Thermodynamics of the formation of composite material structures. A review
Composite material structures at nano-, micro-, and meso- levels have been examined. The application of the Gibbs and Hill thermodynamics has allowed researchers to give the physical interpretation of the Laplace pressure for composite materials; to derive the thermodynamic functions describing the...
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Інститут надтвердих матеріалів ім. В.М. Бакуля НАН України
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irk-123456789-1299152018-02-03T03:02:34Z Thermodynamics of the formation of composite material structures. A review Lisovsky, A.F. Получение, структура, свойства Composite material structures at nano-, micro-, and meso- levels have been examined. The application of the Gibbs and Hill thermodynamics has allowed researchers to give the physical interpretation of the Laplace pressure for composite materials; to derive the thermodynamic functions describing the particles consolidation; to establish the new phenomenon, namely, metal melt imbibition (MMI), and to define its acting forces, to establish the criterion that allows to determine a direction of a liquid phase migration in a composite body, as well as to predict the final structure of composite materials. For the description of composite material structure, which forms under extreme conditions, it has been necessary to use the nonequilibrium thermodynamics. At the extreme conditions the Prigogine concept of local equilibrium is promising. Изучена структура композиционных материалов на нано-, микро- и мезоуровнях. Используя термодинамику Гиббса и Хилла, исследователи дали физическую интерпретацию давления Лапласа для композиционных материалов; получили термодинамические функции, описывающие процесс консолидации частиц; установили новое явление – поглощение металлических расплавов спеченными композиционными телами (phenomen MMI), определили его движущие силы; обосновали критерий, который позволяет определить направление миграции жидкой фазы в композиционных телах; предсказать окончательную структуру композиционных материалов. Для описания структуры композиционных материалов, которые формируются в экстремальных условиях, предложено применять неравновесную термодинамику. Для этих условий перспективной является концепция локального равновесия, разработанная И. Пригожиным. Вивчено структуру композиційних матеріалів на нано-, мікро- і мезорівнях. Використовуючи термодинаміку Гіббса і Хілла, дослідники дали фізичну інтерпретацію тиску Лапласа для композиційних матеріалів; одержали термодинамічні функції, що описують процес консолідації часточок; виявили нове явище – поглинання металевих розплавів спеченими композиційними тілами (phenomenon MMI) та визначили його рушійні сили; обґрунтували критерій, який дозволяє визначити напрямок міграції рідкої фази в композиційних матеріалах та прогнозувати кінцеву їх структуру. Для опису структури композиційних матеріалів, яка формується в екстремальних умовах, запропоновано застосовувати нерівноважну термодинаміку. Для цих умов перспективною є концепція локальної рівноваги, яка розроблена І. Пригожиним. 2015 Article Thermodynamics of the formation of composite material structures. A review / A.F. Lisovsky // Сверхтвердые материалы. — 2015. — № 6. — С. 3-17. — Бібліогр.: 77 назв. — англ. 0203-3119 http://dspace.nbuv.gov.ua/handle/123456789/129915 620.22:536 en Сверхтвердые материалы Інститут надтвердих матеріалів ім. В.М. Бакуля НАН України |
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Получение, структура, свойства Получение, структура, свойства |
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Получение, структура, свойства Получение, структура, свойства Lisovsky, A.F. Thermodynamics of the formation of composite material structures. A review Сверхтвердые материалы |
| description |
Composite material structures at nano-, micro-, and meso- levels have been examined. The application of the Gibbs and Hill thermodynamics has allowed researchers to give the physical interpretation of the Laplace pressure for composite materials; to derive the thermodynamic functions describing the particles consolidation; to establish the new phenomenon, namely, metal melt imbibition (MMI), and to define its acting forces, to establish the criterion that allows to determine a direction of a liquid phase migration in a composite body, as well as to predict the final structure of composite materials. For the description of composite material structure, which forms under extreme conditions, it has been necessary to use the nonequilibrium thermodynamics. At the extreme conditions the Prigogine concept of local equilibrium is promising. |
| format |
Article |
| author |
Lisovsky, A.F. |
| author_facet |
Lisovsky, A.F. |
| author_sort |
Lisovsky, A.F. |
| title |
Thermodynamics of the formation of composite material structures. A review |
| title_short |
Thermodynamics of the formation of composite material structures. A review |
| title_full |
Thermodynamics of the formation of composite material structures. A review |
| title_fullStr |
Thermodynamics of the formation of composite material structures. A review |
| title_full_unstemmed |
Thermodynamics of the formation of composite material structures. A review |
| title_sort |
thermodynamics of the formation of composite material structures. a review |
| publisher |
Інститут надтвердих матеріалів ім. В.М. Бакуля НАН України |
| publishDate |
2015 |
| topic_facet |
Получение, структура, свойства |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/129915 |
| citation_txt |
Thermodynamics of the formation of composite material structures. A review / A.F. Lisovsky // Сверхтвердые материалы. — 2015. — № 6. — С. 3-17. — Бібліогр.: 77 назв. — англ. |
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Сверхтвердые материалы |
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2025-07-09T12:25:49Z |
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2025-07-09T12:25:49Z |
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1837172209058054144 |
| fulltext |
ISSN 0203-3119. Сверхтвердые материалы, 2015, № 6 3
Получение, структура, свойства
UDC 620.22:536
A. F. Lisovsky (Kiev)
lisovsky@ism.kiev.ua
Thermodynamics of the formation of composite
material structures. A review
Composite material structures at nano-, micro-, and meso- levels
have been examined. The application of the Gibbs and Hill thermodynamics has
allowed researchers to give the physical interpretation of the Laplace pressure for
composite materials; to derive the thermodynamic functions describing the particles
consolidation; to establish the new phenomenon, namely, metal melt imbibition (MMI),
and to define its acting forces, to establish the criterion that allows to determine a
direction of a liquid phase migration in a composite body, as well as to predict the final
structure of composite materials. For the description of composite material structure,
which forms under extreme conditions, it has been necessary to use the nonequilibrium
thermodynamics. At the extreme conditions the Prigogine concept of local equilibrium
is promising.
Keywords: composite material, structure, thermodynamics.
INTRODUCTION
The subject of the present study is composite materials consisting
of refractory particles and low-melting point binders; for example, WC–Co, Dia-
mond–WC–Co, Cr3C2–Ni cemented carbides, Ti (C, N)–(Ni, Mo) cermets, W–
(Ni, Fe) heavy alloys, W–Cu, Fe–Cu, Ni–Pb, Cu–(Sn, Pb) constructional materials,
etc. These materials are produced by powder metallurgy. The final structures of
composite materials are formed in the course of sintering. Researchers have well
studied the kinetics and mechanism of sintering at molecular [1–7], micro-, and
macrolevels [8–12], developed mathematical models of sintering powders [13–20].
The authors of [21–23] described the driving forces of sintering, and proposed the
concepts of the Laplace effective pressure, sintering stress, and sintering potential.
The phenomenology, kinetics, driving forces, and mechanism of powder sintering
are analyzed in [24].
The development of the thermodynamic theory of the formation of composite
materials structures at nano- and micro- levels has started only recently and in my
opinion demands further investigation.
Chemical thermodynamics and phase equilibria are widely used at the devel-
opment of composite materials. These areas of thermodynamics are well described
in [25, 26], therefore, in the present review they are not examined.
© A. F. LISOVSKY, 2015
www.ism.kiev.ua/stm 4
The aim of the present review is to generalize the results of thermodynamic
studies of the formation of composite materials structures and define the ways for
the subsequent development of this field of the material science.
INVESTIGATION PROCEDURE
The structure of a composite material is formed of particles. The particle
consists of a core and surface layer. Particles, in which the volume of core Vc is
larger than the volume of surface layer Vl (Vc >> Vl), are macroobjects. Particles, in
which Vc ≈ Vl, are small objects (nanoparticles). To describe thermodynamic state
of the macro object, the Gibbs thermodynamics is used. To make thermodynamic
studies, it is necessary to allocate a system and to determine parameters of its
interaction with the environment. Independent thermodynamic parameters of a
system are the volume, V, pressure, p, temperature, Т, masses of components, m,
chemical potential, μ, and entropy, η. To describe a thermodynamic state of the
system, the following state functions are used: internal energy, U (η, V, m),
enthalpy, Н (η, p, m), Helmholtz free energy, F (T, V, m), and Gibbs free energy, G
(T, p, m). The state functions are total differentials of independent parameters and
supposes linear integration. Using this property, it is possible to determine
independent parameters of a system from state functions, for example:
mTmmPT
i V
Fp
m
G
ij ,,,
; ⎟
⎠
⎞
⎜
⎝
⎛
∂
∂−=⎟
⎠
⎞
⎜
⎝
⎛
∂
∂=μ
≠
, where i and j are the components. Let’s note
the important property of macroobject: its parameters and state functions do not
depend on the particle size, r. Unlike a macroobject, the parameters and state
functions of a small object depend on the size, r [27]. Thus, the particle size, r, is
an independent parameter of a nanoparticle state. For nanoparticles a special
method of thermodynamic research should be developed.
For nanoparticles, it is worthwhile to use the Hill thermodynamic method [28]. Hill
suggested that the system should be considered as an ensemble consisting of a great
number of small objects, n, each containing mi molecules of the i-th kind. Providing
that Σmi = const, the state of this system can be described by any of the known thermo-
dynamic potential, say, internal energy
d(nu) = Td(nη) – pd(nVα) + Σμi d(nmi) + Λdn,
here u, η, Vα, and m belong to a small object α, p is the external pressure, the Λ
magnitude allows for variations of the system energy as the small object sizes vary.
Upon integration and some simplifications of this equation Hill derived an
expression for the internal energy of a small object:
u = Tη – pVα + Σμimi + Λ. (1)
Attention should be drawn to the fact that the surface area, s, and surface
tension, γ, values of a small object are absent in Eq. (1). These values are indirectly
allowed for by the Λ magnitude, which is equal to the work of the small object
formation Λ = γs/3 [29]. For the whole system Hill has derived the equation
U = Tη – pV + Σμimi + Λn.
The Helmholtz free energy, F, and Gibbs free energy, G, are
F = – pV + Σμimi + Λn; G = Σμimi + Λn.
Hill’s method correctly describes the consolidation of nanoparticles.
ISSN 0203-3119. Сверхтвердые материалы, 2015, № 6 5
So, a technique of a thermodynamic research is simple: we allocate a system,
then define the interface parameters, choose a state function, then determine its
change at transition of the system from the initial into final state. The ignoring of
this technique does not allow one to obtain correct results of thermodynamic
studies.
The author [30] used Gibbs thermodynamics to derive the expression of the
Laplace pressure
2
12
1212 V
ApppL δ
δγ=−=Δ ,
where A is the surface area, indices 1, 2 , and 12 indicate that the values they define
refer to the corresponding phase or interface.
It is known from differential geometry that 1
2
1
1
−− +==
δ
δ RRK
V
A , where K is the
surface curvature, R1
and R2 are the main radii of the curvature at the given point.
Taking into account this dependence, the known expression of the Laplace pressure
has been derived
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
+γ=Δ
21
12
11
RR
pL . (2)
There is a contradiction in the expression of the Laplace pressure. Eq. (2)
describes the pressure at a point of a curved surface, which contradicts the physical
essence of the mere notion of the pressure, which is an integral characteristic and
has the meaning only as applied to a certain surface area or to the bulk of the body.
Because of this Eq. (2) may be correctly used only to describe a surface with a
constant curvature, K, i.e., spherical, cylindrical, or plane. Eq. (2) cannot be used
for the definition of the Laplace pressure in a crystal, which has a precise facet.
Expression for the definition of the Laplace pressure in bodies of any geometrical
shape is [30]
v
spL γ=Δ
3
2 ,
where v is the volume of a body.
In [24, 31, 32] the Laplace pressure (Eq. (2)) is used to describe the motive
force of the particle consolidation process. To apply the Laplace pressure for
cavities of an arbitrary geometrical shape, the authors [24, 31, 32] introduced the
so-called effective Laplace pressure and negative Laplace pressure. The authors of
[24, 31, 32] think that at the consolidation of particles the Helmholtz free energy
(F) changes only as a result of the reduction of the solid–gas interface. Thus, the
changes of the Helmholtz free energy in volumetric phases and changes of the
component chemical potentials are ignored. The authors [24, 31, 32] do not give a
clear physical interpretation of the “effective Laplace pressure” and the “negative
Laplace pressure” terms. The application of Eq. (2) gives erroneous results of re-
search [32]. The Laplace pressure can be positive only.
THERMODYNAMICS OF STRUCTURE FORMATION IN TWO-PHASE
COMPOSITE MATERIALS
To study the structure formation of microdispersed composite materials, the
Gibbs thermodynamics was used [33]. The particles consolidation was studied
using a model of a system that consists of n equidimensional particles of solid
phase, α, and one moving phase, ε (a gas or liquid). In its initial state, the system
www.ism.kiev.ua/stm 6
consisted of region I filled with the phase ε and region II that contained n particles
of the phase α and the phase ε. The particles were in point contacts and formed a
regular structure. Regions I and II were separated by a surface III. The passing of
the system from the initial to final state is shown in Fig. 1.
I
III
II
α
ΔV
ε
а б
Fig. 1. A model of a two-phase dispersed system: α is a solid dispersed phase, ε is a moving
phase (gas, liquid); initial (a) and final (b) states [33].
A change of the Helmholtz free energy (ΔF) of the particle consolidation is de-
scribed by the expression
αααεαα Δγ−γ=Δ AgF )2(
3
1 , (3)
where A is the surface area, g is the coefficient, which allows for the variation of
the particle geometry, indices αα and αε show that the indicated values belong to
the appropriate interface.
Thus, the consolidation process proceeds spontaneously when γαα < 2gγαε.
Equation (3) is the thermodynamic potential of the system under consideration.
From this reason, the pressure of the system can be defined from p = – (∂F/∂V)T,m.
This pressure will be further referred to as the shrinkage pressure psh.
( )
V
Agpsh ∂
∂γ−γ= αα
αααε2
3
1 (4)
Equation (4) is basic, using Eq. (4) the author [33] has obtained
( )u
u
u
r
Kp shsh Ф11 3
1
⎟
⎠
⎞
⎜
⎝
⎛ −= ,
where shK = k ( )αααε γ−γg2 , k is the coefficients, r is the size of a particle α, u is
the phase ε volume content of region II (in volume fractions), Ф(u) is a dimen-
sionless parameter, which takes into account changes in the cavity geometry in the
course of the particles consolidation.
It is necessary to note, that the expression of shrinkage pressure psh (4) is an
integral characteristic of the whole group of sintering particles and it is devoid of
contradictions inherent in Eq. (2).
According to Fig. 1, the transformation of the system from state (b) into state
(a) results in the deconsolidation of a refractory skeleton into an ensemble of
particles. Having fulfilled the above requirements for this process, author [33]
ISSN 0203-3119. Сверхтвердые материалы, 2015, № 6 7
obtained the following expression for the change of Helmholtz free energy for this
system:
.)2(
3
1
αααααε Δγ−γ=Δ AgF
Thus, deconsolidation of a refractory skeleton occurs when γαα > 2gγαε. During
the deconsolidation of a refractory skeleton a pressure, Π, generates in region II.
The pressure, Π, has the physical sense of the imbibition pressure. The expression
for the П pressure is [33]:
( )
V
Ag
∂
∂γ−γ= αα
αεαα 2
3
1П ; ( )u
u
u
r
K Φ⎟
⎠
⎞
⎜
⎝
⎛ −= Π
3
1
11П , (5)
where ΠK = k ( )αεαα γ−γ g2 .
In gas medium, the condition γαα < 2gγαε is fulfilled and the consolidation of
particles proceeds spontaneously. When a liquid is a moving phase, the condition
γαα > 2gγαε arises. In these systems the deconsolidation of refractory skeletons into
ensembles of particles takes place. This phenomenon was observed in WC–Co,
WC–Ni, TiC–Ni, Cr3C2–Ni composite materials [34, 35]. In the composite material
having a refractory skeleton, there is a metal melt imbibition by pore-free compos-
ite body (MMI-phenomenon) [36]. The mathematical model of the MMI-
phenomenon is described in [37], where the new law of the mass transfer of a liq-
uid phase in the dispersed structured systems is given. The MMI-phenomenon
takes place only in composite materials, in which refractory particles form non-
equilibrium dihedral angles [38]. According to experimental data [38], composite
materials consisting of refractory particles and a binder can be conventionally di-
vided into two classes. Composites with γαα > 2gγαε form the first class, while the
second class includes those with γαα < 2gγαε. The characteristic feature of the first
class of composites is their ability to imbibe metal melts. In the second class of
composites, equilibrium dihedral angles are formed, and liquid and solid phases are
in mechanical equilibrium. These composites have no pressure, Π (Eq. (5)) and do
not imbibe metal melts.
The Hill thermodynamics was used to study the formation of structures of
nanodisperse composite materials [39]. According to [40], the chemical potential
of a particle component, i, under the equilibrium conditions is not equal to the
chemical potential of a component, i, in the dispersion medium, their interrelation
is described by μr = μ∞ + 2γΩ/3r, where μr and μ∞ are the chemical potentials of a
component in a particle of radius r and in an unlimited volume (gas, vapor, liquid),
respectively, Ω is the atomic volume. This expression has been used in
thermodynamic investigations [39].
A change of the Helmholtz free energy (ΔF) of nanodispersed system is
described by the expression [39]:
∑
=
α
αεαααααεαα γ−γ+Δγ−γ=Δ
k
i
m
ii vm
r
AgF
1
)(
3
2)2(
3
1 , (6)
where m
iv is the molar volume.
The shrinkage pressure, psh, of the nanodispersed composite material is [39]
sh
sh V
Agp
Δ
Δγ−γ= αα
αααε )2(
3
1 ∑
=
α
αααε γ−γ
Δ
+
k
i
m
ii
sh
vm
rV 1
)(
3
21 . (7)
www.ism.kiev.ua/stm 8
A comparison of Eqs. (3) and (4) with (6) and (7) for nanoparticles allows a
conclusion that the additional items ∑
=
α
αεαα γ−γ
k
i
m
ii vm
r 1
)(
3
2 in (6) and
∑
=
α
αααε γ−γ
k
i
m
ii vm
r 1
)(
3
2 in (7) are due to the effect of size of a small object
(nanoparticle).
Features of the deconsolidation of a refractory skeleton in nanodispersed
composite materials are described in [41].
THERMODYNAMICS OF THE STRUCTURE FORMATION
IN THREE-PHASE COMPOSITE MATERIALS
The formation of the structure in a three-phase composite material is a complex
process. A simplified model and the following expression of a change of the free
energy (ΔF) were proposed in [42].
αββεαβαεαβαββββεβββαααεααα Δγ−γ−γ+Δγ−γ+Δγ−γ=Δ AggAgAgF )(
3
1)2(
3
1)2(
3
1 , (8)
where α and β are the particles, ε is the moving phase (gas, liquid).
The particle consolidation process will be spontaneous, if ΔF < 0, i.e., if the
following conditions are met:
2gαγαε > γαα, 2gβγβε > γββ, gαβγαε + gαβγβε > γαβ. (9)
In the gaseous phase these inequalities are always met, in the liquid phase some
of the inequalities (9) may not be met. For instance, if 2gαγαε <γαα contacts between
particles of the phase α do not form in such a system. Similarly, if 2gβγβε < γββ and
gαβγαε + gαβγβε < γαβ contacts β–β and α–β do not form.
Thus, if the inequalities (9) are met, during the particle consolidation a structure
of two interpenetrating skeletons forms from particles of the α and β phases in the
system. If any of the inequalities (9) is not met, aggregates and intergrowths of
several particles form in the system. If all three inequalities are not fulfilled, the
consolidation process does not occur. From Eq. (8) the expression describing
shrinkage pressure in three-phase composite materials have been obtained [42]:
( ) ( ) ( ) ⎥
⎦
⎤
⎢
⎣
⎡
∂
∂
γ−γ+γ+
∂
∂
γ−γ+
∂
∂γ−γ= αβ
αββεαβαεαβ
ββ
βββεβ
αα
αααεα
shshsh
sh V
A
gg
V
A
g
V
Agp 22
3
1 .
The final structures of composite materials are formed during the liquid-phase
sintering. The analysis of physicochemical processes occurring in composite
bodies during the liquid-phase sintering, motive forces, and process mechanisms is
given by German [43]. Thermodynamic studies of sintering of composite materials
with the liquid phase present are described in [44]. The author [44] has investigated
a system, which contained particles α, liquid β, and gas ε in the initial state. In the
final state the particles have formed a refractory skeleton penetrated by a liquid
phase. The change of the Helmholtz free energy in the system transition from the
initial state into the final state is
( ) βεβεαε γ+θ−=Δ AAF cos
3
1 ( ) αααβαα Δγ−γ+ Ag2
3
1 + αβαβΔγ A
3
1 + ∑ μΔ+Δγ αααα
i
iimA
3
1 ,(10)
where θ is the wetting angle.
ISSN 0203-3119. Сверхтвердые материалы, 2015, № 6 9
During liquid phase sintering a number of complex processes take place;
however the thermodynamics does not consider the kinetics and mechanism of
these processes. Equation (10) describes only a change of the Helmholtz free
energy of the system.
In Eq. (10) the ( ) βεβεαε γ+θ AA cos
3
1 term describes the change of the Helmholtz
free energy at the first stage of sintering, in the course of which a complete re-
moval of the gaseous phase from the sintered compact takes place. The main
driving force of this stage is a decrease of the solid–gas and liquid–gas interfaces.
The first term is always below 0, i.e., this process proceeds spontaneously. The
second term of Eq. (10) describes the process of the formation of a skeleton from
particles α. The main driving force of this process is the variation of the particle
contact surfaces, ΔAαα. It follows from Eq. (3) that the formation of the refractory
skeleton is possible with the following relation of surface tensions: γαα < 2gγαβ.
Under these conditions the equilibrium dihedral angles form in the composite
material. If the liquid volume in the composite material is inadequate, the particles
cannot take the equilibrium form and form nonequilibrium dihedral angles. The
structure of such a composite material is also nonequilibrium and as the composite
body contacts the liquid phase, the latter is imbibed to the instant of the formation
of equilibrium dihedral angles and separation of particles by the liquid [35, 36].
The third and fourth terms describe growth of particles during sintering a com-
posite material.
In composite materials, in which nonequilibrium dihedral angles formed, there
is pressure, Π, which has the physical sense of the imbibition pressure [38]. Under
certain conditions stable pores form in these composite materials, which are not
filled by the liquid [45]. The thermodynamics of the filling of isolated pores in
such materials is discussed in [46]. The change of the free energy, F, is described
by the following expression [46]:
( ) Θγ−Δ⎟
⎠
⎞
⎜
⎝
⎛ −γ−γ=Δ αεβαβαα cos
3
12
3
3
1
r
VkV
u
ug
d
KF
cc
, (11)
where K is the coefficient allows for geometrical parameters of a composite body,
ck is the coefficient of the geometric form of a pore, d is the particle size, cV is the
volume of a pore, r is the radius of a pore.
It follows from Eq. (11) that when αβαα γ<γ g2 the ΔF < 0 and hence, an
isolated pore is spontaneously filled with liquid. In composite materials, in which
αβαα γ>γ g2 a spontaneous pore filling with liquid (ΔF < 0) is possible when
( ) Θ
γ
<⎟
⎠
⎞
⎜
⎝
⎛ −γ−γ αε
αβαα cos12
3
1
r
k
u
ug
d
K c .
If ( ) Θ
γ
>⎟
⎠
⎞
⎜
⎝
⎛ −γ−γ αε
αβαα cos12
3
1
r
k
u
ug
d
K c the pore is not filled with liquid. It
follows from these inequalities that depending on the radius, r, the pore can or
cannot be filled with the liquid. The critical pore radius, crr , below which the pore
is filled with the liquid and above which it is not filled, was derived in [46]:
www.ism.kiev.ua/stm 10
( )αβαα
βε
γ−γ
Θγ
⎟
⎠
⎞
⎜
⎝
⎛
−
=
gu
ud
K
kr
c
cr 2
cos
1
3
1
.
The filling of a pore in a solid–liquid nanodisperced system is described in [47],
where the following expression for the thermodynamic function was suggested:
( ) Θγ−Δ⎟
⎠
⎞
⎜
⎝
⎛ −γ−γ=Δ αεβαβαα cos
3
12
3
3
1
r
VkV
u
ug
d
KF
cc
+ ∑
=
α
αααβ γ−γ
k
i
m
ii vm
r 1
)(
3
2 .
To study the thermodynamics of the nanoparticles consolidation in three-phase
composite materials, a system consisting of nα nanoparticles of phase α, nβ
nanoparticles of phase β, and medium ε was used [39]. In the initial state of the
system the particles are separated by phase ε. In the final state the consolidation of
the particles of phases α and β occurred and the particles formed two mutually
penetrating skeletons.The change of the Gibbs free energy in the system transition
from the initial into the final state can be described as the sum of changes of the
Gibbs free energies of two-phase subsystems +Δ+Δ+Δ=Δ αβββαα GGGG
εβα Δ+Δ+ GG [39].
After sintering in three-phase composite materials refractory skeletons are
formed. Features of the formation of refractory skeletons are given in [48].
The structure of a refractory skeleton is described by a contiguity, C, [49] and a
dihedral angle, φ, cos(φ/2) = γss/2γsl [50]. By the thermodynamic investigations
[51] it is established, that dihedral angle, φ, suggested by Smith [50], can be used
only for unbounded contacting surfaces. In composite materials particles are
limited by the closed surfaces which, as has shown by Gibbs [52], influence a state
of a thermodynamic object (a liquid drop, a particle). Considering this position,
expression for a dihedral angle, which is formed by particles, looks like [51]
( )γ+γ
γ=ϕ
122
cos
Ksl
ss ,
where 1K is the coefficient, γ is the averaged surface tension.
The infiltration of refractory skeletons with liquid metals is one of the efficient
methods to produce composite materials with unique properties. During the infil-
tration of the nanodispersed composite materials the important problem is to chose
the prerequisites to the stable existence of metal layers in nanodispersed bodies.
The thermodynamics of the liquid interlayers in nanodispersed composite materials
is described in [53]. The thermodynamic investigations were conducted on the
model of a system shown in Fig. 2.
The change of the Gibbs free energy, ΔG, of the system during the capillary
emptying is described by the equation [53]
=ΔG +Δγ−γ αααααβ A)2(
3
1
1
1
cos)(
3
2 Θγ+γ−γ βε
=
α
αααβ ∑ c
c
c
k
i
m
ii g
d
Vvm
r
.
In this expression when γαα > 2γαβ the first and the second terms are negative
and the ΔG value is defined by the following relations:
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+Δγ−γ αααααβ A)2(
3
1
1
1
cos)(
3
2 Θγ>γ−γ βε
=
α
αααβ ∑ c
c
c
k
i
m
ii g
d
Vvm
r
, ΔG < 0; (12)
+Δγ−γ αααααβ A)2(
3
1
1
1
cos)(
3
2 Θγ<γ−γ βε
=
α
αααβ ∑ c
c
c
k
i
m
ii g
d
Vvm
r
, ΔG > 0. (13)
I II α ε β
а б
Fig. 2. Model of a system in the initial (a) and in the final (b) states, α is a nanoparticle, β –
liquid, ε – gas phase; I indicates a composite body consisting of nanoparticles α and liquid β, II is
the region filled by a gas phase [53].
If Rel. (13) is fulfilled, liquid interlayers of any size are stable, while in Rel.
(12) they are unstable. From Rel. (12) it follows that the capillary empties and the
gas phase occupies the place of the liquid.
THERMODYNAMICS OF MESOSTRUCTURE FORMATION
IN COMPOSITE MATERIALS
A mesostructure consists of a random ensemble of mesoelements (granules),
which are bound by a metal or composite matrix. It is possible to consider the
matrix as a binder phase. The mean sizes of the mesoelements are 30–500 μm. In
[54] mesostructured composite materials have been divided into three types. The
first type is the composites consisting of a metal matrix, for example, Co, Ni,
special alloys, and mesoelements that are formed from composite materials. The
second type includes the composites consisting of mesoelements and a matrix
formed from composite materials. For example, the mesoelements are produced
from WC–6Co cemented carbide and a matrix is formed from W–Fe–Ni heavy
metal. The third type is represented with composites consisting of mesoelements
that are produced from a single crystal and a matrix is made from a composite
material, for example, a diamond mesoelement and a WC–Co composite. In the
composite materials considered above the mesoelements can contact with each
other, forming a three dimensional structure, or may be in regular intervals located
in a matrix without contacts. The structure of a mesocomposite can be described by
the stereology on metallographic specimens [55, 56].
The formation of the mesostructure allows one to produce composite materials
with particular properties by varying sizes and compositions of mesoelements as
well as the composition and properties of a matrix [57, 58].
The thermodynamics of solid–phase sintering of the first type of
mesocomposite materials is described by the following expression
( ) +Δγ−γ=Δ αααεαα AgF 2
3
1 ( ) +γ−γ ββεβα A
3
1 ∑ μΔ
i
ііm , (14)
www.ism.kiev.ua/stm 12
where α is the particles of mesoelement, β is the mesoelement, ε is the gas phase.
If solubility of components in phases is insignificant, expression (14) becomes
( ) +Δγ−γ=Δ αααεαα AgF 2
3
1 ( ) ββεβα γ−γ A
3
1 . (15)
From Eq. (15) the following relationships among the surface tensions are
obtained:
βαβεαααε γ>γγ>γ ;2g , (16)
βαβεαααε γ<γγ>γ ;2g , (17)
βαβεαααε γ>γγ<γ ;2g , (18)
βαβεαααε γ<γγ<γ ;2g . (19)
If Rel. (16) is met, a compact composite material forms. From Rel. (17) it
follows that the consolidation of only particles α occurs; according to Rel. (18) the
consolidation of particles α does not occur; according to Rel. (19) the structure
does not form. Liquid phase sintering of the first type mesocomposite is described
by Eq. (10). During liquid phase sintering in mesostructure composite of the first
class mesoelements imbibe a matrix liquid [38]. In mesostructure composite of the
second class [38] the imbibition of a liquid phase by mesoelements is not observed.
The thermodynamics of the mesostructure formation in a mesocomposite material
of the second type is described by Eq. (10). During the liquid phase sintering in a
mesostructure composite of the first class [38] the mass transfer of the liquid phase
from mesoelements to the matrix is observed. The moving force of this process is a
matrix-to mesoelement pressure gradient, Π, [59]. A special feature of the
formation of the third type mesostructure is the absence of a liquid phase migration
from a matrix into mesoelements. To attain high physico-mechanical properties of
a measocomposition, one should gain a good a matrix-to-mesoelements adhesion
[60]. To achieve this result, it is necessary to study the interaction of phases at the
mesoelement–matrix interface. In some composite materials the mesoelements and
matrix are thermodynamically incompatible; therefore, at the mesoelement–matrix
interface new phases form. In a composite material consisting of WC–Co
mesoelements and W–Fe–Cu matrix at the WC–Co/W–Fe–Cu interface a dense
carbide layer forms (Fig. 3). This layer interferes with a liquid phase penetration
from the W–Fe–Cu matrix into the WC–Co mesoelement [54].
Thermodynamics of the mesostructures formation in nanodispersed composite
materials is given in [61, 62]. In nanodispersed mesostructured composite materials
the consolidation of the nanoparticles is described by Eqs. (6) and (7). In these
materials the nanoparticles dissolve in a dispersion medium and large particles
grow. Large particles destabilize nanodispersed composite materials [62]. Using
composite materials of the first class [38] for the formation of mesostructures in
nanodisperced materials, it is necessary to consider transfer of a liquid phase in
bulks of mesoelements and matrix [63].
From the considered above papers follows, that the surface energies (γαα, γββ,
γαβ, γαε, γβε) affect the formation of composite material structure. To form the re-
quired structure of a composite material, it is necessary to know the values of the
surface energy. At present, there are good methods of research of the structure and
ISSN 0203-3119. Сверхтвердые материалы, 2015, № 6 13
composition of contact and interphase surfaces. They are high resolution transmis-
sion electron microscope equipped with an energy dispersive analyzers of X-rays
(TEM-EDX) and the atom probe tomography (APT) in combination with TEM
[64–67]. Methods of the definition of the surface energy at the liquid/gas interface
are developed, but there are no reliable methods to define the surface energy at the
solid/solid and solid/liquid interface. The last circumstance slows down the de-
velopment of new effective composite materials.
WC–6Co
W–Fe–Cu
Carbide layer
20 мm
Fig. 3. Structure of the WC–6Co/W–Fe–Cu interface [54].
ON THE THERMODYNAMICS OF STRUCTURE FORMATION UNDER
CONDITIONS OF INTENSIVE TECHNOLOGIES
The structures produced by traditional technologies of the powder metallurgy
are described above. At present new methods of the composite material structure
formation are developed: hot pressing (HP), hot isostatic pressing (HIP) [68],
intensive electric sintering (IES) [69], spark plasma sintering (SPS) [70], high
frequency induction heated sintering (HFIHS) [71], rapid omni compaction (ROC)
[72], pulse plasma sintering (PPS) [73], and ultrahigh pressure rapid hot
consolidation (UPRC) [74]. The sintering of WC–Co composites using intensive
technologies is given in the table.
Sintering of WC–Co mixtures on intensive technologies [75]
Sintering
technology
Sintering
temperature,
°С
Holding
time, s
Heating
rate
deg/min
Pres-
sure,
MPa
Compositions
of the initial
powders,
wt %
Size of
WC initial
particles,
nm
Relative
density, %
HP 1300 5400 – 25 WC–11Co 80 100
HIP 1000 3600 – 150 WC–10Co – 100
SPS 1100 600 150 60
WC–12Co–
1VC
40–250 95.94
HFIHS 1200 55 1400 60 WC–10Co 100 99.4
ROC 1000 10 – 830 WC–8Co ~ 100 –
UPRC 1200 900 200 1000 WC–10Co 50 99.4
PPS 1100 300 1200 60 WC–12Co 60 98
IES [47] 1300 10 5000 300 WC–10Co 100–150 98
www.ism.kiev.ua/stm 14
According to HP and HIP technologies, the final structure of a composite mate-
rial is formed at high temperatures for long holding times. Under these conditions
the system achieves an equilibrium state, therefore, for the description of a
composite material structure the Gibbs thermodynamics can be used. The SPS,
HFIHS, ROC, UPRC, PPS, IES technologies use high heating rate and short
holding times. During heating of a powder compact in its volume a temperature
gradient appears, which causes heat streams. Under these conditions the powder
compact is a nonequilibrium system. For the description of the structure formation
by these technologies it is necessary to use nonequilibrium thermodynamics. The
Prigogine concept [76] of local equilibrium is promising in the description of the
structure formation in these powder compacts. According to this concept,
equilibrium thermodynamic relations are true for each small volume, thus intensive
thermodynamic parameters T, p, μ are functions of coordinates and time, t. In a
powder compact the particles are small volumes and thus it is supposed, that the
Gibbs relations are working for each particle.
The local equilibrium theory is applied for an object, which is statistical
ensemble, for example, a micro particle. This theory cannot be used for a
nanoparticle, which is not a statistical ensemble and its parameters are influenced
with fluctuations. In the nonequilibrium thermodynamic theory [76, 77] the
fundamental expressions describing a local production entropy, the equations of
material balance and balance entropy, and conservation of the energy are
developed. This theory can be used for the description of the structure formation of
composite materials produced by intensive technologies.
CONCLUSIONS
The use of the Gibbs and Hill thermodynamics has allowed to describe a
formation of structures in composite materials at nano-, micro-, and meso- levels;
to give a strict conclusion and physical interpretation of the Laplace pressure; to
derive the thermodynamic functions describing process of the particle
consolidation; to prove the MMI phenomenon and to define its moving forces; to
establish the shrinkage pressure, psh, arising during the particles consolidation and
the pressure, П, arising during the deconsolidation of a refractory skeleton; to
prove the existence of the first class of composite materials, in which there is a
pressure, П, and the second class, in which there is no pressure, П, and MMI
phenomenon; to prove the criterion allowing us to determine a direction of the
liquid phase migration in composite materials. These results of thermodynamic
investigations have been achieved for composite materials, whose structure is
formed by traditional methods of the powder metallurgy.
At present, intensive thechnologies of the formation of composite material
structure have been developed. It has been defined that the use of the Gibbs and
Hill thermodynamics for these technologies is incorrect. To describe the structure
formation of composite materials, which are produced by these technologies, it is
necessary to apply the nonequilibrium thermodynamics. The existing theory of
nonequilibrium thermodynamics requires further development for its practical use.
Изучена структура композиционных материалов на нано-, микро- и ме-
зоуровнях. Используя термодинамику Гиббса и Хилла, исследователи дали физическую
интерпретацию давления Лапласа для композиционных материалов; получили термоди-
намические функции, описывающие процесс консолидации частиц; установили новое
явление – поглощение металлических расплавов спеченными композиционными телами
(phenomen MMI), определили его движущие силы; обосновали критерий, который позволя-
ет определить направление миграции жидкой фазы в композиционных телах; предска-
зать окончательную структуру композиционных материалов. Для описания структуры
ISSN 0203-3119. Сверхтвердые материалы, 2015, № 6 15
композиционных материалов, которые формируются в экстремальных условиях, предло-
жено применять неравновесную термодинамику. Для этих условий перспективной явля-
ется концепция локального равновесия, разработанная И. Пригожиным.
Ключевые слова: композиционные материалы, структура, термодина-
мика.
Вивчено структуру композиційних матеріалів на нано-, мікро- і мезорі-
внях. Використовуючи термодинаміку Гіббса і Хілла, дослідники дали фізичну інтерпре-
тацію тиску Лапласа для композиційних матеріалів; одержали термодинамічні функції,
що описують процес консолідації часточок; виявили нове явище – поглинання металевих
розплавів спеченими композиційними тілами (phenomenon MMI) та визначили його рушійні
сили; обґрунтували критерій, який дозволяє визначити напрямок міграції рідкої фази в
композиційних матеріалах та прогнозувати кінцеву їх структуру. Для опису структури
композиційних матеріалів, яка формується в екстремальних умовах, запропоновано за-
стосовувати нерівноважну термодинаміку. Для цих умов перспективною є концепція
локальної рівноваги, яка розроблена І. Пригожиним.
Ключові слова: композиційні матеріали, структура, термодинаміка.
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V. Bakul Institute for Superhard Materials, Received 04.12.14
National Academy of Sciences of Ukraine
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