**3. Kinetic peculiarities of thermodiffusion saturation of titanium with interstitial elements at T>T***α***↔***<sup>β</sup>*

It should be noted that the solubility of nitrogen and oxygen in α-phase is high in comparison with β-phase. At the same time, their diffusion coefficients in α-phase are by two orders lesser than in β-phase (Fedirko & Pohrelyuk, 1995; Fromm & Gebhardt, 1976; Panasyuk, 2007). The solubility and diffusion coefficient of oxygen in α- and β-phases are much higher in compar‐

Modelling of Microstructural Evolution of Titanium During Diffusive Saturation by Interstitial Elements

http://dx.doi.org/10.5772/54626

59

Let us consider the process of isothermal saturation of titanium by nitrogen or oxygen at temperature higher than temperature of allotropic transformation (T>Tα↔β). In this case the initial microstructure of titanium consists of β-phase. According to the thermodynamic analysis, the following scheme of the gas-saturated layer of titanium is suggested (Fig. 12)

**Figure 12.** Scheme of the concentration distribution of interstitial element A (N or O) during saturation of titanium at

During the interaction of titanium with nitrogen or oxygen nitride or oxide layer (0 < x < Y0(τ)) and diffusion zone are formed. The diffusion zone consists of three layers. The layer І (Y0(τ) < x < Y1(τ)), which borders on the nitride layer, is α-phase, significantly enriched in nitrogen or oxygen because of their high solubility in α-phase. This layer is formed and it grows during saturation because of diffusion dissolution of nitrogen or oxygen and structural transforma‐ tions in titanium, because these interstitial elements are α-stabilizers. The layer III (Y2(τ) < x < ∞), which borders on the titanium matrix, at the temperature of saturation consists of β-phase enriched by nitrogen or oxygen. Between the first and third layers the layer II (Y1(τ) < x < Y2(τ)) is formed, which is the dispersed mixture of α- and β-phases, enriched in nitrogen or oxygen.

For analytical description of the process of saturation of titanium by nitrogen or oxygen some model assumptions should be done. The aim of thermochemical treatment of titanium samples is strengthening of their surface layer and as the object of analytical investigation of the kinetics of diffusion saturation of titanium the half-space (0≤x<∞) has been chosen. Nitride or oxide

ison with nitrogen.

(Tkachuk, 2012).

T>Tα↔β

**3.2. Physico-mathematical model**

#### **3.1. Thermodynamic analysis**

According to the phase diagram (Fig. 11 a, b), titanium undergoes allotropic transformation (change of crystal lattice from hcp to bcc) at Tα↔β = 882 0 C (Fromm & Gebhardt, 1976).

**Figure 11.** Ti-N (a) and Ti-O (b) phase diagrams (Fromm & Gebhardt, 1976)

We will be interested in high-temperature (T>Tα↔β) interaction of titanium with the interstitial element A (A – N (nitrogen) or O (oxygen)). Under these conditions, according to the phase diagrams (Fig. 11 a, b), titanium nitrides or oxides (TiAх) as products of chemical reactions and solid solutions of nitrogen or oxygen in α and β-phases of titanium are stable in the system.

In particular, in the concentration range 0<*CA* <*C*<sup>23</sup> solid solution of interstitial element in βphase is stable, while in the concentration range *С*<sup>12</sup> <*CA* <*C*1*S* – solid solution of interstitial element in α-phase. In the concentration range *С*<sup>23</sup> <*CA* <*C*12 solid solutions of interstitial element in α- and β-phases can coexist.

It should be noted that the solubility of nitrogen and oxygen in α-phase is high in comparison with β-phase. At the same time, their diffusion coefficients in α-phase are by two orders lesser than in β-phase (Fedirko & Pohrelyuk, 1995; Fromm & Gebhardt, 1976; Panasyuk, 2007). The solubility and diffusion coefficient of oxygen in α- and β-phases are much higher in compar‐ ison with nitrogen.

#### **3.2. Physico-mathematical model**

**3. Kinetic peculiarities of thermodiffusion saturation of titanium with**

According to the phase diagram (Fig. 11 a, b), titanium undergoes allotropic transformation

We will be interested in high-temperature (T>Tα↔β) interaction of titanium with the interstitial element A (A – N (nitrogen) or O (oxygen)). Under these conditions, according to the phase diagrams (Fig. 11 a, b), titanium nitrides or oxides (TiAх) as products of chemical reactions and solid solutions of nitrogen or oxygen in α and β-phases of titanium are stable in the system.

In particular, in the concentration range 0<*CA* <*C*<sup>23</sup> solid solution of interstitial element in βphase is stable, while in the concentration range *С*<sup>12</sup> <*CA* <*C*1*S* – solid solution of interstitial element in α-phase. In the concentration range *С*<sup>23</sup> <*CA* <*C*12 solid solutions of interstitial

C (Fromm & Gebhardt, 1976).

**interstitial elements at T>T***α***↔***<sup>β</sup>*

58 Titanium Alloys - Advances in Properties Control

(change of crystal lattice from hcp to bcc) at Tα↔β = 882 0

**Figure 11.** Ti-N (a) and Ti-O (b) phase diagrams (Fromm & Gebhardt, 1976)

element in α- and β-phases can coexist.

**3.1. Thermodynamic analysis**

Let us consider the process of isothermal saturation of titanium by nitrogen or oxygen at temperature higher than temperature of allotropic transformation (T>Tα↔β). In this case the initial microstructure of titanium consists of β-phase. According to the thermodynamic analysis, the following scheme of the gas-saturated layer of titanium is suggested (Fig. 12) (Tkachuk, 2012).

**Figure 12.** Scheme of the concentration distribution of interstitial element A (N or O) during saturation of titanium at T>Tα↔β

During the interaction of titanium with nitrogen or oxygen nitride or oxide layer (0 < x < Y0(τ)) and diffusion zone are formed. The diffusion zone consists of three layers. The layer І (Y0(τ) < x < Y1(τ)), which borders on the nitride layer, is α-phase, significantly enriched in nitrogen or oxygen because of their high solubility in α-phase. This layer is formed and it grows during saturation because of diffusion dissolution of nitrogen or oxygen and structural transforma‐ tions in titanium, because these interstitial elements are α-stabilizers. The layer III (Y2(τ) < x < ∞), which borders on the titanium matrix, at the temperature of saturation consists of β-phase enriched by nitrogen or oxygen. Between the first and third layers the layer II (Y1(τ) < x < Y2(τ)) is formed, which is the dispersed mixture of α- and β-phases, enriched in nitrogen or oxygen.

For analytical description of the process of saturation of titanium by nitrogen or oxygen some model assumptions should be done. The aim of thermochemical treatment of titanium samples is strengthening of their surface layer and as the object of analytical investigation of the kinetics of diffusion saturation of titanium the half-space (0≤x<∞) has been chosen. Nitride or oxide film is formed immediately. Surface concentration of nitrogen or oxygen does not change with time and corresponds to stoichiometric titanium nitride (TiN) or oxide (ТіО2). On the interfaces the nitrogen or oxygen concentration, corresponding to equilibrium concentration, according to the phase diagram is constant (Fig. 11 a, b).

The diffusion process in such heterogeneous system will be described by Fick's system of equations:

$$D\_i \partial^2 \mathbb{C}\_i \langle \mathbf{x}, \boldsymbol{\tau} \rangle / \partial \mathbf{x}^2 = \partial \mathbb{C}\_i \langle \mathbf{x}, \boldsymbol{\tau} \rangle / \partial \boldsymbol{\tau} \quad , \quad i = 0, 1, 2, 3. \tag{12}$$

 corresponded to the quasi-stationary state. It is accepted the same distribution law in the first two layers of diffusion zone. It was considered that in the third layer of diffusion zone the

( ) (,) ( ) , (,) ( ) , ( ) () ()


 t

differential equations (12). The following system of equations for calculating the parameters

(j=0,1,2) was obtained by the conditions of mass balance on interfaces (16) and relation (15)

( ) ( ) (,) ( ) , (,) . () () <sup>2</sup>

 t

t


0 1 00 2 11 0 0 10 1 00 21 2 11 3 2

0 0 0 1 00 2 312 2 11 32

= + *AB AB A A*

> *C*1*<sup>S</sup>* −*C*<sup>12</sup> *C*<sup>12</sup> −*C*<sup>23</sup>

Taking the values of these parameters, according to relations (19), the constants *β<sup>j</sup>*

1 1 ( ) 2( ) [ ] 1, 1 , 1 .


( 2/ )/2, /2,

 l p

, *A*<sup>3</sup> =

diffusion coefficients in α- and β-phases, which in turn depend on the temperature. In

oxygen in surface layers of titanium, and equilibrium concentrations of nitrogen and oxygen on interfaces, according to the corresponding phase diagrams (Fig. 11 a, b), are presented in

*A*

b b l

b bl

 l b bl

0 1 0 1 2

*Y Y Y*

Modelling of Microstructural Evolution of Titanium During Diffusive Saturation by Interstitial Elements

 t

(x,τ) satisfy the initial (13) and boundary (14) conditions as well as the

*A A <sup>A</sup>* (18)

/2,

*C*<sup>12</sup> −*C*<sup>23</sup> *C*<sup>23</sup>

depend on concentration of nitrogen or oxygen on interfaces and their

b bl

 ll p

.

C the diffusion coefficients of nitrogen and

l p

2 1 3

0

 t

t

t

t

 t

http://dx.doi.org/10.5772/54626

(17)

61

(19)

for nitrogen

distribution of the interstitial element is realized by Gauss's law:

t

t

The chosen functions Сі

(Tkachuk et al., 2012):

b b

b

where

*A*<sup>0</sup> =

Table 1.

*C*0*<sup>S</sup>* −*C*<sup>01</sup> *C*<sup>01</sup> −*C*1*<sup>S</sup>*

The parameters *β<sup>j</sup>*

, *A*<sup>1</sup> =

and oxygen were calculated (Table 2).

*A*

 l b bl

2

*C*0*<sup>S</sup>* −*C*<sup>01</sup> *C*1*<sup>S</sup>* −*C*<sup>12</sup>

particular, for saturation temperature of Т=950 <sup>0</sup>

*βj*

0 0 0 01 1 1 1 12

*S S S S <sup>x</sup> x Y Cx C C C Cx C C C*

*x Y x Y Cx C C C C x C erfc Y Y <sup>D</sup>*

t

2 12 12 23 3 23

2 () ()

Having solved the system of equations (18), following equations were:

= + - =+

*B* =1 / *A*1*A*2*A*3*λ*0*λ*1*λ*<sup>2</sup> *π* , *λ*<sup>0</sup> = *D*<sup>0</sup> / *D*<sup>1</sup> , *λ*<sup>1</sup> = *D*<sup>1</sup> / *D*<sup>2</sup> , *λ*<sup>2</sup> = *D*<sup>2</sup> / *D*<sup>3</sup> ,

, *A*<sup>2</sup> =

b bl

t

Here Сі (x,τ) and Di are concentration and diffusion coefficients of the interstitial element; index i=0 corresponds to nitride TiNx or oxide TiO2-x layer (0 < x < Y0(τ)); i=1 – α-Ti layer (Y0(τ) < x < Y1(τ)); i=2 – (α+β)-Ti layer (Y1(τ) < x < Y2(τ)); i=3 – β-Ti layer (Y2(τ) < x < ∞)).

Initial conditions (τ=0):

$$\mathbf{C}\_{i}(\mathbf{x}, \mathbf{0}) = \mathbf{0}, \quad \mathbf{Y}\_{i}(\mathbf{0}) = \mathbf{0} \quad \text{for } \mathbf{x} > \mathbf{0}. \tag{13}$$

Boundary conditions (τ>0):

$$\begin{aligned} \mathsf{C}\_{0}(\mathsf{O},\mathsf{\tau}) &= \mathsf{C}\_{0S}, & \mathsf{C}\_{3}(\mathsf{\infty},\mathsf{\tau}) &= \mathsf{0}, & \mathsf{C}\_{0}[\mathsf{Y}\_{0}(\mathsf{\tau}),\mathsf{\tau}] &= \mathsf{C}\_{01}, & \mathsf{C}\_{1}[\mathsf{Y}\_{0}(\mathsf{\tau}),\mathsf{\tau}] &= \mathsf{C}\_{1S}, \\ \mathsf{C}\_{1}[\mathsf{Y}\_{1}(\mathsf{\tau}),\mathsf{\tau}] &= \mathsf{C}\_{2}[\mathsf{Y}\_{1}(\mathsf{\tau}),\mathsf{\tau}] = \mathsf{C}\_{12}, & \mathsf{C}\_{2}[\mathsf{Y}\_{2}(\mathsf{\tau}),\mathsf{\tau}] &= \mathsf{C}\_{3}[\mathsf{Y}\_{2}(\mathsf{\tau}),\mathsf{\tau}] = \mathsf{C}\_{23} \ . \end{aligned} \tag{14}$$

The motion of interfaces will be set by the parabolic dependencies (Lyubov, 1981):

$$Y\_0(\tau) = 2 \cdot \beta\_0 \cdot \sqrt{D\_0 \cdot \tau} \quad \text{ } Y\_1(\tau) = 2 \cdot \beta\_1 \cdot \sqrt{D\_1 \cdot \tau} \text{ } \text{ } Y\_2(\tau) = 2 \cdot \beta\_2 \cdot \sqrt{D\_2 \cdot \tau} \text{ } \text{ }. \tag{15}$$

Here *β<sup>j</sup>* (j=0,1,2) are dimensionless constants (for the specific temperature), which will be determined from the law of conservation of mass on the interfaces. Thus for diffusion fluxes on the interfaces *Yj* (*τ*)are set:

$$\begin{split} & -D\_{0} \frac{\partial \mathbb{C}\_{0}}{\partial \mathbf{x}} \Big|\_{x = \mathbb{V}\_{0}(\mathbf{r}) - 0} + D\_{1} \frac{\partial \mathbb{C}\_{1}}{\partial \mathbf{x}} \Big|\_{x = \mathbb{V}\_{0}(\mathbf{r}) + 0} = (\mathbb{C}\_{01} - \mathbb{C}\_{1S}) \frac{dY\_{0}(\mathbf{r})}{d\mathbf{r}}, \\ & D\_{1} \frac{\partial \mathbb{C}\_{1}}{\partial \mathbf{x}} \Big|\_{x = \mathbb{V}\_{1}(\mathbf{r}) - 0} = D\_{2} \frac{\partial \mathbb{C}\_{2}}{\partial \mathbf{x}} \Big|\_{x = \mathbb{V}\_{1}(\mathbf{r}) + 0}, D\_{2} \frac{\partial \mathbb{C}\_{2}}{\partial \mathbf{x}} \Big|\_{x = \mathbb{V}\_{2}(\mathbf{r}) - 0} = D\_{3} \frac{\partial \mathbb{C}\_{3}}{\partial \mathbf{x}} \Big|\_{x = \mathbb{V}\_{2}(\mathbf{r}) + 0} \end{split} \tag{16}$$

It is difficult to solve the equations system (12) – (16) in analytical form. The method of approximate solution of above mentioned task should be used (Lykov, 1966). It is accepted the linear distribution law of the concentration of the interstitial element in TiAx layer (Fig. 12)  corresponded to the quasi-stationary state. It is accepted the same distribution law in the first two layers of diffusion zone. It was considered that in the third layer of diffusion zone the distribution of the interstitial element is realized by Gauss's law:

$$\begin{split} \mathbf{C}\_{0}(\mathbf{x},\tau) &= \mathbf{C}\_{0S} - (\mathbf{C}\_{0S} - \mathbf{C}\_{01}) \frac{\mathbf{x}}{Y\_{0}(\tau)}, \quad \mathbf{C}\_{1}(\mathbf{x},\tau) = \mathbf{C}\_{1S} - (\mathbf{C}\_{1S} - \mathbf{C}\_{12}) \frac{\mathbf{x} - Y\_{0}(\tau)}{Y\_{1}(\tau) - Y\_{0}(\tau)}, \\ \mathbf{C}\_{2}(\mathbf{x},\tau) &= \mathbf{C}\_{12} - (\mathbf{C}\_{12} - \mathbf{C}\_{23}) \frac{\mathbf{x} - Y\_{1}(\tau)}{Y\_{2}(\tau) - Y\_{1}(\tau)}, \quad \mathbf{C}\_{3}(\mathbf{x},\tau) = \mathbf{C}\_{23} \text{erfc} \frac{\mathbf{x} - Y\_{2}(\tau)}{2\sqrt{D\_{3}\tau}}. \end{split} \tag{17}$$

The chosen functions Сі (x,τ) satisfy the initial (13) and boundary (14) conditions as well as the differential equations (12). The following system of equations for calculating the parameters *βj* (j=0,1,2) was obtained by the conditions of mass balance on interfaces (16) and relation (15) (Tkachuk et al., 2012):

$$\frac{A\_0}{2\beta\_0} \mathbf{l} \frac{1}{\beta\_0} - \frac{1}{A\_1 \boldsymbol{\lambda}\_0 (\boldsymbol{\beta}\_1 - \boldsymbol{\beta}\_0 \boldsymbol{\lambda}\_0)} \mathbf{l} = 1,\\ \frac{(\beta\_1 - \beta\_0 \boldsymbol{\lambda}\_0)}{A\_2 \boldsymbol{\lambda}\_1 (\beta\_2 - \beta\_1 \boldsymbol{\lambda}\_1)} = 1,\\ \frac{2(\beta\_2 - \beta\_1 \boldsymbol{\lambda}\_1)}{A\_3 \boldsymbol{\lambda}\_2 \sqrt{\pi}} = 1. \tag{18}$$

Having solved the system of equations (18), following equations were:

$$\begin{aligned} \beta\_0 = A\_0 (\sqrt{B^2 + 2 \,/\ A\_0} - B) / 2 \, , \quad \qquad \beta\_1 = \beta\_0 \lambda\_0 + A\_2 A\_3 \lambda\_1 \lambda\_2 \sqrt{\pi} \, / 2 \, , \\\beta\_2 = \beta\_1 \lambda\_1 + A\_3 \lambda\_2 \sqrt{\pi} \, / 2 \, , \end{aligned} \tag{19}$$

where

film is formed immediately. Surface concentration of nitrogen or oxygen does not change with time and corresponds to stoichiometric titanium nitride (TiN) or oxide (ТіО2). On the interfaces the nitrogen or oxygen concentration, corresponding to equilibrium concentration, according

The diffusion process in such heterogeneous system will be described by Fick's system of

2 2 ¶ ¶ =¶ ¶ = ( , ) / ( , ) / , 0,1,2,3.

0 0 3 0 0 01 1 0 1 1 1 2 1 12 2 2 3 2 23

The motion of interfaces will be set by the parabolic dependencies (Lyubov, 1981):

 tb

()0 ()0 0 0


*x Y x Y*

tt

0 1 01 1

*<sup>C</sup> <sup>C</sup> dY D D CC xx d CC C <sup>C</sup> DD D D xx xx*

0 0 1

 t

¶¶ ¶ ¶ <sup>=</sup> <sup>=</sup> ¶¶ ¶¶

= - = +

12 2 3 12 2 3

,

0 00 1 11 2 22 *Y D Y DY D* () 2 , () 2 , () 2 .

(0, ) , ( , ) 0, [ ( ), ] , [ ( ), ] , [ ( ), ] [Y ( ), ] C , [ ( ), ] [Y ( ), ] C .

= ¥= = = = = = = *C CC <sup>S</sup> CY C CY C <sup>S</sup>*

t t

tt

*CY C CY C* (14)

(j=0,1,2) are dimensionless constants (for the specific temperature), which will be

determined from the law of conservation of mass on the interfaces. Thus for diffusion fluxes

()0 ()0 ()0 ()0 11 22

= - = + = - = +

It is difficult to solve the equations system (12) – (16) in analytical form. The method of approximate solution of above mentioned task should be used (Lykov, 1966). It is accepted the linear distribution law of the concentration of the interstitial element in TiAx layer (Fig. 12)

*x Y x Y x Y x Y*

=× × × =× × × =× × × (15)

( ) ( ),

*S*

tt

t

t

Y1(τ)); i=2 – (α+β)-Ti layer (Y1(τ) < x < Y2(τ)); i=3 – β-Ti layer (Y2(τ) < x < ∞)).

 tt

i=0 corresponds to nitride TiNx or oxide TiO2-x layer (0 < x < Y0(τ)); i=1 – α-Ti layer (Y0(τ) < x <

*D Cx x Cx i i i <sup>i</sup>* (12)

are concentration and diffusion coefficients of the interstitial element; index

( ,0) 0, (0) 0 0. = = > *C x Y for x i i* (13)

 tt

ttb

t t

> t

> > (16)

to the phase diagram is constant (Fig. 11 a, b).

60 Titanium Alloys - Advances in Properties Control

t

t

 tt

> t

(*τ*)are set:

t

¶ ¶

¶ ¶

equations:

Here Сі

Here *β<sup>j</sup>*

on the interfaces *Yj*

(x,τ) and Di

Initial conditions (τ=0):

Boundary conditions (τ>0):

t

tt

tb

$$\begin{aligned} \mathbf{B} &= \mathbf{1} \left\langle \mathbf{f}\_1 A\_2 A\_3 \lambda\_0 \lambda\_1 \lambda\_2 \sqrt{\pi} \right\rangle, \quad \lambda\_0 = \sqrt{D\_0/D\_1}, \quad \lambda\_1 = \sqrt{D\_1/D\_2}, \quad \lambda\_2 = \sqrt{D\_2/D\_3} \\\ A\_0 &= \frac{\mathbf{C}\_{0S} - \mathbf{C}\_{01}}{\mathbf{C}\_{01} - \mathbf{C}\_{1S}}, \quad A\_1 = \frac{\mathbf{C}\_{0S} - \mathbf{C}\_{01}}{\mathbf{C}\_{1S} - \mathbf{C}\_{12}}, \quad A\_2 = \frac{\mathbf{C}\_{1S} - \mathbf{C}\_{12}}{\mathbf{C}\_{12} - \mathbf{C}\_{23}}, \quad A\_3 = \frac{\mathbf{C}\_{12} - \mathbf{C}\_{23}}{\mathbf{C}\_{23}}. \end{aligned}$$

The parameters *β<sup>j</sup>* depend on concentration of nitrogen or oxygen on interfaces and their diffusion coefficients in α- and β-phases, which in turn depend on the temperature. In particular, for saturation temperature of Т=950 <sup>0</sup> C the diffusion coefficients of nitrogen and oxygen in surface layers of titanium, and equilibrium concentrations of nitrogen and oxygen on interfaces, according to the corresponding phase diagrams (Fig. 11 a, b), are presented in Table 1.

Taking the values of these parameters, according to relations (19), the constants *β<sup>j</sup>* for nitrogen and oxygen were calculated (Table 2).


**Table 1.** Diffusion coefficients of nitrogen and oxygen in the surface layer of titanium (Fedirko & Pohrelyuk, 1995; Fromm & Gebhardt, 1976; Panasyuk, 2007) and equilibrium concentration of nitrogen and oxygen on the interfaces at saturation temperature of Т=950 <sup>0</sup>С


**Table 2.** Calculated constants β<sup>j</sup> and К<sup>j</sup> (j=0,1,2) at saturation temperature of Т=950 <sup>0</sup>С

Taking into consideration the correlation (15), the motion of interfaces will be presented as:

$$Y\_0(\tau) = K\_0 \sqrt{\tau} \quad , \quad Y\_1(\tau) = K\_1 \sqrt{\tau} \quad , \quad \quad Y\_2(\tau) = K\_2 \sqrt{\tau} \quad , \tag{20}$$

The diffusion coefficient of nitrogen or oxygen in β-phase is by two-four orders higher than in α-phase and in nitride or oxide layers, that's why the thickness of β layer is much larger than the thickness of the other layers of diffusion zone (Fig. 13). If the thickness of nitride layer is less than 0.2% and oxide layer is less than 0.1 % of the total thickness of diffusion zone (Y3(τ)), the thickness of α, α + β and β layers will be 16, 8 and 76 % for nitriding and 12, 8 and 80 % for

On the basis of relations (16) the concentration profiles of nitrogen (curves 1) and oxygen (curves 2) in the diffusion zone of

The diffusion coefficient of nitrogen or oxygen in β-phase is by two-four orders higher than in α-phase and in nitride or oxide layers, that's why the thickness of β layer is much larger than the thickness of the other layers of diffusion zone (Fig. 13). If the thickness of nitride layer is less than 0.2% and oxide layer is less than 0.1 % of the total thickness of diffusion zone (Y3(τ)), the

Calculated constants Yі(τ) (і=0,1,2,3) after isothermal exposures of 1 and 5 h during nitriding and oxidation of titanium at Т=950 0C are presented in Table 3. It is clear that according to the assumptions (14) with the increase of processing time the motion of interfaces (Fig. 13 a, b, c, d) occur according to the parabolic dependences proportionally to the corresponding constants of

Figure 13. Kinetics of motion of interfaces Y0(τ) (a), Y1(τ) (b), Y2(τ) (c) and Y3(τ) (d) at nitriding (curves 1) and oxidation (curves 2) of titanium at

**Figure 13.** Kinetics of motion of interfaces Y0(τ) (a), Y1(τ) (b), Y2(τ) (c) and Y3(τ) (d) at nitriding (curves 1) and oxidation

**τ = 1 h τ = 5 h**

**τ = 1 h τ = 5 h** 

Yo, μm

Yo, μm

(τ) (і=0,1,2,3) after isothermal exposures of 1 and 5 h at saturation temperature of Т=950 <sup>0</sup>С

Y1, μm

Y2, μm

Y1, μm

– constants of he parabolic growth of nitride or oxide layer and α, (α+β) layers

http://dx.doi.org/10.5772/54626

63

Modelling of Microstructural Evolution of Titanium During Diffusive Saturation by Interstitial Elements

of diffusion zone stabilized by nitrogen or oxygen. In particular, for saturation temperature of Т=950 0C these calculated constants

Having found the constants of parabolic growth of the layers, and having used the relations (19), it is easy to foresee the kinetics of motion of interfaces: Y0(τ) – interface of nitride or oxide layer (Fig. 13 a), Y1(τ) – interface of solid solution of interstitial element in α-phase (Fig. 13 b), Y2(τ) – interface of mixture of solid solutions of interstitial element in α- and β-phases (Fig. 13 c), Y3(τ) – interface of solid solution of interstitial element in β-phase (Fig. 13 d) at nitriding and oxidation of titanium at Т=950 0C. One could notice that the last interface is identified by the motion of conventional boundary with the specific nitrogen or oxygen

> Y2, μm

Y3, μm

Y3, μm

Y3, μm

Y3, μm

N 0.4 32 47 194 0.85 72 105 433 O 0.5 81 137 658 1.1 182 307 1470

N 0.4 32 47 194 0.85 72 105 433 O 0.5 81 137 658 1.1 182 307 1470

At the same time, the different solubility of nitrogen or oxygen in α and β-phases influences on the distribution of nitrogen or oxygen in the diffusion zone. When the structural phase transformations did not occur in the diffusion zone, the profiles of nitrogen and oxygen in this zone would be with a small gradients because of the low solubility of nitrogen or oxygen in β-phase. In fact, nitrogen and oxygen, being α-stabilizers, stimulate the β→ α phase transfor‐ mation in the layers of the diffusion zone adjacent to nitride or oxide layer. And as the solubility

oxidation respectively.

saturation temperature of Т=950 0C

parabolic growth *Kj* (j=0,1,2).

00 11 22 *YK YK YK* () , () , () ,

where 0 00 1 11 2 22 *K DK DK D* 2 ,2 ,2 

 (19)

> 

concentration, for example С33 = 0.25 аt. %, that is from transcendental equation С3(Y3(τ), τ)=C33.

 

are presented in Table 2.

**Table 3.** Calculated Y<sup>і</sup>

A

Y0, μm

(curves 2) of titanium at saturation temperature of Т=950 0C

Y1, μm

<sup>A</sup>Y0, μm

titanium after nitriding and oxidation during 1 h (Fig. 14 a) and 5 h (Fig. 14 b) are calculated.

Y2, μm

Y1, μm

Table 3. Calculated Yі(τ) (і=0,1,2,3) after isothermal exposures of 1 and 5 h at saturation temperature of Т=950 0С

thickness of α, α + β and β layers will be 16, 8 and 76 % for nitriding and 12, 8 and 80 % for oxidation respectively.

Y2, μm

where *K*<sup>0</sup> =2*β*<sup>0</sup> *D*<sup>0</sup> , *K*<sup>1</sup> =2*β*<sup>1</sup> *D*<sup>1</sup> , *K*<sup>2</sup> =2*β*<sup>2</sup> *D*2 – constants of the parabolic growth of nitride or oxide layer and α, (α+β) layers of diffusion zone stabilized by nitrogen or oxygen. In particular, for saturation temperature of Т=950 <sup>0</sup> C these calculated constants are presented in Table 2.

Having found the constants of parabolic growth of the layers, and having used the relations (20), it is easy to foresee the kinetics of motion of interfaces: Y0(τ) – interface of nitride or oxide layer (Fig. 13 a), Y1(τ) – interface of solid solution of interstitial element in α-phase (Fig. 13 b), Y2(τ) – interface of mixture of solid solutions of interstitial element in α- and β-phases (Fig. 13 c), Y3(τ) – interface of solid solution of interstitial element in β-phase (Fig. 13 d) at nitriding and oxidation of titanium at Т=950 <sup>0</sup> C. One could notice that the last interface is identified by the motion of conventional boundary with the specific nitrogen or oxygen concentration, for example С33 = 0.25 аt. %, that is from transcendental equation С3(Y3(τ), τ)=C33.

Calculated constants Yі (τ) (і=0,1,2,3) after isothermal exposures of 1 and 5 h during nitriding and oxidation of titanium at Т=950 <sup>0</sup> C are presented in Table 3. It is clear that according to the assumptions (15) with the increase of processing time the motion of interfaces (Fig. 13 a, b, c, d) occur according to the parabolic dependences proportionally to the corresponding constants of parabolic growth *Kj* (j=0,1,2).

On the basis of relations (17) the concentration profiles of nitrogen (curves 1) and oxygen (curves 2) in the diffusion zone of titanium after nitriding and oxidation during 1 h (Fig. 14 a) and 5 h (Fig. 14 b) are calculated.

α-phase (Fig. 13 b), Y2(τ) – interface of mixture of solid solutions of interstitial element in α- and β-phases (Fig. 13 c), Y3(τ) – interface of solid solution of interstitial element in β-phase (Fig. 13 d) at nitriding and oxidation of titanium at Т=950 0C. One could notice that the last interface is identified by the motion of conventional boundary with the specific nitrogen or oxygen Modelling of Microstructural Evolution of Titanium During Diffusive Saturation by Interstitial Elements http://dx.doi.org/10.5772/54626 63

– constants of he parabolic growth of nitride or oxide layer and α, (α+β) layers

of diffusion zone stabilized by nitrogen or oxygen. In particular, for saturation temperature of Т=950 0C these calculated constants

Having found the constants of parabolic growth of the layers, and having used the relations (19), it is easy to foresee the kinetics of motion of interfaces: Y0(τ) – interface of nitride or oxide layer (Fig. 13 a), Y1(τ) – interface of solid solution of interstitial element in

00 11 22 *YK YK YK* () , () , () ,

where 0 00 1 11 2 22 *K DK DK D* 2 ,2 ,2 

 (19)

> 

concentration, for example С33 = 0.25 аt. %, that is from transcendental equation С3(Y3(τ), τ)=C33.

 

are presented in Table 2.

**A β<sup>0</sup> β<sup>1</sup> β<sup>2</sup> K0, cm/ sec 1/2 K1, cm/ sec 1/2 K2, cm/ sec 1/2** N 0.183 1.691 0.782 6.328×10-7 5.348×10-5 7.825×10-5 O 0.082 1.481 0.789 8.175×10-7 1.357×10-4 2.288×10-4

**D3, cm2/ sec**

N 3×10-12 2.5×10-10 2.5×10-9 3.2×10-8 50 33 17.5 1.5 0.75 0.25 O 2.5×10-11 2.1×10-9 2.1×10-8 1.6×10-7 66 51 33 4 2 0.25

**Table 1.** Diffusion coefficients of nitrogen and oxygen in the surface layer of titanium (Fedirko & Pohrelyuk, 1995; Fromm & Gebhardt, 1976; Panasyuk, 2007) and equilibrium concentration of nitrogen and oxygen on the interfaces at

**C0S, at. %**

**C01, at. %**

**C1S, at. %**

**C12, at. %**

**C23, at. %**

**C33, at. %**

(j=0,1,2) at saturation temperature of Т=950 <sup>0</sup>С

 tt== = (20)

C these calculated constants are presented in

C. One could notice that the last interface is identified by

C are presented in Table 3. It is clear that according to the

(τ) (і=0,1,2,3) after isothermal exposures of 1 and 5 h during nitriding

Taking into consideration the correlation (15), the motion of interfaces will be presented as:

where *K*<sup>0</sup> =2*β*<sup>0</sup> *D*<sup>0</sup> , *K*<sup>1</sup> =2*β*<sup>1</sup> *D*<sup>1</sup> , *K*<sup>2</sup> =2*β*<sup>2</sup> *D*2 – constants of the parabolic growth of nitride or oxide layer and α, (α+β) layers of diffusion zone stabilized by nitrogen or oxygen. In

Having found the constants of parabolic growth of the layers, and having used the relations (20), it is easy to foresee the kinetics of motion of interfaces: Y0(τ) – interface of nitride or oxide layer (Fig. 13 a), Y1(τ) – interface of solid solution of interstitial element in α-phase (Fig. 13 b), Y2(τ) – interface of mixture of solid solutions of interstitial element in α- and β-phases (Fig. 13 c), Y3(τ) – interface of solid solution of interstitial element in β-phase (Fig. 13 d) at nitriding

the motion of conventional boundary with the specific nitrogen or oxygen concentration, for

assumptions (15) with the increase of processing time the motion of interfaces (Fig. 13 a, b, c, d) occur according to the parabolic dependences proportionally to the corresponding constants

On the basis of relations (17) the concentration profiles of nitrogen (curves 1) and oxygen (curves 2) in the diffusion zone of titanium after nitriding and oxidation during 1 h (Fig. 14 a)

example С33 = 0.25 аt. %, that is from transcendental equation С3(Y3(τ), τ)=C33.

00 11 22 *YK YK YK* () , () , () ,

 tt

**Table 2.** Calculated constants β<sup>j</sup>

saturation temperature of Т=950 <sup>0</sup>С

**D0, cm2/ sec**

**D1, cm2/ sec**

62 Titanium Alloys - Advances in Properties Control

**D2, cm2/ sec**

Table 2.

**A**

and К<sup>j</sup>

tt

particular, for saturation temperature of Т=950 <sup>0</sup>

and oxidation of titanium at Т=950 <sup>0</sup>

and oxidation of titanium at Т=950 <sup>0</sup>

of parabolic growth *Kj* (j=0,1,2).

and 5 h (Fig. 14 b) are calculated.

Calculated constants Yі

Figure 13. Kinetics of motion of interfaces Y0(τ) (a), Y1(τ) (b), Y2(τ) (c) and Y3(τ) (d) at nitriding (curves 1) and oxidation (curves 2) of titanium at saturation temperature of Т=950 0C Calculated constants Yі(τ) (і=0,1,2,3) after isothermal exposures of 1 and 5 h during nitriding and oxidation of titanium at Т=950 0C **Figure 13.** Kinetics of motion of interfaces Y0(τ) (a), Y1(τ) (b), Y2(τ) (c) and Y3(τ) (d) at nitriding (curves 1) and oxidation (curves 2) of titanium at saturation temperature of Т=950 0C

The diffusion coefficient of nitrogen or oxygen in β-phase is by two-four orders higher than in α-phase and in nitride or oxide layers, that's why the thickness of β layer is much larger than the thickness of the other layers of diffusion zone (Fig. 13). If the thickness of nitride layer is less than 0.2% and oxide layer is less than 0.1 % of the total thickness of diffusion zone (Y3(τ)), the thickness of α, α + β and β layers will be 16, 8 and 76 % for nitriding and 12, 8 and 80 % for oxidation respectively. are presented in Table 3. It is clear that according to the assumptions (14) with the increase of processing time the motion of interfaces (Fig. 13 a, b, c, d) occur according to the parabolic dependences proportionally to the corresponding constants of parabolic growth *Kj* (j=0,1,2). On the basis of relations (16) the concentration profiles of nitrogen (curves 1) and oxygen (curves 2) in the diffusion zone of titanium after nitriding and oxidation during 1 h (Fig. 14 a) and 5 h (Fig. 14 b) are calculated. The diffusion coefficient of nitrogen or oxygen in β-phase is by two-four orders higher than in α-phase and in nitride or oxide

layers, that's why the thickness of β layer is much larger than the thickness of the other layers of diffusion zone (Fig. 13). If the thickness of nitride layer is less than 0.2% and oxide layer is less than 0.1 % of the total thickness of diffusion zone (Y3(τ)), the


**Table 3.** Calculated Y<sup>і</sup> (τ) (і=0,1,2,3) after isothermal exposures of 1 and 5 h at saturation temperature of Т=950 <sup>0</sup>С

At the same time, the different solubility of nitrogen or oxygen in α and β-phases influences on the distribution of nitrogen or oxygen in the diffusion zone. When the structural phase transformations did not occur in the diffusion zone, the profiles of nitrogen and oxygen in this zone would be with a small gradients because of the low solubility of nitrogen or oxygen in β-phase. In fact, nitrogen and oxygen, being α-stabilizers, stimulate the β→ α phase transfor‐ mation in the layers of the diffusion zone adjacent to nitride or oxide layer. And as the solubility of nitrogen and oxygen in α-phase is much higher than in β-phase, it can be foreseen that in zone I the profiles of nitrogen and oxygen will have a large gradients (Fig. 14), and respectively the distributions of microhardness in this zone will have a large gradients. It has been confirmed by the literature data (Lazarev et al., 1985) and the experimental investigations' data on nitriding.

**3.4. Results and discussion**

of microhardness.

thermal exposure times: curve 1 – τ = 1 h; curve 2 – τ = 5 h

indicates the increase of its thickness.

The nitride layer of goldish colour is formed on the surface of c.p. titanium after nitriding. Its colour is darkening with the increase of isothermal exposure time in nitrogen atmosphere. It

Modelling of Microstructural Evolution of Titanium During Diffusive Saturation by Interstitial Elements

http://dx.doi.org/10.5772/54626

65

(a) (b)

It is difficult to find the layer II (Fig. 12) in this zone which, according to the phase diagram (Fig. 11 a) has to form. However, two parts of diffusion zone (zone A and zone В) of different structure are clearly identified. Zone A is α-phase formed during nitriding by nitrogen as αstabilizer. Its thickness, according to data of metallographic analysis, increases from 20 to 45 µm with the increase of duration of nitriding from 1 to 5 h. Zone B is α-phase on the basis of solid solution of nitrogen, however formed as a result of β→ α transformation at cooling.

The results of investigation of character of distribution of microhardness on cross section of surface layers of c.p. titanium after nitriding are presented in Fig. 16 a. It is distinguished zone A (layer I, Fig. 12) and zone B (probably, layer ІІ + layer ІІІ, Fig. 12) on curves of distribution

**Figure 16.** Distribution of nitrogen in diffusion zone of titanium (theory) (а) and distribution of microhardness on cross section of surface layer of c.p. titanium (experiment) (b) after its nitriding at Т=950 0С and p=105 Pа for two iso‐

**Figure 15.** Microstructure of surface layer of c.p. titanium for nitriding at τ=1 (а) and 5 h (b) (Т=950 0С, p=105 Pа)

The diffusion zone is formed under titanium nitride layer (Fig. 15 a, b).

**Figure 14.** Concentration profiles of nitrogen and oxygen in diffusion zone of titanium after its saturation at Т=950 0C for two isothermal exposures: a - τ = 1 h; b - τ = 5 h; curves 1 – for nitrogen, curves 2 – for oxygen

It was observed 2.5-3.0 times larger thickness of all layers of diffusion zone after oxidation comparing to nitriding (Fig. 13 b, c, d) as a result of the higher on order diffusion coefficients of oxygen in α- and β-phases compared to the diffusion coefficients of nitrogen (Table 1). Also the larger concentration gradient of oxygen in the layer I adjacent to oxide layer than concen‐ tration gradient of nitrogen in the layer I adjacent to nitride layer was received (Fig. 14). It is caused by higher solubility of oxygen in comparison with nitrogen in α-phase (Table 1).

#### **3.3. Experimental procedure**

Experimental investigation on the example of nitriding of titanium at the temperature of Т=950 0 C was conducted to check the validity of the above elaborated model representations.

Commercially pure (c.p.) titanium samples with dimensions of 10×15×4 mm were investigated. The samples were polished (Rа=0,4 µm), washed with deionized water prior to the treatment. The samples were heated to nitriding temperature in a vacuum of 10-3 Pa. Then they were saturated with molecular nitrogen of the atmospheric pressure at temperature of 950 0 C. The isothermal exposure time in nitrogen was 1 and 5 h. After isothermal exposure the samples were cooled in nitrogen to room temperature.

The microstructure of nitride layers was studied by the use of metallographic microscope "EPIQUANT". Distribution of microhardness on cross section of surface layers of c.p. titanium after nitriding was estimated measuring microhardness at loading of 0.49 N.

#### **3.4. Results and discussion**

of nitrogen and oxygen in α-phase is much higher than in β-phase, it can be foreseen that in zone I the profiles of nitrogen and oxygen will have a large gradients (Fig. 14), and respectively the distributions of microhardness in this zone will have a large gradients. It has been confirmed by the literature data (Lazarev et al., 1985) and the experimental investigations' data

**Figure 14.** Concentration profiles of nitrogen and oxygen in diffusion zone of titanium after its saturation at Т=950 0C

It was observed 2.5-3.0 times larger thickness of all layers of diffusion zone after oxidation comparing to nitriding (Fig. 13 b, c, d) as a result of the higher on order diffusion coefficients of oxygen in α- and β-phases compared to the diffusion coefficients of nitrogen (Table 1). Also the larger concentration gradient of oxygen in the layer I adjacent to oxide layer than concen‐ tration gradient of nitrogen in the layer I adjacent to nitride layer was received (Fig. 14). It is caused by higher solubility of oxygen in comparison with nitrogen in α-phase (Table 1).

Experimental investigation on the example of nitriding of titanium at the temperature of Т=950

Commercially pure (c.p.) titanium samples with dimensions of 10×15×4 mm were investigated. The samples were polished (Rа=0,4 µm), washed with deionized water prior to the treatment. The samples were heated to nitriding temperature in a vacuum of 10-3 Pa. Then they were saturated with molecular nitrogen of the atmospheric pressure at temperature of 950 0

isothermal exposure time in nitrogen was 1 and 5 h. After isothermal exposure the samples

The microstructure of nitride layers was studied by the use of metallographic microscope "EPIQUANT". Distribution of microhardness on cross section of surface layers of c.p. titanium

after nitriding was estimated measuring microhardness at loading of 0.49 N.

C. The

C was conducted to check the validity of the above elaborated model representations.

for two isothermal exposures: a - τ = 1 h; b - τ = 5 h; curves 1 – for nitrogen, curves 2 – for oxygen

on nitriding.

64 Titanium Alloys - Advances in Properties Control

**3.3. Experimental procedure**

were cooled in nitrogen to room temperature.

0

The nitride layer of goldish colour is formed on the surface of c.p. titanium after nitriding. Its colour is darkening with the increase of isothermal exposure time in nitrogen atmosphere. It indicates the increase of its thickness.

The diffusion zone is formed under titanium nitride layer (Fig. 15 a, b).

**Figure 15.** Microstructure of surface layer of c.p. titanium for nitriding at τ=1 (а) and 5 h (b) (Т=950 0С, p=105 Pа)

It is difficult to find the layer II (Fig. 12) in this zone which, according to the phase diagram (Fig. 11 a) has to form. However, two parts of diffusion zone (zone A and zone В) of different structure are clearly identified. Zone A is α-phase formed during nitriding by nitrogen as αstabilizer. Its thickness, according to data of metallographic analysis, increases from 20 to 45 µm with the increase of duration of nitriding from 1 to 5 h. Zone B is α-phase on the basis of solid solution of nitrogen, however formed as a result of β→ α transformation at cooling.

The results of investigation of character of distribution of microhardness on cross section of surface layers of c.p. titanium after nitriding are presented in Fig. 16 a. It is distinguished zone A (layer I, Fig. 12) and zone B (probably, layer ІІ + layer ІІІ, Fig. 12) on curves of distribution of microhardness.

**Figure 16.** Distribution of nitrogen in diffusion zone of titanium (theory) (а) and distribution of microhardness on cross section of surface layer of c.p. titanium (experiment) (b) after its nitriding at Т=950 0С and p=105 Pа for two iso‐ thermal exposure times: curve 1 – τ = 1 h; curve 2 – τ = 5 h

The large gradient of microhardness is characteristic for zone A. It is caused by β→ α trans‐ formation as a result of saturation by nitrogen as α-stabilizer and comparatively its high solubility in α-phase. With increasing distance from surface the microhardness is decreased sharply (Fig. 16 a) that is explained by decrease of nitrogen concentration (Fig. 16 b). The hardness of zone B is considerably less than zone A because of large difference of nitrogen solubility in α- and β-phases. The thickness of these zones is increased with the increase of duration of nitriding (Fig. 16 a). In particular, the thickness of zone A is 34 µm for τ= 1 h and 69 µm for τ=5 h. It can be noticed that this thickness is larger than corresponding thickness, determined by the data of metallographic analysis. The total depth of diffusion zone (zone A + zone В) is 185 µm for τ= 1 h and 425 µm for τ=5 h (Fig. 16 a).

**Author details**

Yaroslav Matychak\*

Ukraine

**References**

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Nauka, Moskow

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67

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The received analytical distribution of nitrogen (Fig. 16 b) and results of microhardness measurements (Fig. 16 a) confirm the correlation between model calculations and experimental data.
