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

Titanium and titanium alloys are the most promising structural materials for the products of the contemporary aircraft and spacecraft engineering, medicine. The complex of characteristics of such products strongly depends on the properties of their surface layers. One of the efficient method of their hardening is the thermodiffusive saturation with interstitial elements, in particular nitrogen or oxygen (Fedirko & Pohrelyuk, 1995; Panasyuk, 2007). Such hightemperature interaction with these interstitial elements is accompanied by not only the formation and growth of a nitride or oxide film, but also the significant dissolution of nitrogen or oxygen in the base metal. The competition of these processes complicates significantly the study of the kinetics and mechanism of such an interaction. In this case, useful information can be obtained from results of an investigation of the high-temperature interaction of titanium in an atmosphere with a decreased nitrogen or oxygen pressure, which simultaneously generates practical interest, because deep diffusion layers without a continuous nitride or oxide film on a titanium surface can be formed (Fedirko & Pohrelyuk, 1995; Panasyuk, 2007). The incubation period of formation of such a film depends to a large degree on the partial gas pressure and saturation temperature. Attempts to choose purposefully an optimal nitrogen or oxygen pressure and temperature–time parameters of such a thermochemical treatment failed. This is due to the complexity and diversity of the interactions of titanium with rarefied gascontaining atmospheres, the absence of data on parameters that characterize surface phenom‐ ena, and a large spread (up to two orders of magnitude) of available data on the diffusion


Introduction
Titanium and titanium alloys are the most promising structural materials for the products of the contemporary aircraft and spacecraft engineering, medicine. The complex of characteristics of such products strongly depends on the properties of their surface layers. One of the efficient method of their hardening is the thermodiffusive saturation with interstitial elements, in particular nitrogen or oxygen (Fedirko & Pohrelyuk, 1995;Panasyuk, 2007). Such hightemperature interaction with these interstitial elements is accompanied by not only the formation and growth of a nitride or oxide film, but also the significant dissolution of nitrogen or oxygen in the base metal. The competition of these processes complicates significantly the study of the kinetics and mechanism of such an interaction. In this case, useful information can be obtained from results of an investigation of the high-temperature interaction of titanium in an atmosphere with a decreased nitrogen or oxygen pressure, which simultaneously generates practical interest, because deep diffusion layers without a continuous nitride or oxide film on a titanium surface can be formed (Fedirko & Pohrelyuk, 1995;Panasyuk, 2007). The incubation period of formation of such a film depends to a large degree on the partial gas pressure and saturation temperature. Attempts to choose purposefully an optimal nitrogen or oxygen pressure and temperature-time parameters of such a thermochemical treatment failed. This is due to the complexity and diversity of the interactions of titanium with rarefied gascontaining atmospheres, the absence of data on parameters that characterize surface phenomena, and a large spread (up to two orders of magnitude) of available data on the diffusion 2. Thermodiffusion saturation of titanium with interstitial elements from a rarefied atmosphere at T<T α↔β 2.1. Physicomathematical model

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 physicochemical 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): a. transport of the interstitial element molecules to a metal surface followed by their physical adsorption, dissociation, and chemisorption (the mass transfer coefficient h, cm/sec); b. segregation of the interstitial element on defects in a contact layer (with a mass capacity ϖ, cm) as a result of the chemical interaction with the metal (the rate of reaction k, cm/ sec); c. diffusion of the interstitial element in α-titanium (diffusion coefficient D, cm 2 /sec).
Processes enumerated in clause a) can be interpreted as a two-stage reaction which consists of a diffusive stage, described by the constant rate h D , and a stage of chemisorption at a constant rate h R . Then, according to the law of summation of kinetic resistances, we have The introduced kinetic parameters (Р і =D, h, k) of the model representation characterize the aforementioned thermoactivated physicochemical processes with the corresponding activation energies (E i ), according to the dependence Р і =Р 0і ехр(-E і /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 (τ 1 and τ 2 ) allow to use a graphical method to compute the mass transfer coefficient h and the surface content (C eq ) of nitrogen which is in equilibrium with the atmosphere (   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.

Mathematical description
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)=C 0 . 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): Here C eq 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 . Using a mathematical procedure 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): or even the simpler condition (D/h →0) of the first kind: 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)=C eq . 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: 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 j 1 of the interstitial element from the environment to the surface (x=-0) and its diffusion flux j = j diff 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 0 0 (0, ) 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). 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): where Specifically, the surface concentration of dissolved interstitial element is Here C(x, τ) = (C(x, τ) − C 0 ) / (C eq − C 0 ) is the relative change of the interstitial element concentration in the solid solution in α-titanium. Its surface concentration C * (0,τ) in the bound state and the total concentration C Σ (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).
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/m 3 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 0 C 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 0 C 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. %.
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).

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 -3 Ра/sec) and temperature-time (750-850 0 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).  (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 0 C and the same exposure time (Fig. 7 a).

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 substantially 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): Then the relative change in the microhardness in the diffusion zone due to dissolved nitrogen (neglecting the contribution of nitride inclusions) is as follows: Here H 0 is the microhardness of the initial titanium sample, H max is the microhardness of titanium at a maximum concentration of dissolved nitrogen C max =C eq , and a is the proportionality coefficient. Relation (11) indicates the possibility to plot the calculated relative concentrations C(x, τ) of nitrogen and experimental data of the relative change in the microhardness H (x, τ), on the same ordinate axis.
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 C Σ (0, τ) (curve 3, Fig. 8 a), increase. 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.
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 microhardness 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. 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 microhardness. 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 С =0.02, which corresponds to a change in the microhardness by an amount H δ = 0.2 GPa, equal to the error in its measurements.

Titanium Alloys -Advances in Properties Control
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 phenomena affect substantially not only the surface concentration of nitrogen (Fig. 10 a) (correspondingly, 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). 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 nonstationary boundary condition (2) indicate that its model representation reflects rather satisfactorily 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 scientifically justified recommendations on external parameters (exposure temperature and time) of nitriding of titanium.

Thermodynamic analysis
According to the phase diagram (Fig. 11 a, b), titanium undergoes allotropic transformation (change of crystal lattice from hcp to bcc) at T α↔β = 882 0 C (Fromm & Gebhardt, 1976). We will be interested in high-temperature (T>T α↔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 (TiA х ) 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 < C A < C 23 solid solution of interstitial element in βphase is stable, while in the concentration range С 12 < C A < C 1S -solid solution of interstitial element in α-phase. In the concentration range С 23 < C A < C 12 solid solutions of interstitial element in α-and β-phases can coexist.

Titanium Alloys -Advances in Properties Control
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 comparison with nitrogen.

Physico-mathematical model
Let us consider the process of isothermal saturation of titanium by nitrogen or oxygen at temperature higher than temperature of allotropic transformation (T>T α↔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>T α↔β
During the interaction of titanium with nitrogen or oxygen nitride or oxide layer (0 < x < Y 0 (τ)) and diffusion zone are formed. The diffusion zone consists of three layers. The layer І (Y 0 (τ) < x < Y 1 (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 transformations in titanium, because these interstitial elements are α-stabilizers. The layer III (Y 2 (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 (Y 1 (τ) < x < Y 2 (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 (ТіО 2 ). On the interfaces the nitrogen or oxygen concentration, corresponding to equilibrium concentration, according to the phase diagram is constant (Fig. 11 a, b).
Initial conditions (τ=0): Boundary conditions (τ>0): The motion of interfaces will be set by the parabolic dependencies (Lyubov, 1981): Here β j (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 Y j (τ)are set: 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 TiA x layer (Fig. 12) Titanium Alloys -Advances in Properties Control 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: The chosen functions С і (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 β j (j=0,1,2) was obtained by the conditions of mass balance on interfaces (16) and relation (15) (Tkachuk et al., 2012): Having solved the system of equations (18), following equations were: where The parameters β j 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 0 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 j for nitrogen and oxygen were calculated ( Table 2).  Table 2. Calculated constants β j and К j (j=0,1,2) at saturation temperature of Т=950 0 С Taking into consideration the correlation (15), the motion of interfaces will be presented as: where K 0 = 2β 0 D 0 , K 1 = 2β 1 D 1 , K 2 = 2β 2 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 0 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: Y 0 (τ) -interface of nitride or oxide layer (Fig. 13 a), Y 1 (τ) -interface of solid solution of interstitial element in α-phase ( Fig. 13 b), Y 2 (τ) -interface of mixture of solid solutions of interstitial element in α-and β-phases ( Fig.  13 c), Y 3 (τ) -interface of solid solution of interstitial element in β-phase (Fig. 13 d) at nitriding and oxidation of titanium at Т=950 0 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 С 33 = 0.25 аt. %, that is from transcendental equation С 3 (Y 3 (τ), τ)=C 33 .
Calculated constants Y і (τ) (і=0,1,2,3) after isothermal exposures of 1 and 5 h during nitriding and oxidation of titanium at Т=950 0 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 K j (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) 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 0 С Titanium Alloys -Advances in Properties Control 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 (Y 3 (τ)), the thickness of α, α + β and β layers will be 16  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 transformation in the layers of the diffusion zone adjacent to nitride or oxide layer. And as the solubility 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 0 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.  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 j K (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 thickness of α, α + β and β layers will be 16 Figure 13. Kinetics of motion of interfaces Y 0 (τ) (a), Y 1 (τ) (b), Y 2 (τ) (c) and Y 3 (τ) (d) at nitriding (curves 1) and oxidation (curves 2) of titanium at saturation temperature of Т=950 0 C 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. Figure 14. Concentration profiles of nitrogen and oxygen in diffusion zone of titanium after its saturation at Т=950 0 C 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 concentration 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 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 (R а =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.

Results and discussion
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). 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. The large gradient of microhardness is characteristic for zone A. It is caused by β→ α transformation 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).
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.

Conclusions
The process of high-temperature interaction of titanium with gaseous medium (nitrogen or oxygen) was modelled at temperatures T<T α↔β and T>T α↔β considering the surface processes and structural phase transformations.
The kinetics of surface processes is reflected by the mass balance equation, which takes into account the interaction of an external flux of impurities to the surface and its chemisorption with diffusion dissolution and segregation on defects as a result of a chemical interaction with titanium atoms.
The kinetics of diffusion saturation of α-titanium by nitrogen under rarefied atmosphere (1 Pa) in the temperature range of 750-850 0 C was investigated experimentally and analytically. The influence of time and temperature parameters on the depth of the nitrided layer and a change of its microhardness was estimated.
It was shown the role of these interstitial elements as α-stabilizers in forming the diffusion zone which contains three layers based on α-phase, α+β-phases and β-phase.
It was received the solution of the formulated task as for diffusion of nitrogen or oxygen in such a heterogeneous medium taking into account the motion of interfaces.
The constants of parabolic growth of layers were calculated. It allowed to foresee the kinetics of their growth and distribution of interstitial elements (nitrogen or oxygen) in diffusion zone.
The adequacy of the proposed model representations was confirmed by the results of experimental investigations on nitriding of titanium at Т=950 0 C. The microstructural evolution (after processing times of 1 and 5 h) of the diffusion zone which is caused by the structural phase transformations during diffusion of nitrogen was examined experimentally.