Combustion synthesis of advanced materials chemistry research and applications series

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-s

C OMBUSTION S YNTHESIS OF

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L IBRARY OF C ONGRESS C ATALOGING - IN -P UBLICATION D ATA

1 Self-propagating high-temperature synthesis 2 Refractory

materials Heat treatment 3 Refractory materials Mathematical models

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

C ONTENTS

Chapter 1 Advances and Challenges in Modeling Combustion

Chapter 2 Modeling Diffusion-Controlled Formation

Chapter 3 Modeling Interaction Kinetics in the CS

Chapter 4 Analysis of the Effect of Mechanical

P REFACE

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 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

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

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

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 TiB 2 (3190), TiB (3350), ZrB 2 (3310), HfB 2 (3320), VB 2 (2670),

VB (2520), NbB 2 (2400), NbB, TaB 2 (2700), TaB, CrB 2 (2470), CrB, MoB 2 (1500), MoB (1800), WB (1700), LaB 6 (2800) Carbides TiC (3210), ZrC (3400), HfC (3900), VC (2400), Nb 2 C (2600),

NbC (2800), Ta 2 C (2600), TaC (2700), SiC (1800), WC, B 4 C,

Cr 3 C 2 , Cr 7 C 3 , Mo 2 C, Al 4 C 3 Aluminides Ni 3 Al, NiAl, Ni 2 Al 3 , TiAl, CoAl, Nb 3 Al, Cu 3 Al, CuAl, FeAl Silicides Ti 5 Si 3 (2500), TiSi (2000), TiSi 2 (1800), Zr 5 Si 3 (2800), ZrSi

(2700), ZrSi 2 (2100), WSi, Cr 5 Si 3 (1700), CrSi 2 (1800), Nb 5 Si 3 (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,

MoSe 2 , WSe 2 Hydrides TiH2, ZrH2, NbH2, CsH2

Nitrides TiN (4900), ZrN (4900), VN (3500), HfN, Nb 2 N (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 (YAlO 2 , MgAl 2 O 4 ), niobates (NaNbO 3 , BaNb 2 O 6 ,

LiNbO 3 ), garnets (Y 3 Al 5 O 12 , Y 3 Fe 5 O 12 ), ferrites (CoFe 2 O 4 , BaFe 2 O 4 , Li 2 Fe 2 O 4 ), titanates (BaTiO 3 , PbTiO 3 ), molybdates (BiMoO 6 , PbMoO 4 ), high-temperature superconductors (YBa 2 Cu 3 O 7-x , LaBa 2 Cu 3 O 7-x , Bi-Sr-Ca-Cu-O) Ternary solid solutions

based on refractory

compounds

TiB 2 -MoB 2 , TiB 2 -CrB 2 , ZrB 2 -CrB 2 , TiC-WC, TiN-ZrN, MoS 2 NbS 2 , WS 2 -NbS 2

-MAX phases Ti2AlC, Ti3AlC2, Ti3SiC2

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

TiC-Cr3C2-Ni, TiC-Cr3C2-Ni-Cr, Cr3C2-Ni-Mo, TiB-Ti, WC-Co, TiN-NiAl-Mo 2 C-Cr

Si 3 N 4 -TiN-SiC, sialons (SiAlO x N y )

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

≈2500 3-4

Solid-solid (formation of

a complex oxide) via a

transient liquid phase

with participation of an

oxidizing gas

3Cu (solid→liquid) + 2BaO2 (solid) + (1/2)Y2O3 (solid) + O2(gas) → YBa 2 Cu 3 O 7-x (solid)

→ C 7 H 14 N 2 O 4 (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, 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

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:

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

Jmol–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

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]

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

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 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]) and metal-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 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 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

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 “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 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

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

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 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 self-ignition 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

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

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

Chapter 2

CS

2.1 INTRODUCTION

CS in the Ti-C system was a subject of extensive theoretical and experimental studies [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 typical configuration 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

Let’s 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 µm for Ti, about 0.1 µm for carbon black and 1

to 30 µm 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)]

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

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<Tm(Ti); (b) rupture of the primary product shell and spreading of molten titanium at T=Tm(Ti); (c and e) growth of the TiC layer on the surface of a carbon particle after titanium melting and spreading at Tm(Ti)≤T≤TCS

A condition for the change of the reaction cell geometry due to titanium melting [Figure 2.2 (b)] is analyzed later

2.2 SCENARIO 1: GROWTH OF S TIC CASE ON THE

TITANIUM PARTICLE SURFACE

In scenario 1, at T<Tm(Ti) a thin uniform layer of TiC is formed on the surface of the Ti particle This is due to fast surface diffusion of C atoms from the Ti/C contact spots in the charge mixture, which agrees with the idea expressed in [88] Because of low carbon solubility in solid β-Ti and high diffusivity of C atoms in the β-phase as compared with that in TiC [90,91], we can consider the

growth of only one product phase-titanium carbide, i.e perform an upper-level

estimate of the product thickness Then the TiC layer growth in the SHS wave is described by a diffusion-type Stefan problem written in spherical symmetry neglecting a volume change at the Ti/TiC interface

, (2.1)

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 mass concentration of carbon, R1(t) is the current position of the TiC/Ti interface, R2 is the outer radius of the Ti particle, and c01, c021 and c023 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

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)=c023 [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 TIC LAYER 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

D[T(t)]

t

2 C

(t) R

C 1

dt

dR )

stage [93,94] Hence we consider that at T≥Tm(Ti) the carbon particles are completely 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

are listed in Table 2.1, wherein the experimental data available in literature 103] for different temperature intervals ΔT are collected It is seen that different data 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 whole temperature 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

[95-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 H

in liquid metals (Fe, Co, Ni, etc.) [104,105]

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, μ=ζρm is the dynamic viscosity, ζ is the kinematic viscosity and ρm is the liquid-phase density For carbon atoms, the covalent radius is

aC=0.077 nm [106] The density of molten Ti is ρm=4.11 g/cm3 [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)×10−2 cm2/s [106] For liquid titanium saturated with carbon, ζ=0.94×10−2 cm2/s at T=Tm(Ti) [107], then DC(m)(Tm) ≈ (4.8−7.2)×10−5

cm2/s For higher temperatures, the value ζ=1.03×10−2 cm2/s at T=2220 K is known [107]; using it at T=TCS gives DC(m)(TCS) ≈ (6.9−10.4)×10−5 cm2/s It should be noted that since Eq (2.6) doesn’t 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

2.5 TEMPERATURE OF THE

REACTION CELL IN THE SHS WAVE

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 T≈TCS) only coalescence and sintering of the product particles occur with minor 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<T<Tm(Ti), where heat release is small, the temperature profile along a sample can be calculated using a known analytical solution [108]

T(ξ) = T0 + (Tm−T0)exp(−vSHSξ/κ), ξ = x − vSHSt, (2.7) where vSHS is the combustion velocity, x is a coordinate along the SHS-sample and κ is the thermal diffusivity To determine T(t) for the reaction cell, a coordinate x0 is chosen for which T(x0, t=0) = T0' = T0+0.01Tm, where T0=298 K Then the heating time, tm, from T0' to Tm is

(2.8)

For a stoichiometric Ti-C mixture (20 wt.% C), κ≈0.04 cm2/s [109] and vSHS=6 cm/s [88,18]

For higher temperatures, Tm≤T≤TCS, we use the spline-approximation of the experimental temperature profile of a steady-state SHS wave in the Ti-C system, which was registered by a micro-thermocouple technique [88] (Figure 2.3) It should be noted that the low-temperature tail [at T<Tm(Ti)] calculated by Eq (2.7) lies slightly above the corresponding part of the experimental curve, which is not shown in Figure 2.3 merely to avoid encumbering So, we use an upper estimate

of temperature (and hence the grown product layer) in the reaction cell In Figure

0 m

m 2

SHS

T 01 0 v

2.3, x=0 corresponds to the melting point of Ti, consequently the heating time from Tm(Ti) to TCS is ΔtCS = tCS−tm = 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 T≤Tm(Ti); 2, cubic-spline

approximation of experimental curve [88] in the range Tm(Ti)≤T≤TCS

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:

−ΔH0

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

dT ) TiC ( c

ad

T

298 p

T

298 p

T

298 p

ad

where Tad is the adiabatic combustion temperature, cp(i) is heat capacity, mi(t) is a current mass of i-th substance, ΔH0

298(TiC1.0) = −3.077 kJ/g is the standard enthalpy of TiC1.0 [110], ΔHm(Ti) = 0.305 kJ/g is the heat of fusion of Ti [110] and I[Tad−Tm(Ti)] is the Heaviside unit-step function The masses of all the substances 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 TIC LAYER 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=R2–R1 << R2, is known [25,111,112] To apply it, we linearize Eqs (2.2) and (2.3) using substitution

Here t varies from 0 to tCS, where tCS is time at which the temperature of the reaction cell reaches its maximal value TCS Then, according to [25,111], the asymptotic solution of Eqs (2.1)-(2.3) with respect to the product layer thickness,

h, looks as

h(τ) = R2 − R1(τ) = βτ1/2 + β1τ/R2 + β2τ3/2/(2R22), (2.11) where

=

τ (t) ∫ D[T( )] d

t

0

π1/2(β/2)exp(β/2)2 erf(β/2) = q, (2.13) which arises in a similar Stefan problem for a semi-infinite sample

For scenario 1 [Figure 2.2 (a)], mTiC(τ) = (4/3)π[R23−R13(τ)]ρTiC, mC(τ) = mC0

− 0.2mTiC(τ), mTi(τ) = (4/3)πR13(τ)ρTi, where ρi is the density of i-th substance For a stoichiometric composition, the initial C-to-Ti mass ratio is mC0/mTi0 = 0.25, where mTi0 = (4/3)πR23ρTi Then, ignoring the temperature dependence of heat capacities in Eq (2.9), the maximal adiabatic heating of the reaction cell, ΔTad =

Tad−298, is estimated as

(2.14)

2.7.2 Results of Calculations for Scenario 1

In the temperature range T0'≤T≤Tm(Ti), equilibrium interfacial concentrations

c01= 0.00138 and c021=0.11 corresponding to the Ti-TiC eutectic temperature

Teu=1918 K [87] were used For higher temperatures, Tm≤T≤TCS, it was assumed that molten titanium remains inside the spherical TiC shell, and the interfacial compositions were taken for an intermediate temperature T=2673 K: c0=0.065,

c021=0.14 [87]; at the C/TiC interface c023=0.2 (maximal solubility of C in the carbide) Calculations have shown that varying the c021 and c01 values along the solidus and liquidus lines of the Ti-C phase diagram in the range T=Tm to TCS has

a negligible effect on the TiC layer thickness and associated heat release

For the temperature range T0'≤T≤Tm the calculations were performed with all the diffusion data listed in Table 2.1 [Figure 2.4 (a and b)] For the whole temperature range, T0'≤T≤TCS, only the data giving D(TCS)< DC(m) (lines 1 to 9 in Table 2.1) were used [Figure 2.4 (c and d)] A maximal TiC layer thickness attained by the time of reaching the titanium melting temperature is small, h(Tm)=0.068 µm [Figure 2.4 (a)], and corresponds to diffusion data No.14 in Table 2.1 The relevant adiabatic heating is insignificant, ΔTad=57 K for R2=10

µm [Figure 2.4 (b)], and decreases with increasing the particle radius For the temperature range T0'≤T≤TCS, the maximal TiC layer thickness corresponds to the set of diffusion parameters No 1 (see in Table 2.1), and this value is small: h(TCS)≈1.6 µm [Figure 2.4 (c)]

)]

( R ]/[R (C)R c 25 0 ) (Ti)R [c (C)]

c 2 (TiC) [c

) (TiC H

1 3 2 3 2 p 3

1 p Ti p p

TiC

TiC 0 0 298 ad

τ

− +

τ ρ

− ρ

ρ Δ

−

=

Δ

.

.

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

The corresponding adiabatic heating is only ΔTad=1064 K for the Ti particle radius of 10 µm 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<Tm(Ti) will most probably form on the surface of solid Ti particles due to fast surface diffusion of C atoms

2.8 RUPTURE OF THE PRIMARY TIC SHELL

The density of solid β-Ti at T=Tm is ρs=4.18 g/cm3 while for molten titanium

at the same temperature ρm=4.11 g/cm3 [107] The conditions for the rupture of the TiC case because of the dilatation of the titanium core during melting can be determined from a continuity equation written for spherical symmetry [113]:

Ur(r) = Ar + B/r2, urr = ∂Ur/∂r = A − 2B/r3, uθθ(r) = Ur/r = A + B/r3,

(2.17)

where urr and uθθ are the radial and shear strain, correspondingly, and A and B are constants which are determined from boundary conditions (2.16)

Hooke’s law for spherical symmetry looks as

σrr = [(1–ν)urr + 2νuθθ], σθθ = (uθθ + νurr),

(2.18)

where σrr and σθθ are the radial and shear stress, correspondingly, Y is the elastic modulus and ν is the Poisson’s ratio [113] Then the solution for σθθ is obtained from Eqs (2.16)-(2.18):

Rupture of the primary TiC shell occurs when the maximal shear stress in the spherical layer (at r=R2) exceeds the ultimate tensile stress σuts Then from Eq (2.19) we obtain a critical thickness, hcr = R2−R1, of the TiC layer:

, ,

The TiC case can burst at h≤hcr This is an upper estimate because we don’t take 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]:

) 2 1 )(

1

(

Y ν

− ν

+ ( 1 )( 1 2 )

Y ν

− ν +

f 1

f f p f 1

f 1 1 2

1

0 r 3

/ 1 m

s

γ +

− γ

− γ +

ρ ν

1

3 2

R 2

R

3

3 2

r 2 R

= R 1

hcr 2

3 / 1

0 uts

0 uts

) p (

p 2

+ σ

− ψ

= ϕ

ρ ν

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 Poisson’s ratio is ν = Y/(2G)−1 = 0.17 As for σuts values for TiC at elevated

temperatures, there are only disembodied data, e.g., σuts(T=1073 K) ≈ 380 MPa,

σuts(T=1273 K) ≈ 280 MPa [90] However, available are data on the bending strength, σb, of titanium carbide over a wide temperature range because it is a typical test for brittle refractory compounds; σb has a maximum of approximately

500 MPa around T=2000 K [96, page 233] Then, using an estimate σuts ~ σb/2 =

250 MPa, from Eq (2.20) we obtain hcr≈0.6R2 Since the calculated value h[T=Tm(Ti)] is very small, for any initial size of Ti particles used in SHS (R2=5 to

100 µm) 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 TIC LAYER ON THE SURFACE OF A SOLID

CARBON 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(τ) = R1(τ)− R0 = βτ1/2 − β1τ/R0 − β2τ3/2/(2R02) (2.22) Here coefficients β, β1 and β2 are defined, as previously, by Eqs (2.12),(2.13) and

τ is determined according to Eq (2.10) where integration is performed over the time range 0≤t≤ΔtCS, which corresponds to the temperature range Tm≤T≤TCS(Figure 2.3)

To calculate adiabatic heating, we turn to Eq (2.9) For the reaction cell shown in Figure 2.2 (c), mTiC(τ) = (4/3)π (R13(τ)−R03)ρTiC, mC(τ) = mC0 − 0.2mTiC(τ), mC0 = (4/3)πR03ρC and mTi(τ) = 4mC0 − 0.8mTiC(τ) Then, ignoring the temperature dependence of heat capacities and neglecting the melting enthalpy of

titanium (because ΔHm(Ti) << |ΔH°298(TiC1.0)| [110]), the adiabatic heating of the reaction cell is estimated as

, (2.23) were subscript “m” denotes melt For calculations, the values of heat capacities (according to [110]) were taken at T=TCS Eq (2.23) refers to incomplete

conversion of carbon into titanium carbide, i.e when 0<ηTiC<1, where the degree

of conversion is expressed as

ηTiC = 1 − mC(τ)/mC0 = 0.2[R13(τ)/R03 − 1]ρTiC /ρC (2.24) For complete conversion ηTiC=1, the maximal adiabatic heating is ΔTad(max) =

−ΔH0

298(TiC1.0)/cp(TiC) = 3095 K, and the adiabatic SHS temperature Tad(max) =

298 + ΔTad(max) = 3393 K It is somewhat higher than the value Tad=3210 K calculated taking into account the temperature dependence of heat capacities [5,114] Thus, Eq (2.23) gives an upper estimate for ΔTad

2.9.2 Results of Calculations for Scenario 2

Numerical results are presented in Figure 2.5 The TiC layer thickness, which can form in the SHS wave with the temperature profile shown in Figure 2.3, was calculated using Eq (2.22) not accounting for the exhaustion of reactants [Figure 2.5 (a)] The maximal value is h≈1.5 µm for a sufficiently large carbon particle size, R0=12.5 µm, at the 1st set of diffusion data in Table 2.1

Adiabatic heating [Figure 2.5 (b)] was calculated taking into account the degree of conversion of carbon into carbide [see Eqs (2.23),(2.24)] A plateau with ΔTad=ΔTad(max) for small R0 values corresponds to complete conversion (ηTiC=1) Thus, from Figure 2.5 (b) it is seen that the diffusion-controlled growth mechanism can provide sufficient adiabatic heating to sustain the SHS process, which results from almost complete conversion, only for small-sized carbon particles: R0<3 µm

This contradicts numerous experimental works where SHS of TiC was performed with coarse-grained graphite: 7 µm [69,115,116], 20 µm [69,115], and

up to 63 µm [117] in diameter

] 1 )/R ( )]/[R (Ti c 4 (C) [c )]

(Ti c 8 0 (C) c 2 (TiC) [c

) (TiC H

0

3 1 m p p C m p p

p TiC

TiC 0

0 298 ad

− τ +

ρ +

−

− ρ

ρ Δ

−

=

Δ

.

.