The impact of a complex microstructure on polycrystalline diffusion is investigated using both numerical and analytical methods. In particular, the diffusion equation is numerically integrated using the finite-difference method to obtain the concentration profile for a diffusant in a simplified microstructural representation. The methodology is first validated for an idealized model of diffusion in a prototypical, single grain-boundary system and then applied to a Voronoi model of a microstructure resulting from homogeneous nucleation and growth. The diffusive behavior is quantified by obtaining uptake curves as a function of time for different ratios of grain boundary to lattice diffusivities. Such curves can be used to estimate an unknown grain-boundary diffusivity, given certain microstructural assumptions. Finally, approximate analytical equations describing a diffusant uptake in polycrystalline microstructural models are developed and found to agree well with the numerical results.

1.
R. T.
Whipple
,
Philos. Mag.
45
,
1225
(
1954
).
2.
T.
Suzuoka
,
J. Phys. Soc. Jpn.
19
,
839
(
1964
).
3.
Y.
Wang
,
J. M.
Rickman
, and
Y. T.
Chou
,
Acta Mater.
44
,
2505
(
1996
).
4.
P. G.
Shewmon
(
McGraw-Hill
, New York,
1963
).
5.
Y. M.
Mishin
and
C.
Herzig
,
Philos. Mag. A
71
,
641
(
1995
).
6.
J. C.
Fisher
,
J. Appl. Phys.
22
,
74
(
1951
).
7.
Y. C.
Chung
and
B. J.
Wuensch
,
J. Appl. Phys.
79
,
8323
(
1996
).
8.
J. W.
Evans
,
J. Appl. Phys.
82
,
628
(
1997
).
9.
H. S.
Levine
and
C. J.
MacCallum
,
J. Appl. Phys.
31
,
595
(
1960
).
10.
B. S.
Bokshtein
,
I. A.
Mogidson
, and
I. L.
Svetlov
,
Phys. Met. Metallogr.
6
,
81
(
1958
).
11.
G. H.
Gilmer
and
H. H.
Farrell
,
J. Appl. Phys.
47
,
3792
(
1976
).
12.
G. H.
Gilmer
and
H. H.
Farrell
,
J. Appl. Phys.
47
,
4373
(
1976
).
13.
Y.
Mishin
and
C.
Herzig
,
Mater. Sci. Eng., A
260
,
55
(
1999
).
14.
I.
Kaur
,
Y.
Mishin
, and
W.
Gust
,
Fundamentals of Grain and Interphase Boundary Diffusion
, 3rd ed. (
Wiley
, New York,
1995
).
15.
W. H.
Press
,
B. P.
Flannery
,
S. A.
Teukolsky
, and
W. T.
Vetterling
,
Numerical Recipes in C: The Art of Scientific Computing
(
Cambridge University Press
, Cambridge, England).
16.
John
Noye
,
Numerical Solutions of Partial Differential Equations
,
Proceedings of the 1981 Conference on the Numerical Solutions of Partial Differential Equations
,
Queens College
, Melbourne University, Australia (
North-Holland
, Amsterdam,
1982
).
17.
C. G.
Ellis
, Med. Biophys. 303/501 class notes.
18.

It was convenient to use both the IMSL numerical libraries (e.g., DGMRES) and MATHEMATICA to solve the systems of equations arising here.

19.
D. J.
Srolovitz
,
Computer Simulation of Microstructure Evolution
.
20.
K. W.
Mahin
,
K.
Hanson
, and
J. W.
Morris
, Jr.
,
Acta Metall.
28
,
443
(
1980
).
21.
J. M.
Rickman
,
W. S.
Tong
, and
K.
Barmak
,
Acta Mater.
45
,
1153
(
1997
).
22.
A.
Okabe
,
B.
Boots
, and
K.
Sugihara
,
Spatial Tessellations: Concepts and Applications of Voronoi Diagrams
(
Wiley
, New York,
1992
).
23.
W. S.
Tong
,
J. M.
Rickman
, and
K.
Barmak
,
J. Mater. Res.
12
,
1501
(
1997
).
24.
E. W.
Hart
,
Acta Metall.
5
,
597
(
1957
).
25.
L. G.
Harrison
,
Trans. Faraday Soc.
57
,
1191
(
1961
).
26.
M. M.
Mezedur
,
M.
Kaviany
, and
W.
Moore
,
AIChE J.
48
,
15
(
2002
).
27.
J.
Hrabe
,
S.
Hrabetova
, and
K.
Segeth
,
Biophys. J.
87
,
1606
(
2004
).
28.
H. S.
Carslaw
and
J. C.
Jaeger
,
Conduction of heat in solids
, 2nd ed. (
Oxford University Press
, Oxford,
1959
).
29.
J.
Crank
,
The Mathematics of Diffusion
, 2nd ed. (
Oxford University Press
, Oxford,
1976
).
You do not currently have access to this content.