MODELING SELF-PROPAGATING HIGH-TEMPERATURE SYNTHESIS: A MICROMETALLURGICAL APPROACH
B.B.Khina1, B.Formanek2 and O.S.Rabinovich3
1Physico-Technical Institute, National Academy of Sciences, Minsk, Belarus
2Silesian University of Technology, Katowice, Poland
3A.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 (1et 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)nexp(m)kexp[E/(RT)], where is the degree of chemical conversion (01), 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) (2et 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. (4,5et 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 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 (8et 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 (3-6et 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 = r2 D(T(t))(r2 c/r)/r,(1)
(c21–c1)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) 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)) = c21,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 = we obtain:
h() = R2 – R1() = 1/2 + 1/R2 + 23/2/(2R22),1= 2/(3+2/2),
(4)
2 = 21 –12[24s/+4(5+6s)+23(1+s)]/[32+60s+2(18+20s)+4(1+s)],
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 = (c23c21)/(c21c1). (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 (10-12et 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/(6a), 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 = 0.9410–2 cm2/s (14), then Dm(Tm) 4.810–5 cm2/s. For higher temperatures the value = 1.0310–2 cm2/s at T= 2220 K is known (14); using it at T=Tc yields Dm(Tc) 6.910–5 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).
Table 1. Available data on diffusion coefficients at TiC layer growth (10-12et al.)
Species / No. / D0, cm2/s /E, kJ/mol
/ D(Tm), cm2/s / D(Tc), cm2/s /Note
C
/ 5102 / 235.6 / 2.3108 / 5.11066.98 / 398.7 / 1.31010 / 1.23106
10 / 438.9 / 1.51011 / 3.7107
45.44 / 447.3 / 4.11011 / 1.2106
114 / 460.2 / 4.61011 / 1.8106
0.1 / 259.4 / 1.0108 / 4.0106
6.5102 / 269.9 / 3.5109 / 1.7106
4.2102 / 307.1 / 2.31010 / 2.6107
220 / 405.8 / 2.6109 / 2.9105 / D(Tc) Dm
370 / 410.0 / 3.4109 / 4.1105 / D(Tc) Dm
1.31103 / 347.3 / 5.8107 / 1.7103 / D(Tc) Dm
77.8 / 338.9 / 5.8108 / 1.4104 / D(Tc) Dm
Ti
/ 4.36104 / 736.4 / 6.51016 / 1.5108 / DTi DCFor 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(xvt) = T0 + (TcT0)exp[v(xvt)/](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, spline-approximation 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) 1.6m for E = 235.6 kJ/mol, D0 = 5102 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) 0.02m 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 boundary conditions:
grad div ur = 0, ur(r=R1) = R1[(s/m)1/31], 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), = 1/3, = [(sm)131], = (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 (10et al.) we obtain an estimate hr (0.70.6)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= 1/21/R023/2/(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 = 1mC()/mC > 0, where mC is the mass of a C particle) is estimated as
Tad =(11)
where i is the density and cp(i) the specific heat of the i-th substance, H298(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) = H298(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 3m) 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.
Fig.1. Calculated adiabatic heating due to diffusion-controlled TiC layer growth on the surface of a carbon particle after titanium spreading in the SHS wave versus the particle radius.
However, as mentioned above, because of the interstitial diffusion mechanism through 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)):
(c21–c1)dR1/dt = DC(T(t))c(R1)/r, (1–c23)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=, iC,Ti, we reduce the problem to an isothermal case which has an analytical solution (19) for the displacement of the phase boundaries
h() = R1()R0 =C C1/2, () = R0()R0 = Ti Ti1/2, (13)
where coefficients C and Ti are found from the following transcendental equations:
1/2Cexp(C/2)2[erf(C/2)+erf(Ti(Ti/C)1/2/2)]/2 = (c23c21)(c21c1)
(14)
1/2Tiexp(Ti/2)2[erf(Ti/2)+erf(C(C/Ti)1/2/2)]/2 = (c23c21)(1c23).
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/(5C)] 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 non-isothermal 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 high-temperature melts, D ~ 105104 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.