Integrated Computational Modelling of Thermochemical Surface Engineering of Stainless Steel

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1 Integrated Computational Modelling of Thermochemical Surface Engineering of Stainless Steel Ömer C. Kücükyildiz, Mads R. Sonne, Jesper Thorborg, Jesper H. Hattel, Marcel A.J. Somers Technical University of Denmark, Department of Mechanical Engineering Produktionstorvet b. 425, 2800 Kgs. Lyngby, Denmark address: Keywords: Low temperature gas nitriding, stainless steel, modelling, diffusion, plasticity, surface reaction Abstract An implicit finite difference method (FDM) based numerical model for the prediction of compositionand stress-depth profiles developing during low temperature gas nitriding (LTGN) of 316 stainless steel is presented. The essential effects governing the kinetics of composition and coupled stress evolution are taken into account in the model: concentration-dependent diffusion of nitrogen atoms, a slow surface reaction, elasto-plastic accommodation of lattice expansion and thermal and mechanical influences on thermodynamics (solubility) and diffusion kinetics. The model is one-dimensional and assumes a plane-stress mechanical state. Huge compressive stress levels and steep stress gradients have previously been suggested to have an influence on the concentration profile. The corresponding large plastic deformation that occurs in the developing case is addressed in the model by isotropic plasticity and force equilibrium. The model is used to explore the role and to assess the kinetics of the surface reaction. Introduction The development of a supersaturated solution of nitrogen in austenitic stainless steel by incorporation of nitrogen at low temperature (<450 C) is associated with the development of composition- and stress-depth profiles. The objective of the present work is to investigate the influence of the kinetics of the surface reaction involved in nitrogen uptake on the overall kinetics of the treatment. The investigation is carried out using an integrated numerical model, which includes the kinetics of the surface reaction(s), nitrogen diffusion in the steel and the composition-induced mechanical effects. The incorporation of nitrogen in iron-based alloys can be considered a by-product of the dissociation of ammonia into hydrogen and nitrogen and is a consequence of the slow dissociation kinetics of ammonia and the slow desorption kinetics of molecular nitrogen [1]. The surface kinetics of the dissociation of ammonia into hydrogen and nitrogen was thoroughly investigated experimentally for pure iron in the 60ies by Grabke ([2], see also [1]). He demonstrated that the flux of nitrogen atoms incorporated into iron is proportional to the hydrogen pressure and the difference between actual and equilibrium nitrogen content. Unfortunately, this pivotal work applies for pure b.c.c. iron only. No data

2 is available for the kinetics of ammonia dissociation on f.c.c. iron, let alone Fe-based alloys. Experimental work on the nitriding kinetics of stainless steel samples has demonstrated that there is a large discrepancy between the observed nitrogen uptake rate and that predicted for local equilibrium between the gas mixture and the stainless steel surface [3]. Furthermore, it also appears that the stationary nitrogen content that is reached at the surface on prolonged nitriding is systematically lower than the equilibrium nitrogen content [3]. While the slower uptake rate was tentatively explained from the kinetics of the surface reaction, the lower stationary nitrogen content was attributed to large compressive stresses in the surface, which lower the equilibrium solubility. The composition-induced compressive stresses influence (reduce) the diffusion coefficient, while the stress-gradients provide an additional driving force for diffusion [4]. Recently, a numerical model to predict concentration- and stress-depth profiles in expanded austenite was presented, applying an explicit finite-difference method, considering stress-assisted concentrationdependent nitrogen diffusion and composition-induced stress, as well as elasto-plastic accommodation of the lattice expansion of austenite by dissolution of large quantities of nitrogen [5]. In an explicit method, for each time step, the numerical error accumulates and the solution converged to is not necessarily the actual solution, because of the strong coupling of several mechanisms, which introduce non-linearities. Besides, due to the stability limit of the explicit method, very small time-steps are required to ensure stability. In order to overcome the accuracy limitation and at the same time the timestep limitation, a fully implicit iterative approach was followed in the present work. The implicit method is stable at greater time-steps and the iterations ensure higher accuracy for the solutions. Another advantage of an implicit method is that it is much faster than an explicit method, such that it enables inverse modelling by varying (and optimizing) over a range of unknown model inputs (not to be confused with fit-parameters). The diffusion equation is solved by an implicit mixed control-volume based-/classical finite difference method, where stress and thermal gradients could constitute additional driving force for diffusion. The stresses are resolved by a plane stress-projected plasticity model, which incorporates the chemical strains while maintaining the in-plane force equilibrium integrated in-depth. Full details of this model will be provided elsewhere [6]. In this work, the implicit model is applied in combination with inverse modelling to reveal information about the kinetics of the surface reaction and its importance for the overall kinetics of nitrogen uptake by stainless steel during low-temperature surface hardening. The starting point for this is formed by experimental nitrogen uptake curves as determined with thermogravimetry for austenitic stainless steel and previously published in Ref. [3]. Systematic variation of selected parameters in the numerical model allows to explore the role of the surface reaction. Firstly, the essential elements of the model are presented where after the results obtained with respect to the kinetics of the surface reaction are given and discussed. Model Physics Pressure and temperature affected diffusion. Nitriding relies on the solid state diffusion of interstitially dissolved nitrogen atoms in the iron-based lattice. The usual assumption of pressureindependent diffusion cannot be justified, since huge compressive stresses are introduced in the surface. These compressive stresses affect the absolute value of the diffusion coefficient (which will be neglected here) and a composition-induced stress gradient induces an additional (positive or negative) driving force for diffusion. Furthermore, in order to resolve any influence of a temperature gradient (for example during heating or cooling of thick samples), the effect of a temperature gradient can also be considered (but will be omitted for the present case of a constant temperature). The diffusive flux induced by a gradient in the chemical potential is defined as:

3 JJ = MM NN cc NN μμ NN where MM NN = DD NN RR TT For a solid solution, including pressure and temperature effects, the chemical potential becomes: μμ NN (aa NN, TT) = μμ NN,0 + RRRR llll(aa NN ) VV NN σσ HH (2) where aa NN is the activity of nitrogen in solid solution, VV NN is the partial molar volume of nitrogen and σσ HH is the hydrostatic stress component (or simply pressure). The activity of nitrogen is defined with reference to N2 at atmospheric pressure, which for a gas mixture of ammonia and hydrogen with nitriding potential KK NN = pp NNNN 3 3/2 pp HH2 (with pj the partial pressure of component j) equals: aa NN = KK TT KK NN (3) where KK TT is the equilibrium constant for ammonia dissociation at temperature T. Substituting Eqs. (2) and (3) in Eq. (1), considering a concentration-dependent diffusion coefficient as was experimentally determined in [7], leads to a non-linear equation with gradient in composition, stress and temperature [5]. (1) Trapping effect. As a consequence of the affinity of chromium for nitrogen, short range ordering of these elements occurs, implying that part of the nitrogen content can be conceived as trapped or captured by chromium atoms [8]. This can alter the concentration profile substantially and affect the achievable nitriding depth. Here, trapping is approached pragmatically by assuming that it is described by an equilibrium constant of the reaction between one atom of Cr and n atoms of N to form CrNn (which does not actually occur as no actual chemical bond is established): KK CCCCNNnn = CCCC γγ NN γγ nn (4) here square brackets indicate the content or concentration of the component concerned. Eq.(4) expresses that for KK CCCCNNnn >0 trapping occurs beyond a minimum nitrogen content, the threshold nitrogen content. Beyond the threshold nitrogen content the amount of nitrogen that can be trapped will be trapped and only the nitrogen content higher than nn CCCC γγ, complying with all locally present Cr atoms has the amount of N that can be trapped is allowed to diffuse. A non-linear iterative approach is used to solve the trapped and residual amounts of nitrogen. Surface kinetics. The chemical reaction that takes place at the surface starts with the dissociation of ammonia gas into hydrogen gas and adsorbed nitrogen atoms, which, if not desorbed after formation of molecular nitrogen, diffuse into the steel. The nitrogen content that occurs at the surface depends on the balance of nitrogen fluxes arriving at the surface (from dissociation) and leaving the surface (by solid state diffusion and/or desorption). Here it is assumed that the desorption reaction is effectively prevented by the relatively low temperature. Following Grabke [2] the net flux of nitrogen arriving at the surface, JJ ssssssss, depends on the dissociation rate and obeys: JJ ssssssss = kk rrrrrrrr cc NN eeee cc NN ss (5) with kk rrrrrrrr being the reaction rate constant for the rate controlling step in ammonia dissociation (which for ferrite depends on the partial pressure of hydrogen gas), cc NN ss the actual surface concentration of N, which depends on the balance of fluxes at the surface, and cc NN eeee the equilibrium concentration, which is

4 achieved only if the steel has a uniform nitrogen content, i.e. the diffusive flux is nil and no N2 develops. The equilibrium concentration of nitrogen in stainless steel was determined experimentally on thin foils as a function of the nitriding potential [9]. For ferrite, which has a very limited solubility of nitrogen, kk rrrrrrrr is independent of the nitrogen content in solid solution. Recognizing that the range of nitrogen solubility in expanded austenite is more than a hundred times larger than in ferrite, it is assumed that kk rrrrrrrr is a function of the surface concentration. The identity of concentration dependence of kk rrrrrrrr is not known a priori, and is exactly the topic to be explored in the present work. Chemically induced strains. The volumetric expansion due to the interstitially dissolved nitrogen atoms is described with a chemical (composition-induced) strain in one of three principal directions. If the chemical strain, εε cch iiii, is assumed isotropic it holds: εε cch iiii = VV(yy NN) 1/3 1/3 VV rrrrrr 1/3 (6) VV rrrrrr where the concentration dependent volume of the iron lattice VV(yy NN ), yy NN being the number of N atoms per metal atom, comes from a polynomial fit to experimental lattice parameter data [5,9]: VV(yy NN ) = yy NN (7) In Eq. (6) the reference volume VV rrrrff represents the nitrogen-free lattice. The partial molar volume in Eq. (2) follows also from Eq. (7) by differentiation towards yy NN. Plane-stress elasto-plasticity. The plane stress model is a projection of the generalized threedimensional model with a von Mises yield surface and isotropic strain hardening [10]. The plane stress model is obtained by applying expressions relating the elastic and plastic strain components in the third direction, which is perpendicular to the plane of the surface, to those in the first and second directions. Thereby only the in-plane stress and strain components are involved. Considering the general stress and strain relations for a plane-stress state, the elastic strain in the direction normal to the surface follows from setting σσ 13 = σσ 23 = σσ 33 equal to zero. For a homogeneous isotropic linear elastic material, this can be expressed in terms of the bulk modulus (K) and shear modulus (G): σσ 33 = KK εε eeee 11 + εε eeee 22 + εε eeee GG εε eeee 33 1 εε 3 11 eeee + εε eeee 22 + εε eeee 33 = 0 (8) Then, the elastic strain normal to the surface becomes: εε eeee 33 = νν εε 11 eeee +εε eeee 22 (νν 1) In the case of pure plastic strain, the relation comes from plastic incompressibility: εε pppp 33 = εε pppp 11 + εε pppp 22 (10) Using the two definitions of the elastic and plastic strain relations, a set of equations can be formulated, where the third (out of plane) direction is omitted, since it is projected onto the in-plane directions. Eqs. 9 and 10 constitute the set of equations representing plane stress. Due to the assumption of isotropic linear elastic behavior and since it is assumed that no bending takes place, the elastic shear strain components related to the third direction are set to zero: (9)

5 εε eeee 13 = εε eeee 23 = 0. For a plane stress state the shear stress components related to the third directions are also zero and therefore the associated plastic shear strain components become: εε pppp 13 = εε pppp 23 = 0. As a special case of plane stress where the stresses in the principal in-plane directions are equal, the in-plane shear stress becomes: σσ 12 = 0. Introducing the composition induced chemical strain, the total strain in the direction normal to the surface can be expressed as εε tttttt 33 = νν εε 1 νν 11 eeee + εε eeee 22 + εε pppp 11 + εε pppp 22 + εε cch 33. Thereby the set of equations used in 2D plane stress with a von Mises yield criterion and isotropic hardening becomes: εε αααα tttttt = εε αααα eeee + εε αααα pppp + εε αααα cch (11) eeee σσ αααα = DD αααααααα eeee εε γγγγ (12) Φ = 3 2 σσ αααασσ αααα 1 3 (σσ αααα) 2 σσ yy (yy NN, εε pppp ) (13) pppp εε αααα ss αααα = γγ 3 (14) 2 σσ γγγγ σσ γγγγ 1 3 σσ γγγγ 2 εε pppp = γγ (15) eeee where αα, ββ, γγ, δδ = (1,2), Φ is the yield function and DD αααααααα are the components of the plane stress elasticity tensor. The equations are iteratively solved by the Newton-Raphson method and thereby the equilibrium equations and the plane stress assumption is solved to a pre-determined level of accuracy (see Ref. [6] for details). Computational Method Integrated computation. An overview of the integrated model is given in Figure 1a and consists of a diffusion module and a mechanical module. The exchange of data between these modules is represented by arrows, corresponding to the data flow of input and output. MECHANICAL MODULE nitriding potential, K_N equilibrium constant, K_T elastic parameters, E, v equilibrium concentration, c_eq reaction rate, k_rate surface flux, J_surf DIFFUSION SOLVER stress, σ elastic strain, ε_el diffusion coefficient, D concentration, c_n chemical expansion, V chemical strain, ε_ch total strain, ε_tot nitrogen trapping, K_NCr yield stress, σ_y work hardening, H plastic strain, ε_pl Figure 1a: System diagram of the integrated model, which consists of a diffusion and mechanical module.

6 Figure 1b: Schematic overview of the main variables of the model being the concentration in depth and the in-plane normal stress components, the (constant) temperature and their relation to the one-dimensional domain. For illustration purpose, the residual stress in the substrate is exagerrated in comparison to that of the case. The main variables considered in the model are the concentration and the in-plane principal stress components (Fig. 1b). The domain of the one-dimensional model spans from the surface towards the centerline (CL) of the material in the depth, z direction. Temperature gradients only play a role during heating and cooling cycles. The concentration-depth profile and stress-depth profile are strongly coupled and force balance in the system determines the level of residual stress in the case and substrate. Mixed control-volume based-/classical finite difference formulation of the implicit diffusion solver. The hydrostatic stress (pressure) and temperature gradient dependent diffusion was previously discretized in [5] by using the classical central finite difference method. A similar approach is used for the pressure and temperature terms here; however the concentration gradients are determined by an implicit control volume based finite difference method [11], which is more stable. Thereby, the numerical method applied becomes an implicit mixed control-volume based-/classical finite difference formulation. In addition, since the diffusion problem is non-linear due to the concentration dependent diffusion coefficient, the diffusion solver is used in an iterative scheme, such that the concentration profile is in equilibrium with the diffusion coefficient, the temperature and the stress state. The controlvolume based finite difference discretization of the mass transfer continuity equation, which only considers the diffusion problem with a concentration dependent diffusion coefficient (Fick s 2 nd law), is solved by the use of resistance terms rr = zz 2 DD : zz ii tt cc ii nn iiiiiiii = cc nnnnnn ii 1 nnnnnn ccii zz ii 1 zz ii + cc nnnnnn ii+1 nnnnnn ccii zz ii+1 2DD nnnnnn+ ii 1 2DD nnnnnn ii 2DD nnnnnn+ zz ii ii+1 2DD nnnnnn ii where the concentration on the new time-step is the sum of the solution at the old time-step and the accumulated incremental contributions cc nnnnnn ii = cc oooooo nn ii + cc with cc = iiiiiiii jj=1 δδδδ being the sum of the incremental solutions of all iterations for a given time-step. The full diffusion problem is then solved iteratively with material properties on the new time level. (16) Force equilibrium. Force equilibrium in depth is determined for the thermo-elastoplastic case, where the elastic stiffness is temperature dependent and the plasticity is for the isotropic case. For 1D, the relationship between force and stress is formulated: FF = LL σσ 0 dddd = 0 (17)

7 σσ = EE εε eeee (18) Considering non-constant element length, a node index is introduced. Furthermore, the force equilibrium has to hold for the full plate thickness, which means that the sum of all force contributions in depth has to equal zero. For this purpose, a numerical summation in the depth direction is carried out. The elastic strain component can be expressed in terms of the total- (εε tttttt ), plastic- (εε pppp ), chemical- (εε cch ), which are substituted into the zero force sum expression: nn nn FF = ii=1 σσ ii dddd ii = EE(TT) ii εε tttttt ii εε pppp ii=1 ii εε cch ii dddd ii = 0 (19) nn The summation of the total strain for all nodes ii=1 EE(TT) ii εε tttttt ii dddd ii is replaced by an average total strain nn εε tttttt dddd, which has an equal magnitude in all nodes and where n is the number of elements and dddd is the average element length. After rearranging, the force equilibrium can be expressed as the average total strain, which is a single-point evaluation for the whole system: εε tttttt = nn EE(TT) oooooo pppp pppp cch ii=1 ii εε ii + εε ii +εεii ddddii nn dddd nn ii=1 EE(TT) ii The onset of plastic yielding can introduce a sudden increase of the average total strain. To maintain the force equilibrium, it is therefore necessary to iterate for an incremental increase of the plastic strain εε ii pppp. Results and discussion The simulations were carried out for three nitriding potentials: KK NN =2.49, 9.38 and infinity, a treatment time of 22 hours at a temperature of 445 C, reflecting the experimental conditions presented in Ref. [3]. The concentration and stress profiles were calculated for two values of the equilibrium constant describing trapping, KK CCCCNNnn. The concentration dependent surface reaction rate constant (kk rrrrrrrr ) was assessed by reverse modelling as follows. The present model calculates the concentration-depth profile of nitrogen including all the couplings mentioned above and displayed in Fig.1a. Then, the total amount of nitrogen incorporated in the sample follows from an integration of the calculated concentration profile with regard to depth. This total nitrogen uptake can be compared to the experimental uptake of nitrogen in the corresponding time step. Optimization of the value for kk rrrrrrrr (or other parameters) is then obtained by a systematic variation of the parameter under investigation until a satisfactory agreement is obtained between calculated and experimental nitrogen uptake. It was verified in the calculations that the surface concentration dependence of kk rrrrrrrr is insensitive for the chosen value for the trapping solubility product (KK CCCCCC ). The dependence of kk rrrrrrrr on the surface concentration of nitrogen and the development of the surface concentration with time are given in Fig. 2a and b, respectively. Evidently, the value of kk rrrrrrrr depends strongly on the nitrogen content at the surface, particularly for higher surface contents of nitrogen. At first glance, this appears counterintuitive as an increase of the content of nitrogen atoms at the surface will make it more difficult for ammonia molecules to adsorb at the surface and dissociate. On the other hand, the dissolution of nitrogen in expanded austenite is accompanied by a pronounced lattice expansion (cf. Eq. (7)) and associated swelling (cf. Ref. [6]), which could lead to more active surface sites for the adsorption and dissociation of ammonia. (20)

8 Figure 2: Dependence of the rate of the surface reaction kk rrrrrrrr, on the nitrogen content at the surface yy NN ss, expressed as the number of nitrogen atoms per metal atom (a.). The insert shows a magnification of development of kk rrrrrrrr up to yy NN ss =0.30. Dependence of nitrogen content at the surface, yy NN ss, expressed as the number of nitrogen atoms per metal atom, on nitriding time, t (b.). The horizontal dashed lines represent the equilibrium nitrogen contents as determined on thin foils [9] for the successive nitriding potentials. The arrows mark the discrepancy between the actual surface content of nitrogen and the equilibrium nitrogen content. Evolution of residual (elastic) in-plane stress (σσ 1111 = σσ 2222 ) and plastic inplane strain (εε pppp ) evolution at the surface with surface concentration yy NN ss (c.). A peculiarity in the evolution of kk rrrrrrrr is observed in the insert in Fig. 2a for all investigated nitriding potentials (KN) at yy NN ss =0.12. The sudden change in kk rrrrrrrr coincides with the reaching of the maximum residual (elastic) stress and an important increase of plastic deformation in the expanded austenite case, implying that the further increase of the yield strength of expanded austenite is not possible and all additional increase in nitrogen content is accommodated plastically. Since the nitrogen content is highest at the surface, plastic deformation is most pronounced in the surface adjacent region. This has been demonstrated to be associated with the appearance of slip lines and an increase of the surface roughness [12]. Consequently, more metal atoms will be accessible at the surface, leading to more adsorption sites. The evolution of the surface content of nitrogen with nitriding time has not reached saturation within the investigated nitriding duration. Moreover, the equilibrium nitrogen content was not reached as a consequence of the compressive composition-induced stresses in the surface (Fig. 2b).

9 Summary The present (first) attempt to elucidate the role of the surface reaction on the overall kinetics of nitriding stainless steel by inverse modelling has demonstrated that the kinetics of the surface reaction determines the overall nitriding kinetics to a very large extent for the investigated laboratory experiments. The value of the reaction rate controlling the transfer of nitrogen from the gas phase to the solid-state increases gradually with a factor 20 with increasing surface concentration of nitrogen atoms. In particular, the plastic accommodation of the lattice expansion due to nitrogen incorporation was observed to play a role of importance. Whether the insight obtained is an actual effect or rather the consequence of various cross-correlations in the integrated computational model is impossible to conclude at this stage. It is recommended to conduct experiments with extremely thin samples such that the role of solid state diffusion, residual stress and plastic deformation can be ruled out and the response of the surface to a sudden change in the gas composition cam be studied, analogous to the seminal work by Grabke on pure iron. Acknowledgement Part of this work has been supported by the Strategic Research Center REWIND Knowledge based engineering for improved reliability of critical wind turbine components, Danish Research Council for Strategic Research, grant no References [1] E.J. Mittemeijer, M.A.J. Somers, Thermochemcial Surface Engineering of Steels, Woodhead Publishing, Woodhead Publishing Series in Metals and Surface Engineering: Number 62, [2] H.-J. Grabke, Reaktion en von Ammoniak, Stickstoff und Wasserstoff an der Oberfäche von Eisen- 1. Zur Kinetik der Nitrierung von Eisen in NH3-H2-Gasgemischen und der Denitrierung, Ber. Bunsenges. phys. Chemie, 72 (1969) [3] T. Christiansen, K. V. Dahl, and M. A. J. Somers, Nitrogen diffusion and nitrogen depth profiles in expanded austenite: experimental assessment, numerical simulation and role of stress, Mater. Sci. Technol., 24 (2008) [4] F. C. Larché and J. Cahn, The effect of self-stress on diffusion in solids, Acta Metall., 30 (1982) [5] F. N. Jespersen, J. H. Hattel, and M. A. J. Somers, Modelling the evolution of composition-and stress-depth profiles in austenitic stainless steels during low-temperature nitriding, Model. Simul. Mater. Sci. Eng., 24 (2016) [6] Ö.C. Kücükyildiz, M.R. Sonne, J. Thorborg, M.A.J. Somers, J.H. Hattel - in preparation. [7] T. L. Christiansen and M. A. J. Somers, Determination of the concentration dependent diffusion coefficient of nitrogen in expanded austenite, Int. J. Mater. Res., 99 (2008) [8] J. Oddershede, K. Ståhl, T.L. Christiansen, M.A.J. Somers, Extended X-ray absorption fine structure investigation of nitrogen stabilized expanded austenite, Scr. Mater., 62 (2010) [9] T. Christiansen, M.A.J. Somers, Controlled dissolution of colossal quantities of nitrogen in stainless steel, Metall. Mater. Trans., 37A (2006) [10] E. de Souza Neto, D. Peric, and D. R. J. Owen, Computational Methods for Plasticity, JohnWiley & Sons Ltd, vol. 55., [11] J. H. Hattel, Fundamentals of Numerical Modelling of Casting Processes, Polyteknisk Forlag, [12] J. C. Stinville,C. Templier, P. Villechaise, L. Pichon, Swelling of 316L austenitic stainless steel induced by plasma nitriding, J. Materials Science, 46 (2011)