Modeling of advanced materials processing on the SHS basis

0098-1354/94 $6.00+0.00 Copyright © 1993 Pergamon Press Ltd

Computers chern. Engng, Vol. 18, Suppl., pp. S253-S258, 1994 Printed in Great Britain. All rights reserved

MODELING OF ADVANCED MATERIALS PROCESSING ON THE SHS BASIS

B.M. KHUSID and B.B. KHINA Heat and Mass Transfer Institute, Byelorussian Academy of Sciences, 15, P.Brovka st., 220072, Minsk, Republic of Belarus

ABSTRACT Mathematical models for heat and mass transfer, phase and structural transformations during the self-propagating hightemperature synthesis of advanced ceramic materials are developed. The main results of computer simulations are presented. SHS-based methods for producing final articles with prescribed structure and properties are briefed. KEYWORDS Self-propagating high-temperature synthesis, reaction kinetics, structure formation, stochastic modeling, autocatalytic effect. INTRODUCTION In the area of Chemical Engineering, computer modeling is being used for a long time. In Materials Science and Engineering, computer models are traditionally employed in restricted areas, e.g. calculations of multi-phase equilibria, plastic deforming of metals, crystallization, solid-state diffusion, etc. The development of novel intensive methods of material processing such as laser, ion-beam and electron-beam treatment of metals and alloys, combustion synthesis (or the so-called selfpropagating high-temperature synthesis (SHS» of advanced composite materials, etc., necessitates elaborating new mathematical models. Therefore, a new expanding area of research, namely computer modeling of intensive high-temperature processes in Materials Science and Engineering, has attracted considerable attention in recent few years. The most important and challenging problem in this area is the mathematical description of intricate physical and chemical phenomena involved, e. g. heat and mass transfer, high-temperature phase, chemical and structural transformations, etc., and their effects on structure, composition, and properties of the material to be obtained. In this lecture, an attempt is made to consider such problems as applied to SHS of advanced ceramic and composite materials. The mathematical models developed are briefed, and their possible applications are outlined. The SHS is based on a wave-like propagation of an exothermic reaction along a charge mixture S253

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(Merzhanov, 1983; Varma and Lebrat, 1992). Typical examples of the SHS are: (a) carbide system: Ti + C ~ TiC, T~2500 °c, wave velocity u=3-4 cm/s; (b) oxide system: 3Cu + 2BaO 2 + ~Y 2 20 3 + 0 2~ YBa 2 Cu 3 0 7-x ,T~1000 °c, u=2-5 mm/s; (c) organic system: C 4 H10 N2 (piperazin)+C 3 H 4 0 4 (malonic acid)~C 7 H14 NO, T=155 °c, u=O.6-1.5 2 4 mm/s (Khusid et al., 1993).

MODELING OF MULTI-STAGE PHENOMENA IN SHS WAVE Numerous physical and chemical phenomena involved in the SHS wave have a complicated multi-stage nature and can proceed via formation of a transient melt, intermediate and/or metastable phases absent in the final product. Therefore, the mathematical description of multi-stage high-temperature interactions in the SHS wave is of primary importance for predicting the SHS wave behaviour as well as the structure and phase constitution of the material to be obtained. Within the traditional frame of the heterogeneous combustion theory (Zeldovich et al., 1985; Merzhanov, 1983), the SHS wave propagation with two sequential reactions A 1 ~A 2 ~A 3 is described as (Khaikin et al., 1968) ( 1)

n

aT//at = F(T/1,T) = (1-T/1) 1 kt"exp[-Ecr/(RT)] aT//at = F(T/1,T/2,T) = (T/1-T/2)n

2

k 2"eXp[-E(1-crE )/(RT)]

(2) (3)

Here p, c, A are the density, specific heat, and thermal conducti vi ty , respecti vely; Q=Q 1+Q2 ' Q1 is the heat release of the i-th reaction (i=1,2); T/ 1 is the conversion degree: T/1=1-a 1 , T/ 2 =a 3 = 1-a 1 -a 2 , where a j ' is the volume fraction of the substance A (j=l to 3) involved in the reactions; n is the reaction J 1 order; cr =Q /Q, (j =E /E, E=E +E , E is the activation energy Q

1

E

1

121

and kl the preexponent for the i-th reaction. However, the appearance of a transient liquid phase in the SHS wave may cause a drastic change in the interaction mechanism and thus influence the final structure of the product. A Model with Microtransfer and Autocatalysis To reveal the effects of melt formation in the SHS wave on the transformation mechanism, a model taking account of the microtransfer and autocatalysis has been developed (Khusid, 1992). For reaction A +B ~(A,B) ~C where the subscripts sand B

BIB

1 denote solid and liquid state, correspondingly, the effect of the conversion degree gradient, VT/, can be significant. For a cluster of particles this effect is connected with diffusion in a melt that equalizes the liquid composition around different particles. Within the phenomenological approach, changes in the

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conversion degree for an isotropic mixture are described as (4)

where r~ is the reaction rate under homogeneous conditions, r is the kinetic parameter, h is the scale of spatial inhomogeneity, or the spatial correlation length. Eq. (4) contains the first term of the expansion of the reaction rate as a series in the spatial derivatives. For a single particle, eq.(4) is identical to the equation for the phase-change kinetics which includes an interface width. It can be derived from the expression for the free energy, F, of an inhomogeneous system such as is used in the Ginzburg-Landau (or Cahn-Hilliard) theory a7)

at

= _rc3F

c3 7)'

af a7)

(5)

where f is the specific free energy of a homogeneo~s state, h is the interface width. In the present case r~104-10 s-! h~10-100 nm. The asymptotic analysis of eqs.(l), (4) and (5) has shown that two different regimes of the SHS wave propagation are possible: (a) if the conversion degree gradient is negligible, the traditional combustion model (Zeldovich et al., 1985), where the SHS wave velocity is determined by the reaction rate at the adiabatic temperature, T ad , can be used; (b) if the effect of the

conversion-degree gradient prevails, the so-called FisherKolmogorov model (Zeldovich et al., 1985), where the reaction wave propagates in nearly isothermal conditions with the velocity u~rh, is valid. In this case, the temperature increase in the SHS-wave preheated zone initiates the autocatalytic chemical reaction that is responsible for the wave propagation. stochastic Model for Multi-stage Interaction in SHS Wave An essential feature of the SHS process is the stochastic nature of high-temperature chemical and structural transformations in a heterogeneous medium due to differences in the particle shape, size, surface structure, etc. Since a relatively small number of particles is present in the SHS wave reaction zone that has a width of 1 to 100 ~m, the stochastic effects may be significant, especially under unstable wave propagation regimes. A stochastic model for SHS with a one-stage reaction has been developed and compared with the classical deterministic model (Astapchik et al., 1993; Khusid and Khina, 1992). It has been shown that continuous generation of two-dimensional disturbances in the stochastic model is responsible for spontaneous origination of the spin combustion regime in the corresponding range of the parameters ~=RT a d IE and 7=cT ad IQ. Also, this provides a gradual transfer from a steady-state regime to the chaotic one and then to the spin mode with changing ~ and 7. To study the effect of the interaction stages on the SHS wave propagation regimes, a stochastic model for mUlti-stage sequential reactions is developed. Each cell is characterized by a temperature, a, and state, 7), the latter attaining M+1 values;

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~=O corresponds to initial state while ~=M refers to the final product. At the (n+1) temporal step the temperature of a cell having the coordinates (k,l) is described as

A 1; B(k,1,n+1) = B(k,l,n) + q(k.l,n+1) + cp h 2 [B(k-1,1,n) + B(k+1,1,n) + B(k,l-l,n) + B(k,l+l,n) - 4B(k,1,n»), q(k.l,n+1) =

(Ql/c)[~(k,l,n+1)

-

~(k,l,n»),

(6)

i=~(k,l,n)

where h is cell size; 1; is a temporal step; q is the temperature rise in a cell due to the i-th reaction, i=l to M. The temperature dependence of the i-th transformation in a given cell is described by the Arrhenius law

In the deterministic model (1)-(3) for two-stage reaction, three regimes are observed (Zeldovich et al., 1985; Khaikin et al., 1968): (a) at 0.5
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developed. (Khusid et al., 1992). Eqs. (1)-(3) for a twodimensional sample are used together with the boundary condition ABT/Byl

y=o

= a(TI y=o - T c )

(8)

where a is the heat exchange coefficient, T is the temperature c of a cooling medium. computer simulations have shown 2 that at 0.5
where ~ is the stress tensor, ~ is the deformation rate tensor, E is the unit tensor, p is the relative density, Il and l; are the shear and bulk viscosities. Computer modeling allows one to determine the regime of die-pressing and extrusion to get a proper density distribution in' the final article (Stelmach et al., 1992).

..

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CONCLUSION The above-listed models demonstrate the vital importance of the computer simulation of complex phenomena involved into SHS processes. Nevertheless, these models might be widely used for many other high-temperature processes of manufacturing composite materials and final articles with the desired structure and properties. REFERENCES Astapchik, A.S., E.P. Podvoiski, !.S. Chebotko, B.M. Khusid, A.G. Merzhanov, and B.B. Khina (1993). Stochastic model for a wavelike exothermal reaction in condensed heterogeneous systems. Phys. Rev. A, 47, No 1. Borovinskaya, I.P., A.G. Merzhanov, A.S. Mukasyan, A.S. Rogachev, B.B. Khina, and B.M. Khusid (1992). Macrokinetics of structure formation during filtration combustion in the titanium-nitrogen system. Dokl. Acad. Nauk SSSR, 322, 912-917. Khaikin, B.M., A.K. Filoneko, and S.I. Khudyaev (1968). Flame propagation with two sequential reactions proceeding in a gas. Fizika Goreniya i Vzryva ~, No 4, 591-600. Khusid, B.M. (1992). Features of models for self-propagating high-temperature synthesis. Int. J. of SHS. ~, 48-54. Khusid, B.M. and B.B. Khina (1992). Mathematical modeling of structure formation during combustion synthesis of advanced materials. In: European Symposium on computer Aided Process Engineering-2 (ESCAPE-2). Supplement to Computers [; Chemical Engineering 17 (D. Depeyre, X.Joulia, B.Kochret and J.-M. Ie Lann, ed.), pp.S245 - S250. Pergamon Press, Oxford. Khusid, B.M, B.B. Khina, Vu DUy Quang, and E.A. Bashtovaya (1992). Numerucal study of quenching the SHS wave with a two-stage reaction. Fizika Goreniya i Vzryva 28, No 4, 76-82. Khusid, B.M., V.A. Mansurov, A.D. Ubortsev, Z.P. Shulman, and A.G. Merzhanov (1993). Phase and chemical transformations in the self-propagating wave of an exothermal reaction in the mixture of organic powders. J. Chem. Soc. Faraday Trans. 89 (in press). Merzhanov, A.G. (1983). Self-propagating high-temperature synthesis. In: Current Topics in Physical Chemistry (Ya.M. Kolotyrkin, ed.), pp.6-45. Khimiya, Moscow. Merzhanov, A.G. (1990). Self-propagating high-temperature synthesis: twenty years of search and findings. In: Combustion and Plasma Synthesis of High-Temperature Materials (Z.A. Munir and J.B. Holt, ed.), pp.1-53. VCH, New York. Stelmach, L.S., A.M. Stolin, and B.M.Khusid (1992). Extrusion rheodynamics for a viscous compressible material. J. Engng. Phys., 61, No 2, 1011-1018. . strunina, A.G., A.V. Dvoryankin, and A.G. Merzhanov (1983). Unstable regimes of combustion of termite systems. Fizika Goreniya i Vzryva, 19, No 2, 30-36. Varma, A. and J.-P. Lebrat (1992). Combustion synthesis of advanced materials. Chem. Engng. Sci. 47, 2179-2194. Zeldovich, Ya.B., G.!. Barenblatt, V.B. Librovich, and G.M. Makhviladze (1985). The Mathematical Theory of Combustion and Explosions. Plenum Press, New York.