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Chapter 15

Gas-Solid Reactions

The reaction between a gas and a solid are like:

Solid1 + gas = solid2 + (G)

A new solid phase is formed; therefore, this kind of reaction involves the whole of the processes of surface nucleation and growth (see Chapter 10). The gas phase that constitutes one of the reactants forms obviously a continuum around the grains of solid 1.

From the experimental point of view, all these reactions are accompanied by a variation of the mass of the condensed phases. The thermogravimetric method will be the best method to follow the extent of the reaction.

We will illustrate with some examples; thus, we will study successively the processes of growth of:

– oxidation of pure metals by a gas like oxygen;

– oxide reduction by hydrogen; and

– metal oxidation by water vapor.

We will generally assume pseudo-steady state modes and usually in the case of a separable rate. The space function is thus given as in Chapter 10, since all the assumptions are retained. For this reason, we will concentrate in this chapter primarily on the mechanisms and reactivities.

544 Handbook of Heterogenous Kinetics

15.1. Classification of gas-solid reactions

We will distinguish two classes for gas-solid reactions: synthesis reactions and double-decomposition reactions that are characterized by a gaseous emission.

15.1.1. Class 1: synthesis reactions

These reactions produce only one solid phase (without gas formation). We can represent this class by the generic reaction:

Solid1 + gas = solid2

Solid1 and solid2 represent pure distinct solid phases.

This class comprises not only all the family of the reactions of oxidation of metals by gases (see section 15.2) but also other reactions such as the hydration of salts:

2 gas 4 4 2H O CuSO CuSO H O+ = ⋅

and lower oxide oxidations:

2 2 312FeO O Fe O2

+ =

Some reaction can use several gases as reactants such as the sulfating of alumina:

2 2 2 3 2 4 333SO O Al O Al (SO )2

+ + =

or the trapping of sulfur dioxide by lime:

2 2 42CaO 2SO O 2CaSO+ + =

15.1.2. Class 2: double-decomposition reactions

These reactions differ from the preceding reactions in appearance, in addition to solid2 one (or several) gas. We will represent them by the generic equation:

Solid1 + Gas1 = Solid2 + Gas2

Gas-Solid Reactions 545

There are several other examples:

– oxidation of a metal by water vapor (see section 15.4) as:

2 2H O Fe FeO H+ = +

– reduction of an oxide by hydrogen (see section 15.3) as:

2 2H FeO Fe H O+ = +

which is the reverse of the preceding reaction, or:

2 3 2 2Fe O + H = 2FeO H O+

– the hydrofluorination of uranium oxide:

2 4 24HF UO UF 2H O+ = +

– the manufacture of cerium dioxide starting from a carbonate:

2 3 2 2 22Ce CO O 4CeO 6CO+ = +

– the oxidation of a non-oxide ceramics:

2 2 3 24NB 5O 2B O 2NO+ = +

Since these reactions are characterized by a gas release, the transport of matter can occur in two manners:

– diffusions of gases through the pores of the formed solid from and toward the solid-solid interface because the release of gas supports the porosity of this solid. Then, the whole of the reactions of growth and in particular the manufacture of the produced gas molecule will proceed at this internal interface. It will be noted that the diffusion of a gas in pores is not an activated phenomenon, the diffusion coefficient varies with T 3/2;

– diffusions of point defects inside the formed solid, with an interface of creation and an interface of consumption of the defects. The produced gas molecule will be then manufactured at the external interface. Diffusion of defect in solid is an activated phenomenon.

These reactions, though important, are less studied than those of class 1 reaction.


15.2. Pure metal gas reactions

We will study the mechanisms of the reaction of oxidation of pure metals by a gas such as:

G M MGg pgp + = [15.R.1]

where M is a pure metal and G a gas of atomicity g. During oxidation, the metal is recovered by a layer of the formed solid, which may be, according to gas, a metallic oxide, a sulfide, a chloride, etc. No other compound is produced by the reaction.

These reactions creating a new solid phase refer clearly the processes of surface nucleation and growth. In the case of oxidation by oxygen, it is very difficult to leave a metal surface not already covered with oxide so that usually only the process of growth occurs. However, studies undertaken under strict conditions showed that sigmoid curves are obtained in the case of metal powders. In the case of sulfurization, the studies of Benard and Oudart highlighted a relatively slow nucleation and a very fast tangential growth so that these reactions usually refer to a two-process model with nucleation and anisotropic growth. The experimental conditions are in general very far away from the equilibrium conditions and thus the latency of germination is very short. As a consequence, gas-metal reactions proved to be ideal to study the process of growth because of little or no disturbance by the nucleation process. The experiments are in general carried out on massive plates and thus the surface of the interfaces remains constant in time and therefore simple space functions are obtained.

The detailed study of these reactions is very important because, in fact, they were historically the first studied heterogenous reactions and were used as a basis for comprehension of the mechanism of the other heterogenous transformations.

Unless otherwise stated, and in particular section 15.2.5 devoted to mixed modes, we will be using the approximation of the pseudo-steady state modes with a rate determining step. Except in certain cases of abnormal diffusions, the rate will be separable and we will be able to devote ourselves to the study of the reactivity, the space functions, of very simple structures in the case of plates, having been determined in Chapter 10.

15.2.1. Experimental data of oxidation of metals

15.2.1.1. Extent and reaction speed

In general the experimental data presented and easily accessible are the variations of the mass of solids with time under isobaric-isothermal conditions


(except perhaps at the first moments) and speed variations with time and/or the mass of solids. Insofar as a pseudo-steady state mode has been checked, it is easy to associate the mass change of the whole of both solids to the fractional extent of the reaction and its variations with the rate (however, many authors use this property without having verified a pseudo-steady state mode). Measurements, in general, are obtained with massive metals and the change of mass is most frequently brought back to the unit of area of the sample. The kinetic law and the speed per unit of area will be expressed by:

0

d( ) andds

m

mm f t vt Δ

Δ⎛ ⎞Δ = = ⎜ ⎟⎝ ⎠

where Δm0 is a selected change of mass so that speed is measured for a given extent (i.e. with constant space function).

We can also characterize the extent of the reaction by the thickness X of the layer of the formed solid and within the framework of pseudo-steady state modes. The mass change is associated to the thickness by:

G

MG

2 pgM Xm

VΔ =

where MG and VMG indicate the atomic mass of the gas G and the molar volume of the formed solid, respectively; p and g are stoichiometric number and atomicity of gas, respectively (reaction [15.R.1]).

15.2.1.2. Principal kinetic laws

We will examine the laws obtained from massive samples in the form of a foil or plate for which the contribution of the lateral faces is negligible. There exist five types of simple kinetic laws [BEN 62, HAU 65, HUN 03, KOF 66, KUB 53, SAR 00]: linear, parabolic, cubic, logarithmic, and reciprocal-logarithmic laws to which we will add two more complex laws, the paralinear law and the general parabolic law. We will express the laws with fractional extent according to time, although the literature generally presents them in terms of change of mass.

15.2.1.2.1. The linear law

In the linear law, the extent is proportional to time, that is, the rate is independent of time:

lk tα =


kl is the kinetic constant or linear constant. This is often called the rate constant improperly, whereas the latter comes from a completely different definition (see section 4.1). It is noted that this kinetic constant is the reactivity and thus varies just as with the intensive parameters (temperature, gas pressure).

15.2.1.2.2. The parabolic law

The law is written in the form:

2pk tα =

where kp is the parabolic constant. The kinetic curve is a concave parabola (Figure 15.1a). The representation is often of the square of the fractional extent versus time that gives a line of slope kp (Figure 15.1b). By derivation, we obtain the rate for a given fractional extent α0:

0

p

02k

α αℜ =

The parabolic constant varies with the physico-chemical variables (temperature, gas pressure) such as for the rate with a fixed fractional extent or the reactivity.

Figure 15.1. Parabolic law

REMARK.– The parabolic law can also be expressed in the following form:

pk tα ′=

We must avoid this form because constant kp' thus defined does not vary as the reactivity with the other variables (pressure, temperature). In particular, we will see

t

α2

t

(a) (b)

α


that speed often follows the Arrhenius law. But this law when applied to constant kp' gives an apparent energy of activation that is twice too low.

15.2.1.2.3. The cubic law

According to this law, the cube of the fractional extent is proportional to time:

3ck tα =

where kc is the cubic constant. The rate for a given fractional extent is then:

0

p203

kα α

ℜ =

The kinetic curve is similar to the parabolic curve, a little more flattened at the end of the reaction but goes up more quickly at the beginning. The best representation is obviously to plot the cube of the fractional extent versus time.

The cubic constant varies with the physico-chemical properties (temperature, gas pressure) as the rate at a given fractional extent, or the reactivity.

15.2.1.2.4. The logarithmic law

This logarithmic law is written as one of the two forms:

0l l

0

log( 1) log

t tk at k

tα

+= + =

This is a law with two constants; the rate for a given extent takes the form:

0

0l

l

expakkαα⎛ ⎞

ℜ = −⎜ ⎟⎝ ⎠

The curve is more flattened than that of the cubic law.

REMARK.– The constant logarithmic curve kl does not have anymore significance for the kinetic constant within the meaning of those defined in the three preceding laws. In particular, it will not vary anymore as the rate constant that progress according to the other variables (temperature, gas pressure).


15.2.1.2.5. The reciprocal or inverse logarithmic law

This law is especially met at low temperatures during the formation of very thin oxide layers. Its analytical expression has the following form:

l 01 logik k tα

= −

This is still a law with two constants. The rate at given extent is expressed by:

0

20 0 l

0

1exp ik kα αα

⎡ ⎤⎛ ⎞ℜ = − −⎢ ⎥⎜ ⎟

⎢ ⎥⎝ ⎠⎣ ⎦

We can apply to this law the same remark as the one formulated for the logarithmic law.

The representative curve is still a little flattened than the curve of the logarithmic law.

15.2.1.2.6. The paralinear law

Such a kinetic curve is made up of two parts: an initial part with concavity turned toward the axis of time and can be qualified sometimes as “parabolic,” although this last law does not fit very well to this part; a second part of the curve is linear and tangent to the preceding part (Figure 15.2).

Figure 15.2. Paralinear law

15.2.1.2.7. The general parabolic law

The general parabolic law has the following mathematical expression:

t

α

Linear part

Pseudo parabolic part


21 2k k ktα α+ =

where k1 and k2 are constants, which, if they cancel one other, lead to either the linear law or the parabolic law. This law is obviously represented by a parabola with concavity turned toward the axis of time (but the representation of the square of the fractional extent versus time is no longer a line, but we have a line if we plot t/α versus α).

In this case, the rate at a given fractional extent is represented as follows:

01 0 22

kk kα α

ℜ =+

15.2.1.3. Interpretation of the kinetic laws according to time

To interpret the kinetic laws we will remember that they were established for thin plate samples and therefore with constant interfacial areas. We will distinguish three categories that belong to the one-parameter models of instantaneous nucleation. They are laws of pure growth:

– the linear law and the parabolic law. The linear law comes simply from a kinetic pure mode with an interfacial reaction as the rate determining step (see Table A.3.5 of Appendix 3). The parabolic law is due to a kinetic pure mode with a traditional diffusion as the rate determining step given in Table A.3.5 of Appendix 3;

– the other simple laws (cubic, logarithmic, and reciprocal logarithmic law), as we will see further, will be explained with modes determined by a non-traditional diffusion: constrained diffusion or under important electric field (thin layers);

– the complex laws (paralinear and general parabolic) law will be explained by mixed kinetics with two rate determining steps (see sections 15.2.5.1 and 15.2.5.2).

15.2.1.4. Influences of temperature on rates

An increase in temperature can act in two different ways on a gas-metal reaction. We will distinguish the action of temperature on a given kinetic law and the law of kinetic change with temperature.

15.2.1.4.1. Influences of temperature for a given kinetic law

If the temperature range is not too wide, the kinetic law often keeps the same form but rate increases with temperature. In many reactions of oxidation of metals,


the experimental rate obeys the Arrhenius law, making it possible to define an apparent energy of activation Ea:

a0 exp

ERT

⎛ ⎞ℜ = ℜ −⎜ ⎟⎝ ⎠

This rule, however, comprises some exceptions, which we will explain later.

15.2.1.4.2. Change of the kinetic law with temperature

It has been noted that very often, in particular with oxygen, the kinetic laws is different from that of another temperature range. At lower temperatures, often both reciprocal logarithmic and logarithmic laws are observed. At medium temperatures, we rather meet parabolic or/and cubic laws, whereas at higher temperatures, we more frequently meet the linear law. The temperature of transition between these laws depends on metal. A characteristic example is the oxidation of tantalum by the di-oxygen, for which the following fields have been observed:

– temperature lower than 100°C: reciprocal logarithmic law;

– temperature between 100°C and 300°C: logarithmic law;

– temperature between 300°C and 450°C: parabolic law;

– temperature ranging between 450°C and 600°C: paralinear law; and

– higher temperature than 600°C: linear law.

We will explain later (see sections 7.11.1 and 10.8.1) the reason of these changes.

15.2.1.4.3. The phenomenon of over-temperature

The reactions considered here are in general very exothermic. The heat produced by the reaction is not evacuated instantaneously, which causes to increase the temperature of the sample. The difference between the real temperature of the sample and the furnace-controlled temperature is called over-temperature. We can evaluate this property according to time.

During time dt, the reaction releases an amount of heat according the following reaction:

0d d( )Q Q m= Δ


where Q0 is the heat released per gram of fixed gas. However, the kinetic law can be put in the following form (by applying the law of Arrhenius to the kinetic constant):

a0 ( )exp

Em k f t

RT⎛ ⎞Δ = −⎜ ⎟⎝ ⎠

and differentiating:

a0 0 2

d ( )d d ( )d exp aE Ef tQ Q k t f t Tdt RTRT

⎡ ⎤ ⎛ ⎞= + −⎜ ⎟⎢ ⎥⎣ ⎦ ⎝ ⎠

This heat can be either evacuated by conduction and radiation or used to raise the temperature of the sample:

4 40 0d ( ) ( ) dQ T T T T cm Tλ σ= − + − +

Hence, the heat balance is written:

4 4a a0 0 0 02

d ( ) d ( )d exp ( ) ( ) dd

E Ef tQ k t f t T T T T T cm Tt RTRT

λ σ⎡ ⎤ ⎛ ⎞+ − = − + − +⎜ ⎟⎢ ⎥⎣ ⎦ ⎝ ⎠

The solution of this equation makes it possible to determine, knowing f(t), the over-temperature (T – T0) according to time.

The experimental results agree with the model. It is noted that over-temperature goes through a maximum, in general, that is achieved quickly in a few minutes, and these value can reach 30°C to 50°C because initial rates are very large. The temperature again becomes constant and equal to the controlled value. Thus, the real speed of the reaction will be increased especially in the first moments and that the kinetic law can be modified accordingly. For example, the parabolic law in isothermal conditions can take a form of the logarithmic or cubic curve. To avoid such an error, it is preferable, on the one hand, to measure the real temperature of the sample and, on the other hand, to take into account, for the determination of the law, the measurement of the non-isothermal condition, which is in general possible because the effect of over-temperature is felt over short time compared with the duration of reactions.

15.2.1.4.4. Influence of gas pressure on the speed

The influence of the gas pressure on speeds is apparently complex. It is not possible to associate a kinetic law with a pressure law. This influence of the pressure


is apparent at the same time on metal, gas, and the observed kinetic law. We find reactions for which the pressure is without influence In other cases, speed is proportional to the pressure or to a power of the latter or follows with the pressure (or its square root) in a homographic or logarithmic law.

In fact, the diversity of the obtained results constitutes a wealth of information so much so that we can affirm that the study of the influence of the pressure is the best way to get the mechanism.

15.2.1.4.5. Nature and structures of the formed products

The formed products of the reactions of oxidation of metals are stoichiometric ionic compounds (Schottky or Frenkel type) or semiconductor, non-stoichiometric compounds. In the last case, usually, we can model these solids in the Wagner approximation of the prevalent defect (see section 2.3.2), which can distinguish the two families:

– n-type semiconductors with interstitial cations or anion vacancies;

– p-type semiconductors with cation vacancies or interstitial anions.

The corresponding defects can be more or less ionized.

15.2.2. Reaction zones and elementary reactions

15.2.2.1. Analysis of a reaction

When oxygen gas comes in contact with a metal plate, the surface of the latter is almost immediately recovered with oxide layer. This means either that the nucleation of this oxide is instantaneous or that the tangential growth of the layer is very large. The choice between the two possibilities is made by following experiments on powders that provide sigmoid curves of fractional extent, which is the sign of a nucleation-growth process. On a massive sample, we will have thus to consider only the growth in the direction perpendicular to the layer of formed product (see section 10.2.1). It seems that this result with oxygen would also be applicable to the reactions of metals with sulfur vapor [BEN 62].

At a given time, our system could thus be represented, as indicated in Figure 15.3, by a layer of the formed solid that grows between the gas and the metal.

To quantify our reasoning, we will consider the framework of a model reaction between a diatomic gas G2 giving G2− anions and a metal M leading to cations with


the same absolute charges as the anions. Thus, our model reaction will be written as follows:

21M G MG2

+ =

These methods will be developed on this reaction and we will consider some examples on how to take into account different valences.

For the MG growth to continue, it is necessary that the reactants come in the contact with one another. We then assume the following steps in the various zones (Figure 15.3):

– adsorption of gas on the MG solid; this is a chemisorption taking into account the temperatures used;

– diffusion of particles, which we call δ, through the MG solid either from gas toward metal or from metal toward gas. The nature of the δ particles, which will be specified further, depends on the nature of the MG solid; and

– reactions at interfaces; we will distinguish two types of interface reactions: - reaction at interface “0”, which will create the diffusing particle. According

to the direction of displacement of δ, interface “0” is either the internal interface, which separates M from MG, or the external one, which separates MG from gas,

- reaction at interface “X”, which consumes the diffusing particle. According to the direction of displacement of δ, interface “X” is either the external interface or the internal interface.

One of the reaction of interfaces ensures the building of the MG network to grow the layer by inward or outward development.

Figure 15.3. Topographies of the phases and zones for a gas-metal reaction

Gas G

Metal M

Produced solid MG

Internal interface

External interface


15.2.2.2. Nature and model of the adsorbed phase

We saw as a first step chemisorption of gas on the MG layer. It is known that if this is the only phenomenon that occurs (see Chapter 6), it leads to an equilibrium where, at a given temperature, a degree of coverage of the surface is associated with the pressure of gas through an isotherm of the form:

( )I Pθ∞ =

The simplest model of the isotherm is the Langmuir isotherm, which corresponds to gas fixing on sites s of the solid having similar energy, with the free and occupied sites forming a surface solution. We will retain this model. Adsorption will thus be modeled by the dissociative elementary reaction:

2G 2s 2G s+ → −

and the corresponding equilibrium gives:

a

a1K P

K Pθ∞ =

+

15.2.2.3. Nature of the diffusing particles and interface reactions

We can imagine two kinds of diffusing particles:

– either the gas diffuses through a porous MG compound;

– or the diffusion in MG occurs via point defects.

In this last case, it is natural to assume that the diffusing entities and thus mobilities will be the same as those that are move under the effect of an electric field and which thus ensure ionic and electronic conductivities, to which we will add the neutral corresponding entities, which if they do not move obviously under the action of an electric field can do it under the action of a concentration gradient.

We will examine the two possibilities and deduce, for each of them, interface reactions.

15.2.2.3.1. Molecular diffusion of gas

If the MG solid is porous, it is possible to consider the diffusion of gas in a molecular state through the pores. Then, there is nothing anymore, but one


interfacial step (since the diffusing species already exists) that will be an interface of type “X” of consumption of gas and that we will write as follows:

2G 2M 2MG+ →

In the beginning, it was thought that the only mode of diffusion that can intervene was deduced by Pilling and Bedworth from a predictive rule in 1923. According to them, if the coefficient of expansion z of the reaction (called sometimes for this reason the Pilling and Bedworth coefficient) were higher than 1, the product MG would not be very porous and thus would be compact. The diffusion would be the rate determining step, which could lead, as we will further see, to the parabolic law. On the contrary, if coefficient z were lower than 1, MG would be very porous, the diffusion would be easy, and thus the reaction of interface would be the rate determining step, which could lead to a linear kinetic law.

This rule is very often taken at fault, so it is necessary to examine other possibilities of diffusions.

15.2.2.3.2. Diffusion of defects in semiconductors

Species, such as point defects, more or less ionized, can be created starting from the reactants and diffuse in the MG solid. We will consider two types of solids:

– semiconductors: non-stoichiometric compounds in which the Wagner approximation is often sufficient to describe the gas-metal reactions. The disorder then consists of a defect carrying matter, more or less ionized and accompanied, if necessary, by charge carriers (electrons or electron holes) compensating for electric neutrality;

– ionic conductors: stoichiometric compounds including disorders with two entities and whose principal ones are the Schottky and Frenkel solids (see section 2.2).

First, we will see the case of non-stoichiometric compounds. Since the Wagner approximation appears sufficient, we will thus take again the four cases of the corresponding classification (see section 2.3.2).

n-type semiconductors with interstitial cations

The disorder in these solids includes interstitial cations and electrons, which, in the case of our solid MG, can be represented by:

[ ]iM 2e'δ = °° +


This disorder can be completely ionized, partially ionized, or not ionized. In our general study, to simplify the expressions, we will consider that the two entities constituting δ remain grouped and thus δ is not ionized.

According to its nature, this disorder will be formed at the interface where there is a metal excess, that is, at the internal interface, which will be thus the “0” interface. The reaction of creation will represent the dissolution of metal MG and will be written as follows:

i M(metal)M V Vδ+ → +

In addition to the defect δ, the reaction creates a metal vacancy in the metal.

We can simplify the preceding reaction by using only the sufficiently diluted species so that their activities are not equal to 1. We will then write as follows:

M(metal)M Vδ→ +

REMARK.– We consider metal M. Although the activity of the metal is equal to 1, these reactions will also be used in the oxidation of the single-phase alloys for which the activity of metal is related to the composition of alloy and is no longer equal to 1 (see section 16.2).

The interstitial cations diffuse and are consumed (as well as the electrons) at the external interface, which will be the “X” interface, where there is an excess of gas in the adsorbed form. So we can write as:

M GG s M G sδ + − → + +

which gives again an adsorption site and building unit of the MG network.

In the same way as that for the internal interface, the reaction can be schematized in the form:

G s sδ + − →

We have seen that the internal interface reaction also produced metal vacancies. These vacancies will tend to diffuse toward the interior of the metal. They will be destroyed either on dislocation or by regrouping to produce a cavity in the metal. This is the heart reaction, which we will write in the following symbolic form:

M(metal)V 0n →


All these reactions are possible in both directions, but during the reaction of oxidation, they will be held overall from left to right.

It is noted that the MG layer develops toward outside, or the layer with outward development.

n-type semiconductors with anion vacancies

The disorders of these solids are anion vacancies and electrons, which, in the case of our solid MG, can be represented by:

GV 2e'δ°°

⎡ ⎤= +⎣ ⎦

This disorder can be completely ionized, partially ionized, or not ionized. In our general study, to simplify the expressions, we will consider that δ is not ionized.

According to its nature, this defect will be formed at the interface where we have a lack of gas, that is, at the internal interface, which will be thus the “0” interface. The reaction of creation will represent the ionization of metal and will be written as follows:

MM Mδ→ +

We can simplify the preceding expression using only the sufficiently diluted species so that their activities are not equal 1. We will then express:

M δ→

Anion vacancies diffuse and are consumed (as well as the electrons) at the external interface, which will be the “X” interface, where there is an excess of gas in the adsorbed form. We will express it as follows:

GG s G sδ + − → +

which gives again an adsorption site and a building unit of the MG network.

In the same way as that for the internal interface, the reaction can be schematized in the form:

G s sδ + − →


It is seen that this case is similar to the preceding case with, however, two differences: on the one hand, the MG layer develops toward inside of the metal–layer with inward development, on the other hand, there is no creation of vacancies in metal, therefore no diffusion inside metal and no heart reaction.

p-type semiconductors with interstitial anions

The disorders of these solids are interstitial anions and electron holes, which, in the case of our solid MG, can be represented by:

[ ]iG 2hδ ′′= + °


According to its nature, this defect will be formed at the interface with an excess of gas (in the adsorbed form), that is, at the external interface, which will be thus the “0” interface. The reaction of creation will be written:

G s sδ− → +

The interstitial anions diffuse and are consumed (as well as the electron hole) at the internal interface, which will be the “X” interface, where there is an excess of the metal, which we will express as follows:

M GM M Gδ + → +

which gives a building unit of the MG network.

This reaction can be schematized in the form:

M 0δ + →

It is noted that the MG layer has an inward development.

p-type semiconductors with cation vacancies

The disorders of these solids are cation vacancies and electron holes, which, in the case of our solid MG, can be represented as follows:

[ ]MV 2hδ ′′= + °



According to its nature, this defect will be formed at the interface where we have a lack of metal, that is, at the external interface, which will be the “0” interface. The reaction of creation will be expressed as follows:

GG s G sδ− → + +

We can simplify the preceding expressions using only the sufficiently diluted species so that their activities are not equal to 1. We will express then:

G s sδ− → +

The cation vacancies diffuse and are consumed (as well as the electron holes) at the internal interface, which will be the “X” interface, where there is an excess of metal, which we will express as follows:

M M(metal)M M Vδ + → +

This reaction can be schematized in the form:

M(metal)M Vδ + →

The metal vacancies diffuse inside the metal and will be destroyed by a heart reaction:

M(metal)V 0n →

It is noted that the MG layer develops toward the outside, the layer with outward development.

REMARK.– We note that all the mechanisms we have described are linear.

Table 15.1 specifies the nature of the interfaces and the directions of development of the layer in the four cases of point defects.

Table 15.2 provides the various reactions at interfaces. It will be noticed that our schematic of the reactions at interfaces depends only on the type of semiconductors: n-type or p-type. The vacancies created in the metal, which occur only in the case of the cation defects, have been placed in brackets.


Semiconductor type for MG Defect

Interface “0”

Interface “X”

Direction of development

n Interstitial cation Internal External Outward

n Anion vacancy Internal External Inward

p Interstitial anion External Internal Inward

p Cation vacancy External Internal Outward

Table 15.1. Nature of the interfaces

Kind of solid Interface “0” Interface “X”

n M(metal)M (V )δ→ + G s sδ + − →

p G s sδ− → + M(metal)M 0 (V )δ + → +

Table 15.2. Reactions at interfaces for the growth of a semiconductor

15.2.2.3.3. Diffusion of defects in stoichiometric solids

If the MG solid is stoichiometric, it has a Schottky or Frenkel disorder but does not show any electronic conductivity (see section 2.2).

Growth of a Frenkel solid

The Frenkel solid presents, for example, interstitial cations and cation vacancies. During oxidation, the first vacancies are thus formed at the interface rich in metal, that is, at the internal interface, whereas the second vacancies are formed at the interface poor in metal, that is, at the external interface. But then the reaction of external interface, which requires electrons to ionize the gas, cannot occur anymore in the absence of electrons. The electrons thus will accumulate on the internal interface (Figure 15.4) and will create an electric field that slows down the diffusion of these cations. The layer, if it is formed, cannot grow anymore.

Growth of a Schottky solid

A Schottky solid presents cation vacancies and anion vacancies. The first vacancies are formed at the external interface and the seconds at the internal interface. But then the external reaction of interface, which requires electrons to ionize the gas, cannot occur anymore in the absence of electrons. The electrons thus will accumulate on the internal interface and will create an electric field that slows


down the diffusion of these cations. As in the preceding case, the layer, if it is formed, cannot grow anymore.

Figure 15.4. Growth of a Frenkel solid

15.2.2.4. Reactivity of the chemical elementary steps

We will apply the rate laws and thus the reactivity for each reaction defined above, each interface reaction being assumed as equivalent to an elementary reaction (see Chapter 4). We will indicate by k′ the rate constant of the reaction i from left to right and ik ′′ from right to left. Ki is the corresponding equilibrium constant ( i i i/K k k′ ′′= ).

15.2.2.4.1. Adsorption step

For the rate of adsorption, we are within the framework of the dissociative adsorption of Langmuir on identical sites, and the rate is thus given by equation [6.12]. Very frequently, we will use this relationship in reactions far from equilibrium and for the low levels of coverage θ << 1. The rate then takes the simple form:

aa

0

d 2 2d

vk P

t sθ ′= =

15.2.2.4.2. Steps at the interfaces

Each reaction at an interface is made up of two elementary opposite steps; thus, the orders are equal to the molecularities. Moreover, normal components of the network being in great amounts, the corresponding concentrations could be presumably constant. Therefore, we can use simplified Table 15.2. Thus, Table 15.3 provides the expressions of the reactivities for the various types of semiconductors. Rates constants k0' and kX' have already been divided by the term “a” and thus have become an areal constant (equation [7.16]).

Gas

+++++++++++++++++

-----------------------------

MG

Metal


Solid MG type External reaction Internal reaction

00C 0

XC

n 0X

X X 0 XX

1C

k s CC

φ θ⎛ ⎞

′= −⎜ ⎟⎝ ⎠

00 0 0

0

1C

kC

φ⎛ ⎞

′= −⎜ ⎟⎝ ⎠

0

V( )KC

X

1K

θθ

−

p 00 0 0 X 0

0

1C

k s CC

φ θ⎛ ⎞

′= −⎜ ⎟⎝ ⎠

( )0X X X Xk C Cφ ′= − 0

1K θ

θ− V

X

( )CK

Table 15.3. Reactivities of interface steps

In Table 15.3, s0 is the number of adsorption sites per unit of area. C0 and CX are the concentrations of the defects that diffuse at the “0” and “X” interfaces. We also find the values 0

0C and 0XC of these concentrations when the corresponding steps

are at equilibrium. θ is the fraction coverage of adsorption. CV is the concentration of metal vacancies at the internal interface. This term is placed between brackets because it intervenes only if the diffusing particle has metal origin (interstitial cations or cation vacancies). In the other cases (anion vacancies and interstitial anions), this term does not appear (or can be taken equal to 1).

15.2.2.4.3. Heart reaction

We saw that when the diffusing particle is of metal origin (interstitial cations or cation vacancies), a heart reaction occurs. However, this reaction is never a rate determining step and thus the expression of the reactivity:

( )0c c c ck C Cφ ′= −

is seldom useful. On the other hand, the expression of concentration 0cC of metal

vacancies in the bulk of metal at equilibrium:

0c

c

1CK

=

will be useful.

15.2.2.5. Pressure of the gas in equilibrium with the metal and the produced solid

The equilibrium pressure P0 of the reaction of oxidation of the metal can be expressed easily according to the equilibrium constants of the various steps Ka, K0, KX, and possibly Kc.


Consider the case of a n-type semiconductor MG (the reasoning is similar for a p-type semiconductor). Thus, under the equilibrium pressure, all the steps are individually in an equilibrium state and that there is no concentration gradient in the zone of diffusion:

– at the external interface, the degree of coverage takes the equilibrium value, that is to say:

a 0

a 01K P

K Pθ∞ =

+

– at the adsorbed gas/MG interface (interface “X”), the concentration of defects is in equilibrium and is thus, taking into account the preceding equilibrium:

0X X

X

1C C

Kθθ

∞

∞

−= =

– at the metal/oxide interface (interface “0”), internal equilibrium results in:

0 00 0

V( )K

C CC

= =

– in the bulk of the metal (if the heart reaction exists), then:

0c c

c

1C CK

= =

In addition, at thermodynamic equilibrium, concentrations gradients are null and thus:

0 00 X 0 XC C C C= = = and 0

c V cC C C= =

Therefore, we obtain:

0 2 2 2a 0 X c

1( )

PK K K K

=

The constant Kc is taken equal to 1 if there is no heart reaction.


15.2.3. Pure modes with interface rate determining step

15.2.3.1. Kinetic laws due to interface rate determining steps

We can now apply the method of resolution of the pure kinetics, with one of the preceding reactions as the rate determining step in pseudo-steady state mode. For each type of solid MG, we will obtain three solutions according to whether the determining step is adsorption, the reaction at internal interface i or at the external interface e (we exclude the modes limited by the heart reaction that we have never encountered). Table 15.4 provides the results obtained in conditions far from equilibrium. To obtain the expression of the reactivity in the opposite case, closed to equilibrium conditions, we have to multiply the preceding relations by the term:

01PP

⎛ ⎞−⎜ ⎟⎝ ⎠

REMARK.– Very often the experimental conditions are very far away from the equilibrium conditions. Thus, for example, at 1,000°C, the equilibrium pressures for nickel oxide and copper oxide are 10−12 and 10−7 atm, respectively, if the studies are undertaken for usual atmospheric pressures. We are obviously very far from equilibrium.

The variations with temperature are included in the rate constants that follow the Arrhenius law and the equilibrium constants that obey the vant’ Hoff law.

MG solid type

Adsorption determining step

Internal interface determining step

External interface determining step

n [ ] a 0a 2k Psφ ′= [ ] 00 kφ ′= [ ]a

X 0 0 cXa

( )1

K Pk s K K

K Pφ ′=

+

p [ ] a 0a 2k Psφ ′= [ ] X 0 aX k K K Pφ ′= [ ]a

0 00a1

K Pk s

K Pφ ′=

+

Table 15.4. Reactivities in pure mode with adsorption or chemical interface reaction as the rate determining step


Within the framework of the formation of a solid MG starting from gas G2, the multiplying coefficients of the steps of adsorption, external interface, and internal interface are ½, 1, and 1, respectively, from which the reactivities in pure modes according to the reactivities of the rate determining step are:

[ ] [ ] [ ] [ ] [ ] [ ]a a a 0 0 0 X X X2 , , andφ φ φ φ φ φ= = =

We see that the preliminary knowledge of the type of solid MG and the influence of the pressure on the reactivity (and thus on the rate for plate samples) make it possible to deduce the rate determining step, knowing that the kinetic law is linear (always for the plate samples).

REMARK.– The homographic law with the root of the pressure comes from equilibrium of the Langmuir isotherm for adsorption. The choice of another isotherm would lead to another law of pressure, for example, the Temkin isotherm would give a logarithm law of pressure for a mode with an external interface reaction as the rate determining step.

15.2.3.2. Example of the oxidation of uranium in interface modes

Now, we will use only the system G2/MG/M. Very often the solid M and the gas G do not have the same electrovalence. It is thus advisable to represent the elementary steps in which only integers can be stoichiometric numbers and their rates with partial orders are equal to these integers. Moreover, the defect is not necessarily neutral and can comprise an ionic entity and one or more electronic entities (the disorder obeying electric neutrality).

To illustrate this, we will take the example of the oxidation of uranium to dioxide uranium:

2 2U O UO+ =

The uranium dioxide is a p-type semiconductor with interstitial oxygen ions iO′′ and uranium hexavalent ions in the place of tetravalent ions (electronic defect UU°° ).

The adsorption of oxygen following the Langmuir model is:

2O 2s 2O s+ → −

the corresponding equilibrium will be as follows:

a

a1K P

K Pθ θ∞= =

+


The diffusion of the defects will occur from external interface where they are created (“0” interface) toward the internal interface where they are consumed (“X” interface).

The reaction of creation is represented as follows:

U i UU O s O U s°°′′+ − → + +

The reactivity is (putting [ ]0 i U0 0O UC °°′′ ⎡ ⎤= = ⎣ ⎦ to respect electric neutrality) as

follows:

20 0 0 0 0 0(1 )k s k C sφ θ θ′ ′′= − −

When this step is at equilibrium, we have:

0 1/ 2 1/ 4 1/ 400 0 0 a1

KC C K K P

θθ

= = =−

Far from equilibrium, the reactivity is:

0 0 0k sφ θ′=

The reaction of consumption of the defects at the internal interface (“X” interface) is:

O i U U OU 4O 2O 2U 3U 6O°°′′+ + + → +

The reactivity (putting [ ]X i UX XUC O °°′′ ⎡ ⎤= = ⎣ ⎦ to respect electric neutrality) is as

follows:

4X X X Xk C kφ ′ ′′− −

When this step is at equilibrium, we have:

0X X 1/ 4

X

1C CK

= =

Far from equilibrium, the reactivity is:

4X X Xk Cφ ′=


We see that the sum of the three steps gives the total reaction on the condition of affecting the external reaction by multiplying a coefficient of 2.

Now let us determine the reactivities for the two pure interface modes.

For a mode with the internal interface reaction as the rate determining step (“X” interface), we will have θ θ∞= and 0

X 0 0C C C= = and thus:

[ ] [ ]2

X 0 aX X X k K K Pφ φ ′= =

For a mode with the external interface reaction as the rate determining step (“0” interface), we will have θ θ∞= and thus:

[ ][ ]0 0 a0

00a2 2 1

K Pks

K P

φφ

′= =

+

We note that the pressure law when the internal interface is the rate determining step is different from that given in Table 15.4, whereas it remains the same when the external interface reaction is the rate determining step. In both cases, the kinetic law remains linear.

15.2.3.3. Conclusions on the reactivities in pure interface modes

When we look at the various results obtained for the reactivities in pure modes of adsorption or interface reactions far from equilibrium, we note:

– these reactivities are obviously independent of time (pseudo-steady state modes);

– these reactivities are much diversified functions of gas pressure; these functions depend on the reaction and the rate determining step;

– the influence of temperature often follows the Arrhenius law. It is not, however, the case when the internal interface reaction is the rate determining step, except if the KaP term is small or large with respect to 1.

15.2.4. Pure diffusion modes

15.2.4.1. Speed per unit of area and rate in diffusion modes

In pure diffusion through the formed product modes, the interfacial steps, adsorption, the diffusion in metal, and the heart reaction, if it exists, are at


equilibrium. The concentrations at the boundaries of the formed layer are thus determined by the corresponding equilibriums.

If the diffusion proceeds from internal interface toward the external one (n-type semiconductor oxides), we have (for a diatomic gas with dissociative adsorption):

0i 0 i2 c( )C C K K= = and

( )e

X X a

1 1CK K K P

θθ

∞

∞

−= = [15.1]

If the diffusion proceeds from the external interface toward the internal one (p-type semiconductor oxides), we have:

0i X

X c

1C CK K

= = and 0 0e 0 0 a1

KC C K K P

θθ

∞

∞

= = =−

[15.2]

The speed per unit of area is then given by the diffusion flux J. The rate is, by indicating by s the area of the selected interface (in general the internal one) and n0 the initial amount of metal that is used:

0

Jsn

ℜ = [15.3]

In the pseudo-steady state mode, the thickness of the formed layer will vary, by indicating by Ve the volume of oxide equivalent to a mole of metal, as follows:

eddX V Jt

= [15.4]

15.2.4.2. Flux of diffusion and electric field

We know that the point defects of the formed product MG actually consist of two charged entities, one ionic (interstitial ions or vacancies) and other electronic (electrons or electron holes). If these two species are not dependant (in the case of the dissociated defect), we understand that these two types of particles with very different sizes do not have the same mobility and that the electronic part with much smaller size can have a larger mobility. It will thus be formed quickly at both interfaces as electric layers of equal charges in absolute values and opposite signs to obey the total electric neutrality. It will thus result from an electric field in the layer, which is advisable to take into account.


Indicate by c + (x) and c − (x) the concentrations of ionic and electronic species, respectively, at a distance x from the interface of creation of the defect (“0” interface). The electric potential V in the solid MG will obey the Poisson equation, which will be written, indicating by e the elementary charge and d the permittivity of the layer, as follows:

[ ]2

2

d 8π ( ) ( )d

V e c x c xdx

= + − −

The concentrations c + (x) and c − (x) obey the following Boltzman distribution:

( ) exp eVc x nkT

⎛ ⎞+ = −⎜ ⎟⎝ ⎠

and ( ) exp eVc x nkT

⎛ ⎞− = ⎜ ⎟⎝ ⎠

from which:

2

2

d 8π shd

V ne eVd kTx

⎛ ⎞= ⎜ ⎟⎝ ⎠

Mott [MOT 40] determined the limit thickness, above which the electric field is so small that its effect can be neglected. To calculate this thickness, it is not necessary to know the solution of the preceding equation, but only its form for the small potentials (V small) is essential. We will thus have on developing the hyperbolic sine and neglecting the higher terms:

2 2

2

d 8πd

V ne VdkTx

=

Assigning:

1/ 2

0 28πdkTX

ne⎛ ⎞= ⎜ ⎟⎝ ⎠

[15.5]

The solution is written as follows:

00

exp XV VX

⎡ ⎤= −⎢ ⎥

⎣ ⎦


This solution shows that for X >> X0, which is for thick layers, the contribution of the electric field will be low. On the contrary for X << X0, the electric field becomes dominating. Mott evaluated X0 around 1,000 Å.

We established equation [5.29], which gives the diffusion flux with the electric field. Because of the impossibility of integrating the relation (Poisson integration) to evaluate the flux, two simplifying but contradictory assumptions were introduced to express the electric field in the layer. These two assumptions provide comparison for the layer of product MG to a plane condenser (we are working only with plates) and it is known that the capacity of such a condenser is inversely proportional to its thickness.

The first assumption is due to Mott and Cabrera [MOT 48], which assumes that the potential difference at the boundaries of the layer MG is constant for any given thickness, the layer is thus comparable with a plane condenser with constant potential difference and thus a charge on the electrodes inversely proportional to its thickness X. This assumption is translated in the form:

ddV Vx X

Δ= −

Then, the flux under the combined action of the field and the concentration gradient is represented while applying [5.29] and [5.5]:

X XX

exp exp exp2 2 2

z Fa V z Fa V z Fa VD CJ C aa RTX RTX x RTX

⎧ ⎫⎡ ⎤⎛ Δ ⎞ ⎛ Δ ⎞ ⎛ Δ ⎞∂⎪ ⎪⎛ ⎞= − − − −⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎨ ⎬⎜ ⎟∂⎝ ⎠⎢ ⎥⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎪ ⎪⎣ ⎦⎩ ⎭

[15.6]

REMARK.– With such an expression of flux, the rate is not separable because we cannot define, starting from flux, a reactivity φ and a geometrical function G, given that φ depends only on the concentration and G depends only on the space.

The second assumption is due to Grimley and Trappnell [GRI 56], which considers that the layer has a constant charge, that is, an electric field that does not depend on the thickness of the layer. Now, we will use the Mott and Cabrera approximation.

15.2.4.3. Normal diffusion: growth of the thick layers

In this case, diffusion flux is due to the concentration gradient and the low electric field (X >> X0 defined by [15.5]). We can take the solution developed in


sections 5.6.2 or 5.6.3 in the case of plates. More generally, the reactivity is given by Table 7.2 and thus:

0 00 X( )(1 )zD C C zφ ε= − + [15.7]

and the rate:

0 0 i0 X

0

( )(1 )zGs

D C C zn

εℜ = − +

For example, for the plates, Table 7.1 gives the expression of G and if the area of the internal interface i is equal to 1 and constant, we have:

0 00 X

0

( )(1 )zD C C zn X

ε− +ℜ =

Switching to the thickness according to [15.4], it becomes

0 00 X

e( )(1 )d

dzD C C zX V

t Xε− +

=

which leads to the traditional parabolic law.

The influence of the gas pressure can be introduced by representing the concentrations 0

0C and 0XC according to equations [15.1] or [15.2]. In many cases,

for conditions far from the equilibrium conditions, we can neglect 0XC with respect

to 00C .

For example, for the oxidation of a metal plate leading to an oxide MO of n-type, we see, according to equation [15.1], that the pressure is without influence on 0

0C and thus we will obtain a parabolic law without the influence of pressure on the rate. On the contrary, if we obtain a p-type oxide MO, equation [15.2] will lead to a parabolic law with a rate that will vary as the square root of the oxygen pressure.

REMARK.– If the diffusing particles are neutral, the term z is zero.

15.2.4.4. Growth limited by a constrained diffusion

We always consider the formation of thick layers (X >> X0 defined by [15.5]), but this diffusion is hindered by barriers (cracks, pores, etc.). In the case of a plate,


we will thus use the expression for the flux given by [5.28]. The variation of the thickness of the layer will be thus, according to [15.4], for neutral particles:

0 0e 0 X( )d exp( )

dV D C CX bX

t X−

= − [15.8]

We notice in this expression that the influence of the gas pressure is identical to the that encountered in the preceding case of normal diffusion through thick layers via 0

0C , indeed the reactivity keeps the same form. It is expression [15.7] and the space function that are modified.

With respect to the kinetic law, we can make approximations in equation [15.8].

If we neglect the variations of X compared with those of the exponential one, we find the differential form of the logarithmic law.

If b is very small, equation [15.8] leads obviously to the traditional parabolic law with normal diffusion.

If a limited development of equation [15.8] after integration is considered, we obtain:

2 3 2 4 3 50 0

e 0 X...( )

2 3 8 30X bX b X b X

V D C C t− = + + + +

and according to the orders of magnitude of X and b, we find, with a good approximation, various power laws such as the cubic law. For this reason, the majority of the non-linear laws are interpreted with diffusion as the rate determining step.

15.2.4.5. Diffusions in intermediate thickness films

We now study layer thickness much lower than X0 (defined by [15.5]). The electric field is thus prevalent and the concentration gradient has a negligible effect. The diffusion flux is thus given by equation [15.6], keeping the Mott and Cabrera assumption. Consider then:

2z Fa V RTXΔ <<

Thickness is then defined as:

1 2z Fa V

XRT

Δ= [15.9]


According to Mott, this thickness would be about 100 Å. Our layer thus has a thickness X such as X1 << X << X0. It comes then for the flux, neglecting the concentration gradient and developing the exponential terms and neglecting the higher terms:

0X 12DC z X

JaX

=

From this, we can define the reactivity and the space function as follows:

0X 12DC z Xa

φ = and 0

sEXn

= [15.10]

Thus, this model gives, for the plates, a parabolic law as that for the thick layers, but it will be noted that the influence of the pressure will intervene through 0

XC , which leads, considering equations [15.1] and [15.2], to an influence of the pressure that is inversely proportional to the root of P for a n-type oxide MO and without an influence of the pressure for p-type oxides MO, which is very different from the growth of thick layers.

15.2.4.6. Growth of thin films

We now study the growth of layers for which the thickness is about X1 (defined by [15.9]). Then, consider again equation [15.6] for the flux, neglecting the concentration gradient, it becomes:

10X

2 shz XDJ C

a X= [15.11]

It is noted that we cannot extract anymore from the flux the function reactivity and the space term G (because of X1, which depends on temperature), so the rate will not be separable.

We can test the corresponding rate equation directly; however, Hurlen [HUR 59] showed that for X of about X1 size, we can compare the variation of the hyperbolic sine to the variations of function A/X2 (A is a constant), which leads by integration to the cubic law for plates and an only apparently separable rate.

15.2.4.7. Growth of very thin films

We now consider film growth with thickness X much lower than X1. We can then again consider equation [15.6] for the flux by neglecting not only the concentration


gradient but also the second exponential term compared with the first one. It then becomes:

10X exp

z XDJ Ca X

=

thus the rate is not separable (because of X1, which depends on T), but to ascertain which kinetic law fits with this case for plates, it is necessary to integrate with respect to time. Integrate in part the function:

1d expd

z XX At X

=

We obtain:

1

1

1 11

1

expexp d

z XX

z XXz X z X

At X z XX Xz X

X

∞

⎛ ⎞−⎜ ⎟⎜ ⎟⎛ ⎞ ⎛ ⎞⎝ ⎠= − −⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠∫

We can definitely recognize the exponential integral function by:

1

1

1 1i

1

expd

XX

z XXz X z X

EX Xz X

X

∞

⎛ ⎞−⎜ ⎟⎜ ⎟⎛ ⎞ ⎛ ⎞⎝ ⎠− − =⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠∫

and as the argument of this function is much higher than 1, the tables show that this function is practically zero and thus integration practically gives:

1expz X

At XX

⎛ ⎞= −⎜ ⎟⎜ ⎟

⎝ ⎠

This function leads, neglecting the variations of X compared with those of the exponential, to practically obtain the reciprocal logarithmic law defined previously in section 15.2.1.2.5.


REMARK.– We see in this example that it is better to test the rate curve versus fractional extent than the thickness or the increase of mass versus time. We avoid integration that often needs new approximations.

15.2.4.8. Summary of the pure diffusion modes

We recapitulated in Table 15.5 the reactivities obtained (column 2) for the various pure diffusion modes. The influences of the pressure are given by corresponding interfacial equilibriums (“0” or “X” interface according to the case of diffusion) through 0 0

0 X or C C .

Layer thickness Reactivity Space function for plates

Kinetic laws for plates

X >> X0 (Fick) 00 (1 )zDC zφ ε= + 0

0

sE

Xn= Parabolic law

X >> X0 (obstructed) 00 (1 )zDC zφ ε= + 0

0

exp( )s

E bXXn

= − Log or cubic law

X1 << X << X0 0

1XDC z Xa

φ = 0

2sEXn

= Parabolic law

X ≈ X1 Non-separable rate Cubic law

X << X1 Non-separable rate Reciprocal log

Table 15.5. Reactivities, space functions, and kinetic laws for plates in the various pure diffusion modes

In Table 15.5, we also provide in the third column the space functions that correspond to the reaction of a plate and finally the last column gives the kinetic law obtained using these space functions.

Ultimately, it is seen that pure kinetic modes (with only a single rate determining step) make it possible to explain the laws that we called simple laws in section 15.2.1.2. They do not give any explanation for the paralinear law and the complete parabolic law.

REMARK.– We obtained various kinetic laws according to the order of magnitude of the thickness of the formed layer. This means that for the thick layers, we should obtain the various laws successively as the thickness grows and not when the formed layers are sufficiently thick. The first moments (for the thin layers) are too short to


give the curve that follows various laws, and the parabolic law fits very well for the whole of the curve. But these distinctions can appear when we change temperature because then the range of thickness is very different.

15.2.4.9. Example of the oxidation of uranium in the normal diffusion mode

We will consider again the case of oxidation of uranium by di-oxygen as we have approached it in section 15.2.3.2. But now, we will consider the calculation of the reactivity in the case of very thick layers with normal diffusion as the rate determining step and leading to a parabolic law.

We calculated (see section 15.2.3.2) that 0 1/ 2 1/ 4 1/ 40 0 aC K K P= . Considering again

equation [15.7] for the reactivity with 2z zε = , it becomes:

1/ 2 1/ 4 1/ 40 a3DK K Pφ =

Thus, the reactivity varies as ¼ of the pressure.

15.2.5. Mixed modes

We saw that the pure modes with a single rate determining step, if they made it possible to explain a certain number of laws, did not cover all of them, in particular the paralinear law and the complete parabolic law (see section 15.2.1.2). From this the idea to complex the solutions utilizing mixed modes in which the kinetics is governed by two rate determining steps, others being constantly at equilibrium.

Taking into account all the elementary steps, we can define five categories of mixed modes:

– mixed modes with adsorption and interface reaction (internal or external) as rate determining steps;

– mixed mode with internal and external interface reactions as rate determining steps;

– mixed mode with adsorption and diffusion as rate determining steps;

– mixed modes with an interface reaction (internal or external) and diffusion as rate determining steps; and

– mixed mode with diffusions at the same time through metal (when it exists) and formed layer as rate determining steps.


We will develop two cases as examples: a mixed mode with internal and external interface reactions as rate determining steps and a mixed mode with external interface reaction and diffusion through the formed layer as rate determining steps.

15.2.5.1. Mixed modes with internal and external interface reactions as rate determining steps

We will treat the case in which the formed product MG is an n-type semiconductor with non-ionized anion vacancies. The other cases give qualitatively equivalent results.

The concerned elementary steps are, in addition to adsorption (G2 + 2s → 2G − s), the two interface steps given in Table 15.2: reaction at “0” interface (internal), which creates the defect and which increases the thickness of the layer, and the reaction at “X” interface (external), which consumes the defect. With these steps, the diffusion of the defect is added through the MG layer.

The two rate determining steps are the internal and external interface reactions. We will consider again the reasoning of section 7.8.

The adsorption step is constantly at equilibrium; thus, the coverage fraction is:

a

a1

K P

K Pθ θ∞= =

+ [15.12]

The diffusion is instantaneous; thus, there does not have any gradient of concentration and the defect concentration [δ] = C is uniform in the whole layer.

We can write the rate of each interface step as:

i i

i

Eφλ

ℜ =

The pseudo-steady state of adsorbed species leads to:

a ae e2

EE

φφ=

and the matter balance of the amount of defect is:

{ } ( )0 i i e e

d δd

n E Et

φ φ= −


However, it can be noticed that on a plate, the two space functions (interfacial surfaces) are always equal with one other and independent of time.

If a indicates the lattice parameter, the amount of defect δ is connected to its concentration C by { }δ Csa= and its variation:

{ }d δ dd d

Cast t

=

Using the expressions of the reactivities of each step, given in Table 15.3, it becomes:

0e

i e 00i

d 1 1d

CC Ca k k s Ct CC

θ∞

⎛ ⎞⎛ ⎞′ ′= − − −⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠

The simplified equation has the following form:

ddC AC Bt

+ =

The significances of A and B, which are independent of time, are obvious with the preceding relationship. The solution of this differential equation with C = C0 at the initial time is:

0 exp( )B BC C AtA A

⎛ ⎞= + − −⎜ ⎟⎝ ⎠

The value chosen for C0 is the defect concentration at the initial time when the layer is formed at the ordinary temperature. This value is always lower than C.

Figure 15.5 gives the variations of the concentration with time. This one tends toward a steady state value:

BCA

∞ =

We will not calculate the rate, which depends on the chosen compounds but not thickness X of the formed layer:

( )0 0 0e i e ii i0 0

i i

d exp( )d

V k V kX B BC C C C Att A AC C

′ ′ ⎡ ⎤⎛ ⎞= − = − − − −⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦ [15.13]


which leads to the kinetic law:

( )0 0e ii0

i

1 exp 1V k B BX C t C At

A A AC′ ⎧ ⎫⎡ ⎤⎛ ⎞ ⎛ ⎞= − + − − −⎡ ⎤⎨ ⎬⎜ ⎟ ⎜ ⎟⎢ ⎥⎣ ⎦⎝ ⎠ ⎝ ⎠⎣ ⎦⎩ ⎭

Figure 15.5. Variation of defect concentration with the concentration according to time

Figure 15.6 gives the shape of the kinetic curve, which reveals an oblique asymptote so that after a transitory mode the law becomes linear according to the following equation:

0 0e ii0

i

1V k B BX C t CA A AC

∞ ′ ⎧ ⎫⎛ ⎞ ⎛ ⎞= − + −⎨ ⎬⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎩ ⎭

The whole of the curve has the paralinear shape as defined in section 15.2.1.2.6. The linear part representing the pseudo-steady state mixed mode is reached at the end of a given time. The “parabolic” part, which is indeed a parabola only for small times, corresponds to the transitory mixed mode and is thus a function of time (although the function of space remains constant).

Figure 15.6. Kinetic curve in mixed-mode interface reactions

X

t

C∞

t

C

C0


The influence of the pressure is expressed using the expressions of A and B in the relation [15.13] and the one of the degree of coverage given by [15.12].

REMARK.– We note in equation [15.13] that the rate is not separable.

15.2.5.2. Mixed mode with diffusion through the formed layer and interface external reaction as rate determining steps

We now consider the case of the mixed mode when diffusion through the formed product and the external interface reaction are rate determining steps during the formation of a n-type semiconductor product MG with non-ionized interstitial cations. The other cases would lead to qualitatively equivalent results. Thus, the rate determining steps are the diffusion through MG (considered as a thick layer within the meaning of section 15.2.4.3) and the interface external reaction (“X” interface), which also causes the growth of the layer. We will thus work on plates, with areas of the interfaces are constant with time.

As a result, adsorption is very fast and is always at equilibrium:

a

a1

PKK P

θ θ∞= =+

[15.14]

The internal interface reaction (“0” interface, see Table 15.2) is also at equilibrium and thus:

0i i i cC C K K= =

Indicate by C0 this concentration of interstitial cations at the external interface at time t.

The speed of diffusion as the unit of area is given by the flux:

( )0i e

i e

D C CJ J J

X

−= = =

The internal interface speed per unit of area (far from equilibrium) is as follows:

e e 0 ev k s Cθ∞′=


The variation of the thickness of the layer is:

m e 0 eddX V k s Ct

θ∞′= [15.15]

Write the balance of the amount of defects at the external interface:

{ } ( ) ( )0 D D e e e C

d δd

n E E J v st

φ φ= − = −

where φD and φe are the reactivities of diffusion and external interface reaction, respectively, and ED and Ee the corresponding space functions.

If a indicates the lattice parameter of MG, the amount of defects at the external interface is associated with their concentration by:

{ } e0

d δ dd d

Cas

t t=

Using speed per unit of area, it becomes:

( )0i ee

e 0 edd

D C CCa k s C

t Xθ∞

−′= − [15.16]

The resolution of the system made up of equations [15.15] and [15.16] would give the kinetic law, but this resolution is not simple. Also, we will be satisfied with a limiting solution.

Taking account of equations [15.15] and [15.16], we can write:

( )0i e

e 0 ee e

m e 0 e

d d d 1d d d

D C Ck s CC C t X

X t X a V k s C

θ

θ

∞

∞

⎡ ⎤−′⎢ ⎥−

⎢ ⎥= =⎢ ⎥′⎢ ⎥⎣ ⎦

[15.17]

Integrating equation [15.17], we would obtain the function Ce(X), which must obey in extreme cases:

If t = 0, X = 0, and 0e eC C= ,

and if e, , and 0t X C→ ∞ → ∞ →


Figure 15.7. Concentration at the internal interface based on the thickness of the layer

The whole of the points for which ed0

dCX

= will be the whole of the extremums

integral curves of [15.17]; then:

0(m) ie

e 0

DCC

k s X Dθ∞

=′ +

The curve is a hyperbola with the following remarkable values:

If X = 0, (m) 0e iC C= , and if (m)

eand 0X C→ ∞ →

The curve Ce(X) starts from 0e eC C= (maximum of the function) and is located

above the preceding hyperbole (Figure 15.7). As these two curves tighten both toward the same limit, we will assumed that they are noncoherent, which is all the more true as X is large or D is small. We will thus assume for [15.17] the approximate solution:

0i

ee 0

( )DC

C Xk s X Dθ∞

=′ +

Substituting into equation [15.15], it becomes for the growth speed of the layer:

0m e 0 i

e 0

dd

V k s DCXt k s X D

θθ

∞

∞

′=

′ + [15.18]

On integrating, we are led to the kinetic law:

20

e 0 m e 0 i2Xk s DX V k s DC tθ θ∞ ∞′ ′+ =

Ce

X

0eC

Approached solution


We recognize the general parabolic law (see section 15.2.1.2.7).

The influence of the pressure is expressed starting from θ∞ in equation [15.14].

REMARKS.–

– We note that the pure modes of diffusion and external reaction inform equation [15.18] are D or ek ′ infinite, respectively.

– We would have ended to the same result by applying the slownesses theorem (see section 7.6.2.2), which indicates that our limiting solution is a pseudo-steady state one.

– The intermediate concentration is not a steady state one, which comes from the fact (see section 7.2.6.3) that a space function (diffusion) is a function of time.

– Expression [15.18] shows that the rate is not separable, which was anticipated because the pseudo-steady state mixed mode is defined by two steps that do not keep space functions constantly equal. This results because one of the external interfaces is independent of time whereas that of diffusion decreases with time.

Mixed modes have allowed us thus to explain two laws (the paralinear law and the complete parabolic one) that we had not explained yet.

15.3. Growth process in the reduction of metallic oxides by hydrogen

The reduction of a metal oxide by hydrogen can lead directly to the metal according to the following form:

2 2MO H M H O+ = +

or to a sub-oxide such as:

2 3 2 2Fe O H FeO H O+ = +

Generally speaking, the molar volume of the formed solid is lower than that of the initial solid (z < 1).

We will stick to the Wagner approximation of the prevalent defect for the initial oxide (and formed oxides if necessary) and thus with the four types of corresponding oxides (see section 2.3.2).


We will consider two types of mechanisms according to the nature of the diffusion:

– diffusion of gases through the pores;

– diffusion of point defects in the solid phase.

15.3.1. Mechanism with diffusion of gases through the pores

15.3.1.1. General mechanism

The most general model has been published by Barret [BAR 73]. It assumes free access of gases to the interface between the initial solid and the formed one. All the reactions proceed then at this interface where the new solid is formed.

The general outline of the mechanism is thus as follows (Figure 15.8): hydrogen diffuses through the pores of the formed solid (pores primarily due to two causes: reduction in molar volume and release of the water vapor) and is adsorbed with dissociation on the surface of the initial oxide.

Adsorbed hydrogen reacts with the oxygen of the network of oxide to form the water molecule in an adsorbed state. The latter is desorbed and diffuses through the pores toward outside.

The network of oxide reorganizes to give the new solid phase.

It is thus seen that all the steps (except diffusions) proceed at the internal interface and there is always an inward development of the layer.

Figure 15.8. Localization steps in the mechanism of gas diffusion during the reduction of an oxide by hydrogen

Initial oxide

Formed solid

Gas 2HOH2


As an example, we will consider the reduction of an oxide MO into the metal (as represented before), for which the law of mass action gives at equilibrium:

2

2

H O

H equil

PK

P⎛ ⎞

= ⎜ ⎟⎜ ⎟⎝ ⎠

and we will consider the four Wagner cases of the oxide MO.

15.3.1.2. n-type oxide with anion vacancies

The defect of MO oxide is of the disorder type OV 2e '°° + , which we will assume associated and thus noted as VO (calculations are obviously to start again with ionized defects).

The first step is thus the diffusion of hydrogen through the pores according to:

2 2,iH H⇔ [15.R.b1]

The second step is the dissociative adsorption of hydrogen on sites s of oxide following:

2,iH 2s 2H s+ ⇔ − [15.R.b2]

The third step is the formation of the water molecule in an adsorbed state with an oxygen ion of the surface of the network:

O 2 O2H s O H O s V s− + ⇔ − + + [15.R.b3]

The fourth step is the desorption of water:

2 2 iH O s H O s− ⇔ + [15.R.b4]

The fifth step is the destruction of the network of oxide to give metal:

O M metalV M M+ ⇔ [15.R.b5]

Formed water diffuses through the pores and gets outside:

2 i 2 extH O H O⇔ [15.R.b6]


To calculate the reactivity of growth, we adopt the following assumptions:

– We look for a pseudo-steady state solution with either an interface reaction or diffusion of water as the rate determining step (diffusion of hydrogen is usually much faster).

– The degree of coverage of surface by hydrogen θΗ will be assumed to be negligible compared with 1 and to water degree of coverage

2H Oθ .

For each step [15.R.bi], the equilibrium constant is Ki and the rate constant for the reaction from left to right is ki'. From thermodynamic equilibrium, we easily calculate that:

2

2

H O2 3 4 5

H equil

PK K K K K

P⎛ ⎞

= =⎜ ⎟⎜ ⎟⎝ ⎠

The second column of Table 15.6 provides the obtained results. The relations reveal, between square brackets, the difference between the ratio of the pressures at equilibrium and the ratio of the experimental pressures.

We note from Table 15.6 that two modes are not discernible by the study of the influence of the gas pressures: the modes with the steps [15.R.b2] and [15.R.b3] as rate determining steps. In addition, these modes are not discernible by the rate law based on time because their space functions are always equal with one other for any given shape of the samples.

15.3.1.3. n-type oxide with interstitial cations

The defect of oxide is of the type presumably associated with iM 2e '°° + and thus noted Mi.

Steps [15.R.b1], [15.R.b2], [15.R.b4], and [15.R.b6] remain identical.

The third step for the production of water molecule is written as follows:

O M i 2 i2H s O M V H O s M s− + + + ⇔ − + + [15.R.c3]

and the fifth step of destruction of the network takes the form:

i metal iM M V⇔ + [15.R.c5]

To calculate the reactivities of growth, we will use the same assumptions as in the preceding case. It is noted that results are identical to those given in Table 15.6, which, thus, can be used with all n-type oxides.


Determining step Growth reactivity φ

2 2

2 2

H O H

H H O equil

1P PP P

φ⎛ ⎞

− ⎜ ⎟⎜ ⎟⎝ ⎠

[15.R.b2] ( )

2 2 2

2 22

22 4 H H O H

2H H O equil4 H O

1k K P P P

P PK P

⎡ ⎤′ ⎛ ⎞⎢ ⎥− ⎜ ⎟⎜ ⎟⎢ ⎥+ ⎝ ⎠⎣ ⎦

( )

2

2

22 4 H

2

4 H O

k K P

K P

′

+

[15.R.b3] or [15.R.c3] ( )

2 2 2

2 22

23 2 4 H H O H

2H H O equil4 H O

1k K K P P P

P PK P

⎡ ⎤′ ⎛ ⎞⎢ ⎥− ⎜ ⎟⎜ ⎟⎢ ⎥+ ⎝ ⎠⎣ ⎦

( )

2

2

23 2 4 H

2

4 H O

k K K P

K P

′

+

[15.R.b4] 2 2 2

2 2 2

4 2 3 5 H H O H

2 3 5 H H H O equil

11k K K K P P P

K K K P P P

⎡ ⎤′ ⎛ ⎞⎢ ⎥− ⎜ ⎟⎜ ⎟+ ⎢ ⎥⎝ ⎠⎣ ⎦

2

2

4 2 3 5 H

2 3 5 H1k K K K P

K K K P

′

+

[15.R.b5] or [15.R.c5]

2 2 2

2 2 2

5 2 3 4 H H O H

H O H H O equil

1k K K K P P P

P P P

⎡ ⎤′ ⎛ ⎞⎢ ⎥− ⎜ ⎟⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

2

2

5 2 3 4 H

H O

k K K K PP

′

[15.R.b6] 2 2 2

2 2

H O 4 H O H

0 H H O equil

1D K P P

l P P

⎡ ⎤⎛ ⎞⎢ ⎥− ⎜ ⎟⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

2H O 4

0

D Kl

Table 15.6. Reactivities for the reduction of an n-type oxide by hydrogen with gas diffusion

15.3.1.4. p-type oxide with cation vacancies

The defect of oxide is of the typepresumably associated with MV 2h°′′ + and thus noted Mi.


The third step for the production of water molecule is written as follows:

O M 22H s O V H O s s− + + ⇔ − + [15.R.d3]


M M metalM V M+ ⇔ [15.R.d5]


Determining step Growth reactivity φ

2 2

2 2

H O H

H H O equil

1P PP P

φ⎛ ⎞

− ⎜ ⎟⎜ ⎟⎝ ⎠

[15.R.d2] ( )

2 2 2

2 22

22 4 H H O H

2H H O equil4 H O

1k K P P P

P PK P

⎡ ⎤′ ⎛ ⎞⎢ ⎥− ⎜ ⎟⎜ ⎟⎢ ⎥+ ⎝ ⎠⎣ ⎦

( )

2

2

22 4 H

2

4 H O

k K P

K P

′

+

[15.R.d3] or [15.R.e3] ( )

2 2 2

2 22

23 2 4 5 H H O H

2H H O equil4 H O

1k K K K P P P

P PK P

⎡ ⎤′ ⎛ ⎞⎢ ⎥− ⎜ ⎟⎜ ⎟⎢ ⎥+ ⎝ ⎠⎣ ⎦

( )

2

2

23 0 4 5 H

2

4 H O

k K K K P

K P

′

+

[15.R.d4] 2 2 2

2 2 2

4 2 3 5 H H O H

2 3 5 H H H O equil

11k K K K P P P

K K K P P P

⎡ ⎤′ ⎛ ⎞⎢ ⎥− ⎜ ⎟⎜ ⎟+ ⎢ ⎥⎝ ⎠⎣ ⎦

2

2

4 2 3 5 H

2 3 5 H1k K K K P

K K K P

′

+

[15.R.d5] or [15.R.e5]

2 2

2 2

H O H5

H H O equil

1P P

kP P

⎡ ⎤⎛ ⎞⎢ ⎥′ − ⎜ ⎟⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

5k ′

[15.R.d6] 2 2 2

2 2

H O 4 H O H

0 H H O equil

1D K P P

l P P

⎡ ⎤⎛ ⎞⎢ ⎥− ⎜ ⎟⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

2H O 4

0

D Kl

Table 15.7. Reactivities for the reduction of a p-type oxide by hydrogen with gas diffusion

To calculate the reactivities of growth, we will use the same assumptions as in the preceding case. The results are given in Table 15.7.

15.3.1.5. p-type oxide with interstitial anions

The defect of oxide is can be of the type presumably associated with iO 2h°° °+ and thus noted VM.


The third step for the production water molecule is written as:

i 2 i2H s O H O s V s− + ⇔ − + + [15.R.e3]



i M O metal iV M O M O+ + ⇔ + [15.R.e5]

To calculate the reactivities of growth, we will use the same assumptions as in the preceding case. It is noted that results are identical to those given in Table 15.7, which thus can be used with all p-type oxides.

15.3.2. Mechanisms with diffusion of defect in the formed solid phase

15.3.2.1. General form of mechanisms

This framework applies more particularly when the reduction leads to a sub-oxide. We will see that in this case the defects of the formed solid are important.

We will have two steps of adsorption of hydrogen and water desorption on the external surface of sub-oxide, accompanied by two reactions at the internal and external interfaces to produce and destroy the defect of sub-oxide and a step of diffusion of this defect from the interface of production toward the one of consumption.

15.3.2.2. Example of the reduction of ferric oxide

Consider as an example the reduction of iron oxide(III) into iron oxide(II). The prevalent defect of iron oxide(II) is formed with an iron vacancy and two electron holes. To simplify, assume the defect is not ionized and thus formed of the only iron VFe vacancies. The first step is the dissociative adsorption of hydrogen at the external gas-FeO interface, following:

2H 2s 2H s+ ⇔ − [15.R.f1]

The second phase will occur at the internal interface (between two oxides). This is the production of the defect of FeO starting from iron oxide(III):

2 3 Fe O FeFe O V 3O 2Fe⇔ + + [15.R.f2]

This step builds up the network of iron oxide(II) at the expense of ferric oxide.

The third step occurs at the external interface, the formation of adsorbed water by the reaction of adsorbed hydrogen with the iron vacancies, following:

O Fe 22H s O V H O s s− + + ⇔ − + [15.R.f3]


REMARK.– This step destroys a building unit of FeO, whereas the preceding step produces three building units.

The fourth step is the desorption of water at this same external interface:

2 2H O s H O s− ⇔ + [15.R.f4]

Of course, it is necessary to assume diffusion of the defect from the internal interface where it is formed toward the external interface where it is consumed:

Fe(internal) Fe(external)V V⇔ [15.R.f5]

In this case, the reaction is still with inward development.

We again consider the same assumptions as in the preceding cases to calculate the reactivities of the various modes. Table 15.8 provides these results.

Rate determining step Growth reactivity φ

2 2

2 2

H O H

H H O equil

1P PP P

φ⎛ ⎞

− ⎜ ⎟⎜ ⎟⎝ ⎠

[15.R.f1] ( )

2 2 2

2 22

21 4 H H O H

2H H O equil4 H O

1k K P P P

P PK P

⎡ ⎤′ ⎛ ⎞⎢ ⎥− ⎜ ⎟⎜ ⎟⎢ ⎥+ ⎝ ⎠⎣ ⎦

( )

2

2

21 4 H

2

4 H O

k K P

K P

′

+

[15.R.f2] 2 2

2 2

H O H2

H H O equil

1P P

kP P

⎡ ⎤⎛ ⎞⎢ ⎥′ − ⎜ ⎟⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

2k ′

[15.R.f3] ( )

2 2 2

2 22 2

23 1 2 4 H H O H

2H H O equil4 H O

1k K K K P P P

P PK P

⎡ ⎤′ ⎛ ⎞⎢ ⎥− ⎜ ⎟⎜ ⎟⎢ ⎥+ ⎝ ⎠⎣ ⎦ ( )

2

2 2

23 1 2 4 H

2

4 H O

k K K K P

K P

′

+

[15.R.f4] 2 2 2

2 2 2

4 1 2 3 H H O H

1 2 3 H H H O equil

11k K K K P P P

K K K P P P

⎡ ⎤′ ⎛ ⎞⎢ ⎥− ⎜ ⎟⎜ ⎟+ ⎢ ⎥⎝ ⎠⎣ ⎦

4 1 2 3 2

1 2 3 21H

H

k K K K PK K K P

′+

[15.R.f5] 2 2

2 2

H O HV2

0 H H O equil

1P PD

Kl P P

⎡ ⎤⎛ ⎞⎢ ⎥− ⎜ ⎟⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

V2

0

DK

l

Table 15.8. Reactivities for the reduction of a p-type oxide by hydrogen with diffusion of point defects


REMARK.– In the case of diffusion in a solid state, the water molecules are formed at the external interface and thus do not cross the produced solid and therefore a less porosity.

REMARK.– We assumed that water was adsorbed in molecular form. If we can choose a mode of dissociated adsorption of water that forms OH − s as is frequently encountered in oxides, then reactions [15.R.f3] and [15.R.f4] are replaced by:

O Fe Fe2H s 2O V 2HO s 2Fe− + + ⇔ − +

and

2 O2HO s H O 2s O− ⇔ + +

The relations so obtained are different.

We considered the case of a sub-oxide with cation vacancies. We can obviously consider in the same manner the three other cases of Wagner defect. Reactions [15.R.f1] and [15.R.f4] remain identical, only steps [15.R.f2], [15.R.f3], and [15.R.f5] are modified. For example, in the reduction of PbO2 into PbO that is with anion vacancies, reactions [15.R.f2] and [15.R.f3] are replaced by the following ones:

2 O Pb OPbO V Pb 2O+ ⇔ +

and

O O 22H s O V H O s s− + ⇔ + − +

15.3.3. Conclusion about the reduction of oxides by hydrogen

We note, from Tables 15.6, 15.7 and 15.8, that the results are very close and that it is very difficult, even impossible, to recognize the diffusing species (gas or point defect) by the study of the influence of the gas pressures on the reactivities. Indeed, we can lead finally only to four expressions for this reactivity that are indexed in Table 15.9, k and k' are constants which follow the Arrhenius law with temperature.

REMARK.– The preceding table is applicable only for non-ionized defects; other expressions would be obtained with ionized defects and that depend on the ionization level.


( )2 2 2

2 22

H H O H2

H H O equilH O

11 '

kP P PP Pk P

φ⎡ ⎤⎛ ⎞⎢ ⎥= − ⎜ ⎟⎜ ⎟⎢ ⎥+ ⎝ ⎠⎣ ⎦

2 2

2 2

H O H

H H O equil

1P P

kP P

φ⎡ ⎤⎛ ⎞⎢ ⎥= − ⎜ ⎟⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

2 2 2

2 2 2

H H H

H H H O equil

11 '

OkP P Pk P P P

φ⎡ ⎤⎛ ⎞⎢ ⎥= − ⎜ ⎟⎜ ⎟+ ⎢ ⎥⎝ ⎠⎣ ⎦

2 2 2

2 2 2

H H O H

H O H H O equil

1P P P

kP P P

φ⎡ ⎤⎛ ⎞⎢ ⎥= − ⎜ ⎟⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

Table 15.9. Reactivities of the reduction of an oxide by hydrogen

15.3.4. Example of the reduction of a uranium oxide

We established models and finally their resolutions in several cases, rather a particular case of an oxide MO with non-ionized point defects.

To leave this framework keeping the screen of the models, we will consider the reduction of uranium oxide U3O8 according to:

4U3O8 + 5H2 = 3U4O9 + 5H2O

The normal structure elements of U3O8 are, for 1 mole, eight oxygen ions, two hexavalent uranium ions, and one tetravalent uranium ion, from which we have the expression (8OO + UU

4+ + 2UU6+). With respect to oxide U4O9, its normal structure

elements, for 1 mole, are as follows: 9OO + 3UU4+ + UU

6+. We will consider as point defects of U4O9: interstitial oxygen anions iO′′ and electron holes trapped on tetravalent uranium ions transforming them into hexavalent ions: 4+U

U .°°

Perrin [PER 02] has shown, starting from kinetic curves and the morphological modeling of the space function, that the preceding reduction proceeds according to a one-process model with instantaneous nucleation with inward development of the formed layer. The rate determining step occurs at the external interface. These data are supplemented by the pseudo-steady state and separable rate tests, which are both satisfied.


The mechanism proposed is integrated in the model that we have presented above. At the external interface, we can represent the dissociative adsorption of hydrogen and desorption of water. The adsorption sites are identified with oxygen ions at the surface of U4O9 network. The respective reactions are thus:

s s2 O U O UH 2O U 2OH U°°+ + ⇔ + [15.R.g1]

s s

''O i 2 O i2OH O H O 2O V+ ⇔ + + [15.R.g2]

Within the U4O9 layer, there is diffusion of the defects [15.R.g3] that are produced at the internal interface according to:

( )6 4 4 6 4

3 8 4 9

''O i OU U U U U

4U O 3U O

4(2U U 8O ) 5U 5O 3U 4U 27O+ + + + +°°+ + ⇔ + + + +

[15.R.g4]

We can certainly doubt this last reaction is elementary, but this is not important since the rate determining step is located at the external interface (as it has been shown while studying the space function). It can thus be only the adsorption of hydrogen or the desorption of water.

The experimental study is carried out at the temperature of 500°C under partial pressures of hydrogen and water vapor of about some kilopascals. However, the equilibrium ratio is very large:

2

2

H O 6

H equil

10PP

⎛ ⎞>⎜ ⎟⎜ ⎟

⎝ ⎠

The experiments are thus carried out very far from the equilibrium condition, and we can thus neglect the reverse reaction of the rate determining step.

Table 15.10 gives the expressions of the reactivity for the two pure modes, the first with hydrogen adsorption as the rate determining step and the second with water desorption as the rate determining step.

The experiments, led by the switch method, have shown that the water vapor is without influence and the expression of the mode with water desorption as the rate determining step is compatible with the experimental effect of hydrogen with an excellent correlation coefficient (r = 0.999).


We will thus conclude that the reaction of reduction of U3O8 into U4O9 by hydrogen is, at around 500°C, as determined by the step of water desorption.

Rate determining step Reactivity φ

Hydrogen adsorption ( )2

2

1 2 4 H21/2 1/ 20

H O 2 1

k K K P

P K K

′

+

Water desorption ( )2

2

1/52 1 4 H

21/2 1/201 4 H1

k K K P

K K P

′

+

Table 15.10. Pure modes with adsorption or desorption as the rate determining step in the reduction of U308 into U409 by hydrogen

15.4. Growth process of oxidation of metals by water vapor

15.4.1. General approach of mechanism

The oxidation of metals by water vapor is carried out in experiments fixing the partial pressures of water and hydrogen (produced gas). The simultaneous presence of these two gases involves, by the action of the chemical equilibrium in gas phase, a partial pressure of oxygen.

Some authors considered that the oxidation of metal by water vapor is, in fact, an oxidation by oxygen essentially present. But many experiments have shown that it is not true and that the kinetic curves and the influences of pressures obtained with water were not as those obtained with pure oxygen under the same partial pressure. Water, without a doubt, has a direct reaction with the metal.

In all exactness, we must consider that the system will be the seat of two parallel oxidation reactions, one by oxygen and the other by water. If hydrogen and water pressures are sufficient, the reaction of oxygen is very low and if its pressure has an influence on the reactivity, we can neglect its oxidizing action. We will thus consider the specific oxidation of metal by water.

The general mechanism will be roughly similar to that developed in section 15.2 for the oxidation of metals by oxygen with adsorption of water on the surface of oxide, two interface reactions that create and consume the defect and the diffusion of the defect. With these steps, it will be necessary to add desorption of the formed hydrogen at the external interface.


We will solve the mechanisms by adopting the approach of the prevalent Wagner defect and considering only pure modes with only a single rate determining step. The experimental conditions are in general far from equilibrium and we will not have to take into accounts the reverse reaction of the rate determining step.

The examples are related to the following reaction:

M + H2O = MO + H2

We will assume that defects are not ionized.

The transposition with other reactions or the cases of ionized defects can be done without difficulty but requires making calculations again, the obtained laws for the reactivities not being identical to the cases considered here.

In all the cases, we will have steps of adsorption of water and desorption of hydrogen on the surface of oxide, which will be:

2 OH O 2s 2OH s+ + ⇔ − [15.R.h1]

22H s H 2s− ⇔ + [15.R.h6]

REMARK.– In certain cases, part of formed hydrogen can diffuse (jump from one oxygen ion of the network to another) through oxide and dissolve in the metal. However, in general, the small concerned amounts and the low molar mass of hydrogen have no effect on the thermogravimetric measurements. On the other hand, in other cases, the formed gas can dissolve in oxide in considerable amounts, such as in the case of the reduction of U4O11 by ammonia, which forms nitrogen whose considerable part is dissolved in the formed UO2 and takes part in the mass of the solid phases.

15.4.2. n-type formed oxide with interstitial cations

The defect of oxide will be Mi, and the internal reaction of interface that creates the defect will be:

i(int) MM M V⇔ + [15.R.h2]

where VM indicates a metal vacancy. We will assume that its diffusion in the metal and the heart reaction that eliminates it are never the rate determining steps.



Water adsorption 2

2

2 2 22 4 5 6 H O

1 21/2 1/ 24 5H 2 6

( )

( )

K K K K Pk

P K K K K′⎡ ⎤+⎣ ⎦

Internal interface 2k ′

Diffusion 2 5( )DK K

External interface 2

2

4 2 5 1 H

1 H

( )

1O

O

k K K K P

K P

′

+

Hydrogen desorption ( )2

2

2 2 26 1 2 4 5 H O

2

1 H O

( )

1

k K K K K P

K P

′

+

Table 15.11. Reactivities of oxidation of a metal by water forming an n-type oxide

The interstitial cation migrates from the internal interface toward the external interface:

i(int) i(ext)M M⇔ [15.R.h3]

where it reacts with adsorbed hydroxyls as follows:

i(ext) M OM OH s H s M O+ − ⇔ − + + [15.R.h4]

The oxide film is formed with outward development.

We thus have five modes with rate determining steps: [15.R.h1], [15.R.h2], [15.R.h3], [15.R.h4], and [15.R.h6]. The results are given in Table 15.11.

It will be noted that we cannot distinguish the modes given by steps 2 and 3, but the kinetic laws will be different because the space function is not the same, step 3 being diffusion.

15.4.3. n-type formed oxide with anion vacancies

The defect of oxide is VO. Reactions [15.R.h2], [15.R.h3] and [15.R.h4] are replaced by the following ones:


At the internal interface; creation of the defect and construction of the network:

M O(int)M M V⇔ + [15.R.i2]

The defect diffuses from the internal interface toward the external one:

O(int) O(ext)V V⇔ [15.R.i3]

At the external interface; consumption of the defect:

O(ext) OV OH s H s O+ − ⇔ − + [15.R.i4]

The layer has an inward development.

The results are given in Table 15.11, with k5 = 1.

15.4.4. p-type formed oxide with cation vacancies

The defect of oxide is VM. Reactions [15.R.g2], [15.R.g3] and [15.R.g4] are replaced by the following ones.

At the internal interface; consumption of the defect:

M(int) M mM V M V+ ⇔ + [15.R.j2]

The defect diffuses from the internal interface toward the external interface:

M((ext) M(int)V V⇔ [15.R.j3]

At the external interface; creation of the defect and construction of the network:

O M(ext)OH s H s O V− ⇔ − + + [15.R.j4]

The layer has an outward development.

The results are still given by Table 15.12.

We will note as previously that we cannot distinguish the modes given by steps 2 and 3, but the kinetic laws will be different because the space functions are not identical, step 3 being diffusion.



Water adsorption 2

2

2 2 22 4 5 6 H O

1 21/2 1/ 2H 2 4 5 6

( )

( )

K K K K Pk

P K K K K′⎡ ⎤+⎣ ⎦

Internal interface 2

2

H O1/ 2 1/22 1 4 6

H

Pk K K K

P′

Diffusion 2

2

H O1/ 2 1/ 21 4 6

H

PDK K K

P

External interface 2

2

4 1 H O

1 H O1

k K P

K P

′

+

Hydrogen desorption ( )2

2

2 2 26 1 2 4 5 H O

1 H O 2

( )

1

k K K K K P

K P

′

+

Table 15.12. Reactivities of the oxidation of a metal by water forming a p-type oxide

15.4.5. p-type formed oxide with interstitial anions

The defect of oxide is an interstitial anion Oi. Reactions [15.R.h2], [15.R.h3], and [15.R.h4], are replaced by the following ones:

At the internal interface, creation of the defect and construction of the network occur:

i(int) M OM O M O+ ⇔ + [15.R.k2]

The defect diffuses from the internal interface toward the external one:

i(ext) i(inter)O O⇔ [15.R.k3]

At the external interface, consumption of the defect occurs:

i(ext)OH s H s O− ⇔ − + [15.R.k4]


The layer has inward development.

The results are given by Table 15.12, with k5 = 1.

With these examples, we did not have the ambition to provide all the possible quantitative results because they are numerous and take into account the intervention of more or less ionized defects and stoichiometric numbers of the total reaction. However, the methods that we have presented make it possible to carry out calculations in each particular case.

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