Analytically Solvable Differential Diffusion Equations Describing the Intermediate Phase Growth

Abstract

Analytical method to solve differential diffusion equations of describing the growth of the phase wedge during the intermetallic compound formation with a narrow concentration range of homogeneity in bicrystals is proposed. A model of describing the diffusion phase growth from point sources inside polycrystals grains is regarded. Analytical method to solve differential diffusion equations for such model is suggested. Parabolic, cubic, fourth power diffusion regimes for different scales from nanometers to micrometers and millimeters are analyzed.

Key words: diffusion, reaction, phase growth law, intermetallic compounds, grain boundaries.

Full text

ДЕФЕКТЫ КРИСТАЛЛИЧЕС
КОЙ РЕШЁТКИ
PACS numbers: 61.72.Cc, 64.75.Op, 66.30.Dn, 66.30.Ny, 66.30.Pa, 68.35.Fx, 68.35.Rh
Analytically
Solvable Differential Diffusion Equations
Describing
the Intermediate Phase Growth
M.
V. Yarmolenko
Kyiv
National University of Technologies and Design, Cherkasy Branch,
Faculty
of Market, Information and Innovation Technologies,
241/2
V. Chornovola Str.,
UA
-18028 Cherkasy, Ukraine
Analytical
method to solve differential diffusion equations describing the
growth
of the phase wedge during the intermetallic-compound formation
with
a narrow concentration range of homogeneity in bicrystals is proposed.
A
model describing the diffusion phase growth from point source inside the
polycrystal
grains is regarded. Analytical method to solve differential diffu-
sion
equations for such a model is suggested. Parabolic, cubic, and fourth
power
diffusion regimes for different scales from nanometers to micrometers
and
millimeters are analysed.
Key
words: diffusion, reaction, phase-growth law, intermetallic compounds,
grain
boundaries.
Зап
ропоновано аналітичну методу розв’язування диференційного рів-
няння,
що описує кінетику утворення інтерметалевої фази вздовж межі
між
зернами з одночасним проникненням у самі зерна. Розглянуто мо-
дель,
який описує кінетику утворення інтерметалевої фази з точкового
джерела
всередині полікристалічних зерен. Запропоновано відповідну
аналітичн
у методу розв’язування диференційного рівняння такого моде-
л
ю. Проаналізовано дифузійні режими (параболічний, кубічний, четвер-
того
степеня) для різних масштабів — від нанометрового до мікрометро-
вого
та міліметрового.
Ключові
слова: дифузія, реакції, закон зростання фази, інтерметалеві
сполуки, міжфазні межі.
Corresponding author: Mykhaylo Viktorovych Yarmolenko
E-mail: yarmolenko.mv@knutd.edu.ua
Citation: M. V. Yarmolenko, Analytically Solvable Differential Diffusion Equations
Describing the Intermediate Phase Growth, Metallofiz. Noveishie Tekhnol., 40, No. 9:
1201–1207 (2018), DOI: 10.15407/mfint.40.09.1201.
Ìåòàëëîôèç.
íîâåéøèå
òåõíîë.
/
Metallofiz.
Noveishie
Tekhnol.
2018
, т. 40, № 9, сс. 1201–1207 / DOI: 10.15407/mfint.40.09.1201
Îттиски
доступнû непосредственно от издателя
Ôотокопирование
разрешено только
в
соответствии с лицензией

2018 ÈÌÔ (Èнститут металлофизики
им. Ã. Â. Êурдюмова ÍÀÍ Óкраинû)
Íапечатано в Óкраине.
1201

1202 M. V. YARMOLENKO
Предлагается
аналитический
метод
решения
дифференциального
урав-
нения,
которое описûвает кинетику образования интерметаллического
соединения
вдоль границû между зёрнами с одновременнûм проникнове-
нием
в сами зёрна. Рассматривается модель, которая описûвает кинетику
образования
интерметаллического соединения из точечного источника
внутри
поликристаллических зёрен. Предлагается соответствующий ана-
литический
метод решения дифференциального уравнения такой модели.
Àнализируются
диффузионнûе режимû (параболический, кубический,
четвёртой
степени) для разнûх масштабов — от нанометрового до микро-
метр
ового и миллиметрового.
Ключевые
слова: диффузия, реакции, закон роста фазû, интерметалли-
ческие
соединения, межфазнûе границû.
(Received March 12, 2018)
1. INTRODUCTION
Analytical method of interdiffusion problems was presented in [1]. The
researchers analysed concentration profile of Zn in the diffusion re-
gion of Zn–Cu alloy (α-brass, solid-state solution, concentration of Zn
was less than 30%). This system has several intermediate phases too
(β-brass, concentration of Zn is about 50%, γ-brass, concentration of
Zn is about 68%, ε-brass, concentration of Zn is about 84%). These
phases are formed between α-brass and Zn during diffusion. Approxi-
mation of constant diffusion flux along the diffusion direction within
the width of each phase is used (so-called constant flux method) for de-
scribing the growth kinetics of the phases which was theoretically
grounded in [2]. This technique necessitates no allowance for the con-
centration dependence of D(C). Deviations from the parabolic law of
phase growth in cylindrical and spherical samples were analysed in [3]
using this method. This method was applied for describing the growth
kinetics of thin γ-brass and ε-brass layers in a cylindrical sample at
400°C (Cu was in the centre of the cylindrical specimens). The γ-brass
layer grew slower and the ε-brass layer grew more rapidly than in the
planar sample [4]. Model of the growth of an intermediate phase be-
tween low-soluble components on diffusion at grain boundaries involv-
ing outflow was suggested in [5] and criteria for a transition from the
Fisher regime t1/4
to a parabolic one were established. It was proved in
[6] that perpendicular grain boundaries do not influence phase growth
kinetics in B-regime. This result allows us to use the well-known model
of a polycrystal as a 3D array of grain boundaries to be perpendicular
to the interface for describing the phase growth. There were no expla-
nations in [5, 6] how one can solve differential diffusion equations be-
cause of a very complicated method. The formalism suggested was ex-
tended to the case of the growth of a solid-state solution with an expo-

ANALYTICALLY SOLVABLE DIFFERENTIAL DIFFUSION EQUATIONS 1203
nential concentration dependence of the diffusion coefficient. Analyt-
ical solution and Monte Carlo modelling of the Kirkendall effect were
suggested in [7]. Grain boundary (GB) diffusion parameters determi-
nation using A-kinetics of intermetallic layer formation was proposed
in [8]. Experimental data on Cu5Zn8 (γ-brass) diffusion growth kinetics
were used for separate determination of the volume diffusion activa-
tion enthalpy and the GB activation enthalpy. Alternative models of
competition of voiding and Kirkendall shift during compound growth
in reactive diffusion were analysed in [9]. One can improve the meth-
ods to solve the diffusion equations for the growth of intermediate
phase in bicrystals, polycrystals and inside grains.
2. MODELS AND METHODS
Model 1. The model of the phase layer growth during the intermetallic
compound formation with a narrow concentration range of homogenei-
ty, DC1, in bicrystals is based on the following assumptions [5, 6]:
1. An intermediate phase forms at first on the base of the grain
boundary; the latter, transforming from the boundary A–A to the
boundary 1–1, remains, due to easy influx with a diffusion coefficient
Db and having a thickness of δ ≈ 1 nm (i.e., the GB is not overgrown
with a new phase and does not bifurcate).
2. Formed phase 1 broadens normally to the GB due to volume diffu-
sion with a diffusion coefficient D << Db.
3. At all the points of the formed 1–A phase boundary between the
broadening phase 1 and the matrix A the concentration of the compo-
nent B is C1 on the side of phase 1 and is zero on the side of phase A
(solubility of B in A is ignored).
4. Outflow from the GB is the same at all GB points:
11
1
2
() , ( ,0) .
(, ) (,0)
C DC
C Cx xt t
x xt y xt C
DD
∂D
= = =
∂
(1)
5. A flow in the volume of a phase wedge normal to the GB is con-
stant along x (a corresponding property is proved in [2]) in a reference
system associated with the moving nose of the wedge, y(t).
The equation for y(t) has such a form [5, 6]:
() ()
,
()
dyt A yt
B
dt y t t
= −
(2)
where
11 11
/ , (1 / ) 2 / .
b
ADCCB DCC
=D =δD
There were no explanations in [5, 6] how equation (2) can be solved
as a very complicated method was used. A simpler method can be point-
ed out.

1204 M. V. YARMOLENKO
Method 1. One can simplify equation (2) by the following way
() 2
2 ( ),
dz t B
A zt
dt t
= −
(3)
where
2
0
() () () ().zt utvt y t= =
One can transform equation (3) into
() () 2
() () () 2 .
du t dv t B
vt ut vt A
dt dt t

+ +=


(4)
Assumption
() 2 () 0
dv t B vt
dt t
+=
leads to
( ) exp( 4 ).vt B t= −
Next step gives:
4 44
0
2
() 2 .
4
Bt Bt Bt
A AA
u t A e dt te e C
BB
B
= = −+
∫
(5)
General solution of Eq. (3) is as follows:
4
0
2
() .
4
Bt
A AA
zt t Ce
BB
B
−
= −+
(6)
Using initial conditions z(t = 0) = 0 one can obtain finally:
2
( ) (1 ex p( 4 ))
4
AA
zt t B t
BB
= − −−
(7)
or
2
( ) (1 ex p ( 4 )) .
4
AA
yt t B t
BB
= − −−
(8)
Equation (8) shows the Fisher diffusion regime:
22
1
4
1
() 2
b
DC
yt t
DC
δD
=
(9)
for
() .
22
bb
DD
yt
DD
δ< < δ
Model 2. A model of the phase layer growth during the intermetallic

ANALYTICALLY SOLVABLE DIFFERENTIAL DIFFUSION EQUATIONS 1205
compound formation with a narrow concentration range of homogenei-
ty inside grains is based on the following assumptions:
1. An intermediate phase 1 forms inside grains from a point source
of substance A that is surrounded by substance B. The point source has
a diameter of δ ≈ 1 nm. The dislocations steps can be the point sources
in nanometers scale.
2. Dislocation pipe is easy path for A-atoms to go from substance A
to the dislocations steps with a diffusion coefficient Dd ≈ Db and a di-
ameter of δ ≈ 1 nm.
3. Formed spherical phases 1 broadens in 3D space from the disloca-
tions steps due to diffusion with a diffusion coefficient D1
(D < D1 < Dd).
Method 2. One can use constant flux method [3] to get differential
equation for intermetallic compound growing inside polycrystals
grains from a point source and forming small spherical particles
(which form the polycrystals [10] and 3-dimensional integrated cir-
cuits [11]):
2
11
sph 1
4 () ()
4 () , ()
2 () 2
RtD C dR t
J R tC Rt
R t dt
pδ D δ
= = p >>
−δ
(10)
or
211
1
() () 2
DC
dR t Rt
dt C
D
= δ
, (11)
and the solution
11
3
1
3
() .
2
DC
Rt t
C
Dδ
=
(12)
3. ANALYSIS
It was proved in [6] that perpendicular grain boundaries do not influ-
ence phase growth kinetics in B-regime. This result allows us to use the
well-known model of a polycrystal as a 3D array of grain boundaries to
be perpendicular to the interface for describing the phase growth. The
growth phase layer law in polycrystals for diffusion time
22
1
11 3
1
42 8
b
DC
tDC
→
δ
>D
(13)
is parabolic because volume diffusion is more pronounced than GB dif-
fusion.
Parabolic diffusion regime is valid (in micrometers and millimetres

1206 M. V. YARMOLENKO
scales [4]) for y(t) > (Dbδ)/2D [5, 6]:
1
1
2
() .
DC
yt t
C
D
=
(14)
Parabolic diffusion regime is valid in nanometres scale and the
growth phase layer law is as follows:
1
1
2
() .
b
DC
yt t
C
D
=
(15)
A comparison of Eqs. (12) and (15) show that
23
1
11 5
1
23
.
2
b
CD
tCD
→
δ
≈D
(16)
One can find:
22
1
11 3
11
34 6
b
DC
tDC
→
δ
≈D
and
23
11
1
34
.
2
b
D
yt D
→


≈δ




(17)
4. SUMMARY
The growth law of the phase layer during the intermetallic compound
formation with a narrow concentration range of homogeneity is para-
bolic for diffusion time
23
1
5
1
.
2
b
CD
tCD
δ
<D
The growth phase layer law inside polycrystals grains is proportion-
al to
3
t
in about 100 nanometres scale for diffusion time
2 3 22
11
53
111
.
26
bb
C D DC
t
CD DC
δδ
<<
DD
The growth phase layer law in bicrystals in B-regime is the same as
the Fisher solution: the phase wedge is proportional to
4
t
for diffusion
time:

ANALYTICALLY SOLVABLE DIFFERENTIAL DIFFUSION EQUATIONS 1207
δδ
<<
DD
22 22
11
33
11 1
.
68
bb
DC DC
t
DC DC
The phase wedges and roughness are smoothed during phase growth
[4, 6]. Smoothing rate is the more pronounced, the smaller the rough-
ness radius [3]. The growth phase layer law in polycrystals in microme-
ters and millimetres scales for diffusion time
22
1
3
1
8
b
DC
tDC
δ
>D
is parabolic because volume diffusion is more pronounced than GB dif-
fusion.
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