Surface Tension Gradient as Additional Driving Force for Grain Boundary Diffusion. Equilibrium and Non-Equilibrium Cases
Classic Fisher model of grain-boundary diffusion with leakage into bulk is generalized by consideration of the surface-tension change along the grain boundary. Класична Фішерова модель для зерномежової дифузії з відсмоктуванням в об’єм зерен узагальнено з урахуванням зміни поверхневого натягу вздовж...
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Інститут металофізики ім. Г.В. Курдюмова НАН України
2013
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| Cite this: | Surface Tension Gradient as Additional Driving Force for Grain Boundary Diffusion. Equilibrium and Non-Equilibrium Cases / B. Bokstein, A. Rodin // Металлофизика и новейшие технологии. — 2013. — Т. 35, № 9. — С. 1223-1230. — Бібліогр.: 21 назв. — рос. |
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| citation_txt | Surface Tension Gradient as Additional Driving Force for Grain Boundary Diffusion. Equilibrium and Non-Equilibrium Cases / B. Bokstein, A. Rodin // Металлофизика и новейшие технологии. — 2013. — Т. 35, № 9. — С. 1223-1230. — Бібліогр.: 21 назв. — рос. |
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| description | Classic Fisher model of grain-boundary diffusion with leakage into bulk is generalized by consideration of the surface-tension change along the grain boundary.
Класична Фішерова модель для зерномежової дифузії з відсмоктуванням в об’єм зерен узагальнено з урахуванням зміни поверхневого натягу вздовж межі зерна.
Классическая модель Фишера для зернограничной диффузии с отсосом в объём зёрен обобщена с учётом изменения поверхностного натяжения вдоль границы зерна.
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| first_indexed | 2025-12-07T17:59:08Z |
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1223
PACS numbers: 61.72.Mm, 61.72.S-, 68.35.Fx
Surface Tension Gradient as Additional Driving Force
for Grain Boundary Diffusion.
Equilibrium and Non-Equilibrium Cases
B. Bokstein and A. Rodin
National University of Science and Technology «MISiS»,
4 Leninskiy Prospekt,
119049 Moscow, Russia
Classic Fisher model of grain-boundary diffusion with leakage into bulk is
generalized by consideration of the surface-tension change along the grain
boundary.
Класична Фішерова модель для зерномежової дифузії з відсмоктуванням
в об’єм зерен узагальнено з урахуванням зміни поверхневого натягу
вздовж межі зерна.
Классическая модель Фишера для зернограничной диффузии с отсосом в
объём зёрен обобщена с учётом изменения поверхностного натяжения
вдоль границы зерна.
Key words: diffusion, grain boundary, surface tension, chemical potential.
(Received June 21, 2013)
1. INTRODUCTION
It is well known that diffusion flux of i-th component is governed
mainly by the gradient of its chemical potential. Consequently, at con-
stant temperature, the driving forces usually discussed are the concen-
tration gradient in an ideal solution or thermodynamic activity gradi-
ent in non-ideal solution [1, 2]. Stress gradient is an example of addi-
tional driving force, which can change the diffusion process [3]. These
forces fit for bulk and interface diffusion.
During round table discussion at the International Conference DSS-
2010 [4], Prof. L. Klinger stated new idea that if grain boundaries (GB)
energy is not constant for all points of the GB, its gradient provides
Металлофиз. новейшие технол. / Metallofiz. Noveishie Tekhnol.
2013, т. 35, № 9, сс. 1223—1230
Оттиски доступны непосредственно от издателя
Фотокопирование разрешено только
в соответствии с лицензией
2013 ИМФ (Институт металлофизики
им. Г. В. Курдюмова НАН Украины)
Напечатано в Украине.
1224 B. BOKSTEIN and A. RODIN
the additional and independent driving force for atoms motion in GB
(by analogy with Marangoni effect).
Some later, L. Klinger and Eu. Rabkin [5] applied this idea to the
penetrative wetting of GB’s.
The main goal of this paper is to develop the model of GB diffusion
taking into account the surface tension gradient as an additional driv-
ing force.
2. DESCRIPTION OF THE MODEL
Grain boundaries are the paths of fast mass transport in polycrystal-
line solids. Mainly, the macroscopic description of GB diffusion is
based on the Fisher’s model [6]. According to this model (Fig. 1), GB is
a thin plate with the thickness ( 510
9
m) and diffusion coeffi-
cient, which is much higher than the bulk diffusivity (Db Dv). The
model takes into account the fast diffusion flux from the surface
(y 0) along GB and diffusion cleavage from GB to the bulk.
Later Gibbs [7] developed this model for the case of heterodiffusion
and took into account enrichment coefficient s, which gives the rela-
tion between the GB and bulk concentrations
b
v /2
.
x
C
s
C
(1)
Fig. 1. Schematic description of GB diffusion model according to Fisher with
condition Cb(y, t) Cv(x /2, y, t).
SURFACE TENSION GRADIENT AS DRIVING FORCE FOR DIFFUSION 1225
The simplest solution was obtained with following approximations:
concentration on the surface is constant; GB concentration weakly de-
pends on time ( 0b tC , quasi-stationary approximation); diffusion
flux along y-axis in the grain bulk is neglected; segregation isotherm is
linear (s does not depend on concentration).
This solution can be represented as:
b b
v b
( , ) (0) exp( / ),
( , , ) ( , ) erfc /(2 ) ,
C y t C y L
C x y t C y t x Dt
(2)
where
1/2 1/2
b v
( [ /(4 )] )L s D t D , and Fisher’s solution corresponds to
s 1 (self-diffusion).
It should be noted that later the exact solutions for the GB diffusion
problem were obtained by Whipple [8]. Different adsorption models
were used to describe the effect of segregation on GB diffusion (see,
e.g., [9—13]).
The important point is that if GB adsorption takes place, the surface
tension becomes dependent on concentration. The gradient of surface
tension is an additional driving force for the mass transport along the
interface. To take it into account, the Fisher’s model can be combined
with Zhukhovitskii theory of surface phenomena [14].
According to this theory, GB chemical potential of a solute in an ide-
al solution can be written as
,ln b
st
b fCRT (3)
where f is the area corresponding to 1 mole of atoms, f / ( is a mo-
lar volume for the solvent).
The diffusion flux along GB can be written as:
,),(grad),( b
bbbb
y
MtyMtyj
(4)
where Mb is a diffusant mobility at GB.
For complete understanding of the problem, we need to divide two
different states of GB’s: equilibrium and non-equilibrium.
Equilibrium State. It is well known that GB’s are the non-equilibrium
defects of crystals. From thermodynamic point of view, there are no
conditions for equilibrium between GB and the bulk. Nevertheless, due
to geometrical restrictions, it is valid to assume that GB’s are in meta-
stable equilibrium with the adjacent bulk region [15, 16]. In this case,
we have to use the condition of equality of chemical potentials for each
component at GB and in the bulk:
vb ii , (5)
1226 B. BOKSTEIN and A. RODIN
and, consequently, the equation (4) can be rewritten for a dilute solu-
tion as
.),( v
v
bv
bb
y
C
C
RTM
y
Mtyj
(6)
For the linear segregation isotherm (see Eq.(1)) with constant s, we
can obtain:
.),( b
b
b
b
b
b
y
C
D
y
C
C
RTM
tyj
(7)
Here, the relation b b
/ constD MRT C was used.
Thus, the same equations, as in Fisher—Gibbs model, will be re-
tained, neglecting dependence of on Cb [17].
Non-Equilibrium State. Let us assume now that GB chemical potential
for the diffusant is not equal to its value in the bulk. On the other
hand, we will keep the condition of equality of chemical potentials for
the matrix element at GB and in adjacent bulk.
The GB concentration change with time can be described as
.
2
div
2
div
2
),(div
2
bb
b
2
b
b
b
b
2
b
b
xxxx
xx
jD
yRT
fC
y
C
DjD
y
fM
y
C
C
RTM
jDtyj
t
C
(8)
For demonstration of this effect, the approximations indicated above
are kept.
The assumption of equality of chemical potential of solvent at GB
and in the bulk, according [14], leads to the following concentration
dependence of GB surface tension:
1 1b 1
( / ) ln( / ),RT f a a (9)
where and 1 are the GB surface tensions of the solution with given
concentration of solute (Cv) and of pure solvent, a1 and a1b are the
thermodynamic activities of the solvent in the bulk and in GB. With
the use of ideal dilute solution approximation (ai Xi) and assumption
that Eq. (1) is still valid, we arrive to
1b 1 1b 1 b b b b
ln( / ) ln ln ln(1 ) ln(1 ) ( /s).a a X X X X X X X X
The expression (9) can be rewritten as
1 b
( / )( 1)/X RT f s s (10)
SURFACE TENSION GRADIENT AS DRIVING FORCE FOR DIFFUSION 1227
or, taking into account that f / and C X/,
1 b
( 1)/ .C RT s s (11)
Using the substitution:
y
C
RT
s
s
y
C
Cy
bb
b
1
(12)
and the solution for bulk diffusion along x-axis from the constant
source, one can obtain:
).0(
21
)0(
21
v
b
b2
b
2
b
v
bbb
b
b
xC
t
D
y
C
s
s
C
yy
C
D
xC
t
D
y
C
RT
s
s
RT
fC
y
C
y
D
t
C
For the quasi-stationary regime, we finally get the following equation:
0
s
21
b
bb
b2
b
2
tD
CD
y
C
s
s
C
yy
C
(13)
or
,0)1(
2
b
2
b
2
b
2
b
L
C
y
C
A
y
C
AC (14)
where ( 1)/A s s , and
2 1/2
b
s [ /(4 )]L D t D , or
.0
)1()1(
2
b
b
2
b
b
2
b
2
LAC
C
y
C
AC
A
y
C
(15)
3. RESULTS OF CALCULATIONS
Solution of this equation with boundary condition Сb(y 0) Cb0 const
and Сb(y ) 0 can be written as an integral:
b
b0
1/2
(1 )
.
(1 2 /3)
C
C
L A
y d
A
(16)
It is easy to see that, if A 0 (s 1), we get Fisher’s solution. To illus-
trate this solution, the diffusion parameters were taken suitable to Cu
self-diffusion [18]. For bulk diffusion,
5 196800/( )10 RTD e m
2/sec. For
grain boundary diffusion,
15 85700/( )
b
1,2 10 RTD e m
2/sec. In Figure
1228 B. BOKSTEIN and A. RODIN
2, the GB-concentration profiles for different s values in comparison
with Gibbs’ solution are shown.
The results of calculation show that: GB surface tension gradient, as
an additional driving force for GB diffusion, alters the GB diffusion
profile; the concentration profile in this case is not linear in semi-
logarithmic coordinates; the effect of GB surface tension gradient is
the most important in the case of negative segregation, while for the
s 1, it is negligible.
One can see that effective Fisher length L (y value, corresponding to
b b0
ln( / ) 1C C ) decreases with decreasing of s. Note that the less is
the value of s, the less is L. Consequently, the concentration gradient
and the gradient of surface tension increase. That is the reason why the
additional force connected with surface tension gradient is important
for negative adsorption only (s 1).
4. APPLICATION TO THE Fe GB DIFFUSION IN Cu
In study of Fe diffusion in Cu [19], very strange result was obtained.
With the use of the microprobe analysis, it was shown that in a wide
temperature range (550—800C) the difference between concentration
profiles near and far from GB is very small. The authors could not ex-
plain that. It should be noted that the temperature range 550—600C
corresponds to regime «B» of GB diffusion [20], if annealing time is
about 100 h. Let us take as approximation that GB diffusion coeffi-
cient of Fe is not very different in comparison with Cu GB diffusion
coefficient (both the atomic size and the bulk diffusion coefficient are
almost the same) and use the model described above. To compare the
a b
Fig. 2. Calculated GB-concentration profile for 100 h of annealing at 600C
(Db 1.410
7
and Dv 1.810
12
cm
2/s) for different s values (dashed lines cor-
respond to the Gibbs solution).
SURFACE TENSION GRADIENT AS DRIVING FORCE FOR DIFFUSION 1229
penetration depth, we can compare the depth (y
*), at which the concen-
tration in e
2
times smaller in comparison with surface concentration
(ln(C0/C) 2). In the bulk for 100 h, y
*
17 m, and, in GB if s 1,
y*
35 m (Fig. 2). Depending on s value, the depths will be as follow:
y*(s 0.1) 7 m, y
*(s 0.5) 20 m.
Unfortunately, we do not know the segregation factor for this sys-
tem, but according to surface tension measurements: surface(1200 K)
1.8 J/m2
for pure Cu and surface(1200 K) 1.9 J/m2
for Cu—0.72 at.%
Fe [21]. Using the estimation 3/surfaceGB , the enrichment coeffi-
cient can be calculated from Eq. (10) ( 7.1310
6
m
3/mole):
1 1
b
6
2 9
1 1
3 3
7.13 1.8 1.9 10
1 1 0,67 0,33.
3 0.72 10 8.31 1220 5 10
s
s
X RT XRT
Of course, it is very roughly estimated value. However, the tendency
can be clearly seen. If the segregation factor is less than 0.5, the esti-
mated in Fisher’s model penetration depth along GB is less than bulk
diffusion depth.
Finally, we can conclude that GB surface tension gradient acts as an
additional driving force and it must be taken into account in the case of
negative adsorption (s 1).
The work was performed with financial support of Russian Ministry
of Education and Science, Contract No. 16.513.12.3009. Special
thanks to Prof. L. Klinger (Technion, Haifa) for his very useful re-
marks.
REFERENCES
1. J. Philibert, Atom Movements: Diffusion and Mass Transport in Solids (Les
Editions de Physique: 1991).
2. B. S. Bokstein, M. I. Mendelev, and D. J. Srolovitz, Thermodynamics and Ki-
netics in Material Science (Oxford University Press: 2006).
3. L. M. Klinger and E. Rabkin, Defect and Diffusion Forum, 309—310: 19 (2011).
4. B. Bokstein, A. Rodin, and B. Straumal, Proc. of the International Workshop
‘Grain Boundary Diffusion, Stresses, Segregation—DSS-2010’ (June 1—4, 2010,
Moscow).
5. L. Klinger and Eu. Rabkin, Scripta Mat., 62: 918 (2010).
6. J. C. Fisher, J. Appl. Phys., 22, No. 1: 74 (1951).
7. G. B. Gibbs, physica status solidi, 16, No. 1: 27 (1966).
8. R. T. P. Whipple, Philos. Mag., 45, No. 371: 1225 (1954).
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Balluffi) (Metals Park, OH: ASM: 1980), p. 239.
10. B. S. Bokstein, V. E. Fradkov, and D. L. Beke, Philos. Mag. A, 65, No. 2: 277
1230 B. BOKSTEIN and A. RODIN
(1992).
11. B. S. Bokstein, A. S. Ostrovsky, and A. O. Rodin, Philos. Mag. A, 72, No. 4: 829
(1995).
12. V. Esin, B. Bokstein, and A. Rodin, Defect and Diffusion Forum, 309—310: 29
(2011).
13. Yu. Mishin and Chr. Herzig, J. Appl. Phys., 73: 8206 (1993).
14. A. A. Zhukhovitsky, J. Phys. Chem., 15: 296 (1944).
15. B. S. Bokstein, Ch. V. Kopetsky, and L. S. Shvindlerman, Thermodynamics and
Kinetics of Grain Boundaries in Metals (Moscow: Metallurgy Press: 1986).
16. G. Gottstein and L. S. Shvindlerman, Grain Boundary Migration in Metals (Bo-
ca Raton: CRC Press: 2010).
17. L. Klinger, Private communication.
18. T. Surholt and Chr. Herzig, Acta Mater., 45: 3817 (1997).
19. D. Prokoshkina, V. Esin, and A. Rodin, Defect and Diffusion Forum, 323—325:
171 (2012).
20. L. G. Harrison, Trans. Faraday Soc., 57: 119 (1961).
21. S. Zhevnenko and E. Gershman, Private communication.
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/HRV (Za stvaranje Adobe PDF dokumenata najpogodnijih za visokokvalitetni ispis prije tiskanja koristite ove postavke. Stvoreni PDF dokumenti mogu se otvoriti Acrobat i Adobe Reader 5.0 i kasnijim verzijama.)
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/NLD (Gebruik deze instellingen om Adobe PDF-documenten te maken die zijn geoptimaliseerd voor prepress-afdrukken van hoge kwaliteit. De gemaakte PDF-documenten kunnen worden geopend met Acrobat en Adobe Reader 5.0 en hoger.)
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| id | nasplib_isofts_kiev_ua-123456789-104213 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1024-1809 |
| language | English |
| last_indexed | 2025-12-07T17:59:08Z |
| publishDate | 2013 |
| publisher | Інститут металофізики ім. Г.В. Курдюмова НАН України |
| record_format | dspace |
| spelling | Bokstein, B. Rodin, A. 2016-07-04T16:29:51Z 2016-07-04T16:29:51Z 2013 Surface Tension Gradient as Additional Driving Force for Grain Boundary Diffusion. Equilibrium and Non-Equilibrium Cases / B. Bokstein, A. Rodin // Металлофизика и новейшие технологии. — 2013. — Т. 35, № 9. — С. 1223-1230. — Бібліогр.: 21 назв. — рос. 1024-1809 PACS numbers: 61.72.Mm, 61.72.S-, 68.35.Fx https://nasplib.isofts.kiev.ua/handle/123456789/104213 Classic Fisher model of grain-boundary diffusion with leakage into bulk is generalized by consideration of the surface-tension change along the grain boundary. Класична Фішерова модель для зерномежової дифузії з відсмоктуванням в об’єм зерен узагальнено з урахуванням зміни поверхневого натягу вздовж межі зерна. Классическая модель Фишера для зернограничной диффузии с отсосом в объём зёрен обобщена с учётом изменения поверхностного натяжения вдоль границы зерна. The work was performed with financial support of Russian Ministry of Education and Science, Contract No. 16.513.12.3009. Special thanks en Інститут металофізики ім. Г.В. Курдюмова НАН України Металлофизика и новейшие технологии Дефекты кристаллической решётки Surface Tension Gradient as Additional Driving Force for Grain Boundary Diffusion. Equilibrium and Non-Equilibrium Cases Градиент поверхностного натяжения как дополнительная движущая сила для зернограничной диффузии. Равновесный и неравновесный случаи Article published earlier |
| spellingShingle | Surface Tension Gradient as Additional Driving Force for Grain Boundary Diffusion. Equilibrium and Non-Equilibrium Cases Bokstein, B. Rodin, A. Дефекты кристаллической решётки |
| title | Surface Tension Gradient as Additional Driving Force for Grain Boundary Diffusion. Equilibrium and Non-Equilibrium Cases |
| title_alt | Градиент поверхностного натяжения как дополнительная движущая сила для зернограничной диффузии. Равновесный и неравновесный случаи |
| title_full | Surface Tension Gradient as Additional Driving Force for Grain Boundary Diffusion. Equilibrium and Non-Equilibrium Cases |
| title_fullStr | Surface Tension Gradient as Additional Driving Force for Grain Boundary Diffusion. Equilibrium and Non-Equilibrium Cases |
| title_full_unstemmed | Surface Tension Gradient as Additional Driving Force for Grain Boundary Diffusion. Equilibrium and Non-Equilibrium Cases |
| title_short | Surface Tension Gradient as Additional Driving Force for Grain Boundary Diffusion. Equilibrium and Non-Equilibrium Cases |
| title_sort | surface tension gradient as additional driving force for grain boundary diffusion. equilibrium and non-equilibrium cases |
| topic | Дефекты кристаллической решётки |
| topic_facet | Дефекты кристаллической решётки |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/104213 |
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