Sha W., Malinov S. Titanium Alloys: Modelling of Microstructure, Properties and Applications

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Nitriding: modelling of hardness profiles and the kinetics 513

for the prediction of microhardness profiles with high accuracy within the

range of the data used for the models.

The influence of the nitriding processing parameters on the microhardness

profiles, calculated using both models, is explained from a metallurgical

point of view, showing good agreement with the fundamental diffusion theory

and the data from the literature.

The results obtained with the models are quite similar so it can be assumed

that they both may be used for simulations of microhardness profiles with

sufficient accuracy. The first model can potentially be used effectively for

optimisation of the processes of gas and plasma nitriding and it can predict

microhardness profiles of titanium alloys, nitrided at temperatures between

700 and 1100 °C for periods of time in the range of 1–100 hours. The second

model can be used for the same purpose but only for gas nitriding in the

same temperature and time ranges.

These models can be used by titanium users, saving them both experimental

time and cost. Similar models can be created for the simulation and prediction

of different materials’ properties and characteristics.

18.2 Kinetics of gas nitriding

In this section, the objective is to model the kinetics of formation and growth

of nitrided layers in commercial titanium alloys under different thermodynamic

conditions. The modelling can help in optimisation of the processing parameters

in order to achieve a desirable combination of microstructure and properties.

The gas nitriding of titanium and titanium alloys is a diffusion process.

Hence, the kinetics of formation and growth of the surface nitrided layer can

be modelled by appropriate application of the fundamental diffusion theory.

For this purpose, analytical and numerical solutions are introduced, based on

the physical model discussed below.

18.2.1 Phase transformations during the

process of nitriding

The formation of nitrided layers in titanium alloys is a complicated process

and involves several reactions taking place simultaneously at the boundary

between the gas atmosphere and the metal and within the substrate. The

kinetics of the diffusion process of nitriding have been studied in the past. A

simplified physical model for the formation and growth of nitrided layers

during gas nitriding in titanium is suggested and described by Malinov et al.

(2003). The model is based on reaction diffusion rules and is applicable for

nitriding temperatures below the β-transus. If the titanium material is in an

active nitrogen containing environment at high temperature, a nitrogen mass

transfer from the medium to the solid occurs. The nitrogen absorbed at the

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Titanium alloys: modelling of microstructure514

surface diffuses into the titanium, forming an interstitial solution of nitrogen

in the hcp α titanium phase (Fig. 18.11 top). The surface layer formed is

called the diffusion zone, α(N). This process can continue as long as the α

titanium matrix can dissolve nitrogen at the nitrogen medium–solid interface,

where the nitrogen concentration is the highest. If the concentration of nitrogen

at the interface becomes higher than the α phase is able to retain in interstitial

solution, a reaction at the interface occurs leading to the formation of a new

phase – Ti

N (Fig. 18.11 middle). There is a concentration jump of nitrogen

at the surface and, as a result, the total nitrided layer consists of a compound

layer (Ti

N) on the top and a diffusion zone underneath. Following the same

rules, when the concentration of nitrogen at the interface becomes higher

than the one acceptable in Ti

N, there is a phase transformation at the surface

and the Ti

N transforms to TiN (Fig. 18.11 bottom). The sub-layer with

titanium nitrides only (TiN and Ti

N) forms the compound layer, while α(N)

is the diffusion zone.

For real cases of nitriding of titanium alloys, the alloying elements present

can cause different deviations and modifications. The presence of alloying

elements in the general case might result in:

• simultaneous formation of two or more titanium nitrides;

TiN

Compound

layer

Diffusion zone

Increasing thickness

Reduced nitrogen concentration

Base

material

Time of nitriding

α(N)-Ti α-Ti

α-Ti

α(N)-Ti

18.11

A schematic presentation of the kinetics of formation and

growth of surface layers during gas nitriding of titanium.

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Nitriding: modelling of hardness profiles and the kinetics 515

• formation of complex stable or metastable nitrides of the alloying elements;

• change of the composition ranges of existence of the different phases.

Despite these possible influences, the alloying additions in the classical

titanium alloys are small in quantity and usually are dissolved in the hcp α-

titanium matrix by forming a substitutional solid solution. Hence, dramatic

changes to the kinetics of formation and the phase compositions of nitrided

layers in titanium alloys are not very likely. This has been confirmed by

experimental studies of the microstructure and the phase compositions of

nitrided layers in different titanium alloys (Lakshmi et al., 2002).

18.2.2 Physical model

In most of the cases, the main phases observed on the surface after nitriding

of different titanium alloys at different thermodynamic conditions are TiN

and Ti

N (Chapter 16) (Shashkov, 2001; Song et al., 2002; Spies et al.,

2001). Hence, the model suggested in Section 18.2.1 can be adopted with

sufficient accuracy to model the formation of nitrided layers in titanium

alloys. The development of the mathematical model based on this physical

model for solving the diffusion equation in appropriate conditions will allow

quantitative simulations of the kinetics of nitride layer formation under various

processing conditions. The mathematical models are discussed in the following

sections.

It should be admitted that the presence of TiO

is not taken into account

in the models. The oxygen may have significant influence on the kinetics of

formation and growth of nitrided layers and should be a subject for further

studies. At present, it is difficult to model simultaneous formation of titanium

nitrides and titanium oxides. The models will work with better accuracy

when there is no formation of oxides or the oxygen is a very small amount.

18.2.3 Diffusion coefficients

Precise data on the diffusion coefficients are necessary for accurate

mathematical modelling of any diffusion process. A literature survey on the

diffusion coefficients of nitrogen in α titanium, however, reveals some

discrepancies between diffusion coefficients given by different authors. Various

values of A and Q have been given, based on the general form of the equation,

Eq. [16.1].

Samsonova (1976) gave parameters for the diffusion coefficient for nitrogen

in α-Ti in the temperature range 900–1400 °C as follows: A = 1.2 × 10

–6

/sec and Q = 189.45 kJ/mole. Similar values were given by Fromm and

Gebhardt (1980) (A = 1.2 × 10

–6

/sec and Q = 178 kJ/mole) for the same

temperature interval and by Smithells et al. (1992) (A = 1.2 × 10

–6

/sec

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Titanium alloys: modelling of microstructure516

and Q = 176.9 kJ/mole) for the temperature range 900–1570 °C. Different

data for the diffusion parameters were suggested by Metin and Inal (1989),

A = 9.6 × 10

–5

/sec and Q = 214.7 kJ/mole. The diffusivity of nitrogen in

α-Ti is calculated using Eq. [16.1] and the parameters reported in various

sources (Fig. 18.12).

The calculations are carried out for a temperature range of 850–1000 °C,

the usual temperature interval for nitriding of titanium alloys. Some of the

results are in agreement, but there are also significant differences between

the diffusion coefficients suggested by the various authors. For further

comparison, the diffusivity of nitrogen in α-Ti is calculated for a temperature

of 950 °C using the data from various authors and the results are compared

Metin and Inal (1989)

Smithells

et al

. (1992)

Fromm and Gebhardt (1980)

Samsonova (1976)

× 10

–13

1.6

1.4

1.2

0.8

0.6

0.4

0.2

Diffusion coefficiet (m

/sec)

850 900 950 1000

(°C)

18.12

Diffusion coefficients of nitrogen in α-Ti according to various

authors.

Table 18.1

Diffusion coefficients of nitrogen in α-Ti at

950 °C according to different authors

Source

, m

/s (×10

–14

)

Samsonova (1976) 0.973

Fromm and Gebhardt (1980) 2.999

Smithells

et al

. (1992) 3.341

Metin and Inal (1989) 6.496

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Nitriding: modelling of hardness profiles and the kinetics 517

in Table 18.1.

There is up to six times difference between calculated diffusion coefficients

based on different sources. The modelling results will be different depending

on which one of the diffusion coefficients is used in the calculations. In the

models here, experimental data are compared with modelling results using

different diffusion coefficients.

18.2.4 Analytical solutions

Mathematical modelling of the diffusion processes in materials is carried out

by solving the fundamental differential equation of diffusion for the appropriate

initial and boundary conditions. If the diffusion is one-dimensional, i.e. there

is a gradient of concentration only along one axis (the case in nitriding of

titanium) and if the diffusion coefficient is constant, the general diffusion

equation can be simplified to:

∂

[18.1]

where C is the concentration of the diffusing element (nitrogen), t is time and

x is the space coordinate.

Both analytical and numerical approaches are used, depending on the

nature of the problem. Analytical solutions are simpler but are applicable

only for ideal initial and boundary conditions. In spite of this, they can be

applied to model different phenomena in materials science. One significant

disadvantage is that analytical solutions are difficult to find for moving

boundary problems, and where there is a point of interruption. Here, analytical

solutions are used for the first and approximate calculation of the nitrogen

gradient in the diffusion zone and the layer thickness under different

thermodynamic conditions. In order to apply a simple analytical solution, it

is assumed that there is no compound layer formed. This assumption is quite

reasonable for approximate calculations, and has experimental justification.

Experimental results with nitrided layers in titanium show that the thickness

of the compound layer is within a range of 1–20 µm, while the diffusion

zone is in the range of hundreds of micrometers, depending on the temperature

and time of nitriding. Thus, the diffusion zone contributes to more than 95%

of the entire layer. Hence, if the concentration profile and the thickness of

the diffusion zone are modelled, the results can be used for estimation of the

layer thickness. The first simple solution can be obtained assuming simple

initial and boundary conditions from first order:

C(x, 0) = C

[18.2a]

C(0, t) = C

[18.2b]

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Titanium alloys: modelling of microstructure518

The first condition, Eq. [18.2a], means that the initial concentration of nitrogen

in the alloy is C

. This value can be obtained from the alloy bulk composition

analysis. The second condition, Eq. [18.2b], implies that the concentration of

nitrogen at the surface accepts value C

from the very beginning of the

nitriding and is maintained at this constant value throughout the process. The

immediate jump of the concentration of nitrogen is not justified because, in

reality, the concentration of nitrogen on the surface gradually increases from

to C

during the early period of the process. However, the condition of

maintaining the concentration at the surface at constant value C

(after it has

been reached) is correct. It should be reiterated here that the above analytical

solution concerns only the diffusion zone. The concentration of nitrogen at

the top of the diffusion zone (at the interface between the compound layer

and the diffusion zone) is kept constant and is equal to the maximum solubility

of nitrogen in α-Ti. This value can be taken from the Ti–N phase diagram.

Solving Eq. [18.1] in initial and boundary conditions Eq. [18.2], one can

obtain:

C(x, t) = C

+ (C

– C

)(1 – erf(η)) [18.3a]

where

η =

[18.3b]

Using these equations, the nitrogen profile in the diffusion zone is calculated

after nitriding at 950 °C for 1, 3 and 5 hours. For C

, a value of 6 wt.% is

used, which is taken from the Ti–N phase diagram. Nitrogen concentration

profiles are calculated for diffusion coefficients suggested by different authors

(see Section 18.2.3) and the results are compared with experimental results

for the thickness of the nitrided layer after nitriding at the same temperature

and time (Fig. 18.13). The experimental values are obtained from microhardness

profiles.

The best correspondence between experimental and calculated values for

the layer thicknesses is after using the diffusion coefficient given by Metin

and Inal (1989). This coefficient is therefore used in the further simulations.

A more realistic analytical solution can be derived by applying boundary

conditions from third order. If the flux of nitrogen atoms across the boundary

interface (J

) is proportional to the difference between the equilibrium (C

)

and the current (C(0, t)) concentration of nitrogen at the boundary, the boundary

condition can be written as:

CCt

= –

(0, )

= ( – (0, ))

∂

[18.4]

where α is a coefficient in m/s.

This condition traces the realistic evolution of the nitrogen profile in the

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Nitriding: modelling of hardness profiles and the kinetics 519

Nitrogen concentration (wt.%)

Metin and Inal (1989)

Smithells

et al

. (1992)

Samsonova (1976)

Experimental

data ranges

Ti-8-1-1

Ti-6-2-4-2

Ti-6-4

0 50 100 150

Depth (µm)

(a)

Nitrogen concentration (wt.%)

Metin and Inal (1989)

Smithells

et al

. (1992)

Samsonova (1976)

Experimental

data ranges

Ti-8-1-1

0 50 100 150 200

Depth (µm)

(b)

Ti-6-2-4-2

Ti-6-4

18.13

Calculated profiles for nitrogen concentration in the diffusion

layer using different diffusion coefficients compared with

experimental data, for layer thickness. Nitriding temperature 950 °C.

Nitriding time: (a) 1 h; (b) 3 h; (c) 5 h.

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Titanium alloys: modelling of microstructure520

diffusion zone. The nitrogen concentration at the boundary increases gradually

from zero (or any other value) to the equilibrium concentration (C

), and is

kept at this value afterwards. These are the real conditions for the evolution

of the diffusion zone during nitriding of titanium alloys. One disadvantage of

this boundary condition is that the coefficient α is unknown. It depends on

many factors and varies significantly for different conditions. To obtain the

α coefficient, experimental data are necessary.

The analytical solution of Eq. [18.1] at boundary condition Eq. [18.4] is:

Cxt C

( , ) = erfc( ) – exp +

erfc +

αη





































[18.5]

This solution is used to model the evolution of nitrogen in the diffusion zone

during nitriding of Ti-6Al-4V, Ti-6Al-2Sn-4Zr-2Mo and Ti-8Al-1Mo-1V alloys.

The value of α is obtained by fitting the modelling results to experimentally

observed nitrided layers. The best correspondence between modelling and

experimental results is obtained for α = 2.55 × 10

–8

m/sec. The simulation

results after nitriding at 900 and 950 °C for different periods of time are

shown in Figs. 18.14 and 18.15, respectively. The concentration of the nitrogen

at the surface gradually increases as the concentration gradient develops.

Nitrogen concentration (wt.%)

Metin and Inal (1989)

Smithells

et al

. (1992)

Samsonova (1976)

Experimental

data ranges

Ti-8-1-1

0 50 100 150 200

Depth (µm)

(c)

Ti-6-2-4-2

Ti-6-4

18.13

Continued

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Nitriding: modelling of hardness profiles and the kinetics 521

The concentration of nitrogen at the surface reaches a value of 4.5 wt.%

after nitriding for 10 min at both temperatures, but is still not in equilibrium.

After nitriding for one hour, again at both temperatures, the nitrogen

concentration in the α phase at the surface already reaches the equilibrium

value of 6 wt.%. This means that titanium nitride should be formed already.

These simulation results are in agreement with the experimental observations

from the X-ray analysis. The diffraction patterns show the presence of titanium

nitride after nitriding at 950 °C for one hour. Good correspondence between

modelling and experimental results regarding the layer thickness evolution is

observed.

Finally, it should be stated once again that the analytical solutions described

above can be used to describe the evolution of the diffusion zone only. It is

possible to apply some analytical solutions to model the reaction diffusion

and formation of the entire (including compounds) layers in certain conditions.

However, it is preferable to use numerical simulation for modelling the

entire physical model.

18.2.5 Numerical simulations

In addition to the analytical solutions, a mathematical model and program

package for numerical simulation of the nitriding process in titanium alloys

have been developed. The mathematical model, based on the physical model

Nitrogen concentration (wt.%)

0 50 100 150

Depth (µm)

10 min

60 min

180 min

300 min

18.14

Calculated profiles for nitrogen concentration in the diffusion

layer for different times of nitriding at 900 °C.

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Titanium alloys: modelling of microstructure522

Nitrogen concentration (wt.%)

0 50 100 150 200

Depth (µm)

10 min

60 min

180 min

300 min

18.15

Profiles for nitrogen concentration in the diffusion layer for

different times of nitriding at 950 °C, compared with experimental

micrographs, of Ti-6Al-2Sn-4Zr-2Mo nitrided at the same conditions.

180 min

60 min

300 min

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