1 thermodynamics and kinetics of gas an 2015 thermochemical surface engine

At typical temperatures for the above-mentioned thermochemical processes, ammonia therefore tends to decompose into its elementary gaseous constituents, according to the following reacti

Thermodynamics and kinetics

of gas and gas–solid reactions

J T Slycke 1 , E J Mittemeijer 2 , M A J Somers 3

1Consultant, The Netherlands; 2Max Planck Institute for Intelligent Systems

and Institute for Materials Science, University of Stuttgart, Stuttgart, Germany;

3Technical University of Denmark (DTU), Kongens Lyngby, Denmark

1.1 Introduction

Thermochemical surface treatments of steel components are of great importance for industry; the mechanical (hardness, wear and fatigue resistances) and chemical (corrosion resistance) properties of the surface layer of treated parts can be improved considerably as a result of the chemical and structural changes induced

by the processes in a surface region of the treated parts Today the most important

of these processes are nitriding/nitrocarburising and carburising/carbonitriding, where interstitially dissolved nitrogen and/or carbon diffuse(s) into the surface of the component Nitriding and nitrocarburising of heat treatable steels are carried out at relatively low temperatures where the steel is essentially ferritic, whereas carburising and carbonitriding are carried out at higher temperatures where the steel is essentially austenitic and, after the treatment, usually experiences a desired martensitic transformation upon quenching Besides these nitriding and carburising processes and their variants, also boriding involves the dissolution of interstitials,

in this case boron, into the host material and brings about a hard and wear-resistant surface layer onto the steel surfaces Especially the nitriding/nitrocarburising treatment, which in practice can be carried out according to quite a number of process variants (see Part Three: Nitriding, nitrocarburising and carburising), can be considered as the most versatile surface engineering treatment available today, since it enables,

by tuned application of this process, improvement, to a great extent, of the fatigue, corrosion, wear and tribological performance of steel components Furthermore, as compared to processes that are based on a martensitic transformation, such as classical heat treatment of carbon and alloyed steels, very small dimensional changes of the component occur Nitriding/nitrocarburising is therefore the surface engineering process that is today most widely applied (see Mittemeijer, 2013)

In order to obtain the desired properties, it is necessary to have (i) full process control and (ii) knowledge of the relationship between composition, temperature and pressure of the medium used in the thermochemical treatment and the resulting microstructure of the surface layer produced Although such process control, as determined by freely chosen, specific values of process parameters is perfectly feasible in the laboratory, in (commercial) practice, and particularly as regards nitriding/nitrocarburising, a largely phenomenological approach has been adopted

Thermochemical Surface Engineering of Steels http://dx.doi.org/10.1533/9780857096524.1.3

until now, i.e one has relied more on experience with individual equipments than

on knowledge, steering and understanding of the effect of the fundamental process parameters The possibility of using some ‘probe’ in gaseous media, not only in carburising/carbonitriding but now also in nitriding/nitrocarburising, can change this

situation fundamentally The use of the oxygen probe and other sensors in practical

heat treatment for control of the nitriding/nitrocarburising process is discussed in Chapter 3 and in Mittemeijer and Somers (1997)

In this chapter thermodynamic and kinetic fundamentals are presented for gaseous processing, since the gaseous treatment allows the largest flexibility for controlled thermochemical surface engineering (i.e., in principle the chemical potentials of the components, in the gas atmosphere, can be set at specific values) More important, for basic understanding, is the recognition that gaseous processing can be understood

on the basis of elementary chemical reactions in the gas and at the surface of the component

Each reaction, whether this is an interchange reaction between different gas species (equilibria of gas exchange reactions are discussed in Section 1.2), or a mass transfer reaction between the gas phase and the solid iron-based substrate at its surface (equilibria of such gas–solid reactions are discussed in Section 1.3), proceeds at its own rate The kinetics of these reactions are discussed in Sections 1.4 (gas-exchange reactions) and 1.5 (gas–solid reactions) In many processing atmospheres more than one mass transferring reaction for a given species is possible Thus a ‘competition’

of the different mass transfer paths occurs, a situation that the researcher and the process engineer have to be aware of

Practical heat treatment involves processing in closed chambers or reactors, through which the process gas flows The process gas only resides within the reactor for a finite period of time As a consequence, a given gas phase reaction may, or may not, reach (or closely approach) its equilibrium, depending on the reaction rates between the gas components A fast reaction, for example the water–gas exchange reaction (cf Section 1.2.3), may attain the corresponding equilibrium state in most cases, whereas a reaction that proceeds relatively slowly, for example the decomposition

of ammonia, will generally not reach the corresponding equilibrium state (Section 1.4) Yet, and very remarkably, it is in particular this relatively slow progress of the ammonia decomposition reaction that provides the basis for several different industrially important surface engineering processes such as nitriding, nitrocarburising and carbonitriding

Control of the process parameters is a required, but not a sufficient prerequisite

to realise tuned thermochemical processing The relation between the process parameters and the composition and structure of the surface layer produced must

be known as well (cf Section 1.6) On the one hand, it should be simply admitted that, as regards the low temperature processes of nitriding/nitrocarburising,

insufficient data is available on layer-growth kinetics to be able to employ in practice model descriptions, as given in Somers (2011) and Woehrle et al (2013)

On the other hand, the relations governing the thermodynamics of the equilibria

of the (gaseous) media for nitriding/nitrocarburising as well as for carburising/carbonitriding (gas-exchange reactions; see Section 1.2) and the gas–solid equilibria

possibly (closely) realised at the surface of the solid substrate (see Section 1.3) are known.

The ‘nitridability’ and the ‘carburisability’ of a medium (here ‘outer gas atmosphere’), i.e the ability of that medium to nitride respectively carburise, are thermodynamically characterised by its chemical potential of nitrogen, and that of carbon, respectively Equilibrium with the component requires that the chemical potential concerned must be the same in both the process medium and in the workpiece

at the surface Particularly in the initial stage of the treatment, considerable deviations from this equilibrium occur (for the example of nitriding, see the kinetic calculations

and experimental data in Rozendaal et al., 1983; Friehling et al., 2001; and Stein

et al., 2013) The chemical potentials are not only used to describe the equilibrium,

but, although less well founded, also to describe the kinetics of processes involving the components (here nitrogen and carbon) concerned It can therefore be concluded that control of the chemical potentials of nitrogen and carbon is a prerequisite for controlled and targeted nitriding/nitrocarburising and carburising/carbonitriding The chemical potentials of nitrogen and carbon in the iron-based substrate correspond with so-called nitrogen and carbon activities, respectively, which, then

as well, prescribe both the phase present and its composition (nitrogen and carbon solubility) At present the numerical data for the relations between the chemical potential, or activity, for nitrogen and the nitrogen content in an iron substrate (without carbon) and for the relations between the chemical potential, or activity, for carbon and carbon content in an iron substrate (without nitrogen) are well known If nitrogen and carbon are simultaneously present, as in the carbonitride layer produced

by nitrocarburising, or in the austenitic surface layer formed by carbonitriding, then, although the activities of nitrogen and carbon in the substrate can be known,

it is presently not possible to accurately calculate the carbon and nitrogen contents, because the activity coefficients that describe the proportionality between activity and composition are not yet well known for the case of simultaneous presence of

nitrogen and carbon in the various iron-based phases (see also Leineweber et al.,

2012) Modelling tools and related thermochemical and kinetic databases are under development, which in principle can allow estimates of the establishment of phase equilibria and the development of phase compositions also for quite complex alloy systems However, sufficiently accurate data to reliably apply such models are currently lacking

The above consideration leads to the conclusion that well-controlled nitriding/nitrocarburising and carburising/carbonitriding at least require that the chemical potentials, and the corresponding nitrogen and carbon activities as imposed by the applied media, should be known and controllable The introduction of the various types of probes and analysing systems in the last decades makes this possible also

in heat treatment practice (where gaseous media are used, see below) It appears that considerable confusion exists concerning the use of thermodynamic terms such

as activity, chemical potential and equilibrium as well as kinetic terms associated with chemical reaction rates This chapter aims at providing a basic understanding

of the thermodynamic and chemical reaction rate concepts involved in gaseous nitriding, carburising, nitrocarburising and carbonitriding The restriction to gaseous

processes is not essential, but it is based on the following consideration Of the known treatment media, salt baths do indeed allow excellent reproducibility, but controlled modifi cation of the imposed chemical potentials of nitrogen and carbon

is not (yet) possible Similarly, plasma or low-pressure (‘vacuum’) processes prove worthwhile in practice, but they do not (so far) allow specifi c, controlled change

of the imposed chemical potentials of nitrogen and carbon, but rely rather on the control of mass transfer rates Gaseous media allow nitrogen and carbon potentials

to be determined during the treatment (by use of appropriate probes) as well as to

be controlled (by altering the composition of the gas mixture introduced into the furnace) When it is important to produce surface layers with well-defi ned compositions and microstructures as well as to maintain fl exibility to adapt the treatment to the desired treatment-induced property of the workpiece, gaseous media appear to be most suitable to date

1.2 Equilibria for gas-exchange reactions

Basic thermodynamic relations

The chemical potential, that is, the partial molar Gibbs energy, m I,s, of an element

(or compound) I, dissolved in the solid matrix of an element (or compound) M, is

defi ned by:

a I,s = activity of I in dissolved state, with respect to the reference state In

the reference state a I,s = 1

R = the gas constant (here: R = 8.3145 J mol–1 K–1)

T = the absolute temperature.

There are no prerequisites for the selection of the reference state Therefore, the relevant reference state should always be specifi ed when activities are discussed

The chemical potential m I,g of a gas constituent I in a gas mixture, is defi ned by:

f I = fugacity of I in the gas mixture.

p0I = pressure of I in the reference state.

Normally, the pressure of the reference state is chosen as 1 atm (p0I = 1 atm) Here,

only ideal gases will be considered, or it is assumed that the fugacity coeffi cient concerned is constant (in the latter case, the fugacity coeffi cient is incorporated in

where: p I = partial pressure of I.

Consider the equilibrium between the components A, B, C and D (elements or

compounds; solids or gases) at constant temperature and constant pressure:

where the different c I parameters represent the stoichiometric coeffi cients for the reactants and products The equilibrium condition (here assuming all four components are solids; for gaseous constituents, see discussion around Eq [1.6b] below) yields:

G cmmmmmmmI s I s I s I s I s0 0,,, w, withithithmmmmmmm0 0 with representing the chemical potentials of the

constituents I (that is, the constituents A, B, C and D in Reaction [1.3]) in their reference state, and c I > 0 for products and c I < 0 for reactants in [1.3] (see Eq

[1.4b]) The equilibrium constant K n , where the subscript n indicates the reaction name or number (in this general case n refers to Reaction [1.3]),1 is defi ned as:

and is by defi nition dimensionless, because activities are dimensionless!

If a component is gaseous, its ‘activity’ is replaced by p I /p0

I (cf Eq [1.2b]; for

a non-ideal gas, p I is replaced by f I, cf Eq [1.2a]) Hence, if constituents B and

1 In this chapter, the various gas-exchange and gas–solid reactions of relevance to thermochemical surface engineering are, for convenience, given descriptive names These can

be established names in the literature such as ‘WGS’, for the ‘water-gas shift gas-exchange reaction’ (also called ‘homogeneous water-gas reaction’) occurring in the gaseous phase, or

‘HWG’, for the ‘heterogeneous water gas reaction’ for transfer of carbon to a solid substrate

In other cases, names such as ‘NH3(g)’ are used to indicate the gas-exchange reaction for decomposition (and formation) of ammonia in the gaseous phase, or ‘NH3(s)’ for the classic gas–solid ammonia nitriding reaction indicating nitrogen transfer to a solid substrate.

D in Reaction [1.3] are gaseous and A and C are solids, the equilibrium constant becomes:

K

p p p p n

D c

ÊÁ

ÊËÁË ˆ¯˜ˆ˜ˆ¯˜¯

ÊËÁ

ÊÁ

ÊËÁË ˆ¯˜ˆ˜ˆ¯˜¯

In Eq [1.5a], DG0 = DH0 – T · DS0, where DH0 and DS0 represent the enthalpy and entropy

for the concerned reaction, respectively, and T is the absolute temperature.

Since, for many reactions, DH0 and DS0 are practically constant over a large temperature range, Eq [1.5a] can conveniently be written as:

=d K d Klnln(1

b n values for a number of gas-exchange reactions and gas–solid reactions (see

Table 1.1 Gas exchange, or gas shift, reaction equilibria relevant

to nitriding, nitrocarburising, carburising and carbonitriding

0 –2

· (p · (p )0 – 0 – )

0 1/2

· (p · (p )0 1 0 1 )

0.6255 –5906

The equilibrium constant K n is defi ned in Eqs [1.5a], [1.6a] and [1.6b] in Section 1.2 The parameters a n and b n are defi ned in Eq [1.5b] The values for the equilibrium constants have been evaluated, using Thermo-Calc (2012), in the temperature range 600–1300 K (327–1027°C), but can with good accuracy be used also outside this range.

Section 1.3), respectively, relevant to nitriding, nitrocarburising, carburising and carbonitriding processes.

In the following, all gases are treated as ideal and the same reference pressure is

used, that is, 1 atm total pressure Then, p0

I can be replaced by p0

1.2.1 Ammonia decomposition

Ammonia, NH3, is an important component in thermochemical process atmospheres dedicated to nitriding, nitrocarburising and carbonitriding Within this context, its reactions with other gas species and iron surfaces will be discussed in detail in this chapter It should be noted that ammonia is an unstable molecule at temperatures exceeding approximately 0°C (273 K) At typical temperatures for the above-mentioned thermochemical processes, ammonia therefore tends to decompose into its elementary gaseous constituents, according to the following reaction:

Table 1.2 Equilibria for gas–solid reactions relevant to nitriding, nitrocarburising, carburising and carbonitriding

0 1/2 2

· ·

The equilibrium constant K n is defined in Eqs [1.5a], [1.6a] and [1.6b] in Section 1.2 The parameters a n and b n are defined in Eq [1.5b] The column ‘Reference state’ indicates the condition (at the temperature concerned) at which

a N,s = 1, or a C,s = 1 The values for the equilibrium constants have been evaluated, using Thermo-Calc (2012), in the temperature range 600–1300 K (327–1027°C), but can with good accuracy be used also outside this range.

2NH3(g)  N2(g) + 3H2(g) [1.7]The equilibrium constant for this gaseous equilibrium (here named ‘NH3(g)’, where (g) indicates that a gas-exchange reaction is concerned) is:

temperature) of the resulting gas mixture will, at most, be around a N,s ≈ 0.5 (this

activity parameter is discussed in Section 1.3.1), as shown in Figure 1.1 (the calculation procedure for this diagram is discussed in Section 1.2.6) Such an equilibrium atmosphere consisting of practically fully, so-called dissociated, or cracked, ammonia,

is, by itself, not suitable for thermochemical (nitriding, nitrocarburising) processing Instead, dissociated ammonia is often used as a protective atmosphere, for example, during annealing or brazing, either by itself, or in mixtures with (additional) N2gas:2 The relatively high H2 content3 will, via its buffering effect4 against oxidising species (cf Reaction [1.11] and the discussion in Section 1.5.5), effectively prevent oxidation or decarburisation that otherwise may occur when using only N2 gas as protective atmosphere, since practical heat treatment furnaces never will be free from traces of O2, H2O and CO2 Mixtures of dissociated ammonia and pure ammonia are often used as nitriding atmospheres, allowing control of the nitriding process via the partial pressures of ammonia and hydrogen gas (see Section 1.3.1 for details on gaseous nitriding with ammonia)

Fortunately, Reaction [1.7] proceeds at a limited rate (see discussion in Section 1.4.1), implying that under typical thermochemical process conditions, with a fi nite

2 For completeness it is mentioned that at high temperature (above 900°C) annealing in N2 containing atmospheres can lead to signifi cant dissociation of N2 and atomic nitrogen then is dissolved in a solid iron-based substrate (see Section 1.3.2), which for solids such as high- alloyed steels, e.g stainless steels, leads to CrN and/or Cr2N precipitation (sensitisation).

-3 In the absence of oxidising species (see footnote 4), the presence of signifi cant amounts of H2 in the protective atmosphere may, at high temperatures, lead to decarburisation of steel

by formation of CH4 (see Section 1.3.5) However, as discussed in Section 1.5.3, the rate of this reaction is very slow at typical annealing or brazing temperatures and decarburisation

by this reaction is therefore negligible.

4 Firstly, the presence of H2 removes any traces of free O2 by formation of H2O Secondly, the surplus (after removal of O2 by H2O formation) of H2 reduces the oxygen ‘activity’,

expressed as p H2O /p H2, and reduces, via the water–gas shift reaction (see Section 1.2.3), also the

p CO2O /p CO ratio Thereby the rate by which dissolved carbon, in the surface-adjacent region of

an iron-based substrate, reacts to gaseous CO, via the rapid heterogeneous water-gas reaction (cf the rate controlling reaction step II and Eq [1.104] in Section 1.5.5), is reduced and thus decarburisation is counteracted.

atmosphere residence time in the furnace chamber, the reaction will not attain equilibrium and a considerable ammonia partial pressure and a corresponding relatively high nitrogen ‘activity’ (see discussion around Eq [1.6b]) therefore can be built up in the atmosphere It is for this reason (‘this twist of nature’) that nitriding

is possible at all! This is further discussed in Sections 1.4 and 1.5

1.2.2 Oxidation reactions in the gas phase

The presence of oxygen-carrying species like CO2 or H2O entails that the atmosphere has certain oxidising properties, which can be characterised by a partial pressure of gaseous molecular oxygen In the presence of carbon monoxide (CO), the following reaction (here named ‘CO/O2’) can occur:

The temperature dependence of this equilibrium constant is given in Table 1.1

In the presence of hydrogen gas (H2) the following reaction (here named ‘H2/

O2’) is possible:

Figure 1.1 Equilibrium composition for ammonia decomposition, at a total pressure of 1 atm,

according to Reaction [1.7] and corresponding nitrogen ‘activity’ (see discussion around Eq [1.6b] and Eq [1.27a], or Eq [1.19a]), as a function of temperature.

H2(g) + ½O2(g)  H2O(g) [1.11]with the equilibrium constant:

The temperature dependence of this equilibrium constant is given in Table 1.1

The p O2 values that can be obtained from Eqs [1.10] and [1.12] in nitrocarburising, carburising and carbonitriding atmospheres typically have extremely low values, meaning that the likelihood of fi nding free oxygen molecules in the atmospheres is extremely low Still, the ability to control the oxidation potential, or oxygen activity,

of the atmosphere is of profound importance in thermochemical processing The oxygen activity of the atmosphere can be measured using an oxygen probe that can serve as a means of accurate process control (see Chapter 3)

It can be shown simply that, if the water–gas exchange reaction (CO+ H2O 

CO2 + H2; see Section 1.2.3) is in equilibrium, then this is equivalent with both

Reactions [1.9] and [1.11] being in equilibrium with the same p O2

1.2.3 Water–gas-exchange reaction

The water–gas-exchange reaction, also often called the (homogeneous) water–gas (shift) reaction, is an important gas-exchange reaction in thermochemical process atmospheres: it facilitates the balance between the four gaseous species, CO, CO2,

H2 and H2O As discussed in detail in this chapter, these constituents play central roles in the nitrocarburising, carburising and carbonitriding processes The water–gas exchange equilibrium (here named ‘WGS’ (after the traditional designation; ‘water–gas shift’ reaction)) takes the form:

The temperature dependence of this equilibrium constant is given in Table 1.1

A combination of the Reactions [1.9] and [1.11] yields directly [1.13] (see the end of Section 1.2.2), implying that a corresponding combination of Eqs [1.10] and [1.12] gives Eq [1.14a]:

As discussed in Section 1.4.3 the WGS equilibrium is rapidly established and therefore

in practice this equilibrium is often (but not always) realised, implying that often (but not always) Reactions [1.9] and [1.11] are at, or close to, equilibrium as well

1.2.4 Methane gas-exchange reaction

In actual carburising atmospheres, carbon is transferred from the atmosphere to the surfaces of the components being treated, which is discussed in Section 1.3 As a consequence, carburising species (such as CO) are consumed and oxidising species (such as H2O and CO2) are formed In order to preserve the carburising properties

of the atmosphere, small quantities of an enrichment gas, typically methane (natural gas), or propane, are added This invokes additional, different gas-exchange reactions, involving those hydrocarbons that reform CO and H2 from the oxidising species Several reaction paths are possible (also depending on the type of enrichment gas)

An important reaction is the following one (here named ‘CH4(g)’, where (g) indicates that a gas-exchange reaction is concerned):

The temperature dependence of this equilibrium constant is given in Table 1.1

1.2.5 Hydrogen cyanide gas-exchange reaction

The simultaneous presence of carburising species, such as CO or CH4, and the nitriding species NH3 in the atmosphere, allows direct reaction between these species

to form HCN gas Here the reaction of NH3 with CO, thereby producing gaseous HCN and gaseous H2O, is presented due to its potential importance This gas-exchange reaction, here named ‘HCN(g)’, where (g) indicates that a gas-exchange reaction is concerned, takes the form:

in nitrocarburising and carbonitriding atmospheres: HCN potentially provides an additional path for the mass transfer of nitrogen and carbon from the gas phase to the iron surface (see discussions in Sections 1.3.6 and 1.5.6) Table 1.3 gives indications

of expected equilibrium contents of HCN under selected process conditions

As shown in Table 1.3, the volume fractions of HCN formed in the HCN exchange reaction are very small (note that it is presupposed here that the equilibrium

gas-[1.17] is realised); also relatively with respect to the NH3 content in the case of nitrocarburising at a typical temperature as 580°C However, for carbonitriding conditions (at 820°C or higher), the HCN equilibrium content exceeds the NH3content (the ‘residual ammonia’ content) of the atmosphere.

1.2.6 Examples

In this section, the basic thermodynamics for evaluating reaction equilibria have been presented and a number of important gas exchange reactions have been reviewed in this sense It must be emphasised that although the discussion concerned ‘gas phase’ reactions, each of the reactions mentioned, in fact occurs at available surfaces in the furnace chamber or reactor Such a surface can be the (hot) surface of the load itself, the furnace brickwork or the insulation material, the heating elements, or other constructional material surfaces The character of the available surfaces, by their catalytic action, determines the rate by which the reaction in question proceeds and thereby how near the reaction will approach its equilibrium state These aspects are discussed in Section 1.4

Genuine equilibrium will seldom occur in practical thermochemical processing, typically involving flowing gas streams through the furnace chamber Yet, the equilibrium concept is a powerful tool (as it indicates the state the system strives for)

in process control and in troubleshooting problems related to gaseous atmospheres and processes

In the following, three examples of gas-equilibrium calculations are given in order

to illustrate how the composition of process atmospheres can be estimated

Example 1: Equilibrium calculation for dissociation of ammonia

Task

Derive the degree of ammonia dissociation at equilibrium, as a function of temperature

for the decomposition of pure ammonia at constant total pressure, p tot

Table 1.3 Characteristics of the HCN gas-exchange equilibrium [1.17] under different initial conditions

0.250 0.002 0.002

0.045469 0.198776 0.198658

0.245469 0.000776 0.000658

0.004531 0.001224 0.001342

0.004531 0.001224 0.001342 The calculations were made for Reaction [1.17] in isolation, thus disregarding other gas reactions such as the water-gas exchange (Reaction [1.13]) and the ammonia decomposition reactions (Reaction [1.7]) The atmosphere compositions are given as partial pressures, with a total gas pressure of 1 atm with the balance consisting of a non- reactive gas.

Ammonia decomposition is accompanied by an increase in the number of gas molecules, resulting (at constant pressure) in a volume increase (cf Reaction [1.7]) Principally, the ammonia decomposition at equilibrium is only partial: some ammonia always remains Consider the partial decomposition of one mole of ammonia at equilibrium:

to a varying total pressure) The total pressure is:

where the reference pressure p0 = 1 atm

Inserting the partial pressures into the equilibrium equation yields:

( )g

1/2 3

The temperature dependence for K NH3(g) is given in Table 1.1 On this basis, q has been

plotted versus temperature in Figure 1.2 for three different total gas pressures

It follows that, at any given temperature, an increasing total pressure suppresses

the ammonia decomposition (reduced q) This is a direct consequence of the ammonia

decomposition reaction being accompanied by a volume increase A pressure increase therefore tends to shift (‘push’) the decomposition reaction [1.7] to the left

Knowing q, the different partial pressures and other derived quantities, for example

the nitrogen ‘activity’ (see discussion around Eq [1.6b]; also see Section 1.3), can

be calculated An example is shown in Figure 1.1, for p tot = 1 atm

Example 2: Equilibrium calculation for a carburising atmosphere

Task

Find the equilibrium composition for a carburising atmosphere based on an ingoing atmosphere with known volume fractions of CO, CO2 H2, H2O and CH4, and with

an inert gas (IG) as balance (nitrogen gas, N2, is in practical heat treating often seen

as non-reactive, which strictly speaking is not correct; see Section 1.3.2) at a given

temperature, under constant total pressure, p tot This example is based on a similar

example given in Darken et al (1953).

The partial pressure, p¢I , for an (ideal) gas species, I, is proportional to its volume fraction, u¢I, as follows:

p¢I = u¢I · p tot

The ingoing atmosphere can therefore be characterised by the following six partial

pressures: p CO¢ , p CO2¢ , p H2¢ , p H2O¢ , p CH4¢ , and p IG,g¢ , where the subscript IG,g indicates the

Figure 1.2 Fractional degree of (equilibrium) dissociation for gaseous ammonia, q, as a

function of the temperature at three different levels for the total gas pressure, p tot, expressed

in [atm].

inert gas and the prime signs refer to the ingoing (non-equilibrium) values of the

partial pressures

The partial pressures of the species in the atmosphere parcel considered can be converted to a number of moles for each of the ingoing gas species, using the relation (valid for ideal gases):

where n I¢, is the ingoing number of moles of gas constituent I and n¢tot is the total

number of moles of ingoing gas in the atmosphere parcel studied For n tot¢ we are free to choose any value It is convenient to consider one mole of ingoing gas, that

is, n tot¢ = 1 mol

From this information the number of moles of atoms for each of the elements (C,

H, O and IG) present in the ingoing atmosphere parcel (having constant pressure)

can be determined: n C , n H , n O and n IG Hence:

of inert gas in the ingoing gas mixture

At this point the initial conditions have been detailed Now a side step is made The strategy followed in solving the problem posed is based on application of Gibbs’ phase rule (see, for example, Atkins and de Paula (2006, pp 175–6); or Mittemeijer (2010, Section 7.4)) In the following text box, Gibbs’ phase rule is described such that it allows a direct application to the case considered here, which is shown subsequently

5 Note that the inert gas, IG, here is treated as if it is monatomic (e.g., Ar(g)) If the ‘inert’ gas instead is diatomic (e.g., N2(g)) the last atom balance equation (above) for the ingoing

atmosphere would read: n IG = 2n¢ IG,g Similarly, the atom balance for the diatomic inert gas in

the equilibrium atmosphere (cf Eq [IV] below) would then read: n IG = 2n IG,g In the problem solution process, the fi rst of these relations is inserted into the second, yielding the equality:

n IG,g = n¢ IG,g, which is the same result as found for the monatomic case (see text under Eq [VIa] below) The number of atoms in the inert gas molecule therefore has no consequence for the outcome of the calculations.

Gibbs’ phase rule

The phase rule is a general relation between the number of degrees of freedom,

f, the number of components, n, and the number of phases at equilibrium, r, for

a system of any composition, as follows:

f = n – r + 2

Here, the number of degrees of freedom (f) represents the number of choose intensive state variables, i.e the number of mathematically independent

free-to-variables, such as temperature, pressure and composition parameters for solid,

liquid and gaseous phases (partial pressures for a gaseous phase), which do not

depend on the size of the system under consideration

The value n represents the number of components, or more precisely, the minimum number of (mathematically) independent components necessary to

define the composition of all phases of the system Note that a phase can be composed of a number of constituents (species), with each constituent composed

of a number of elements with fixed ratios (a constituent thus has a fixed, specified composition; e.g., a gas phase composed of CO2 and H2 molecules, with CO2and H2 as constituents and C, O and H as elements) The constituents of a phase are the (dependent) components

The number of independent components is found from the number of dependent

components (constituents (or species)), minus the number of relations (or constraints) linking the components (constituents (or species)) mathematically

(such as atom balances) minus the number of independent chemical reactions

involving the (active) components Here an independent reaction (in equilibrium)

is one that contains one or more active components (constituents (or species)) not yet taking part in other reactions already specified and which components (constituents (or species)) have not been constrained/defined by earlier specifications of degrees of freedom (specified intensive state variables, given constraints, etc.) All independent reactions together must comprise all active components This will be illustrated by Examples 2, 3 and 4

The phase rule applied to Example 2

The system involves only one phase, the gas phase, entailing that r = 1.

The number of dependent components (constituents) in this phase (the gaseous species) is six, that is, CO, CO2, H2, H2O, CH4 and IG These dependent components (constituents or species) are based on, in total, four different elements (C, H, O and IG) and consequently four atom balances, defining the number (of moles) of

atoms of each kind (n C , n H , n O and n IG; see further below), can be defined to link the dependent components (constituents (species)) However, only three of these atom balances are independent, since the total number (of moles) of atoms remains constant The fourth atom balance can thus be found from the other three and the knowledge of the total number (of moles) of atoms This means that the number of

independent components, n = 6 – 3 = 3, without, at this stage, having considered the

constraint that ‘the number of independent chemical reactions involving the (active) components’ (see at the end of the text box above) has to be subtracted as well (see below)

Inserting now these numbers into Gibbs’ phase rule, we find the number of remaining degrees of freedom to be:

f = 3 – 1 + 2 = 4

In order to solve the problem (here determination of the composition of the gas

phase in equilibrium) it is required that f = 0, that is, that there must be no remaining

degrees of freedom Then the system is fully defined and the problem is solvable

Since the intensive state variables temperature (T) and the total pressure (p tot) of the atmosphere here are predefined/given, the remaining number of degrees of freedom

is reduced to f = 2

Until this stage ‘the number of independent chemical reactions involving the (active) species’ (see at the end of the text box above) has not been subtracted in the

above calculation of f Hence, it here remains to identify two independent equilibrium

reactions, that must chemically link active dependent components (constituents (or species)) in the gas phase (i.e., the active components not yet taking part in other reactions already specified and which constituents have not been constrained/defined

by earlier specifications of degrees of freedom; both these limitations do not pertain

to this example and therefore the reactions in equilibrium to be specified must contain

all active components) As a result, two corresponding equilibrium equations (the

equilibrium constants) that mathematically link all (active) gas constituents must

be formulated and thereby the number of degrees of freedom has been reduced to

f = 0.

Detailing the relationships

The atmosphere parcel considered cannot exchange mass with its surroundings This means that the mass or number of moles of atoms of each element contained in the atmosphere parcel remains unchanged during the reactions towards equilibrium

For the equilibrium gas species then the following four atom balance equations

introducing this additional parameter and replacing one of the mass balances above with this relationship, the use of the four relationships [I–IV] given here is chosen

in the calculation procedure presented here, but recognising that only three of these

are independent The values for n C , n H , n O and n IG (and thereby also n∑) are known from the ingoing gas-atmosphere composition and have been specifi ed earlier At

this stage, our system has four remaining degrees of freedoms, f = 4 (see above

application of Gibbs’ phase rule to this example)

With specifi ed values for the two intensive state variables temperature (T) and total gas pressure (p tot ), the number of degrees of freedom is further reduced to f = 2.

In the general case, the total number of moles of gaseous species, n tot (not to be

confused with the constant total number of moles of atoms, n∑) contained in the

atmosphere parcel will be variable (it can increase or decrease), depending on the

character of the governing gas-exchange reactions leading towards equilibrium The

total number of moles of gaseous species at equilibrium, n tot, is thus unknown, but complies with the following equation:

n tot = n CO + n CO2 + n H2 + n H2O + n CH4 + n IG,g

This equation makes clear that in total six unknowns have to be determined to

defi ne the equilibrium: n CO , n CO2 , n H2 , n H2O , n CH4 and n IG,g This means that six mathematical relationships, linking these unknowns, are required in order to solve the problem Besides the relations [I – IV] given above, two additional relationships are therefore required

In the furnace, the active gaseous species may react towards equilibrium according

to different gas-exchange reactions, such as the water–gas reaction, and different gas-exchange reactions involving methane, CH4 Although several reactions/equilibria may be defi ned, Gibbs’ phase rule commands that, at this stage, only two degrees of freedom are left and therefore (see at the end of the text box above and the text on pages 18–19) only two, independent, gas-exchange reactions/equilibria involving the (fi ve) active species can be defi ned (consideration of further additional gas-exchange reactions/equilibria will therefore not add any new, i.e independent, information)

This will bring the number of degrees of freedom to f = 0

The freedom therefore exists to choose two independent chemical reactions (in

equilibrium), to bring f to 0, out of the minimal set of reactions (in equilibrium),

that together comprise all active components (here this minimal set comprises two reactions (see what follows), which is just the number of reactions needed (this is different for Example 4 presented in Section 1.3.7)) Firstly, the water–gas exchange reaction ‘WGS’, i.e Reaction [1.13], is taken here:

production from fossil fuels often called ‘steam reforming’ of methane), i.e Reaction [1.15]:

CH4(g) + H2O(g)  CO(g) + 3H2(g)

Evidently, this reaction is accompanied by an increase in the number of moles of gas (from two to four) and will thus lead to a volume increase when proceeding to the right, and vice versa The equilibrium constant for this reaction is:

case the variable n IG,g only appears in Eq [IV] For the inert gas, the mass balance relation in Eq [IV] is trivial since IG does not take part in any reactions and thus

remains invariant This means that n IG,g ∫ n¢ IG,g , where n¢IG,g and n IG,g are the ingoing and equilibrium number of moles of inert gas, respectively, such that Eq [IV] can

be disregarded and the known value of n IG,g can be used in the solution process Then three atom-balance equations (I–III) remain, where the gas species are expressed as number of moles, and two equations ([Va] and [VIa]) where the gas species are expressed as partial pressures Therefore it must be decided if partial pressures or number of moles are used during the solution process Here, the number

of moles is chosen and thus the partial pressures are converted into a number of moles using the relation (valid for ideal gas species):

CH g

CO

n n CO

n n H CH

n CH n

n n H O

tot tot

ÊËÁË ˆ¯˜ˆ˜ˆ¯˜¯ · (p · (ppp0 –20 –0 –)) [VIb]

Solving the problem

Now fi ve unknowns; n CO , n CO2 , n H2 , n H2O , n CH4 (n IG,g is known; see above, and n tot, appearing in Eq [VIb] is simply substituted by the sum of the number of moles

of all gas species, see above) occur and fi ve equations [I, II, III, Vb and VIb] are

available So the problem is solvable However, to solve a system of fi ve, in part, nonlinear equations, is not trivial It is, though, possible to express three unknowns

as a function of the remaining two unknowns, using the linear equations [I–III] (note that the values of n C , n H , n O (and n IG) are known) This leaves a nonlinear system of two equations and two unknowns, which can be solved using iterative techniques For problems like this one, powerful computer-based solvers are available in different mathematical toolboxes.

Numerical examples

Consider the following ingoing gas composition (volume percent): 19% CO, 1%

CO2, 38% H2, 2% H2O and 2% CH4, with the balance (38%) consisting of N2 (which

here is treated as inert), intended for gas carburising, at given temperature (T) of 930°C (1203K) and given total pressure (p tot) of 1 atm

• The equilibrium partial pressures of the gas species in this carburising atmosphere are

shown in Figure 1.3(a) as a function of temperature (T) in a wide temperature range at a constant total pressure (p tot) of 1 atm The diagram also shows the carbon ‘activity’ (see below Eq [1.6a]) imposed by the equilibrium atmosphere on a substrate surface, which can be determined from any of the relevant mass-transfer reactions discussed in Section 1.3

• The partial pressures of the gas species in this equilibrium carburising atmosphere are

shown in Figure 1.3(b) as a function of the total pressure (p tot) in a wide pressure range

for a constant temperature (T) of 930°C (1203K) Again, the carbon ‘activity’ (see below

Eq [1.6a]) imposed by the atmosphere on a substrate surface, has been shown as well.

Example 3: Equilibrium calculation for a nitrocarburising atmosphere

Task

Find the equilibrium composition for a nitrocarburising atmosphere based on an ingoing atmosphere with known volume fractions of CO, CO2 H2, H2O, NH3 and HCN, with the balance consisting of an inert gas (such as nitrogen gas, N2, which may here be treated as non-reactive, which strictly speaking is not correct; see Sections 1.3.2 and 1.5.2 and see also the beginning of the text of Example 2 in Section 1.2.6)

for a given temperature and under constant total pressure, p tot It will be assumed that ammonia decomposition and formation of hydrocarbons (such as CH4) can be disregarded These restrictions are justified for low-temperature nitrocarburising atmospheres and at high gas flow velocity (as can be maintained in a tube furnace),

or in case of a high gas-volume exchange rate (as can occur in a box furnace) Note that in reality equilibrium may not be realised, which can particularly pertain to the case of nitrocarburising (Mittemeijer, 2013) Here it is assumed that all gas-exchange reactions supposed to occur (cf above text) do reach equilibrium

The partial pressure, p I , for an (ideal) gas species, I, is proportional to its volume fraction, u I, as follows:

p I = u I · p tot

The ingoing atmosphere can therefore be characterised by the following seven partial

Figure 1.3 (a) Example of the equilibrium partial pressures of the gas components (logarithmic

scale) of a carburising atmosphere under the specified process conditions, as a function

of temperature The corresponding carbon ‘activity’, a C,s, imposed by the atmosphere on

a substrate surface, on the right-hand scale is given in relation to that of graphite at the

temperature concerned (for which a C,s ∫ 1); (b) example of the equilibrium partial pressures

of the gas components (logarithmic scale) of a carburising atmosphere under the specified process conditions as function of the total pressure (logarithmic scale) The corresponding

carbon ‘activity’, a C,s, imposed by the atmosphere on a substrate surface, on the hand scale is given in relation to that of graphite at the temperature concerned (for which

right-a C,s ∫ 1).

pressures: p CO¢ , p CO2¢ , p H2¢ , p H2O¢ , p NH3¢ , p HCN¢ and p IG,g¢ , where subscript IG,g indicates the

inert gas and the prime sign indicates that the ingoing (non-equilibrium) quantities are concerned

The partial pressures of the species in the atmosphere parcel considered can be converted to number of moles for each of the ingoing gas species, using the relation (valid for ideal gases):

where n¢I is the ingoing number of moles of gas constituent I and n¢tot is the total

number of moles of ingoing gas in the atmosphere parcel studied For n¢tot we are free to choose any value It is convenient to consider one mole of ingoing gas, that

is, n¢tot = 1 mol

From this information it is possible to determine the number of moles, present

in the ingoing atmosphere parcel under consideration, for each of the elements (C,

H, O, N and IG), n C , n H , n O , n N and n IG, as follows:

in the ingoing gas mixture In the (typical) case when the ingoing atmosphere does

not contain HCN, n¢HCN = 0 mol

At this point the initial conditions have been detailed Again, to arrive at the strategy for solving the problem, Gibbs’ phase rule is applied (see the text box provided with Example 2 in Section 1.2.6)

The phase rule applied to Example 3

The system involves only one phase, the gas phase, entailing that r = 1.

The number of dependent components (constituents) in this phase (the gaseous species) is seven, that is, CO, CO2, H2, H2O, NH3, HCN and IG These dependent components (constituents (species)) are based on, in total, fi ve different elements (C, H, O, N and IG) and consequently fi ve atom balances, defi ning the number of

(moles of) atoms of each kind (n C , n H , n O , n N and n IG), can be defi ned to link the dependent components (constituents (species)) However, only four of these atom

6 See the discussion on the effect of monatomic and diatomic (inert) gases on this number (n IG), in footnote 5 on page 17 above.

balances are independent, since the total number (of moles) of atoms remains constant The fifth atom balance can thus be found from the other four and the knowledge of the total number (of moles) of atoms This means that the number of independent

components, n = 7 – 4 = 3, without, at this stage, having considered the constraint

that ‘the number of independent chemical reactions involving the (active) components’ (see at the end of the text box above) has to be subtracted as well (see below) Inserting now these numbers into Gibbs’ phase rule, we find the number of remaining degrees of freedom to be:

f = 3 – 1 + 2 = 4

In order to solve the problem (here determination of the composition of the gas

phase in equilibrium) it is required that f = 0, that is, that there must be no remaining

degrees of freedom Then the system is fully defined and the problem is solvable

Since the intensive variables temperature (T) and the total pressure (p tot) of the atmosphere here are predefined/given, the remaining number of degrees of freedom

is reduced to f = 2.

Hence, two independent chemical reactions (in equilibrium) have to be identified

in order to chemically link all active dependent components (constituents (or species))

in the gas phase (see at the end of the text box in Example 2 and the text on pages 18–19) As a result, two corresponding equilibrium equations (the equilibrium constants) that mathematically link all (active) gas constituents can be formulated

and thereby the number of degrees of freedom has been reduced to f = 0.

Detailing the relationships

The atmosphere parcel considered cannot exchange mass with its surroundings This means that the mass, or number of moles of atoms, of each element contained in the atmosphere parcel, remains unchanged during the reactions towards equilibrium For the equilibrium gas species then the following five mass balance equations hold:

above, only four of these mass balances are independent since n∑ = n C + n H + n O +

n N + n IG , where n∑ represents the total number of moles of atoms However, instead

of introducing this additional parameter and replacing one of the mass balances above with this relationship, the use of the five relationships [Ia–IVa and V] given above is chosen, realising that only four of these are independent The values for

these fi ve atom balances (and thereby also n∑) are known from the ingoing gas atmosphere composition and have been specifi ed earlier At this stage, our system

has four remaining degrees of freedoms, f = 4 (see above).

With specifi ed values for the two intensive state variables temperature (T) and total gas pressure (p tot), the number of degrees of freedom is further reduced to

f = 2.

In the general case, the total number of moles of gaseous species, n tot (not to be

confused with the constant total number of moles of atoms, n∑), contained in the

atmosphere parcel, will be variable (it can increase or decrease) depending on the

character of the governing gas-exchange reactions leading towards equilibrium The

total number of moles of gaseous species at equilibrium, n tot, is thus unknown, but complies with the following equation:

n tot = n CO + n CO2 + n H2 + n H2O + n NH3 + n HCN + n IG,g

This equation makes clear that in total seven unknowns have to be determined to

defi ne the equilibrium: n CO , n CO2 , n H2 , n H2O , n NH3 , n HCN and n IG,g This means that seven mathematical relationships, linking these unknowns, are required in order to solve the problem Besides the relations [Ia–IVa and V] given above, two additional relationships are therefore required

In the furnace, the active gaseous species may react towards equilibrium according

to different gas-exchange reactions, such as the water–gas reaction, and different gas-exchange reactions involving ammonia (NH3) and hydrogen cyanide (HCN) Although several reactions/equilibria may be defi ned, Gibbs’ phase rule commands that, at this stage, only two degrees of freedom are left and therefore (see at the end

of the text box in Example 2 and the text on pages 18–19) only two, independent, gas-exchange reactions/equilibria involving (here) all (six) active species can be defi ned (consideration of further additional gas-exchange reactions/equilibria will therefore not add any new, i.e independent, information) This will bring the number

of degrees of freedom to f = 0

The freedom therefore exists to choose two independent chemical reactions (in

equilibrium), to bring f to 0, out of the minimal set of reactions (in equilibrium),

that together comprise all active components (here this minimal set comprises two reactions (see what follows), which is just the number of reactions needed (this is different for Example 4 presented in Section 1.3.7)) Firstly, the water-gas exchange reaction ‘WGS’, i.e Reaction [1.13], is taken here:

with the equilibrium constant:

In this case the variable n IG,g only appears in Eq [V] For the inert gas, the mass balance relation in Eq [V] is trivial since IG does not take part in any reaction and

thus remains invariant This means that n IG,g ∫ n¢ IG,g , where n¢IG,g and n IG,g are the ingoing and equilibrium number of moles of inert gas, respectively, such that Eq

[V] can be disregarded and the known value of n IG,g can be used in the solution process

Further, the two gas-exchange reactions considered here do not involve any change

in the number of moles of gas This means that the volume of the atmosphere parcel remains unchanged (at any given reference temperature) upon establishment of the

equilibria The consequence of this is that n tot ∫ n¢ tot and thus not only does n tot not appear in the expressions for the equilibrium constants (Eqs [VI] and [VII]; this was different in Example 2: cf Eq [VIb] in Example 2), but the known value of

n tot can be directly used in the variant of the solution process considered here (see Eqs [Ib–IVb] below)

Now, since n IG ∫ n¢ IG (see above), four atom-balance equations ([Ia–IVa]) remain, where the gas species are expressed as number of moles, and two equations ([VI] and [VII]) where the gas species are expressed as partial pressures Therefore it must be decided if partial pressures or number of moles are used during the solution process Here, different from the treatment followed in Example 2, partial pressures are used and thus now the number of moles is converted into partial pressures using the relation (valid for ideal gas species):

Solving the problem

Now six unknowns, p CO , p CO2 , p H2 , p H2O , p NH3 and p HCN, occur and six equations [Ib–IVb] and [VI] and [VII] are available So the problem is solvable However, to solve a system of six, in part, nonlinear equations, is not trivial It is, though, possible

to express four of the unknowns as a function of the remaining two unknowns, using

the linear equations [Ib–IVb] This leaves a nonlinear system of two equations and

two unknowns, which can be solved using iterative techniques For problems like this one, powerful computer-based solvers are available in different mathematical toolboxes

Numerical examples

Consider an ingoing base gas atmosphere, intended for nitrocarburising, with the following composition (volume percent): 5% CO, 5% CO2, 10% H2, 2% H2O and 35% NH3, with the balance consisting of N2 (which here is treated as inert) at given

temperature (T) of 580°C (853K) and given total pressure (p tot) of 1 atm

• The equilibrium partial pressures of the gas species of a nitrocarburising atmosphere are shown in Figure 1.4(a) as a function of the added amount of ammonia (NH3) at 580°C (853K) and at a constant total pressure of 1 atm The diagram also shows the carbon and nitrogen ‘activities’ (see below Eq [1.6a]) imposed by the atmosphere on a substrate surface (see Section 1.3).

• The equilibrium partial pressures of the gas species for a nitrocarburising atmosphere are shown in Figure 1.4(b) as a function of the addition of carbon monoxide (CO) at 580°C (853K) and at a constant total pressure of 1 atm Again, the carbon and nitrogen ‘activities’ (see below Eq [1.6a]) imposed by the atmosphere on a substrate surface, are shown as well.

1.3 Equilibria for gas–solid reactions

In this section the equilibria for the gas–solid reactions of importance for mass transfer of nitrogen and carbon during nitriding, nitrocarburising, carburising and carbonitriding will be reviewed The thermodynamic relations, by which these equilibria can be described, are the same as those given in Section 1.2 and the reader

is referred to this section for details on the theoretical basis

It should be noted that when a nitrogen or carbon activity is mentioned in the text, this pertains to the activity value imposed on the substrate M at its surface by the particular gas–solid reaction under discussion in the text and is then denoted by

a N,s or a C,s, respectively In some cases a more detailed notation is used to avoid ambiguity For example, in Section 1.3.1, when discussing the nitrogen activities imposed on the substrate by the two nitriding reactions [1.19] (reaction ‘N2(s)’) and [1.21] (reaction ‘NH3(s)’), the more elaborate notations N s

Figure 1.4 (a) Example of the equilibrium partial pressures for selected gas components (note:

logarithmic scales) in a nitrocarburising atmosphere under the specifi ed process conditions as a

function of the amount of added ammonia The diagram also shows the carbon ‘activity’, a C,s

(relative to graphite at the temperature concerned) and the nitrogen ‘activity’, a N,s (relative to nitrogen gas at 1 atm at the temperature concerned) imposed by the atmosphere on a substrate surface It should be noted that the partial pressures for the gas components not shown in the

diagram (p H2 ª 0.0953 atm, p H O H O H O22 ª 0.0260 atm and p CO2ª 0.0453 atm) only exhibit small

changes, while N2, being the balancing gas (p N2 = 0.430 atm), shows a gradual decrease,

with increasing volume fraction of ingoing NH3 The numeric values given here are valid for an ammonia addition of 35 vol.%; (b) example of the equilibrium partial pressures for selected gas components (note: logarithmic scales) of a nitrocarburising atmosphere under the specifi ed process conditions as a function of the addition of carbon monoxide The diagram

also shows the carbon ‘activity’, a C,s (relative to graphite at the temperature concerned) and

the nitrogen ‘activity’, a N,s (relative to nitrogen gas at 1 atm at the temperature concerned) imposed by the atmosphere on a substrate surface It should be noted that the partial pressures for the gas components not shown in the diagram (p H2 ª 0.0953 atm, p H O H O H O22 ª 0.0260 atm

and p CO2 ª 0.0453 atm) only exhibit small changes, while N2, being the balancing gas (pN2

= 0.430 atm), shows a gradual decrease, with increasing volume fraction of ingoing CO The numeric values given here are valid for a carbon monoxide addition of 5 vol.%.

1.3.1 Nitrogen transfer by ammonia

The nitriding of the solid M in a NH3/H2 gas mixture can formally be conceived

as the result brought about by N2 gas in contact with M under a certain pressure This statement is derived from the Gibbs energy (and thus the chemical potential) being a state variable, which means that the path to achieve a certain state is without importance for the value of the Gibbs energy in this state Therefore, the Gibbs energy change involved in nitriding in a NH3/H2 gas mixture according to reaction [1.21]

is identical to the Gibbs energy change for the path followed by the (hypothetical) reactions [1.19] and [1.20]:

been established In theory this means that a homogeneous solid phase is everywhere

in equilibrium with a homogeneous gas Of course, for a gas–solid reaction this is not possible In practice, a so-called (closely approached) ‘local equilibrium’ can occur

at the workpiece surface,7 i.e the interface between the gas and the solid phase The equilibrium constant for the equilibrium [1.19] is:

is achieved for the interstitial components only, but not for the substitutional components, as

a consequence of distinct differences in mobility between these components This defi nition, with this background, is not applicable at all to gas–metal interactions.

% CO (b)

and for equilibrium [1.21]:

3

3

2 3

· (p · (p )0 –1/))))–1/ 2 2 a a p p · (p

N denotes the nitrogen activity in the solid substrate at its surface

as imposed by the N2(s) reaction, Reaction [1.19], under the given (hypothetical) conditions (temperature and pressure) and NH

s a

3 ( ) ( )s s

,

N is the nitrogen activity in the solid substrate at its surface as imposed by the actually occurring NH3(s) reaction, Reaction [1.21] Since the three Reactions [1.19]–[1.21] are assumed to be in equilibrium, the nitrogen activities imposed on the (iron) substrate by Reactions [1.19] and [1.21] are identical

The numerical values for the equilibrium constants are independent of the unit selected for indicating the value of the gas pressures, but do depend on the reference state adopted for nitrogen dissolved in the (iron) substrate (see further below) The pressure of the (hypothetical) N2 gas (as present in Reactions [1.19] and [1.20]) can thus be calculated from the equilibrium equation [1.20a]:

where p p p p p p p NH22//p p p p p p p H3/22 pertains to the (actually occurring) equilibrium [1.21]

In accordance with Eq [1.4a] and Reaction [1.19]:

( )

N s,

N s,

It is now the moment to select the value for m0

N,s and to do this in such a way (we

do have this freedom; see discussion of Eqs [1.1] and [1.2]), that:

mmmmmmmmmmmmmmmmmmmmN s N s N s N s N s N s0 0 , , , ∫ 1 1 2 2mmmmmmmmmmmmmmmmmmmmN g N g N g0 0N g N g2, [1.25]Equation [1.25] only requires that the reference state of the nitrogen dissolved in the (iron) matrix is equated with the reference state of nitrogen gas We now select as reference state: nitrogen gas at 1 atm and at the temperature concerned, leading to

the equilibrium constant (Eq [1.19a]) for Reaction [1.19], K N2(s) ∫ 1, independent

of temperature (see Table 1.2)

Consequently, Eqs [1.24] and [1.25] yield:

Often, particularly in the German literature, the nitriding potential is denoted as

K N: ‘Nitrierkennzahl’ This is an unfortunate name: the nitriding potential is not

a ‘number’ (‘Zahl’ in German), suggesting it to be dimensionless, but is rather an intensive state variable with a pressure-related unit (for example, [atm–1/2], or [bar–1/2]

are often used in practice) Therefore, and also because the variable K N is used in

another meaning in Sections 1.4 and 1.5, in this text the notation nitriding potential will be used as well as its symbol r N (compatible with most English literature) Note that the ‘nitriding potential’ should not be confused with the ‘nitrogen potential’,8

also used in applied heat treatment processing

Considering that the equilibrium constant for the N2(s) reaction (Reaction [1.19]),

K N2(s) = 1, the numerical value for the nitrogen activity, NH3(s) a N,s, according to Eq [1.27a] can be interpreted as the square root of the pressure of the hypothetical nitrogen gas occurring in Reactions [1.19] and [1.20], which, at a given temperature, would bring about the same solubility of nitrogen in the (iron) matrix as ammonia according to Reaction [1.21], that is, NH3(s) a N,s = (p N2 /p0)1/2 It therefore becomes clear that, an activity of N in M, as defi ned here, can attain numerical values (much) larger than 1 It can also be concluded that at a constant temperature (and

therefore constant K NH3(g) and K NH3(s)) the nitrogen activity, and consequently also the nitrogen content, at the surface of the nitrided surface layer, is determined by the nitriding potential

The selection of nitrogen gas at 1 atm and the occurring temperature as reference state is entirely arbitrary It has been done to indicate that under normal nitriding conditions, nitrogen gas is the most stable form of nitrogen For example, compounds such as Fe4N1–x (g¢ iron nitride) or Si3N4 could just as well have been selected as reference state for nitrogen This would have dramatic consequences for the numerical

values for a N,s!

8 The concept ‘nitrogen potential’, N pot, is used in practical (austenitic) carbonitriding (and austenitic nitriding) This quantity is a property of the nitriding atmosphere, defi ned as the mass percentage of nitrogen taken up by a thin foil of pure iron upon equilibration with the atmosphere.

For equilibrium between nitrogen gas at 1 atm and pure iron (a N,s = 1), ferrite can dissolve at most (at 900°C (1173K)) about 0.004 mass-% nitrogen and austenite can dissolve at most (also at 900°C (1173K)) about 0.031 mass-% nitrogen (these two points (composition, temperature) represent an eutectoid reaction between ferrite,

austenite and nitrogen gas (1 atm) in the stable phase diagram for the Fe-N system)

The solubility of a component at constant chemical potential (activity) varies from phase to phase and depends on the temperature The activity corresponding to

nitrogen gas of 1 atm (a N,s = 1) is not suffi cient to produce g¢-Fe4N1-x and e-Fe2N1-y

at the surface of iron workpieces To this end, a nitriding medium corresponding to

a N,s >> l in the substrate is necessary An NH3/H2 gas mixture is such a medium (as are cyanide/cyanate salt baths, nitrogen plasmas, etc.) However, it should be noted here that, at the normal nitriding temperatures (and at 1 atm), thermal decomposition

of the ammonia at equilibrium is almost complete (see Figure 1.1) As a consequence

of the slow kinetics of this thermal dissociation in the gas phase, in combination with

a suffi ciently high gas exchange rate, signifi cant nitriding potentials (Eq [1.28]) can still be achieved in the furnace atmosphere (see further the discussion in Sections 1.4 and 1.5 and particularly Figure 4 in Mittemeijer and Somers, 1997) In this sense gaseous nitriding should be considered a (fortunate) by-product of the slow ammonia dissociation, which is catalysed at the surface of the workpiece and at other surfaces

inside the furnace The well known metastable phase diagram for the Fe-N system (cf the above remark on the stable Fe-N phase diagram) could thus be determined

and in fact pertains to equilibria (at the surface of the solid, nitrided iron substrate) with such a NH3-H2 gas mixture of variable chemical potential (see Mittemeijer and Somers, 1997; and Mittemeijer, 2013)

1.3.2 Nitrogen transfer by nitrogen gas

As indicated in the previous section, also molecular nitrogen gas can interact with the solid phase M (e.g., ferritic iron) and thereby transfer of atomic nitrogen from the gas phase into the surface of the solid can occur, and vice versa This reaction has already been named ‘N2(s)’ (see previous section), where (s) indicates that a gas–solid reaction is concerned:

N s( )

where a N,s denotes the nitrogen activity (complete notation, N2(s) a N,s) in the solid substrate at its surface as imposed by the N2(s) reaction [1.19] under the given conditions (temperature and pressure) As shown in the previous section (see discussion

with respect to Eq [1.26]), this equilibrium constant is by defi nition K N2(s)∫ 1 and

is independent of temperature (see Table 1.2)

In the previous section, this reaction was given a high nitrogen activity via a

hypothetical high nitrogen gas pressure Under actual and more typical, lower, (partial)

pressures, for example p N2 ≤ 1 atm, the nitrogen activity of Reaction [1.19] will be much lower than what can be achieved with (non-equilibrium) ammonia-hydrogen gas mixtures Indeed, using pure nitrogen gas at an arbitrary pressure and at any temperature, it follows from Reaction [1.19] that its nitrogen activity is given by

a N,s = (p N2 /p0)1/2, which under the indicated conditions (p N2 ≤ 1 atm) is lower than,

or equal to, unity (cf Figure 1.1)

In nitriding, nitrocarburising and carbonitriding atmospheres the activity of nitrogen in the iron substrate at its surface typically can be given a high value in reactions with gases such as ammonia (Reaction [1.21]) and hydrogen cyanide (Reaction [1.49]) As discussed above, equilibrium with nitrogen gas (cf Eq [1.19]) normally corresponds with a low value for the nitrogen activity in the iron substrate

at its surface and therefore this gas–solid exchange reaction rather tends to remove nitrogen from the iron surface (de-nitriding/nitrogen desorption) by the formation

of molecular nitrogen gas along the direction from the right to the left of Eq [1.19] The corresponding rate of nitrogen loss is low, but not negligible for temperatures larger than, say, 460°C (733 K), also depending on the nitrogen content of the solid Particularly at carbonitriding temperatures (>800°C (>1073 K)) the effect of nitrogen formation (nitrogen desorption) is substantial This effect is discussed in more detail

in Section 1.5

1.3.3 Carbon transfer via the Boudouard reaction

Carburising of the solid phase M in a CO/CO2 gas mixture, Reaction [1.31], can formally be conceived as the result brought about by graphite (CGr) in contact with

M This statement is a consequence of the Gibbs energy (and thus the chemical potential) being a state variable, implying that the Gibbs energy of a (final) state

is independent of the path followed to arrive at this state (cf the beginning of Section 1.3.1) Therefore, carburising in a CO/CO2 gas mixture, in accordance with Reaction [1.31], can be conceived as the sum of the (hypothetical) Reactions [1.29] and [1.30]:

at the workpiece surface

The equilibrium constants of equilibria [1.29]–[1.31] are:

given (hypothetical) conditions (temperature and pressure), and therefore:

p Boud

K Boud K

C s a a C s p p CO CO Gr

The numerical values for the equilibrium constants are independent of the unit selected for indicating the value of the gas pressures but do depend on the reference state adopted for carbon dissolved in the (iron) solid substrate (see further below) The activity of the (hypothetical) graphite (as present in Eqs [1.29a] and [1.30a]) can thus be calculated from the equilibrium [1.30]:

CO /p CO2 follows from the (actually occurring) equilibrium [1.31]

In accordance with Eq [1.4a] and Reaction [1.29]:

It is now the moment to select the value for m0

C,s and to do this in such a way (we

do have this freedom; see the discussion of Eqs [1.1] and [1.2]) that:

equilibrium constant (Eq [1.29a]) for Reaction [1.29], K CGr(s) ∫ 1, independent of temperature (see Table 1.2)

Consequently, Eqs [1.34] and [1.35] yield:

where the right-hand equality often is written as:

where Boud r C is defi ned as the carburising potential (with the unit [pressure]), imposed

by the Boudouard reaction [1.31], as follows:

to the role of the hypothetical nitrogen gas in Eqs [1.19a] and [1.20a] The numerical value of the carbon activity according to Eq [1.37a] can then be interpreted as the activity of the hypothetical graphite that, at the same temperature but at a different pressure, would bring about the same solubility of carbon in the (iron) substrate

It follows that the thus defi ned activity of C in M can attain numerical values both larger and smaller than 1 It can also be said that, at a constant temperature (and

therefore constant K Boud/CGr and K Boud), the carbon activity, and consequently the carbon content, in the carburised component at its surface are determined by the carburising potential

The pressure of the hypothetical N2 gas (as present in Eqs [1.19] and [1.20])

is normally not equal to the pressure of the reference state of N2 gas (p0 = 1 atm) (see Eq [1.22] and the discussion under Eq [1.28]) Likewise, the pressure of the hypothetical graphite (as present in Reactions [1.29] and [1.30]) is normally not

equal to the pressure of the reference state of graphite (p0 = 1 atm) (see Eq [1.32] and the above discussion) Synthetic diamond manufacturing involves a practical application of activity change of graphite by means of pressure change Subjecting graphite to a pressure of approximately 70000 atm at approximately 2000 K leads to

9 Unlike the nitriding potential used in nitriding and nitrocarburising, the notion carburising potential has not found application in industrial carburising Instead the ‘carbon potential’ is widely used in practical carburising and carbonitriding (see Chapter 13) This quantity is a property of the carburising atmosphere, defi ned as the mass percentage of carbon taken up by a thin foil of pure iron upon equilibration with the atmosphere (see also footnote 8 on page 32).

a value of a C,Gr of about 7.7 (Mittemeijer and Slycke, 1996) Although it is possible

to vary the carbon activity by altering the pressure of the graphite, if this had been chosen as carburising agent, this is not a method of general practical interest It is much more convenient to control the carbon activity through the composition of CO/

CO2 gas mixtures (Reaction [1.31]), or CO/H2/H2O gas mixtures (the heterogeneous water–gas reaction; see Reaction [1.39] and Section 1.3.4) at (or close to) ambient pressure (1 atm)

The selection of graphite at 1 atm at the occurring temperature as reference state

is entirely arbitrary It has been done to indicate that under normal carburising conditions, graphite is the most stable form of carbon For example, compounds such as Fe3C (cementite), or SiC, could equally well have been selected as reference state for carbon This would lead to considerable changes of the numerical values

for a C,s

In the equilibrium between graphite at 1 atm and carbon dissolved in iron

(a C,Gr ∫ CGr(s) a C,s = 1, cf Reaction [1.29]), ferrite can dissolve at most (at 738°C (=1011 K)), about 0.018 mass-% C, and austenite can dissolve at most (at 1154°C (=1427 K)), about 2.03 mass-% C (Mittemeijer and Slycke, 1996) The solubility of a component at constant chemical potential/constant activity varies from phase to phase and depends on the temperature The activity of graphite at 1 atm

(a C,Gr = 1) in equilibrium with a pure iron surface (CGr(s) a C,s = 1) is not suffi cient

to produce cementite (the iron carbide, Fe3C) or other iron carbides on the iron

surface To this end, a carburising medium imposing a carbon activity a C,s > l at the iron surface is required A CO/CO2 gas mixture is such a medium (as are cyanide/cyanate salt baths, hydrocarbon gases, carbon plasmas, etc.) However, it should be noted here that at 1 atm, in principle, graphite (soot) formation according to Eq [1.30a] cannot be excluded If the gas-exchange rate for the CO/CO2 gas mixture

is suffi ciently high with respect to the kinetics of the establishment of equilibrium [1.30] at 1 atm, considerable carburising potentials (Eq [1.38]) can be obtained in the furnace atmosphere This is further discussed in Section 1.5

1.3.4 Carbon transfer via the heterogeneous water–gas

reaction

In gaseous nitrocarburising, carburising and carbonitriding the atmospheres do contain (besides other species) a mixture of H2, H2O, CO and CO2 In such atmospheres another important carburising reaction, the heterogeneous water–gas reaction (here called ‘HWG’) can occur:

where a C,s here has been used to denote the carbon activity (complete notation:

HWG a C,s) imposed by the HWG reaction Accordingly:

The heterogeneous water–gas reaction can be seen as a combination of the Boudouard

reaction [1.31] and the water–gas shift (WGS) gas-exchange reaction [1.9] (also

called homogeneous water–gas reaction), and its equilibrium constant can thus be given as: K HWG = K Boud /K WGS In thermochemical processing the HWG reaction proceeds much faster than the Boudouard reaction Hence, in the gas atmospheres considered, carburising proceeds preferentially according to the HWG reaction [1.39] and the associated carbon activity follows from Eq [1.40] and is (in the presence

of hydrogen and water vapour) not given by [1.37a] or [1.37b]

1.3.5 Carbon transfer by hydrocarbons

Most atmospheres for nitrocarburising, carburising and carbonitriding contain more

or less methane, CH4 This component is often a residue from the atmosphere generation process and/or a result of addition of enrichment gases, such as natural gas, methane, or propane, for control of the carbon activity (carburising potential) Methane can transfer carbon to the solid phase along the following reaction (here called ‘CH4(s)’, where (s) indicates that a gas–solid reaction is concerned):

where a C,s here has been used to denote the carbon activity (complete notation:

CH4(s) a C,s) imposed by the CH4(s) reaction The temperature dependence of the equilibrium

constant K CH4(s) for Reaction [1.44] is given in Table 1.2 Accordingly:

is the carburising potential (with the unit [pressure–1]) for the methane gas–solid reaction

As discussed in Section 1.5, the methane gas–solid reaction is quite slow at typical thermochemical processing temperatures This means that Reaction [1.43] will usually not attain equilibrium The methane content in, for example, a carburising atmosphere is often several times higher than what could be expected on the basis

of the prevailing carbon activity (e.g as governed by the heterogeneous water–gas reaction, Eq [1.40]; see end of Section 1.3.4)

For gaseous carburising (and occasionally nitrocarburising) processes acetylene (ethyne, C2H2) or other unsaturated hydrocarbons, generally denoted CmHn, have been shown to have carburising ability For atmospheres based on CmHn–H2 gas mixtures the carburising reaction (here called ‘CmHn(s)’, where (s) indicates that a gas–solid reaction is concerned) can be formulated as:

where a C,s here has been used to denote the carbon activity (complete notation:

CmHn(s) a C,s) imposed by the CmHn(s) reaction As in the previous sections, equilibrium equations can be formulated for different hydrocarbon gases, for defi nition of carburising potentials and activities Gas–solid reactions such as [1.47] are normally not used for process control of carburising atmospheres When a hydrocarbon is used as carburising medium in practice, for example in low-pressure (‘vacuum’) carburising, control of the thermodynamic parameters governing the carburising is not exercised Instead, the process is governed by mass-transfer control

1.3.6 Carbon and nitrogen transfer by hydrogen cyanide

As mentioned in Section 1.2.4, hydrogen cyanide (HCN) will form in atmospheres containing species such as carbon monoxide (CO) and ammonia (NH3) Hydrogen cyanide has in principle the ability to transfer both carbon and nitrogen from the gas to the solid at the iron surface during nitrocarburising and carbonitriding, in accordance with the following reaction (here called ‘HCN(s)’, where (s) indicates that a gas–solid reaction is concerned):

activities (complete notations: HCN(s) a C,s and HCN(s) a N,s, respectively) imposed by the HCN(s) reaction The temperature dependence of this equilibrium constant is given in Table 1.2 The reaction kinetics for this reaction is discussed in Section 1.5.6.

1.3.7 Example

Equilibrium calculations, leading to definition of the composition of a, to be composed

or resulting, gas atmosphere on the basis of the thermodynamics for the important gas–solid reactions involved, are essential for process control This is illustrated in the following example

Example 4: Equilibrium composition of a carburising atmosphere with

Task

The determination of the equilibrium (composition) for a carburising atmosphere

based on an ingoing gas mixture with known (fixed) volume fractions of CO, CO2

H2, H2O, CH4 and with an inert gas (such as nitrogen gas; note that N2 is in practical heat treating often seen as non-reactive, which strictly speaking is not correct, see Section 1.3.2) as the balance, has been dealt with in Example 2 in Section 1.2.6

This equilibrium gas atmosphere imposes a certain carbon activity (a C,s) in an iron

substrate at its surface, at a given temperature (T) and total pressure (p tot) Starting

from this situation, in order to impose a prescribed a C,s , the composition of the ingoing gas atmosphere can be changed by, as one possibility, small additions of (‘enrichment’ with) either a hydrocarbon or an oxidising agent This reflects what

is done in commercial practice The task in this example, then, is, for a given

a C,s, (i) to determine the type and to calculate the amount of the enrichment gas component to be added and (ii) to calculate the composition of the final equilibrium gas atmosphere

The desired variation of the composition of the ingoing atmosphere, as realised (see above) with addition of either a small amount of a hydrocarbon or a small amount

of an oxygen-containing gas medium (e.g., air or pure oxygen), is realised, in this example, with the possible addition of either (extra) CH4 or O2 (i.e the addition

of (extra) CH4 and the addition of O2 are handled as mutually excluding!) Which one of the enrichment gases to use and in what amounts depend on the ingoing gas composition and the specified operating conditions and are not known until the calculation of the final equilibrium state has been made During the calculation the amount (and type) of enrichment gas is therefore unknown, which has an impact on the amount of balancing inert gas, since it, obviously, also depends on the amount of added (enrichment) gas (see below) In the following calculation a special precaution

is made to express the mutually excluding nature of the addition of either (extra)

CH4 or O2

The partial pressure, p¢I , for an ingoing gas species, I, is (for an ideal gas) proportional to its volume fraction, u I¢, as follows:

p I¢ = u¢I · p tot

The ingoing atmosphere can therefore be characterised by the following partial

pressures: p¢CO , p CO2¢ , p¢H2 , p H2O¢ , p CH4¢ , p IG,g¢ and p¢XCH4 , or p XO2¢ Here subscript IG,g

indicates the inert gas and the prime signs indicate that the ingoing (non-equilibrium) partial pressures are concerned Again: the addition of (extra) CH4 and the addition

of O2 (corresponding with p XCH4¢ and p¢XO2) are mutually excluding As such, the

amounts of these additions, as expressed by p¢XCH4 and p XO2¢ , are unknowns, but for any given situation, one of them is always zero, while the other one must be determined This means that, while the partial pressures of the ingoing active species (except for the added (enrichment) gas component) are fi xed, the partial pressure

of the ingoing inert gas, p IG,g¢ , depends on the amount of enrichment gas needed to fulfi l the carbon activity requirement

The partial pressures of the species in the atmosphere parcel considered can be converted to number of moles for each of the ingoing gas species, using the relation (valid for ideal gases):

where n I¢ is the ingoing number of moles of gas constituent I and n tot¢ is the total

number of moles of ingoing gas in the atmosphere parcel studied For n tot¢ one can choose any value It is convenient to consider one mole of ingoing gas, that is,

in the ingoing gas mixture Here, the value of n IG,g¢ depends on the type and amount

of added (enrichment) gas component and complies with the following equation:

n IG,g¢ = n tot¢ – n CO¢ – n CO2¢ – n¢H2 – n H2O¢ – n¢CH4 – n XCH4¢ – n¢XO2

where, as stated, one of the variables n XCH4¢ and n XO2¢ always is zero, while the other

10 See the discussion on the effect of monatomic and diatomic (inert) gases on this number

(n IG), in footnote 5, on page 17.

takes a certain (to be determined in the calculation) value, larger than zero It should

be noted that none of the four atom mass balances given above is numerically defined until the type and amount of added (enrichment) gas component is known (i.e., determined by the following calculation)

Since it is unknown, at this stage of the calculation, which (enrichment) gas component (of the two: CH4 and O2) and what amount (either n¢XCH4 or n XO2¢ ) has to

be added, an ‘enrichment variable’, n X¢, now is introduced for which it holds:

for n X¢ > 0: n¢XCH4 = n¢X and n XO2¢ = 0;

for n X¢ < 0: n¢XCH4 = 0 and n¢XO2 = –n X¢;

These relations can be written as n¢ XCH4 = max(n¢ X , 0) and n¢ XO2 = max(– n¢ X, 0)

respectively, and are valid for all values of n¢ X (whether positive or negative) Evidently, if O2 is the gas component to be added, n¢ X must be negative The function

max(value1, value2) returns the largest of the two arguments This simplifies the

solution procedure in a (computer) algorithm, as shown below The ingoing atom balances can then be rewritten as:

n C = n CO¢ + n CO2¢ + n CH4¢ + max(n X¢, 0)

n H = 2n H2¢ + 2n H2O¢ + 4n CH4¢ + 4 · max(n X¢, 0)

n O = n CO¢ + 2n CO2¢ + n H2O¢ + 2 · max(–n X¢, 0)

n IG = n¢ IG,g

where the value of n IG,g¢ depends on the amount of enrichment gas (expressed as the

absolute value of the enrichment variable n X¢), as follows:

n IG,g¢ = n tot¢ – n CO¢ – n CO2¢ – n¢H2 – n H2O¢ – n¢CH4 – |n¢X |.

At this point the initial conditions have been detailed Similarly as in Examples

2 and 3 in Section 1.2.6, now Gibbs’ phase rule is used to arrive at the strategy for solving the problem (see the text box provided with Example 2 in Section 1.2.6)

In the following, the treatment first aims at determining the composition parameters

of the equilibrium for a fixed value of a C,s for the case that the composition parameters

of the gas at the inlet are known, i.e., also n X¢ is, for the moment, dealt with as a

known parameter

The phase rule applied to Example 4

The problem aims at finding the composition of the gas phase in equilibrium with carbon in solid solution in the surface of a solid substrate, e.g an iron phase In this case there are two phases (gas and solid) involved in the problem Thus, for the

calculation considered, r = 2.

The number of dependent components (constituents, or species) involved in the equilibrium between the gas phase (the gaseous species) and the solid phase (carbon in solid solution), is seven: six in the gas phase, CO, CO2, H2, H2O, CH4