Crystallographic Orientation Dependence of Nitrogen Mass Transport in Austenitic Stainless Steel

: The lattice stress-induced di ﬀ usion of nitrogen and hydrogen in austenitic stainless steel, taking place during nitriding in nitrogen / hydrogen plasma, is analyzed in the presented work. Stress-induced di ﬀ usion has an anisotropic nature and depends on the orientation of the crystal lattice. However, during simulations, it is not enough to take into account only the anisotropy of stress-induced di ﬀ usion, since this leads to contradictory results when comparing with experimental data. The problem is the surface concentration of nitrogen. Processes on the steel surface such as adsorption, desorption and heterogeneous chemical reactions are also very important. In the presented work, it is shown that these surface processes also have anisotropic natures, and it is very important to take this anisotropy into account during simulations. The inﬂuence of anisotropic surface processes on austenitic steel nitriding is analyzed in this study. It is shown that the nitrogen di ﬀ usion is anisotropic due to the e ﬀ ects of the anisotropic stress gradient and the anisotropic e ﬀ ects on the steel surface.


Introduction
Austenitic stainless steels (ASS) are often used as engineering materials in many industrial applications among which are food and chemical processing, the automotive industry, and some surgical implants due to very high corrosion resistance in many aggressive environments because of a passive Cr 2 O 3 surface film. Whereas the corrosion resistance of these materials is excellent, their hardness (with a surface hardness of~200-300 HV) and wear resistance are relatively low. Many attempts have therefore been made to increase the wear resistance without deteriorating the corrosion resistance. Plasma nitriding is one of the well-known processes for the surface modification of metals and alloys. Thus, this technology is rapidly gaining acceptance for the treatment of ASSs in various industry sectors to achieve combined improvements in wear and corrosion resistance.
The low-temperature plasma nitriding (performed at temperatures up to 450 • C) of ASS results in improved properties, such as a greatly increased surface hardness, wear resistance, fatigue resistance, and corrosion resistance [1][2][3][4][5][6][7][8][9][10][11][12][13]. Such a hardening process is related to the lattice distortion of the fcc austenitic phase (γ phase), leading to the formation of nitrogen-rich expanded austenite (γ N phase), with a hardness close to 1500 HV and no loss of corrosion resistance [6,14]. The high nitrogen content dissolved in the γ N crystalline lattice leads to a noticeable expansion of the lattice, giving rise to high compressive residual stresses [9][10][11][12]. Intensive studies on γ N phase structure and formation have been undertaken by various researchers in the past several decades, and a comprehensive review of the scientific literature regarding the formation, characteristics and properties of the γ N phase is presented

Mass Transport Model
A generalized formalism of the stress-assisted diffusion model presented in our previous work [23][24][25][26][27] is presented hereafter, allowing both the isotropic and anisotropic nitriding of ASS to be simulated. Based on Fick's law the interstitial atom (nitrogen and/or hydrogen) diffusion through the austenite phase can be described as: where C int is the interstitial concentration and J int is the interstitial atom flux. In the case of the interstitial atom's diffusion into the austenite phase, the flux of this species can be written as [29,30]: Metals 2020, 10, 615 3 of 13 where L int is the Onsager coefficient and µ int (C int , σ) is the chemical potential of interstitials in a metallic lattice, which at constant temperature depends on the interstitial concentration C int (symbolizes the relative concentration of interstitials in a metallic lattice) and stress σ. In general, the gradient of the chemical potential is the main driving force for interstitial atom diffusion in the metallic lattice. The stress effects on the interstitial's diffusion are reflected in the chemical potential, which incorporates chemical and mechanical energies. For a system under a stress σ, the chemical potential of interstitial atoms in lattice sites is given by [29,30]: where V int σ is the stress-dependent part of chemical potential; (µ 0 int + RT ln( f int ·C int )) denotes the chemical potential of interstitial atoms in the stress-free state (σ = 0); µ 0 int is the standard chemical potential; f int is the activity coefficient; V int is the partial molar volume of interstitials in the solid matrix; and R and T are the universal gas constant and the absolute temperature, respectively.
Thus, a phenomenological consideration of the hydrostatic stress' effects on interstitial behavior in metals yields the equation of stress-assisted diffusion under isothermal conditions (and by assuming an ideal solid solution so that ∂ ln(C int ) = 0) as follows [24,27,29,30]: is the diffusion coefficient of interstitial atoms in a stress-free solid. Substituting Equation (4) into Equation (1), the diffusion equation including the coupling effect of stresses and diffusion is obtained by: To evaluate the compressive stress and the compositional strain induced by the interstitial atom diffusion in the expanded austenite, a mechanical model recently developed in our previous work [26] was used. The dependence of compositionally induced biaxial compressive residual stresses in the [h k l] direction on the concentration of interstitials is obtained by [26,31,32]: The anisotropic stress factor X stress (hkl) in any crystallographic direction [h k l] depends on the so-called orientation factor P hkl = h 2 ·k 2 + h 2 ·l 2 + k 2 ·l 2 · h 2 + k 2 + l 2 −2 [31] and on the Young's , where h, k and l are Miller indexes; and S 11 , S 12 and S 44 are independent intrinsic single crystal elastic constants (elastic compliances), which are related to the non-zero stiffness tensor components C ij by the relationships: S 11 = (C 11 +C 12 ) (C 11 −C 12 )(C 11 +2C 12 ) , S 12 = −C 12 (C 11 −C 12 )(C 11 +2C 12 ) and S 44 = 1 C 44 . In addition, a linear relationship exists between the total strain in the [h k l] direction (which is linearly related to the stress through Hooke's law, i.e., ε(hkl) = 1 E hkl ·σ(hkl)) and the mean concentration of interstitials in the expanded austenite, i.e., ε(hkl) = β int a 0 ·C int [26], where β int is the Vegard's constant for interstitial atoms dissolved in austenite and a 0 is the lattice parameter of the austenite (strain-free). It is important to note that for nitrided layers containing the γ N phase, various types of strain (thermal, compositional, elastic and plastic) can be considered. However, we only take into account anisotropic compositional strain in our model. Hence, the incorporation of the Metals 2020, 10, 615 4 of 13 compositional, thermal, plastic and elastic contributions to the strain, induced in the austenite lattice due to the formation of the γ N phase, is the main goal for our future investigations.
Introducing Equation (6) into Equation (5) as the diffusion is one-dimensional along the x-direction (the x axis is the diffusion axis), the diffusion governing equation considering the effect of stress-assisted anisotropic diffusion takes the following form: where ∇ x is x-axis projection of the gradient and Φ 0 int-hkl is the adsorption term, which describes the process of interstitial atom adsorption on the steel surface (i.e. Φ 0 int (x 0,t) = 0 and Φ 0 int (x = 0,t) 0). The process of nitriding is influenced not only by the processes in the bulk but also by processes on the surface. Experimental studies in [33][34][35] and our earlier theoretical studies [25,36] have shown that hydrogen has an important role in the nitriding process. A detailed description of the kinetic model of nitrogen and hydrogen adsorption in ASS taking place during plasma nitriding is proposed in our previous work [25]. The processes of adsorption, desorption, and surface reactions occurring on the steel surface during interaction with N 2 -H 2 plasma were represented by the following steps: (1) the adsorption and dissociation of nitrogen and hydrogen molecules, (2) the formation of water molecules and desorption (oxygen chemical etching by hydrogen), (3) the adsorption, dissociation and desorption of oxygen from residual gas during the nitriding process, (4) the adsorption and dissociation of NH radicals formed in the N 2 -H 2 mixture, and (5) the diffusion of nitrogen and hydrogen into the bulk of steel. Furthermore, it is well known that adsorption, desorption and surface reactions depend on surface energy. According to the thermodynamic theories of crystal growth [37,38], the surface energy is generally dependent on the orientation of the surface, i.e., anisotropy of the surface energy leads to anisotropy of the adsorption term Φ 0 int-hkl in Equation (7). For fcc metals, as predicted by a bond-breaking model for the surface energies of solids in work [39] and by the embedded-atom method in [40], the close-packed (111) surface has the lowest surface energy. The surface energies for the other surface orientations increase linearly with an increasing angle between the surfaces (hkl) and (111). Thus, based on the principle of surface energy minimization, the adsorption/desorption probabilities and surface reactions rates on the (111) surface have the lowest values. The anisotropy of the surface free energies is exactly described in terms of the Miller indices in [39] (see [39], Table 4).
It was shown that surface free energy of fcc crystals is directly proportional to h 2 + k 2 + l 2 −1/2 and consistently supports the trend γ(111) < γ(110) < γ(100). Thus, the developed model equations (see Equation (1) in [25]) for the adsorption, desorption and chemical surface reactions of nitrogen and hydrogen were modified by including the crystal orientation factor A hkl = h 2 + k 2 + l 2 −1/2 : For a steel surface layer (x = 0): Metals 2020, 10, 615 5 of 13 where C H2 , C H , C NH , C O2 , C O , C N2 , C Fe and C N are the relative concentrations of H 2 , H, NH, O 2 , O, N 2 , Fe and N respectively; α NH , α O , α H , and α N are the sticking coefficients of the NH radical, O 2 , H 2 and N 2 , respectively, to the corresponding steel surface components (possible adsorption sites); i k is the relative flux of the k-th component to the surface; R NH,H2O are the reaction rate constants; des k and dis k are the desorption and dissociation probabilities of the k-th component, respectively; h is the thickness of the layer; and D is diffusion coefficient. The process of the plasma nitriding of ASS (due to interaction with N 2 -H 2 plasma) can be viewed as diffusion in a vapor-solid diffusion couple, as shown schematically in Figure 1.
where CH2, CH, CNH, CO2, CO, CN2, CFe and CN are the relative concentrations of H2, H, NH, O2, O, N2, Fe and N respectively; αNH, αO, αH, and αN are the sticking coefficients of the NH radical, O2, H2 and N2, respectively, to the corresponding steel surface components (possible adsorption sites); ik is the relative flux of the k-th component to the surface; RNH,H2O are the reaction rate constants; desk and disk are the desorption and dissociation probabilities of the k-th component, respectively; h is the thickness of the layer; and D is diffusion coefficient.
The process of the plasma nitriding of ASS (due to interaction with N2-H2 plasma) can be viewed as diffusion in a vapor-solid diffusion couple, as shown schematically in Figure 1.

Results and Discussion
This work aims to study the impacts of the elastic anisotropy of ASS, as well as the anisotropy of the surface free energies, on the phenomenon of anisotropic nitrogen penetration during low temperature nitriding. The governing equations (Equations (7) and (8)) were solved by a finite-difference method. The material properties of AISI 316L austenitic stainless steel and nitriding parameters used in the numerical simulation are listed in Table 1.  A key element of the presented stress-assisted diffusion model is the anisotropic stress factor, X stress(hkl) , which depends on the crystallographic orientation-dependent elastic modulus of ASS. Recently, in our previous work [26], it was found that the stress anisotropy (taking to account the anisotropy in the Young's modulus for the fcc crystal) has a significant impact on the nitrogen concentration evolution during the expanded austenite formation. As mentioned above, the anisotropic stress factor, X stress(hkl) , depends on the orientation factor, P hkl , and Young's modulus, E hkl (see Equation (6)). Furthermore, X stress(hkl) is the greatest for (100), the smallest for (111) and has intermediate values for the other crystal orientations. In work [26], we have shown that the values of the nitrogen penetration depths correlate well with the values of the anisotropic stress factor, X stress(hkl) , i.e., they are the lowest for (111), the deepest for (100) and intermediate for the other orientations. According to the experimental results in [6,8,[16][17][18][19], it can be concluded that the nitrogen surface concentration also increases in the order (111) < (110) < (100) for different crystallographic orientations. However, as we have shown in [26], if not taking into account the anisotropic nature of the adsorption, desorption and surface reactions, the calculated results show the opposite trend, i.e., the surface nitrogen content increases in the order (100) < (110) < (111). Thus, the main reason for current study is to solve this discrepancy between the calculation results and experimental observations.
The solution of this problem was split into two aspects. Firstly, experimental studies in [33][34][35] and our earlier theoretical studies [25,36] have shown that hydrogen has an important role in the nitriding process. Thus, the effects of the hydrogen on the adsorption, desorption and surface reaction processes occurring on the ASS surface due to interaction with N 2 -H 2 plasma, as well as the hydrogen stress-assisted diffusion processes into the bulk of steel, should be considered. Secondly, adsorption, desorption and surface reaction processes are expected to be greatly affected by the surface tension or surface energy. Generally, the crystalline nature of materials leads to the anisotropy of the physical quantities, such as surface energy or surface tension, which must be taken into account in theoretical studies. Thus, the anisotropy of adsorption term, Φ 0 int-hkl , in Equation (7) was described in terms of the Miller indices according to the predicted results for fcc metals in [39,40], where the relationship between the values of the surface energies of low index planes was summarized as γ(111) < γ(110) < γ(100).
For sequential coupling analysis, both isotropic and anisotropic surface effects (due to interaction with N 2 -H 2 plasma) were compared in a single analysis. In order to validate the effects of isotropic hydrogen and nitrogen adsorption, desorption and surface reaction processes on the nitrogen distribution resulting from low temperature plasma nitriding, the simulation was performed using a model based on Equations (7) and (8) with the anisotropic X stress(hkl) term but with the isotropic adsorption term Φ 0 int-hkl (i.e., with A hkl = 1 for all crystallographic orientations). The results are presented in Figure 2 in the 1.1-1.3 curves (for different orientation single crystals, No ADS hkl + DIF hkl curves). It is found that the nitrogen penetration depth is highest for the (100) oriented crystal and smallest for the (111) oriented crystal and that the nitrogen penetration depth follows the order (100) > (110) > (111). The same order is shown for the experimentally measured nitrogen diffusion depths [6,8,[16][17][18][19]. However, as is clearly seen in Figures 2 and 3 (square points, No ADS hkl + DIF hkl model), the surface concentration of nitrogen shows an opposite tendency, i.e., the surface nitrogen content increases in the order (100) < (110) < (111), and these results are not consistent with the experimental observations. Hence, the evaluation of isotropic hydrogen/nitrogen adsorption, desorption and surface reaction processes on the ASS surface does not allow the results of calculations to coincide with the experimental observations. To illustrate the effects of anisotropic surface processes, additional simulation was performed using a model based on Equations (7) and (8) with the anisotropic X stress(hkl) term and with the anisotropic adsorption term Φ 0 int-hkl (i.e., with different values of A hkl for different crystallographic orientations). The nitrogen depth profiles in the (100), (110) and (111) oriented single crystals were calculated and are presented in Figure 2 (2.1-2.3 curves, ADS hkl + DIF hkl model). For this case, the values of nitrogen penetration depth and surface concentration are given in Figure 3 (circular points, ADS hkl + DIF hkl model). It is seen that the penetration depth and surface concentration follow the order (100) > (110) > (111), and this is in qualitative agreement with the experimental results. By comparing the results shown in Figures 2 and 3, it can be concluded that the nitrogen diffusion is anisotropic due to effects of the anisotropic stress gradient, nitrogen/hydrogen concentration gradient and anisotropic effects on the steel surface.
2.3 curves, ADShkl + DIFhkl model). For this case, the values of nitrogen penetration depth and surface concentration are given in Figure 3 (circular points, ADShkl + DIFhkl model). It is seen that the penetration depth and surface concentration follow the order (100) > (110) > (111), and this is in qualitative agreement with the experimental results. By comparing the results shown in Figures 2  and 3, it can be concluded that the nitrogen diffusion is anisotropic due to effects of the anisotropic stress gradient, nitrogen/hydrogen concentration gradient and anisotropic effects on the steel surface.   To understand the role of anisotropic stress-assisted nitrogen and hydrogen diffusion and anisotropic effects on the steel surface on nitrogen transport, additional analysis was carried out. First, the simulation was performed using a model based on Equations (7) and (8) with an isotropic Xstress(hkl) term and with an isotropic adsorption term Φ0 int-hkl (i.e., the values of Xstress(hkl) and A hkl were kept constant for all crystallographic orientations, No ADShkl + No DIFhkl model). Second, calculations were performed with anisotropic Xstress(hkl) and Φ0 int-hkl terms (ADShkl + DIFhkl model). The results of these calculations are shown in Figure 4. Generally, the steel surfaces exhibit an oxide coverage before the nitriding, which depends on the pre-history of the steel sample. During nitriding, the oxygen content of the surface is reduced due to the hydrogen etching mechanism [33,35]. Thus, as the initial To understand the role of anisotropic stress-assisted nitrogen and hydrogen diffusion and anisotropic effects on the steel surface on nitrogen transport, additional analysis was carried out. First, the simulation was performed using a model based on Equations (7) and (8) with an isotropic X stress(hkl) term and with an isotropic adsorption term Φ 0 int-hkl (i.e., the values of X stress(hkl) and A hkl were kept constant for all crystallographic orientations, No ADS hkl + No DIF hkl model). Second, calculations were performed with anisotropic X stress(hkl) and Φ 0 int-hkl terms (ADS hkl + DIF hkl model). The results of these calculations are shown in Figure 4. Generally, the steel surfaces exhibit an oxide coverage before the nitriding, which depends on the pre-history of the steel sample. During nitriding, the oxygen content of the surface is reduced due to the hydrogen etching mechanism [33,35]. Thus, as the initial condition for calculations, the natural oxide layer of 0.5 µm thickness was taken. As can be seen in Figure 4, the shape of the oxygen concentration profiles is determined by the oxygen diffusion into the steel bulk and towards the steel surface, where it is removed by the hydrogen assistance. By comparing the nitrogen and hydrogen profiles shown in Figure 4, it was observed that coupled analysis with anisotropic effects showed a higher level of nitrogen penetration as compared to that without anisotropic effects. This clearly demonstrates that the nitrogen diffusion is enhanced by the anisotropic stress-assisted hydrogen diffusion and anisotropic effects on the surface due to interaction with N 2 -H 2 plasma. Thus, it is important to study the kinetics of the surface processes.   (2) anisotropic (ADShkl + DIFhkl) surface processes and diffusion.
As mentioned above, all the steps of adsorption, desorption and surface reaction processes occurring on the steel surface due to interaction with N2-H2 plasma are shown schematically in Figure  1. The kinetics of the mentioned processes were analyzed, and it was found that the main impacts on nitrogen transport are from the nitrogen/hydrogen adsorption processes and water molecule formation and subsequent desorption. The formation and desorption of water molecules causes oxygen elimination, i.e., a chemical etching of the adsorbed oxygen by hydrogen. Thus, the values of the corresponding kinetic parameters αH, αN and RH2O are important for the transport of nitrogen as well as for the dynamic surface concentration change. Those parameters are constants at given experimental conditions but can be varied by changing the experimental conditions, e.g., temperature or the energy of ions arriving from plasma [42]. The influences of these parameters on nitrogen penetration depth and surface concentration are shown in Figures 5-8. As can be seen in , with increases in the hydrogen sticking coefficient, αH, and water molecule formation reaction rate, constant, RH2O, the process of removing oxygen from the steel surface is enhanced and reaches a maximum. As a result, the amount of adsorbed and diffused nitrogen increases. Moreover, with a further increase in the hydrogen sticking coefficient, αH, the surface concentration of nitrogen and nitrogen penetration depth decrease ( Figure 6). It can be explained by the fact that more adsorption sites are occupied by hydrogen and, as a result, the amount of adsorbed nitrogen decreases; consequently, the surface concentration and the penetration depth of nitrogen decrease. The dependencies of the nitrogen surface concentration and penetration depth on the nitrogen sticking coefficient, αN, are presented in Figure 8. The nitrogen surface concentration and penetration depth As mentioned above, all the steps of adsorption, desorption and surface reaction processes occurring on the steel surface due to interaction with N 2 -H 2 plasma are shown schematically in Figure 1. The kinetics of the mentioned processes were analyzed, and it was found that the main impacts on nitrogen transport are from the nitrogen/hydrogen adsorption processes and water molecule formation and subsequent desorption. The formation and desorption of water molecules causes oxygen elimination, i.e., a chemical etching of the adsorbed oxygen by hydrogen. Thus, the values of the corresponding kinetic parameters α H , α N and R H2O are important for the transport of nitrogen as well as for the dynamic surface concentration change. Those parameters are constants at given experimental conditions but can be varied by changing the experimental conditions, e.g., temperature or the energy of ions arriving from plasma [42]. The influences of these parameters on nitrogen penetration depth and surface concentration are shown in Figures 5-8. As can be seen in Figures 5-7, with increases in the hydrogen sticking coefficient, α H , and water molecule formation reaction rate, constant, R H2O , the process of removing oxygen from the steel surface is enhanced and reaches a maximum. As a result, the amount of adsorbed and diffused nitrogen increases. Moreover, with a further increase in the hydrogen sticking coefficient, α H , the surface concentration of nitrogen and nitrogen penetration depth decrease ( Figure 6). It can be explained by the fact that more adsorption sites are occupied by hydrogen and, as a result, the amount of adsorbed nitrogen decreases; consequently, the surface concentration and the penetration depth of nitrogen decrease. The dependencies of the nitrogen surface concentration and penetration depth on the nitrogen sticking coefficient, α N , are presented in Figure 8. The nitrogen surface concentration and penetration depth increase and approach steady state values.      In this study, it is shown that the anisotropic nature of lattice stresses influences the different penetration of nitrogen and hydrogen in the differently orientated grains. The grain orientationdependent penetration of nitrogen and hydrogen is also influenced by the anisotropic nature of the processes on the surface such as adsorption and heterogeneous chemical reactions. The most important aspect is that the inclusion of the anisotropy of the surface processes into the model  In this study, it is shown that the anisotropic nature of lattice stresses influences the different penetration of nitrogen and hydrogen in the differently orientated grains. The grain orientationdependent penetration of nitrogen and hydrogen is also influenced by the anisotropic nature of the processes on the surface such as adsorption and heterogeneous chemical reactions. The most important aspect is that the inclusion of the anisotropy of the surface processes into the model In this study, it is shown that the anisotropic nature of lattice stresses influences the different penetration of nitrogen and hydrogen in the differently orientated grains. The grain orientation-dependent penetration of nitrogen and hydrogen is also influenced by the anisotropic nature of the processes on the surface such as adsorption and heterogeneous chemical reactions. The most important aspect is that the inclusion of the anisotropy of the surface processes into the model explains the experimental observations, i.e., that both the penetration depth and the surface concentration of nitrogen in ASS increase in the order (100) > (110) > (111). The model also explains the role and influence of hydrogen in the plasma nitriding process. Hydrogen enhances nitrogen diffusion by reducing the diffusion barrier formed by the oxygen adsorbed on the steel surface during the