Combustion_Synthesisof Advanced Materials

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CHEMISTRY RESEARCH AND APPLICATIONS SERIES

COMBUSTION SYNTHESIS OF

ADVANCED MATERIALS

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CHEMISTRY RESEARCH AND APPLICATIONS SERIES

COMBUSTION SYNTHESIS OFADVANCED 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 2010620.1'43--dc22

2009052731

Published by Nova Science Publishers, Inc. New York


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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


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CONTENTS

Preface xi

Chapter 1 Advances and Challenges in Modeling Combustion

Synthesis 1

Chapter 2 Modeling Diffusion-Controlled Formation

of TiC in the Conditions of CS 13Chapter 3 Modeling Interaction Kinetics in the CS

of Nickel Monoaluminide 39

Chapter 4 Analysis of the Effect of Mechanical

Activation on Combustion Synthesis 75

References 93

Index 105


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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, non-

equilibrium 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 non-

isothermal 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 toCS;

modeling of phase formation kinetics for two typical SHS reactions,Ti+CTiC (CS of an interstitial compound) and Ni+AlNiAl (CS of an


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B. B. Khinaxii

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 systemsand phase-formation mechanism maps;

analysis of the physicochemical mechanism of the experimentally knownstrong 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.


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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 aprescribed 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.


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B. B. Khina2

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 Compounds and adiabatic combustion temperature, K (in brackets)

Borides 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, WSe2Hydrides 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

based on refractory

compounds

TiB2-MoB2, TiB2-CrB2, ZrB2-CrB2, TiC-WC, TiN-ZrN, MoS2-

NbS2, WS2-NbS2

MAX phases Ti2AlC, Ti3AlC2, Ti3SiC2

Cermets TiC-Ni, TiC-Cr, TiC-Co, TiC-Ni-Cr, TiC-Ni-Mo, TiC-Fe-Cr, TiC-

Cr3C2-Ni, TiC-Cr3C2-Ni-Cr, Cr3C2-Ni-Mo, TiB-Ti, WC-Co, TiC-

TiN-NiAl-Mo2C-Cr

Composites and

functionally-graded

materials

TiC-TiB2, TiB2-Al2O3, TiC-Al2O3 (2300), TiN-Al2O3, B4C-Al2O3,

MoSi2-Al2O3 (3300), MoB-Al2O3 (4000), Cr3C2-Al2O3, 6VN-

5Al2O3 (4800), ZrO2-Al2O3-2Nb, AlN-BN, AlN-SiC, AlN-TiB2,

Si3N4-TiN-SiC, sialons (SiAlOxNy)


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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

Combustion wave

velocity, cm/s

Solid-solid (formation of

a carbide) via a transient

liquid phase (melting of a

metallic reactant) [17,18]

Ti (solidliquid) + C(solid) TiC (solid)

2500 3-4


a complex oxide) via atransient liquid phase

with participation of an

oxidizing gas

3Cu (solidliquid) +

2BaO2 (solid) +(1/2)Y2O3 (solid) + O2

(gas) YBa2Cu3O7-x(solid)

1000 0.2-0.5

Solid-gas with or without

melting of a metallic

reactant [13]

Ti (solidliquid) +(1/2)N2 (gas) TiN(solid)

1600-2000 0.1-0.2


a carbide) via

intermediate gas-transportreactions [12]

Ta (solid) + C (solid)

TaC (solid)2600 0.5-2

Liquid-liquid in organic

systems with the

formation of a solid

product [15]

C4H10N2 (liquid,

piperazine) + C3H4O4

(liquid, malonic acid)

C7H14N2O4 (solid,salt)

155 0.06-0.15

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, whichmay bring about unusual reaction routes: (i) high temperature, up to 3500 C, (ii)a high rate of self-heating, up to 10

6K/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 viauncommon 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


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B. B. Khina4

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:

cp T/t = (T) + Q/t (1.1)

/t = (1)nexp(m)kexp(E/RT), (1.2)

where T is temperature, is density, cp is heat capacity, is thermal conductivity,Q is the heat release of exothermal reactions, is the degree of chemicalconversion (from 0 in the unreacted state to 1 for complete conversion), R=8.314

Jmol1

K1

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/tdenotes 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 sharply increases and the heat release rate reaches itsmaximum and starts decreasing while the temperature almost reaches theadiabatic 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 =1and T=Tad, and the heat release rate, Q/t, falls down to zero. The temperatureof 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 zonewhile 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.


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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 tocorrespond 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].


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B. B. Khina6

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 andTi (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 theroughness 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


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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 thediffusion 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 CS-

systems fast interaction begins after fusion of a lower-melting-point reactant [3-

5,31] 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]) andmetal-metal (Ni-Al [21,23]) systems gave rise to an qualitative notion of a non-

traditional 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 thiscase, 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 nickeland 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 liquid-phase 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


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B. B. Khina8

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 forthe 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 solid-

state 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 simpleestimates 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 thefollowing 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.


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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 isa 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 andexperimental 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 moltenAl) 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. Khina10

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 cross-section 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 issometimes 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 self-

ignition in the Ni-Al system at low heating rates, dT/dt


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formed below the melting point of aluminum Tm(Al)=660 C [67]. At a thickNiAl3 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 andthe 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 growingphase 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=4.8102

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) = 4.6109

cm2/s. Lets compare it with experimental data

on reaction diffusion in the Ni-Al system. For NiAl, D=(2.53.6)1010 cm2/s atT=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=5.41010 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 measuredresults 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. Khina12

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 growthand 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.


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Chapter 2

MODELING DIFFUSION-CONTROLLED

FORMATION OF TIC IN THE CONDITIONS OF

CS

2.1.INTRODUCTION

CS in the Ti-C system was a subject of extensive theoretical and experimentalstudies [53,17,18,62,69] because of industrial significance of the product, titanium

carbide, which is used for a wide range of applications because of its high melting

point, hardness and chemical stability. It is a suitable candidate for theoretical

investigation for the following reasons: (i) the Ti-C phase diagram [87] (see

Figure 2.1) contains only one binary compound TiC whose melting temperature

Tm(TiC)=3423 K exceeds the experimental SHS temperature TCS=3083 K [88]

and (ii) numerous diffusion data for titanium carbide are available in literature

[89-91]. We consider the case of spherical symmetry which better fits a typicalconfiguration of reacting particles in CS. With respect to the phase diagram, here

the solid-solid-liquid mechanism [situation C(solid)/TiC(solid)/ Ti(liquid)] is

quasi-equilibrium and the solid-liquid mechanism [situation C(solid)/Ti(liquid)] is

truly non-equilibrium.

Lets consider solid-state diffusion-controlled formation of the product,

titanium carbide, during heating of the Ti+C charge mixture in the SHS wave.

Typical particle radii are 5 to 100 m for Ti, about 0.1 m for carbon black and 1

to 30 m for milled graphite [17,18,69,88]. Two scenarios with a different

geometry of a unit reaction cell are examined: (1) a solid Ti particle surrounded

by carbon particles in a stoichiometric mass ratio at temperatures below the Ti

melting point, Tm(Ti)=1940 K [Figure 2.2 (a and d)], and (2) a solid carbon

particle surrounded by liquid titanium at T>Tm(Ti) [Figure 2.2 (c and e)].


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B. B. Khina14

Figure 2.1. The equilibrium Ti-C phase diagram [87] and experimental SHS temperature.


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Modeling Diffusion-Controlled Formation of TiC 15

Figure 2.2. Schematic of an elementary reaction cell in the SHS wave in the Ti-C system

(a and c) and corresponding concentration profiles for solid-state diffusion (d and e) [60]:

(a and d) growth of the TiC layer on the surface of a titanium particle at T


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B. B. Khina16

, (2.1)

,(2.2)

where D is the chemical diffusion coefficient in TiC, which is usually associated

with the diffusion coefficient of carbon in the carbide layer, cC is the massconcentration of carbon, R1(t) is the current position of the TiC/Ti interface, R2 is

the outer radius of the Ti particle, and c01, c

021 and c

023 are the interface

concentrations [Figure 2.2 (d)] according to the equilibrium phase diagram. The

boundary (at r=R2) and initial conditions to Eq. (2.1) are

cC(t, R2) = c023, cC(t, r=0) = c

01, cC[t, R1(t)] = c

021, R1(t=0) = R2. (2.3)

In [62,63] the Stefan-type boundary condition, Eq. (2.2), was posed at both

Ti/TiC and TiC/C interfaces. We should outline that in interstitial compounds

such as nitrides, carbides and many borides, the partial diffusion coefficient of

nonmetal species exceeds that of metal atoms by several orders of magnitude,

which is due to the interstitial diffusion mechanism. Hence, the growth of

titanium carbide occurs at the Ti/TiC interface and is controlled by the diffusion

of C atoms across the TiC layer. But at the C/TiC interface the growth of TiC at

the expense of graphite, which requires the supply of Ti atoms, cannot occur.

Thus, the first-kind boundary condition, cC(t, R2)=c0

23 [see Eq. (2.3)] is used for

the C/TiC interface, which actually denotes an ideal diffusion contact of carbon

particles with the outer surface of the growing TiC layer due to fast surface

diffusion of the C atoms from the C/TiC contact spots.

2.3.SCENARIO 2:GROWTH OF A TICLAYER ON THE

SURFACE OF SOLID CARBON PARTICLES

The physical background for scenario 2 [Figure 2.2 (c)] is the following.

Spreading of molten titanium towards solid carbon in the SHS wave was observed

experimentally [92,93]. Since it is accompanied with chemical interaction, for a

sufficiently small C particle size the spreading velocity is not the rate-limiting

=

r

cr

rr

D[T(t)]

t

cC2

2

C

(t)R

C10

1

0

21

1

r

cD[T(t)]

dt

dR)cc(

=


30/123


stage [93,94]. Hence we consider that at TTm(Ti) the carbon particles arecompletely enveloped with liquid titanium, and a thin TiC layer forms at the Ti/C

interface separating the parent phases. The product growth occurs at the

Ti(melt)/TiC interface, i.e. outwards, due to diffusion of carbon atoms across the

TiC layer [Figure 2.2 (e)]. Since the diffusion coefficient of carbon in the melt is

at least an order of magnitude higher than in the carbide (Table 2.1), it is

reasonable to presume that the titanium melt is saturated with carbon (otherwise

the TiC layer will be dissolving). Then the boundary condition at r=R0 to

diffusion equation (2.1) and initial conditions to Eqs. (2.1),(2.2) look as

cC(t, R0) = c023, cC(t, r>R1) = c01, R1(t=0) = R0, (2.4)

where R0=const is the initial radius of the carbon particle.

2.4.DIFFUSION DATA FOR TIC

The parameters for calculating the diffusion coefficient in TiC in the

Arrhenius form

D = D0 exp[E/RT(t)] (2.5)

are listed in Table 2.1, wherein the experimental data available in literature [95-

103] for different temperature intervals T are collected. It is seen that differentdata sources give substantially different values of both activation energy and

preexponential factor, thus it seems necessary to select the parameters values

suitable for numerical calculations. Since the extrapolation of D to the wholetemperature range of SHS may bring about overestimated values, the diffusion

coefficients in TiC calculated at T=Tm(Ti) and TCS must be compared with the

diffusion coefficient in molten titanium: it is obvious that the value of D in a solid

metal-base refractory compound is at least an order of magnitude lower than in a

melt of the corresponding metal.

Because of the absence of experimental data, the diffusion coefficient of C in

molten Ti is estimated by a simple Stokes-Einstein (or Sutherland-Einstein)

formula, which was used for assessing the diffusion parameters of C, N, O and Hin liquid metals (Fe, Co, Ni, etc.) [104,105]

Di(m)

= kBT/(nai), (2.6)


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B. B. Khina18

where numerical factor n=4 for substantially differing atomic radii of the melt

components and n=6 for close radii, ai is the atomic radius of i-th diffusing

species in the melt, =m is the dynamic viscosity, is the kinematic viscosityand m is the liquid-phase density. For carbon atoms, the covalent radius isaC=0.077 nm [106]. The density of molten Ti is m=4.11 g/cm

3[107]. A typical

value of the kinematic viscosity for such liquid metals as Al, Fe, Co, Ni et al. near

the melting point is (0.5-1)102 cm2/s [106]. For liquid titanium saturated withcarbon, =0.94102 cm2/s at T=Tm(Ti) [107], then DC

(m)(Tm) (4.87.2)10

5

cm2/s. For higher temperatures, the value =1.03102 cm2/s at T=2220 K is

known [107]; using it at T=TCS gives DC(m)

(TCS) (6.910.4)105

cm

2

/s. Itshould be noted that since Eq. (2.6) doesnt account for chemical interaction in the

melt, which may be substantial for the Ti-C system, these DC(m)

values are upper

estimates. Then the values of diffusion coefficients in TiC, which are close to or

higher than the upper estimate of DC(m)

(TCS), are excluded from consideration

(lines 10 to 14 in Table 2.1).

Table 2.1. Diffusion data for titanium carbide

Species No. D0, cm /s E,kJ/mol

T, K D[Tm(Ti)],cm

2/s

D(TCS),cm

2/s

Refs. Note

C

1 5102 235.6 2073-2973 2.3108 5.1106 [90,91,95,101]2 6.98 398.7 1723-2973 1.31010 1.23106 [89,95,96,98]

(TiC0.97)3 10 438.9 1873-2573 1.51011 3.7107 [95,96] (TiC0.9)4 45.44 447.3 1723-2553 4.11011 1.2106 [96,98]

(TiC0.887)

5 114 460.2 2018-2353 4.61011 1.8106 [96,99](TiC0.67)

6 0.1 259.4 1553-17731.010

8

4.0106

[90,91]7 6.5102 269.9 1673-1973 3.5109 1.7106 [90,102]8 4.2102 307.1 1983-2573 2.31010 2.6107 [90,91] (TiC0.9)9 0.48 328.42 1473-2023 6.91010 1.3106 [103] (TiC1.0)10 99.48 328.42 1473-2023 1.4107 2.7104 [103] (TiC0.5) D(TCS) >

DC(m)

(TCS)11 77.8 338.9 1473-1673 5.8108 1.4104 [89] D(TCS) >

DC(m)

(TCS)12 220 405.8 2200-2600 2.6109 2.9105 [100] (TiCx,

x=0.86-0.91)D(TCS) DC

(m)(TCS)

13 370 410.0 2200-2600 3.4109 4.1105 [100] (TiCx,

x=0.86-0.91)

D(TCS)

DC(m)

(TCS)14 1.31103 347.3 1173-1473 5.8107 1.7103 [90] D(TCS) >>

DC(m)

(TCS)

Ti 4.36104 736.4 2193-2488 6.51016 1.5108 [90,97] (TiCx,x=0.67-0.97)

DTi


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2.5.TEMPERATURE OF THE

REACTION CELL IN THE SHSWAVE

Self-heating from ambient temperature, T0, to TCS during the combustion

synthesis is due to the adiabatic heat release of chemical reactions which are

almost accomplished when maximal temperature is reached, and in the after-burn

zone (at TTCS) only coalescence and sintering of the product particles occur withminor heat release [3-5]. Hence calculations of the product layer thickness and

relevant heat release should be done in the time interval [0, tCS] corresponding to

the attainment of TCS.To perform calculations, we have to know the time dependence of

temperature in the reaction cell, T(t). We consider a steady-state combustion

regime. For a low-temperature portion of the SHS wave, T0


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B. B. Khina20

2.3, x=0 corresponds to the melting point of Ti, consequently the heating time

from Tm(Ti) to TCS is tCS = tCStm = x(TCS)/vSHS.

Figure 2.3. Temperature profile of the SHS wave in the Ti-C system [60]: 1, analytical

solution for a steady-state SHS wave [Eq. (2.7)] for TTm(Ti); 2, cubic-splineapproximation of experimental curve [88] in the range Tm(Ti)TTCS.

2.6.ADIABATIC HEAT RELEASE IN THE REACTION CELL

Having the heating law of the reaction cell, we can calculate the heat release

due to diffusion-controlled phase layer growth in non-isothermal conditions and

thus the maximal temperature attained, and then compare it with experimental

TCS. In adiabatic conditions, a heat balance equation for the formation of

stoichiometric TiC1.0 is written as:

H0298(TiC1.0)mTiC(t) = mTiC(t) + mC(t) +

mTi(t) , (2.9)

dT)TiC(cadT

298

p dT)C(cadT

298

p

+ )Ti(H)]Ti(TT[IdT)Ti(c mmadT

298p

ad


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where Tad is the adiabatic combustion temperature, cp(i) is heat capacity, mi(t) is a

current mass of i-th substance, H0298(TiC1.0) = 3.077 kJ/g is the standardenthalpy of TiC1.0 [110], Hm(Ti) = 0.305 kJ/g is the heat of fusion of Ti [110]and I[TadTm(Ti)] is the Heaviside unit-step function. The masses of all thesubstances are determined using a solution of the Stefan problem for particular

geometry of the reaction cell, and then Tad is calculated from Eq. (2.9).

2.7.MODELING OF TICLAYER GROWTH ON THE TITANIUM

PARTICLE SURFACE

2.7.1. Analytical Solution to Scenario 1

Problem (2.1)-(2.3),(2.5) is non-linear and in a general case can be solved

only numerically. However, for a similar linear problem (with D=const) an

asymptotic solution for the growth of a spherical phase layer, which is valid for a

small layer thickness h=R2R1


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36/123


Figure 2.4. Thickness of the TiC layer formed on the surface of a titanium particle by the

time of attainment of Tm(Ti) (a) and TCS (c), and relevant adiabatic heating (b and d) [60].

Numbers at curves correspond to diffusion data sets in Table 2.1.


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B. B. Khina24

The corresponding adiabatic heating is only Tad=1064 K for the Ti particleradius of 10 m and sharply drops with increasing R2 [Figure 2.4 (d)]. Thus, heat

release due to product growth is insufficient to sustain the SHS wave ( i.e. to reach

TCS=3083 K).

The obtained result, viz. a small thickness of TiC grown in the temperature

range below Tm(Ti), qualitatively agrees with experimental data [17,18]: in

rapidly cooled samples almost no interaction was observed in the so-called

preheating zone of the SHS wave.

However, at the attainment of T=Tm(Ti) the melting of titanium can bring

about the rupture of the primary TiC shell and the spreading of the metallic melt.

It should be noted that in [69] the diffusion-controlled TiC formation was

assessed using 6 different sets of the diffusion data, but only an isothermal

situation below the titanium melting point was examined. Besides, the TiC layer

growth was considered on the surface of a carbon particle whereas, as mentioned

above, the initial TiC film at T


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where urrand u are the radial and shear strain, correspondingly, and A and B are

constants which are determined from boundary conditions (2.16).

Hookes law for spherical symmetry looks as

rr= [(1)urr+ 2u], = (u + urr),

(2.18)

where rr and are the radial and shear stress, correspondingly, Y is the elastic

modulus and is the Poissons ratio [113]. Then the solution for is obtainedfrom Eqs. (2.16)-(2.18):

(r) = , f = , fr= ,

= . (2.19)

Rupture of the primary TiC shell occurs when the maximal shear stress in the

spherical layer (at r=R2) exceeds the ultimate tensile stress uts. Then from Eq.(2.19) we obtain a critical thickness, hcr= R2R1, of the TiC layer:

, ,

. (2.20)

The TiC case can burst at hhcr. This is an upper estimate because we donttake into account partial dissolution of TiC in molten titanium due to the eutectic

reaction at 1645 C.To calculate the hcr value, we have to determine the mechanical properties of

TiC at the melting temperature of titanium. The temperature dependencies of the

elastic modulus, Y, and shear modulus, G, for TiC are known in the following

form [96]:

)21)(1(

Y

+ )21)(1(Y

+

f1

ffp

f1

f11

21

Y r0

r

3/1

m

s

+

++

31

32

R2

R3

32

r2

R

+

21

1

=1

Rh 2cr

3/1

0uts

0uts

)p(

p2

+

+=

= 1

21

Y33/1

m

s


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B. B. Khina26

Y(T) = Y0 bYTexp(T0/T), G(T) = G0 bGTexp(T0/T), (2.21)

where T0=320 K, Y0=461 GPa, bY=0.0702 GPa/K, G0=197 GPa and bG=0.0299

GPa/K. Then at Tm(Ti)=1940 K we have Y=346 GPa and G=148 GPa, thus the

Poissons ratio is = Y/(2G)1 = 0.17. As foruts values for TiC at elevatedtemperatures, there are only disembodied data, e.g., uts(T=1073 K) 380 MPa,uts(T=1273 K) 280 MPa [90]. However, available are data on the bendingstrength, b, of titanium carbide over a wide temperature range because it is atypical test for brittle refractory compounds; b has a maximum of approximately

500 MPa around T=2000 K [96, page 233]. Then, using an estimate uts ~ b/2 =250 MPa, from Eq. (2.20) we obtain hcr0.6R2. Since the calculated valueh[T=Tm(Ti)] is very small, for any initial size of Ti particles used in SHS (R2=5 to

100 m) melting of the titanium core will inevitably bring about the rupture of the

primary TiC shell and spreading of the melt. This changes the geometry of a unit

reaction cell as shown in Figure 2.2 (a-c).

2.9.GROWTH OF A TICLAYER ON THE SURFACE OF A SOLIDCARBON PARTICLE

2.9.1. Analytical Solution to Scenario 2

For scenario 2 [Figure 2.2 (c and e)], an asymptotic solution to Eqs.

(2.2),(2.3)-(2.5) with respect to the TiC layer thickness, h, can be obtained

similarly to Eq. (2.11) [25,111,112]:

h() = R1() R0 = 1/2

1/R023/2

/(2R02). (2.22)

Here coefficients , 1 and 2 are defined, as previously, by Eqs. (2.12),(2.13) and is determined according to Eq. (2.10) where integration is performed over thetime range 0ttCS, which corresponds to the temperature range TmTTCS(Figure 2.3).

To calculate adiabatic heating, we turn to Eq. (2.9). For the reaction cell

shown in Figure 2.2 (c), mTiC() = (4/3) (R13

()R03

)TiC, mC() = mC0

0.2mTiC(), mC

0= (4/3)R0

3C and mTi() = 4mC0 0.8mTiC(). Then, ignoring the

temperature dependence of heat capacities and neglecting the melting enthalpy of


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titanium (because Hm(Ti)


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B. B. Khina28

Figure 2.5. Formation of the TiC layer on the surface of a carbon particle in the SHS wave

after spreading of molten titanium (Tm(Ti)TTCS): (a) product layer thickness and (b)corresponding adiabatic heating vs. carbon particle radius [60]. Numbers at curves

correspond to diffusion data sets in Table 2.1.

The results obtained regarding the above concept suggest that fast and

complete conversion of reactants into the final product providing the required heat

release can be achieved via a different route (without diffusion control of the

product formation). For further analysis of the diffusion model it makes sense toestimate a structural parameter of the product, viz. porosity. To do this, it is

necessary to evaluate the displacement of the C/TiC interface.


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2.9.3. Displacement of the C/TiC Interface in the Emptying-Core

Mechanism

Above we have calculated the thickness of a spherical product layer formed

due to interstitial diffusion of C atoms through titanium carbide, i.e. outward

growth of TiC on the surface of carbon particles. Hence the TiC particles formed

after complete conversion of the reactants will be hollow. This pattern of

diffusion-controlled product formation is sometimes called the emptying-core

mechanism [115]. Lets estimate the displacement of the C/TiC interface, i.e.

inward growth of the product layer due to diffusion of Ti atoms across TiC.

From Figure 2.5 (a) it is seen that at R05 m the effect of curvature is minor:raising R0 from 5 to 12.5 m increases the TiC thickness by less than 10%. Thus

the diffusion problem can be considered for a semi-infinite rod. The diffusion

equations are written for both C and Ti atoms

, i C,Ti. (2.25)

The Stefan-type boundary conditions to Eq. (2.25) are formulated at

interfaces Ti(melt)/TiC (r=R1) and C/TiC (r=R0) taking into account that here

R0=R0(t) and cC + cTi =1

, . (2.26)

The initial conditions are

R0(t=0) = R1(t=0) = R00. (2.27)

Here DC and DTi are the partial diffusion coefficients of C and Ti atoms in TiC

(see Table 2.1), r is the radial coordinate and R00 is the initial position of the

C/Ti(melt) interface at which a thin TiC layer originates at t=0. Using substitution

, i C,Ti, (2.28)

2

i

2

ii

r

c)]t(T[D

t

c

=

(t)R

CC

10

1

0

21

1

r

c[T(t)]D

dt

dR)cc(

=

(t)R

CTi

00

32

0

r

c[T(t)]D

dt

dR)c1(

=

= d)][T(D(t)t

0ii


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B. B. Khina30

the non-isothermal problem (2.25)-(2.27) is reduced to an isothermal (linear) case

which has an analytical solution [118] for the displacement of phase boundaries

Ti/TiC (h) and C/TiC ():

h(C) = R1(C) R00 = C C

1/2, (Ti) = R0(Ti) R

00 = Ti Ti

1/2. (2.29)

The coefficients C and Ti are determined from transcendental equations:

1/2(C/2)exp(C/2)2{erf(C/2) + erf[Ti(Ti/C)

1/2/2]} = (c

023c

021)/(c

021c

01)

1/2(Ti/2)exp(Ti/2)2{erf(Ti/2) + erf[C(C/Ti)

1/2/2]} = (c

023c

021)/(1c

023).

(2.30)

The calculated displacement of the C/TiCx interface during interaction in the

SHS wave (at TmTTCS) is negligibly small: =4.7 nm


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[114,116] or by short-term compaction of a hot pellet immediately after the

completion of SHS [3,4 and literature cited therein]. If as-synthesized TiC

particles were hollow, which follows from the above described route, the

attainment of such relative density would require prolonged pressure sintering at a

high temperature.

Thus the structural characteristic of the SHS product emerging from the

diffusion model disagrees with experimental observations, which therefore

supports an idea of a dissolution-precipitation route capable of producing dense

TiC particles, which includes dissolution of carbon in molten titanium and

subsequent crystallization of the product grains.

It should be noted that a conclusion in favor of the diffusion-controlledgrowth of hollow TiC shells on the surface of carbon particles having a size of

2R0=7 m (initial porosity of a sample was 0=0.2) and 20 m (0=0.4) was madein [115] basing solely on the porosity measurements and microstructures of as-

synthesized specimens. Lets analyze these experimental data. For all of the

samples the initial temperature, T0, was 293 K, and the total porosity measured

after SHS was almost the same, t(m)

=0.460.5. As shown above, SHS of TiC viathis mechanism is possible for small-sized carbon particles, R03.5 m, but closed

porosity will be cl=0.33 which greatly exceeds the measured valuecl

(m)=0.060.08 [115]. Besides, total porosity of an as-synthesized sample for the

formation of TiC1.0 is estimated as

t = 1 (10)[TiC(0.8/Ti+0.2/C)]1

(2.31)

implying that the specimen volume doesnt change during SHS, which is true for

strongly compacted green pellets (as in [115]). Here Ti=4.51 g/cm3

[106] is the

density of initial -Ti particles. Then for samples with 7 m diameter carbonparticles the formation of dense TiC grains (TiC=4.91 g/cm

3) yields the total final

porosity t=0.41, which is close to experimental data. But if hollow TiC particlesare formed via the diffusion mechanism, then, substituting into Eq. (2.31) eff=3.3g/cm

3instead ofTiC we obtain t'0.13. In this case t' signifies the fraction of

pores between the hollow particles. But this value is less than 0.1540.005 (theScher-Zallen criterion), which is required by the percolation theory [120,121] for

the existence of open porosity. Thus, the sample will contain only closed pores

(inside the TiC particles and between them) whereas in [115] high open porosity

was observed: op(m)

= t(m)cl

(m) 0.4.

For larger carbon particles (2R0=20 m), as demonstrated above, SHS via the

diffusion mechanism is impossible (for T0298 K) because of low heat release per


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B. B. Khina32

unit reaction cell [Figure 2.5 (b)]. If dense TiC particles are formed, from Eq.

(2.31) for 0=0.4 we have t=0.56 which is close to experimental porosityt

(m)=0.460.5. For the formation of hollow particles (cl=0.33), Eq. (2.31) gives

t'=0.34, and then the total porosity will be t = t' + cl = 0.67, which substantiallyexceeds the measured value. Maximal closed porosity in the experiments was

cl(m)

=0.22 for samples with the particle diameter 17 m (Ti) and 20 m (carbon)[115]. Since SHS was performed under isostatic gas pressure, the origin of closed

pores should be ascribed to partial sintering of dense TiC grains (presumably

precipitated from melt) around voids formed on the sites of outflown titanium

particles. This is supported by the fact that the closed porosity was noticed to

increase with raising the gas pressure (from cl(m)

=0.12 at 1 bar to 0.22 at 70 bar)

while the total porosity remained almost the same, t(m)

=0.470.5 [115].

2.10.ANALYSIS OF THE SHRINKING-COREMECHANISM IN

THE TI-CSYSTEM

Lets discuss a dissolution-precipitation route of the TiC formation, whichcan produce 100% dense particles. According to the idea first proposed in [122]

and used for studying SHS in the Ni-Al [67] and Nb-C [123] systems, as soon as a

metallic melt spreads and engulfs solid particles, a thin film of an intermediate

phase (here TiC) forms around them instantaneously. In this interaction pattern,

the phase layer sequence in the reaction cell corresponds to the equilibrium phase

diagram (Figure 2.1). Then the product particles (TiC) precipitate from the

saturated melt due to diffusion of carbon atoms across this film. The film

thickness remains constant: it is believed that its growth rate at the C/TiCinterface is equal to the dissolution rate at the melt/TiC interface. Thus, the TiC

film shrinks to the center of the carbon particle as the latter dissolves. This pattern

is sometimes called the shrinking-core mechanism. It corresponds to the solid-

solid-liquid mechanism which, for the Ti-C system, is truly quasi-equilibrium.

However, the film thickness has not been previously estimated using realistic

diffusion data.

The concentration profile of carbon in the reaction cell is similar to that

shown in Figure 2.2 (a) but with R0=R0(t); final TiC particles precipitate from the

melt in domain [R1(t), R2]. In a general case, the displacement of the melt/TiC and

C/TiC interfaces is determined by Eqs. (2.25)-(2.27) with the only difference that

Eq. (2.25) should be written in spherical symmetry. But since the TiC film

thickness is small and DC>>DTi, outdiffusion of carbon atoms through the film is


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not the rate-limiting stage. Thus the process is controlled by indiffusion of Ti

atoms across the TiC layer, and radial shrinking of the film is described as

, R0(t=0) = R00, h0 = R1(t)R0(t) = const,

(2.32)

where h0 is the layer thickness. For a thin film, a steady-state concentration profile

can be used to determine the concentration gradient at r=R0(t) in Eq. (2.32):

cC(r) = c023R0/r + (1 R0/r) (R1c

021 R0c

023)/h0, R0(t) r R1(t).

(2.33)

Using Ti defined by Eq. (2.28) and introducing z = R0/R00, from Eqs.

(2.32),(2.33) we obtain:

. (2.34)

By the attainment of the maximal SHS temperature TCS, which corresponds to

time t=tCS, the carbon particle completely dissolves, i.e. R0(tCS)=0. Then,

integrating Eq. (2.34) from 0 to tCS, we receive a non-linear equation linking the

initial radius of the carbon particle R00 with the thickness of the TiC film

. (2.35)

The results of the numerical solution of Eq. (2.35) are shown in Figure 2.6. It

is seen that the thickness of the TiC film for the initial radius of carbon particles

R0

0=0.5 m is close to the crystal lattice period: h0=0.5 nm ~ aTiC=0.4327 nm

[89], and still decreases with increasing R

0

0. Hence the aforesaid quasi-equilibrium solid-solid-liquid mechanism loses its physical meaning: a minimal

thickness of a crystalline phase must be about the size of a critical nucleus which

is typically of the order of 10 lattice periods.

(t)R

CTi

00

32

0

r

c[T(t)]D

dt

dR)c1(

=

20

0

0

23

0

21

0

23

Ti0

0

0))(Rc1(

ccddz

1/hzRz

=+

( )[ ]0

23

0

21

0

23SHSTi0

0

00

0

00c1

cc)(t1hRhRh

=+ /ln


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B. B. Khina34

Figure 2.6. Calculated thickness of a titanium carbide film vs. carbon particle radius fordiffusion-controlled dissolution-precipitation (solid-solid-liquid) mechanism [60].

2.11.PHASE-FORMATION-MECHANISM MAP FOR NON-

ISOTHERMAL INTERACTION IN THE TI-CSYSTEM

The above-presented consistent analysis of the solid-solid-liquid (diffusion-

controlled) mechanism, which was performed using available experimental data

on both solid-state diffusion in the product phase and characteristics of the SHS

wave, has demonstrated that this widely used concept is actually not applicable to

modeling SHS of titanium carbide. This is because the physical meaning of the

results obtained within this approach (e.g., the product structure and density)

disagrees with experimental data.

It is shown that formal calculation of the product-layer thickness and

associated heat release for small particles of a nonmetallic reactant (for scenario

2) can bring about numerical data supporting the diffusion model. Thus the

comparison of theoretical and experimental results should be performed usingstructural characteristics of SHS products, e.g., porosity.

Therefore, only the solid-liquid mechanism, which for the Ti-C system is

truly non-equilibrium, can operate during SHS to produce dense product particles.


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It involves direct contact of solid carbon with molten titanium without the

presence of a continuous TiC interlayer. The product formation occurs via

dissolution of carbon in liquid titanium at the C/Ti(melt) interface and

precipitation of TiC grains. Because of fast diffusion in high-temperature melts

(DC(m)

~105104 cm2/s), diffusion in liquid is not the rate-limiting stage and the

phase-forming process responsible for major heat release is crystallization of the

product particles. Because of the presence of the solid/liquid interface and strong

chemical interaction between the C and Ti atoms, crystallization of the TiC

particles will occur via heterogeneous nucleation at the C/Ti(melt) interface rather

than through homogeneous nucleation in the melt. Besides, the activation energy

for the former is generally lower than for the latter. The nucleated TiC grains must

detach from the carbon particle surface, otherwise a thin TiC layer separating the

parent phases will form and the situation will reduce to the solid-solid-liquid

(quasi-equilibrium) pattern considered above. This process continues until

complete consumption of solid carbon is achieved. Further, in the after-burn zone

of the SHS wave, growth and coalescence of the TiC particles in the metallic melt

can occur. In this case, the final size of product particles will depend on the

conditions of crystallization and subsequent coalescence/sintering but not on the

size of initial reactants. An important factor is the melt lifetime which depends onthe Ti-to-C ratio in the charge, structure of a green pellet determining the melt

spreading conditions and heat exchange with the environment.

This solid-liquid mechanism qualitatively agrees with the results obtained in

experiments on arresting SHS wave in the binary Ti-C [18] and multi-component

Ti-C-Ni-Mo [19] systems where in rapidly quenched samples the formation of

small uniform-sized TiC particles was observed in the molten metal around a

graphite particle, which were apparently detaching from the surface of carbon

particles [18]. This mechanism may also be valid for other interstitial compoundssuch as carbides, borides etc. for which the SHS temperature exceeds the melting

point of a metallic reactant but is below the melting temperatures of the non-metal

and product, and for certain intermetallics.

In this situation, the critical thickness, hcr, of the layer of a primary product

formed on the surface of a metal particle before the attainment of the melting

temperature, which is determined by Eq. (2.20), acquires a precise physical

meaning. This is a criterion for the changeover from the solid-solid-liquid

(diffusion-controlled) mechanism (a slow route of product formation) to thesolid-liquid mechanism (a fast rout). As stated above, in the wave propagation

mode this thickness is small (h


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B. B. Khina36

melting reactant may be small, down to ~110 K/min [67,83], and hence asufficiently thick case of the primary product can be formed on a metal-particle

surface to prevent the liquid core from spreading.

Basing on the above, a diagram of interaction routes can be constructed for

non-isothermal heterogeneous interaction in the Ti-C system. Consider linear

heating with rate vT, which corresponds to CS in the TE mode. Then is the time tm

corresponding to the attainment of melting temperature of titanium is determined

as

tm = [Tm(Ti)T0]/vT. (2.36)

The thickness of a primary TiC case (see scenario 1) corresponding to the

changeover of interaction pathways, is written as

hTiC[(tm)] = hcr, (2.37)

where is determined by formula (2.10) and hcr is calculated according to Eq.(2.20).

Solving Eq. (2.37) together with Eqs. (2.10)-(2.13) and (2.36) permits

obtaining the aforesaid criterion for the changeover of interaction mechanisms at

different heating rates vT and initial particle sizes of the metallic reactant

(titanium) R0

0. A typical radius of titanium particles used in CS is 1-50 m; incalculations the R

00 value was varied within 0.5-150 m. For numerical

calculations, we use three data sets, viz. Nos. 1, 11 and 14 from Table 2.1 for the

following considerations. As seen in Figure 2.4 (a), sets No. 1 and 11 give a

reasonable thickness of the TiC layer grown on the titanium particle surface

during heating in the SHS wave while data set No. 14 yields a maximal (probablyoverestimated) value. The results are presented in Figure 2.7 as a map of phase

formation mechanisms in coordinates heating rate and initial radius of a

titanium particle where lines 1-3 refer to different sets of the diffusion

parameters in the titanium carbide.

Parametric domain I, which lies below the line corresponding to the

attainment of the critical thickness, hcr, of the primary refractory product (TiC),

refers to the diffusion-controlled growth mechanism. This is a slow, quasi-

equilibrium route typical of diffusion annealing in weakly non-isothermalconditions. Here, a sufficiently thick shell of a primary refractory product (in our

system, TiC) is grown of the metal particle surface during heating to Tm so that

after melting the metallic reactant remains inside the shell (hTiC>hcr) and further


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interaction (at T>Tm) proceeds slowly since the rate-limiting stage is still the

solid-state diffusion across this spherical layer.

Figure 2.7. Diagram of phase formation mechanisms for synthesis of titanium carbide in

non-isothermal conditions: domain I is the solid-state diffusion-controlled TiC growth, or

slow route typical of furnace synthesis; domain II is the non-equilibrium crystallization

mechanism, or fast route typical of CS. Calculated using different data sets for diffusionin TiC (see Table 2.1): data set No. 1 (line 1), No.11 (line 2) and No. 14 (line 3).

Thus, the models of CS employing this approach [62-66,68,69] are valid inthis range of parameters (the heating rate and metal particle size). From Figure 2.7

it is seen that for small-sized metal particles (R000.5 m) the quasi-equilibrium

diffusion-controlled growth of the refractory product can proceed at high heating

rates typical of the SHS wave, vT~104-10

5K/s. This agrees qualitatively with

certain results observed during SHS in mixtures of small (nanosized) particles and

in mechanically activated SHS [124].

Domain II, in its physical meaning, corresponds to a fast route typical of CS

where the non-equilibrium dissolution-precipitation mechanism operates toprovide fast completion of the reaction. Here, the refractory product layer formed

during heating from T0 to the melting point of the metallic reactant is thin

(hTiC


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B. B. Khina38

direct contact with the solid non-metal particles, which is a necessary condition

for the dissolution-precipitation interaction route to occur. In this parametric

domain, the model [57-59] is applicable.

The diagram clearly demonstrates the difference, in terms of reaction

mechanisms, between CS (domain II), where fast conversion of reactants into the

products takes place, and traditional furnace synthesis (domain I) where the

interaction proceeds slowly because of a low heating rate and/or large particle size

of the metallic reactant.


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Chapter 3

MODELING INTERACTION KINETICS IN THE

CS OF NICKEL MONOALUMINIDE

3.1.INTRODUCTION

The CS of nickel monoaluminide has gained much attention in literature

[21,23,67] because this compound possesses a unique combination of strength

properties and resistance to gas corrosion at high temperatures, and is used in a

variety of applications as a structural material [125,126] and protective coating

[127]. Besides, of substantial interest is CS in multilayer thin-film Ni-Al system

where the layer thickness, h, varies from ~1 m to ~10 nm and in stacked foilswith h~10-100 m [128]. The former process is used for joining of metallicglasses [129-131], welding of a pure NiAl layer to high-strength superalloys

[132], welding/soldering of microscopic objects such as electronic components,

and similar applications [133-135] while the latter can be used for near-net-shape

manufacturing of NiAl articles [136]. Earlier [137], SHS in laminated Ni/Al foils

was used for experimental modeling of the reaction mechanism in Ni-Al powder

mixtures. In both cases, CS in this system is a subject of extensive experimental

and theoretical investigation.

However, an intricate physical mechanism of phase and structure formation

during CS of NiAl is not well understood yet. It has been demonstrated

experimentally that during SHS in powder mixtures [21,23] and in lamellar Ni-Al

systems [136] with h~10-100 m the dissolution-precipitation (DP) mechanism

takes place: at heating above the Al melting temperature solid nickel dissolves inliquid aluminum and NiAl grains crystallize from the supersaturated melt. This

interaction route may have non-equilibrium nature since the Ni-Al phase diagram

[138,139] contains four compound phases: NiAl3 (melts at 1127 K), Ni2Al3


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B. B. Khina40

existing at T


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Modeling Interaction Kinetics in the CS of Nickel Monoaluminide 41

Besides, there is a difference in interaction patterns between SHS in thin (h


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Modeling Interaction Kinetics in the CS of Nickel Monoaluminide 43

substantially below Tad because of heat removal into the substrate, thus the

formation of only solid final product, NiAl, during CS is possible.

As is known [21,23,136,137], fast interaction during SHS in the given system

begins after melt formation, i.e. above the aluminum melting temperature,

Tm(Al)=933 K, or the eutectic temperature Teu(Al-NiAl3)=913 K (Figure 3.1).

According to the Gibbs phase rule, in a binary system the contact of a solid phase

layer (here pure nickel or Ni-base solid solution) with a two-phase region (here

solid NiAl particles dispersed in Al-base melt at T


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B. B. Khina44

Hence, this situation is close to CS in the TE mode. In this case, the NiAl

phase layer growth is due to solid-state interdiffusion across this layer. The role of

interface kinetics, which was studied in [81], is ignored, and the Al concentrations

at phase boundaries 1/2 and 2/3 correspond to equilibrium values.

3.2.2. Phase Composition of the Reaction Zone

The experimental data concerning the phase formation sequence in Ni-Al

thin-film diffusion couples during isothermal annealing or upon heating at low

rates, vT ~ 1 K/s, are contradictory. According to [163,164], the first phase to form

is NiAl3, and the other equilibrium phases of the Ni-Al system can appear only

after the NiAl3 layer exceeds a certain thickness or after complete consumption of

one of the starting metals. The final phase composition corresponds to the initial

Al-to-Ni thickness ratio of the films. However, in [165,166] the growth of B2-

NiAl with a metastable concentration of about 63 at.% Al as the first phase was

observed. In [167], the formation of metastable -phase (Ni2Al9) during annealingof Ni/Al multilayers was detected, which quickly transformed into stable NiAl3

due to interaction with nickel. At high heating rates, which are typical of CS, theconditions for the first phase nucleation at the Ni/Al interface may be different

from those at slow heating.

For constructing the model, the following basic assumption is made. It is

considered that a thin continuous NiAl layer nucleates at the initial Ni/Al interface

at T0 = Teu(Al-NiAl3) = 913 K. During the growth of NiAl compound in the whole

temperature range T0 T Tad=Tm(NiAl), the interlayers of other equilibriumphases of the Al-Ni system (Figure 3.1) [138,139], viz. NiAl3 (Tm=1127 K),

Ni2Al3 (Tm=1406 K) and Ni3Al (Tm=1668 K), are absent, and metastableequilibria at interfaces Al-base melt/solid NiAl and NiAl/Ni(s) are described

by the corresponding equilibrium solubility-limit lines, viz. lines GFE/LK and

HIJ/ABC, respectively (Figure 3.1). The latter presumption is only for the sake of

simplicity, in order to avoid cumbersome calculation of

Combustion_Synthesisof Advanced Materials

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