MODELING SELF-PROPAGATING HIGH-TEMPERATURE SYNTHESIS: A
MICROMETALLURGICAL APPROACH
B.B.Khina1, B.Formanek2 and O.S.Rabinovich3
1
Physico-Technical Institute, National Academy of Sciences, Minsk, Belarus
2
Silesian University of Technology, Katowice, Poland
3
A.V.Luikov Heat/Mass Transfer Institute, National Academy of Sciences,
Minsk, Belarus
Within the frame of the wide-spread approach to modeling SHS, viz., the solid-state
diffusion-controlled product growth concept, thermodynamic and kinetic estimates are
made using the known values of diffusion coefficients for the SHS of TiC taking into
account a change of the geometry of a unit reaction cell. It is demonstrated that this
mechanism cannot provide heat release necessary for sustaining the SHS wave, and
the structure of the final product will not agree with the experimental results reported
in literature. It is shown that only a non-equilibrium mechanism implying a direct
contact of a metallic melt with a non-metal can provide the required interaction rate
and bring about the experimentally observed product structure, and a new
“micrometallurgical” model of phase and structure formation during SHS is proposed.
INTRODUCTION
Self-propagating high-temperature synthesis (SHS) has proved to be a
promising cost-effective method for producing a wide range of compounds and
composite materials due to (i) short processing time, (ii) low energy consumption, (iii)
high purity due to volatilization of impurities, and (iv) unique structure and properties
of the final products. Also, SHS presents a substantial scientific interest for Physical
Metallurgy because of non-traditional phase and structure formation mechanisms
associated with extreme conditions in SHS waves (temperature up to 3500 C, heating
rate up to 106 K/s, temperature gradient up to 105 K/cm, rapid cooling after synthesis,
up to 100 K/s, and fast accomplishment of conversion, ~1 to maximum 10 s). Hence
experimental studies in this area are usually combined with theoretical researches
which traditionally play a significant role in the investigations of phase and structure
formation in SHS waves and the development of SHS-based technologies (1 et al.).
In modeling SHS, a formal model based on the classical combustion theory has
acquired a wide use, in which the reaction rate and associated heat release is described
as /t = (1)nexp(m)kexp[E/(RT)], where  is the degree of chemical
conversion (01), n, m and k are formal parameters and E is the activation energy.
This approach permits studying the dynamic regimes of SHS waves (e.g., oscillating
and spin combustion) (2 et al.) but it is not linked to process-specific phase formation
mechanisms. In many SHS-systems, e.g., Ti-C, apparent E values calculated from the
dependence of the SHS wave velocity, v, on temperature (v/Tc vs. 1/Tc where Tc is the
combustion temperature) appear to be close to the activation energy for solid-state
diffusion (3 and other work). Thus, a “diffusion-controlled growth” concept has
gained acceptance for modeling SHS: a quasi-equilibrium product phase forming on
the particle surface separates the initial reactants, and its further thickening is
controlled by solid-state diffusion (see, e.g. (4,5 et al) for the Ti-C system).
However, in many works employing this approach the values of diffusion
coefficients used for calculations are not reported, and modeling is performed with
2 - 73
either dimensionless parameters, which are varied in a certain range, or with an
apparent activation energy estimated from experimental data without taking into
account the preexponent (4,5). Besides, the approaches known in literature don’t take
into account a change of the spatial arrangement of reacting particles due to melting
and spreading of the metallic reactant; in most models the effect of melting is reduced
to a change of the equilibrium interfacial concentrations and the ratio of diffusion
coefficients in contacting phases (5). Since diffusion coefficients for many refractory
compounds are known in literature, it seems reasonable to verify the validity of such
models. In Ref. (6), numerical assessment of the titanium carbide formation during
SHS was made within the frame of the “diffusion-controlled growth“ concept using
experimental values of the diffusion coefficients. However, only isothermal
conditions at relatively low temperatures (below the Ti melting point, Tm) were
examined, and the TiC layer growth was considered on the surface of spherical carbon
particles (6) whereas, due to faster surface diffusion of C atoms from the Ti/C contact
areas in the charge mixture, the primary TiC layer at T<Tm will form on the surface of
the titanium particles. Besides, recent works on studying SHS using microscopic
high-speed video recording (7 et al.), in situ time-resolved X-ray diffraction (8 et al.),
which have revealed new features of transformations in SHS waves, necessitate a new
insight into the phase formation mechanisms operating during the SHS.
Hence, the goal of this work is to analyze, using available data on diffusion
coefficients, the applicability limits of the “diffusion-controlled growth” concept for
modeling SHS on the example of the most studied Ti-C system for non-isothermal
conditions taking into account a change of the geometry of reacting particles due to
melting of the metallic reactant, and to develop a new qualitative model of phase and
structure formation during SHS.
The Ti-C system is chosen because of the availability of the diffusion data for
the TiC growth, a large body of theoretical and experimental research (3-6 et al.), and
also due to the fact that at the SHS temperatures the equilibrium Ti-C phase diagram
contains only one solid phase, viz. titanium carbide TiCx.
FORMATION OF TiC LAYER ON THE SURFACE OF Ti PARTICLE
As a first situation, we consider solid-state diffusion-controlled formation of the
primary product, titanium carbide, during heating of the Ti+C charge mixture in the
SHS wave preheat zone. A typical Ti particle radius is 10 to 100 m, while for C
particles the radius is about 0.1 m for carbon black and 1 to 10 m for milled
graphite. At temperatures below the Ti melting point, Tm(Ti)1940 K, a thin uniform
layer of titanium carbide is formed on the surface of spherical Ti particles. Because of
low carbon solubility in -Ti and high diffusivity of C atoms in solid solution as
compared with that in TiC, we can neglect the diffusion flux of C atoms in solid -Ti,
and consider the formation of only one product phase, titanium carbide. Then the TiC
layer thickening in the SHS wave is described in spherical symmetry using a
diffusion-type Stefan problem neglecting the volume change at the Ti/TiC interface
c/t = r2 D(T(t)) (r2 c/r)/r,
(1)
(c21–c1)dR1/dt = D(T(t)) c(R1(t))/r,
(2)
where D is the chemical diffusion coefficient in TiC, which is usually associated with
carbon diffusion through the carbide layer, c is the mass concentration of carbon, R1(t)
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is the current position of the TiC/Ti interface, R2 is the outer radius of a Ti particle,
co1, co21 and co23 are equilibrium concentrations according to the Ti-C phase diagram..
The initial and boundary conditions to diffusion equation (1) are
c(t, R2) = co23,
c(t, r=0) = co1,
c(t, R1(t)) = c21,
R1(t=0) = R2.
(3)
Since in Refs.(4,5) the Stephan-type boundary condition (Eq.(2)) was postulated
at both Ti/TiC and TiC/C interfaces, it should be outlined that in interstitial
compounds such as nitrides, carbides, borides, etc., the partial diffusion coefficients of
the metal and non-metal species differ by orders of magnitude because the atoms
diffuse on their own sublattices. Hence the TiC layer grows at the Ti/TiC interface
due to the lattice transformation of -Ti into TiC, which is controlled by the diffusion
transport of C atoms across the TiC layer, while at the C/TiC interface no lattice
transformation of the graphite into TiC can occur. Thus, the first-kind boundary
condition, c(t, R2) = co23, 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 fact surface diffusion of the C atoms from the Ti/C contact spots.
Non-linear problem (1)-(3) can be solved only numerically, however, for a
similar linear problem with D=const, an asymptotic solution the phase layer
thickening is known (9) which is valid for a small product layer thickness, h << R2.
Linearizing Eqs.(1),(2) using substitution =Error! we obtain:
h() = R2 – R1() = 1/2 + 1/R2 + 23/2/(2R22),
1 = 2/(3+2/2),
(4)
2 = 21 – 1 [24s/+4(5+6s) + 2 (1+s)]/[32+60s+ (18+20s)+ (1+s)],
2
3
2
4
where  is a solution of a transcendental equation arising from a similar problem for a
semi-infinite sample,
1/2(/2) exp(/2)2 erf(/2) = s,
s = (c23c21)/(c21c1).
(5)
The parameters for calculating the diffusion coefficient in the Arrhenius form,
D=D0exp[E/(RT(t))], where E is the activation energy and D0 the preexponential
factor, are presented in Table 1 (10-12 et al.). Since extrapolation of D to the whole
temperature range of SHS may bring about overestimated values, the diffusion
coefficients in TiC calculated at T=Tm and the maximum SHS temperature, Tc = 3083
K (13) were compared with those in molten titanium. The diffusion coefficient of C
atoms in the titanium melt was estimated using Stokes-Einstein formula Dm =
kBT/(6a), where kB is the Boltzmann constant, a is the atomic radius in the melt, 
= m is the dynamic viscosity,  is the kinematic viscosity, m is the density of the
liquid phase. For C atoms the covalent radius is a= 0.077 nm, the density of molten
Ti is  = 4.11 g/cm3 (14). The kinematic viscosity of molten Ti saturated with carbon
at T=Tm is  = 0.9410–2 cm2/s (14), then Dm(Tm)  4.810–5 cm2/s. For higher
temperatures the value  = 1.0310–2 cm2/s at T= 2220 K is known (14); using it at
T=Tc yields Dm(Tc)  6.910–5 cm2/s. It should be noted that these Dm values are the
upper-limit estimates since the Stokes-Einstein formula doesn’t include the chemical
interaction in the melt, which for the Ti-C system may be substantial. Then the
diffusion coefficients in TiC, which are close to or higher than the upper estimate of
Dm, were excluded from consideration (lines 9-12 in Table 1).
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Table 1. Available data on diffusion coefficients at TiC layer growth (10-12 et al.)
Species No. D0, cm2/s E, kJ/mol D(Tm), cm2/s D(Tc), cm2/s
Note
235.6
1.
5102
2.3108
5.1106
10

6.98
398.7
2.
1.310
1.23106
10
438.9
3.
1.51011
3.7107
45.44
447.3
4.
4.11011
1.2106
11

114
460.2
5.
4.610
1.8106
0.1
259.4
6.
1.0108
4.0106
C
269.9
7.
6.5102
3.5109
1.7106
2
10


307.1
8.
4.210
2.310
2.6107
220
405.8
9.
2.6109
2.9105
D(Tc)  Dm
9
5


370
410.0
10.
3.410
4.110
D(Tc)  Dm
347.3
11. 1.31103
5.8107
1.7103
D(Tc)  Dm
8
4


12.
77.8
338.9
5.810
1.410
D(Tc)  Dm
4
16
8


Ti
736.4
4.3610
6.510
1.510
DTi  DC
For calculating , the time dependence of the temperature of a Ti particle, T(t),
must be evaluated. For relatively low temperatures, T0 < T < Tm, the temperature
profile was calculated using the analytical solution for a narrow reaction zone (15):
T(xvt) = T0 + (TcT0) exp[v(xvt)/]
(6)
where v is the combustion wave velocity, x is a coordinate along the SHS-sample,  is
the thermal diffusivity. To determine T(t) for an individual Ti particle, a coordinate x1
was chosen where T(x1,t=0) = T0+0.01Tm, T0=298 K. Then the heating time to Tm is
tm = (/v2) ln[0.01Tm/(Tm–T0)]. For a stoichiometric Ti-C mixture (20 wt.% C),  
0.04 cm2/s and v = 6 cm/s (13,16). For higher temperatures, Tm  T  Tc, splineapproximation of the experimentally registered temperature profile (13) was used.
Calculations have shown that the maximal TiC layer thickness, h, grown on the
titanium particle surface at the attainment of the maximal SHS temperature, Tc, is
small: h(Tc)  1.6 m for E = 235.6 kJ/mol, D0 = 5102 cm2/s (line 1 in Table 1) at
the assumption that molten titanium retains inside the TiC case. The heat release
calculated for the TiC1.0 formation without taking into account the temperature
dependence of the specific heat (i.e., an upper-level estimate), is insufficient to sustain
the SHS wave propagation: the adiabatic temperature rise Tad = 1064 K < Tc for the
Ti particle radius R2 = 10 m, and sharply drops with increasing R2. The TiC layer
thickness attained by reaching the titanium melting temperature is still smaller: h(Tm)
 0.02 m for R2 = 10 m, and the associated adiabatic heating is insignificant, Tad
= 14 K. However, at the attainment of T=Tm titanium melting can result in the rupture
of the primary TiC case and spreading of the metallic melt.
RUPTURE OF PRIMARY TiC CASE AND SPREADING OF MOLTEN Ti
The density of solid -Ti at T=Tm is s=4.18 g/cm3 while for molten titanium at
the same temperature m=4.11 g/cm3 (14). The conditions for the rupture of the
primary TiC case because of the dilatation of the titanium core during melting are
determined from the continuity equation for spherical symmetry with relevant
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boundary conditions:
grad div ur = 0,
ur(r=R1) = R1[(s/m)1/31],
rr(r=R2) = p0,
(7)
where ur is the radial displacement, rr is the radial stress, p0 = 0.1 MPa is the outer
pressure. Since the plasticity of TiC is low, we consider only elastic deformation.
Then, using the theory of elasticity (17), we obtain a criterion for the rupture of the
primary TiC case, which can occur when the maximal tangential stress at r=R2
exceeds the ultimate tensile stress:
hr = R2(1),  = Error!1/3,  = Error![(sm)131],  = Error!
(8)
where B is tensile strength,  the Poisson ratio, M the elastic modulus. Rupture of the
TiC spherical layer occurs at h > hr. Using the data on the mechanical properties of
titanium carbide at T=Tm (10 et al.) we obtain an estimate hr  (0.70.6)R2.
Since the calculated value h(Tm) is very small, for any size of Ti particles used
in SHS (R2=10 to 100 m) melting of the titanium core of a particle in the SHS wave
will inevitably bring about the rupture of the primary TiC case and spreading of the
melt. This will change the geometry of a unit reaction cell composed of a Ti particle
and carbon in a stoichiometric ratio for the TiC1.0 formation.
GROWTH OF TiC LAYER ON THE SURFACE OF A CARBON PARTICLE
The spreading of molten titanium towards solid carbon is accompanied with
chemical interaction, and for small-sized C particles the spreading velocity is not the
rate-limiting stage (18). Hence we consider that at T = Tm the carbon particles in the
SHS wave are enveloped with liquid Ti, and product formation occurs on the surface
of a carbon particle due to diffusion of the C atoms across the growing TiC layer
towards the Ti(melt)/TiC interface, where the build-on of the TiC crystal lattice
occurs. In this case, initial and boundary (at r=R0) conditions to Eqs.(1),(2) look as
c(t, R0) = co23, c(t, r>R1) = co1, R1(t=0) = R0,
(9)
where R0 is the initial radius of the carbon particle. Here it is implied that the carbon
content of the melt is constant corresponding to the solubility limit. Using the
approach developed in Ref.(9), for this situation we obtain, similarly to Eqs.(4),(5), an
asymptotic solution to Eqs.(1),(2),(9) with respect to h():
h() = R1() – R0 = 1/2  1/R0 23/2/(2R02),
(10)
where coefficients , 1 and 2 are calculated as previously.
The maximal calculated product layer thickness formed in the SHS wave for a
sufficiently large carbon particle size, R0 = 10 m, is h = 1.5 m at the set of
diffusion parameters No.2 in Table 1.
Assuming uniform composition of the product phase, TiC1.0, through the layer
thickness and neglecting the temperature dependency of heat capacities, maximal
adiabatic heating of a reaction cell (a carbon particle enveloped with molten titanium
with a stoichiometric C-to-Ti mass ratio, 0.2) for incomplete conversion of carbon (C
= 1mC()/mC > 0, where mC is the mass of a C particle) is estimated as
2 - 77
Tad = Error!
(11)
where i is the density and cp(i) the specific heat of the i-th substance, H298(TiC) is
the standard enthalpy, subscript m refers to the melt. For complete conversion of
carbon into carbide (C=0), the maximal adiabatic temperature rise is max(Tad) =
-H298(TiC)/cp(TiC).
Calculated adiabatic heating, Tad, for different diffusion coefficients in TiC is
shown in Fig.1; the numbers at lines correspond to the sets of parameters in Table 1.
A plateau at small R0 values corresponds to the complete conversion of carbon into
titanium carbide. From Fig.1 it is seen that for small-sized carbon particles (R0 
3m) the mechanism of diffusion-controlled growth can provide complete conversion
of reactants into the final product and sufficient adiabatic heating and heat release rate
to sustain the SHS wave in the Ti-C system.
Tad, K
Fig.1. Calculated adiabatic
heating due to diffusioncontrolled TiC layer growth on
the surface of a carbon particle
after titanium spreading in the
SHS wave versus the particle
radius.
4000
3000
1
2000
1000
5
7
6
2,4
However, as mentioned
above, because of the interstitial
0
5
10
15 diffusion mechanism through
R0, m
the TiC layer the latter grows on
the Ti/TiC interface. Hence the
TiC particles formed after the complete conversion of the reactants will be hollow.
Let’s estimate the displacement of the C/TiC interface, i.e., growth of the reaction
product towards carbon which occurs due to diffusion of Ti atoms across the TiC
layer. Then the Stefan-type boundary conditions to Eq.(1), which is written for
diffusion of both C and Ti atoms for the case of a semi-infinite rod, are formulated at
the interfaces Ti(melt)/TiC (r=R1(t)) and C/TiC (r=R0(t)):
(c21–c1)dR1/dt = DC(T(t))c(R1)/r, (1–c23)dR1/dt = DTi(T(t))c(R0)/r,
(12)
where DC and DTi are the partial diffusion coefficients of C and Ti atoms in TiC.
Using substitution i=Error!, iC,Ti, we reduce the problem to an isothermal case
which has an analytical solution (19) for the displacement of the phase boundaries
h() = R1()  R0 = C C1/2,
() = R0()  R0 = Ti Ti1/2,
(13)
where coefficients C and Ti are found from the following transcendental equations:
1/2C exp(C/2)2[erf(C/2) + erf(Ti(Ti/C)1/2/2)]/2 = (c23c21)(c21c1)
(14)
2 - 78
1/2Ti exp(Ti/2)2[erf(Ti/2) + erf(C(C/Ti)1/2/2)]/2 = (c23c21)(1c23).
Calculations have shown that due to the smallness of the Ti partial diffusivity,
DTi << DC (Table 1), the displacement of the C/TiCx interface during all the heating
period in the SHS wave is vanishingly small:  = 4.7 nm << h. Thus, we can estimate
the apparent density of hollow TiC1.0 particles formed in the SHS wave by the above
described quasi-equilibrium diffusion-controlled growth mechanism: eff = 1/[1/TiC +
1/(5C)]  3.3 g/cm3 (for TiC = 4.91 (10-12), C  2 g/cm3), which is 67% of the
theoretical density of the titanium carbide TiC1.0. But, according to numerous
experimental data, the relative density of the titanium carbide particles produced by
SHS is close to 100%.
A MICROMETALLURGICAL MODEL OF PHASE FORMATION IN SHS
The above-stated consistent consideration of the phase formation in nonisothermal conditions taking into account experimental data on the diffusion
coefficients, boundary conditions characteristic of the growth of interstitial phases, the
thermal profile of the SHS wave and a change of spatial arrangement of reactants due
to melting and spreading of a metallic reactant have demonstrated that widely used
model based on the concept of quasi-equilibrium diffusion-controlled phase layer
growth actually is not applicable to modeling SHS of interstitial compound such a
carbides, borides, nitrides etc. This is because the physical meaning of the results (the
product structure and corresponding density) obtained within this approach disagree
with experimental data. It should be noted that formal calculation of the product layer
thickness and corresponding heat release for small-sized particles of a non-metallic
reactant can bring about numerical results supporting the existing model.
Therefore, only a non-equilibrium mechanism involving a direct contact of solid
carbon with molten Ti without a continuous TiC interlayer separating the reactants
can operate in the SHS wave. In this situation, the formation of dense product can
occur via partial dissolution of carbon in liquid titanium at the C/Ti(melt) interface
and subsequent crystallization of TiC grains. Then diffusion mass transfer in the
liquid phase is not a rate-limiting stage because of high diffusion coefficients in hightemperature melts, D ~ 105104 cm2/s, and the phase-forming process responsible
for major heat release is heterogeneous crystallization of the product particles. This
mechanism qualitatively agrees with the experimental results on phase formation
during SHS in the Ti-C system obtained in Ref.(16) on rapidly quenched samples.
Hence it seems reasonable to employ the Kolmogorov-Avrami formalism (20)
for the description of the crystallization kinetics and relevant heat release in the SHS
wave. For a 3-dimentional case, the heat transfer and kinetic equations look as
cp T/t = (T) + Q /t,
 = 1  exp[fError!(Error!)3d]
(15)
where  is the thermal conductivity, Q = H298(product) is the heat release per unit
mass,  is the volume fraction of the crystallized product, f is the shape factor, t0 is
time instant when the product crystallization starts at a given point, I is the
heterogeneous nucleation rate and G is the crystal growth rate.
CONCLUSION
The consistent analysis performed in this work has demonstrated the limitations
2 - 79
of the quasi-equilibrium concept of solid-state diffusion-controlled phase layer growth
with respect to the formation of interstitial compounds in SHS waves. It is shown that
formal calculations of the phase layer thickness and heat release within the frame of
the quasi-equilibrium model for small-sized particles of a non-metallic reactant may
support this approach while thorough consideration of the results referring to the
product structure suggests that only a non-equilibrium interaction mechanism can
accord with experimental data. A qualitative physical mechanism is proposed which
implies direct contact of particles of a solid non-metal with a metallic melt without the
presence of an equilibrium interlayer of a solid product. Fully-dense product particles
are formed through dissolution of carbon in the melt and subsequent heterogeneous
crystallization of the compound phase. A mathematical description of phase formation
in SHS wave involving the Kolmogorov-Avrami formalism is proposed.
The work was partly supported by the Belarussian Fundamental Research
Foundation (grant Х02Р-037).
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