SCIENTIFIC RESEARCH AND DEVELOPMENT
NUMERICAL METHOD FOR SIMULATING SINTERING
A. V. Galakhov
Translated from Novye Ogneupory, No. 5 pp. 30 – 37, May 2009.
Original article submitted February 20, 2009.
A numerical method is proposed in order to simulate sintering, based on fundamental equations of diffusion
theory (Fick equation). The method makes it possible to consider actual geometry of particles from which a
powder compact is composed, and it may be used for an assembly of particles of another shape, dimensions
and reciprocal position. A boundary element method is used for numerical realization. Results are presented
for simulating sintering of Al
particles of different shape and sizes. Dependences are presented for the ef
fect of different material characteristics, in particular dihedral angle (it specifies the relationship of free sur
face energy and intergranular boundary energy), on sintering kinetics and the value of intergranular boundary
Keywords: sintering, simulation, boundary element method, dihedral angle.
Sintering is the main operation in powder technology.
This explains the attention of scientists and practical engi-
neers towards sintering. As a rule, use of experimental meth-
ods for studying sintering is connected with significant ex-
penditure of material and time resources, and therefore the
predominance of theoretical studies in this field is not sur
prising. The overwhelming number are devoted to develop
ing analytical dependences and numerical methods for pre
dicting changes in powder contact characteristics and its
structure during a thermal process. Currently there are two
main cardinally different approaches to describing sintering:
classical physical sintering theory  and phenomenological
. The first rests on physical constants at the level of indi
vidual particles, and the second is based on continuum equa
tions for solid (viscous medium) mechanics, and it requires
for its implementation presence of empirical coefficients and
it is used for describing macroscopic problems of powder
At the basis of physical sintering theory is the idea of dif
fusion mass transfer in an assembly of particles in contact.
This mass transfer is due to a tendency to reduce the surface
energy of powder packing and it is accompanied by changes
in particle shape and their reciprocal position. As basic ana
lytical dependences it is normal to use equations describing
the shape change of two particles in contact. The form of
these equations is proposed by the founders of the theory Ya.
I. Frenkel’  and G. C. Kuczinski ; in recent works these
equations have interpreted changes connected mainly with
the geometry of diffusion mass transfer and its mechanism.
Currently, in order to describe the main mass transfer mecha
nisms, taking part in sintering, there are several definitive
dependences for the rate of movement of internal points and
points of particle boundaries (in this context free and inter
granular boundaries are similar). The general form of these
for mass transfer within the body of a particle it is volu
DP , (1)
is displacement velocity vector for a point within a
particle m/sec, within which there is a pressure gradient DP,
is a coefficient containing values connected with
the geometry of mass transfer, m
is a volumetric dif
fusion coefficient (as it is in the form of a coefficient of
self-diffusion of vacancies within the body of a particle),
for mass transfer along free and intergranular boundaries
it is surface and grain-boundary diffusion
Refractories and Industrial Ceramics Vol. 50, No. 3, 2009
1083-4877/09/5003-0191 © 2009 Springer Science+Business Media, Inc.
A. A. Baikov Institute of Metallurgy and Materials Science, Rus
sian Academy of Sciences, Moscow, Russia.