A Kinetic Model for Metal Nonmetal Reactions
A.M. KANURY In attempts to mathematically model the process of gasless selfpropagating hightemperature synthesis (SHS) of materials, the existing chemical kinetic rate relations are found to be unable to explicitly account for the dependence of the reaction rate on reactant particle size, reactant proportions in the supply mixture, and presence of an inert. A rate expression is developed in this article based on the following mechanism: the metal melts and flows around the nonmetal particle; an intermediate (liquid) complex is formed; the metal ions diffuse from the melt, across the complex, to readily react with the nonmetal at its surface; the product of reaction is the complex itself, which dissolves on the metalmelt side at such a rate that the thickness of the complex is proportional to the instantaneous diameter of the nonmetal particle. As the concen tration of the metal in the outer melt decreases with time, the nonmetal particle gradually be comes smaller. Coupling these two temporally varying quantities, a relation is obtained for the instantaneous rate as dependent on temperature, supply particle size, compaction density, reac tant proportions, and the inert content. I. INTRODUCTION CONVERSION of chemical energy into thermal en ergy is the main objective of ordinary combustion pro cesses. The products of combustion then are of concern only in the contexts of combustion efficiency, thermal radiation from the flames, and pollution effects of the exhaust. Turning the problem around, certain reaction processes which behave much like ordinary combustion may very well be suited to economically produce certain desired products. Combustive synthesis or self propagating hightemperature synthesis (SHS) is just such a method of producing a solid product. Transition met als, as well as other metals, such as magnesium, man ganese, iron, cobalt, and nickel, are now well known to react with nonmetals, such as boron, carbon, silicon, phosphorus, and sulfur, in a selfsustainingly exothermic manner to produce numerous important products. For obvious reasons, SHS reactions of the type [solid + solid ~ solid] are also termed gasless combustion, al though truly gasless systems are indeed rare. Self propagating hightemperature synthesis reactions in which gaseous reactants are involved to produce nitrides and hydrides are also possible. The present article deals only with gasless combustion. The concept of SHS has been formulated and proven by Merzhanov l~] and coworkers with efforts spanning over two decades. Over the past few years, there is a great deal of SHS research activity in this country and elsewhere, t2] Both the science and technology of com bustive synthesis, however, are but in an emerging stage to date. A number of important issues remain to be re solved. One such issue is related to the very definition and nature of a reaction in solid state. A theoretical model describing the chemical kinetics
A.M. KANURY, Professor, is with the Department of Mechanical Engineering, Oregon State University, Corvallis, OR 973316001. This article is based on a presentation made in the symposium Reaction Synthesis of Materials presented during the TMS Annual Meeting, New Orleans, LA, February 1721, 1991, under the auspices of the TMS Powder Metallurgy Committee.
of reaction in a homogeneous mixture of particles of a melting solid and a nonmelting solid is presented in this article. The model reaction considered is that of titanium reacting with carbon to form titanium carbide. It appears that the present development can be easily extended to other similar reactions. We specifically seek to develop an equation for the intrinsic reaction rate (kilograms or kmoles of carbon consumed per unit time per unit vol ume of the compact) as dependent on the temperature, particle size, compaction density, reactant proportions, and dilution with the inert product. Such a rate equation is necessary as a constitutive relation for inclusion in the oretical models seeking to predict the rate of propagation of the SHS reaction wave in a reactant compact of any arbitrary configuration. The reasons for the unavailability of such an intrinsic rate equation in the existing literature can only be sur mised. For reactions involving only gaseous reactants, the concept of collision frequency offers a reasonable starting point to arrive at a rate equation. For solidstate reactions, however, this concept is of little or no use. Whereas gas molecules are of the order of angstroms in size and are relatively free to translate in space, solid particles are generally of the order of tens of microns in size, usually constrained in a crystal matrix which can be excited vibrationally but not translationally. The rate of an SHS reaction is intuitively expected to be higher at higher temperature and to become zero when either of the reactants is completely depleted. Most of the existing theoretical analyses of the SHS process con trive a rate equation to account for these trivial depend encies on the basis of the extensive experience either with gas reaction kinetics or with metal tarnishing and rusting reactions. The temperature dependency is usually taken to be described by an Arrheniustype exponential. This turns out to be correct, since diffusion in condensed phase follows an exp
[E/(RT)]
type of temperature depen dence. [3 4 5] The influence of reactant concentrations C~ is taken, again based on the experience with gas reac tions, as: rate ~
CTC ~
r C ~, where n and m are positive constants known as the orders of the reaction with re spect to species 1 and 2, respectively. Since the particles
METALLURGICAL TRANSACTIONS A VOLUME 23A, SEPTEMBER 19922349
are large compared to typical gas molecules, the defi nition of carbon concentration is itself prone to some ambiguity. The concentrations are replaced in some works with (1  r/), where ~/is called the extent of reaction, which is zero at the start of the reaction and unity upon com plete reaction. The expression (1  7/) may be defined in a number of ways; it may be the ratio of the particle diameter to its initial diameter; the particle volume to the initial volume; the particle mass to its initial mass; the metal mass to its initial mass; or the metal concentration in the melt to its initial concentration. The term ~7 may also be defined as the mass or concentration of the prod uct in the melt normalized by its final value. Not all investigators appear to use the same definition of 77. We will note later in this article that there is no single def inition of the extent of reaction which would serve well under all conditions. Even more importantly, the exist ing literature appears to shed little light on the depen dence of the intrinsic reaction rate on variables such as the reactant particle size, the degree of dilution with the inert product, the excess reactant in the initial mixture, and the density to which the compact is compressed. Perhaps the most elegant available experimental study of the chemical kinetics of a reaction involving a melting reactant and a nonmelting one is that of Aleksandrov and Korchagin.161 Employing an electron microscope and Xray phase analysis with a synchrotron radiation diffracto meter, these authors have witnessed that the metal melts and flows around the nonmetal particle (Figure 1). A layer of lowmeltingpoint intermetallide complex is pro duced by reaction at the nonmetal surface. This inter metallide is presumably some form of a distinct metastable phase. Reference 6 contains experimental evidence in dicating that the layer of intermetallide complex, diffu sional transport across which is an essential element of the reaction process, arises not only in the thermite re action of Fe203A1 and intermetallic reaction of NiA1 but also in reactions involving Ti, Nb, or Ta with C and Ti, Nb, Ta, and Hf with B. It is assumed that metal ions and atoms diffuse from the outlying melt across the shell of the intermetallide complex to the nonmetal surface. Reaction at this sur
TI * TIC mixture melt in which the
concentr tion of tit nium gr du lly decre ses
with time. CTi liquid complex of thickness ~
cross
which titanium
c tions diffuse
to the carbon surface. Carbon
p rticle whose
diameter gradually
decre ses
with time due to reaction at its surface. Fig. 1Schematic of the Aleksandrov/Korchagin model.
face is assumed to produce the intermetallide at the same rate as it is dissolved at the outer surface into the metal melt. The intermetallide thus quickly attains a quasi stationarily constant thickness. As the reaction pro gresses gradually, the diameter of the nonmetal decreases with time; in the outlying melt, the metal concentration decreases, and the product concentration increases with time. Assuming that the rate of reaction at the nonmetal sur face is far higher than the rate of metal diffusion across the intermetallide complex, Aleksandrov and Korchagin t61 heuristically infer that the reaction rate is proportional to the instantaneous surface area S of the nonmetal particle and to the instantaneous concentration C of metal in the outlying melt. For a stoichiometric mixture in which the nonmetal particles are spherical, if ~7 is the extent of re action, based on the carbon particle volume or mass, S oc (1  ,/)2/3 and C ~ (1  r/) so that the rate
dTi/ dt oc
(1  r/) 5/3. The proportionality constant is set to be a function of temperature, t is time. By extension, for any mixture composition and particle shape,
drl/ dt ~
(1  /zT/) (1  ~7) (i~)/jl. The term/x is the ratio of the metal s stoichiometric molar coefficient to its ac tual molar coefficient such that/zr/ 1. The indicesj = 1, 2, and 3, respectively, stand for planar, cylindrical, and spherical nonmetal particles. It is obvious that if /z < 1, the mixture is metalrich, all the nonmetal gets consumed, and ~/goes from 0 to 1. If/z > 1, the mixture is nonmetalrich, and r/goes from 0 to 1//z. The mean ings of both 7/ and /z in this later case are not entirely clear. The effect of possible dilution of the initial mix ture with the inert product is not addressed by the heu ristic model of Aleksandrov and Korchagin. t61 II.
TH ORY
A Stoichiometry and Thermodynamics
If only for clarity of this presentation, let the metal be titanium and the nonmetal be carbon. Thus, the reaction of concern is 1 kmol of C + a kmol of Ti + b kmol of TiC > (b + 1) kmol of TiC +(a 1)kmolofTi: ifa>l > (b + a) kmol of TiC +(1a) kmolofC: ifa<l [1] a < 1, = 1, and > 1 represent carbonrich, stoichio metric, and titaniumrich initial mixtures, respectively. (We note that a is the inverse of Aleksandrov s/z.) The expressions b = 0 and >0 represent mixtures undiluted and diluted with the product, respectively. The present manner, evident from the left side of Eq. [1], of de scribing the starting mixture composition in terms of the mole numbers a and b is desirably unambiguous. Assuming complete conversion of Ti and C to TiC,
2350VOLUME 23A SEPTEMBER 1992 METALLURGICAL TRANSACTIONS A
the adiabatic reaction temperature can be obtained from the first law of thermodynamics as follows:
Tr,ad :
To + {hc/[ a

1 CpT
+ (1 +
b)CpTiC]}:
if a> 1
= To + {ahr  a)Cpc + a
+ b CpTiC]}:
ifa < 1 [2] where To is the supply temperature, hc is the enthalpy of combustion, and
Cpi
is the specific heat of ith species.
B. Kinetics
Although based on an experiment which is perhaps the most elegant and detailed available, the rate equation has been obtained by Aleksandrov/Korchagin in a rather qualitative way. The description of the starting mixture and the definition of the extent of reaction have been left with some clarity desired. In the following, we construct a simple theory based on Aleksandrov/Korchagin ex perimental observations. Our development follows the fundamental principles of conservation of species and overall mass in a system involving diffusional transport of the metal to the nonmetal surface at which the reac tion takes place. The reaction itself is assumed to be much faster than the diffusion process. This appears to be a reasonable assumption in many solidstate reactions aris ing in SHS, intermetallic alloying, metal + gas, as well as semiconductor processes.t7.8.9] In the context of Figure 1, let us denote the radial coordinate by r, the instantaneous spherical carbon par ticle diameter by dp, and the intermetallide thickness by 6. Let Cri be the concentration of titanium at any radial location in the complex,
CTim
the concentration in the The boundary conditions stipulate that the titanium con centration is zero at the particle surface due to the fast reaction: r =
dp/2,
Cvi = 0. It is assumed that titanium remains uniform at all times in the outlying melt so that its concentration is
CTim
(a function of time yet to be determined). Thus, at the outer edge of the melt product complex,
r = 6 + dpl2),
CTi = CTi m
Solution of Eq. [3] gives the molar diffusion rate of titanium across the melt complex to the carbon surface to be Ti molar diffusion rate (for spherical carbon particles)
= 7rdp dp + 26)DCTim/6
[4] The carbon molar consumption rate can be written in terms of the rate of decrease of the particle volume as C molar consumption rate (for spherical particles) = (1/Mc)
d/dt) [pcTrd3/6]
[5] where Mc and Pc are the molecular weight [kg/kmol] and density of carbon [kg/m3], respectively. The stoi chiometry indicated in Eq. [1 ] dictates that the two rates given by Eqs. [4] and [5] be equal, thus leading to one of the two equations needed for the two unknown func tions of time
CTim t )
and
dp t). 1/Mc) d/dt) [pcrrd3/6] =  rrdp dp + 26)DCr~m/8
[ ]
At time t = 0, the carbon particle diameter is dpo and the mixture composition is [1 kmol C + a kmol Ti + b kmol TIC]. At any time t > 0, the mixture contains by kmols C [ 7rd3/6]
[pc/Mc]
Ti
[Zrpc/ 6Mc)]
[(a  1)d~o + d3]: TiC
[rrpc/ 6Mc)]
[(b + 1)d~o  d3] by m 3 [~rd3/61 [(7r/6)
pc/Mc) MTffPTi)]
[(a  1)d~o + d3] [(7r/6)
pc/Mc) MTic/RTic)]
[(b + 1)d~o  d 3] With the kmols of Ti and volumes of Ti and TiC melts thus known, the concentrations can be easily found in terms of
dp t)
to close the problem with Eq. [6].
CTi m
or CTiCm
[(a
 1
+
dJdpo) 3]
or [(b +
1
dpldpo) 3]
[(a  1) +
dJdpo) 31
MTi/Pri) h
[(b + 1)

dp/dpo) 3]
(MTic/PliC) [7] melt.
d e
and CTim are unknown functions of time. Let D = Do exp
[E/ RT)]
be the coefficient of diffusion of titanium across the intermetallide complex, T tem perature, E activation energy, and R the universal gas constant. If the diffusion of titanium across the complex occurs much slower than the reaction at the carbon sur face, CT~(r) is governed by (1/r 2)
d/dr) [Dr 2 dCxi/dr]
= 0 [3] We can also find the melt diameter
d m
as
a function of time.
dm/dpo) 3 = dp/dpo) 3 + pe/Mr
9 {[(a  l) +
dJdpo) 3]
(MTi/Pri) + [(b + 1) 
dp/dpo) 3]
(Mvic/Pnc)} [8]
METALLURGICAL TRANSACTIONS A VOLUME 23A, SEPTEMBER 19922351
With the nondimensional variables and parameters defined as
A = dp/dpo; r
~
CTim/fTimO; * = D/Do /3 = 6/dp; r* = Dot/d~
Eqs. [6] and [7] transform to the following form:
dA3/d~ * = {(6Mc/Pc)
[(1 +
2/3)/fl]CT~.,o}A6A
[9]
~) : l/CTimO)
[(a  1) + A 31  {[(a  l) + A31
MTi/PTi)
+ [(b + 1)  A3I
(MTic/PTic)}
[10l The initial concentration of Ti in the melt is simply
CTim0 m
a/[a(MTi/PTi ) b b(MTic/P.ric) ]
[l 1] Solutions of Eqs. [9] and [10], with the initial conditions A(0) = 1 and ~b(0) = 1, give A and ~b as functions of It*; A, a, b,/3]. The molar volumes
(M/p)i are,
of course, constants. Before proceeding further with the theory of spherical particle, let us digress briefly to examine the effect of particle shape. If the particles are cylindrical pellets of length ~ and diameter dp rather than spheres of diameter dp, Eq. [3] can be written in cylindrical coordinates and solved (with the assumption of negligible lengthwise diffusion) to obtain the molar diffusion rate of titanium across the complex as
27r~DCTim/ln [(dp + 2~)/dp].
For ~
<dp,
this rate relation can be safely simplified to Ti molar diffusion rate (for cylindrical carbon particles)
= r
+ r162
[12] Thus, while Eq. [4] gives the titanium diffusion rate per unit area of the particle surface as [1 +
(28/dp)]DCTim/~
when the particles are spherical, Eq. [ 12] gives this rate as [1 +
(~/dp)]DCTim/~
when the particles are cylin drical. The effect of curvature is, thus, clearly more se rious when the particles are spherical. If the curvature is vanishing,
i.e., ~/dp > O,
both these rate relations lead to the planar particle limit in which the diffusion rate per unit area is
DCTi,n/t%
The equations corresponding to Eqs. [7] through [11] can be easily derived for the cy lindrical and planar particles. Since no new concepts are involved, we shall not carry out this derivation. Instead, we return to the spherical particle considerations. The physical meaning of the nondimensional variables A, ~b, and r* and parameters h and/3 needs to be men tioned here. The definition of nondimensional time r* is trivial in that the physical time t is measured in units of the diffusional time
(d2po/Do).
We note that the initial size of the carbon particle is embedded in the definition of nondimensional time. The parameter A, being of the form of the wellknown Arrhenius exponential, embod ies the temperature dependency of the diffusion rate. The particle diameter A is unity at time ~* equal to zero. If the supply mixture is titaniumrich,
i.e., the
mole number a > 1, all the carbon will be consumed in due time so that A > 0 as ~* > ~. As such, an extent of reaction can be defined as proportional to (1

A 3) so that it varies from zero initially to unity finally. Such an extent of reaction represents the volume fraction or mass fraction of carbon yet to be consumed at any given time. In this case of a > 1, some titanium will remain uncon sumed at infinite time. The titanium concentration ~b is unity at time r* equal to zero. If the supply mixture is titaniumdeficient,
i.e.,
the mole number a < 1, all the titanium will be con sumed in due time so that ~b ~ 0 as r* > o0. As such, an extent of reaction can be defined as proportional to
1 
~b) so that it varies from zero initially to unity fi nally. Such an extent of reaction represents the concen tration fraction of titanium yet to be consumed at any given time. In this case of a < 1, some carbon will re main unconsumed at infinite time. In order to see the conditions under which the extent of reaction defined on the basis of A above is equal to that defined on the basis of th, differentiate th(t) given by Eq. [10]. If the molar volumes of Ti and TiC are equal, we obtain
dqb/dT*
= (i/a)
dA3/d'r *
[13] independent of the dilution coefficient b. The solution of this equation is th = (A 3
+ a  1)/a.
If a = 1, ~b = A 3, so that the two foregoing definitions of the ex tent of reaction are equivalent only when (1) the molar volumes of Ti and TiC are equal and (2) the initial mix ture is stoichiometric. Finally, /3 is the thickness of the intermediate melt complex measured in units of the instantaneous diameter of the carbon particle. The thickness of the intermediate complex itself has been observed by Aleksandrov and Korchagin t6j to be about 0.1 /xm when the carbon par ticle is 40 /zm in diameter. Thus, under the observed conditions of titanium + carbon reaction,/3 is of the order of 0.0025. It is assumed that/3 is an experimentally de termined characteristic of the reaction mechanism. Since /3 is quite small, the factor [(1 +
2/3)//3]
in Eq. [9] may safely be approximated as equal to [1//3]. Thus,
dA3/dt o: qbA//3 o~
qbA 2. This result is entirely consistent with the argument that the rate is proportional to the product of the instantaneous titanium concentration and carbon particle surface area. This consistency motivates us to set/3
= 8/dp
= a constant for a given reaction so that as
dp
decreases with time, so also does & Aleksandrov and Korchagin appear to suggest that 8 itself is probably a constant characteristic of the studied reaction.
III. RESULTS AND DISCUSSION
A. Thermodynamics
If Ti and TiC are in solid state, the JANAF tables give hc = 183.3 + 8.4 MJ/kmol. The specific heats depend on temperature. However, for approximate estimation of the reaction temperature, they can be taken to be about 55, 40, and 25 kJ/(kmol K), respectively, for TiC, Ti, and C. Thus, for undiluted stoichiometric mixtures with To = 300 K, Eq. [2] gives the adiabatic reaction tem perature to be roughly 3630 K. Mixtures with a r 1 and b ~ 0 obviously lead to lower reaction temperatures.
2352VOLUME 23A, SEPTEMBER 1992 METALLURGICAL TRANSACTIONS A