Mathematical modelling of the sintering process of iron-based metal-glass materials
Based on the study of the mechanisms of diffuse coalescence and coagulation, we review mathematical methods of description and construction of models for sintering process of the metal–ceramic materials. These models are represented by a set of nonlinear differential equations including bulk, grain-...
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Інститут металофізики ім. Г.В. Курдюмова НАН України
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| Цитувати: | Mathematical modelling of the sintering process of iron-based metal-glass materials / T.G. Jabbarov, O.A. Dyshin, M.B. Babanli, I.I. Abbasov // Progress in Physics of Metals. — 2019. — Vol. 20, No 4. — P. 584-619. — Bibliog.: 122 titles. — eng. |
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Jabbarov, T.G. Dyshin, O.A. Babanli, M.B. Abbasov, I.I. 2020-04-17T09:11:33Z 2020-04-17T09:11:33Z 2019 Mathematical modelling of the sintering process of iron-based metal-glass materials / T.G. Jabbarov, O.A. Dyshin, M.B. Babanli, I.I. Abbasov // Progress in Physics of Metals. — 2019. — Vol. 20, No 4. — P. 584-619. — Bibliog.: 122 titles. — eng. 1608-1021 DOI: https://doi.org/10.15407/ufm.20.04.584 https://nasplib.isofts.kiev.ua/handle/123456789/167938 Based on the study of the mechanisms of diffuse coalescence and coagulation, we review mathematical methods of description and construction of models for sintering process of the metal–ceramic materials. These models are represented by a set of nonlinear differential equations including bulk, grain-boundary, and surface diffusion coefficients, and correspond to a sequence of the temperature stage levels increasing with a certain rate and having different durations. By adjusting the levels, rates and durations of temperature regimes, technical parameters. На основі дослідження механізмів дифузійної коалесценції та коаґуляції оглядаються математичні методи опису та побудови моделів процесу спікання металокерамічних матеріалів. Ці моделі представлено системою нелінійних диференційних рівнянь, що включають коефіцієнти об’ємної, зерномежової та поверхневої дифузії, а також відповідають послідовності стадійних рівнів температури, зростаючих із певною швидкістю та що мають різні тривалості. На основе исследования механизмов диффузионной коалесценции и коагуляции обозреваются математические методы описания и построения моделей процесса спекания металлокерамических материалов. Эти модели представлены системой нелинейных дифференциальных уравнений, включающих коэффициенты объёмной, зернограничной и поверхностной диффузии, а также соответствуют последовательности стадийных уровней температуры, возрастающих с определённой скоростью и имеющих разные продолжительности. en Інститут металофізики ім. Г.В. Курдюмова НАН України Успехи физики металлов Mathematical modelling of the sintering process of iron-based metal-glass materials Математичне моделювання процесів спікання металоскляних матеріалів на основі заліза Математическое моделирование процесса спекания металлостеклянных материалов на основе железа Article published earlier |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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| title |
Mathematical modelling of the sintering process of iron-based metal-glass materials |
| spellingShingle |
Mathematical modelling of the sintering process of iron-based metal-glass materials Jabbarov, T.G. Dyshin, O.A. Babanli, M.B. Abbasov, I.I. |
| title_short |
Mathematical modelling of the sintering process of iron-based metal-glass materials |
| title_full |
Mathematical modelling of the sintering process of iron-based metal-glass materials |
| title_fullStr |
Mathematical modelling of the sintering process of iron-based metal-glass materials |
| title_full_unstemmed |
Mathematical modelling of the sintering process of iron-based metal-glass materials |
| title_sort |
mathematical modelling of the sintering process of iron-based metal-glass materials |
| author |
Jabbarov, T.G. Dyshin, O.A. Babanli, M.B. Abbasov, I.I. |
| author_facet |
Jabbarov, T.G. Dyshin, O.A. Babanli, M.B. Abbasov, I.I. |
| publishDate |
2019 |
| language |
English |
| container_title |
Успехи физики металлов |
| publisher |
Інститут металофізики ім. Г.В. Курдюмова НАН України |
| format |
Article |
| title_alt |
Математичне моделювання процесів спікання металоскляних матеріалів на основі заліза Математическое моделирование процесса спекания металлостеклянных материалов на основе железа |
| description |
Based on the study of the mechanisms of diffuse coalescence and coagulation, we review mathematical methods of description and construction of models for sintering process of the metal–ceramic materials. These models are represented by a set of nonlinear differential equations including bulk, grain-boundary, and surface diffusion coefficients, and correspond to a sequence of the temperature stage levels increasing with a certain rate and having different durations. By adjusting the levels, rates and durations of temperature regimes, technical parameters.
На основі дослідження механізмів дифузійної коалесценції та коаґуляції оглядаються математичні методи опису та побудови моделів процесу спікання металокерамічних матеріалів. Ці моделі представлено системою нелінійних диференційних рівнянь, що включають коефіцієнти об’ємної, зерномежової та поверхневої дифузії, а також відповідають послідовності стадійних рівнів температури, зростаючих із певною швидкістю та що мають різні тривалості.
На основе исследования механизмов диффузионной коалесценции и коагуляции обозреваются математические методы описания и построения моделей процесса спекания металлокерамических материалов. Эти модели представлены системой нелинейных дифференциальных уравнений, включающих коэффициенты объёмной, зернограничной и поверхностной диффузии, а также соответствуют последовательности стадийных уровней температуры, возрастающих с определённой скоростью и имеющих разные продолжительности.
|
| issn |
1608-1021 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/167938 |
| citation_txt |
Mathematical modelling of the sintering process of iron-based metal-glass materials / T.G. Jabbarov, O.A. Dyshin, M.B. Babanli, I.I. Abbasov // Progress in Physics of Metals. — 2019. — Vol. 20, No 4. — P. 584-619. — Bibliog.: 122 titles. — eng. |
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584 ISSN 1608-1021. Prog. Phys. Met., 2019, Vol. 20, No. 4
© T.G. JAbbAroV, o.A. DyShIN, M.b. bAbANlI, I.I. AbbASoV, 2019
https://doi.org/10.15407/ufm.20.04.584
T.G. JabbaroV, o.a. DyShin, m.b. babanli, and i.i. abbaSoV
Department of Mechanical and Materials science engineering,
azerbaijan state oil and industry University,
16/21 azadliq ave., az-1010 Baku, azerbaijan
MatheMatical Modelling
of the sintering Process
of iron-Based Metal-glass Materials
based on the study of the mechanisms of diffuse coalescence and coagulation, we
review mathematical methods of description and construction of models for sinter-
ing process of the metal–ceramic materials. These models are represented by a set
of nonlinear differential equations including bulk, grain-boundary, and surface dif-
fusion coefficients, and correspond to a sequence of the temperature stage levels
increasing with a certain rate and having different durations. by adjusting the lev-
els, rates and durations of temperature regimes, technical parameters of the charge,
it is possible to control the sintering process online. The description of the kinetics
of liquid-phase sintering under pressure is performed based on the rheological the-
ory of sintering using the diffusion–viscous flow mechanism. According to this
mechanism, there are a tangential slippage along the grain boundaries and a de-
crease of the volume of pores due to the ejection of vacancies to the surface. After
the formation of the liquid phase during sintering of the powder solid, (generally)
firstly, there is a growth of grains, and then, a compaction of the obtained alloy.
The process of sintering of the iron, cast iron, and sitall (glassceramic) powders is
considered as the mutual diffusion of two (quasi)binary alloys: cast iron (iron +
carbon) and fayalite (iron + sitall). The calculation of the interdiffusion coefficient
of the resulting alloy is carried out according to the Darken formula. A number of
features characterize sintering of multicomponent systems. The sintering of dis-
similar materials (with different melting points) is a complex eutectic process, in
which, along with self-diffusion, causing the mass transfer to the region of particle
contact, there is an interdiffusion, which provides homogenization of the composi-
tion via equalization of the concentrations of dissimilar atoms within the sample.
Under conditions of limited solubility or complete insolubility of the components,
ISSN 1608-1021. Usp. Fiz. Met., 2019, Vol. 20, No. 4 585
1. introduction
Mathematical Modelling of the Sintering of Fe-Based Metal-Glass Materials
sintering of the system is complicated by isolating homogeneous particles from mu-
tual contact, hindering the flow of self-diffusion, and thereby, worsening the sinter-
ing conditions. For the numerical solution of the problem, a fourth-order runge–
Kutta method with a variable integration step is used. A software package for
solving the problem is developed, the calculation results are given on the example
of an alloy of a powder mixture of iron, cast iron, and sitallized glass.
Keywords: metal-ceramic and metal-glass materials, solid- and liquid-phase sinter-
ing, self- and interdiffusion, sintering rheology.
The last decades are characterized by progress in the field of creating
new types of powder materials with enhanced tribological properties
that cannot be obtained by casting. Sintering can be called as one of the
main technological operations of powder metallurgy. The complexity
and versatility of the physical and physicochemical processes occurring
during sintering make this problem one of the actual, albeit difficult, in
the materials science.
The searches for ways to improve wear resistance led to the creation
of iron-based materials with filling the pores by a sitallized glass. Solid
inclusions of glass play the role of solid load-bearing inclusions held by
a metal plastic matrix. The resulting metal-glass (MG) materials differ
from their analogues with increased wear-resistance and low friction
coefficient.
high brittleness of glass does not adversely affect the performance
characteristics of products obtained from MG materials since small in-
clusions of glass melted and fill the pores during sintering are less sus-
ceptible to brittle fracture than large ones [1–7].
Antifriction properties of iron graphite with a pearlite-ferritic
structure containing 6 vol.% of glass are studied. It is shown that the
material containing glass has a lower coefficient of friction, less wear
and a longer service life [8–11]. however, with a high demand for MG
materials, they remain poorly understood.
To improve the interfacial interaction and the intensification of dif-
fusion phenomena, and, consequently, increase the mechanical proper-
ties and wear resistance of iron-glass materials, they have to be alloyed
with components more wetted by glass [12]. This effect can be caused,
e.g., by the use of cast iron powder in the composition of the charge,
since cast iron contains a relatively larger amount of silicon and man-
ganese, which have the property of formation of difficult-to-recover
oxides when heated. It should be expected that these oxides in the sin-
tering process will be well moistened with glass and thus contribute to
its strong connection with the metal base. As a result of the studies in
refs. [13, 14], the best composition of the ‘iron–cast iron–glass’ powder
586 ISSN 1608-1021. Prog. Phys. Met., 2019, Vol. 20, No. 4
T.G. Jabbarov, O.A. Dyshin, M.B. Babanli, and I.I. Abbasov
mixture is established to reach highest hardness and strength: 44% of
iron, 50% of cast iron, and 6% of glass. We can consider the 1200 °С
as an optimum sintering temperature for ‘iron–cast iron’-type materials.
There are there stages of the sintering [15–17]: (i) temperature in-
creasing — heating (non-isothermal, solid-phase sintering); (ii) expo-
sure at a constant temperature (isothermal, liquid-phase sintering); (iii)
temperature decreasing — cooling.
Solid-phase sintering of the powder solid is carried out by the diffu-
sion mechanism and occurs without the formation of a liquid phase. At
that, the following main processes occur: bulk and surface diffusion of
atoms, compaction, recrystallization of the metal powder solid (growth
of some grains due to others of the same phase), and the transfer of
atoms of matter through the gas phase due to the bulk and surface dif-
fusion, viscous flow caused by external loads during sintering under the
pressure. This type of sintering is accompanied by the impact and devel-
opment of bonds between particles (diffusion along grain boundaries
and phases), the formation and growth of contacts (necks), the ‘healing’
of pores (closing through porosity) with their enlargement by spheroidiza-
tion, compaction of the workpiece and its compaction. The latter occurs
during heating and is mainly due to the bulk deformation of particles,
carried out with bulk self-diffusion of atoms and adsorption of atoms
(adatoms) on the surface (surface diffusion).
The fundamental works [18–22] are devoted to the description of
the process of solid-phase sintering of metal solids, including powder
ones. The theory of solid-phase sintering was further developed in the
works [23–39] and others.
While the processes of the sintering of metal powders occurring in
the solid phase are studied in detail (currently, there are theories ex-
plaining these processes), the sintering in the presence of the liquid
phase is poorly studied. The presence of a liquid phase in the sintered
system leads to a number of additional phenomena as compared with
sintering in the solid phase, e.g., processes of dissolution of the solid
phase in the liquid and then solidification of the solid components, vis-
cous flow of the liquid, etc. [40].
Some regularities of sintering in the presence of the liquid phase
were firstly formulated in refs. [41, 42], where the sintering process in
the W–Ni–Cu system was investigated. Authors [41, 42] showed that
the compaction during sintering is carried out in three stages. The first
stage consists in the liquid flow towards the rearrangement of grains of
the solid phase in the direction of their denser packing. The second in-
volves the dissolution of small particles and the supercooling of the
material on larger grains. The third phase may also include an addi-
tional compaction due to the coalescence of the grains between them-
selves, which has to obey the laws of sintering in the solid phase. Devel-
ISSN 1608-1021. Usp. Fiz. Met., 2019, Vol. 20, No. 4 587
Mathematical Modelling of the Sintering of Fe-Based Metal-Glass Materials
oping these ideas in ref. [43], author presented theoretical calculation
to the formula describing the processes of compaction during sintering
for the first two stages. Authors of refs. [44, 45] considered sintering
with the liquid phase from the point of view of surface phenomena and
gave an analysis of their quantitative laws.
Further development of the theory of sintering with a liquid phase
was obtained in the works [46, 47], where authors stressed on the main
differences occurring during sintering in liquid and solid phases. Along
with the manifestation of capillary forces in the contact of particles
(wetting and spreading of the liquid in solid particles, capillary pres-
sure, penetration of liquid into the capillary slots between solid parti-
cles), the surface area of interaction between the components increases
substantially. This dramatically accelerates the process of alloy forma-
tion and leads to the immeasurable manifestation of the associated ther-
mal and bulk effects.
The diffusion-caused processes in alloys based on different types of
crystal lattice (e.g., f.c.c. [48, 49], b.c.c. [50] or h.c.p. [51, 52]), espe-
cially nanocrystalline (NC) and nanostructured (NS) alloys, are inten-
sively studied and developed. This is because diffusion processes play an
important role in the tuning properties of the materials, including pro-
cesses of degradation and return of the form and structure of nanoma-
terials. In a number of experimental studies it was found that at a dif-
fusion along the grain boundaries (Gb) in NC and NS metals and alloys,
the grain-boundary diffusion coefficients are several orders of magni-
tude greater than those in coarse-grained materials [29–30, 37].
The goal of this review work is the construction of mathematical
models for porosity and graininess during sintering of a multicompo-
nent mixture of powders of metal and ceramic materials. These models
are represented by a closed system of differential equations correspond-
ing to a sequence of temperature–time sintering regimes increasing
with a certain rate. The scientific basis of the models are the mecha-
nisms of diffusion coalescence kinetics and coagulation, occurring in a
dispersed medium [53–57] and the rheological basis of pressure sinter-
ing kinetics using diffusion viscous flow mechanisms and heterodiffu-
sion (interdiffusion) in alloys [23–24, 28, 46–47, 58–61]. Continuous
‘fitting’ of solutions at the joints of coupled temperature regimes is car-
ried out using a specially developed set of computer programs for con-
struction (runge–Kutta-method-based) numerical solutions of a system
of differential equations (at all temperature stages of sintering) with
variable integration step. The results of calculations are presented on an
example of solid-phase sintering of the pressed briquettes of a mixture
of metal powders (iron and cast iron) containing sitallized, high-silicon-
earthy glass powder of low concentration. The sintering process of this
mixture is considered as mutual diffusion of two binary alloys: iron +
588 ISSN 1608-1021. Prog. Phys. Met., 2019, Vol. 20, No. 4
T.G. Jabbarov, O.A. Dyshin, M.B. Babanli, and I.I. Abbasov
+ carbon (grey cast iron) and iron + sitall (metal-glass material Fe2Sio4,
known as fayalite [62]). The coefficient of mutual diffusion in binary
systems is calculated taking into account the Darken relation [63, 64].
The mathematical models available in the literature and methods for
describing the kinetics of sintering processes of powder materials main-
ly scoped on one or two component systems [1, 3, 27, 65–70, 71, 72].
Sintering processes and their computer simulation for multicomponent
systems consisting of several metal and ceramic powder and glass mate-
rials are studied not enough due to difficulties in the calculations of the
interdiffusion coefficients [73, 74–78].
2. Theoretical Technique
2.1. Solid-Phase Sintering Kinetics of Powder Materials
During the period of rapid temperature rise, used before the isothermal
treatment of active powders in the sintering process, a considerable
compaction is commonly achieved with rates much higher than the rate
after transition to isothermal (liquid phase) sintering.
For the mathematical description of compaction, the choice of the
dependent variable becomes crucial. As such a dependent variable, au-
thor of ref. [25] suggested to use the relative volume of pores υc/υn,
where υn and υc are the pore volumes and solid porosity before and after
sintering.
As follows from ref. [28], during sintering of solids with different
densities, values of the ratio of porosity of solids пc/пn after and before
sintering (п is a porosity defined as a fraction of the volume of the po-
rous solid) are similar. At a significant change of the density of the
porous solid before sintering, dn [g/cm3], the пc/пn ratio cannot be con-
stant during the sintering process, which follows from the dependencies:
υc /υn = [dn(dk − dc)]/[dc(dk − dn)], (1)
пc /пn = (dk − dc)/(dk − dn), (2)
where dc is sintered solid density, dk is density of a compact matter
(without pores).
From eqs. (1) and (2), we find that
(υc/υn)/(пc/пn) = dn/dc. (3)
From eq. (2), after simple algebraic transformations, we obtain the
following relation between the porosity пc and the volume shrinkage
(compaction) ΔV/V0:
пc = 1 − (1 − п0)/[1 − (ΔV/V0)
2]. (4)
The theoretical relationship between the volume shrinkage ΔV/V0
and the maximum value of interparticle contacts for sintering for sin-
ISSN 1608-1021. Usp. Fiz. Met., 2019, Vol. 20, No. 4 589
Mathematical Modelling of the Sintering of Fe-Based Metal-Glass Materials
tering powders achieved by
sintering is obtained based
on analysis of the geometry
of the near-contact zone.
When sintering by the mech-
anism of bulk or surface self-
diffusion, the formation of
contact is not accompanied
by the convergence of parti-
cles, i.e. volume shrinkage in
this case is absent. If the sin-
tering is carried out due to a viscous flow or diffusion of vacancies in
the contact of the boundary, capable of absorbing vacancies indefinitely,
then the following relation between the relative approach of the parti-
cles is obtained, i.e. linear shrinkage Δl/l0 and bulk shrinkage ΔV/V0 [28]:
ΔV/V0 = 3Δl/l0. (5)
In idealized sintering kinetics models, it is commonly assumed that
any one mass transfer mechanism works during sintering and that mass
transfer occurs at the point of contact between two identical spheres.
however, we have to mean that during sintering of real samples, par-
ticles can have different shapes and sizes, each particle contacts several
particles at once, contact in practice is not perfect (point), and in places
of contact particles are usually flattened during compaction. however,
most importantly, mass transfer in real samples is not controlled by any
single process, but by several.
The initial stage of sintering is characterized by the formation and
growth of the neck between the particles, followed by shrinkage of the
sample, except the cases when mass transfer occurs only due to evapora-
tion/condensation or bulk and surface diffusion. Shrinkage in the ini-
tial stage of sintering reaches only a few percent and occurs as long as
the pores in the sample are almost completely interconnected and pene-
trate the entire volume.
Authors of ref. [79] attempted to determine the parameters of
grain-boundary diffusion by measuring shrinkage at the initial stage of
sintering, based on a simplified model of sintering of two spheres (Fig. 1)
with modified equations to describe the geometry of the resulting neck.
In the works [19, 80, 81], the following relations were assumed to be
satisfied:
Δr = ρ0 = x2/4rg, (6)
where Δr is approach of the centres of particles, ρ0 is the radius of cur-
vature of the surface of the neck, rg is the radius of the particle.
In the paper [79], there were proposed modified equations for the
shape of the neck, the distance between the centres of approaching par-
Fig. 1. Scheme of the idealized sin tering model
of two spheres [79]
590 ISSN 1608-1021. Prog. Phys. Met., 2019, Vol. 20, No. 4
T.G. Jabbarov, O.A. Dyshin, M.B. Babanli, and I.I. Abbasov
ticles Δr and the neck radius x:
x = (5π/9)rg (Δr/rg)
0.46, ρ0 = (7π/8)rg (Δr/rg)
0.5. (7)
Using the geometric relations (7) and assuming that
Δl/l0 = Δr/rg, (8)
they obtained equation for shrinkage kinetics [79]:
Δl/l0 = (50γΩbDb/7πkBTrg
4)0.31t0.31, (9)
where γ is surface tension associated with surface pressure σ (also called
capillary pressure) in the case of a spherical particle of radius rg by ratio
σ = 2γ/rg, b is the width of the boundary between the particles, kB is the
boltzmann constant, T is the absolute temperature, Ω is the atomic volume.
From eq. (9), it follows that the diffusion permeability of grain
boundaries (bDb) can be determined from the slope of the shrinkage
curve (i.e., the dependences Δl/l0 on t0.31):
bDb = [−∂(Δl/l0)/∂t0.31]1/0.31
(7πkBTrg
4/50γΩ). (10)
It should be noted that eq. (10) is valid only for x < 0.3rg or close
to this inequality, since it is assumed that the equation for the driving
force in [79] assumed that x ≤ rg.
There was shown [35] that the geometry of the neck, described by
eq. (7), is still far from the exact formula obtained by numerical meth-
ods. It was assumed that the mutual penetration of the spheres that the
volume of the lenticular area of overlapping spheres is equal to the
newly formed volume around the point of contact. however, this as-
sumption is exactly fulfilled if there are no transfer mechanisms that
do not shrink (e.g., surface diffusion and evaporation or condensation).
The paper [82] deals with analysis of the mass transfer with an ac-
count of grain-boundary and bulk diffusions, where authors suggested
improved equations of the geometry of the neck obtained dynamically
(without using the graphical method):
x = 1.92rg (Δr/rg)
0.49, ρ0 = 2.32rg (Δr/rg)
1.14. (11)
The final shrinkage equation, based on relations (11), has the
form:
[ ]2.1
0 0 3 4
0
2
( / ) ( / )/
2
V b
B g B g
D bDl
l l d l l dt
lk Tr k Tr
γΩ γΩ∆
∆ ∆ = + . (12)
From the last formula, it follows that the curve of shrinkage kinet-
ics in the coordinates (Δl/l0)
2.1[d (Δl/l0)/dt] and Δl/l0 should be a straight
line, the slope of which gives the coefficient diffusion DV, and the seg-
ment cut off from the ordinate axis bDb is the product for grain bound-
ary diffusion.
The most complete description of the sintering model of two spheres,
which takes into account both the energy of grain boundaries and its
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effect on the neck surface, and additional volumetric diffusion from the
neck surface, was given in [33], where exact solutions were used to de-
scribe the neck geometry. For this, along with ρ0, the parameter Ar was
introduced, which is the area on the neck surface, which atoms can
achieve through bulk diffusion.
Ar = 4πρ0[(C − (π − B)/2)(rg + ρ0) cos C − ρ0(sin C − sin((π − B)/2))], (13)
where
C = sin–1 [rg − Δr + ρ0 sin((π − B)/2))/(rg + r0)]. (14)
In the last two expressions, B is the angle of the solution of the
grain-boundary groove in the neck between the particles, which is
formed to achieve local equilibrium between the surface tension of the
grain boundary and the free surface of the neck.
Assuming again the valid relation (8), the author obtained the fol-
lowing equation of shrinkage kinetics [33]:
[X3ρ0 /(rg X + ρ0 cos ((π − B)/2))] d(Δl/l0)/dt =
= (2γΩDV/πkBTrg
3)XAr/rg
2 + 4γΩbDb/kBTrg
4, (15)
where X = x/rg is a specific neck size. It did not take into account the
existence of transfer mechanisms that did not contribute to shrinkage
(such as surface diffusion and transfer through the gas phase).
As follows from (15), measuring the shrinkage rate, the neck size
and the angle of the groove solution located in the neck at the grain
boundary, one can obtain as a result of sintering experiments the values
of the coefficients of volume and grain boundary diffusion by con-
structing the dependence [X3ρ0 [d(Δl/l0)/dt]/[rg X + ρ0 cos ((π − B)/2)] from
Ar/rg
2X. This results in a straight line with a slope λ, giving the co-
efficient of bulk diffusion DV, and a segment K, cut off on the axis
Ar/rg
2 = 0, giving the product bDb for grain boundary diffusion:
DV = (πkBTrg
3/2γΩ) λ, bDb = (kBTrg
4/4γΩ)K. (16)
If there is no mass transfer from the surface, then, eq. (15) becomes
simpler and takes the form
2.06 1.030
0 03 4
( / ) 2.63 0.70
( / ) ( / )V b
B g B g
d l l D bD
l l l l
dt k Tr k Tr
∆ γΩ γΩ
∆ = ∆ + (17)
for the first 3.5% of shrinkage. expression (17) only slightly differs
from eq. (12), obtained by neglecting surface diffusion and bulk diffu-
sion from the surface, as well as using the approximation of expression
(11) for the geometry of the neck.
If the bulk diffusion Db dominates over the grain boundary diffu-
sion Db, eq. (17) has a solution with respect to y = Δl/l0:
y ≅ (5.34γ Ω DV/kBTrg
3)0.49t0.49. (18)
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In the opposite case, when grain-boundary diffusion dominates over
bulk diffusion, equation (17) has the solution:
y ≅ (2.14γ Ω bDb/kBTrg
4)0.33t0.33. (19)
Denoting y = Δl/l0 and substituting (5) in (4), we obtain an equation,
which relates пc /п0 with Δl/l0:
0
2
1
1
(1 3 )c y
− Π
Π = −
−
.
Solving this equation for y, we obtain
1/2
011
1
3 1 c
y
− Π
= − − Π
.
Differentiating with respect to t both sides of the equation for пc
and taking into account the expression for y, we obtain
3/2
1/2
0
6(1 )
(1 )
c cd dy
dt dt
Π − Π
= −
− Π
. (20)
The last equation can now be written as
3/2 2.1
0
2.1 3 4
0
0
16(1 ) 3 2
13 21
1 1
1
c c V b
cB g B g
c
d D bD
dt k Tr k Tr
− ΠΠ − Π ⋅ γΩ γΩ = − +
− Π − Π
− Π −
− Π
. (20′)
For the first 3.5% of shrinkage, we will use the kinetic equation
(17), which can be written, taking into account the above relation пc
with y, as a differential equation:
1.03
3/2 2.06
0
2.06 1.03 3 4
0
0
16(1 ) 3 2.63 7
.
13 101
1 1
1
c c V b
cB g B g
c
d D bD
dt k Tr k Tr
− ΠΠ − − Π ⋅ γΩ γΩ = +
− Π − Π − Π −
− Π
(20′′)
For a topologically continuously transforming structure of a powder
material in the sintering process, a quantitative analysis of the kinetics
of changes of the characteristic linear parameter of the porous micro-
structure L (L is an average grain size of the sample) can be performed
using the general theory of diffusion coalescence of disperse systems
[83]. If the main mechanism of mass transfer is a surface diffusion,
then kinetic equation for L (dependent on time t) is [84]:
L4 = L4
0 + (B1γDS δ4/(kBT)) t, (21)
where DS is the surface diffusion coefficient, ΔL = L − L0 is the thick-
ness of the layer, where surface diffusion takes place, of the order of
atomic diameter δ, γ is the surface tension, B1 is a numerical constant
≈30, L0 is the initial value of L before the sintering.
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The last equation satisfies many experimental data for oxide and
metal powders [32, 35, 84]. For metal island-type films, the coalescence
occurs predominantly via the mechanism of surface diffusion of ada-
toms [22, 85].
Differentiating (21) with respect to t, we obtain the kinetic differ-
ential equation for L:
dL/dt = B1γDS δ4/(4L3kBT). (22)
2.2. Iron Self-Diffusion and Carbon Diffusion in Iron
The subject of our consideration is the diffusion mechanism of sintering
a powder mixture, which is based on iron (44%) and grey cast iron
(50%). Depending on the grade, the composition of grey iron includes
C (2.9–3.7%), Si (1.2–2.6%), Mn (0.5–1.1%), P (more than 0.2–0.3%),
and S (0.12–0.15%). There is permission for alloying grey cast iron
with Cr, Ni, Cu, and some other components.
Generally, the diffusion coefficient D, which is an important quan-
tity that determines the diffusion rate, depends on the nature of the
alloyed and alloying atoms (see, e.g., refs [86–88]) as well as the struc-
tural factors (grain size, density and distribution of lattice defects), but
most strongly depends on temperature. The dependence of the diffusion
coefficient D on temperature obeys the Arrhenius law [89, 90]:
D = D0 exp (−ΔH/(kBT)) (23)
or
D = D0 exp (−Q/(RT)), (23′)
where R is a universal gas constant, D0 is a pre-exponential factor dif-
ferent from the value of D0 in eq. (23), Q is the activation energy. De-
pendence (23) is experimentally confirmed for many systems with high
D for diffusion in interstitial alloys (e.g., carbon in α- and γ-Fe) and
substitution alloys (e.g., Au in Ag).
The exponential temperature dependence of D is explained by the
origin of diffusion due to thermally activated atomic motion, which is
always described by the exponential exp (q/kBT), where q is the activa-
tion energy of the elementary act of atom displacement.
Table 1 [89] contains characteristics of the parameters of self-diffu-
sion of ferrite or α-iron (Fe below 912 °C), and austenite or γ-iron (Fe
above 912 °C); while Table 2 [89] exhibits carbon diffusion coefficient
in them. The activation energy of self-diffusion of metals ΔHD relates
to the melting temperature Tm: ΔHD /Tm ∼− 1.5 · 10–3 eV/K and the ratio
ΔHD /Tm ∼− 15 with the heat of fusion Lh. These empirical rules allow
us to estimate the value of the diffusion coefficient in the case of the
measurement impossibility. however, there are some exceptions, e.g.,
for Ge.
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Atoms in austenite are arranged into an f.c.c. crystal structure.
heating the ferrite up to 912 °C leads to the formation of the smallest
austenite grains at the boundaries of ferritic grains. Further heating
leads to the growth of new austenitic grains with the complete replace-
ment of old ferritic grains with new austenitic ones — transformations
occur in iron: the crystalline structure of iron spontaneously transforms
from b.c.c. into f.c.c. At the cooling below 912 °C, transformations of
the same type occur in Fe but in the opposite direction.
The activation energy Q is expressed in terms of the activation en-
thalpy ΔH [89],
Q = ΔH + RT, (24)
and obeys the Arrhenius law [91],
r = A exp (−Q/RT), (25)
where r is the reaction rate for sum of reactants A1 + A2 to be converted
into product A1A2, A is a pre-exponential factor.
The activation energy Q and pre-exponential factor A can be calcu-
lated from the experimental data [91]. Consider an elementary second-
order reaction in which two different particles A1 + A2 → products with
different concentrations of the substance A1 and A2. We assume that
with an excess of A2, the order of reaction for substance A2 is zero.
Then, the expression for the rate of this reaction has the form −dc/dt =
= rt. Integrating the right side of this equation in the range [0, t] and
the left one in the range [c0, c] allows calculating the chemical reaction
constant r:
r = (1/t) ln (c0/c), (26)
where c0 is the initial concentration of the initial reactant A1 at the mo-
ment of time t = 0, c is the concentration of this reactant at the moment
of time t.
Table. 1. Diffusion coefficient of iron [89]
Self-diffusion in pure metals
Metal Structure D0, cm
2/s ∆HD, eV ∆HD/Tm, eV/K ∆HD/Lh
α-Fe b.c.c. 2 2.5 1.4 ⋅ 10−3 15.5
γ-Fe f.c.c. 0.4 2.8 1.6 ⋅ 10−3 17.4
Table 2. The diffusion coefficient of carbon in ferrite and austenite [89]
Impurity diffusion in metals
Type of solution Metal Impurity D0, cm
2/s ∆H, eV D/D* at 1000 K
Interstitial solid
solution
α-Fe C 0.004 0.83 1.44 ⋅ 106
γ-Fe C 0.67 1.6 3.87 ⋅ 106
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From eq. (25), we obtain
Q = [R (T2T1)/(T2 − T1)] ln (r2/r1), (27)
where
rj = (1/tj) ln (c0/cj) (j = 1, 2). (28)
Two different formulas (23) and (23′) for diffusion coefficient
yield
ΔH/Q = kB/R, (29)
which results to
ΔH = Q/NA, (30)
where the relation kB = R/NA (NA — Avogadro’s number) is used.
Sometimes, there is a deviation from the Arrhenius equation, which
can be interpreted as inconstancy of the (effective) reaction activation
energy at different temperatures.
To determine experimentally the distribution of the A diffusant
concentration in a substitutional solid solution B (in our case of a cast
iron, A — graphite, B — iron), a diffusion pair (bi-pair) is applied. It
consists of two semi-infinite samples (A and B) interconnected by a flat
boundary through which diffusion occurs from one sample (A) with an
initial diffusant concentration c2 to another sample (B) with an initial
diffusant concentration c1 (c1 < c2).
From a practical point of view, to obtain the distribution of the
carbon concentration in iron, the most important is to build the distri-
bution of carbon concentration after the so-called cementation (surface
saturation of the material with carbon) [92]. Commonly, the cementa-
tion is carried out at 910–950 °C, and sometimes to accelerate the pro-
cess — at 1000–1050 °C, i.e. at the temperature of chemical heat treat-
ment (ChT), the diffusion zone at any time moment has an austenitic
structure with a variable concentration of carbon, decreasing with dis-
tance from the surface into the depth of the product.
In the surface layer of the product, the carbon concentration is usu-
ally 0.8–1.2 mas.% and does not reach the limit of solubility at the
carburizing temperature. Therefore, the Fe3C is not formed at this tem-
perature, and the surface layer (as well as in the bulk) is in the auste-
nitic state. The transformation of ferritic grains into austenitic ones
occurs as follows. When ferritic iron is heated to 912 °C, the old com-
position of ferritic grains changes to a new composition of austenitic
grains, i.e. α-iron transforms into β-iron. Firstly, there is a formation of
new, very fine, austenitic grains, which overlap old boundaries of fer-
ritic grains thereby forming the so-called perlite. Then, these grains
grow until all the old ferritic grains disappear and cementite appears.
Transformation of the ferrite into austenite is accompanied with vol-
ume changes. The density of austenite is 2% higher than that of ferrite;
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it means that the austenite atom occupies a smaller volume than the
ferrite atom [93].
on the region of the diffusion zone adjacent to the surface, and hav-
ing a carbon concentration more than 0.8 mas.%, at room temperature,
there is cementite + perlite; while the region with a carbon concentra-
tion less than 0.8 mas.% is occupied by excess ferrite + perlite (Fig. 2,
a), [92]). Moving from the surface, due to the continuous decreasing of
C concentration, secondary cementite gradually disappears, and the
structure becomes pure pearlitic, and then the amount of pearlite grad-
ually decreases and the volume fraction of ferrite increases.
Thus, at a high ChT temperature, the diffusion zone (cemented
layer) is a single-phase; after slow cooling, it becomes a two-phase (fer-
rite + cementite) over the whole section. observing the surface structure
of carburized iron (e.g., via an optical microscope), using the relation for
different structural components, one can approximately construct the
distribution of the carbon concentration cC (Fig. 2, b) [92]).
When the liquid metal, obtained after the melting temperature Tm
(for Fe, Tm = 1540 °C), is cooled to the point of solidification, then, when
Fig. 2. Plotting the carbon concentration distribution after cemen-
tation according to optical microscope data [64]
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a liquid state transforms into a solid one, a phase transformation occurs
between the liquid and the solid phases with a heat release. When cooling
the austenite it transforms into ferrite, it is also a phase (γ → α) transfor-
mation; however, in this case, the solid state transforms into the solid
one (also with a heat release). In case of a heating, the opposite process
occurs: α-phase of Fe transforms into the γ-phase (with heat absorption).
The cementation is an intermediate operation, followed by quench-
ing (rapid cooling). The hardness of the resulting material is propor-
tional to the final carbon content. After the quenching of the cementite,
we can analyse the hardness over the sample cross-section, moving away
from the surface to the sample centre. Using analysed experimental
data, we can plot the functional dependence of hardness on the distance
from the surface: the resultant hardness–distant curve acts as a concen-
tration curve of carbon content in the alloy.
2.3. Application of Sitallized Glass: Main Advantages
Sitallized glass (the so-called sitall) is an artificial polycrystalline mate-
rial, obtained by crystallization of glass of the appropriate chemical
composition, possessing higher physicochemical properties as compared
with this glass. ‘sitall’ is a combination of two words: ‘glass’ and ‘crys-
tal. one can say that sitall is something like hybrid of glass and ceram-
ics. There are distinctions from glass and ceramics: (granular) crystal-
line structure and much smaller crystals with monolithic structure,
respectively. An important stage of the transformation of glass into
sitall is the formation of crystallization centres, which can occur spon-
taneously (homogeneous mechanism) or due to the introduction of for-
eign outer particles (heterogeneous nucleation) [94].
In a process of sintering of iron with sitall, a metal-glass material
Fe2Sio4 is formed called as fayalite [95]. Moving over the metal, the
molten sitall accumulates mainly in the grain boundaries, is saturated
with metal absorbing its oxides, and crystallizes.
Metal-glass materials should be distinguished from amorphous ma-
terials, in which, unlike crystalline, there is no a long-range order in
the arrangement of atoms. Such materials can be obtained by solidifica-
tion of a drop of metal melts under conditions of intensive heat remov-
al by the ‘shooting method’ and are called as amorphous metal alloys or
metal glasses [96].
The nature of the properties of glass ceramics has much in common
with the nature of the same properties of the original glass and similar
ceramic materials, and at the same time significantly exceeds these
indicators. Sitall is several times stronger than glass, most ceramic
materials and even some metals, its bending strength can reach 2500–
3000 kg/cm2 and for some experimental samples — 4200–5600 kg/cm2,
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i.e. exceeds the strength of quartz glass, stainless steel, and titanium.
Sitall has a dense microcrystalline structure similar to the structure of
ceramic materials based on pure oxides, characterized by rather small sizes
of randomly oriented crystals and the absence of porosity. For phosphate-
containing sitall, crystal sizes range from 0.5 to 6 µm; while 20–50 µm for
conventional ceramics. The density of sitall is 2.5–2.72 g/cm3 providing inden-
tation hardness close to the hardness of hardened steels and greater than
the hardness of fused quartz, brass, cast iron, granite, and glass.
Iron-based metal-glass materials, as a rule, have a heterogeneous
structure, contributing to the improvement of mechanical properties
and wear resistance. The role of glass is that it wets the metal base,
strengthening the metal frame by dissolving metal oxides in the glass.
To intensify the processes of interfacial interaction, and as a result,
increase the mechanical properties and wear resistance of iron-glass
materials obtained by cold pressing and subsequent sintering, they have
to be doped with well-wetted glass components. In this regard, it seems
appropriate to use the iron powder as an alloying component of iron-
glass materials.
For this purpose, we used grey cast iron, which (depending on the
grade) includes C, Si, Mn, P, S. The presence of silicon and manganese
contributes to the formation in the process of heating of the difficultly
reducible oxides. It should be assumed that in the process of sintering,
these oxides will be well wetted with glass and, to some extent dissolv-
ing in the glass, will promote adhesion of the metal frame to the glass.
The results of work [96, 97] show that in the microstructure of
metal-glass composite materials, in particular, based on Fe, after sinter-
ing, in addition to the metal base, pores and inclusions of glass, a new
phase such as fayalite and hematite may appear as a result of metal-
glass interaction.
The study of the structure and properties of composite materials
based on mixtures of iron and cast iron powder showed [99, 100] that
an increase in the content of cast iron powder in the mixture leads to a
slight increase in the porosity of the samples at the same pressing pres-
sure. With decreasing particle size of cast iron powder, the density of
compressed pellets increases and the value of elastic aftereffect decreases.
2.4. Pre-Exponential Factor of Diffusion Coefficient
in the Fe-Alloy and Sitallized Glass
The observed values of the pre-exponential factor of diffusion coeffi-
cient D0 for amorphous alloys have a much greater variation (10−15–
1015 m2/s) than the corresponding values for crystalline alloys (10−6–
102 m2/s) and, therefore, the values of D0 for amorphous alloys and
crystalline systems with the same ∆H can differ by several orders of
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magnitude [101]. For the experimental values of D0 and the enthalpy of
diffusion activation ∆H of amorphous alloys, the following relation was
established [102]:
D0 = A eΔH/B, (31)
where A and B are constants. This relationship is universal in the sense
that it satisfies not only amorphous alloys [103, 104], but also the D0
value for self-diffusion and impurity diffusion of crystalline alloys, in-
cluding displacements of atoms into the substitutional or interstitial
lattice sites [105, 106].
The comparison of the values of the parameter A (10−7 m2/s) for
crystalline alloys with the corresponding values (10−19–1020 m2/s) for
amorphous alloys and the values of B (0.41 eV) for crystalline alloys and
(0.055 eV) for amorphous alloys shows that these parameters are abso-
lutely different [107].
For the diffusion of vacancies or cracks in crystals, the constants A
and B are calculated using the zener theory for the pre-exponential fac-
tor [108]. According to this theory, the pre-exponential factor D'0 (iden-
tical to D0 for crystals) reads as
D'0 = ga2 fν0e
ΔS/kB, (32)
where g — geometric factor, a — effective distance between atomic
hops (jumps), ν0 — effective frequency of jumps (hops), f — correlation
factor, and ∆S — entropy of diffusion.
For crystals,
ΔS = −βµ ΔH, (33)
where βµ represents the temperature dependence of the shear modulus
[104, 105]. As follows from eqs. (32) and (33), A in eq. (31) is
A = ga2 fν0, B = −kB/βν, (34)
which is well confirmed by experimental data for the diffusion in crys-
tals [104]. Using eqs. (32)–(34) for crystals, we obtain
D0 = ga2 fν0 e
−ΔHβµ/kB. (35)
For metal systems, a = 3 · 10–10 m and ν0 ≈ 1012 1/s, we can assume
fg ≈ 1, so the parameter A is approximately equal to 10−7 m2/s with a
deviation for different metals no more than one order [104].
For crystals, the correlation dependence between D0 and Q (as noted
above, Q ≈ ΔH) was investigated previously in ref. [109]. Using the
thermodynamic approach there were obtained the expression
ΔS ≈ βµQ, (36)
where
βν = −∂ (ln µ)/∂T (37)
is temperature dependence of the shear modulus. As a result, the follow-
ing relationship was obtained for diffusion of crystals (so-called zener
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ratio):
lnD0 = lnA + βµQ/kB; (38)
particularly, βµ ≈ 2.2 · 10–4 1/K for Cu, Ni, Pd, while βµ = 3.5 · 10–4 1/K
for Ag [110].
Thus, the pre-exponential factor D0 for metal alloys (in particular,
for fayalite Fe2Sio4) can be determined by the formula
D0 = A e−ΔHβµ/kB, (39)
where A = 10–7 m2/s for all metal alloys, and βµ = 2.85 · 10–4 1/K
is the
characteristic (average) value of the parameter βµ for metals.
Thus, the diffusion coefficient of Fe2Sio4 can be written as the Ar-
rhenius formula (23) with a pre-exponential factor D0 defined by for-
mula (39), i.e.
D = A−ΔHβµ/kB e−ΔHβµ/(kBT) (40)
with A = 10–7 m2/s and βµ = 2.85 · 10–4 1/K. The value of activation en-
thalpy ∆H is calculated via the formula ∆H = Q/NA, and the activation
energy Q (or E as denoted in ref. [34]) is determined by formula (30).
In the works [102, 103, 111, 112], an attempt was made to establish
a correlation dependence of lnD0 on Q for self-diffusion and impurity
diffusion of amorphous alloys using formula (31) and the zener’s ratio
with parameter B = kB/βµ. however, the value of βµ = 1.6 · 10–3 1/K, thus
obtained deviated from the known average values of elasticity by a
value of approximately one order of magnitude [107]. This is explained
by the fact that, under the conditions of boundary diffusion between
clusters of a heterogeneous amorphous alloy, the values of the parame-
ters A and βµ strongly depend on the properties of the boundary region,
which are unknown in most cases. Consequently, the correlation depen-
dence of lnD0 on Q cannot be obtained using a relation similar to eq.
(36) in the case of amorphous alloys.
2.5. Calculation of Diffusion Coefficients
from Experimental Data on Annealing
A systematic quantitative analysis of grain-boundary diffusion became
possible only after the appearance of the article [113]. Almost all earlier
works on grain boundary diffusion were qualitative, since they did not
provide numerical data that serve as rigorous evidence of the true coef-
ficients of grain boundary and bulk diffusion. however, even in that
time it was clear that solutions of equations for bulk diffusion, corre-
sponding to diffusion in an inhomogeneous medium, could not be di-
rectly used to determine the grain-boundary diffusion coefficients. This
is because the process is heterogeneous and is complicated by the fact
that diffusion along the grain boundaries is always accompanied by re-
moval of diffusant from the boundaries to the bulk of grains, i.e. bulk
diffusion appears with a coefficient of DV.
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As was suggested [113], the grain boundary is a thin homogeneous
layer with a thickness δ of several interatomic distances in a semi-infi-
nite crystal, and the grain-boundary coefficient diffusion Db over this
layer is much larger than the coefficient of volume diffusion DV beyond
its limits. This problem was solved [113] taking into account the re-
moval of the diffusant from the grain boundary to the crystal bulk, but
neglecting direct bulk diffusion from the diffusant source on the sur-
face. As shown [113], if the grain-boundary diffusion dominates, the
logarithm of the average concentration c– in such a layer (parallel to the
source of the diffusant) varies linearly with the depth of penetration z,
in contrast to the parabolic dependence for bulk diffusion. however,
this conclusion [113] was based on a set of approximations and was not
exact. The exact solution of the same problem, i.e. for the model of a
uniform layer at the grain boundary, was obtained in refs. [114, 115]
for diffusion from an ‘instantaneous’ source (an infinitely thin layer on
the surface). exact solution [114] for the model in ref. [113] was too
complicated for experimenters for using in practice.
Following above-mentioned theoretical works, another article [116]
appeared where authors used completely different approach to grain-
boundary diffusion based on the consideration of the entire polycrystal-
line solid. They showed that, in the region of grain-boundary diffusion,
lg c– varies linearly not with z, but with z6/5. Just after the work [116],
experimenters had the opportunity for using exact solutions of the prob-
lem of grain boundary diffusion in order to process the obtained data.
The model of grain-boundary diffusion proposed in ref. [116] al-
though is far from reality, is quite suitable for describing experiments
under certain conditions [37]. The grain boundary is a semi-infinite iso-
tropic layer of the same thickness δ with a high diffusion coefficient
(restricted in a semi-infinite perfect crystal with a low diffusion coef-
ficient) with the grain boundary perpendicular to the surface on which
there is a diffusant.
let the y-axis be perpendicular to the grain boundary layer, and the
z-axis perpendicular to the free surface. At the diffusion annealing of
sample with a temperature T and annealing time τ, material diffuse
along the z-axis into the bulk of the grain as well as along the grain
boundary. The diffusion coefficient at the boundary is much larger than
elsewhere, and therefore the bulk diffusion of matter begins from the
walls of the grain boundary into the surrounding crystals. let DV be the
bulk diffusion coefficient describing diffusion in the grains, and Db be
the diffusion coefficient in the grain boundary layer. If Db >> DV, the
direct diffusion contribution from the surface decreases with distance
much faster than the grain boundary contribution. Therefore, in the
region far from the surface, diffusant enters to the bulk practically
from the grain boundaries only. The derivation of the diffusion equa-
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T.G. Jabbarov, O.A. Dyshin, M.B. Babanli, and I.I. Abbasov
tion is based on the following assumptions: (1) the diffusion laws are
satisfied both in the crystal and in the grains; (2) the diffusion coeffi-
cients DV and Db are isotropic and do not depend on the concentration,
coordinates and time; (3) the diffusant concentration c (z, τ) and its flux
Jz in the z-direction, determined by the first Fick’s law,
Jz = –DV ∂ c (z, τ)/∂ τ, (41)
are continuous on the walls of the grain boundary, i.e. at y = ±δ/2, at
that, the flow Jz is continuous at the interface between the boundary
and the grain; (4) the grain boundary thickness δ is so small that a
change in the concentration across the boundary (i.e., in the y-direction)
can be neglected.
An approximate solution of the problem of grain-boundary diffu-
sion for the case with a constant concentration source at z = 0 was ob-
tained in ref. [113], where the diffusant concentration in all other
points of the system was assumed zero. The exact solution for diffusion
from a source with a constant concentration was obtained in ref. [114].
The solutions in refs. [113, 114] are valid only for fine grain boundar-
ies, when Db >> DV, and when the diffusion coefficients Db and DV do not
depend on concentration, coordinates, and time, and the condition of
continuity of inflow on the interface between the grain boundary layer
and the crystal is satisfied.
To describe the grain-boundary diffusion analytically, the dimen-
sionless variables ε, η, β (corresponding to y, z, τ), and the dimension-
less parameter ∆,
1/2
/2
( )V
y
D
− δ
ε =
τ
,
1/2( )V
z
D
η =
τ
,
1/2 3/2 1/2
( 1)
2( ) 2
b
V V
D
D D
δ∆ − δ
β = =
τ τ
, b
V
D
D
∆ = , (42)
are introduced.
The physical meaning of the η parameter is in that it describes the
influence of direct bulk diffusion from a source into a crystal at a given
depth z: the stronger this influence, the lesser η. The parameter ε (de-
pendent on a given distance from the plane of the grain boundary) de-
scribes the contribution of bulk diffusion from the grain boundary to
the crystal. The parameter ∆ shows how many times the diffusion along
the grain boundaries is greater as compared to the bulk diffusion. At
the solid-phase sintering, carried out at low and medium temperatures,
the coefficient of grain boundary diffusion Db is significantly greater
than the coefficient of bulk diffusion DV and, as a rule, ∆ = 103–105
[116]. The diffusion coefficient along the grain boundaries (DGb), which
includes diffusion parallel to grain boundaries (Db) and surface diffu-
sion (DS), is not a characteristic of the material, but depends on the
properties of the boundary, i.e. its structure. For low-angle grain bound-
aries (misorientation <15 °), DGb differs weakly from DV. The value of
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Mathematical Modelling of the Sintering of Fe-Based Metal-Glass Materials
DGb (at a given axis of rotation) increases with increasing angle of rota-
tion. For some boundaries with a highly ordered (bulk [50–52] or planar
[117–120]) structure and close-packed grains, extremely low diffusion
coefficients DGb are observed. For a direction, parallel to the dislocation
lines, i.e. parallel to the axis of rotation, diffusion is faster than in the
perpendicular direction, where diffusion, as a rule, is a bulk. Such
anisotropy is confirmed experimentally. With an increase in the angle
of rotation, the anisotropy decreases, but with a misorientation of more
than 15°, a significant anisotropy of the diffusion coefficient conserves.
Comparison of the bulk, parallel to grain boundaries, and surface diffu-
sion, showed that the smallest activation energy is observed for the
surface diffusion, while the greatest one for bulk diffusion. If assume
that the pre-exponential coefficient D0 has approximately the same val-
ue independently on the diffusion type, then in accordance with the
Arrhenius dependence the highest diffusion coefficient is observed for
the surface diffusion, and the lowest for the bulk diffusion [85].
At the same temperature, not all atoms have the same mobility.
Atoms located on the surface of the particles and, especially on their
protrusions (humps), have greater mobility. Therefore, surface atoms
(especially those located at the humps) possess the greatest displace-
ments in the initial sintering stage. Such atoms possess the greatest
surface energy and can easily leave their places, trying to occupy more
stable positions in the valleys (hollows, cavities) of the particles. highly
mobile atoms are concentrated in narrow areas of the interparticle space
and begin to belong simultaneously to several particles. Such surface
diffusion leads to an increase in interparticle contacts and to the hard-
ening of powder solids. The most effective manifestation of surface dif-
fusion is observed at low and medium sintering temperatures.
At high temperatures, a significant role belongs to the bulk diffu-
sion, which results to decrease of the mechanical strength of particles,
increase of plasticity, and their ability to flow in a metal under the ac-
tion of surface tension forces [121]. With an increase of the annealing
time, the grain-boundary peaks in the concentration distribution graph
are gradually smoothed and, as a result, it is becoming more difficult to
observe experimentally accelerated diffusion along the grain boundar-
ies. The optimal conditions for determining the parameters of grain
boundary diffusion are that the value β has to be at least more than 10,
in order to determine accurately coefficient of the grain boundary dif-
fusion Db from the experiments on different layers. The duration of
diffusion annealing should be chosen in accordance with this require-
ment [37].
In the work [114], there is no expression for an average diffusant
concentration c–; therefore, the solution in ref. [115] will be further
used. This solution is obtained for diffusion from an instant source —
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T.G. Jabbarov, O.A. Dyshin, M.B. Babanli, and I.I. Abbasov
diffusion from a thin-film source on the sample surface with a very
small film thickness δ.
The equation for the concentration of diffusant in the grain cg (y, z, τ)
reads in a short form as
cg (ε, η, β) = c1 (η) + c2 (ε, η, β), (43)
where c1 is the contribution of direct bulk diffusion from a source with
a constant concentration to the bulk in the z-direction, and c2 is the
contribution of the grain boundary.
The exact solution for the diffusion equation from an ‘instanta-
neous’ source (or thin-film) was obtained in ref. [115]. Since c1 decreas-
es much faster than c2, then in the area far from the source, c1 << c2.
Therefore, almost the entire diffusant located in this region comes there
via the bulk diffusion from the grain boundary and, consequently, far
from the diffusion source, c– = c–2.
The coefficient of grain-boundary diffusion is substantially greater
than the bulk diffusion coefficient and, as a rule,
Δ = Db/DV. (44)
From the second equality in eq. (42), we have:
β = δDb/(2DV
3/2τ1/2) = (δ/2)(Db/DV)(DVτ)–1/2. (45)
Considering the approximate equality DV ≅ 10–4Db and Arrhenius
formula Db = Db0 exp (−Eb/RT), where an activation energy of grain
boundary diffusion Eb = 200–300 kJ/mole for most metal alloys, we find
that even at sufficiently high temperatures T:
β >> 102. (46)
let us consider two cases: (i) 102 < β < 104 and (ii) β >> 104.
In ref. [37], for these cases, simplified versions of the equations for
calculating the product δDb for an ‘instantaneous’ source are reported.
For the first (i) case:
δDb = 0.2968D585/119τ–121/133(−∂ lg c–/∂z6/5)–200/119; (47)
for the second (ii) case:
δDb = 2 (DV τ)–1/2(−∂ lg c–/∂z6/5)–5/3(0.775)5/3, (48)
where c– is the average concentration of the diffusant, and z is the depth
of its penetration into the grain from the instantaneous source (this is
how [115] the diffusion from the thin-film source on the sample surface
with a very small film thickness δ is of the order of the atomic diame-
ter). Using the advantage of the approach in ref. [116], there is the
following relation for a source of finite thickness:
−∂ lg c–/∂z6/5 = 0.314β–0.592. (49)
Substituting the expression from (42) into eq. (47) and assuming
0.9849 ≈ 1, we obtain for the first (i) case:
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Mathematical Modelling of the Sintering of Fe-Based Metal-Glass Materials
δDb = 0.2968 DV
4.916τ0.5084(0.314)–1.68δDb/(2DV
1.5τ0.5), (50)
where one can find out
DV = 0.9932 τ0.295. (51)
Since Db/DV = Δ, eq. (50) can be written as
Db = 0.1484DV
4.416
τ–1.0084(0.314)–1.68Δ, (52)
from which, taking into account eq. (51) for DV, we obtain
Db = 1.0090 Δ τ0.295. (53)
In the second (ii) case, eqs. (48) and (49) result to
δDb = 0.32587 DV
0.5τ–0.5(0.314)–1.67β0.9866. (54)
Substituting β from eq. (42) and assuming 0.9866 ≈ 1, we obtain
δDb = 0.32587 DV
0.5τ–0.5(0.314)–1.67(δ Db/2DV
1/5τ0.5), (55)
and
DV = 1.1262 τ–1. (56)
eq. (54) can be written in the form
δDb = 0.32587 DV
0.5τ–0.5(0.314)–1.67(δ/2)(Δ/(2DV
0/5τ0.5)), (57)
which results to
Db = 1.1262 τ–1. (58)
relations (50) and (53) as well as (56) and (58) have a physical
meaning, if the dependence of τ on T is known. We use formulas (51)
and (53) for numerical integration of the eqs. (15)–(19):
DV, ij = 0.9932 τ ij
0.295, Db, ij = 1.0089 τij
0.295, (59)
where
τb = hi
j = ti, j − ti − 1, j = (Ti, j − Ti − 1, j)/wTj
.
Sintering of multicomponent systems is characterized by a series of
features consisting in the fact that sintering of dissimilar materials is
a more complex eutectic process, in which, along with self-diffusion,
causing mass transfer to the particle contact area, an interdiffusion
(heterodiffusion) should occur, providing balancing of the concentra-
tions of opposite atoms within the sample. With limited solubility or
complete insolubility of the components, the sintering system is compli-
cated by isolating homogeneous particles from mutual contact, which
prevents self-diffusion from occurring and worsens sintering conditions.
We study mixture of powders consisting of iron (44%), grey cast
iron (50%), and 6% of sitallized glass (sitall). Iron compound with sitall
(Fe2Sio4 — fayalite) will be considered as component A, and iron com-
pound with carbon (cast iron) — component B. The heterodiffusion coef-
ficient D of the binary system of components A and B is expressed in
terms of the partial diffusion coefficients [23]:
D
∼
= cBDA + cADB; (60)
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T.G. Jabbarov, O.A. Dyshin, M.B. Babanli, and I.I. Abbasov
we consider the case when concentrations of the components A and B are
equal, cA = cB = 0.5. Darken’s theory has been criticized many times;
however, it is in quite good agreement with experiments [23].
The diffusion coefficient determined in experiments is a certain
average value of the diffusion coefficient along the grain boundaries DGb
(including Db and DS) and the bulk diffusion coefficient [116]:
D
∼
= (Db + DV + DS)/3. (61)
The coefficient DA is defined in eq. (40); DB is contained in Table 2.
Knowing Db, DV and D
∼
, we can calculate DS using eq. (61).
2.6. Simulation Approach for Sintering of Powder Mixture
Constructing a model of the powder sintering process, we accept as-
sumptions [71] following below.
• material grains are single crystal and the average grain size is
taken as the structural parameter of the material;
• compaction of the material at the stage of sintering in vacuum is
carried out by the mechanism of diffusion-viscous flow with thermal
slipping along the grain boundaries and decreasing the volume of pores
due to the pushing of voids onto the surface;
• the growth of material grains occurs due to thermally activated
diffusion coalescence of dispersed particles with the redistribution of
the small particles along the surface of larger particles under the action
of surface self-diffusion, localized in the surface (one atomic diameter δ
thick) layer;
• in the rheological description of the porous structure, the Macken-
zie–Shuttleworth model [61] is used, in accordance with which the phase
of void is localized in an ensemble of interacting equal spherical pores
displaced with respect to each other so far that the porous material can
be represented as a set of elements, each of which includes a pore sur-
rounded by a layer of incompressible matter;
• compaction at the stage of sintering under pressure (acting on the
solid as a comprehensive compression) is carried out according to the
mechanism of viscous flow;
• temperature (Tj, j = 1, …, n) does not vary over the volume of the
sintering space with a temperature equal to the temperature of the ma-
terial.
let us define Δ1j = (tj – 1, tj) (j = 1, …, n) as the time intervals of non-
isothermal sintering with temperature Tj supported on the interval Δ1j
of duration τ1j, where tj = Σj
l = 1τ1j. Define Δ2 = (tn, tn + τ2) as the time in-
terval of isothermal sintering with duration τ2 with the constant tem-
perature TC. Denote the average sizes of grains and porosity in the in-
terval Δ1j ( j = 1, …, n) or Δ2 as п1j and L1j or п2 and L2, respectively.
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Mathematical Modelling of the Sintering of Fe-Based Metal-Glass Materials
To describe a stage of the solid phase sintering, we use a model in-
cluding mathematical equations and conditions following below.
• equation of the kinetics of compaction of the material at the j-th
temperature stage (see also eqs. (20), (20′) and (20′′)):
( )
3/2 2.1
1, 1,
2.11/21/2
0 0 1,
1/2
, ,0
3 4
1, 1,
6(1 ) 3
(1 ) 1 (1 ) (1 )
2 1
1 ;
13 2
j j
j
V j b j
jB j g B j g
d
dt
D bD
k Tr k T r
Π − Π ⋅
= − ×
− Π − − Π − Π
γΩ γΩ− Π × − + − Π
(62)
for the first 3.5% of shrinkage, the kinetic equation reads as
( )
3/2 2.06
1, 1,
2.061/21/2
0 0 1,
1.031/2
, ,0
1.03 3 4
1, 1,
6(1 ) 3
(1 ) 1 (1 ) (1 )
2.63 0.71
1 ,
13 2
j j
j
V j b j
jB j g B j g
d
dt
D bD
k Tr k T r
Π − Π ⋅
= − ×
− Π − − Π − Π
γΩ γΩ− Π × − + − Π
(62′)
where Db, j and DV, j are the coefficients of grain-boundary and bulk dif-
fusion (m2/s), kB is the boltzmann constant in units of J/K, Tj is the
temperature at the j-stage of sintering (K), п0 is the porosity before the
start of sintering, п1j porosity at the j-stage of sintering.
• equation of the kinetics of grain growth (see also eq. (22)):
dL1, j/dt = B1γ DS, j δ4/(4L3
1, j kTj ); (63)
here DS, j is the surface diffusion coefficient, L1, j − L0 is the thickness of
the layer (of the order of the atomic diameter δ) in which surface diffu-
sion takes place, γ is the surface tension, Tj is the absolute temperature
at the j-stage of sintering, numerical constant B1 ≈ 30.
• equation of temperature rise:
dTj/dt = wTj
, wTj
= (Tj − Tj − 1)/τ1j, tj − 1 < t ≤ tj. (64)
• Initial conditions:
п|t = 0 = п0, L|t = 0 = L0, T|t = 0 = T0. (65)
• Conditions of conjugation of temperature stages in time:
п1, j|t = tj–1
= п1, j − 1, L1, j|t = tj–1
= L1, j − 1, T|t = tj–1
= Tj − 1, j = 2, ..., n. (66)
The conjugation conditions (66) at the junctions of the temperature
stages carry out the continuous join of solution obtained for the indi-
vidual temperature stages. The values obtained in this way for porosity
пt and graininess Lt at t = tn will be used as initial data for liquid-phase
sintering.
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T.G. Jabbarov, O.A. Dyshin, M.B. Babanli, and I.I. Abbasov
Note that at low and medium annealing temperatures, Db >> DV. In
this case, we can use the kinetic equation (19) applicable if the grain-
boundary diffusion dominates over the bulk diffusion. Taking into ac-
count relation between dпc/dt and dy/dt, we obtain the following equa-
tion for the kinetics of compaction at the j-th temperature regime
(stage):
0.331.5
1, 1, , 0.67
0.5 4
0 1,
6(1 ) 2.14
0,33
(1 )
j j b j
B j g
d bD
t
dt k T r
−
Π − − Π γΩ
= ⋅ − Π
,
which can be used for all t ∈ Δ1j starting with j = 2 to j = n.
The main difference between liquid-phase sintering and solid-phase
sintering is that (along with a change of capillary forces in the contact
of particles) when a melt (wetting the particles) appears, the surface
area of interacting components increases significantly. This circum-
stance strongly accelerates the process of alloy formation, and the ther-
mal and bulk effects (associated with it) become apparent immediately
[46]. At once of the contact with the solid phase, liquid phase atoms
begin to diffuse into the solid surface. Such diffusion in the first stage
of interaction between the solid and liquid phases causes an increase in
the volume of the particles of the component (which is the basis of the
powder solid) and the separation of the particle centres (which leads to
an increase in the linear dimensions of the entire melt). The subsequent
dissolution of particles of the solid phase in the liquid is accompanied
by a decrease of their volume, the approaching of their geometric cen-
tres due to the impact of capillary forces and, as a result, an increase of
the shrinkage of the powder solid. Thus, after the formation of a liquid
phase during sintering of a powder solid, in a general case, firstly, the
growth emerges, and then it is followed by the shrinkage (compaction) [47].
The kinetics of compaction at the stage of liquid-phase sintering
will be analysed based on the rheological description of the deformation
of the porous structure considering diffusion-viscous flow and compac-
tion of the porous solid under the uniform stressed state in conditions
of the uniform compression [24].
Such formulation of the problem characterizes the sintering process
with simultaneous effects of both laplace (capillary) pressure Pc and
hydrostatic pressure P applied externally and obtained by summation of
the capillary pressure Pc with the inert gas pressure on the material Pg:
P = Pc + Pk.
Following the hydrodynamic analogy of the theory of elasticity, it
can be assumed that in an isotropic medium, the viscosity tensor has
two components: the bulk (χ) and shear (η) viscosity coefficients. They
are similar to those used in the mechanics: compression (K) and shear
(G) moduli. however, in contrast to a non-porous solid, in which there
are no internal degrees of freedom for the mutual change of its parts
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Mathematical Modelling of the Sintering of Fe-Based Metal-Glass Materials
and the modules K and G are large, in a porous solid, where there is
such a freedom, the values of χ and η coefficients are significantly
smaller and depend on porosity.
To describe the dimensionless bulk deformation under the uniform
compression, one can use the relation [47]:
ΔV/V = εii = −P/χ, (67)
where εii is the sum of the diagonal terms of the strain tensor.
based on the hydrodynamic analogy of the theory of elasticity, since
the state-of-the-art version of the rheological sintering theory [59] is
based on this analogy, we can write the following viscous flow equation
for a compressible solid [24]:
V·/V = ε·ii = 3ε·rr = −P/χ; (68)
since, in the above formulation of the problem, the strain rate tensor is
spherically symmetric and can be reduced to one radial component ε·rr.
Taking into account the obvious relation between the volume of the po-
rous solid, V, and the volume of the matter contained in it (without
pores), VB, V = VB(1 − п), we obtain
V·/V = п· (1 − п); (69)
here п is a dimensionless porosity of the solid.
In the rheological description of the porous structure (χ, η), the
Mackenzie–Shuttleworth model is accepted, according to which the
phase of void can be represented as localized in an ensemble of non-in-
teracting equal-size pores, which are distant from each other. Moreover,
they are distant do far that one can introduce into consideration an ele-
ment of a porous solid consisting of a pore radius R surrounded by a
layer of an incompressible solid with radius R° with п = (R/R°)3.
From the hydrodynamic analogy, the relationship between χ and η
is given by the relation
χ = 4η (1 − п)/3п. (70)
From relations (68) and (69), we obtain the following law of compac-
tion of a porous solid:
dп/dt = − (1 − п)P/χ (71)
or using eq. (70)
dп/dt = − 3пP/(4η). (72)
Taking into account the dependence of the coefficient of shear vis-
cosity of a porous solid η on porosity п established in refs. [59–61],
η = η0(1 − п)5/3, (73)
the differential equation of a porous solid (72) can be written as
dп/dt = − 3пP/[4(1 − п)5/3η0]. (74)
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T.G. Jabbarov, O.A. Dyshin, M.B. Babanli, and I.I. Abbasovvvv
For small porosities, we can restrict ourselves to the linear depen-
dence of η on п, assuming (1 − п)5/3 ∼− 1 − (5/3)п. Then, we obtain the
following law of compaction of a porous solid [59]:
dп/dt = − (3/4)[п/(1 − 5п/3)](P/η0). (75)
equations for the mathematical description of the stage of isother-
mal (liquid phase) sintering under pressure P includes:
equation of the kinetics of compaction of the material (see eq. (74)):
dп2/dt = − [п2/(1 − п2)
5/3] 3P/(4η0), t ∈ Δ2; (76)
rheological model of porous material (see equations (70), (73)):
χ = 4η (1 − п2)/3п2, η = η0 (1 − п2)
5/3; (77)
equation of grain growth kinetics of the material:
dL2/dt = B1γDs, nδ4/(4 L3
2 kTc), t ∈ Δ2, (78)
where Ds, n relates with activation energy of the grain-boundary diffu-
sion (Qs),
Ds, n = Ds, 0 exp (− Qs/RTc). (79)
The capillary pressure Pc applied to the pore surface and the instan-
taneous radius of pore Rp are determined by the equations [71]
Pc = 2п2 σ/Rp, Pp = Rp°(1 − п1, n)/[п1, n(1 − п2)], (80)
where σ = 2π/Rp° is the surface pressure, Rp° is the radius of the grains
before sintering; initial conditions read as
п2|t = tn
= п1, j|t = tn
, L2|t = tn
= L1, j(t)|t = tn
, Tc = Tn. (81)
The density of the material at the stages of the process is calcu-
lated depending on its current porosity using the equations:
ρ1, j = (1 − п1, j (t))ρ0, (j = 1, …, n), ρ2 = (1 − п2(t))ρν. (82)
Fig. 3. Temperature dependence of
porosity
Fig. 4. Temperature-dependent average
grain size of the powder
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Mathematical Modelling of the Sintering of Fe-Based Metal-Glass Materials
Volumetric shrinkage of the stages is calculated by the equations:
ΔV1, j/V0 = (V0 − m/ρ1, j)/V0, (j = 1, …, n), ΔV2/V0 = (V0 − m/ρ2)/V0, (83)
where ΔV1, j = V0 − V1, j, ΔV2 = V0 − V2, and m is mass of the sample.
The quality index (residual porosity пp, average grain size Lp, and
density ρp ) of the alloy can be determined from the relations:
пp = п2(τ1 + τ2), Lp = L2(τ1 + τ2), ρp = ρ2/(τ1 + τ2).
If we know D, DV, and Db, one can determine Ds as a function of T
and τ, which gives a possibility to calculate (via the fourth order runge–
Kutta method) porosity п and graininess in discrete points tij (i = 1, …,
M1; j = 1, …, n) over all time interval of the solid-phase sintering. rep-
resentative results in Fig. 3, Fig. 4 are plotted for 102 < β < 104 and ∆ = 102.
3. Summary
For the mathematical description of the process of sintering a mixture
of powders, a system of nonlinear differential equations of the kine -
tics of compaction and grain growth for sequential temperature regimes
is constructed. by fitting the solutions of these equations at the junc-
tions of temperature regimes and integrating the resulting system of
equations using the fourth order runge–Kutta method, the numeri -
cal time discretization of continuous porosity and grain curves is
const ructed.
The investigated mixture of iron powders, grey cast iron, and sital-
lized glass (sitall) is considered as a system of two components. A com-
pound of iron with a sitall (fayalite) represents the first component; the
second component is a compound of iron with carbon (cast iron). The
Darken formula is applied to calculate the interdiffusion coefficient of
this (quasi)binary system.
In the rheological description of the sintering kinetics under pres-
sure (uniform compression), the mechanism of diffusion-viscous flow
was used within the framework of the Mackenzie–Shuttleworth model.
In accordance with this mechanism, thermal slippage occurs along the
grain boundaries and a decrease of the pore volume due to the ejection
of vacancies on the surface.
During the process of formation of a liquid phase, the capillary
forces at the contact of the particles vary. When a melt (wetting the
particles) appears, the surface area of interaction between the compo-
nents increases significantly that accelerates the sintering process. At
the same time, in the process of sintering a powder solid, one can ob-
serve a decrease of porosity and an increase of graininess. Then, the
shrinkage and stabilization of both porosity and graininess occurs.
The proposed algorithm for numerical simulation of compaction ki-
netics for solid-phase sintering of highly dispersed and, particularly,
612 ISSN 1608-1021. Prog. Phys. Met., 2019, Vol. 20, No. 4
T.G. Jabbarov, O.A. Dyshin, M.B. Babanli, and I.I. Abbasov
nanosize powders allows analysing and controlling the technical param-
eters of the process of solid-phase sintering of metal-ceramic alloys. The
calculated values of these parameters (obtained as a result of solid phase
sintering, are used as their input values in the process of liquid-phase
sintering [122].
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received May 5, 2019;
in final version, october 8, 2019
Т.Г. Джаббаров, О.А. Дишін, М.Б. Бабанли, І.І. Аббасов
Кафедра машинобудування та матеріалознавства,
Азербайджанський державний університет нафти і промисловості,
просп. Азадлиг, 16/21, 1010 баку, Азербайджан
мАТемАТичНе моДелюВАННя процеСІВ СпІКАННя
меТАлоСКляНих мАТерІАлІВ НА оСНоВІ ЗАлІЗА
На основі дослідження механізмів дифузійної коалесценції та коаґуляції огля-
даються математичні методи опису та побудови моделів процесу спікання ме-
тало керамічних матеріалів. ці моделі представлено системою нелінійних ди фе-
ренційних рівнянь, що включають коефіцієнти об’ємної, зерномежової та
поверхневої дифузії, а також відповідають послідовності стадійних рівнів тем-
ператури, зростаючих із певною швидкістю та що мають різні тривалості.
реґулюючи рівні, швидкості та тривалості температурних режимів, технічні па-
ра метри шихти, можна контролювати процес спікання в режимі онлайн. опис
кінетики рідкофазового спікання під тиском проведено на основі реологічної тео-
рії спікання з використанням механізму дифузійно-в’язкої течії, відповідно до
якого відбувається танґенційне проковзування вздовж меж зерен і зменшення
об’єму пор через виштовхування вакансій на поверхню. після утворення рідкої
фази при спіканні порошкового тіла (в загальному випадку) спочатку виявляється
зростання зерен, а потім — усадка одержуваного стопу. процес спікання порош-
ків заліза, чавуну і ситалу розглянуто як взаємну дифузію двох (квази) бінарних
стопів: чавуну (залізо + вуглець) і фаяліту (залізо + ситал). розрахунок коефіцієнта
взаємної дифузії результуючого стопу проведено за Даркеновою формулою. Спі-
кання багатокомпонентних систем характеризується рядом особливостей, які
полягають у тому, що спікання різнорідних матеріалів (з різними температурами
топлення) є складним евтектичним процесом, в якому, поряд із самодифузією,
яка зумовлює перенесення маси в область контакту частинок, відбувається вза-
ємна дифузія, що забезпечує гомогенізацію складу шляхом вирівнювання кон-
центрацій різнойменних атомів у межах зразка. В умовах обмеженої розчинности
або пов ної нерозчинности компонентів спікання системи ускладнюється ізолю-
ванням однорідних частинок від взаємного кон такту, перешкоджаючи перебігу
самодифузії та тим самим погіршуючи умови спікання. Для чисельного розв’язу-
вання задачі використано методу рунґе–Кутти четвертого порядку точности зі
змінним кроком інтеґрування. розроблено програмний комплекс розв’язання
задачі, а результати розрахунку наведено на прикладі стопу порошкової суміші
заліза, чавуну та сита лізованого скла.
Ключові слова: металокерамічні та металоскляні матеріали, твердофазове та
рідкофазове спікання, само- та взаємна дифузія, реологія спікання.
ISSN 1608-1021. Usp. Fiz. Met., 2019, Vol. 20, No. 4 619
Mathematical Modelling of the Sintering of Fe-Based Metal-Glass Materials
Т. Г. Джаббаров, О. А. Дышин, М. Б. Бабанлы, И. И. Аббасов
Кафедра машиностроения и материаловедения,
Азербайджанский государственный университет нефти и промышленности,
просп. Азадлыг, 16/21, 1010 баку, Азербайджан
мАТемАТичеСКое моДелироВАНие процеССА СпеКАНия
меТАллоСТеКляННых мАТериАлоВ НА оСНоВе желеЗА
На основе исследования механизмов диффузионной коалесценции и коагуляции
обозреваются математические методы описания и построения моделей процесса
спекания металлокерамических материалов. Эти модели представлены системой
нелинейных дифференциальных уравнений, включающих коэффициенты объём-
ной, зернограничной и поверхностной диффузии, а также соответствуют после-
довательности стадийных уровней температуры, возрастающих с определённой
скоростью и имеющих разные продолжительности. регулируя уровни, скорости
и продолжительности температурных режимов, технические параметры шихты,
можно контролировать процесс спекания в режиме онлайн. описание кинети-
ки жидкофазного спекания под давлением проведено на основе реологической
теории спекания с использованием механизма диффузионно-вязкого течения,
в соответствии с которым происходит тангенциальное проскальзывание вдоль
границ зёрен и уменьшение объёма пор из-за выталкивания вакансий на по-
верхность. после образования жидкой фазы при спекании порошкового тела (в
общем случае) вначале обнаруживается рост зёрен, а затем — усадка получаемо-
го сплава. процесс спекания порошков железа, чугуна и ситалла рассмотрен как
взаимная диффузия двух (квази)бинарных сплавов: чугуна (железо + углерод) и
фаялита (железо + ситалл). расчёт коэффициента взаимной диффузии результи-
рующего сплава проведён по формуле Даркена. Спекание многокомпонентных
систем характеризуется рядом особенностей, заключающихся в том, что спека-
ние разнородных материалов (с различными температурами плавления) явля-
ется сложным эвтектическим процессом, в котором, наряду с самодиффузией,
обуславливающей перенос массы в область контакта частиц, происходит вза-
имная диффузия, обеспечивающая гомогенизацию состава путём выравнивания
концентраций разноимённых атомов в пределах образца. В условиях ограничен-
ной растворимости или полной нерастворимости компонентов спекание системы
осложняется изолированием однородных частиц от взаимного контакта, препят-
ствуя протеканию самодиффузии и тем самым ухудшая условия спекания. Для
численного решения задачи использован метод рунге–Кутты четвёртого порядка
точности с переменным шагом интегрирования. разработан программный ком-
плекс решения задачи, а результаты расчёта приведены на примере сплава по-
рошковой смеси железа, чугуна и ситаллизованного стекла.
Ключевые слова: металлокерамические и металлостеклянные материалы, твер-
дофазное и жидкофазное спекание, само- и взаимная диффузия, реология спе-
кания.
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