Competition of Voiding and Kirkendall Shift during Compound Growth in Reactive Diffusion–Alternative Models
The simultaneous growth of both the phase-layer thickness and the void sizes during the intermetallic-compound formation with different mobilities of components and with a narrow concentration-range of homogeneity is described. This is done with account of competition for extra vacancies between dis...
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| Опубліковано в: : | Металлофизика и новейшие технологии |
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Інститут металофізики ім. Г.В. Курдюмова НАН України
2016
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| Цитувати: | Competition of Voiding and Kirkendall Shift during Compound Growth in Reactive Diffusion–Alternative Models / T.V. Zaporozhets, N.V. Storozhuk, A.M. Gusak // Металлофизика и новейшие технологии. — 2016. — Т. 38, № 10. — С. 1279-1292. — Бібліогр.: 17 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859650535500546048 |
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| author | Zaporozhets, T.V. Storozhuk, N.V. Gusak, A.M. |
| author_facet | Zaporozhets, T.V. Storozhuk, N.V. Gusak, A.M. |
| citation_txt | Competition of Voiding and Kirkendall Shift during Compound Growth in Reactive Diffusion–Alternative Models / T.V. Zaporozhets, N.V. Storozhuk, A.M. Gusak // Металлофизика и новейшие технологии. — 2016. — Т. 38, № 10. — С. 1279-1292. — Бібліогр.: 17 назв. — англ. |
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| container_title | Металлофизика и новейшие технологии |
| description | The simultaneous growth of both the phase-layer thickness and the void sizes during the intermetallic-compound formation with different mobilities of components and with a narrow concentration-range of homogeneity is described. This is done with account of competition for extra vacancies between dislocation steps and interfaces (K-sinks leading to Kirkendall shift) and voids (F-sinks providing Frenkel voiding). Three alternative models for three alternative places of preferential voids’ formation are formulated and compared. Possibilities of control over Kirkendall shift versus Frenkel voiding competition are discussed.
Описано одночасний ріст товщини прошарку фази та розмірів пор у процесі формування інтерметалевої сполуки у системі з відмінними рухливостями компонентів і з вузьким концентраційним інтервалом гомогенности. При цьому враховано конкуренцію зайвих вакансій між дислокаційними сходинками та міжфазними інтерфейсами (K-стоками, які приводять до Кіркендаллового зміщення) і порами (F-стоками, які забезпечують Френкелеве пороутворення). Запропоновано і порівняно три альтернативні моделі для трьох альтернативних місць переважного зародження пор. Обговорюються можливості контролю конкуренції Кіркендаллового зміщення і Френкелевого пороутворення.
Описан одновременный рост толщины слоя фазы и размеров пор в процессе формирования интерметаллического соединения в системе с различными подвижностями компонентов и с узким концентрационным интервалом гомогенности. При этом учтена конкуренция избыточных вакансий между дислокационными ступеньками и межфазными интерфейсами (K-стоками, которые приводят к киркендалловскому смещению) и порами (F-стоками, обеспечивающими порообразование по Френкелю). Проведён сравнительный анализ трёх альтернативных моделей для трёх альтернативных мест преимущественного зарождения пор. Обсуждаются возможности контроля конкуренции киркендалловского смещения и порообразования по Френкелю.
|
| first_indexed | 2025-12-07T13:32:58Z |
| format | Article |
| fulltext |
1279
ДЕФЕКТЫ КРИСТАЛЛИЧЕСКОЙ РЕШЁТКИ
PACS numbers:61.72.Bb, 61.72.jd,61.72.Qq,64.75.Op,66.30.Ny,68.35.Dv, 68.35.Fx
Competition of Voiding and Kirkendall Shift during Compound
Growth in Reactive Diffusion–Alternative Models
T. V. Zaporozhets, N. V. Storozhuk, and A. M. Gusak
Bohdan Khmelnytsky National University of Cherkasy,
81 Shevchenko Blvd.,
18031 Cherkasy, Ukraine
The simultaneous growth of both the phase-layer thickness and the void sizes
during the intermetallic-compound formation with different mobilities of
components and with a narrow concentration-range of homogeneity is de-
scribed. This is done with account of competition for extra vacancies between
dislocation steps and interfaces (K-sinks leading to Kirkendall shift) and
voids (F-sinks providing Frenkel voiding). Three alternative models for three
alternative places of preferential voids’ formation are formulated and com-
pared. Possibilities of control over Kirkendall shift versus Frenkel voiding
competition are discussed.
Key words: diffusion, reaction, phase-growth law, voids, intermetallic com-
pounds.
Описано одночасний ріст товщини прошарку фази та розмірів пор у про-
цесі формування інтерметалевої сполуки у системі з відмінними рухливо-
стями компонентів і з вузьким концентраційним інтервалом гомогеннос-
ти. При цьому враховано конкуренцію зайвих вакансій між дислокацій-
ними сходинками та міжфазними інтерфейсами (K-стоками, які приво-
дять до Кіркендаллового зміщення) і порами (F-стоками, які забезпечу-
ють Френкелеве пороутворення). Запропоновано і порівняно три альтер-
нативні моделі для трьох альтернативних місць переважного зародження
пор. Обговорюються можливості контролю конкуренції Кіркендаллового
зміщення і Френкелевого пороутворення.
Corresponding author: Tetyana Vasylivna Zaporozhets
E-mail: zaptet@ukr.net
Please cite this article as: T. V. Zaporozhets, N. V. Storozhuk, and A. M. Gusak,
Competition of Voiding and Kirkendall Shift during Compound Growth in Reactive
Diffusion–Alternative Models, Metallofiz. Noveishie Tekhnol., 38, No. 10:
1279—1292 (2016), DOI: 10.15407/mfint.38.10.1279.
Металлофиз. новейшие технол. / Metallofiz. Noveishie Tekhnol.
2016, т. 38, № 10, сс. 1279—1292 / DOI: 10.15407/mfint.38.10.1279
Оттиски доступны непосредственно от издателя
Фотокопирование разрешено только
в соответствии с лицензией
2016 ИМФ (Институт металлофизики
им. Г. В. Курдюмова НАН Украины)
Напечатано в Украине.
1280 T. V. ZAPOROZHETS, N. V. STOROZHUK, and A. M. GUSAK
Ключові слова: дифузія, реакції, закон росту фази, пори, інтерметалева
сполука.
Описан одновременный рост толщины слоя фазы и размеров пор в процес-
се формирования интерметаллического соединения в системе с различ-
ными подвижностями компонентов и с узким концентрационным интер-
валом гомогенности. При этом учтена конкуренция избыточных вакансий
между дислокационными ступеньками и межфазными интерфейсами (K-
стоками, которые приводят к киркендалловскому смещению) и порами
(F-стоками, обеспечивающими порообразование по Френкелю). Проведён
сравнительный анализ трёх альтернативных моделей для трёх альтерна-
тивных мест преимущественного зарождения пор. Обсуждаются возмож-
ности контроля конкуренции киркендалловского смещения и порообра-
зования по Френкелю.
Ключевые слова: диффузия, реакции, закон роста фазы, поры, интерме-
таллические соединения.
(Received July 29, 2016)
1. INTRODUCTION
Reactive diffusion (growth of single or several intermediate phase lay-
ers in the contact zone of two materials) is often accompanied by void-
ing–formation and growth of voids inside these layers or at their in-
terfaces or in the neighbouring material. Sometimes, this phenomenon
can be useful–as for fabrication of hollow nanoshells for drug delivery
and other applications [1—5]. Yet, typically, voiding is harmful and
eventually leads to failures of joints under repeating loadings and
stresses. For example, in soldering, the reaction of copper with tin-
based solder leads to formation of two intermetallic phases Cu6Sn5 and
Cu3Sn [6]. In particular, the growth of the latter (Cu3Sn), typically,
leads to formation and growth of multiple voids. Voids formation, in
principle, can be related to molar volume changes during phase trans-
formations. Yet, voiding during Cu3Sn growth is usually called the
Kirkendall voiding. It proceeds because diffusivity of copper inside the
growing Cu3Sn phase is much larger than that of tin [7]. It leads to
large vacancy flux inside the growing Cu3Sn phase layer, directed from
tin to copper. Hence, supersaturation with vacancies is formed in the
vicinity of interface Cu3SnCu.
The relaxation of vacancy supersaturation can proceed via at least
two ways. The first way is voiding because of vacancy relaxation at the
so-called F-sinks (terminology of Ya. E. Geguzin [8]), which means just
joining of vacancies into voids (pores). The second way is vacancy re-
laxation without voiding, at the so-called K-sinks, according to the
same terminology of Ya. E. Geguzin (at dislocation kinks or at the
grain boundaries, leading eventually to Kirkendall shift). The compe-
COMPETITION OF VOIDING AND KIRKENDALL SHIFT IN REACTIVE DIFFUSION 1281
tition of the K- and F-sinks during interdiffusion was studied by
Geguzin’s group in experiments with diffusion couples at relatively
low pressures [8, 9]. In particular, the pressure of up to 100 atm was
shown to almost completely suppress the void formation, although it
almost did not change the frequencies of diffusion jumps. Thus, the
Frenkel effect practically vanished during interdiffusion, whereas the
Kirkendall effect (marker shift due to lattice shift) became stronger.
Intuitively, pressure is not a unique method of voiding suppression.
One can also try to increase a density of the K-sinks (say, dislocation
density) or/and decrease a density of heterogeneous void nucleation
sites. Main equations describing the competition of the K- and F-sinks
were published in [10] for homogeneous alloys as well as for inter- and
reactive diffusion [11]. As we now can see, the models presented in [10,
11], were too limited. Here, we will develop three alternative models of
K-sinks versus F-sinks competition in reactive diffusion. These three
models differ by the place of void nucleation and, hence, by the mecha-
nism of growth.
In the intermediate phase Cu3Sn of the binary couple CuSn, the
main diffusant is copper
* *
Sn Cu
( )D D . Diffusion of copper through the
phase layer generates the inverse vacancy flux directed from tin (or
from neighbouring phase Cu6Sn5) to copper. This vacancy flux is essen-
tially stopped somewhere in the vicinity of interface Cu3SnCu (ІМСВ).
If extra vacancies are consumed by the K-sinks, it gives just the
Kirkendall shift–phase layer moves as a whole towards copper filling
the emptiness generated by outgoing vacancy flux. Yet, if, by some
reasons, the K-sinks are not powerful enough, voids nucleate and
grow. At that, we consider three cases. CASE 1: Voids can appear in-
side the phase layer, closer to interface Cu3SnCu (narrow ІМС-layer in
M1 in Fig. 1). This version correlates with the fact that the intermedi-
Fig. 1. Three models of possible voids’ formation due to vacancy supersatura-
tion in the vicinity of interface IMC|B–in some zone (of width W) of effective
vacancy sinks: M1–inside ІМС, M3–directly at the interface ІМСB, M2–in
pure B.
1282 T. V. ZAPOROZHETS, N. V. STOROZHUK, and A. M. GUSAK
ate phase contains 25 percent of the low-melting tin, so that vacancy
formation and migration energies are lower than those in copper are.
Indeed, in the SEM images, many voids can be noticed inside the phase
layer (Fig. 2, a). On the other hand, many voids can be seen directly at
the interface Cu3SnCu. It gives us CASE 3–model M3 in Fig. 1. This
case of heterogeneous nucleation looks like more probable (Fig. 2, b),
especially at the joints of interface with grain boundaries in the grow-
ing phase or in copper. CASE 2 (nucleation at the copper side) is also
possible since the vacancy flux compensates the flux of copper from
copper side–model M2 in Fig. 1. CASE 2 is realized in the formation of
hollow nanoshells and was described qualitatively by Goesele et al. in
the bridge model [3] and mathematically (for the case of one surviving
central void) in [4, 5] (Fig. 2, c).
One can find experimental evidence for each of three possible voids
locations [12, 13]. Yet, the location of the void inside the growing
phase layer does not guarantee that this void nucleated within this lay-
er (alternatively, it can be incorporated into the growing phase from
the parent phase). During phase layer growth, the void can (a) migrate
in the concentration gradient, (b) move together with surrounding lat-
tice, (c) move together with (be pinned and dragged by) moving inter-
face. Therefore, classification of cases is directly related only to initial
stages of the void.
Below, we present the phenomenological models of void formation
for the abovementioned three cases (structures).
2. MODELS
We treat voiding as a result of heterogeneous nucleation at some suita-
Fig. 2. Morphology of reaction zone with different void location: SEM image
of diffusion couple ‘electroplated Culiquid Sn’, after 188 hours of annealing
at 210C (inserted–96 hours) (a); SEM image of diffusion couple ‘Cu after
recrystallization annealingliquid Sn’, after 188 hours of annealing at 210C
(inserted–96 hours) (b); TEM image of cylindrical core-shell system
ZnOAl2O3 after 15 hours at 600C [3] (c).
COMPETITION OF VOIDING AND KIRKENDALL SHIFT IN REACTIVE DIFFUSION 1283
ble sites. To simplify the analysis, we consider a net of the K- and F-
sinks. We treat all F-sinks as the same in one cross-section (so far we do
not take into account the size distribution and ripening of voids). K-
sinks are the steps and kinks at interfaces and dislocations. Their ef-
fectiveness is characterized by the mean free path length LV or means
relaxation time V
2
V V V
( )L D . Concrete meaning of LV in models M1,
M2 and M3 may be different: for models M1 and M2, the value of LV,
most probably, is determined by the dislocations density in the bulk of
the IMC or in pure B, for model M3 the LV is determined by the mor-
phology of interface.
Let us consider the growth of IMC with mean stoichiometric compo-
sition
IMC
B
c and narrow homogeneity range
IMC
B
c . If the tracer diffu-
sivities
* *
A B
,D D significantly differ,
* *
A B
D D , then the vacancy flux
is significant and directed from A to B (as shown for M1 in Fig. 1).
Basic equations. Growth kinetics of the intermediate phase layer of
width X because of reaction between two almost pure materials with
small mutual solubilities is described by the equation [8]:
BIMC IMC
B B
1
.
(1 )
d X
J
dt c c
(1)
Here, is an average atomic volume, the flux density JB is determined
in the laboratory reference frame, it is almost the same in all sections
(JB/x 0), but of course, decreases with time due to a decrease of
mean average concentration gradient. It means a steady-state approx-
imation. This approximation works the better–the less is the concen-
tration range of IMC [14, 15].
In general case, when system is still characterized by the nonequilib-
rium vacancy redistribution, the flux density in the laboratory refer-
ence frame contains two terms–first term proportional to the concen-
tration gradient cB (or chemical potential gradient B) of one of the
basic components, second term–proportional to the vacancy concen-
tration gradient cV (or vacancy chemical potential gradient
eq
V V V
( ln( / )) :kT c c
* *
IMC IMC B A
B D B B B V
( )
(1 ) .
D D
J D c c c
kT
(2)
Here,
IMC * IMC *
D B B B A
((1 ) )D c D c D –Darken interdiffusivity with the
thermodynamic factor
2
B B A B
2
B B
.
c c c g
kT c kT c
Vacancy chemical po-
tential
eq
V V V
ln( / )kT c c becomes zero when vacancy concentration
reaches an equilibrium value
eq
V
c .
In most cases, the components of diffusion couple have different
1284 T. V. ZAPOROZHETS, N. V. STOROZHUK, and A. M. GUSAK
melting points. Therefore, the properties of compound interfaces with
two parent phases differ significantly. The interface between the com-
pound and low-melting component, most probably, does not cause any
delay with supply of vacancies and crossing of interface barriers by the
migrating atoms. Respectively, in our model, we will treat vacancies at
the left interface AІМС to be always in equilibrium and their chemical
potential to be zero. The main deviation of vacancies from equilibrium
is expected in the vicinity of interface ІМСB. As mentioned above,
here one can distinguish three versions of vacancy sinks working in the
vicinity of this interface.
Let W be the width of the vacancy sinks zone (to be specified in each
case) and S–the cross-section area. Then, the total vacancy number
V V
WN c WS in the sinks zone changes according to the following
the balance equation:
K F
V V V
V
( )
0 .
Wd c dN dN
SW j S
dt dt dt
(3)
In equation (3), zero in the right-hand side demonstrates that the va-
cancy flux enters the sinks zone but does not leave it. All incoming va-
cancies go to K- or, alternatively, to F-sinks. In case of small deviation
from equilibrium, the action of K-sinks is traditionally described in
the frame of relaxation approximation:
K eq
V V V
V
V
( )
.
dN c t c
N dt
dt
(4)
At the initial stages of reactive diffusion, the vacancy supersatura-
tion may be significant. In this case, it seems reasonable to use the dif-
ference of chemical potentials (instead of deviation of vacancy concen-
tration) as a true driving force of vacancy relaxation. Let us make
some ‘inverse simplification’:
eq eq eq eq
eq eqV V V V V V V V V
V V V V 2 2
V V V V V
( ) ( ) / 1 ln( ( ) / )
.
c t c c t c c t c D c
D c D c
D L L kT
Then,
K eq
V V V V
2
V
.
W WdN D cWS
dt kTL
(5)
One can expect that equation (5) is more general and is reduced to
equation (4) at small deviations from equilibrium.
Rate of vacancy elimination at growing voids is proportional to the
number of void heterogeneous nucleation sites per unit interface area
ns:
COMPETITION OF VOIDING AND KIRKENDALL SHIFT IN REACTIVE DIFFUSION 1285
F
V s void .
dN n S dV
dt dt
(6)
Here, void
/dV dt –void volume growth rate (void being treated as a
sphere with radius r in models M1 and M2 or lens-type with cross-
section of radius r along interface in model M3).
The void growth rate is affected by the action of K-sinks around the
void. This problem was considered recently in [8]. Modifying result of
[8] in terms of vacancy chemical potential, one can write:
IMC * IMC * V
B A B B
V
1 1
((1 ) ) .
W
dr
c D c D
dt kT r L
(7)
Gradient of vacancy chemical potential is included in the vacancy
flux:
* * V
V B A
( ) .
WWc D
j D D
X X kT
(8)
This expression can be used on the right-hand side of equation (3). The
left-hand side of equation (3) can be treated as zero, in the frame of
steady-state approximation for vacancies. This approximation is the
result of vacancy high mobility. Thus, after substitution of equations
(5), (6), (8) into equation (3) with zero in the left-hand side one obtains
an equation for determination of the unknown V
W kT .
Evaluation of effective sinks zone width W. The width of the vacancy
sinks zone depends on the nature of sinks. For the model M3, it is rea-
sonable to consider both K- and F-sinks located at the interface. There-
fore, in this model, we take W , where is an interface width (about
nanometre).
In the case of models M1, M2, the sinks are present, theoretically,
all over the phase layer. Yet, as we already discussed before equation
(3), essential deviation of vacancies from equilibrium is expected only
in the vicinity of interface ІМСB. Let us find the vacancy chemical po-
tential along the phase layer. For this, we use the steady-state approx-
imation not only for vacancies but as well for base components:
B
B
eq
V V V
V
V
( ) 0,
( )
( ) 0.
c
J
t x
c c x
j
t x kT
(9)
(Correctness of steady-state approximation for compounds with nar-
row homogeneity range was first proved in [15].) In the second equa-
tion (9), the relaxation of vacancies at K-sinks is taken into account.
1286 T. V. ZAPOROZHETS, N. V. STOROZHUK, and A. M. GUSAK
Substitution of equation (8) into second equation (9) leads to equation:
* * 22
V B A B
2 eq 2
V V V V
( ) ( )1
.
x D D c
D kTx c D x
(10)
The first equation (9) means that the flux of the main component
within the phase layer is almost constant along х. Then, using equation
(2) for this flux, one can express 2cB/x2
in terms of vacancy chemical
potential:
2 IMC IMC * * 2
VB B B B A
2 2
D
( )(1 )( )
.
xc c c D D
D kTx x
(11)
The combination of equations (10), (11) gives the resulting equation
for vacancy chemical potential:
2
NG V V
2 2
D V
( ) ( )1
.
D x x
D kT kTx L
(12)
Here, * * IMC * IMC *
NG A B B B B A
/( (1 ) )D D D c D c D is a Nazarov—Gurov
(Nernst—Planck) diffusion coefficient [16, 14].
Equation (12) is a typical screening equation with screening length
V NG D
L D D , which can be treated as effective vacancy mean free path
ef
V
L with account of the inverse Kirkendall effect: the local flux of va-
cancies to sink generates local segregation due to the difference of
components mobilities, segregation effect influences the vacancy flux
and makes this flux less. Note that the ratio DNG/DD tends to zero when
one of the components is much faster than another one, and in this
case, the effective mean free path becomes significantly less. At equal
mobilities of the components, the
ef
V
L tends to LV. Thus, the width W of
effective sinks zone should be taken as
ef
V
L in the model M1, LV in the
model M2. In short, ef
M1 V
W L , WM2 LV, WM3 .
Kinetics of intermediate phase growth and void growth. We use equa-
tions (3), (5), (6), (8) for determination of vacancy chemical potential,
equation (1)–for determination of phase width, equation (7)–for de-
termination of void radius. At that, in each model, we use the corre-
sponding diffusivity D
W
(D
IMC, D
B, D
int
respectively for the models M1,
M2, M3). We omit the long algebra and give only final expression after
the transition to dimensionless variables.
In the case of models M1, M2, we used dimensionless parameter
3
s V
/( 4 ),Q n L dimensionless phase width X/X*, void radius
R/R*
and time tDDc/(X*)2, where characteristic sizes are as
COMPETITION OF VOIDING AND KIRKENDALL SHIFT IN REACTIVE DIFFUSION 1287
follow:
* 2
V
/X L , characteristic phase width corresponding to
crossover from linear to parabolic phase growth kinetics [17],
* 2 * 1 2 *
s V s
/ ( 4 ) (4 ) /R n L n X Y X
(Y is an average lateral distance between voids), Q R
*/LV.
As a result, for the model M1, we get the expressions for V
/( )W kT ,
for phase growth rate d/d and for void growth rate d/d:
* *
V B A
IMC
( ) 1
,
1 (1 (1 ))
W D D c
kT QD
(13)
NG
D
IMC IMC
B B
(1 (1 ))
1 1
,
1 (1 (1 ))(1 )
D
Q
d D
d Qc c
(14)
2
4* *
s VB A
D 2
4( ) (1 ) 1
.
1 (1 (1 ))
n LD Dd Q
d QD
(15)
For the model M2, one gets:
* *
V B A
IMC B
IMC
( ) 1
,
1 (1 (1 ))
W D D c
kT D D
Q
D
(16)
NG B
2
D IMC
IMC IMC
VB B
IMC
(1 (1 ))
1 1
,
(1 )
1 (1 (1 ))
B
D D
Q
d D D
d Lc c D
Q
D
(17)
2* * B
B A
D IMC B
IMC
( ) 1 (1 ) 1
.
1 (1 (1 ))
D Dd D Q
d QD D D
Q
D
(18)
In the case of model M3, mainly equation (7) is changed–we consid-
er the 2D supply of vacancies along 2D interface for the construction
of 3D (lens-type) void. Simple but long algebra with introduction of
non-dimensional parameter P (LV/R
max)2, where
max 1/2
s( )R n (about
this threshold voids will fill the total cross-section), R/Rmax
gives:
* *
V B A
IMC int
IMC 2
( ) 1
,
2
1 1
(1 ) ln(1/ )
W D D c
kT D D P
D
(19)
1288 T. V. ZAPOROZHETS, N. V. STOROZHUK, and A. M. GUSAK
NG int
D IMC 2
IMC IMC
B B
2
2
1
(1 ) ln(1/ )1 1
,
(1 ) 2
1 1
(1 ) ln(1/ )IMC
D D P
D Dd
d c c D P
D
int (20)
2
22* * int
VB A
D IMC 2int
s
IMC 2
2 2
1
ln(1/ ) (1 ) ln(1/ )( ) 1
,
2
1 1
(1 ) ln(1/ )
P
PPLD Dd D
d nD D D P
D
(21)
where
3
3
4 2 1
cos cos .
3 3sin
(22)
3. ANALYSIS
Below, we suggest some numerical and asymptotic analysis to assess
the feasibility of the proposed models.
First, we solve numerically equations (13)—(21) with the calculation
of annihilation at the K- and F-sinks,
K
V
N ,
F
V
N , according to equations
(5), (6). An example of numeric calculation in the case of M1 is demon-
strated in Fig. 3. Calculations were made for various values of mean
free path of vacancies LV at fixed density of void nucleation sites ns
(Fig. 3, a). Alternatively, calculations were made for various densities
ns of void nucleation sites at fixed LV (Fig. 3, b). Evidently, the growth
laws for compound layer thickness X and for average void radius R
are no universal for different stages. Therefore, at the right of Fig. 3,
we demonstrate the exponents of power dependencies km,
m dln/dln and q, q dln/dln.
Second, asymptotic behaviour at various stages is studied. For ex-
ample, for model M1, we can distinguish some asymptotic regimes. At
DNG/DD 1 and 1, the following subregimes with different power
laws km, q
can be found (see Table 1).
In general, asymptotic analysis of model M1 gives 6 principal re-
gimes, differing with time exponents of the dependencies
m dln/dln, q dln/dln at m 1/2, 1, 3/2, 2 and q 1/2, 1/4, 1/6
(Fig. 4).
Regimes I, II, III correspond to phase growth with an account of va-
cancies redistribution but practically without feedback from voiding.
In Regime I, the parabolic growth of phase layer is controlled by the
slower diffusant (via Nernst—Planck, or Nazarov—Gurov, combination
of diffusivities, DNP DNG DADB/(cADA cBDB) [14]). The growth of
COMPETITION OF VOIDING AND KIRKENDALL SHIFT IN REACTIVE DIFFUSION 1289
voids is also parabolic and is controlled by the difference of diffusivi-
ties. Regime II is a transient one (with linear phase growth) to another
parabolic regime but controlled by the faster component (via Darken
combination of diffusivities, DD cBDA cADB). In Regimes III and IV,
the phase layer growth kinetics remains parabolic but the void growth
kinetics is changed. Regimes V and VI are most exotic but have not too
many chances of realization in the real systems (large voids in the thin
phase layer are difficult to imagine).
In the case of model M2, the regimes and time exponents, similar to
the case of model M1, can be obtained at the intervals
B IMC/D D
NG D
/ ,D D
B IMC
NG D
/ / 1,D D D D
B IMC/ 1,D D along axis
Fig. 3. Example of numeric calculations for the case of model M1 (regime IV at
Fig. 4) at ns 1010
m
2, LV 10
8—10
5
m (a) and LV 1 m, ns 109—1012
m
2
(b).
The parameter –void fraction (ratio of the total voids volume to the phase
layer volume).
1290 T. V. ZAPOROZHETS, N. V. STOROZHUK, and A. M. GUSAK
and intervals 1, 1 1/Q, 1/Q along the axis .
One can see that at various stages of reactive growth the time expo-
nents of layer growth and especially of void growth may change rather
significantly. Thus, the experimental systematic study of voiding in
various regimes looks promising. Of course, one should remember
about natural constraints for void growth (like R2ns 1).
4. SUMMARY
Three alternative models of void formation during reactive growth of
TABLE 1.
DNG/DD DNG/DD 1 1
d/d NG
i i
B B D
1 1
(1 )
D
c c D
i i
B B
1 1
(1 ) 2c c
i i
B B
1 1
(1 )c c
NG
i i
B B D
2
(1 )
D
c c D
i i
B B
1
2 (1 )c c
i i
B B
2
(1 )c c
m 1/2 1 1/2
* * *
B A
D
2( )D D R
D
* * *
B A
D
2( )D D R
D
* * *
i i 1/4B A
B B
D
2( )
2 (1 )
D D R
c c
D
q 1/2 1/2 1/4
Fig. 4. Map of asymptotic regimes for model M1.
COMPETITION OF VOIDING AND KIRKENDALL SHIFT IN REACTIVE DIFFUSION 1291
compound layer were suggested taking into account Kirkendall-type
vacancies’ sinks (dislocations, grain boundaries) as well as Frenkel-
type vacancies’ sinks (voids). For each case, the self-consistent set of
equations compound growth and void growth kinetics are formulated
and partially analysed. To choose the right model for given system, ad-
ditional experiments are needed. The vacancy mean free migration
path length LV appears to be especially important for void growth and,
in principle, can be used as one of the parameters controlling the void-
ing kinetics. Another major parameter is a density of heterogeneous
void nucleation sites. If one provides more effective work of K-sinks,
one can expect less voiding (less activity of F-sinks). Providing more
effective work of K-sinks can be realized by increasing of dislocation
density, if this effect will not be eliminated by recrystallization. For
this, one needs to anneal at sufficiently low temperature.
This research was supported by a Marie Curie International Re-
search Staff Exchange Scheme Fellowship within the 7th
European
Community Framework Programme (Ref. 612552) and by Ministry of
Education and Science of Ukraine (project 0115U000638 and
0116U004691). Authors are grateful to Csaba Cserhati and Zoltan Er-
délyi (University of Debrecen, Hungary) for fruitful discussions, to
Oleksii Liashenko (University of Grenoble) for sharing experimental
evidence, and also to Lab for Joining Technologies and Corrosion
(EMPA, Switzerland) for the opportunity of SEM study.
REFERENCES
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<<
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|
| id | nasplib_isofts_kiev_ua-123456789-112616 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1024-1809 |
| language | English |
| last_indexed | 2025-12-07T13:32:58Z |
| publishDate | 2016 |
| publisher | Інститут металофізики ім. Г.В. Курдюмова НАН України |
| record_format | dspace |
| spelling | Zaporozhets, T.V. Storozhuk, N.V. Gusak, A.M. 2017-01-23T18:48:54Z 2017-01-23T18:48:54Z 2016 Competition of Voiding and Kirkendall Shift during Compound Growth in Reactive Diffusion–Alternative Models / T.V. Zaporozhets, N.V. Storozhuk, A.M. Gusak // Металлофизика и новейшие технологии. — 2016. — Т. 38, № 10. — С. 1279-1292. — Бібліогр.: 17 назв. — англ. 1024-1809 DOI: 10.15407/mfint.38.10.1279 PACS: 61.72.Bb, 61.72.jd, 61.72.Qq, 64.75.Op, 66.30.Ny, 68.35.Dv, 68.35.Fx https://nasplib.isofts.kiev.ua/handle/123456789/112616 The simultaneous growth of both the phase-layer thickness and the void sizes during the intermetallic-compound formation with different mobilities of components and with a narrow concentration-range of homogeneity is described. This is done with account of competition for extra vacancies between dislocation steps and interfaces (K-sinks leading to Kirkendall shift) and voids (F-sinks providing Frenkel voiding). Three alternative models for three alternative places of preferential voids’ formation are formulated and compared. Possibilities of control over Kirkendall shift versus Frenkel voiding competition are discussed. Описано одночасний ріст товщини прошарку фази та розмірів пор у процесі формування інтерметалевої сполуки у системі з відмінними рухливостями компонентів і з вузьким концентраційним інтервалом гомогенности. При цьому враховано конкуренцію зайвих вакансій між дислокаційними сходинками та міжфазними інтерфейсами (K-стоками, які приводять до Кіркендаллового зміщення) і порами (F-стоками, які забезпечують Френкелеве пороутворення). Запропоновано і порівняно три альтернативні моделі для трьох альтернативних місць переважного зародження пор. Обговорюються можливості контролю конкуренції Кіркендаллового зміщення і Френкелевого пороутворення. Описан одновременный рост толщины слоя фазы и размеров пор в процессе формирования интерметаллического соединения в системе с различными подвижностями компонентов и с узким концентрационным интервалом гомогенности. При этом учтена конкуренция избыточных вакансий между дислокационными ступеньками и межфазными интерфейсами (K-стоками, которые приводят к киркендалловскому смещению) и порами (F-стоками, обеспечивающими порообразование по Френкелю). Проведён сравнительный анализ трёх альтернативных моделей для трёх альтернативных мест преимущественного зарождения пор. Обсуждаются возможности контроля конкуренции киркендалловского смещения и порообразования по Френкелю. This research was supported by a Marie Curie International Research Staff Exchange Scheme Fellowship within the 7th European Community Framework Programme (Ref. 612552) and by Ministry of Education and Science of Ukraine (project 0115U000638 and 0116U004691). Authors are grateful to Csaba Cserhati and Zoltan Erdélyi (University of Debrecen, Hungary) for fruitful discussions, to Oleksii Liashenko (University of Grenoble) for sharing experimental evidence, and also to Lab for Joining Technologies and Corrosion (EMPA, Switzerland) for the opportunity of SEM study. en Інститут металофізики ім. Г.В. Курдюмова НАН України Металлофизика и новейшие технологии Дефекты кристаллической решётки Competition of Voiding and Kirkendall Shift during Compound Growth in Reactive Diffusion–Alternative Models Конкуренція пороутворення і Кіркендаллового зміщення під час росту сполуки при реакційній дифузії: альтернативні моделі Конкуренция порообразования и киркендалловского смещения во время роста соединения при реакционной диффузии: альтернативные модели Article published earlier |
| spellingShingle | Competition of Voiding and Kirkendall Shift during Compound Growth in Reactive Diffusion–Alternative Models Zaporozhets, T.V. Storozhuk, N.V. Gusak, A.M. Дефекты кристаллической решётки |
| title | Competition of Voiding and Kirkendall Shift during Compound Growth in Reactive Diffusion–Alternative Models |
| title_alt | Конкуренція пороутворення і Кіркендаллового зміщення під час росту сполуки при реакційній дифузії: альтернативні моделі Конкуренция порообразования и киркендалловского смещения во время роста соединения при реакционной диффузии: альтернативные модели |
| title_full | Competition of Voiding and Kirkendall Shift during Compound Growth in Reactive Diffusion–Alternative Models |
| title_fullStr | Competition of Voiding and Kirkendall Shift during Compound Growth in Reactive Diffusion–Alternative Models |
| title_full_unstemmed | Competition of Voiding and Kirkendall Shift during Compound Growth in Reactive Diffusion–Alternative Models |
| title_short | Competition of Voiding and Kirkendall Shift during Compound Growth in Reactive Diffusion–Alternative Models |
| title_sort | competition of voiding and kirkendall shift during compound growth in reactive diffusion–alternative models |
| topic | Дефекты кристаллической решётки |
| topic_facet | Дефекты кристаллической решётки |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/112616 |
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