R  2 t  l  1  
R t  l  1 
6
l  1  6

(52)
Putting Eqns. (50) and (51) into Eqn. (52), one obtains:
R t  l   R t  
b
a

   60 J 2 R03 N 0 R  2 t  l  1   60 J 2  0 1 R t  l  1 
l  1 

l
1
3
(53)
Eqns. (52) and (53) determine R t  l  as a result of step-by-step calculations
for sequential values of l. The obtained expression of R t  l  can be put into Eqn.
(25) to find the yield strength changed by the sequence of l laser pulses.
23
8. CHANGE THE AVERAGE GRAIN RADIUS IN
ABSENCE OF EXTERNAL FACTORS
It is possible that a sample of metal or another polycrystal is not in absolute
thermodynamic equilibrium state, but in a conditional equilibrium corresponding to a
maximum of entropy obtained under certain constraints. Then the sample structure
may change in time. Consider this phenomenon supposing that it is a result of the free
electron resonance scattering by boundary surfaces between neighbor grains that
come into contact.
In our case the kinetics developed above remains valid also in absence of any
external influence, and Eqns. (24) and (26) describe the average grain radius and the
yield strength dependence on time. Eqns. (47) and (48) allows one to determine the
coefficients a and b:
a  a 0 i  f b Q 

(54)
and
b  b0 i  f  d Q 
  0 1
(55)
From Eqns. (54) and (55) one finds also

a a 0  0 1  0 1  f b

i Q
b b0
f d 0 1
(56)
In Eqns. (54), (55) and (56)  f b and  f  d are widths of final channels
corresponding to a bond creation and decay, respectively, a0 and b0 are constants. If
a  0 and b  0, then lim R t   R0 3
a0
t 
24
b0

 0 1
3
Q
 0 1
3

   0 1
 f 3b  f  d
3
.
The coalescence and disintegration of grains lead to the metal ageing, if
lim R t   R0 .
t 
9.
CONCLUSIONS
The author began this research trying to find the mechanism(s) of the observed
electric current influence on the mechanical properties of conducting polycrystal
surfaces (see, for example, [16], [17], [18] and [19]). The well known connection
(Eqn. (25)) between the average size of grains and the yield strength [3] has set the
author thinking that, possibly, the electric current provokes the grains coalescence and
disintegration in a conducting polycrystal. These processes change the distribution of
grains with respect to their size and, as a consequence, the average grain radius is
changed. It leads to the change of the yield strength and, apparently, hardness. In the
case of the electric current action, its role, mainly, is to bring into temporary contact
neighbor grains separated each from other by narrow split.
In the review [1] the grain growth is considered, which is not due by the
coalescence, but by the motion of elementary crystallites and their clusters. It
corresponds to the model of polycrystal as a "liquid" of elementary monocrystallites.
The grain sizes change provoked by this process is included in our consideration.
In the present work the grain coalescence, disintegration and growth kinetics is
considered as an isolated problem, a "problem in itself", but not as a part of the
research concerning the effect of electric current upon mechanical properties of a
conducting polycrystal surface. The main results are fit for their applications to
different problems. They are valid independently of concrete microscopic mechanism
responsible for grains coalescence and disintegration. At the same time we tried to
find its possible mechanism and used it to determine kinetic coefficients. Possibly, the
25
capture of conductance electrons to quantum bound states on the boundary between
two contacting neighbor grains can serve as such a mechanism when the electric
current action is considered. However, in the case when there are enough hot
electrons produced, for example, by the laser light, the resonance scattering of
conductance electrons by the boundary surface between two neighbor grains, may
lead to the creation or decay of a bond between these grains.
The system of kinetic equations describing grain coalescence and disintegration
was obtained for a certain model (Sec. 2). It was rewritten in the continuous
approximation when the grain radius is considered as a continuous variable. In this
case grain growth was included because it is essentially the process of the continuous
change of the grain size. Note that, for short, we write on the grain growth, but mean
the both processes, increase and decrease of the grain size, produced by the same
mechanism [1].
Concrete results were obtained in the average grain radius approximation (Secs.
3, 4, 7 and 8). These results led to the conclusions that the average grain radius
decreases (therefore, the yield strength and hardness increase) at small values of the
electric current and increases (therefore, the yield strength and hardness decrease) at
large values. These conclusions can be checked experimentally.
The developed theory is applied to the effect of ultra-short laser pulses upon the
ensemble of conductance electrons that leads to the grains coalescence and
disintegration processes (Sec.7). It leads to the same change of surface properties as in
the case of the electric current. This means, laser pulses of relatively small intensity
can be used to process the conducting polycrystal (e. g., metal) surface.
Grains can coalesce and disintegrate, but relatively slowly, even without any
external factor influence (Sec. 8). The average grain radius always slowly changes (in
26
exception of one very special case) as a consequence of these processes. This is a kind
of metal ageing.
27
Acknowledgment.
The author is thankful to Samuel Goldsmith, Reuven (Ray) Boxman, Boris
Alterkop and Nahum Parkansky for the important help and fruitful discussions.
28
9.
1.
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30