Combustion synthesis of advanced materials chemistry research and applications series

CHEMISTRY RESEARCH AND APPLICATIONS SERIES

COMBUSTION SYNTHESIS OF
ADVANCED MATERIALS

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CHEMISTRY RESEARCH AND APPLICATIONS SERIES

COMBUSTION SYNTHESIS OF
ADVANCED MATERIALS

B. B. KHINA

Nova Science Publishers, Inc.
New York

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LIBRARY OF CONGRESS CATALOGING-IN-PUBLICATION DATA
Khina, B. B. (Boris B.)
Combustion synthesis of advanced materials / author, B.B. Khina.
p. cm.

Includes bibliographical references and index.
ISBN 978-1-61324-254-4 (eBook)
1. Self-propagating high-temperature synthesis. 2. Refractory
materials--Heat treatment. 3. Refractory materials--Mathematical models.
I. Title.
TP363.K46 2010
620.1'43--dc22
2009052731

Published by Nova Science Publishers, Inc. † New York


DEDICATION
To the memory of Professor Zinoviy P. Shulman (1924-2007) and Professor
Leonid G. Voroshnin (1936-2006) who had taught me the scientific meaning of
old Russian proverb, “trust, but verify”.
The important thing in science is not so much to obtain new facts as to
discover new ways of thinking about them.
Sir William Bragg



CONTENTS
Preface
Chapter 1
Chapter 2
Chapter 3
Chapter 4

xi

Advances and Challenges in Modeling Combustion
Synthesis

1

Modeling Diffusion-Controlled Formation
of TiC in the Conditions of CS

13

Modeling Interaction Kinetics in the CS
of Nickel Monoaluminide

39

Analysis of the Effect of Mechanical
Activation on Combustion Synthesis

75

References

93

Index

105

PREFACE
Self-propagating high-temperature synthesis (SHS), or combustion synthesis
(CS) is a phenomenon of wave-like localization of chemical reactions in
condensed media which permits efficiently synthesizing a wide range of
refractory compounds (carbides, borides, intermetallics, etc.) and advanced
composite materials. CS, where complex heterogeneous reactions proceed in
substantially non-isothermal conditions, brings about fine-grained structure and
novel properties of the target products and is characterized by fast
accomplishment interaction, within ~0.1-1 s, whereas traditional furnace synthesis
of the same compounds in close-to-isothermal conditions may take several hours
for the same particle size and close final temperature. Uncommon, nonequilibrium phase formation routes inherent of SHS, which have been revealed
experimentally, are the main subject of this book.
The main goal of this book is to describe basic approaches to modeling nonisothermal interaction kinetics during CS of advanced materials and reveal the
existing controversies and apparent contradictions between different theories, on
one hand, and between theory and experimental data, on the other hand, and to
develop criteria for a transition from traditional solid-state diffusion-controlled
phase formation kinetics (a “slow”, quasi-equilibrium interaction pathway) to
non-equilibrium, “fast” dissolution-precipitation route.
Features:
•
•

analysis of the physicochemical background of modeling approaches to
CS;
modeling of phase formation kinetics for two typical SHS reactions,
Ti+C→TiC (CS of an interstitial compound) and Ni+Al→NiAl (CS of an


B. B. Khina

xii

•
•

intermetallic compound), in strongly non-isothermal conditions using the
diffusion approach and experimentally known values of the diffusion
parameters;
novel criteria for the changeover of interaction routes in these systems
and phase-formation mechanism maps;
analysis of the physicochemical mechanism of the experimentally known
strong influence of preliminary mechanical activation of solid reactant
particles on SHS in metal-based systems.

It is anticipated that the book will serve the scientists, engineers, graduate and
post-graduate students in Solid-State Physics and Chemistry, Heterogeneous
Combustion, Materials Science and related areas, who are involved in the research
and development of CS-related methods for the synthesis of novel advanced
materials.


Chapter 1

ADVANCES AND CHALLENGES IN MODELING
COMBUSTION SYNTHESIS
1.1. APPROACHES TO MODELING NON-ISOTHERMAL
INTERACTION KINETICS DURING CS
Combustion synthesis (CS), or self-propagating high-temperature synthesis
(SHS), also known as solid flame, is a versatile, cost and energy efficient method
for producing refractory compounds (carbides, borides, nitrides, intermetallics,

complex oxides etc.) and advanced composite materials possessing fine-grain
structure and superior properties. Extensive research in this area was initiated by
A.G.Merzhanov in Chernogolovka, Moscow district, Russia, in mid 1960es [1,2],
who is internationally recognized as a pioneer of SHS. The advantages of CS
include short processing time, low energy consumption, high product purity due
to volatilization of impurities, and unique structure and properties of the final
products. Besides, CS can be combined with pressing, extrusion, casting and other
processes to produce near-net-shape articles [3-10]. Despite vast literature
available in this area, CS is still a subject of extensive experimental and
theoretical investigation.
Combustion synthesis can be carried out in the wave propagation mode, or
“true SHS”, and in the thermal explosion (TE) mode. In the former case, a
compact reactive powder mixture is ignited at one end to initiate an exothermic
reaction which propagates through the specimen as a combustion wave leaving
behind a hot final product [3-10]. In the latter case, a pellet is heated up at a
prescribed rate (typically 1-100 K/s) until at a certain temperature called the
ignition point, Tign, an exothermal reaction becomes self-sustaining and the
temperature rises to its final value, TCS, almost uniformly throughout the sample.


B. B. Khina

2

Typically, the value of Tign is close to the melting point of a lower-melting
reactant or to the eutectic temperature.
Examples of CS products are listed in Table 1.1, and characteristics of SHS
reactions in certain systems are presented in Table 1.2.
Table 1.1. Examples of compounds and materials produced by combustion
synthesis [3-14]

Type of material
Borides

Compounds and adiabatic combustion temperature, K (in brackets)
TiB2 (3190), TiB (3350), ZrB2 (3310), HfB2 (3320), VB2 (2670),
VB (2520), NbB2 (2400), NbB, TaB2 (2700), TaB, CrB2 (2470),
CrB, MoB2 (1500), MoB (1800), WB (1700), LaB6 (2800)
Carbides
TiC (3210), ZrC (3400), HfC (3900), VC (2400), Nb2C (2600),
NbC (2800), Ta2C (2600), TaC (2700), SiC (1800), WC, B4C,
Cr3C2, Cr7C3, Mo2C, Al4C3
Aluminides
Ni3Al, NiAl, Ni2Al3, TiAl, CoAl, Nb3Al, Cu3Al, CuAl, FeAl
Silicides
Ti5Si3 (2500), TiSi (2000), TiSi2 (1800), Zr5Si3 (2800), ZrSi
(2700), ZrSi2 (2100), WSi, Cr5Si3 (1700), CrSi2 (1800), Nb5Si3
(3340), NbSi2 (1900), MoSi2 (1900), V5Si3 (2260), TaSi2 (1800)
Intermetallics
NbGe, TiCo, NiTi
Sulfides and selenides MgS, MnS (3000), MoS2 (2900), WS2, TiSe2, NbSe2, TaSe2,
MoSe2, WSe2
Hydrides
TiH2, ZrH2, NbH2, CsH2
Nitrides
TiN (4900), ZrN (4900), VN (3500), HfN, Nb2N (2670), NbN
(3500), Ta2N (3000), TaN (3360), Mg3N2 (2900), Si3N4 (4300),
BN (3700), AlN (2900)
Carbonitrides
TiC-TiN, NbC-NbN, TaC-TaN, ZrC-ZrN
Complex oxides

Aluminates (YAlO2, MgAl2O4), niobates (NaNbO3, BaNb2O6,
LiNbO3), garnets (Y3Al5O12, Y3Fe5O12), ferrites (CoFe2O4,
BaFe2O4, Li2Fe2O4), titanates (BaTiO3, PbTiO3), molybdates
(BiMoO6, PbMoO4), high-temperature superconductors
(YBa2Cu3O7-x, LaBa2Cu3O7-x, Bi-Sr-Ca-Cu-O)
Ternary solid solutions TiB2-MoB2, TiB2-CrB2, ZrB2-CrB2, TiC-WC, TiN-ZrN, MoS2based on refractory
NbS2, WS2-NbS2
compounds
MAX phases
Ti2AlC, Ti3AlC2, Ti3SiC2
Cermets
TiC-Ni, TiC-Cr, TiC-Co, TiC-Ni-Cr, TiC-Ni-Mo, TiC-Fe-Cr, TiCCr3C2-Ni, TiC-Cr3C2-Ni-Cr, Cr3C2-Ni-Mo, TiB-Ti, WC-Co, TiCTiN-NiAl-Mo2C-Cr
Composites and
TiC-TiB2, TiB2-Al2O3, TiC-Al2O3 (2300), TiN-Al2O3, B4C-Al2O3,
functionally-graded
MoSi2-Al2O3 (3300), MoB-Al2O3 (4000), Cr3C2-Al2O3, 6VN5Al2O3 (4800), ZrO2-Al2O3-2Nb, AlN-BN, AlN-SiC, AlN-TiB2,
materials
Si3N4-TiN-SiC, sialons (SiAlOxNy)


Advances and Challenges in Modeling Combustion Synthesis

3

Table 1.2. Features of combustion synthesis waves for certain typical
reactions [3-18]
Type of interaction

Reaction

Experimental
combustion
temperature, °C
≈2500

Solid-solid (formation of
a carbide) via a transient
liquid phase (melting of a
metallic reactant) [17,18]
Solid-solid (formation of
a complex oxide) via a
transient liquid phase
with participation of an
oxidizing gas
Solid-gas with or without
melting of a metallic
reactant [13]
Solid-solid (formation of
a carbide) via
intermediate gas-transport
reactions [12]
Liquid-liquid in organic
systems with the
formation of a solid
product [15]

Ti (solid→liquid) + C
(solid) → TiC (solid)

3-4

3Cu (solid→liquid) +
2BaO2 (solid) +
(1/2)Y2O3 (solid) + O2
(gas) → YBa2Cu3O7-x
(solid)
Ti (solid→liquid) +
(1/2)N2 (gas) → TiN
(solid)
Ta (solid) + C (solid)
→ TaC (solid)

≈1000

0.2-0.5

≈1600-2000

0.1-0.2

≈2600

0.5-2

C4H10N2 (liquid,
piperazine) + C3H4O4
(liquid, malonic acid)
→ C7H14N2O4 (solid,
salt)

155

0.06-0.15

Combustion wave
velocity, cm/s

The unique features of the obtained products, e.g., high purity, small and
uniform grain size, etc., are ascribed to extreme conditions inherent in CS, which
may bring about unusual reaction routes: (i) high temperature, up to 3500 °C, (ii)
a high rate of self-heating, up to 106 K/s, (iii) steep temperature gradient in SHS
waves, up to 105 K/cm, (iv) rapid cooling after synthesis, up to 100 K/s, and (v)
fast accomplishment of conversion, from about 1 s to the maximum of 10 s [3-6].
It should be noted that traditional furnace synthesis of refractory compounds
requires a much longer time, ~1-10 h, for the same initial composition, particle
size and close final temperature. It has been demonstrated experimentally [16-23]
that in many systems phase and structure formation during CS proceeds via
uncommon interaction mechanisms from the point of view of the classical
Physical Metallurgy [24,25].
Modeling and simulation traditionally play in important part in the
development of CS and CS-related technologies (see reviews [3-5,11,26-29] and


B. B. Khina

4

references cited therein). An adequate mathematical model is supposed to
describe both heat transfer in a heterogeneous reactive medium and the interaction
kinetics, which is responsible for heat release during CS.

In modeling CS, a quasi-homogeneous, or continual model [30,31], which is
based on classical combustion theory, is widely used. Heat transfer, which is
considered on the volume-averaged basis, and the reaction rate in a sample are
described as follows:
ρcp ∂T/∂t = ∇(λ∇T) + Q ∂η/∂t

(1.1)

∂η/∂t = (1−η)n exp(–mη) k exp(−E/RT),

(1.2)

where T is temperature, ρ is density, cp is heat capacity, λ is thermal conductivity,
Q is the heat release of exothermal reactions, η is the degree of chemical
conversion (from 0 in the unreacted state to 1 for complete conversion), R=8.314
J mol–1K–1 is the universal gas constant, n (the reaction order), k (preexponential
factor) and m are formal parameters and E is the activation energy; term Q∂η/∂t
denotes the heat release rate.
The thermal structure of a combustion wave according to Zeldovich and
Frank-Kamenetskiy [32] is shown schematically in Figure 1.1. Typically, three
zones are distinguished: (i) the preheating zone where almost no reaction occurs
and the main processes are heat and mass transfer accompanied with evaporation
of volatile impurities; in Russian literature it is often termed as “the Michelson
zone” after V.A.Michelson (1860-1927) who described the temperature profile
ahead of the moving combustion front [32], (ii) zone of thermal reaction where
the conversion degree η sharply increases and the heat release rate reaches its
maximum and starts decreasing while the temperature almost reaches the
adiabatic value, and (iii) the after-burn, or post-reaction zone where the
interaction terminates. The latter zone is characterized by a slow increase in both
conversion degree and temperature, which finally attain their maximal values η=1

and T=Tad, and the heat release rate, Q∂η/∂t, falls down to zero. The temperature
of the reaction front, Tf, corresponds to the onset of fast thermal reaction. In
regard to combustion synthesis of materials, it is believed that complex
heterogeneous reactions, which may proceed via uncommon (fast) mechanisms
and are responsible for major heat release, occur in the thermal reaction zone
while the after-burn zone, where the heat release rate is minor, is dominated by
the processes bringing about the formation of final structure of the product, such
as Ostwald ripening, recrystallization etc.


Advances and Challenges in Modeling Combustion Synthesis

5

Figure 1.1. Schematic of the thermal structure of a combustion wave.

The approach formulated in Eqs. (1.1) and (1.2) permitted modeling dynamic
regimes of SHS, e.g., oscillating [30] and spin combustion [33,34]. It was also
used for studying the effect of intrinsic stochasticity of heterogeneous reactions,
which can be attributed to a difference in the surface morphology, impurity
content and hence reactivity of solid reactant particles, on the dynamic behavior
of a solid flame for a one-stage [35] and multi-stage reaction [36] employing the
cellular automata method.
It should be outlined that this model is not linked to any process-specific
phase formation mechanism and hence is referred to as a formal one. When
applying this approach to modeling CS in a particular system, the value of the
most important model parameter, viz. activation energy E, is supposed to
correspond to the apparent activation energy of the CS as a whole. The latter is
determined from experimental graphs “the combustion wave velocity vs.
temperature” plotted in the Arrhenius form, and in its physical meaning

corresponds to a real rate-limiting stage of phase formation during CS, which may
be different in different temperature ranges. For example, below the melting
temperature, Tm, of a metallic reactant E always refers to solid-state diffusion in
the product while at T>Tm it can refer to processes in the melt (diffusion or
crystallization) [37]. This method for choosing the E value was used when
studying numerically the conditions of arresting a high-temperature state of
substances in the SHS wave by fast cooling for the cases of a one-stage [38] and
two-stage exothermal reaction [39].


6

B. B. Khina

In recent papers [40,41], this formal model [see Eqs. (1.1) and (1.2)] was
employed for studying the SHS of a NiTi shape memory alloy. The activation
energy used in calculations was E=113 kJ/kg, which is equivalent to 12.05 kJ/mol
(because the molar mass of NiTi is 106.6 g/mol). This is an extraordinary low
value for a reaction in a condensed system and can correspond only to diffusion in
a transient melt formed in the CS wave. However, according to reference data
[42], the activation energy for diffusion in some pure liquid metals is the
following: Li, E=12 kJ/mol; Sn, E=11.2 kJ/mol; Zn, E=21.3 kJ/mol; Cu, E=40.7
kJ/mol; Fe, E=51.2 kJ/mol. Thus the value of E used for calculations in [40,41] is
close to that for diffusion in low-melting metals such as Li or Sn, and is by the
factor of 4 lower than for iron whose melting point, Tm, lies between Tm of Ni and
Ti (the activation energy for diffusion in liquid metals is known to be proportional
to Tm [43]). All the more, this E value is incomparably lower than a typical
activation energy for diffusion in intermetallic compounds. Hence in this case the
most important parameter of the formal model, E, appears to be physically
meaningless.

Recently, new features of SHS were observed experimentally [44-47]. First,
microscopic high-speed video recording [44,45] and photographing [46]
demonstrated micro-heterogeneous nature of SHS which revealed itself in the
roughness of the combustion wave front, chaotic oscillations of the local flame
propagation rate and new dynamic behaviors such as relay-race, scintillation and
quasi-homogeneous patterns. Second, the formation of non-equilibrium structure
and composition of SHS products was examined experimentally and interpreted
qualitatively in terms of relationships between characteristic times of reaction tr,
structuring ts and cooling tc [47]. These features were attributed to two main
factors: inhomogeneous heat transfer in the charge mixture and a specific reaction
mechanism [46].
These results gave rise to new, heterogeneous models [48-51] involving heat
transfer on the particle-to-particle basis [48-50] and percolation phenomena in a
system of chaotically distributed reactive and inert particles [51]. However, in
these models the traditional formal kinetics for a thermal reaction [Eq. (1.2)] was
employed. Thus, an urgent and still unresolved problem in CS is an adequate
description of fast interaction kinetics in a unit reaction cell containing particles or
layers of dissimilar reactants whose composition corresponds to the average
composition of a charge mixture.
The most widely used kinetic model, which is connected to a particular phase
forming mechanism, is a “solid-state diffusion-controlled growth” concept first
applied to CS in [52] for planar symmetry and in [53] for spherical symmetry of
an elementary diffusion couple. As in a charge mixture there are contacts of


Advances and Challenges in Modeling Combustion Synthesis

7

dissimilar particles, a layer of an intermediate or final solid product forms upon

heating thus separating the initial reactants. The growth rate of the reaction
product and associated heat release necessary for sustaining combustion is
controlled by solid-state diffusion through this layer. Then, the diffusion-type
Stefan problem is formulated instead of Eq. (1.2). However, as demonstrated
below in more detail, in most cases modeling was performed not with real
diffusion data, which are known for many refractory compounds, but using either
dimensionless coefficients varied in a certain range or fitting parameters chosen to
match the calculated and measured results of the SHS temperature profile and
velocity. It should be emphasized that Diffusion in Materials is a well-developed
cross-disciplinary area within Materials Science and Solid State Physics, and the
diffusion parameters for many of the phases produced by CS (carbides, nitrides,
intermetallics etc.) have been measured experimentally at different temperatures,
and these data are supposed to be used in modeling. Besides, in most of the CSsystems fast interaction begins after fusion of a lower-melting-point reactant [35,31] but within this approach melting does not alter the phase layer sequence in
an elementary diffusion couple [52,53].
A number of experimental results obtained by the combustion-wave arresting
technique in metal-nonmetal (Ti-C [17,18], (Ti+Ni+Mo)-C [19], Mo-Si [20]) and
metal-metal (Ni-Al [21,23]) systems gave rise to an qualitative notion of a nontraditional phase formation route. It involves dissolution of a higher-melting-point
reactant (metal or non-metal) in the melt of a lower-melting-point reactant and
crystallization of a final product from the saturated liquid.
Besides, there is much controversy over the presence of an intermediate solid
phase in the dissolution-precipitation route. In [21] it is concluded that during
SHS in the Ni-Al system, solid Ni dissolves in liquid Al through a solid interlayer
separating aluminum from nickel, which agrees with the phase diagram. In this
case, the rate-limiting stage is solid-state diffusion across this layer. But in [23]
for the same system it is found that above 854 °C a solid interlayer between nickel
and molten Al is absent; then the overall interaction during CS is controlled by
either diffusion in the melt or crystallization kinetics.
Such a situation is considered in recent models [54-59], where a solid reactant
(nickel [54-56] or carbon [57-59]) dissolves directly in the liquid based on a
lower-melting component (Al and Ti, respectively) and product grains (NiAl and

TiC, correspondingly) precipitate from the melt; the rate-limiting stage is liquidphase diffusion [54-56] or crystallization kinetics [57-59]. However, within these
approaches the fundamental problem of the existence of a thin solid-phase
interlayer at the solid/liquid interface is not discussed nor a criterion is obtained
for transition between the solid-state diffusion-controlled mechanism and the


8

B. B. Khina

dissolution-precipitation route with or without a thin interlayer. Hence, the
applicability limits of the existing modeling approaches have not been clearly
determined so far. The role of high heating rates, which are intrinsic in CS, in
most of the models is not accounted for in an explicit form.
Thus, adequate description of the interaction kinetics in condensed
heterogeneous systems in non-isothermal conditions of CS is an urgent problem in
this area of science and technology, and the absence of a comprehensive model
hinders the development of new CS-based processes and novel advanced
materials.
Hereinafter the situation where a reaction between condensed reactants
proceeds through a solid layer, i.e. solid reactant (C for the Ti-C system or Ni for
the Ni-Al system)/solid final or intermediate product (TiC or one of intermetallics
of the Ni-Al system, respectively)/liquid (Ti or Al melt), will be provisionally
called “solid-solid-liquid mechanism” since the interaction occurs at both
solid/solid and solid/liquid interface. This term will be used both for the “solidstate diffusion-controlled growth” pattern where the product layer is growing and
for dissolution-precipitation route where the interlayer remains very thin. As the
diffusion coefficient in a melt is much higher than in solids, the rate-limiting stage
in this mechanism is diffusion across the solid interlayer. The second route, viz.
dissolution-precipitation without an interlayer, can be referred to as “solid-liquid
mechanism” since the interaction of condensed reactants (solid C or Ni with

molten Ti or Al, respectively) occurs at the solid/liquid interface while the product
(TiC or NiAl) crystallizes from the melt. However, up to now the solid-liquid
mechanism has not been validated theoretically, nor the applicability limits of the
solid-solid-liquid mechanism based on solid-state diffusion kinetics have ever
been determined with respect to strongly non-isothermal conditions typical of CS.
Thus, the main goal of this work is to develop a system of relatively simple
estimates and evaluate the applicability limits of the “solid-solid-liquid
mechanism” approach to modeling CS and determine criteria for a change of
interaction routes basing the calculations on experimental data to a maximum
possible extent [60,61]. Below, a brief discussion of the diffusion concept of CS is
presented. Then, calculations for particulars system, viz. Ti-C and Ni-Al, are
performed using available experimental data on both the diffusion coefficients in
the growing phase and thermal characteristics of CS.
The choice of these binary systems for a modeling study is motivated by the
following reasons. First, those are typical SHS systems which have been a subject
of extensive experimental investigation (see reviews [3-11,16] and references
cited therein). Second, the synthesis products, viz. TiC and NiAl, have a wide
industrial application because of their unique physical and mechanical properties.


Advances and Challenges in Modeling Combustion Synthesis

9

Hence a large number of parameters needed for numerical calculations can be
found in literature. Third, both of these substances are typical representatives of
wide classes of chemical compounds that have different properties connected with
their intrinsic structural features. Titanium carbide is a typical interstitial
compound (like many carbides, nitrides and certain borides) wherein the
diffusivities of constituent atoms, Ti and C, differ substantially. Hence the growth

of TiC in an elementary diffusion couple Ti/TiC/C during CS is dominated by the
diffusion of carbon atoms in the TiC layer and proceeds mainly at the Ti/TiC
interface. The experimentally measured parameters such as the chemical diffusion
coefficient or the parabolic growth-rate constant for TiC are connected with the
partial diffusion coefficient of carbon in this compound. Nickel monoaluminide is
a typical substitutional compound with an ordered crystalline structure (like many
intermetallics) where the rates of diffusion of Ni and Al atoms are comparable.
Thus its growth during CS occurs at both sides of a NiAl layer and can be
characterized by a single parameter, namely the interdiffusion coefficient, which
is measured experimentally.
For the Ti-C system, different situations are considered that can arise during
CS within the frame of the above concept and, wherever possible, a quantitative
and/or qualitative comparison between the outcome of calculations and
experimental results is drawn. Emphasis is made on the structural characteristics
of the CS product, titanium carbide, that emerge from this approach. The
conditions for a change of the geometry of a unit reaction cell in the SHS wave
due to melting of a metallic reactant (titanium) are analyzed and a
micromechanistic criterion for the changeover of interaction pathways is derived.
For the Ni-Al system, calculations within the frame of the diffusion-controlled
growth kinetics are performed taking into account both the growth of the product
phase, NiAl, and its dissolution in the parent phases (solid or liquid Ni and molten
Al) due to variation of solubility limits with temperature according to the
equilibrium phase diagram. Finally, the “solid-liquid mechanism” concept for CS
is justified and phase-formation mechanism maps for these two systems in
strongly non-isothermal conditions are plotted.


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B. B. Khina

1.2. BRIEF REVIEW OF DIFFUSION-BASED
KINETIC MODELS OF CS
The interaction kinetics controlled by solid-state diffusion was used for
numerical [52,53,62-69] and analytical [70] study of CS for the case of planar
diffusion couples (alternating lamellae of reactants) [52,63,64,66,70] and
spherical symmetry (growth of a product layer on the surface of a spherical
reactant particle) [53,65,67-69].
Inherent in this concept are two basic assumptions: (i) the phase composition
of the diffusion zone between parent phases corresponds to the isothermal crosssection of an equilibrium phase diagram, i.e. the nucleation of product phases
occurs instantaneously over all contact surfaces and (ii) the interfacial
concentrations are equal to equilibrium values. This results in the parabolic law of
phase layer growth [71-73].
It should be noted that in many diffusion experiments the phase layer
sequence deviates from equilibrium: the absence of certain phases was observed
in solid-state thin-film interdiffusion [74,75] and in the interaction of a solid and a
liquid metal (e.g., Al) [76,77]. These phenomena were ascribed to a reaction
barrier at the interface of contacting phases [78] without considering the
nucleation rate of a new phase. The effect of a nucleation barrier was examined
theoretically using the thermodynamics of nucleation [79,80] and the kinetic
mechanism of phase formation in the diffusion zone [81], and it was shown that in
the field of a steep concentration gradient the formation of an intermediate phase
is suppressed [79-81]. This effect has never been considered in the diffusion
models of CS. As in the theory of diffusion-controlled interaction in solids the
nucleation kinetics is not included and it is assumed that critical nuclei of missing
phases continuously form and dissolve [72,73], this qualitative concept is
sometimes used in interpreting the results of CS [21].
It will be fair to say that deviation of phase-boundary concentrations from
equilibrium due a reaction barrier was examined qualitatively for SHS [64] in the
case of planar geometry. This effect is noticeable only in the low-temperature part

of the SHS wave, and at high temperatures a strong barrier can only slightly
decrease the combustion velocity [64]. Also, the influence of such barrier on selfignition in the Ni-Al system at low heating rates, dT/dt<60 K/min, was studied
quantitatively using experimental data on both thin-film interdiffusion in the
NiAl3 compound [82], which is the first phase to form in Ni-Al diffusion couples,
and bulk diffusion data [83]. Similar calculations were performed using an
experimental temperature profile of SHS to determine the NiAl3 layer thickness


Advances and Challenges in Modeling Combustion Synthesis

11

formed below the melting point of aluminum Tm(Al)=660 °C [67]. At a thick
NiAl3 layer (low heating rates) the reaction barrier is of little significance, but it
can slow down the interaction for thin layers (higher heating rates) [83,67]: e.g., at
dT/dt > 35 K/min the formation of the primary product can be suppressed [67].
But, as noted in [83,67], these results refer not to the SHS itself but only to a
preliminary stage (i.e. the preheating zone of the SHS wave) because fast
interaction begins at T>Tm(Al), the combustion temperature reaches 1400 °C and
the final product is NiAl [67].
It should be outlined that in many works using the diffusion model of CS the
calculations were performed with dimensionless (relative) parameters varied in a
certain range. A known or estimated value of the activation energy for diffusion in
one of the phases was used only as a scaling factor and thus the results obtained
revealed only qualitative characteristics of the process [52,53,62,66]. Besides,
many of the modeling attempts [52,53] did not account for a change in the spatial
configuration of reacting particles due to melting and spreading of a metallic
reactant. The effect of melting was reduced to a change of interfacial
concentrations and the ratio of diffusion coefficients in contacting phases [62].
In more recent papers [67,68], the parameter values (the activation energy E

and preexponent D0) used for calculating the diffusion coefficient in a growing
phase were presented. However, those were not the real values measured in
independent works on solid-state diffusion but merely fitting parameters
calculated from the characteristics of CS. For example, the formation of NiAl
above 640 °C was modeled using D0=4.8×10–2 cm2/s and E=171 kJ/mol [67]. As
noted in [67], this E value was the experimentally determined activation energy
for the CS process as a whole. Then the diffusion coefficient in NiAl at T=1273 K
is D = D0exp(−E/RT) = 4.6×10−9 cm2/s. Let’s compare it with experimental data
on reaction diffusion in the Ni-Al system. For NiAl, D=(2.5–3.6)×10−10 cm2/s at
T=1273 K [84]. The parameters for interdiffusion in this phase are E=230 kJ/mol
and D0=1.5 cm2/s [85], hence at T=1273 K D=5.4×10−10 cm2/s. Thus, the
diffusion coefficient used in modeling SHS exceeds the experimental value by an
order of magnitude.
SHS wave in the Ti-Al system with the Ti-to-Al molar ratio of 1:3 in the
charge mixture was modeled using E=200 kJ/mol and D0=4.39 cm2/s for phase
TiAl3 [68]. This E value was obtained from experiments on combustion synthesis
using Arrhenius plots, and D0 was chosen to match the calculated and measured
results of the propagation speed. Again, these values refer to the SHS wave as a
whole but not to interdiffusion in TiAl3. However, experimental data on SHS of
TiAl3 for the same starting composition, which were analyzed using the classical


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B. B. Khina

combustion model [see Eqs. (1.1),(1.2)], gave a substantially higher activation
energy: E=483 kJ/mol [86]. If solid-state diffusion in the TiAl3 layer is really the
rate-limiting stage of the process, then the values of apparent activation energy
ought to agree (within an experimental error) regardless of the particular form of a

model.
Diffusion coefficients are measured experimentally within a rather wide
margin of error using a variety of techniques, and typically various methods yield
different values. But since diffusion parameters for many refractory compounds,
which can be produced by combustion synthesis, can be found in literature, it
appears possible to verify the validity of the diffusion-based kinetic model of SHS
employing a somewhat opposite approach: estimating the product layer growth
and heat release using the experimental characteristics of SHS and independent
diffusion data. The models, parameter values and results of simulations for two
classical CS-systems, viz. Ti-C and Ni-Al, will be considered in more detail in the
subsequent chapters.