MODELING OF DEFORMATION-ENHANCED DIFFUSION
MASS TRANSFER DURING MECHANICAL ALLOYING IN
BINARY METALLIC SUBSTITUTIONAL SYSTEMS
Boris B. Khina
Physico-Technical Institute of the National Academy of Sciences of Belarus,
Minsk, Belarus,
Mechanical alloying (MA) is known as a versatile method for producing in situ
dispersion-strengthened composites and far-from equilibrium materials such as
nanocrystalline alloys and compounds, supersaturated solid solutions and
quasicrystalline and amorphous phases, which possess unique properties. Enhanced
solid-state diffusion, which was observed experimentally in many systems under the
action of intensive periodic plastic deformation (IPPD) during MA, has received
substantial interest and was a subject of debates in literature. However, the physics of
this complex phenomenon is not entirely understood yet, which impedes further
development of MA and novel advanced materials by MA-based technologies. In this
work, a mathematical model of enhanced diffusion in a binary substitutional system
subjected to IPPD is developed. The model includes a number of physical factors such
as generation of non-equilibrium point defects by gliding screw dislocations during
deformation, interaction of point defects with edge components of dislocation loops
during intervals between collisions, cross-term interaction of diffusion fluxes of
lattice atoms and vacancies, interaction of the vacancy flux with a non-coherent phase
boundary, etc. Numerical simulations performed for systems Cu-Al and Ni-Cr using
realistic parameter values under the deformation conditions typical of MA in a
vibratory mill have revealed the mechanism of this phenomenon.
1. INTRODUCTION
The phenomenon of deformation-enhanced solid-state diffusion in metals and
alloys is known experimentally for years [1]. It was observed during mechanical
alloying (MA), which is used as a versatile method for producing in situ
dispersion-strengthened composites and far-from-equilibrium materials such as
nanocrystalline alloys and compounds, supersaturated solid solutions,
quasicrystalline and amorphous phases [2, 3]. Besides, enhanced diffusion takes
place in repetitive cold rolling, equal-channel angular pressing (ECAP) and other
metal processing methods that involve intensive periodic plastic deformation
(IPPD). However, the physical mechanism of this intricate phenomenon is not
well understood yet, which hinders further development and commercialization of
the so-called mechanically driven alloys (MDA) produced by a variety of methods
based on IPPD and, in particular, elaboration of novel advanced materials by MAbased technologies.
The role of diffusion processes in the formation of metastable phases during
MA has been a subject of hot debates in literature [4–8]. Some authors completely
deny the role of diffusion in MA and ascribe the observed effects to the
1-42
“mechanical intermixing” of atoms under shear straining, but the physical
meaning of this notion is not explained in detail nor numerical estimates for the
fast intermixing rate are presented [5, 8]. A theory for decomposition of a solid
solution (the demixing phenomenon) under IPPD has been developed [6], where
only the rotational modes of deformation of nanograins in the course of MA are
considered. In this case, the disclinations at triple grain boundary junctions are
considered as the sources of non-equilibrium vacancies. The latter penetrate into
the grains of a supersaturated solid solution (SSS) and thus cause enhanced
diffusion of solute atoms towards the grain boundaries, which results in demixing.
The considered situation corresponds to late stages of MA when the SSS has
already formed, i.e., the mechanism of SSS formation is not considered while in
most cases the goal of MA is to produce a metastable phase but not to decompose
it. Besides, as outlined in Ref. [8], within this theory the mechanism of vacancy
generation by disclinations is not described on the physical level nor a formula (or
even a simple estimate) for the vacancy production rate is given. It should also be
noted that, according to recent experimental observations, nanoparticles (e.g., Ni
nanograins obtained by cold rolling) can contain dislocations whose density is
high,  = 4.71011-1.31012 cm2 [9]. Therefore, the deformation mechanism via
dislocation glide, which is preferable from the energy point of view, can take
place in nanomaterials during IPPD.
In modeling attempts [10,11], only separate aspects were studied, e.g.,
diffusion along curved dislocation pipes [10] or a change in geometry of an
elementary diffusion couple due to deformation [11] using the trivial Fickian
diffusion equation without any additional terms accounting for the
generation/annihilation of non-equilibrium point defects during IPPD. Besides, the
whole processing time of MA was taken as the time of deformation-enhanced
diffusion (from 1 h [10] to 50 h [11]) while it is known that during MA the
deformation time during a ball-powder-ball or ball-powder-wall collision is much
shorter than a time interval between collisions [12,13]. Also, the values of
diffusion coefficients D used for calculations in Ref. [11] were fitted to match the
experimental data on the alloying degree instead of estimating them
independently.
As was outlined earlier [14], this situation in the MA science and technology
is due to a lack of theoretical understanding of diffusion mechanisms under IPPD
on the background of a large amount of experimental data accumulated in
literature. In connection with the above, the goal of this work is to develop a
rigorous mathematical model of enhanced diffusion in a binary substitutional
system subjected to IPPD and perform computer modeling using the
experimentally known (e.g., diffusion coefficients that are measured under
equilibrium conditions) or independently estimated parameter values (e.g.,
deformation conditions in a vibratory mill during MA and the generation rate of
non-equilibrium point defects). The model is based on certain ideas of the theory
of diffusion in irradiated alloys [15, 16].
2. FORMULATION OF THE MODEL
During early stages of MA, fracturing and cold welding of initial metal particles result
in the formation of composite (lamellar) particles with characteristic lamella thickness
L=0.5-0.1 m [2,3]. The physical situation considered in this work is an elementary
1-43
diffusion couple “phase 1 (pure metal A)/phase 2 (metal B)” in a substitutional alloy
system where, in equilibrium conditions, diffusion occurs via the vacancy mechanism,
hence JA+JB+JV=0. As ball-powder-ball or ball-powder-wall collisions in the course
of MA are chaotic [2, 3, 12], changes in the shape and size of this couple inside a
composite particle is neglected. According to the aforesaid, the dislocation
mechanism of plastic deformation dominates. Periodic deformation brings about the
generation of point defects (non-equilibrium vacancies and interstitial atoms of two
sorts: Ai and Bi) in the crystal lattice due to jog dragging by gliding screw dislocations
according to the Hirsch-Mott theory [17]. This process in non-conservative: formation
of a vacancy actually denotes an increase of the lattice site number while the
formation of a self- interstitial atom decreases the number of lattice sites by one.
Besides, non-conservative is also the process of adsorption of non-equilibrium point
defects by edge dislocation, which is accompanied by the climb of the latter [18].
Hence we have to depart from the assumption about the constancy of the lattice site
density, which is traditional in the classical solid-state diffusion theory.
Basing of our previous works [19–21], the diffusion equations for lattice
species (atoms A, B and vacancies V) and interstitials of both sorts, Ai and Bi, are
written as
C k
t
  div J k   k  C k 
N
,
t
(1)
J k    D kn grad Ñn , J V  J B  J A  0, k, n  A, B, V ,
(2)
n
where Ck is the concentration of k-th species, Jk is the diffusion flux, k are
sink/source terms for k-th species (generation/annihilation rate), N is the lattice site
density,  = a03  1/N is the average volume per one lattice site, a0 is the lattice
period, Dkn the are components of the matrix of diffusion coefficients. The latter can
be determined using the theory of diffusion in alloys containing non-equilibrium
vacancies [22] taking into account the fact that when the vacancy concentration CV
substantially exceeds the equilibrium value CV0, the diffusion coefficients increase by
the factor of CV/CV0:
DBB  DB = DB*(gBB–gAB)CV/CV0, DBV = –DB*CB/CV0, DVB = DA–DB,
DAA  DA = DA*(gAA–gBA)CV/CV0, DVV = DV+DA, DV = [CBDB*+CADA*]/CV0.
(3)
Here DA* and DB* are the self-diffusion coefficients of atoms A and B, which are
measured in quasi-equilibrium conditions (i.e. at CV=CV0), g is the thermodynamic
factor: gkk = 1 + (ln k)/(ln Ck), gkn=(Ck/Cn)(ln k)/(ln Cn), k,n=A,B, kn,  is the
activity coefficient. For the sake of simplicity, hereinafter the ideal solid solution is
assumed, i.e. gAA=gBB=1, gAB=gBA=0. The values of DA* and DB* obey the Arrhenius
law D*=D0exp[–E/(RT)], where E is the activation energy and D0 is the preexponent.
The sink/source terms  in the right-hand side (RHS) of Eq. (1) are due to
generation of point defects by gliding screws during deformation, annihilation of
excess vacancies and interstitial atoms in the bulk, and interaction of non-equilibrium
vacancies and interstitials Ai and Bi with edge components of dislocation loops, which
act as volume-distributed sinks. The former takes place during the deformation while
1-44
the latter two processes occur mainly during a pause between sequential deformation
events (collisions) in the course of MA. Then, basing on Eqs. (1)-(3), the following
reaction-diffusion equations for lattice species (atoms B and vacancies) and
interstitials of two sorts are formulated:
C B
C V 
C B
 
N
 D BB
  C B 
 D BV
 Pi C B
x 
x
x 
t

t
(4)
  B D B C B  e   iv (C V  C 0V )C B ,
i
C V
i
i
C V 
C B
 
N
 D VB
  C V 
 D VV
 Pv
x 
x
x 
t

t
i
(5)
  V D V (C V  C 0V ) e   iv (C V  C 0V )(C B  C A ),
i
C B
i
t

C B
 
i
DB
x  i x
i

  C  N  P C
Bi
i B

t

(6)
  B D B C B  e   iv (C V  C 0V )C B ,
i
C A
t
i

i
i
i
C A
 
i
DA
i

x 
x

  C  N  P C
Ai
i A

t

(7)
  A D A C A  e   iv (C V  C 0V )C A ,
i
i
i
i
CB + CA +CV = 1,
(8)
where Pi and Pv are the production rate of interstitials and vacancies, correspondingly,
e is the density of edge dislocations, k are the coefficients describing the efficiency
of edge dislocations as sinks for point defects of the k-th sort, k ~ 1 [15,16], iv is the
annihilation rate of Frenkel pairs Ai-V and Bi-V. In Eqs. (6),(7) the cross-link terms
for diffusion of interstitial atoms Ai and Bi are neglected.
The term N/t is determined by the rates of non-conservative processes, viz.
generation of non-equilibrium point defects by gliding screws and interaction of the
former with edge components of dislocation loops. Thus it can be defined as
N
 Pi  Pv   e [ A D A C A   B D B C B   V D V (C V  C 0V )] .
i
i
i
i
i
i
t
(9)
The annihilation rate of Frenkel pairs is determined as [15,16]
iv = (4r0/)(Di + DV).
(10)
Here Di D A or D B and r0 is the capture radius, r0 = b/2 [15,16] where b is the
i
i
Burgers vector (the shortest spacing between atoms in the crystal lattice). The value of
DV is defined above by Eq. (3) while the diffusion coefficients of self-interstitial
1-45
atoms in substitutional alloys are not known and it is typically assumed that D A , D B
i
i
~ (101-103)DV [15,16] because the activation enthalpy for migration of selfinterstitials H im 0.1 eV [16] is substantially lower than that for vacancies (e.g.,
=0.62 eV for vacancy migration in Al and 1.35 eV for Ni [24]).
H m
v
The equilibrium vacancy concentration is calculated as
CV0 = exp[–HV/(kBT)],
(11)
where HV is the vacancy formation enthalpy and kB is the Boltzmann constant.
The rate of point defect generation according to the Hirsch-Mott theory [17] is
Pk =  (b/2)(fks)1/2, ki,v,
(12)
where  is the strain rate (during MA,  =101-104 s1 [12,13]), 0.5 is the fraction of
forest dislocations, s  e = /2 is the density of screw dislocations,  is the total
dislocation density, fv and fi are the fractions of vacancy and interstitial producing
jogs on a gliding screw, fv+fi = 1 and typically fv>fi [17].
The boundary conditions to reaction-diffusion equations (4)-(7) look as
Jk(x=0) = 0, Jk(x=L) = 0, kB,V,Bi,Ai.
(13)
Besides, typically the boundary between phases 1 and 2 with coordinate L/2 is a
localized sink for non-equilibrium vacancies, hence an additional boundary condition
is to be posed for vacancy diffusion:
CV(x=L/2) = CV0.
(14)
The initial conditions are:
C A ( x )  C B ( x )  0 , CV(x)=CV0 at 0<x<L, CB(x)=1(x–L/2),
i
(15)
i
where 1(x–L/2) is the Heaviside step function.
3. RESULTS OF COMLUTER SINULATIONS AND DISCUSSION
The formulated non-linear problem (3)-(15) was solved numerically using the finitedifference method. The parameter values for computer simulations for systems Cu
(metal A, phase 1)-Al (metal B, phase 2) and Ni (metal A, phase 1)-Cr (metal B,
phase 2) are listed in Tables 1 and 2, correspondingly.
Table 1. Values of diffusion parameters for the Al-Cu system used in modeling [23].
phase 2 (Al-based)
phase 1 (Cu-based)
Hv,
D0,
E,
D*(TMA)
D0,
E,
D*(TMA)
[24]
eV
cm2/s kJ/mol , cm2/s
cm2/s kJ/mol , cm2/s
Al 1.71
142.3
0.3 196.8 8.610–29
0.76
2.010–
20
Cu 0.647
7.8102
135
0.2
0
6.610–
196.4
29
1-46
1.17
cv0(TM
A)
5.41
011
1.61
16

0
Table 2. Values of diffusion parameters for the Ni-Cr system at TMA=373 K [25, 26].
phase 2 (Cr)
phase 1 (Ni)
HV [24],
CV0(TMA) b, cm
2
2
D*(TMA), cm /s D*(TMA), cm /s eV
Cr
1.38
1.010–31
3.410–26
2.710–19 2.5010–8
Ni
1.40
6.410–31
1.510–26
1.310–19 2.4910–8
A cyclic process is considered: generation of point defects during collision and
diffusion and point defect relaxation during the rest period between collisions. For the
Cu-Al system, the parameters corresponded to MA in vibratory mill “SPEX 8000”
with the oscillation frequency =20 Hz: the collision duration t~104 s [12], the
cycle duration tc = 1/(2) = 0.025 s, and the rest period tr = tct = 0.0249 s; the
deformation rate  ~10 s1 was taken as a lower-level estimate. Since the local shorttime heating due to a head-on collision is small [12,13], the particle temperature
during MA was assumed to be TMA=373 K (100 C), and the dislocation density
=1011 cm–2 was taken as a plausible estimate. For the Ni-Cr system, the MA
parameters corresponded to the vibratory mill designed in the Belorussian-Russian
University (Mogilev): =25 Hz, t=2.510–5 s [13], then tr=1,997510–2 s, and the
strain rate estimated using the concept of Hertzian collision was  =3.6104 s1 [13].
The results of simulations are presented in Figs.1-3. Modeling of enhanced diffusion
in the Cu-Al system during MA was performed for two possible situations: (i) the
phase boundary 2/1 cannot act as a vacancy sink (as was suggested in Ref. [27] on a
merely qualitative level, at a certain stage of IPPD of nanomaterials, the boundaries of
nanograins may be saturated with vacancies and cannot act as vacancy sinks), i.e.
boundary condition (14) is not included in simulation (Fig. 1), and (ii) the interface
2/1 acts as a localized sink for non-equilibrium vacancies, i.e. Eq. (14) is taken into
account (Fig. 2). It is seen that within the developed model substantial alloying can
occur after a relatively short time of MA, 4000 s (line 5 in Fig. 1(a)). A steady-state
concentration profile of vacancies establishes in a short time (250 s for Fig. 1 and
about 1200 s for Fig. 2). The enhancement of diffusion is due to accumulation of nonequilibrium vacancies and increase in the diffusion coefficients (see Eqs. (3)), slow
relaxation of vacancy concentration during pauses between collisions, and the linkage
of diffusion fluxes of lattice atoms and vacancies via the off-diagonal diffusion
coefficients, as seen from Eqs. (3)-(5). Simulation has revealed that interaction of
vacancies with the phase boundary may have a selective effect on the diffusion of
atoms A and B in the conditions of MA: strong flux of vacancies towards the interface
enhances the flux of foreign atoms in the opposite direction in phase 2 (compare Fig.
1 and Fig. 2). The latter is sometimes called “the inverse Kirkendall effect”. Thus,
interaction of diffusion fluxes with the phase boundary appears to be an important
factor, which has not earlier received substantial attention in literature. The synergetic
influence of the above factors on diffusion mass transfer brings about the formation of
a wide zone of solid solution in phase 2 within a reasonably short time of MA, t=4000
s (line 5 in Fig. 2(a)). The occurrence of a small peak on the concentration profile of
atoms A near the 2/1 phase boundary (Fig. 2(a), line 5) is due to the fact that the
vacancy concentration in this point corresponds to the equilibrium value, hence the
values of diffusion coefficients DA and DB are close to those for self diffusion D*, i.e.
the phase boundary, which is considered as a sink for non-equilibrium vacancies, can
act as a barrier for deformation-enhanced diffusion. The concentrations of interstitial
atoms Ai and Bi is much less than that of vacancies, which is due to a lower rate of
their generation by jogs on gliding screw dislocations and fast annihilation of the
1-47
Frenkel pairs.
1
(a)
CB
0.5
3
0
4
1
2
phase 2 (Al)
0.2
0.4
0.6
0.8
x/L
10
lg(CV-CV0)
phase 1 (Cu)
5
CA , 10
-2
(b)
C B ,10
i
1
1
3,4
i
2
(с)
5
4
1.5
-4
0.5
2-5
1
2
10
(d)
5
2 3
0.5
-6
0.5
0
0.5
1 0
x/L
x/L
1
0.5
0
1
x/L
Fig. 1. Concentration profiles of atoms B (a), vacancies (b) and interstitials Ai (c) and
Bi (d) obtained by modeling without boundary condition (14) for different durations
of MA: 1, t=0; 2, t=250 s (10000 deformation-rest cycles); 3, t=1250 s (50000 cycles);
4, t=2000 s (80000 cycles); 5, t=4000 s (160000 cycles).
1
(a)
phase 2 (Al)
CA
5
0.5
4
3
0
1
2
0.2
0.4
lg CV
-2
4
(b)
-4
0.6
C A , 1010
x/L
0.8
1
i
6
(c)
CB ,1010
i
3-5
(d)
5
4
3-5
2
-6
phase 1 (Cu)
4
2
2
2
-8
3
2
-10
-12
0
0.5
x/L
1
0.5
0
1-48
x/L
1
0
0.5
x/L
1
Fig. 2. Concentration profiles of atoms A (a), vacancies (b) and interstitials (c and d)
for modeling with boundary condition (14). MA durations are the same as in Fig. 1.
CA
1.0
(a)
phase 2 (Cr)
0.8
5
0.6
0.4
4
phase 1 (Ni)
3
0.2
1
2
0
0
0.1
0.2
0.3
0.4
0.5
x/L
CV
10
8
2
0.7
0.8
0.9
i
5
4
6
phase 1 (Ni)
3
10
10
4
phase 1 (Ni)
3-5
15
10
20
0
0.2
0.4
2
2
1
phase 2 (Cr)
10
1.0
C A , 109
(c)
(b)
3-5
10-5
0.6
0.6
x/L
0.8
1.0
0
1
phase 2 (Cr)
0.2
0.4
2
0.6
x/L
0.8
1.0
Fig. 3. Profiles of atoms A (Ni) (a), vacancies (b) and interstitials (c) for modeling
with boundary condition (14): 1, t=15 s (750 cycles); 2, t=100 s (5000 cycles); 3,
t=400 s (20000 cycles); 4, t=1100 s (55000 cycles); 5, t=1800 s (90000 cycles).
Computer modeling of diffusion in the Ni-Cr system under the conditions of
IPPD was performed using boundary condition (14), i.e. considering the phase
boundary 2/1 as a sink for vacancies. In this situation, atoms A diffuse into phase 2,
and a wide zone of supersaturated solid solution forms in a relatively short time of
MA, t=1100-1800 s (55000-90000 deformation-rest cycles) (lines 4 and 5 in Fig.
3(a)). The reasons for strong enhancement of diffusion are as described above.
Besides, noteworthy is a steep concentration gradient of non-equilibrium vacancies in
phase 2 in the vicinity of the interface (Fig. 3(b)). This causes a strong flux of foreign
atoms A from the phase boundary into phase 2 (the inverse Kirkendall effect). Far
from the interface, the concentration of non-equilibrium vacancies in phase 2 is high,
but their gradient is small. Hence diffusion of atoms A inside this phase occurs due to
their own concentration gradient but with a high partial diffusion coefficient because
CV>>CV0 (see Eqs. (3)). The steady-state profile of vacancies is established after a
1-49
relatively short time, t=400 s, or 20000 cycles (lines 3-5 in Fig. 3(b)). The interstitial
atoms Ai are accumulated mainly in phase 1 because in this phase the annihilation rate
of Frenkel pairs, iv, appears to be higher than in phase 2.
4. CONCLUSIONS
Computer simulation within the frame of the developed model using the physically
grounded parameter values have demonstrated that enhanced solid-state diffusion in
binary metallic systems during IPPD in the conditions of MA is caused by a
synergetic action of several physical factors: (i) generation of non-equilibrium
vacancies due to jog dragging by gliding screw components of dislocation loops
during deformation, (ii) slow relaxation of point defects during rest periods between
collisions in the milling device, (iii) increase in the diffusion coefficients due to
accumulation of non-equilibrium vacancies, (iv) interaction of diffusion fluxes with
the phase boundary, which may act as a sink for vacancies thus changing the direction
of their diffusion, and (v) cross-term interaction of diffusion fluxes of different
species (lattice atoms and vacancies) via the off-diagonal diffusion coefficients.
The outcomes of simulation have revealed the physical mechanism of enhanced
diffusion in metals under the action of IPPD and provide an explanation to numerous
experimental data on the formation of supersaturated solid solutions during MA in
binary substitutional systems [2-4]. The obtained results agree qualitatively with
theoretical work [28] where the evolution of vacancy concentration during MA was
studied using a different approach. Besides, the calculated concentrations of nonequilibrium vacancies generated by IPPD in the conditions of MA agree
quantitatively with those measured experimentally in copper after equal channel
angular pressing (ECAP), repetitive cold rolling and other deformation methods [29].
It should be noted that the accumulation of non-equilibrium vacancies during
MA causes a distortion of the crystal lattice and increases the free energy of the alloy,
and above a certain threshold concentration it can bring about a metastable phase
transition, e.g., solid-state amorphization [30].
REFERENCES
1. Larikov L N, Falchenko V M, Mazanko V F, Gurevich S M, Kharchenko G K,
Ignatenko A I: Doklady Akademii Nauk SSSR 1975 221 (5) 1073-1075 (in Russian).
2. Suryanarayana C. Progress in Materials Science 2001 46 (1-2) 1-184.
3. Zhang D L: 'Processing of advanced materials using high-energy mechanical
milling'. Progress in Materials Science 2004 49 (3-4) 537-560.
4. Skakov Yu A: Metal Science and Heat Treatment 2004 46 (3-4) 137-145.
5. Shtremel' M A:. Metal Science and Heat Treatment 2004 46 (3-4) 146-147.
6. Gapontsev V L, Koloskov V M: 'Metal Science and Heat Treatment' 2007 49 (1112) 503-513.
7. Skakov Yu A: Metal Science and Heat Treatment 2007 49 (11-12) 514-516.
8. Shtremel' M A: Metal Science and Heat Treatment 2007 49 (11-12) 517-518.
9. Wu X-L, Ma E: 'Dislocations in nanocrystalline grains'. Applied Physics Letters
2006 88 (23) 231911 (3 pp.).
10. Rabkin E, Estrin Y: Scripta Materialia 1998 39 (12) 1731-1736.
11. Mahapatra T K, Das D, Manna I, Pabi S.K: Acta Materialia 1998 46 (10) 35013510.
12. Maurice D R, Courtney T H: Metallurgical Transactions A 1990 21 (2) 289-303.
1-50
13. Lovshenko G F, Khina B B: Friction and Wear 2005 26 (4) 434-445 (in Russian).
14. Khina B B, Froes F H: 'Modeling mechanical alloying: advances and challenges'.
Journal of Metals (JOM) 1996 48 (7) 36-38.
15. Mansur L K: Journal of Nuclear Materials 1979 83 109-127.
16. Johnson R A, Lam N Q: Physical Review B 1976 13 (10) 4364-4375.
17. Nabarro F R N, Basinski Z S, Holt D B: Advances in Physics 1964 50 193-323.
18. Novikov I I: Defects of crystalline structure in metals. Moscow, Nauka, 1983 (in
Russian).
19. Khina B B, Solpan I, Lovshenko G F: Journal of Materials Science 2004 39 (1617) 5135-5138.
20. Khina B B, Lovshenko F G, Konstantinov V M, Formanek B: Metal Physics and
Advanced Technologies 2005 27 (5) 609-623 (in Russian).
21. Khina B B, Formanek B: Defect and Diffusion Forum 2006 249 105-110.
22. Gurov K P, Kartashkin B A, Ugaste U E: Mutual diffusion in multiphase metallic
systems. Moscow, Nauka, 1981 (in Russian).
23. Brandes E A, Brook G B (editors): Smithells metals reference book, 7th edition.
Oxford, Butterworth-Heinemann, 1992.
24. Bokshtein B S: Diffusion in metals. Moscow, Metallurgiya, 1978 (in Russian).
25. Drits M E (editor): Properties of elements: a handbook. Moscow, Metallurgiya,
1985 (in Russian).
26. Grigor’ev I S, Meylikhov E Z (editors): Physical parameters: a handbook.
Moscow, Energoatomizdat, 1991 (in Russian).
27. Nechaev Yu S: Physics 2001 65 (10) 1507-1514 (in Russian).
28. Zhang B Q, Lu L, Lai M O:. Physica B: Condensed Matter 2003 325 120-129.
29. Ungar T, Schafler E, Hanak P, Bernstorff S, Zehetbauer M: Materials Science and
Engineering A 2007 462 (1-2) 398-401.
30. Fecht H J: Nature 1992 356 (6365) 133-135.
1-51