**2. Thermodiffusion saturation of titanium with interstitial elements from a rarefied atmosphere at T<T***α***↔***<sup>β</sup>*

### **2.1. Physicomathematical model**

coefficient of nitrogen or oxygen in titanium (Panasyuk, 2007; Metin, 1989; Kofstad, 1966). This is why investigations (experimental and theoretical) aimed at elucidating the kinetic regular‐ ities and peculiarities of the distribution of interstitial elements in a surface layer, which

Diffusive processes determine changes of properties of surface layers of the structural materials in many cases, for example, in the process of their thermochemical treatment or in the conditions of operation at high temperature. However, the diffusion in solids is often accompanied by the structural phase transformations. These processes are interconnected and interdependent: diffusion of the elements can stimulate structural phase transformations, and the latter change the conditions of diffusion. It is difficult to describe these processes analyti‐

Gebhardt, 1976), is interesting for such theoretical and experimental investigations, in particular its high-temperature interaction with nitrogen or oxygen. Due to high affinity of these elements with titanium nitride or oxide layer forms and grows on the surface. Unlike many alloying elements, in particular vanadium, molybdenum, which are β-stabilizers, above mentioned interstitial elements are α-stabilizers, which can stimulate structural phase transformations in titanium. The microstructural evolution during α↔β phase transformation as a result of migration of β-stabilizers is presented in (Malinov et al., 2003). However, the authors did not take into consideration the role of nitrogen as α-stabilizer in the structural transformations. It was demonstrated (Matychak, 2009) in the studies of interconnection of nitrogen diffusion and structural phase transformations during high-temperature nitriding that, in particular, under the rarefied atmosphere, the continuous nitride layer on the surface

**•** to establish the kinetic peculiarities of interaction of titanium with the interstitial element A (nitrogen or oxygen) at the temperature lower and higher than temperature of allotropic

**•** to investigate experimentally and model analytically the process of diffusive saturation of α-titanium with the interstitial element from a rarefied atmosphere taking into account the

**•** to model the interdependence of the processes of external supply of the interstitial element to the surface and its chemosorption with diffusive dissolution and segregation on defects,

**•** to estimate the influence of temperature-time parameters of treatment on the depth of

**•** to establish the kinetic peculiarities of diffusive saturation of titanium with the interstitial

caused by the chemical interaction with the titanium atoms;

element caused by the structural phase transformations.

diffusion zone and change of its microhardness;

С (Fromm &

determines the changes of its physicomechanical characteristics, are urgent.

cally. Titanium, which undergoes the polymorphic transformation at Tα↔β = 882 0

was absent for a long time.

46 Titanium Alloys - Advances in Properties Control

transformation Tα↔β;

surface phenomena;

The aim of work is:

#### *2.1.1. Phenomenology of surface phenomena*

Let us consider the interaction of α-titanium with a rarefied gas atmosphere in a temperature range which is below the temperature of the α↔ β allotropic transformation. In such a system, peculiarities of the interaction predominantly manifest themselves on the titanium surface as a result of adsorption, chemisorption, chemical reactions, generation of point defects, and the formation of two-dimensional structures. Along with phase formation, which includes these processes on the surface, the transfer of the interstitial element in the depth of titanium, i.e., its diffusive saturation, plays an important role. Experimental data indicate that, for rather long exposures, at certain rarefaction of the interstitial element, only islands of a nitride or oxide film, rather than a continuous film, are formed on the titanium surface (Fedirko & Pohrelyuk, 1995). In this case, the kinetics of saturation is sensitive to the interstitial element transfer to the surface of titanium and the intensity of surface processes. Thus, the surface interstitial element concentration depends on time. The defectiveness of the metal and its influences on the diffusion activity and reactivity of the interstitial element also play an important role. Due to lattice defects, in particular vacancies, dislocations of the surface layer, the probability of inequilibrium segregations of the interstitial element increases as a result of the chemical interaction with titanium, which introduces changes in the diffusive saturation of titanium with the interstitial element. That is why it is incorrect to describe analytically the kinetics of saturation with the known Fick's equation by setting constant values of the surface concentration (the first boundary-value task). This indicates the actuality and importance of an adequate choice of boundary conditions for the formulation of the corresponding diffusion problem. To do this, it is necessary to have a clear notion of the interrelation of the physico‐ chemical processes on a surface and near it.

The interaction of titanium with a rarefied gas atmosphere can be schematically illustrated by following processes with relevant parameters characterizing them (Fig. 1):


Processes enumerated in clause a) can be interpreted as a two-stage reaction which consists of a diffusive stage, described by the constant rate hD, and a stage of chemisorption at a constant rate hR. Then, according to the law of summation of kinetic resistances, we have

Let us represent the inequilibrium processes mentioned in clauses (a) – (c) in the formulation of the diffusion problem through the adequate setting of the boundary conditions of mass

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

Since the aim of the diffusive saturation of a titanium sample is primarily to harden its surface layer, as an object of the analytic investigation of the kinetics of this process, we chose a halfspace (0 ≤ x < ∞) with the initial (τ=0) interstitial element concentration C(x,τ=0)=C0. For the calculation of the concentration of dissolved interstitial element in the titanium sample it is need to solve Fick's diffusion equation considering initial and boundary conditions (Matychak

<sup>0</sup> *D C x x C x for x C x C C* ¶ ¶ =¶ ¶ > < <¥ = ¥ = ( , ) / ( , ) / 0, 0 , ( ,0) ( , ) ,

0

Here Ceq is a quasiequilibrium surface concentration of the interstitial element, which depends

The boundary condition (2) was proposed on the basis of notions of a contact layer with a thickness 2δ between the metal and the environment, in which processes of migration of an impurity and the chemical reaction of the first order (Fig. 1) occur (Prytula et al., 2005). Using a mathematical procedure (Fedirko et al., 2005; Matychak, 1999), this layer was replaced by an imaginary layer of zero thickness (2δ→ 0) with a mass capacity ω. For such a transition, we introduced averaged characteristics of the contact layer, specifically the surface concentration of the impurity C(+0,τ). Note that neglecting the contact layer ω=0, and, correspondingly, k = 0, from Eq.(2) we obtain the typical boundary condition of mass exchange of the third kind

/ [ (0, )] <sup>0</sup> -¶ ¶ = -

 t lim ®+

Let us point at the characteristic peculiarities of the proposed generalized boundary condition (2), which distinguishes it from the quasistationary boundary condition (3). The latter one reflects to a certain extent the real situation of the asymptotic approximation of the surface concentration to its equilibrium value. At the same time, according to condition (3), all atoms

<sup>=</sup> = = *eq <sup>x</sup>*

<sup>=</sup> *D C x hC C*

0 (0, ) ( , )

× = - - - + ¶ ¶ =+ / ( )( ) / 0. *eq dC d h C C k C C D C x for x* (2)

t

*C C x C const* (4)

*x eq* (3)

t

(1)

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

49

 t

 tt

or even the simpler condition (D/h →0) of the first kind:

t

exchange on the surface.

et al., 2007):

*2.1.2. Mathematical description*

2 2

t

w

(Raichenko, 1981):

 t

on its partial pressure in the atmosphere.

**Figure 1.** Scheme of mass fluxes in the Ti–A system

*<sup>h</sup>* <sup>−</sup><sup>1</sup> <sup>=</sup>*<sup>h</sup> <sup>D</sup>* <sup>−</sup><sup>1</sup> <sup>+</sup> *<sup>h</sup> <sup>R</sup>* −1 . The introduced kinetic parameters (Рі =D, h, k) of the model representation characterize the aforementioned thermoactivated physicochemical processes with the corre‐ sponding activation energies (Ei ), according to the dependence Рі =Р0іехр(-Eі /RT). The effective parameters h and k depend not only on temperature, but also largely on the partial pressure of interstitial element and defectiveness of the material. That is why they are usually calculated from specific experimental data of the kinetics of saturation. In particular, the experimental data for the spatial distribution of the interstitial element in surface layer of titanium after various exposure time (τ1 and τ2) allow to use a graphical method to compute the mass transfer coefficient h and the surface content (Ceq) of nitrogen which is in equilibrium with the atmos‐ phere (Fig. 2) (Matychak et al., 2009).

**Figure 2.** Graphical method for the determination of the h and Ceq parameters

Let us represent the inequilibrium processes mentioned in clauses (a) – (c) in the formulation of the diffusion problem through the adequate setting of the boundary conditions of mass exchange on the surface.

#### *2.1.2. Mathematical description*

*<sup>h</sup>* <sup>−</sup><sup>1</sup> <sup>=</sup>*<sup>h</sup> <sup>D</sup>*

<sup>−</sup><sup>1</sup> <sup>+</sup> *<sup>h</sup> <sup>R</sup>* −1

sponding activation energies (Ei

**Figure 1.** Scheme of mass fluxes in the Ti–A system

48 Titanium Alloys - Advances in Properties Control

phere (Fig. 2) (Matychak et al., 2009).

. The introduced kinetic parameters (Рі

**Figure 2.** Graphical method for the determination of the h and Ceq parameters

characterize the aforementioned thermoactivated physicochemical processes with the corre‐

parameters h and k depend not only on temperature, but also largely on the partial pressure of interstitial element and defectiveness of the material. That is why they are usually calculated from specific experimental data of the kinetics of saturation. In particular, the experimental data for the spatial distribution of the interstitial element in surface layer of titanium after various exposure time (τ1 and τ2) allow to use a graphical method to compute the mass transfer coefficient h and the surface content (Ceq) of nitrogen which is in equilibrium with the atmos‐

), according to the dependence Рі

=D, h, k) of the model representation

/RT). The effective

=Р0іехр(-Eі

Since the aim of the diffusive saturation of a titanium sample is primarily to harden its surface layer, as an object of the analytic investigation of the kinetics of this process, we chose a halfspace (0 ≤ x < ∞) with the initial (τ=0) interstitial element concentration C(x,τ=0)=C0. For the calculation of the concentration of dissolved interstitial element in the titanium sample it is need to solve Fick's diffusion equation considering initial and boundary conditions (Matychak et al., 2007):

$$\text{'}\,\text{D}\widehat{\text{C}}\{\text{x},\text{\text{\textdegree}}\}/\left\langle\text{\textdegree\text{x}}^{2}=\text{\textdegree C}\{\text{x},\text{\textdegree\text{}}\}/\left\langle\text{\textdegree\text{\textdegree}}\quad\text{for}\,\,\text{\textdegree}\,\text{ }\pi>0,\,\text{\textdegree}<\text{\textdegree}\,\text{C}\{\text{x},\text{\textdegree}\}=\text{C}\{\text{\textdegree\text{}},\text{\textdegree}\right\}=\text{C}\_{\text{0}}\,\text{}\,\tag{1}$$

$$
\rho \cdot d\mathbf{C} / d\tau = h(\mathbf{C}\_{eq} - \mathbf{C}) - k(\mathbf{C} - \mathbf{C}\_0) + D\hat{\mathbf{c}}\mathbf{C} / \hat{\boldsymbol{\alpha}}\tag{2}
$$

Here Ceq is a quasiequilibrium surface concentration of the interstitial element, which depends on its partial pressure in the atmosphere.

The boundary condition (2) was proposed on the basis of notions of a contact layer with a thickness 2δ between the metal and the environment, in which processes of migration of an impurity and the chemical reaction of the first order (Fig. 1) occur (Prytula et al., 2005). Using a mathematical procedure (Fedirko et al., 2005; Matychak, 1999), this layer was replaced by an imaginary layer of zero thickness (2δ→ 0) with a mass capacity ω. For such a transition, we introduced averaged characteristics of the contact layer, specifically the surface concentration of the impurity C(+0,τ). Note that neglecting the contact layer ω=0, and, correspondingly, k = 0, from Eq.(2) we obtain the typical boundary condition of mass exchange of the third kind (Raichenko, 1981):

$$-\text{D}\partial\mathcal{C} \;/\left.\partial\mathbf{x}\right|\_{\mathbf{x}=\mathbf{0}} = \hbar[\mathbf{C}\_{eq} - \mathcal{C}(0,\tau)]\tag{3}$$

or even the simpler condition (D/h →0) of the first kind:

$$\mathcal{C}(0,\tau) = \lim\_{\mathbf{x} \to \mathbf{0}} \mathcal{C}(\mathbf{x}, \tau) = \mathcal{C}\_{eq} = const \tag{4}$$

Let us point at the characteristic peculiarities of the proposed generalized boundary condition (2), which distinguishes it from the quasistationary boundary condition (3). The latter one reflects to a certain extent the real situation of the asymptotic approximation of the surface concentration to its equilibrium value. At the same time, according to condition (3), all atoms adsorbed on the surface diffuse into the metal and are distributed in compliance with the law of diffusion. That is why, according to condition (3), when D→ 0, we have C(0,τ)=Ceq. That is, the surface concentration becomes equilibrium instantaneously and is independent of time. Thus, from the proposed generalized nonstationary boundary condition (2), in the absence of diffusion D→ 0 of the impurity in the volume of the metal, we have the following time dependence of its surface concentration:

$$\text{C}(0,\tau) = \frac{h\text{C}\_{\text{eq}} + k\text{C}\_{0}}{h+k} - \frac{h\text{(C}\_{\text{eq}} - \text{C}\_{0})}{h+k} \exp\left[-\frac{(h+k)\tau}{\alpha}\right],\tag{5}$$

The evolution of spatial distribution of dissolved nitrogen in titanium (Fig. 3) during its thermodiffusive saturation gives a solution of the equations (1), (2), which in the analytic form

> 1 1 1 22 11 ( , ) ( ) / (2 ) ( , ) ( , ) / ( ),


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

tt

= +- - × D (8)

 t

=× =× (9)

 t

C (below the temperature of allotropic

C for 1, 5, and 10 h in a rarefied (to

C/h).

C/sec. After an isothermal exposure, the

(0,τ) in the bound state

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51

t

*Dτ*) ⋅ *erfc q*<sup>1</sup> *Dτ* + *x* / (2 *Dτ*) ,

*Dτ*) ⋅ *erfc q*<sup>2</sup> *Dτ* + *x* / (2 *Dτ*) ,

Specifically, the surface concentration of dissolved interstitial element is

2 21 1 *C hhk f q f q h D* (0, ) / ( ) [ ( ) / ( ) / ] / ( ) ,

tt

Here *<sup>C</sup>*¯(*<sup>x</sup>*, *<sup>τ</sup>*)=(*C*(*x*, *<sup>τ</sup>*)−*C*0) / (*Ceq* <sup>−</sup>*C*0) is the relative change of the interstitial element concen‐

The obtained results for the diffusive saturation of titanium with nitrogen under low partial

The samples (10×15×1 mm) of VT1-0 commercially pure titanium were investigated after an

1 Pa) dynamic nitrogen atmosphere (the specific inleakage rate was 7×10-3 Ра/sec). Before treatment, samples were ground to Ra = 0,4 µm, washed in acetone and alcohol, and dried.

Upon loading the samples in an ampoule, the system was pumped down to a pressure of 10-3 Ра, then the nitrogen was blown through, and required parameters of the gas medium

samples were furnace-cooled in nitrogen (the mean cooling rate was 100 0

transformation) were confirmed by the experimental results (Matychak et al., 2009).

2 2 1 1 12 2 2 *f q D erfc q D f q D erfc q D* ( ) exp( ) ( ) , ( ) exp( ) ( ).

 tt

tration in the solid solution in α-titanium. Its surface concentration C\*

and the total concentration CΣ(0,τ) are determined by formulas (6).

is as follows (Matychak et al., 2007):

2

2

t

 t

pressure (1 Pa) in the temperature range of 750-850 0

isothermal exposure at temperatures of 750, 800, and 850 0

were set. Heating was performed at a rate 0.04 0

**2.2. Technique and results of experimental tests**

*q*<sup>1</sup> =(1 + *Δ*)/ 2*ω* , *q*<sup>2</sup> =(1−*Δ*)/ 2*ω* , *Δ* = 1−4*ω*(*h* + *k*) / *D* .

t

*F*1(*x*, *τ*)=exp(*q*1*x* + *q*<sup>1</sup>

*F*2(*x*, *τ*)=exp(*q*2*x* + *q*<sup>2</sup>

t

where

where

*2.2.1. Methods*

which is determined by the intensity of surface processes. One more peculiarity of proposed non-stationary condition (2) concerns the action of the operator d/dτ, which describes the kinetics of accumulation of the interstitial element in the vicinity of the interface. In particular, the difference between the flux j1 of the interstitial element from the environment to the surface (x=-0) and its diffusion flux j = jdiff in the metal (x=+0) determine the kinetics of accumulation (segregation) of the interstitial element in the vicinity of the interface as a result of the chemical interaction (Fig. 1). The interstitial element is accumulated in the contact layer on defects modeled as "traps" for the diffusant. Then its concentration in the surface layer in the bound state (in nitride or oxide compounds) and its total concentration (in the solid solution and compounds) in the vicinity of the surface are computed from the relations

$$\mathbf{C}^\*(0,\tau) = \left(\frac{k}{\alpha}\right) \Big| \begin{bmatrix} \mathbf{C}(0,t) - \mathbf{C}\_0 \end{bmatrix} dt, \qquad \mathbf{C}\_{\Sigma}(0,\tau) = \mathbf{C}(0,\tau) + \mathbf{C}^\*(0,\tau). \tag{6}$$

Thus, only a part of all adsorbed interstitial element atoms dissolves in the metal and diffuses in the volume. The remaining nitrogen atoms segregate in the form of compounds near the surface (Fig. 3).

**Figure 3.** Evolution of the interstitial element distribution during diffusive saturation: (a: τ= τ1, b: τ= τ2; τ1<τ<sup>2</sup>

The evolution of spatial distribution of dissolved nitrogen in titanium (Fig. 3) during its thermodiffusive saturation gives a solution of the equations (1), (2), which in the analytic form is as follows (Matychak et al., 2007):

$$\overline{\mathbf{C}}(\mathbf{x},\tau) = h(\mathbf{h}+\mathbf{k})^{-1} \text{erfc}\left[\mathbf{x} \mid \left(\mathbf{2\sqrt{D\tau}}\right)\right] - h\left[q\_2^{-1}\mathbf{F}\_2(\mathbf{x},\tau) - q\_1^{-1}\mathbf{F}\_1(\mathbf{x},\tau)\right] / \left(\mathbf{D}\Lambda\right) \tag{7}$$

where

adsorbed on the surface diffuse into the metal and are distributed in compliance with the law of diffusion. That is why, according to condition (3), when D→ 0, we have C(0,τ)=Ceq. That is, the surface concentration becomes equilibrium instantaneously and is independent of time. Thus, from the proposed generalized nonstationary boundary condition (2), in the absence of diffusion D→ 0 of the impurity in the volume of the metal, we have the following time

eq 0 ( ) eq 0 ( ) (0, ) exp ,

compounds) in the vicinity of the surface are computed from the relations

0

\* \*

(0, ) [ (0, ) ] , (0, ) (0, ) (0, ).

= - =+ ç ÷

**Figure 3.** Evolution of the interstitial element distribution during diffusive saturation: (a: τ= τ1, b: τ= τ2; τ1<τ<sup>2</sup>

å

Thus, only a part of all adsorbed interstitial element atoms dissolves in the metal and diffuses in the volume. The remaining nitrogen atoms segregate in the form of compounds near the

tt

è ø<sup>ò</sup> *<sup>k</sup> <sup>C</sup> C t C dt C C C* (6)

 t

0

w

æ ö

t

t

surface (Fig. 3).

<sup>+</sup> - é ù <sup>+</sup> =- -ê ú + + ê ú ë û

which is determined by the intensity of surface processes. One more peculiarity of proposed non-stationary condition (2) concerns the action of the operator d/dτ, which describes the kinetics of accumulation of the interstitial element in the vicinity of the interface. In particular, the difference between the flux j1 of the interstitial element from the environment to the surface (x=-0) and its diffusion flux j = jdiff in the metal (x=+0) determine the kinetics of accumulation (segregation) of the interstitial element in the vicinity of the interface as a result of the chemical interaction (Fig. 1). The interstitial element is accumulated in the contact layer on defects modeled as "traps" for the diffusant. Then its concentration in the surface layer in the bound state (in nitride or oxide compounds) and its total concentration (in the solid solution and

*hC kC hC C h k*

t

w

*hk hk* (5)

dependence of its surface concentration:

50 Titanium Alloys - Advances in Properties Control

*C*

t

*F*1(*x*, *τ*)=exp(*q*1*x* + *q*<sup>1</sup> 2 *Dτ*) ⋅ *erfc q*<sup>1</sup> *Dτ* + *x* / (2 *Dτ*) , *F*2(*x*, *τ*)=exp(*q*2*x* + *q*<sup>2</sup> 2 *Dτ*) ⋅ *erfc q*<sup>2</sup> *Dτ* + *x* / (2 *Dτ*) , *q*<sup>1</sup> =(1 + *Δ*)/ 2*ω* , *q*<sup>2</sup> =(1−*Δ*)/ 2*ω* , *Δ* = 1−4*ω*(*h* + *k*) / *D* .

Specifically, the surface concentration of dissolved interstitial element is

$$\overline{\mathbb{C}}(0,\tau) = h \left/ \left( h + k \right) - \left[ f\_2(\tau) \left/ q\_2 - f\_1(\tau) \right/ q\_1 \right] \cdot h \left/ \left( D\Delta \right) \right. \tag{8}$$

where

$$f\_1(\tau) = \exp(q\_1^2 D\tau) \cdot \text{erfc}(q\_1 \sqrt{D\tau}) \quad f\_2(\tau) = \exp(q\_2^2 D\tau) \cdot \text{erfc}(q\_2 \sqrt{D\tau}).\tag{9}$$

Here *<sup>C</sup>*¯(*<sup>x</sup>*, *<sup>τ</sup>*)=(*C*(*x*, *<sup>τ</sup>*)−*C*0) / (*Ceq* <sup>−</sup>*C*0) is the relative change of the interstitial element concen‐ tration in the solid solution in α-titanium. Its surface concentration C\* (0,τ) in the bound state and the total concentration CΣ(0,τ) are determined by formulas (6).

The obtained results for the diffusive saturation of titanium with nitrogen under low partial pressure (1 Pa) in the temperature range of 750-850 0 C (below the temperature of allotropic transformation) were confirmed by the experimental results (Matychak et al., 2009).

#### **2.2. Technique and results of experimental tests**

#### *2.2.1. Methods*

The samples (10×15×1 mm) of VT1-0 commercially pure titanium were investigated after an isothermal exposure at temperatures of 750, 800, and 850 0 C for 1, 5, and 10 h in a rarefied (to 1 Pa) dynamic nitrogen atmosphere (the specific inleakage rate was 7×10-3 Ра/sec). Before treatment, samples were ground to Ra = 0,4 µm, washed in acetone and alcohol, and dried.

Upon loading the samples in an ampoule, the system was pumped down to a pressure of 10-3 Ра, then the nitrogen was blown through, and required parameters of the gas medium were set. Heating was performed at a rate 0.04 0 C/sec. After an isothermal exposure, the samples were furnace-cooled in nitrogen (the mean cooling rate was 100 0 C/h).

Commercially-pure gaseous nitrogen was used, which, according to a technical specification, contained not more than 0.4 vol. % of oxygen and 0.07 g/m3 of water vapor. Before feeding in the reaction space of a furnace, nitrogen was purified from oxygen and moisture by passing through a capsule with silica gel and titanium chips heated to a temperature higher by 50 0C than the saturation temperature. After every 3–4 tests, to restore the efficiency of the system for purification of nitrogen, silica gel was annealed at 180 0C for 3–4 h, and titanium chips were replaced by new ones. Due to this, the oxygen concentration in nitrogen ranged from 0.01 to 0.03 vol. %.

α-solid solution, the layer of which increases in thickness as the temperature–time parameters

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

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53

The surface microhardness of titanium changes as a result of nitride formation. After nitriding at 750 and 800 0C for 5 h and at 850 0C for 1 h when nitride formation is not very intensive, which is evidenced by reflexes of relative intensity of the nitride of lower valence (Ti2N), the surface microhardness of titanium ranges from 4.4 to 7.3 GPa (Fig. 6). As the exposure time increases to 5 – 10 h at 850 0C when nitride islands cover a major part of the surface of the alloy, it rises to 10–13 GPa. The temperature–time nitriding parameters affect the surface micro‐ hardness of titanium and the depth of the nitrided layer, which increases monotonically in thickness with the temperature and time of exposure in a nitrogen-containing atmosphere (Fig. 6 c, d). Temperature influences analogously the surface microhardness for a given exposure time (Fig. 6 a). The effect of the time of saturation at 850 0C is somewhat different. As the exposure time increases from 1 to 5 h, the microhardness increases 2.5 times, and as the

(a) (b) (c)

**Figure 5.** Microstructure of surface layers of VT1-0 titanium nitrided in a rarefied dynamic nitrogen atmosphere (1 Pa):

Curves of the distribution of the microhardness over a cross-section of the hardened surface layers shift in the direction of higher values of the hardness with increases in the saturation temperature (Fig. 7 a) and saturation time (Fig. 7 b). During nitriding at a temperature of 850

C for 5 h, the surface hardening of titanium is more significant than those at 750 and 800 0C

increase; its grains are etched less than the matrix (Fig. 5).

exposure time increases from 5 to 10 h, it rises only by 1.67 GPa (Fig. 6 b).

(d) (e)

(a) 750 0C, 5 h;(b) 800 0C, 5 h; (c) 850 0C, 1 h; (d) 850 0C, 5 h; (e) 850 0C, 10 h

and the same exposure time (Fig. 7 a).

0

The microstructure of "oblique" microsections of samples was studied with a "Epiquant" microscope equipped with a camera and a computer with digital image analysis software.

The surface hardening was assessed based on the microhardness measured with a PMT-3M unit under a load of 0.49 N. As the depth of a nitrided layer, the depth of a zone was accepted in which the microhardness was higher than that of the core by δН=0.2 GРа (Fedirko & Pohrelyuk, 1995).

### *2.2.2. Results of experimental investigations*

An analysis of experimental data of the influence of the partial nitrogen pressure on the saturation of titanium alloys during nitriding indicates that, in the range of rarefaction of the active gas 0.1– 10 Pa (the specific inleakage rate ranged from 7×10-2 to 7×10-4 Ра/sec), the kinetics of nitriding is sensitive to processes related to the nitrogen feed to the gas–metal interaction zone (Fedirko & Pohrelyuk,1995). Under such conditions, in a certain time range, which depends on the nitriding temperature, one can maintain the dynamic equilibrium between the adsorbed nitrogen and nitrogen transported by diffusion in the depth of the titanium matrix and shift significantly in time the beginning of the formation of a continuous nitride film. Metallographic analysis of the surface of VT1-0 titanium samples nitrided in this range of gas-dynamic (1 Pa; 7×10-3Ра/sec) and temperature–time (750-850 <sup>0</sup> C; 5 h) parameters confirmed the absence of a continuous nitride film on their surfaces (Fig.4). Instead of it, we observe the initiation and growth of nitride islands, predominantly between grains (Fig. 4 a-c).

**Figure 4.** Surface of VT1-0 titanium after nitriding in a rarefied dynamic nitrogen atmosphere (1 Pa): (a) 750 0C, 5 h; (b) 800 0C, 5 h;(c) 850 0C, 5 h; (d) 850 0C, 10 h

After an exposure for 10 h at 850 0C (Fig. 4 d), almost all grain boundaries contain nitrides, which favor the formation of a surface network from nitride inclusions and the formation of the corresponding surface topography. The dissolution of nitrogen in titanium stabilizes an α-solid solution, the layer of which increases in thickness as the temperature–time parameters increase; its grains are etched less than the matrix (Fig. 5).

Commercially-pure gaseous nitrogen was used, which, according to a technical specification,

the reaction space of a furnace, nitrogen was purified from oxygen and moisture by passing through a capsule with silica gel and titanium chips heated to a temperature higher by 50 0C than the saturation temperature. After every 3–4 tests, to restore the efficiency of the system for purification of nitrogen, silica gel was annealed at 180 0C for 3–4 h, and titanium chips were replaced by new ones. Due to this, the oxygen concentration in nitrogen ranged from 0.01 to

The microstructure of "oblique" microsections of samples was studied with a "Epiquant" microscope equipped with a camera and a computer with digital image analysis software.

The surface hardening was assessed based on the microhardness measured with a PMT-3M unit under a load of 0.49 N. As the depth of a nitrided layer, the depth of a zone was accepted in which the microhardness was higher than that of the core by δН=0.2 GРа (Fedirko &

An analysis of experimental data of the influence of the partial nitrogen pressure on the saturation of titanium alloys during nitriding indicates that, in the range of rarefaction of the active gas 0.1– 10 Pa (the specific inleakage rate ranged from 7×10-2 to 7×10-4 Ра/sec), the kinetics of nitriding is sensitive to processes related to the nitrogen feed to the gas–metal interaction zone (Fedirko & Pohrelyuk,1995). Under such conditions, in a certain time range, which depends on the nitriding temperature, one can maintain the dynamic equilibrium between the adsorbed nitrogen and nitrogen transported by diffusion in the depth of the titanium matrix and shift significantly in time the beginning of the formation of a continuous nitride film. Metallographic analysis of the surface of VT1-0 titanium samples nitrided in this range of gas-dynamic (1 Pa; 7×10-3Ра/sec) and

film on their surfaces (Fig.4). Instead of it, we observe the initiation and growth of nitride islands,

(a) (b) (c) (d)

**Figure 4.** Surface of VT1-0 titanium after nitriding in a rarefied dynamic nitrogen atmosphere (1 Pa): (a) 750 0C, 5 h;

After an exposure for 10 h at 850 0C (Fig. 4 d), almost all grain boundaries contain nitrides, which favor the formation of a surface network from nitride inclusions and the formation of the corresponding surface topography. The dissolution of nitrogen in titanium stabilizes an

C; 5 h) parameters confirmed the absence of a continuous nitride

of water vapor. Before feeding in

contained not more than 0.4 vol. % of oxygen and 0.07 g/m3

0.03 vol. %.

Pohrelyuk, 1995).

*2.2.2. Results of experimental investigations*

52 Titanium Alloys - Advances in Properties Control

temperature–time (750-850 <sup>0</sup>

predominantly between grains (Fig. 4 a-c).

(b) 800 0C, 5 h;(c) 850 0C, 5 h; (d) 850 0C, 10 h

The surface microhardness of titanium changes as a result of nitride formation. After nitriding at 750 and 800 0C for 5 h and at 850 0C for 1 h when nitride formation is not very intensive, which is evidenced by reflexes of relative intensity of the nitride of lower valence (Ti2N), the surface microhardness of titanium ranges from 4.4 to 7.3 GPa (Fig. 6). As the exposure time increases to 5 – 10 h at 850 0C when nitride islands cover a major part of the surface of the alloy, it rises to 10–13 GPa. The temperature–time nitriding parameters affect the surface micro‐ hardness of titanium and the depth of the nitrided layer, which increases monotonically in thickness with the temperature and time of exposure in a nitrogen-containing atmosphere (Fig. 6 c, d). Temperature influences analogously the surface microhardness for a given exposure time (Fig. 6 a). The effect of the time of saturation at 850 0C is somewhat different. As the exposure time increases from 1 to 5 h, the microhardness increases 2.5 times, and as the exposure time increases from 5 to 10 h, it rises only by 1.67 GPa (Fig. 6 b).

**Figure 5.** Microstructure of surface layers of VT1-0 titanium nitrided in a rarefied dynamic nitrogen atmosphere (1 Pa): (a) 750 0C, 5 h;(b) 800 0C, 5 h; (c) 850 0C, 1 h; (d) 850 0C, 5 h; (e) 850 0C, 10 h

Curves of the distribution of the microhardness over a cross-section of the hardened surface layers shift in the direction of higher values of the hardness with increases in the saturation temperature (Fig. 7 a) and saturation time (Fig. 7 b). During nitriding at a temperature of 850 0 C for 5 h, the surface hardening of titanium is more significant than those at 750 and 800 0C and the same exposure time (Fig. 7 a).

0 0 *Hx H a Cx C* (,) [(,) ]

0 max 0 *Hx Hx H H H Cx* ( , ) [ ( , ) ]/[ ] ( , )

= - -º

 t

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

Then the relative change in the microhardness in the diffusion zone due to dissolved nitrogen

Here H0 is the microhardness of the initial titanium sample, Hmax is the microhardness of titanium at a maximum concentration of dissolved nitrogen Cmax=Ceq, and a is the proportion‐ ality coefficient. Relation (11) indicates the possibility to plot the calculated relative concen‐ trations *<sup>C</sup>*¯(*<sup>x</sup>*, *<sup>τ</sup>*) of nitrogen and experimental data of the relative change in the microhardness

The roles of the time parameter and temperature are illustrated by analytic curves of the nitrogen content (Fig. 8, Fig. 9, Fig. 10), constructed from relations (6) – (8), and experimental data (see Fig. 4) using relation (11). For analytic calculations the following parameters were

Diffusion coefficients of nitrogen were calculated according to the dependence

surface concentration of dissolved nitrogen C(0, τ) (curve 1) and its concentration in nitride

**Figure 8.** Time dependence of the surface nitrogen content ( (a): curve (1) in the solid solution*C*(0, τ); (2) in nitride inclusions *<sup>C</sup>* <sup>∗</sup>(0, <sup>τ</sup>); (3) total content *C*∑(0, <sup>τ</sup>)and its distribution in the surface layer of titanium (b) for different expo‐ sure times: (1), τ = 1 h; (2) τ = 5 h, (3) and (4) τ = 10 h (curve (4) was constructed for the condition *C*(0, τ)=const,*h* →∞)

at a nitriding temperature T = 850 0C. Marks correspond to experimental data (H(x, τ))

(0, τ) (curve 2), as well as its total content CΣ(0, τ) (curve 3, Fig. 8 a), increase.

/sec, h=1∙10-8 cm2

= +× - (10)

t

(11)

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

55

/sec,

/ sec; for T=800 0C – D = 3.4∙10-11 cm2

/sec, h=1∙10-7 cm/sec; ω = 10-5 cm, k/h = 0.002.

/sec, Е=214.7 kJ/mole (Metin & Inal, 1989). An analysis

C in nitrogen, with increase in the saturation time, both the

t

 t

(neglecting the contribution of nitride inclusions) is as follows:

t

C – D =1∙10-11 cm2

of these curves gives grounds for the following conclusions.

h = 3∙10-8 cm/sec; for T = 850 0C – D=1∙10-10 cm2

*D* =*D*0exp(− *E* / *RT* ), where D0=0.96 cm2

For an isothermal exposure T=850 0

*<sup>H</sup>*¯(*<sup>x</sup>*, *<sup>τ</sup>*), on the same ordinate axis.

used: for T=750 0

inclusions C\*

**Figure 6.** Dependences of the surface microhardness (a, b) and the depth of the hardened zone (c, d) on the tempera‐ ture–time nitriding parameters of VT1-0 titanium in a rarefied dynamic atmosphere (1 Pa)

**Figure 7.** Distribution of the microhardness over the cross-section of surface layers of VT1-0 titanium nitrided in a rare‐ fied dynamic atmosphere (1 Pa) depending on the temperature (a) and time of an isothermal exposure (b): (1) 750 0C, 5 h; (2) 800 0C, 5 h; (3) 850 0C, 5 h; (4) 850 0C, 1 h; (5) 850 0C, 10 h

#### **2.3. Assessment of the temperature–time parameters of nitriding and analysis of results**

It is known that the profile of nitrogen concentration in the surface layer of titanium substan‐ tially affects its physicomechanical properties. For experimental investigations of hardened nitrided layers, the method of layer-by layer testing of microhardness, which substantially depends on the content of dissolved nitrogen in titanium, is widely used. Let us use a known linear dependence of change of the microhardness on the concentration of an interstitial impurities in titanium (Korotaev et al., 1989):

Modelling of Microstructural Evolution of Titanium During Diffusive Saturation by Interstitial Elements http://dx.doi.org/10.5772/54626 55

$$H(\mathbf{x}, \tau) = H\_0 + a \cdot [\mathbf{C}(\mathbf{x}, \tau) - \mathbf{C}\_0] \tag{10}$$

Then the relative change in the microhardness in the diffusion zone due to dissolved nitrogen (neglecting the contribution of nitride inclusions) is as follows:

$$\overline{H}(\mathbf{x},\tau) = \left[H(\mathbf{x},\tau) - H\_0\right] / \left[H\_{\text{max}} - H\_0\right] \equiv \overline{\mathbf{C}}(\mathbf{x},\tau) \tag{11}$$

Here H0 is the microhardness of the initial titanium sample, Hmax is the microhardness of titanium at a maximum concentration of dissolved nitrogen Cmax=Ceq, and a is the proportion‐ ality coefficient. Relation (11) indicates the possibility to plot the calculated relative concen‐ trations *<sup>C</sup>*¯(*<sup>x</sup>*, *<sup>τ</sup>*) of nitrogen and experimental data of the relative change in the microhardness *<sup>H</sup>*¯(*<sup>x</sup>*, *<sup>τ</sup>*), on the same ordinate axis.

The roles of the time parameter and temperature are illustrated by analytic curves of the nitrogen content (Fig. 8, Fig. 9, Fig. 10), constructed from relations (6) – (8), and experimental data (see Fig. 4) using relation (11). For analytic calculations the following parameters were used: for T=750 0 C – D =1∙10-11 cm2 /sec, h=1∙10-8 cm2 / sec; for T=800 0C – D = 3.4∙10-11 cm2 /sec, h = 3∙10-8 cm/sec; for T = 850 0C – D=1∙10-10 cm2 /sec, h=1∙10-7 cm/sec; ω = 10-5 cm, k/h = 0.002. Diffusion coefficients of nitrogen were calculated according to the dependence *D* =*D*0exp(− *E* / *RT* ), where D0=0.96 cm2 /sec, Е=214.7 kJ/mole (Metin & Inal, 1989). An analysis of these curves gives grounds for the following conclusions.

For an isothermal exposure T=850 0 C in nitrogen, with increase in the saturation time, both the surface concentration of dissolved nitrogen C(0, τ) (curve 1) and its concentration in nitride inclusions C\* (0, τ) (curve 2), as well as its total content CΣ(0, τ) (curve 3, Fig. 8 a), increase.

**Figure 7.** Distribution of the microhardness over the cross-section of surface layers of VT1-0 titanium nitrided in a rare‐ fied dynamic atmosphere (1 Pa) depending on the temperature (a) and time of an isothermal exposure (b): (1) 750 0C,

**Figure 6.** Dependences of the surface microhardness (a, b) and the depth of the hardened zone (c, d) on the tempera‐

ture–time nitriding parameters of VT1-0 titanium in a rarefied dynamic atmosphere (1 Pa)

54 Titanium Alloys - Advances in Properties Control

**2.3. Assessment of the temperature–time parameters of nitriding and analysis of results**

It is known that the profile of nitrogen concentration in the surface layer of titanium substan‐ tially affects its physicomechanical properties. For experimental investigations of hardened nitrided layers, the method of layer-by layer testing of microhardness, which substantially depends on the content of dissolved nitrogen in titanium, is widely used. Let us use a known linear dependence of change of the microhardness on the concentration of an interstitial

5 h; (2) 800 0C, 5 h; (3) 850 0C, 5 h; (4) 850 0C, 1 h; (5) 850 0C, 10 h

impurities in titanium (Korotaev et al., 1989):

**Figure 8.** Time dependence of the surface nitrogen content ( (a): curve (1) in the solid solution*C*(0, τ); (2) in nitride inclusions *<sup>C</sup>* <sup>∗</sup>(0, <sup>τ</sup>); (3) total content *C*∑(0, <sup>τ</sup>)and its distribution in the surface layer of titanium (b) for different expo‐ sure times: (1), τ = 1 h; (2) τ = 5 h, (3) and (4) τ = 10 h (curve (4) was constructed for the condition *C*(0, τ)=const,*h* →∞) at a nitriding temperature T = 850 0C. Marks correspond to experimental data (H(x, τ))

The nitrogen content in the surface layer and the depth of the diffusion zone change additively as the exposure time (curves 1 – 3, Fig. 8 b) and the temperature of the isothermal exposure (curves 1 – 3, Fig. 9) change.

An analysis of these curves confirms an adequate increase in the nitrogen content over the whole depth of the diffusion zone and the increase in the depth of this zone as the treatment time and temperature increase (curves 1 – 3, Fig. 10). It was found that, in the statement of the first boundaryvalue problem (*h* →0 , *C*(0, *τ*)=*Const*),overestimated values of the nitrogen concentration were obtained (curves 4 in Fig. 8 b Fig. 10). The calculated data show that the surface phenom‐ ena affect substantially not only the surface concentration of nitrogen (Fig. 10 a) (corresponding‐ ly, the surface hardness of titanium), but also the nitrogen content in layers more remote from

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

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

57

**Figure 10.** Time dependence of the content of dissolved nitrogen on the surface (a) and at a depth x = 10 μm (b), and the depth of the nitrided layer (c) for different saturation temperatures: (1) 750 0C; (2) 800 0C; (3, 4) 850 <sup>0</sup>C (curves 4

Thus, the presented results indicate the critical role of the surface phenomena (adsorption and chemisorption) in the kinetic regularities of nitriding of titanium in a rarefied atmosphere. The calculated data obtained on the basis of the solution of the diffusion task using the nonsta‐ tionary boundary condition (2) indicate that its model representation reflects rather satisfac‐ torily the main tendencies of the high-temperature interaction of titanium with rarefied nitrogen. For the provision of a specified hardened layer, the proposed model gives scientifi‐ cally justified recommendations on external parameters (exposure temperature and time) of

were constructed for the condition *C*(0, τ) = const, h → ∞)

nitriding of titanium.

the surface (Fig. 10 b), and the depth of the hardened diffusion zone (Fig. 10 c).

On the whole, the analytic calculations of content profiles correlate well with the experimental results of relative changes in the microhardness of the surface layer (Fig. 8 b, Fig. 9). The corresponding curves have a monotonic character; the microhardness over the cross-section of the sample decreases gradually in the depth of the metal until it attains values characteristic for titanium. At the same time, there are insignificant disagreements between the theoretical and experimental results. In particular, for short exposures, the zone of change of the micro‐ hardness extends to a larger depth than the value which follows from the nitrogen distribution (Fig. 8 b). This can be explained by an insignificant content of oxygen, which is characterized by a larger diffusion mobility than nitrogen. For larger exposure times when the percentage of nitrogen is larger than that of oxygen, this effect is leveled.

**Figure 9.** Distribution of nitrogen in the surface layer of titanium after an exposure τ= 5 h for different saturation tem‐ perature: (1), 750 0C; (2), 800 0C; (3), 850 0C. Marks correspond to experimental data *H*¯(*x*, <sup>τ</sup>)

Some disagreement between the calculated nitrogen distribution and experimental data of change in the microhardness is also observed near the surface, particularly as the time (Fig. 8 b) and temperature (Fig. 9) of the treatment increase. In our opinion, this is due to the influence of nitride inclusions, the content of which increases under such conditions, on the microhard‐ ness. That is why it is more expedient to use the modified dependence (11) with allowance for such an influence.

Not only data of the surface concentration of nitrogen (correspondingly, the hardness as well), but also data of its concentration at a certain distance from the surface and the depth of the nitrided layer depending on the temperature–time parameters are of practical interest. The corresponding curves (Fig. 10) were constructed for the same parameters as in the preceding figures. It should be noted that the depth of the diffusion zone was determined behind the front of propagation of the relative nitrogen concentration *<sup>С</sup>*¯=0.02, which corresponds to a change in the microhardness by an amount Hδ = 0.2 GPa, equal to the error in its measurements.

An analysis of these curves confirms an adequate increase in the nitrogen content over the whole depth of the diffusion zone and the increase in the depth of this zone as the treatment time and temperature increase (curves 1 – 3, Fig. 10). It was found that, in the statement of the first boundaryvalue problem (*h* →0 , *C*(0, *τ*)=*Const*),overestimated values of the nitrogen concentration were obtained (curves 4 in Fig. 8 b Fig. 10). The calculated data show that the surface phenom‐ ena affect substantially not only the surface concentration of nitrogen (Fig. 10 a) (corresponding‐ ly, the surface hardness of titanium), but also the nitrogen content in layers more remote from the surface (Fig. 10 b), and the depth of the hardened diffusion zone (Fig. 10 c).

The nitrogen content in the surface layer and the depth of the diffusion zone change additively as the exposure time (curves 1 – 3, Fig. 8 b) and the temperature of the isothermal exposure

On the whole, the analytic calculations of content profiles correlate well with the experimental results of relative changes in the microhardness of the surface layer (Fig. 8 b, Fig. 9). The corresponding curves have a monotonic character; the microhardness over the cross-section of the sample decreases gradually in the depth of the metal until it attains values characteristic for titanium. At the same time, there are insignificant disagreements between the theoretical and experimental results. In particular, for short exposures, the zone of change of the micro‐ hardness extends to a larger depth than the value which follows from the nitrogen distribution (Fig. 8 b). This can be explained by an insignificant content of oxygen, which is characterized by a larger diffusion mobility than nitrogen. For larger exposure times when the percentage

**Figure 9.** Distribution of nitrogen in the surface layer of titanium after an exposure τ= 5 h for different saturation tem‐

Some disagreement between the calculated nitrogen distribution and experimental data of change in the microhardness is also observed near the surface, particularly as the time (Fig. 8 b) and temperature (Fig. 9) of the treatment increase. In our opinion, this is due to the influence of nitride inclusions, the content of which increases under such conditions, on the microhard‐ ness. That is why it is more expedient to use the modified dependence (11) with allowance for

Not only data of the surface concentration of nitrogen (correspondingly, the hardness as well), but also data of its concentration at a certain distance from the surface and the depth of the nitrided layer depending on the temperature–time parameters are of practical interest. The corresponding curves (Fig. 10) were constructed for the same parameters as in the preceding figures. It should be noted that the depth of the diffusion zone was determined behind the front of propagation of the relative nitrogen concentration *<sup>С</sup>*¯=0.02, which corresponds to a change in the microhardness by an amount Hδ = 0.2 GPa, equal to the error in its measurements.

perature: (1), 750 0C; (2), 800 0C; (3), 850 0C. Marks correspond to experimental data *H*¯(*x*, <sup>τ</sup>)

of nitrogen is larger than that of oxygen, this effect is leveled.

(curves 1 – 3, Fig. 9) change.

56 Titanium Alloys - Advances in Properties Control

such an influence.

**Figure 10.** Time dependence of the content of dissolved nitrogen on the surface (a) and at a depth x = 10 μm (b), and the depth of the nitrided layer (c) for different saturation temperatures: (1) 750 0C; (2) 800 0C; (3, 4) 850 <sup>0</sup>C (curves 4 were constructed for the condition *C*(0, τ) = const, h → ∞)

Thus, the presented results indicate the critical role of the surface phenomena (adsorption and chemisorption) in the kinetic regularities of nitriding of titanium in a rarefied atmosphere. The calculated data obtained on the basis of the solution of the diffusion task using the nonsta‐ tionary boundary condition (2) indicate that its model representation reflects rather satisfac‐ torily the main tendencies of the high-temperature interaction of titanium with rarefied nitrogen. For the provision of a specified hardened layer, the proposed model gives scientifi‐ cally justified recommendations on external parameters (exposure temperature and time) of nitriding of titanium.
