On the Theory of Interphase Interaction in a Mixture of Reacting Metal Particles.pdf

Combustion, Explosion, and Shock Waves, Vol. 38, No. 6, pp. 655{664, 2002

On the Theory of Interphase Interaction

in a Mixture of Reacting Metal Particles

O. B. Kovalev 1

and V. M. Fomin 1

UDC 536.46

Translated from Fizika Goreniya i Vzryva, Vol. 38, No. 6, pp. 55{65, November{December, 2002.

Original article submitted January 21, 2002.

A mathematical model of interphase interaction in a mixture of powders is constructed

on the basis of the principles of mechanics of multiphase, multispecies, and hetero-

geneous media, as applied to SHS processes. The evolution of structural transfor-

mations, chemical reactions, and phase transitions are considered at the level of a

mesocell of the mixture. Analytical solutions are presented for conjugate thermod-

iusion problems on single particles and in the mesocell, which close the equations

of conservation laws. By an example of a single-stage chemical reaction in a mixture

of nickel and aluminum powders (Ni + Al = NiAl), the problem of the structure of

an SHS heat wave propagating over a nite-thickness semi-innite rod is solved nu-

merically. It is shown that the wave structure is strictly nonstationary and contains

inections and isothermal sectors at the melting point of the components, which vary

periodically in time. To solve the problem of thermal explosion in a mixture of metal

powders in a three-dimensional domain of complex geometry, a numerical algorithm

with the use of the method of dummy domains was developed.

Key words: high-temperature synthesis, intermetallides, thermal explosion, mathe-

matical simulation.

INTRODUCTION

of SHS. Kinetic models without elementary acts of in-

teraction between atoms are unsuitable for this purpose.

Reacting gases have all the components of the mixture

present at each point of space, whereas the initial com-

ponents in grainy solids are separated in space at the

level of individual particles and can react via contact

surfaces only. The absence of mixing constricts the re-

action rate depending on the degree of development of

the contact surface through which reagents are supplied

due to diusion processes.

The evolution of structural, phase, and chemical

transformations in reacting powders was considered in

[10] from the viewpoint of mechanics of multiphase me-

dia. A physical model of macrostructural transforma-

tions was proposed, which is based on the analysis of

evolution of the mesocell of the mixture with increas-

ing temperature, and equations of SHS mechanics were

obtained, which allow one to consider macrostructural

transformations under compression by pressures that

admit the assumption of incompressibility of compo-

nents in the solid and liquid states. We consider the

equations of conservation laws for masses of each com-

Modeling of the processes of self-propagating high-

temperature synthesis (SHS) was initially developed on

the basis of the thermal theory of gasless combustion

[1, 2] with a formal kinetic description of chemical inter-

action. Development of models with diusion kinetics

[3{7] signicantly complicated the procedure of numeri-

cal simulation because it became necessary to obtain so-

lutions of conjugate problems with moving boundaries

at the level of the mesocell of the mixture. Oering a

detailed description of diusion processes in a cell, ther-

mal explosion models [8, 9] ignore the temperature dis-

tribution over the entire specimen, which does not make

it possible to study the formation of a spatial structure,

which is not uniform in real processes and largely aects

the properties of the material obtained.

At the moment, there exists the problem of a sys-

tematic mathematical description of chemical reactions

1 Institute of Theoretical and Applied Mechanics,

Siberian Division, Russian Academy of Sciences,

Novosibirsk 630090; kovalev@itam.nsc.ru.

0010-5082/02/3806-0655 $27.00 c 2002 Plenum Publishing Corporation

655

656

Kovalev and Fomin

ponent, including solid and liquid components, with

right sides that take into account mass transfer due to

phase transitions and chemical reactions. The equa-

tions of momentum and energy in the condensed phase

and in the gas are written in a compact form without

clear identication of temperature zones of structural

transformations. Within model concepts, formation of

the structure of the nal product, consisting of closed

(spherical) and open pores, with certain graininess and

chemical composition is studied. The one-temperature

approximation of the model of [10] was considered in

[11]. The SHS model [10, 11] developed by the authors

diers from the known equations of the thermal the-

ory and homogeneous and multiphase models by the

fact that it allows one to reject the assumption of in-

stantaneous phase transitions, which are usually de-

scribed by means of introducing the heat capacity via

the Dirac function. The thermophysical parameters

of the mixture (density, heat capacity, thermal conduc-

tivity, porosity, etc.) depend on the composition of the

medium, which varies in time and space.

In the present work, based on the temperature-

homogeneous approximation of equations of SHS me-

chanics [10], the structure of the heat wave is studied,

and the problem of electrothermal explosion of a pow-

der mixture in a spatial domain of complex geometry is

solved.

the general case on porosity and temperature, ij is the

mean density, and i is the true density of the com-

ponent. We recall that the volume concentrations of

each component are represented as a sum of the solid

ad liquid components: n 1 = n 11 + n 12 , n 2 = n 21 + n 22 ,

n 3 = n 31 + n 32 , and the specic internal energies

of solid (e j1 ) and liquid (e j2 ) phases are written as

e j1 = c j T + e j1 and e j2 = c j T + e j1 + L j , where c j

is the specic heat capacity and L j is the heat of the

phase transition of the jth component of the mixture

(j = 1; 2; 3). To calculate the composition of intermedi-

ate phases and the nal product, we represent the law of

mass conservation for each component in the following

form:

dt j n j = ' j ; j = 1; 2; 3

(2)

3 3 J; ' 2 = 2 2

3 3 J; ' 3 = J

;

dt j n j2 = j ; j = 1; 2; 3

(3)

3 3 J; 2 = I 2 2 2

3 3 J; 3 = I 3 + J

Here I 1 , I 2 , and I 3 are the mass velocities of melting

of the components, J is the mass velocity of synthesis,

1 , 2 , and 3 are the molecular weights of the sub-

stances A, B, and AB, respectively, and 1 , 2 , and 3

are stoichiometric coecients.

It can easily be seen that the law of conservation

for the entire mass of the mixture is valid:

1 n 1 + 2 n 2 + 3 n 3 = 1 n A + 2 n B :

Here n A and n B are the volume concentrations of the

initial components A and B. The initial porosity is

m 0 = 1 n A n B . Then, the current porosity of

the mixture is m 2 = 1 n 1 n 2 n 3 . Assuming

that the particles and the cell of the mixture have a

spherical shape and the particle materials are incom-

pressible ( j = const), we express the mass veloci-

ties of melting and synthesis (A + B = AB) in terms of

the changes in the corresponding boundaries of phase

[R 1 (t);R 2 (t);R 3 (t)] and chemical [r 1 (t);r 2 (t)] transfor-

mations:

FORMULATION OF THE PROBLEM

AND GOVERNING EQUATIONS

Since the chemical reactions of synthesis proceed

without participation of the gas phase, the assumption

of inertness and neutrality of the gas in pores allows us

to neglect the gas inuence on heat- and mass-transfer

processes in the mixture. In the absence of a dynamic

action, which could set the medium into motion, we can

assume that the velocity equals zero and neglect convec-

tive terms in the equations of [10]. Following the aver-

aging technique in mechanics of heterogeneous media

[12], we introduce the mean temperature T and write

the law of conservation of energy of the mixture in the

following form:

@t =

3 X

@x k e @T

@x k ;

I 1 = 3 1 n A z 1

dt ; I 2 = 3 2 n B z 2

dz 2

dt ;

(1)

(4)

k=1

dz 3

dt ; J = 3 n B d

I 3 = 3 3 n B z 3

dt (y 2 y 1 );

3 X

2 X

3 X

2 X

i n ij e ij :

E =

ij e ij =

R A ; z 2 = R 2

R B ; z 3 = R 3

i=1

j=1

i=1

j=1

R AB ;

Here E is the total energy of the mixture consisting

of the substances A and B and the product AB, e

is the eective thermal conductivity, which depends in

y 1 = r 1

R B ; y 2 = r 2

R B

' 1 = 1 1

1 = I 1 1 1

dz 1

z 1 = R 1

On the Theory of Interphase Interaction in a Mixture of Reacting Metal Particles

657

(R A , R B , and R AB are the radii of the particles A and

B and the product AB).

With allowance for the initial conditions

t = t 0 : n 1 (t 0 ;x) = n A ; n 2 (t 0 ;x) = n B ;

n 3 (t 0 ;x) = 0;

(5)

One known method for calculating the eective thermal

conductivity, which is based on the formula proposed by

Odelevskii [15], is considered below. The calculation is

performed in two stages. First, the eective thermal

conductivity is calculated for a grainy layer of particles

of the substance A with gas pores:

y 1 (t 0 ;x) = y 2 (t 0 ;x) = 1;

(6)

e1 = A

m 0 2

1=(1 0 ) (1 m 0 2 )=3

;

Eqs. (2){(4) after integration yield

n 1 = n A n B 1 1

3 3

(y 2 y 1 );

(7)

A =

11 ; T 6 T A ;

12 ; T > T A ;

n 2 = n B n B 2 2

3 3

(y 2 y 1 );

(8)

m 2 + n 1 :

Note, g is the thermal conductivity of the gas, 11 and

12 are the thermal conductivities of the substance A in

the solid and liquid states, respectively, and T A is the

melting point for particles of the substance A. Consider-

ing the substance A with pores as a matrix and particles

of the substance B as spherical inclusions in this matrix,

we obtain the eective thermal conductivity of a porous

mixture of the substances A and B:

0 = g

A ; m 0 2 = n 1

n 3 = n B (y 2 y 1 ): (9)

The mean heat capacity of the mixture can be written

3 X

1 + n B c s

c 0

c j j n j = c 0 0

y 2 y 1

;

j=1

where c 0 0 = c 1 1 n A + c 2 2 n B = const, 0 is the mean

density of the initial mixture, 0 = (m 0 + A = 1 +

B = 2 ) 1 , and c 0 = c 1 A + c 2 B is the mean heat ca-

pacity. If the composition of the mixture is specied in

mass fractions A and B , it is easy to express the vol-

ume concentrations through them: n A = 0 A = 1

;

e2 = e1

m 0 2

1=(1 00 ) (1 m 0 2 )=3

and

e1 ; B =

21 ; T 6 T B ;

22 ; T > T B ;

m 0 2

= n 2

n B = 0 B = 2 .

Using the equations of continuity (2) and (3), we

rewrite the law of conservation of energy of the mixture:

( 21 and 22 are the thermal conductivities of the sub-

stance B in the solid and liquid states).

The eective thermal conductivity, which is used in

(10) and changes in the process of phase and chemical

transformations, can be calculated by the formula

e = (1 n 3 ) e2 + n 31 31 + n 32 32 ; (13)

where 31 and 32 are the thermal conductivities of the

product AB in the solid and liquid states.

The values of R 1 (t), R 2 (t), R 3 (t), r 1 (t), and r 2 (t)

are determined by solving the corresponding problems

of heat and mass transfer at the level of single par-

ticles of the substances A and B and at the level of

the mesocell of the mixture, with allowance for the cor-

responding kinetics of phase and chemical transforma-

tions, which are considered below.

0 c 0

1 + n B 3 c s

0 c 0 (y 2 y 1 )

3 X

@x k e @T

@x k J(c s T + Q s )

3 X

I j L j ; (10)

k=1

j=1

Q s = e 3 + L 3

3 3 (e 1 + L 1 )

3 3 (e 2 + L 2 ); (11)

c s = c 3 1 1

3 3 c 1 2 2

3 3 c 2

(12)

(Q s is the heat of the process).

Despite the important role of thermal conductiv-

ity in combustion processes in pressed powder systems,

there are only some scattered data on its dependence

on compacting pressure or porosity, thermal conductiv-

ity of components, and other factors [13{15]. There are

no recommendations on evaluating thermal conductiv-

ity in the general case, the use of the generalized conduc-

tivity theory [13] encounters many problems and does

not yield satisfactory results, and experimental data are

presented in an extremely limited number of papers [14].

MELTING KINETICS

OF THE METALLIC PHASE

The mean temperature of the mixture is constant

within a selected macrovolume and varies only accord-

ing to the averaged law of energy conservation (10). The

necessary condition of the averaging technique is that

00 = B

1 1

2 2

658

Kovalev and Fomin

the particle diameter and the size of the chosen macro-

volume should be incomparable. In other words, the

macrovolume should contain a large number of single

particles. The melting process begins when the temper-

ature T reaches the melting point of the metal T A . All

particles in the macrovolume undergo the phase transi-

tion simultaneously.

We consider the temperature distribution T p inside

a single particle of radius R A , located in a medium with

a temperature T.

When the melting point T A is reached, the phase-

transition boundary R 1 (t) appears inside the parti-

cle. The temperature distribution T p satises the heat-

conduction equation

a T

Fig. 1. Melting kinetics of spherical particles: =

1=6 (z 2 =2 z 3 =3).

@T p

@t = @ 2 T p

@T p

We introduce the dimensionless variables z =

R 1 =R A and = (tt A )=t 0 , which are self-similar for

Eq. (17) if

@r 2

(14)

1 c 1

a T =

; 0 6 r 6 R A

;

L 1

c 1 (T T A )

1 c 1 R A

for t = t A ,

t 0 =

T p (t A ;r) = T A ; 0 6 r 6 R A ;

R 1 (t A ) = R A ;

Equation (17) takes the form

d =

(1 z)z ; z(0) = 1:

for t > t A ,

(18)

@T p

@r = 0;

r = R A : T p = T;

r = 0:

The solution of Eq. (18) in the region > 0, 0 6

z 6 1 yields a universal curve for particle melting

= 1=6 (z 2 =2 z 3 =3) (Fig. 1). In practical calcu-

lations, this curve is approximated by a polynomial and

can be used to calculate the melting kinetics of individ-

ual components of the powder mixture.

r = R 1 (t):

@R 1

= 1

1 L 1

@T p

r=R 1 (t) ;

T p (t;R 1 (t)) = T A :

(15)

Here t A is the time when the temperature of the ambient

medium becomes higher than the melting point or equal

to it (T > T A ).

According to the energy equation (10), the temper-

ature T at the moment of particle melting cannot be

signicantly dierent from the melting point T A , since

the entire heat supplied to the macroobject is spent on

melting. Taking into account additionally the smallness

of the particle radius (5{10 m), we may assume that

the temperature distribution on these small scales sat-

ises the steady-state heat-conduction equation whose

solution has the form

DIFFUSION KINETICS

OF INTERMETALLIDE FORMATION

: (16)

In considering SHS in a mixture of Ni and Al

metal powders, we follow a simplied interaction pat-

tern (Ni + Al = NiAl), based on the physical concepts of

this phenomenon known from [16]. With appearance of

the melt of aluminum particles in the cell of the mix-

ture at a temperature T = 933 K, solid particles of

nickel start to dissolve in the melt. We assume that

the particle surface retains the concentration of nickel,

which is equilibrium for this temperature, and the ve-

locity of the process is limited by diusion of nickel into

the ambient melt. The mathematical formulation of the

problem in the cell reduces to the following:

D e

R A R 1

1 R 1

1 R A

T p =

TR A

T A R 1

Then, Eq. (15) yields the equation of motion of the

phase-transition boundary R 1 (t):

@R 1

= 1

1 L 1

T T A

R A R 1

R A

R 1 ;

@t = @ 2 c

@r ; R 2 6 r 6 +1; (19)

@r 2

for t = t e ,

R 2 = R B ; c = 0; R B 6 r 6 +1;

T > T A ; t = t A : R 1 = R A :

(17)

On the Theory of Interphase Interaction in a Mixture of Reacting Metal Particles

659

for t > t e and r = R 2 (t),

c = c e ; (1 c e ) @R 2

= D e @c

r=R 2 :

An innitely thin layer of the product AB increas-

ing with time arises in the cell of the mixture in the

temperature range between the melting points of the

metal components, at the contact boundary of the solid

particle B and the melt layer A at the initial time. The

equilibrium concentrations C 1 and C 2 weakly dependent

on temperature are established at the layer boundaries.

The distribution of concentration of the substance B in

the layer satises the diusion equation.

The mathematical formulation of the problem re-

duces to the following:

Without allowance for surface tension and kinet-

ics of surface processes, this problem was formulated

and solved in [17]. The nonstationary analytical solu-

tion obtained in [17] in the form of a power series in t

requires cumbersome calculations of the coecients of

the series. For simplicity, we assume that the concen-

tration distribution diers little from the steady-state

distribution. Then, the solution c = c e R 2 =r satises

the diusion equation (19) in the stationary case, and

the dissolution velocity is determined from the equation

(1 c e ) @R 2

@r ; r 1 6 r 6 r 2 : (21)

Here D = D 0 exp(E A =RT) is the coecient of mutual

diusion of the substances A and B in the layer of AB

with an activation energy E A (D 0 is a preexponent).

The initial and boundary conditions for (21) are written

in the following form:

for t = t A ,

@r 2

= D e (T)c e

R 2

;

(20)

which, under conditions of a weakly changing tem-

perature, in the dimensionless variables z = R 2 =R B

and = (tt e )=t 0 has the solution = 0:5(1 z 2 )

(0 6 6 0:5). Here

t 0 = R B (1 c e )

D e c e

r 1 = r 2 = R B ;

C = 1; 0 6 r 6 R B ; C = 0; R B 6 r 6 R cell ; (22)

for t > t A and r = r 1 (t),

C = C 1 ; (1 C 1 ) @r 1

According to [18], nickel dissolution in liquid aluminum

is accompanied by heat release Q e = 60:7 kJ/mole. The

diusion kinetics of dissolution can be represented in

the following form: D e (T) = D e0 exp(E e =RT), where

D e0 = 4:8 10 6 m 2 /sec and E e = 74:94 kJ/mole [18].

From Eq. (10), we can easily obtain the charac-

teristic time of melting of a nickel particle of radius

R B = 20 m:

@t = D @C

r=r 1 ;

(23)

for t > t A and r = r 2 (t),

C = C 2 ; C 2 @r 2

@t = D @C

r=r 2 :

(24)

Here R cell is the radius of the mesocell of the mixture

[11].

t liq = R B 2 L 2

2 T B

= 10:5 sec:

Because of the small thickness of the layer of the

product AB, we assume that the concentration distribu-

tion satises the steady-state diusion equation whose

solution has the form

C = r 1 r 2 (C 1 C 2 )

(r 2 r 1 )r

On the other hand, the characteristic time of nickel dis-

solution in the Al melt is

t p = R B (1 c e )

D e (T B )

sec:

+ r 2 C 2 r 1 C 1

r 2 r 1

: (25)

This means that the dissolution velocity of the nickel

particle is much higher than the velocity of its melting.

Reactive interaction of nickel and aluminum begins

when the solution reaches the saturation concentration

with instantaneous formation of a layer of products. De-

tailed notions on the mechanism of SHS reactions and

formation kinetics of intermediate phases in a mixture

of Ni and Al powders are currently unavailable. With-

out loss of generality of the approach, we conne our-

selves to the case of a one-stage pattern of the chem-

ical reaction in the mixture of metal powders A and

B: A + B ! AB. This may serve as a generic interac-

tion pattern of nickel and aluminum (Al + Ni = NiAl),

titanium and nickel (Ni + Ti!TiNi), titanium and alu-

minum (Al + Ti!TiAl), et al. [1, 16].

Substituting the resultant solution into (23) and (24),

we obtain the equations of motion of interphase bound-

aries

dr 1

dt = D

C 1 C 2

1 C 1

r 2

(r 2 r 1 )r 1 ; (26)

dr 2

dt = D

C 1 C 2

C 2

r 1

(r 2 r 1 )r 2 ; (27)

t = t A : r 1 = r 2 = R B :

We assume that the temperature either changes

weakly in time or is constant, which is possible in the

case of long-time annealing. In the dimensionless vari-

ables y 1 = r 1 =R B , y 2 = r 2 =R B , and = (tt A )=t d ,

Eqs. (26) and (27) take the form

@t = @ 2 C

= 1:5 10 2

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