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Chapter 14

Reactions Between Solids

The reaction between solids is used in the synthesis of many products, as well as in the heavy industry such as cement and concrete and in the fine chemistry to prepare products with high added value.

In the cement industry, the main components of this type of reaction are calcium silicates prepared by the action of silica on calcium carbonate (calcareous) according to:

3 2 3 2CaCO SiO CaSiO CO+ = +

3 2 2 4 22CaCO SiO Ca SiO 2CO+ = +

3 2 3 5 23CaCO SiO Ca SiO 3CO+ = +

The electronics industry requires the manufacture of ceramics such as barium aluminate as lightning protectors by the action of alumina on barium carbonate or of the dielectric material as condensers such as barium titanate, which is prepared by a solid-solid reaction and is more economical [NIE 94] than other methods, by the following reaction:

3 2 3 2BaCO TiO BaTiO CO+ = +

Chapter written in collaboration with Gérard Thomas.

490 Handbook of Heterogenous Kinetics

The preparation of catalysts, such as nickel molybdate or cobalt tungstate, is often carried out from precursors and frequently involves a final stage between two solids such as:

3 4NiO MoO NiMoO+ =

The products used in pyrotechnics (e.g., fireworks, explosives) are the result of a reaction between a reducer and a powerful oxidant; for example, boron and potassium nitrate, zirconium and potassium chlorate, and tungsten and lead chromate. These reactions can release 1,200 to 8,000 J/g of mixture of the reactants during a few milliseconds of reactions between the solids such as:

4 2 3 3W 2PbCrO 2PbO Cr O WO+ = + +

14.1. Classification of the reactions between solids

Reactions between solids can be classified into various categories: simple additions, addition involving decomposition or redox reactions, and exchange reactions or double decompositions.

14.1.1. Simple addition reactions

The reaction can be expressed as:

solid solid solidA B AB+ =

As an example, we can represent the synthesis of nickel tungstate as follows:

3 4NiO WO NiWO+ =

14.1.2. Addition reactions involving decomposition

Addition reactions involve decomposition of one of the solids. The reaction can be represented as follows:

solid solid solid gasA C B G+ = +

Reactions Between Solids 491

As an example, for this type of reaction, we have the synthesis of barium aluminate:

3 2 3 2 4 2BaCO Al O BaAl O CO+ = +

14.1.3. Addition reactions involving a redox reaction

In the addition reaction involving a redox reaction, two ions present in one of the solids undergo redox reaction. The reaction has the same general form as that of the preceding reaction. As an example, we can write:

2 2UO 4C UC 2CO+ = +

or

3 4 3 4 21Co O 3MoO 3CoMoO O2

+ = +

14.1.4. Exchange reactions or double decompositions

These reactions involve the exchange of cations and anions between two solids. The reaction can be represented in the following form:

solid solid solid solidAC BD AD BC+ = +

Refer to the reaction between lithium bromide and potassium fluoride:

LiBr KF LiF KBr+ = +

and also that between copper metal and silver chloride, which belong to the same type of reaction:

Cu AgCl CuCl Ag+ = +

It should be noted that the reactions between two solids are sometimes accompanied, at least temporarily, by the production of secondary compounds.

Weak modification of the reaction conditions, such as temperature, partial pressure of a gas component, morphology of a solid, can also direct the reaction.


Thus, the reaction between silicon and cuprous chloride(I) can lead to the formation of not only copper metal but also copper silicides according to:

4Si 4CuCl SiCl 4Cu+ = +

4 37 4Si 4CuCl SiCl Cu Si3 3

+ = +

14.2. The modeling assumptions

In the case of a single solid as product (which excludes the reactions of double decompositions that we will approach in section 14.4.2.4), we can write the reaction between two solids in the general form:

solid C solid B solid G gasA C B ( G )ν ν ν+ = +

taking into account the formation of a new solid B and possibly of a gas G (νG can be null).

To define the fractional extent of this reaction, we consider as a reference the initial amount n0 of the least abundant solid reactant (from the stoichiometric point of view). Assume A is the reactant, which means that:

C(0) C 0n nν>

The fractional extent with respect to A will be then:

{ }0A

0

Ann

α−

= [14.1]

where { }A is the amount of A present at time t.

The fractional extent with respect to C will be:

{ }C(0)C

C 0

Cnn

αν

−= [14.2]


We assume that the system evolves according to a pseudo-steady state mode and thus the various fractional extents are equal. The common value is the fractional extent α of the reaction (see section 7.4.3).

We also generally assume that, between two massive solids (or two grains in the case of a powder), the rate of growth is separable (which we can verify thereafter) and thus we will have:

dd

Etα φℜ = = [14.3]

where φ is the reactivity of growth, depending only on the intensive variables (temperature, partial pressures of gases) and independent of time if these variables are kept constant. E is the space function.

To model the evolution of the rate of a powder (mixture of grains) with time, it is necessary to introduce design parameters that define the relative placement of the grains in space.

14.3. The experimental measure of the extent of the reactions

The experimental measure of the extent of the reactions between solids according to time often presents delicate problems because we cannot generally follow the uninterrupted evolution and we must carry out the measurement of each point one after the other (which turns problematic for all measurements based on the switch experiment). The method is most usually based on the X-ray diffraction analysis of the phases (see sections 1.8.2.3, 11.2.4, and 11.2.5).

However, in a certain number of cases, we can carry out uninterrupted measurements by thermogravimetry. This will be the case if the reaction is accompanied by a gaseous emission representing a variation in the mass of the solid phase. It will also be the case each time the reaction consumes or produces a solid with a specific magnetic property (ferromagnetic or paramagnetic) very different from that of the other involved solids. Thermomagnetometry is then used [THO 83a] (see section 1.8.2.2).

The use of microcalorimetry for the measurement of heat flow is sensitive. For this, the two cold solids are mixed and then introduced in the calorimeter set at the reaction temperature. It should be noted that the reactions between solids that do not emit any gases are not associated with a very high thermal effect.


The morphological control of the powder is ensured in a precise manner by microscopic observation, the determination of specific areas, and sizes of the grains in order to know the relative distribution of the grains, called design parameters of the powder. This design is the result of the operation of initial mixing of powders, an operation that has to be methodically carried out in order to lead to definite and reproducible designs.

14.4. Reactivities of reactions between solids

Taking into account the complexity of the design parameters that can make the rate non-separable, it is desirable to determine the mechanism in order to model the reactivity in a case in which the rate will be separable.

14.4.1. Position of the problem and experimental approach

Reactions between solids proceed basically via the two processes of surface nucleation and growth and thus their study is based on the assumptions and methods developed in Chapters 8 to 11.

For the reaction between massive solids, from the experimental knowledge of the kinetic law, it is theoretically possible to deduce the specific nucleation frequency and the growth reactivity, with the condition of having modeled the space function.

In the case of reactions between powders, the problems of relative placement of grains of the initial solid in the space (design of the powder) seriously complicate the obtained laws and the total rate is not always separable.

For the reaction mechanism, that is, to study the reactivity, we can use a method that fixes in a simple way the space functions. For this, we carry out the reaction between two plates A and C (for this, we press each of the two initial powder pellets slightly to assemble two cohesive pellets). Then, these two assembled pellets are placed in contact in a matrix under high pressure to carry out the reaction.

After the reaction, product B is formed at the interface between the plates. For example, Figure 14.1 shows the reaction that proceeds by development of the layer of B at the expense of that of A, with an expansion coefficient of B of 1 compared with that of A.

If the experiment thus carried out leads to a rate independent of time, this means that an interface step is rate determining at interface A/B or B/C whose areas are invariable. If the rate is a function of time, this means that diffusion through the formed solid B is the rate determining step.


Figure 14.1. Relative placement of the phases for the study of the reactivity

Indeed, we can write the rate as follows:

D0

sGn

φℜ = [14.4]

where the area of the initial plates (or at least the smallest of both) is s. GD is equal to 1 for an interface reaction as the rate determining step and 1/X for diffusion according to the Fick law as given in Table 7.1.

The mechanisms that we describe in plane symmetry will thus have as an aim to understand the transport phenomenon and the reactions that lead to the expressions of the reactivity as a function of intensive variables (temperature and partial pressures of gas).

Under such conditions, we observe that generally a one-process model with instantaneous nucleation and slow growth is enough to describe the reaction (see Chapter 10).

14.4.2. Structures of the reaction mechanism of growth

We examine four types of reactions: simple addition reactions, reactions accompanied by a redox reaction, reactions accompanied by decomposition, and finally double decompositions.

14.4.2.1. Study of the simple addition reactions

We illustrate this case by the reaction between two oxides represented as follows:

2AO BO ABO+ =

A

B

C

x

eC(0)

eA(0)


The obtained solid ABO2 forms a continuous layer, including point defects, ensuring the transport of matter. We assume, in our example, that the initial oxides present a very weak vapor tension that prohibits the transport of cationic species by a gas phase. On the other hand, these oxides can present a dissociation tension so that the transport of oxygen can also be ensured by the gas phase. Moreover, it is legitimate to assume that these oxides can present anion defects in the presence of a gas phase (oxygen, for example) and thus the transport of oxygen can also be carried out by the solid phase via anion defects.

The partial pressure of oxygen is kept constant during the entire reaction.

If we initially consider the growth reaction of ABO2, assuming that, for one reason or another (one-process model with instantaneous nucleation or fast surface diffusion), interface AO/BO is immediately entirely covered with a fine layer of ABO2, it is possible to determine the kinetic laws relating to the pure modes that will be either with a constant rate if an interface reaction is the rate determining step or a function of time if diffusion is the rate determining step.

14.4.2.1.1. Growth of the new phase without oxygen short circuit

Figure 14.2 gives the relative placement of the phases at time t. There are four types of interfaces:

– interface a between AO and the gas phase;

– interface b between AO and ABO2;

– interface c between BO and ABO2; and

– interface d between BO and the gas phase.

The point defects of solid ABO2 must ensure:

– exclusively the transport of species A (interstitial cations or cation vacancies of A) and oxygen anions, in which case, the diffusing cationic species is created at interface b or c;

– or exclusively the transport of species B (interstitial cations or cation vacancies of B) and oxygen anions, in which case, the diffusing cation species is created at interface c or b;

– or both simultaneously with a counter-diffusion process (see section 14.5.3.3).

We consider, as an example, the case in which solid ABO2 has anion vacancies and interstitial cations A. To simplify, we consider neutral point defects.


The defects make it possible to diffuse species A and O and to bring them in contact with BO to ensure the growth of ABO2, which is thus formed at interface c. B remains a constantly motionless species by assuming the absence of defects related to B in ABO2.

At interface b, cation A jumps from its site in the initial crystalline AO toward its arrival site (an interstitial position of ABO2) according to:

i 2 A i 2 AV (ABO ) A (AO) A (ABO ) V (AO)+ → +

The interstitial cation moves from the A-rich zone (interface b) toward the A-poor zone (interface c). The arrival of A causes the creation of a building unit ABO2, with destruction of BO following:

O 2 i 2 O 2 2O (ABO ) A (ABO ) BO V (ABO ) ABO+ + → +

The oxygen vacancy thus created diffuses from the rich vacancy zone (interface c) toward the poor vacancy zone (interface b), where the exchange of oxygen with AO lattice occurs as follows:

O O 2 O 2 OO (AO) V (ABO ) O (ABO ) V (AO)+ → +

Figure 14.2. Growth of a single layer in an addition reaction

In solid AO, the oxygen vacancy thus created in the first step of vacancy A are destroyed by an opposite Schottky reaction after a diffusion step of vacancies O and A from b in AO:

O AV (AO) V (AO) 0+ →

This step is in fact the destruction of the AO building unit at an interface.


It is noted that the sum of the six steps gives the total reaction (all the multiplying coefficients are equal to 1).

In addition, two interfaces are mobile (a and c) and two are fixed (b and d).

Other cases of mechanisms can also be studied. The defects of the formed solid are used as a guide to locate the interfaces where they are created and consumed and give the development direction of the layer.

14.4.2.1.2. Nucleation of the new phase

Consider the above-mentioned example, at the beginning of reaction, solids AO and BO are in direct contact and they will form point defects in BO, for example, ions A in interstitial positions according to:

i O O 2A (BO) O (AO) BO V (AO) ABO+ + → +

This condensation reaction probably proceeds in several steps and is accompanied by the annihilation of the vacancies in AO (after diffusion in AO):

O AV (AO) V (AO) 0+ →

14.4.2.1.3. Growth of the new phase with oxygen short circuit

Some experimental results showed an influence of oxygen pressure that cannot be interpreted by the preceding mechanism. A mechanism for the diffusion of oxygen in gas phase was then proposed. The model is completely conceivable if the formed product is porous.

REMARK.– This mechanism can be written on the basis of precedence and by replacing diffusion of the oxygen vacancies by two reactions, one at interface c that fixes the oxygen gas and the other that releases it at interface b, according to the following reactions:

Interface c: O 2 2 O 21V (ABO ) O O (ABO )2

+ →

Interface b: O 2 O 2 21O (ABO ) V (ABO ) O2

→ +

Between the two interfaces, diffusion of oxygen occurs in gas phase that ensures the transport of oxygen from b toward c, short-circuiting the diffusion in solid phase ABO2.


14.4.2.1.4. Reactions with counter-diffusion

As the cations have sometimes (in certain temperature ranges) very close diffusion coefficients, it is possible that the diffusion of the anion, which is perhaps very fast, and the diffusion of both cations A and B, each one in two opposing directions through the solid layer of ABO2, are the two rate determining steps. (These are parallel mechanisms and if the diffusion coefficients are different, it is the fastest one that is the rate determining step). We can say that we have a counter-diffusion of modes A and B.

Model a system with instantaneous nucleation and the diffusion of two cations is the rate determining step for the growth.

Consider again Figure 14.2. The point defects of solid ABO2 allow in addition to the diffusion of the anions, diffusion of a cation relative to A and diffusion of a cation relative to B. The diffusing species (interstitial cations or cation vacancies) of each species are created and consumed by reactions at the appropriate interfaces.

If one of the cation diffusions is very fast compared with the other, we are brought back to a mechanism of simple diffusion, with diffusion only of a cation (the fastest) as the rate determining step.

If both diffusions have comparable speeds, we will have, in fact, two parallel mechanisms leading to the same reaction (see Appendix 8). But these two steps occur in zones that are identical for the interfacial reactions and identical (the layer of formed product) for the diffusions so that in the case of a double rate determining step, the space function will be the same and then the reactivity will be the sum of the two reactivities obtained for each mechanism.

Consider the case of the diffusion of interstitial cations A and B.

– For path I with diffusion of A, we have the following steps:

At interface b: creation of ions A in interstitial positions according to:

i 2 A i 2 AV (ABO ) A (AO) A (ABO ) V (AO)+ → +

Diffusion of interstitial cations A from interface b toward interface c.

At interface c: consumption of interstitial cations A and building of the network of ABO2:

O 2 i 2 2 i 2 O 2O (ABO ) A (ABO ) BO (ABO ) V (ABO ) V (ABO )+ + → + +


– For path II with diffusion of B, the steps will be as follows:

At interface c: creation of ions B in interstitial positions according to:

i 2 B i 2 BV (ABO ) B (BO) B (ABO ) V (BO)+ → +

Diffusion of interstitial ions B from interface c toward interface b.

At interface b: consumption of interstitial ions B building the network of ABO2:

O 2 i 2 2 i 2 O 2O (ABO ) B (ABO ) AO (ABO ) V (ABO ) V (ABO )+ + → + +

In these reactions, we have to add the creation and consumption of the carrying oxygen defects (e.g. oxygen vacancies with very fast diffusion) for each of the two paths according to the following form:

O O 2 O 2 OO (AO) V (ABO ) O (ABO ) V (AO)+ → +

and

O O 2 O 2 OO (BO) V (ABO ) O (ABO ) V (BO)+ → +

and, of course, the annihilations of the vacancies in AO and BO for each path:

O AV (AO) V (AO) 0+ →

O BV (BO) V (BO) 0+ →

The total rate is the sum of the rates according to the following two paths:

I IIℜ = ℜ + ℜ

Examine the case in which both cation diffusions are the rate determining steps. Thus, equation (14.4) can be written as:

I DI c II DII b

0 0

G s G sn n

φ φℜ = +

but

DI DII c band G G s s= =


The rate is thus separable and the reactivity of the reaction is the sum of the reactivity of the two paths:

I II A A B BD C D Cφ φ φ= + = Δ + Δ

where A Band C CΔ Δ are fixed by the corresponding interface equilibrium.

14.4.2.1.5. Formation of several compounds

When diffusion is the rate determining step, there are chemical potential gradients that can involve the formation of several phases at least in a transitory way.

Figure 14.3. Formation of two compounds during an addition reaction

We study the example of the formation of a compound ABO2 (with anion vacancies and interstitial cation A) (the thermodynamically stable phase) with the formation of an intermediate compound A2BO3 (having the same point defects). This phase is obviously created close to oxide AO since it is richer in A. Figure 14.3 shows the relative placement of the phases and the concerned interfaces.

Phase A2BO3 appears because the chemical potential of oxygen interface b is such that this phase is locally stable. This means that a concentration gradient exists and so only a diffusion mode is expected

In the same manner, at interface b, an ion A of AO jumps into interstitial position in A2BO3 (phase 2) according to the following form:

i A i AV (2) A (1) A (2) V (1)+ → +


The oxygen vacancies that will be formed further are consumed there according to following expression:

O O O OO (1) V (2) O (2) V (1)+ → +

We will find in AO, the reverse of the Schottky reaction, which destroys anion and cation vacancies.

At interface c, A2BO3 is generated at the expense of ABO2 by transforming the defects of a phase into defects of the other phase as follows:

A i B O A B O OA (3) A (2) B (3) 2O (3) 2A (2) B (2) O (2) V (2)+ + + → + + +

At interface d, the production of ABO2, which also produces oxygen vacancies, takes place as follows:

i B O A B O OA (3) B (4) O (4) A (3) B (3) O (3) V (3)+ + → + + +

Thus, oxygen vacancies diffuse from interface d toward interface b and interstitial ions A diffuse from b toward d. As a result, the three interfaces move in the direction of the arrows in Figure 14.3. Also, it is noted that the thickness of layer of A2BO3 increases at interface c whereas that of of AO decreases at an interface. When reactant AO is completely consumed, phase A2BO3 is, in turn, consumed according to the final reaction:

2 3 2A BO BO 2ABO+ →

so that at the end of the reaction, there remains only one product, ABO2, the only one stable under the experimental conditions.

We can find such a case for the formation of barium titanate by the action of barium carbonate on titanium dioxide, with intermediate formation of di-barium titanate (Ba2TiO4) [NIE 90].

14.4.2.2. The study of addition reactions with redox reactions

We can write such a reaction in the following way:

2 3 2 3 21AO B O AB O O2

+ = +


In such a reaction, the reduction of cation B proceeds as described previously. This reduction can be carried out, for example, by the electrons associated with the diffusing particles and created with interface steps. Thus, for a mixed n-type oxide, the arrival of interstitial cation A is equivalent to one of two electrons and gives at interface c (Figure 14.2) a succession of steps representing the desorption of oxygen, reduction of B, destruction of B2O3, and creation of AB2O3:

i O O iA O s V O s A°° °°+ + → + − +

22O s 2s O− → +

B BB e B′ ′+ →

B O O i A B O O2B V 2O A A 2B 2O V°°′ + + + → + + +

The other steps (diffusion of oxygen vacancies and the Schottky reaction) are similar to those described previously (see section 14.2.2.1.4).

Such a reaction is used in the manufacture of cobalt molybdate, in which Co3+ ions are reduced to Co2+ as follows:

3 4 3 4 21Co O 3MoO 3CoMoO O2

+ = +

14.4.2.3. Study of the reactions with decomposition

Consider the example of a reaction expressed in the following form:

3 2 2AO BCO ABO CO+ = +

The reactions of this type include nitrate, oxalate, and carbonate decomposition and are very much used because they are much easier to carry out than with corresponding oxides alone. They do not consist of decomposition followed by a reaction between oxides because these reactions are observed already at temperatures lower than 200°C to 300°C of the corresponding decomposition temperatures. It is thus perhaps that the diffusing particle tackles the complex anion (e.g. carbonate) and allows the reaction between the solids.

If the formed solid is, for example, a n-type compound with interstitial cation A, the reaction still proceeds by the jump of an ion A from the oxide into an interstitial position of ABO2 and after diffusion causes the decomposition of carbonate ion and


thus the elimination of carbon dioxide. The whole of the steps can be represented by the following sequence:

CO 33i 3 CO 2 iA CO s O CO s A+ + → + − +

2 2CO s CO s− → +

3B CO i A B O OB O A A B O V+ + → + + +

The other steps (diffusions and the Schottky reaction) are similar to those described previously.

The synthesis of barium titanate starting from barium carbonate and titanium dioxide with carbon dioxide release illustrates this type of reaction.

14.4.2.4. Double decomposition reactions

We consider as an example the following reaction:

′ ′ ′ ′CA + C A = CA + C A

The space distribution of the phases in the course of reaction will depend on the relative speeds of anions and cations. If, in fact, anions A and A′ diffuse more quickly, CA′ will be formed in contact with CA and CA′ in contact with C′A′ according to Figure 14.4. (If the cations diffuse more quickly than the anions, the order of the formed phases would be reversed.)

Figure 14.4. Space design in a double-decomposition reaction

It remains to represent the reactions at the various interfaces a, b, and c, which depend on the nature of the point defects, and then to a choose rate determining step that can be one of the interface reactions or diffusion through one or the other formed phases (or both).

CA CA′ C′A C′A′

(a) (b) (c)


14.4.3. Expression of the reactivities, reaction of titanium dioxide with barium carbonate

After representing the steps of the mechanism, the reactivity of each presumably elementary step is expressed according to its rate constant and the concentration of the various species that intervene there. Thus, we can represent, as in section 7.2.3, the various expressions of balances, conservation of electric charges, etc. Because the resolution of the system of differential equations obtained is too complex, we consider for simplified solutions that are pseudo-steady state modes with one rate determining step and possibly two by the application of the theorem of slownesses (when its conditions for application are fulfilled, see sections 7.5 and 7.6).

We illustrate the case for the formation of barium titanate through the reaction of titanium dioxide with barium carbonate according to following reaction:

2 3 3 2TiO BaCO BaTiO CO+ = +

The reaction is assumed to be carried out at a constant temperature, with carbon dioxide and oxygen under partial pressures

2COP and 2OP kept constant.

14.4.3.1. Growth mechanism

The thermodynamic conditions are such that the intermediate compound Ba2TiO4 cannot be obtained.

Solid BaTiO3 has two kinds of prevalent defects: oxygen vacancies and barium vacancies. Figure 14.5 gives the relative placement of the phases (noted from 1 to 3) and of the interfaces (noted from a to d). The concentration of species X in phase j close to interface i will be [ ]X( )ji .

We know that the mechanism must envisage transfer of a cation; in our case, this will be the barium cation via its vacancies that are formed in the area with a low barium concentration, that is, at interface c. In the same way, it is necessary to envisage a transfer of oxygen, and we will choose transfer by the gas phase.

At interface c, barium ions leave phase 2 and form BaTiO3 and two vacancies, one of barium and the other of oxygen, as follows:

2 Ba O 3 Ba OTiO Ba (2c) O (2c) BaTiO V (2c) V (2c)+ + → + + [14.Eq.1]

So interface c moves inward toward titanium oxide.


To simplify these expressions, we assume neutral defects. The barium vacancies thus formed diffuse through phase 2 from interface c toward interface b [14.Eq.2]. The oxygen vacancies are filled by the gas at interface c as follows:

O 2 O2V (2c) O (gas,(2c)) 2O (2c)+ → [14.Eq.3]

Figure 14.5. Placement of the phases during the preparation of barium titanate

At interface b, cation vacancy coming from interface c by diffusion is filled by a barium ion of phase 1 as shown in the following reaction:

Ba Ba Ba BaV (2b) Ba (1b) Ba (2b) V (1b)+ → + [14.Eq.4]

Carbonate ion breaks up, and neglecting the adsorbed species, we can write:

CO 333 2 2 CO2CO (1b) CO (gas(1b)) O (gas,(1b)) 2V (1b)→ + + [14.Eq.5]

Finally, at an interface, carbonate and barium vacancies are destroyed in phase 1 (after diffusion from interface b toward interface a) according to the following reaction:

3Ba COV (1a) V (1a) 0+ → [14.Eq.6]

By adding the six quasi-chemical steps and diffusion, after having multiplied (14.3) and (14.5) by 1/2, we can obtain the final reaction.

BaTiO3

TiO2

BaCO3

c interface Phase 2

Phase 3

a interface

b interface

d interface

Phase 1 VBa

VBa

VO


14.4.3.2. Reactivity and influence of the partial pressures of gases

We calculate the reactivity in the pseudo-steady state mode by considering diffusion of barium vacancies in phase 2 from interface c toward interface b as the rate determining step. If D is the coefficient of diffusion of these vacancies, the reactivity as given in Table 7.1 is expressed as follows:

[ ] [ ]( )Ba BaV (2c) V (2b)Dφ = −

To calculate the vacancy concentrations, we can assume that all the steps, (14.Eq.1) and from (14.Eq.3) to (14.Eq.6), are at equilibrium (Ki is the equilibrium constant of the step (14.Eq.i)) and that the partial pressures of carbon dioxide and oxygen are the same at all the points and especially at interfaces b and c.

After having noticed that the global equilibrium pressure of carbon dioxide is given by:

2

0 1/ 2 1/ 2CO 1 3 4 5 6P K K K K K=

we thus obtain:

2

2

2

CO1/ 2 1/ 21 3 O 0

CO

1P

DK K PP

φ⎛ ⎞

= −⎜ ⎟⎜ ⎟⎝ ⎠

We get a reactivity that varies as the square root of the oxygen pressure and that decreases linearly with the carbon dioxide pressure.

REMARK.– We have seen that in the reaction between solids, diffusion of two species, one originating from anions and the other originating from cations, is necessary and it is the slowest one that is the rate determining step. We will see, on the other hand that, in the gas-metal or, more generally, gas-solid reactions (Chapter 15), for which the diffusion of only one species is necessary, this is the fastest step that is likely to be the rate determining step. It is probably for this reason that we most frequently meet modes with diffusion as the rate determining step in the reaction between solids.

REMARK.– If both the diffusing species have almost the same velocity, we can consider the pseudo-steady state mixed mode with two diffusions as rate determining steps by applying the theorem of slownesses (see section 7.6.2). But as


the zones of diffusion of these two species have the same sizes, the rate remains separable, and we only have to add the reverse of the reactivities.

14.5. Rates of the reactions between powders

The reaction between mixed powders A and C is characterized by the fact that it can be held, at least at the beginning, only around the contact between grains of different nature. The number and relative packing of these contacts are determined by the choice of the mixing parameters, thus defining the design of the granular stacking.

The design obtained by mixing depends primarily on the following parameters:

– the initial amounts (in moles) of both powders n0 for A and C(0)n for C, or one of these amounts and one of the following ratios:

0A

C(0)

nn

σ = or C(0)C

0

nn

σ =

– size and shape of grains, which we will choose as spherical with radius:

A(0)r and C(0)r ; and

– homogenity of the mixture, function of the mixing mode, and the granular distribution and nature of the grains.

14.5.1. Problems of designs

Consider a simple addition reaction between two solids, which we can schematize as follows:

A + C = AC

From a traditional point of view, this reaction between grains of A and C compared with spheres will have to be studied in spherical symmetry assuming that small grains of C, for example, are surrounded by the bigger grains of A.

At the beginning, all the spheres of A and C have their initial radius. To simplify the reasoning, we can say that species A and C are those that ensure transport inside AC (which is sufficient for the study of the space functions).


Two families of models can give an account of the configuration of the mixture of grains A and C.

14.5.1.1. Spatial models of the stacking of grains of a mixture

Figure 14.6 provides the scheme of a stacking (represented without contact to distinguish the grains) of big grains Ai (i varying from 1 to NA) and of small grains Cj (j varying from 1 to NC). We now consider two ways to model this stacking:

– The M1 model is a model of average field type. The small grains C are compared with a continuum C around A.

– The M2 model, on the contrary, postulates specific contacts between grains of A and C, with a discrete distribution of these grains.

14.5.1.1.1. The M1 model with individual shells

How should this be shared in the M1 model with the continuum of C around the amount of grains A inside C?

The solution consists of considering that the whole of the powder is equivalent to NA identical cells, each cell consisting of a grain surrounded by an amount of C defined by the mix design Cσ .

Figure 14.6. Real stacking of grains of two populations of A and C

Matter C around grain A can then be distributed either:

– in a uniform way with a constant thickness layer of C; this is the M11 model in Figure 14.7 (population A);

– or with a law of distribution for thickness, as represented in Figure 14.7 (population B); this is the M12 model.


Figure 14.7. The M1 models with continuum of C around grains A: (a) M11; (b) M12

The M12 model gives a priori a better representation of reality since the fine grains are generally placed into interstitial spaces showing a network made up of the coarse grains, and the complex shape of the pores leads to a law of rather broad distribution. Nevertheless, the M11 model with uniform distribution thickness seems a useful simplification of the M12 model.

For the M11 approach, some authors have proposed kinetic laws on the basis of the M12 model, assuming the Poisson law of distribution to describe the fraction of pile with a thickness, ieC, made up of small grains of C with diameter eC on the surface of the big spherical grains.

If Ns represents the number of piles that we can put on the surface of grain A and Ni the number of piles with thickness ieC, we will have:

ii

s

NP

N= and

( )ssexp( )

!

i

i

aNP aN

i= −

where a is a parameter of standardization such as:

s ii

aN i P= ∑

The product aNs is a function of composition σΑ, molar masses MA and MC, densities ρA and ρC of the reactants, and their average sizes, characterized for species C by thickness (or diameter) eC.

(a) (b)

Cell Cell


For example, Thomas and Ingrain [THO 83b] obtained for large grains of molybdenum trioxide of 250-μm diameter and thin grains of cobalt oxide Co3O4, according to the Poisson law for size distribution, a good agreement between the theoretical initial rate and that found by the experiment as a function of composition (see section 14.5.4.2).

Figure 14.8. The M2 model for stacking with discrete distribution

14.5.1.1.2. The M2 model with a discrete distribution

This model seems a better approach in reality for the mixture of solids (Figure 14.8). However, it involves original problem of nucleation/growth because nucleation can occur only at the contact points and not on the whole surface of a grain.

For a given model of stacking, it is necessary to define the method to be adopted in order to model the phenomenon intervening the evolution of the solid-solid system and in particular the influence of the design parameters on the rate.

14.5.1.2. Method for the kinetic study

We wish to find solutions for the fractional extent-time or rate-time functions with any space packing of grains.

To study these functions, three study levels must be examined:

– The scale of a contact between two grains A/C.

– The scale of a cell of a grain A surrounded by its neighbors.

– The scale of the mixture appearing as an assembly of cells.

Cell


We are considering a solution α(t) for the scale of a contact between two grains and then for a cell. The result can then be extended to the whole powder.

14.5.1.2.1. First level: contact between two grains

We consider two possibilities according to whether the reactant is plastic or not (i.e. more or less close to a temperature of change of state: fusion, sublimation, polymorphic tranformation).

If one of the components is rather plastic so that a deformation occurs during heating, the contact interface will then have a surface whose area will be the total area of the nonplastic grains. The problem is then brought back to the M1 model. At the level of the cell, a continuum of one of the phases surrounds the other. As for the gas-solid reaction, the solution has been already examined in Chapter 10.

If none of the components is plastic, then after the period of nucleation, the reaction of growth between two grains A and C in contact can exhibit two types of behavior: anisotropic growth or isotropic growth starting from a point, which we have already examined in Chapter 10.

It is noted that in both cases, in pure modes, the rate between two grains remains separable, as we have assumed in section 14.4.1.

14.5.1.2.2. Second level: the cell

Now consider a spheric grain A surrounded with grains Ai and Cj of both reactants.

In the M1 model, the central grain is surrounded by a continuum of C. We consider the traditional case in which the whole surface of A is in contact with the whole surface of the other reactant C (being either solid or gas), which we have already examined in Chapter 10.

In the case of a collection of definite grains in a discrete way, we consider a cell with a collection of grains made up of a central grain and its close neigbors. The surounding is extended to a scale defined according to the composition of the powder, and the amount of the second reactant solid C around A is sufficiently representative of the total composition of the powder.

Two series of cells can thus be defined, either modifying the local composition or modifying the design.

We define cell of average local composition as a cell with the same composition as that of the mixture of powders.


We define cell of average design as a discrete cell presenting the composition nearest to that of the mixture of powders.

We define average cell collection as a collection of elementary cells that are representative of all the cells.

An elementary cell is a cell defined by particular interactions between grains A and C.

For example, for a random collection AC (Figure 14.9a), the elementary cell is given by a random distribution of grains C and A around grain A by taking into account the composition of the cell. This cell will be then single and identical to the local cell of average composition.

Figure 14.9. Design of cells of a stoichiometric mixture A/C (rA = rC )

To represent a collection AC with very strong attraction between A and C, any given elementary cell should include more contacts AC for the same global composition of the mixture and less contacts A-A and CC for the cell of average local composition. To respect the global composition, it is necessary to build several elementary cells, each one with a statistical weight fi, definite so that the average of these inhomogenous elementary cells leads to a global composition equal to that of the macroscopic mixture. Thus, Figures 14.9b and c show a mixture with infinite attraction between A and C, with the same amount of both reactants in two elementary cells and a ratio (1.1).

To manage the reactions between the central grain A and the surrounding grains Ai and Cj of the cell, we consider the reactions at each contact A/C, knowing that the new phase AC could nucleate at any contact and at any time.

Two corollaries rise from these considerations:

– When a fast tangential growth occurs, a central grain A as well as a neighboring grain Ai is covered with AC. The problem then is very simplified and is reduced to the one-process model with radial growth on only grain A (see section 10.2.1).

A A

A

C C

C A

AC

C

C C

C C

C A

A

A A

AA

(a) (b) (c)


– The local composition of the studied cell can then lead to a reaction that will consume A completely or partially even if A is the least abundant component (in the stoichiometric sense).

14.5.1.2.3. Third level: the whole powder.

We assume that mixture M is equivalent to a collection (Si) of elementary cells of real stacking: M = (S1, S2, … , Si, … , Sn).

The average composition calculated starting from the fi fraction of cells of σi composition on the series (S) must be the average composition of the mixture:

1

n

i ii

fσ σ=

= ∑

The function fractional extent α(t) relating to the mixture is obtained from a median value calculated on all the expressions αι(t) correspondent with each cell i with the statistical weight fi.

However, this can lead to the following problems:

– Each cell is not independent of its neighbors, and it would be necessary to follow the variations in geometry of the mixture during the reaction (consumption of A and C, expansion if the molar volume of the formed solid is higher than the sum of molar volumes of the reactants).

– If large non-homogeneities appear in the mixture, it will be more difficult to choose the series of cells and, in this case, the definition of the statistical weights fi can be arbitrary if we do not use analytical methods to evaluate the homogenity of the mixture.

Thus, it appears that the study of space modeling of the reaction between solids must successively take into account:

1. The possibility, or not, of a fast tangential growth, or a layer of A or C on the other reactant.

In the event of slow tangential growth or in the absence of layers, it will be necessary to define the nucleation laws and to specify the homogenity of the mixture to build the cells.

In the event of fast tangential growth, it is necessary to define an average cell as a grain surrounded by a continuum of C with or without instantaneous nucleation on


the whole external surface of A. We thus find that the two-process model with nucleation leads to an anisotropic growth (see section 10.5).

For a layer, the average cell is the same as previously mentioned (A + continuum of C), but nucleation, which is not inevitably instantaneous, has a probability of occurring at any given point, but at different time points, and can be followed as an isotropic growth. We thus find that the two-process model with nucleation gives an isotropic growth (see section 10.6).

2. It is necessary to establish the space function E(t) or E(α) for two grains A and C.

3. It is necessary to generalize the rate either to the collection of the elementary cells or to the average cells.

4. From the cells, it is necessary to calculate the total rate of the powder.

REMARK.– We do not know a priori whether the rate is separable or not at the cell and powder levels, so we cannot be satisfied to combine the space functions.

14.5.2. Rates of a two-grain level

At a two-grain level, only one nucleus at the contact initiates the reaction whose extent is counted from this time (Figure 14.10).

If the diffusion of a reactant (C, for example) on the surface of the product of the reaction (path OH on Figure 14.10) AC is easy, the tangential growth of AC on the surface of grain A will occur very quickly as soon as a nucleus is formed at the contact. In this case, we can find the anisotropic growth of AC on the surface of A.

Figure 14.10. Isotropic growth at the contact of two grains

If surface diffusion is very slow, then the growth becomes isotropic and only an interface step can be rate determining.

A

O

C

H

AC


We must recognize that very often the reactions between solids have diffusion as the rate determining step. In consequence with different energies of activation, the velocity of surface diffusion is faster at moderate temperature and it is only at very high temperatures that the diffusion in the bulk is faster than the diffusion at the surface. It is obviously the fastest of both diffusions (which are two parallel steps) that is essential.

The growth is counted starting from the moment when the nucleus is formed at the contact point.

If one of the two grains (C, for example) covers the whole surface of the other, the problem is brought back to that of the M1 model, as seen in Chapter 10.

If the tangential growth is fast, one of the grains is surrounded very quickly with a spherical crown of AC. A shell is formed, and we are brought back to the case of nucleation giving an anisotropic growth (see section 10.4).

If no coating and no surface diffusion intervene, then the rate determining step is the interface reaction causing the advance into A. The calculation of the space function relative to this case has already been carried out (see section 10.2).

14.5.3. Rate of a granular cell

For a two-grain scale, only one nucleus at each contact initiates the reaction. But in a cell, if no surface diffusion intervenes, the various contacts nucleate generally at different time points and all the isotropic types of growth could be considered a priori.

However, the one-process model with slow nucleation and instantaneous growth is forever highlighted in the reactions between solids. This is due undoubtedly to the fact that the network of grains A and C is almost always connected (except, perhaps, for the systems with very diluted C), whereas for the solid-gas systems, for example, the grains are independent of each other and react individually with the gas.

In addition, it is known that the isotropic growth with a bulk diffusion as the rate determining step is not conceivable.

If the tangential diffusion of one reactant (C, for example) on the surface of product AC is easy, the tangential growth of AC can be carried out very quickly as soon as a nucleus of AC is formed at one of contacts of the powder A/C; the connectivity of the network of the inter-grains bonds distributes quasi-instantaneously the matter of AC on the surface of all the grains of A, leading to shell growth.


In the case of two-process models in which the growth starts from a point, only those with an interface reaction as the rate determining step will be examined at the cell level. On the other hand, the one-process models with slow growth will be studied in detail.

A final point must be underlined before approaching the study of the cells. The system is not symmetrical with respect to A and C at the level of the cell and powder. Asymmetry comes from the size and the composition. A cell centered on A surrounded with C and a cell centered on C surrounded with A will behave differently. For given sizes of A and C and composition, a representative collection of various cells will have to be defined to represent the powder.

14.5.3.1. One-process models with slow growth and interface reactions as rate determining steps

14.5.3.1.1. Analytical treatment for a cell of M1 type

For M1 cells (matter continuum around one of the reactants), the one-process models with slow growth and interface reactions as rate determining steps have already been considered (see section 10.2) and the results are given in Appendix 3.

14.5.3.1.2. Analytical treatment for a cell of M2 type

If all the diffusions are infinitely fast (on the surface and in the bulk), the transport of matter onto the interfaces are instantaneous whatever the design (sizes, relative placements of the grains) and we are thus brought back to the case of M1 cells.

If the growth is isotropic, then the problem quickly becomes very complicated. However, it is possible to consider the case of a cell made up of a central spherical grain A (with radius rA(0)) surrounded by n grains of the other reactant C tangent to the central grain and equidistant from each other, as in the case of a pure mode with an internal interface reaction (in A) as the rate determining step and with inward development (in A). If 2β0 indicates (Figure 14.11a), the angle between the center lines of two successive grains C, we have:

220

4π4n

β =

It follows:


0πn

β =

Figure 14.11. Distribution of one reactant around the other

The maximum number of grains C around A is given as follows (Figure 14.10b):

2

max 20m

πint n

β⎛ ⎞

= ⎜ ⎟⎝ ⎠

with C(0)0m

A(0) C(0)

sinr

r rβ =

+

Considering C(0)C

A(0)

rr

ρ = , we have then:

C0m

C

sin1

ρβ

ρ=

+

At each contact point C/A, a sphere sector with radius mAr V tφ= is formed, from which a reduced time is obtained as follows:

mA

A(0) A(0)

V t rr r

φθ = =

The fractional extent of the transformation can be expressed with respect to A and it cannot reach the end value of 1, except if there is sufficient reactant C for which it is necessary that:

β0

C

C

A

β0 C

C

A rA(0)

rC(0)

(a) (b)


3 3C(0) C A(0)

mC mA

nr rV V

ν>

If this inequality is not obeyed, the final fractional extent will be as follows:

3C(0) C mA

3mC A(0)

f

nr VV r

να =

To express the fractional extent at any time t, we cut out grain A into spherical layers with radius R (0 < R < rA(0)) and thickness dR. Call dVA the transformed volume in such a layer, then the amount of transformed A in this layer is thus

A mAd /V V and the fractional extent is:

A ( 0)

A3 0A(0)

3 d4π

rV

rα = ∫

At a given time, all the sectors of AC have the same size. If we look at all the circles with radius R traced inside grain A, two categories can occur:

– those that cut only non-secant AC sectors;

– those that cut only secant AC sectors.

We can thus divide grain A into four areas:

– if 0 < R < rA(0) − r = R1: the circle with radius R never cuts grains of product formed AC;

– if R1 < R < R2, the sphere with radius R cut non-secant portions of sectors (not covering between grains AC, along the sphere with radius R);

– if R2 < R < R3, the sectors cut by the sphere with radius R are secant (they are recovered); and

– if R3 < R < rA(0), the sphere with R can again cut portions of non-secant sectors.

We then calculate the values of R2 and R3.

From Figure 14.12b, we obtain:

A(0) 0HP cos cosr R rβ γ= − =


Figure 14.12. Relative positions of the circle with radius R and grains

Moreover, in the triangle OMP, we have:

0sinsinR r

βγ=

and we deduce:

( )A(0) 20

A(0)

cos 12 2r R

R rβ θ= − +

We examine the cases in which these solutions exist and, if so, they are acceptable.

For the existence of R2, we express P1H (Figure 14.13a):

1 A(0) 0P H sinr β= [14.5a]

thus R2 exists if 0 1sin .θ β θ> =

P

R

r

rA(0)

P′β0

A

AC

AC

C

C O P

M

H β0 γ

R r

(a) (b)


For the existence of R3, from Figure 14.13b, we have:

2 2 21 1P M HM P H= + and A(0) A(0) 0HM OH (1 cos )r r β= − = − [14.5b]

Therefore, 01 0P M 2 sin

2r

β=

Figure 14.13. Curves with existence of R2 and R3

The condition so that R3 exists is PM r< ; therefore,

032sin

2β

θ θ> =

We have A(0)

R yr

= and thus 1

A(0)

1R

yr

θ= = − .

On considering the fractional extent, we can have three periods:

– If 0πsin sinn

θ β< =

We will have A( 0)

2

23

A(0)

3 d4π

r

Rn R R

rα = Ω∫ with 02π(1 cos )βΩ = − and thus:

( )21 2

1

13 1 d2 2 2n y y y

yθ

θα

−

⎡ ⎤−⎢ ⎥= − +⎢ ⎥⎣ ⎦

∫

P1

P2

H M

β0

rA(0)

P1

P2

H M β0

rA(0)

(a) (b)


– If π πsin 2sin2n n

θ< <

( )

( )

32A A (0)

( 0)

2

A ( 0)

3

A ( 0)

22 2

1

21 2

13 1 dy 3 d2 2 2

13 1 d2 2 2

RRr r

Rr

Rr

n y y y yy

n y y yy

θ

θα

θ

−

⎡ ⎤−⎢ ⎥= − + +⎢ ⎥⎣ ⎦

⎡ ⎤−⎢ ⎥+ − +⎢ ⎥⎣ ⎦

∫ ∫

∫

– If π2sin2 n

θ >

( )2A ( 0)

2

212 2

1

13 1 d 3 d2 2 2

Rr

R

n y y y y yyθ

θα

−

⎡ ⎤−⎢ ⎥= − + +⎢ ⎥⎣ ⎦

∫ ∫

where R2 and R3 are obtained from equation [14.5a and b].

We can use these three expressions to compute the whole of the kinetic curve.

14.5.3.2. One-process models with slow growth and diffusion as the rate- determining step

It is known that the one-process models with slow growth and diffusion as the rate-determining step are conceivable only for the anisotropic growth. The literature especially considered the case of diffusion of a component in M1-type cells with a central spherical grain A surrounded by a continuum of the other reactant. In fact, the results for other geometries other than spherical grains A are given in section 10.2 and assembled in the column “diffusion” of tables of Appendix 3.

14.5.3.3. One-process models with slow growth and counter-diffusion of two components as the rate determining step

For the counter-diffusion modes, we can consider two types of cells M1 and M2 according to whether the surface diffusion is very fast or very slow.

14.5.3.3.1. Analytical treatment of M1 cells, spherical crown

We can consider the following assumptions:

– The system is made up of a spherical grain A and very fine grains C forming a matter continuum in spherical crown around A for a given composition of mixture.


– The nuclei born at the initial time at all contact points A/C and velocities of surface diffusion (or tangential growth) are the same as those at the initial time when sphere A is surrounded by a layer of the formed product AC (Figure 14.14).

– The rate determining steps for the AC growth are two parallel diffusion processes: diffusion of A in one direction and diffusion of C in the opposite direction through the AC layer.

Figure 14.14. Counter-diffusion in an M1 cell

We can consider in the same way as for a single diffusion assuming a pseudo-steady state mode in spherical symmetry by setting the two mechanisms in parallel.

Call rA(0) the initial radius of the grains A, re(0) is the initial radius of the cell including grain A and the spherical crown of the continuum of C, ra2 is the radius of interface C/AC, ra1 is the radius of interface A/AC and ra3 is the radius of the empty sphere created by the heart reaction of annihilation of the vacancies of A (Figure 14.13). Let us indicate by φA and φC the reactivities of diffusions of A and C, expressed by:

A A AD Cφ = Δ and C C C D Cφ = Δ

If zA and zC are the coefficients of expansion AC with respect to A and C, the parameter of composition σΑ is defined as the ratio of the initial amount of C to that of A in the cell with initial radius re(0):

( )3

A A(0)A 3 3

C e(0) A(0)

z r

z r rσ =

−

A

C

AC

ra1 ra3

ra2


We can write fractional extent with respect to A in two different ways:

– The ratio of the amount of AC formed to the initial amount of A.

– The ratio of the disappeared amount A to the initial amount of A.

These two points of view can be expressed as follows:

3 3a 2 a1

3A A(0)

r rz r

α−

= [14.6]

3 3 3A(0) a1 a3

3A(0)

r r rr

α− +

= [14.7]

In addition, the movement of interface A/AC is due only to the diffusion of C whereas that of interface heart/A is due only to the diffusion of A, which we can write by the following movement equations (see Chapter 9):

2 2a1a1 A C D a1

dd mr

r V G rt

φ= − or a(0) a1 a 2a1mA C

a 2 a1

dd

r r rrV

t r rφ= −

− [14.8]

And 3

a(0) A12 a3a3 A A

a 2 a1

dd m

r rrr V

t r rφ=

− [14.9]

By solving the system with four equations [14.6], [14.7], [14.8], and [14.9] thus obtained, the counter-diffusion law is expressed as follows:

A C A C A(0) A C A C A(0)( , , , , , ) ( , , , , , )f z z r g t z z rα φ φ φ φ=

with

( )A C C A 2A(0)

2 1 tg z zr

φ φ⎡ ⎤= − +⎣ ⎦

and

( )

( )

2 / 3

A A C A C A

C A A C A

2 / 3A A C A C A A C

A C C A A A

1 1 1

1 1 1

C

z z z zf

z z

z z z z zz z z

φ φ αφ φ σ

φ φ φ α αφ φ φ σ

⎡ ⎤− +⎢ ⎥= − +

+⎢ ⎥⎣ ⎦

⎡ ⎤− + ⎛ ⎞⎢ ⎥+ − −⎜ ⎟+⎢ ⎥⎝ ⎠⎣ ⎦


We can verify that the laws with only one centrifugal or centripetal diffusion as the rate determining step result from this expression by considering σΑ = 1 and by expressing the limited development with respect to φΑ /φC.

A limit of these models is to assume a continuum of C, that is, a ratio of the sizes of initial grains of A and C as infinite, with the interest to study a more general case by rejecting the system and by eliminating this restrictive assumption.

REMARK.– If the two determining steps of parallel diffusions occur in the same zone (layer of B), then the rate is separable.

14.5.3.3.2. Treatment of cells of M2 type, sphere-sphere

We study the case of a cell made up of a central sphere surrounded by spherical grains of C with counter-diffusion of A and C. Because the surface diffusion is very fast in a cell where all the grains of A and C are in contact, the system will evolve almost instantaneously in a system in which each spherical grain is surrounded by a shell of AC.

REMARK.– The ratio of A and C radii is considered in the range from 1 to 3. For a ratio from 0.3 to 1, we invert the roles of A and C.

The ordered or random mixture of grains A and C in a given cell whose composition is fixed leads to the same kinetic laws, the design of the mixture is without effect, and the law α(t) does not depend on the relative placement of grains A and C.

In what follows, we thus propose to calculate the α(t) law if the ratio of the grain radius varies and in which the product, as a result of diffusion, moves simultaneously inward to grains A and C.

On the assumption of a very large tangential growth, the reaction, homogenous or not, cell is a cell of average local composition in which grain A is surrounded by NC grains C (Figure 14.15). The relative placement of grains C compared with the central grain A does not have any importance.

We define the parameter σC in the average cell as the ratio of the initial amount of matter of A to that of C in this cell:

3A A(0)

C 3C C(0) C

z rz r N

σ = [14.10]


Figure 14.15. The M2 model of counter-diffusion for a cell centered on grain A surrounded with NC grains C

We consider the fractional extent with respect to C (by taking the initial amount of C as a reference) Compound C is assumed to be the least abundant. This fractional extent is equal to the ratio of the amount of AC formed to the initial amount of C:

3 3 3 3a 2 a1 C c2 c1

C A C3C C C(0)

( ) ( )r r N r rN z r

α α σ− + −

= = [14.11]

At time t, the amount of matter of A having disappeared by the reaction with C to form the crown around A is equal to the amount of AC present on the grain A or:

3 3 3 3a 2 a1 A A(0) a1( )r r z r r− = − [14.12]

In the same way, the amount of AC formed around grain C gives:

3 3 3 3c2 c1 C c(0) c1( )r r z r r− = − [14.13]

rc1

rc2

Initial C grain

C

AC

C

A

AC

AC

ra2

ra3

Initial A grain

ra1

rc3


The change in the volume of A due to the formation of AC around A by diffusion of C is written:

C(0) a2 a12 a1a1 C mA

a2 a1

d

dr r rr

r Vt r r

φ= −−

[14.14]

The change of volume due to the formation of AC on C will be:

C(0) c2 c12 c1c1 C mC

c2 c1

d

dr r rr

r Vt r r

φ= −−

[14.15]

The reactivities being defined by:

C AD cφ = Δ and A CD cφ = Δ .

Combining equations [14.10] to [14.15], we obtain an expression between αC and t via parameters zA, zC, φΑ, φC, rA(0), and rC(0).

The graphs in Figures 14.16a and b show the influence, on the one hand, of the ratio of the reactivities of diffusion of A and C and, on the other hand, that of the ratio of the initial radius of grains A and C. These graphs have been plotted for zA = zC = 1.5, VmAφArC(0) = 1. In graph b, we also considered VmCφCrC(0) = 1.

REMARK.– The two zones of diffusion of A and C being different, the process with two parallel rate-determining steps studied here does not lead to a separable rate.

Figure 14.16. Kinetic curves with counter-diffusion in M2 cells


We can examine two simple one-process models, defined by the relative values of the coefficients of diffusion of A and C.

If the coefficient of diffusion of A is much lower than that of C, the formation of AC is then ensured by the only diffusion of C. Figure 14.17a specifies the spatial design of the system. This design is equivalent to a completely spherical geometry with a continuum of C around A. The rate becomes separable and we get:

2 / 3 2 / 3

C C C mAA A A A 2

A A A(0)

1 ( 1) ( 1) 1 2(1 )V

z z z z tr

α α φσ σ

⎡ ⎤ ⎛ ⎞+ − + − − = + −⎜ ⎟⎢ ⎥

⎣ ⎦ ⎝ ⎠

This formula is a generalization of the Valensi-Carter law [VAL 31] (see section 10.2.2), with a stoichiometry ratio of the mixture other than 1, and in the cases in which grains C can have any size and position with respect to grains A.

If the coefficient of diffusion of C is much lower than that of A, the formation of AC is then ensured only by the diffusion of A. Figure 14.16b specifies the spatial design of the system. The rate becomes separable, and we get:

[ ]2 / 3

2 / 3 A mCC A C A C C 2

C(0)

1 ( 1) ( 1)(1 ) 2(1 )V

z z z z tr

φα α+ − + − − = + −

This is identical to the Valensi-Carter law.

Figure 14.17. One-process model with counter-diffusion as the rate determining step in M2 cells

A study of the reaction of chromium trioxide with dicobalt tetraoxide highlights well the relative positions of the phases [JAN 97].

(a) (b)

AC AC

C C A A

Direction of diffusion Direction of diffusion


The laws that we have just established would transpose easily to other shapes of the grains (cylinders, plates, etc.). The rates in such laws are separable.

14.5.3.4. The case of nucleation growth with inward development and an internal interface reaction as the rate determining step for growth

To approach the study of the systems in two-process model of nucleation and growth, a method of simulation by finite elements makes it possible to represent the reaction process in a cell of the M2 type.

The method consists of cutting out space into identical elementary volumes and applying the extent laws locally.

No obvious analytical solution is possible if the number of contacts is high. Also, we resort to a simulation whose validity is guaranteed by the fact that the result so obtained on a simple system with two diametrically opposite contacts (Figure 14.18) with nucleation at different times is identical to that given by a possible analytical solution in this simple case.

14.5.3.4.1. Calculation with two contacts

Thus, we consider the case of a spherical grain A with radius rA(0) surrounded by two spherical grains C. We assume that an internal interface reaction on A is the rate determining step and an inward development into A. φ is the reactivity and VmA is the molar volume of A. The ratio of the initial radii of A and C is such that the composition is stoichiometric. We assume that two nuclei occur at contact points O1 and O2 (Figure 14.17) at times τ1 and τ2. At any time t, the radii of the interfaces are r1 = O1M and r2 = O2M, such that:

1 mA 1( )r V tφ τ= − and 2 mA 2( )r V tφ τ= −

Figure 14.18. Cell with two diametrically opposite contacts

C C O1 O2 H1

M

M′

H

K1

H2

K2

K′2K′1

β2β1

β ′1

A


As long as time A(0)1 2

mA

2rt

Vτ τ

φ< + + , the two caps centered at O1 and O2 are never

superimposed, their surfaces are s1 and s2 (Figure 14.18), and the rate that is separable is considered:

2 2mA 1 1 2 21 2

3A(0) A(0)

1 21 2

A(0) A(0)

3 2π (1 cos ) (1 cos )( )dd 4π

3 1 1 2 2 2

V r rs st n r

r rr r

r r

β βα φ φ

φ

′ ′⎡ ⎤− + −+ ⎣ ⎦= =

⎡ ⎤⎛ ⎞ ⎛ ⎞= − + −⎢ ⎥⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦

[14.16]

If time A(0)1 2

mA

2rt V τ τφ> + + , the two caps are secant and the only effective

surfaces are the portions of caps K1M and K'1M' for nucleus 1 and K2M and K'2M' for nucleus 2. The rate that is separable is thus:

{ }

21 2

A(0)

2 2mA 1 1 1 2 2 2

3A(0)

( ' ' )d 4d

3 2π (1 cos ' ) (1 cos ) (1 cos ' ) (1 cos )

4π

s sb ac

t n

V r r

r

α φ

β β β βφ

+= −

⎡ ⎤ ⎡ ⎤− − − + − − −⎣ ⎦ ⎣ ⎦=

[14.17]

where cos β1 and cos β2 are solutions of the system deduced from triangles O1MH and O2MH, respectively:

1 1 2 2

1 1 2 2 A(0)

sin sincos cos 2

r rr r r

β ββ β

⎧ =⎪⎨ + =⎪⎩

We get:

2 2 2A(0) 2 1

11 A(0)

4cos

4r r r

r rβ

− += and

2 2 22 A(0) 1

22 A(0)

4cos

4r r r

r rβ

+ −=

To solve this, we consider the following dimensionless properties:

mA

A(0)

( )i iV

tr

θ φ τ= − and mA

A(0)i i

Vr

η φτ=


We obtain after integration:

– If A(0)1 2

mA

2rt

Vτ τ

φ< + +

3 4 3 41 1 2 2

1 23 3

( )2 16 2 16

θ θ θ θα θ θ = − + −

– if A(0)1 2

mA

2rt

Vτ τ

φ> + +

23 4 3 4 2 2 21 1 2 2 1 2 1

1 2 2 2

22 2 2 2 2 2 2 32 1 1 2 1 2 2 1 2 1 2 2

1

3 3 1( ) 1 2 12 16 2 16 4 4 4 4

31 1 14 4 4 16 16 8 2 2 2 4 4

θ θ θ θ θ θ θα θ θ θ θ

θ θ θ θ θ θ θ θ θ θ θ θθ

⎛ ⎞ ⎛ ⎞= − + − − − + − − −⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠

⎛ ⎞ ⎛ ⎞− − + − − − + − + + − + −⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠

14.5.3.4.2. Simulation with three contacts; role of the design

We simulate by the finite element method configurations with three contacts.

For a given design of a system with three grains of C around a grain of A, modifying the order of appearance of the nucleus with ηi, the result led to differentiated curves (Figure 14.19a).

Figure 14.19. Simulation of a cell with three contacts


It is noted that the order of nucleation and the relative placement of the contacts have a notable influence on the kinetic curve. The effects are especially sensitive to the effects of time when the caps are superimposed.

14.5.4. Rates on the scale of the powder

14.5.4.1. Switching from the cell to the powder

On switching from the cell to the powder, two possibilities arise:

– either we define a single cell representing the whole system;

– or an average can be carried out with various elementary cells.

The first case with a single cell is adopted implicitly by all the authors having considered the case of M11 cells; thus, all the corresponding laws are valid for the powder. The fractional extent calculated for a cell is identical to that for the whole mixture with composition σΑ. The rates are separable.

In the second case, the mixture is the superposition of different cells

– by composition x (x varying from 0 to ∞);

– by the relative position of the x fixed spheres.

If the maximum probability of finding the average cell for a composition close to σΑ fixed at the beginning, then the stoichiometric composition x of each elementary cell is no longer taken equal to the average composition σΑ, with the distribution of the probabilities P(i) of finding a cell x being assumed to obey a given law. The average fractional extent calculated for all the cells is then considered as follows:

1

1

( )

( )

ni

n

P i

P i

αα = ∑

∑

However, we have seen in a cell scale that various designs for the same composition can give different results. The mixture of powders must thus be represented for each composition by a second collection of cells, with NA grains of A and NC grain of C, differing by the relative placement of the spheres of both reactants A and C around the central grain, which can be either A or C.

Thus, generally, it is not enough to stipulate the number of contacts A/C to define the reaction rate between two solids A and C. With various designs j, Ω


possibilities can be met with the probability fj, the expression of the fractional extent becomes more complex.

To simplify, the stoichiometric composition of each elementary cell can be taken equal to the average composition: the powder is then a collection having the same composition but with different designs. The law of weighting of the designs is a priori a function of the homogenity of the mixture. This being a function of the attraction and repulsion between grains.

The two-process model with nucleation and growth and diffusion in the bulk as the rate determining step with the M2 cell is not studied here considering its low probability and the difficulty of its treatment. On the other hand, we will consider the cases of two-process models with pure interface modes.

14.5.4.1.1. Models with a fast surface diffusion

For the one-process models with a slow growth by simple diffusion (M11 cell with a central spherical grain surrounded by a continuum of C), the assumption of a fast tangential growth on the surface involves two outcomes:

– a thin layer of product C forms at the initial time a shell around the zone that has not reacted;

– the relative position of the grains of both reactants does not have any importance; the design problems involved in the homogenity of the powder are cancelled.

In the powder, as soon as a grain of A is attacked, all the other grains of A are also attacked at the same time. The flow of A around the grains of C and that of C around the grains of A are assumed to be perfectly ensured in these models.

For all the M11 and M2 models, the method of the average cell is the only valid method. The method of distribution of cells based on the composition cannot be adopted because the network of grains of A and C is completely connected and the product AC in the spherical crown is also very quickly surrounded.

The composition is a total parameter, not a local one. The homogenity is not a kinetic factor of the mixture. Also, the stoichiometric composition of each elementary cell is taken equal to the average composition, and the powder is represented by a collection of identical cells: the calculated fractional extent for a cell is then identical to that of the total mixture and the rate is separable.

14.5.4.1.2. Models with a slow surface diffusion

This case is quite rare.


The evolution of the system is then very different from that of the preceding case because the network of formed product AC is not connected: the reaction develops around the contacts and then we can consider:

– either with an average cell, in a very approximate way;

– or with a composition distribution of cells;

– or with a place distribution of cells;

– or with both composition and place distributions of cell.

The case of a composition distribution: if the distribution of C around A is rather wide, considering an average design, we can choose the Poisson or Gauss law of composition.

For a discrete Poisson law, the median value aNS is a function of σΑ:

( )SSP( ) exp( )

!

iaNi aN

i= − and ( ) 1

iP i =∑

For the normal Gauss law, P(X) represents the fraction of composition cells that is lower than or equal to X. The density of normalized probability is given by:

22

d ( ) 1( ) expd 22π

xP xq xx

σσσ−⎡ ⎤= = −⎢ ⎥⎣ ⎦

where Aσ is the average ratio of the amount of matter and σ is the standard deviation of the distribution, parameter which characterizes the homogenity of the cells and thus of mixture. The fractional extent of the powder is then given by:

A20

1( ) exp d22π

xx x

σα α

σσ

∞ ⎡ ⎤−= −⎢ ⎥

⎢ ⎥⎣ ⎦∫

If σ is low, we are subjugated to the case of a single average cell.

Case of a place distribution: we now assume that all the cells have the same composition.

To represent the influence of the fluctuation kinetics on the placement, we must consider the parameters that would define the respective positions of spheres C or A around a sphere of A, and in particular:


– the maximum number of grains of C (species j) or A (species i) that we can place around central A (species i), also called co-ordination number zij of the spheres j around a sphere A (species i) with a size ratio /i i jk r r= or the co-ordination number zii;

– the distribution of considered positions around A, which depends on the nature of the interactions between grains A and C. Two limit mixtures are often considered: random mixtures and ordered mixtures.

For the random mixtures, the co-ordination number is then 0ijz . It is given, for

example, by the Suzuki-Oshima model [SUZ 83a, SUZ 83b]. The model defines a contact sphere (Sc) surrounded exclusively with C (Figure 14.20) and whose center coincides with that of sphere i and whose radius is:

2S 2i i jr r r r= +

Figure 14.20. Contact sphere according to Suzuki and Oshima

The area of the contact sphere divided by the area of the surface cut (on Sc) by a sphere of co-ordination number j defines 0

ijz .

A geometrical calculation makes it possible to obtain:

0C

2(1 )1 (2 )

iij

i i i

kz

k k kε

+=

+ − +

Contact sphere

ri rj

rSc


Constant εC is obtained from the average co-ordination number z in the stacking of spheres having the same size (ki = 1).

z is the average co-ordination number of a stacking. This is practically a constant that is approximately 6.2 for a given stacking.

Neglecting the collective effects of stacking, we obtain εC = 0.402, with zii = 6.

Then, the partial co-ordination number zij of the sphere ri in a given stacking of spheres ri and rj is assumed to be proportional to 0

ijz and to the fraction of the area of j in the mixture:

20

2 2j j

ij ijj j i i

n rz z

n r n r=

+

where ni is the numerical fraction of grain i (connected to σΑ by a simple relation).

For a given composition σC of the mixture, we can evaluate median number NC of C that we are able to place around grain A for a fixed size ratio of grains.

3A A(0)

C 3C C(0) C

z rz r N

σ =

For a random mixture, we can postulate that the average cell has a composition close to that of the total composition and the fractional extent can be simply calculated as that for the numerical average of the fractional extent of the average cells on all the possible places 0

ijz by permuting the NC spheres with each other. Each position has as much probability to be reached as another since the mixture is assumed to be without interaction:

C1

1( , ) ( , )C ii

t tα σ α σΩ

=

=Ω ∑

For the average cell, we can define around each central grain A an environment, such that the composition of the cell thus described will be equal to the average composition of the mixture preserving the number of contact average A/C. Indeed, it is kinetically dissimilar to take an average cell having 2NC half-grains of C surrounding A and a cell having NC grains of C surrounding A.


Also, the volume of each C grain belonging to the partition will be taken equal to a fraction y of the volume of the grain. In a first approximation, the spherical grains of A possibly in contact with the central grain of A will not count in the evaluation of the extent. The cell thus will consist of a grain of A in the center, surrounded by NC/y = zAC pieces of C.

Consider y: each grain of C takes part in several contacts with other grains of A. If the number of grains C in contact with a central grain of A is zij, then the fraction y of volume C to be taken into account:

mC A A mCAC Ccell

mA AC C C C mA

V N V VVV z yV N V V

σ σ= = =

that is,

C

AC A

Ny

z N=

The calculation of y is reduced to the evaluation of zAC, which we can estimate by the Suzuki model. This partition is possible only when the reaction process is with simple diffusion of species C toward the species A as the rate determining step. It is appropriate to take as a central grain the species in which the diffusion occurs. The form of spieces C will be without influence and account for only the volume of the spieces and the site of the contact point.

For the ordered mixtures, simply two limit cases can be considered:

– when grains C on the surface of A present infinite repulsions;

– when attractions are infinite.

If repulsions C-C are infinite (as in Figure 14.21), the only possible configuration will be that where spheres C are most distant from each other. The presence of strong interactions would result in describing the total system rather by a distribution of composition of a single cell.

If attractions C-C are infinite, only one configuration is to be retained, the one in which grains C are in contact around A.

Reaction rate in the case of infinite repulsions is much higher than that obtained in infinite attractions such as shown in our preceding study of a sphere surrounded by three grains.

We can in the same way consider attractions A/C (infinite) and repulsions A/C.


Figure 14.21. Mixture A/C with infinite repulsions C-C

14.5.4.2. Comprehensive approach, influence of the design parameters

The influences of the design parameters, particle shapes and sizes, composition, places have been examined on various levels. Comprehensive approaches are also possible and, in particular, models making it possible to explain the influence of the size of grains and composition on the initial rate have been developed (see, e.g. [COU 81] and [KOM 64]).

The influence of the mixture composition on the initial rate, in the case of diffusion of only one species, can be approached overall in the following way.

Two zones of over-stoichiometric compositions of C are analyzed, one (named p) not very rich in C and the other (named r) very rich in C.

It is known that the rate is proportional to the area of contact sac and the number of AC contacts NAC. We finally assume infinite attractions A-C of the grains.

Calculation is performed by considering grains C as larger than grains A.

In zone p, NA grains of A are all in contact with the surface of grains C and all are likely to react. This occurs until a limit composition σC is reached, for which all the grains of C entirely cover the surface of A. Beyond σC, the surplus grains A will be placed into interstitial positions and will no longer remain in direct contact with grains C (zone r). They will also no longer react at the initial times.

Assume the case of an excess of C. The fractional extent is defined with respect to A. If φ is the reactivity of growth, the rate for spheres will be as follows:

mA AC AC3

A A(0)

3d Ed 4π

V N st N r

φα φ= =

C

A A A

C

C C

C


In the case of a very fast nucleation of the new solid phase of AC around each contact and if each grain is surrounded in the same way, this expression is true at any given time or whatever the fractional extent.

In the opposite case, this expression will be valid only at the very first times.

Evaluate sAC and NAC as a function of the parameters: grain radius and composition.

A contact is established at the nearest distance δ (in micrometers).

– if rA(0) < δ < rC(0), the contact area can be estimated as the average between the area of the projected sphere on a plan and the area of a half-sphere A, that is:

2A(0)

AC

3π2r

s =

– if δ < rC(0) < rA(0), then we can consider:

A(0) C(0)AC

A(0) C(0)

2πr r

sr r

δ=+

REMARK.– If Nt is the total number of contacts in the powder:

( )A C 2tz

N N N= +

We can calculate NAC within the framework of random models of mixtures according to the numerical fraction NA of grains A and their co-ordination number zA:

( )A C A CAC A C AC2

22 t

n n z z zN N N t Nz

= + =

where tAC is the fraction of contacts A/C and z is the average co-ordination number of the mixture.

It is then necessary to evaluate zA with the help of a model such as the Dodds model [DOD 75], developed by Bideau [BID 82].

We develop here the NAC calculation for mixtures in which attractions between the two types of grains are very strong. This case is very frequent when one of the


two species consists of very small grains (in micrometers). We consider 2

A(0)AC

3π2r

s = (grains A are very small).

– initially consider zone r (very rich in C): σC > σClim.

Grains A, of which there are very few, will be in contact with grains C whatever the number of these grains. The number of contacts A/C in the powder is thus equal to the number of grains A:

AC

A

1NN

= ; hence, mAA(0)

d 9 1d 8

Vt rα φ

⎛ ⎞= ⎜ ⎟⎜ ⎟

⎝ ⎠

– now consider zone p (not very rich in C): 1 < σC < σClim

All the grains of A that cannot be placed on the surface of C will be put into interstitial positions. The number NAC takes a limiting value, 0

CAz . The partial co-ordination number 0

CAz is estimated simply by the ratio of the area of grain C and the area of a large circle of grain A and thus:

2C(0) mC A(0)AC

C2A A(0) mA C(0)

4 4r V rNN r V r

σ= = , so CmC

C(0)

d 9d 2

Vt r

σα φ⎛ ⎞

= ⎜ ⎟⎜ ⎟⎝ ⎠

– finally the intermediate zone, which is characterized by σC = limC .σ

The passage from one zone to another is carried out for the σClim composition verifying the two preceding relations, or:

lim

mA C(0)C

mC A(0)4V rV r

σ =

These expressions fit well in the case of titanium carbonate/barium dioxide mixtures [COU 80]. The curve of the rate versus the composition (σC) thus obtained fits with two line segments. The grinding of barium carbonate (solid A) really increases the rate if σC >

limCσ and the grinding of titanium oxide (solid C) increases the slope of the curve of the rate versus the composition in zone p. The two line segments were also obtained for the reaction between molybdenum trioxide and trichromium tetraoxide [ING 86].


14.6. Conclusion

The problem of the kinetics of reaction between powder solids is strongly complicated by the design of the mixtures, which can lead to complex laws and even fractional extent that never reach the value of 1 at the end of the reaction. Fortunately, there are several examples of reactions between solids that follow the one-process model with instantaneous nucleation and slow growth, that is, a model of M1 cells and generally with diffusion as the rate determining step. Their application and results are given in the tables of Appendix 3. This undoubtedly explains few studies on the problem of nucleation in this family of reactions.

handbook of heterogenous kinetics (soustelle/handbook of heterogenous kinetics) || reactions between...

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