The Influence of Stress on Interstitial Diffusion - Carbon Diffusion Data in Austenite Revisited

Thomas L. Christiansena, Marcel A.J. Somersb

Technical University of Denmark, Department of Mechanical Engineering Kgs. Lyngby, Denmark

atch@mek.dtu.dk, bsomers@mek.dtu.dk

Keywords: stress; diffusion coefficient, Fe-austenite, carbon, expanded austenite Abstract. The present paper addresses the influence of chemical induced stresses on diffusion in interstitial systems. This is exemplified by simulations of carbon diffusion in austenite at high temperatures and it is shown that old well established literature data is flawed by the occurrence of composition induced stress. For the technological relevant system of expanded austenite the diffusion can be dramatically affected by composition induced stress. Introduction The diffusion of interstitially dissolved carbon in austenite is commonly encountered in many systems. The concentration dependent diffusion coefficient of carbon in Fe-austenite was assessed more than half century ago by Wells et al. [1]. The data obtained by Wells et al. is well established and the validity has hitherto been unquestioned. In the original experiments by Wells et al. two iron specimens with different carbon contents were welded together and subsequently annealed at temperatures in the range 1000-1300°C for extended periods of time. Hence a redistribution of carbon atoms occurred. Based on the redistribution of carbon atoms the concentration dependent diffusion coefficient of carbon in Fe-austenite was obtained by the Boltzmann-Matano analysis of the carbon profile. Effectively, carbon is moving from the specimen with high carbon content to the specimen with low carbon content. In this setup, the lattice parameter of the austenite changes with changing carbon content, which means that composition induced stresses (chemical stresses) will occur after redistribution of carbon atoms. Figure 1 depicts the carbon concentration before and after annealing in the two iron specimens and the regions where tensile and compressive stress inevitably will arise upon redistribution of the carbon atoms. Consequently, diffusion occurs also under the influence of a self-induced stress gradient. This stress gradient will have an influence on the diffusing carbon atoms, i.e. there will be a contribution from stresses on the chemical potential. The present paper examines the effect of such stresses on the assessed concentration dependent diffusion coefficient of C in Fe-austenite by means of numerical simulation.

Defect and Diffusion Forum Vols. 297-301 (2010) pp 1408-1413© (2010) Trans Tech Publications, Switzerlanddoi:10.4028/www.scientific.net/DDF.297-301.1408

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Figure 1. Concentration profiles of carbon in a diffusion couple of Fe-austenite before and after annealing for 19.5 hours at 1127°C (from [1]). The slabs contain initially 7.67 and 0.07 atomic percent carbon for the left and right, respectively. The development of composition induced stresses is indicated in the figure. The simulations are not inherent to the system of carbon in austenite, but can be of relevance to other (technically important) systems, viz. expanded austenite. Expanded austenite is obtained when nitrogen and/or carbon atoms are inserted into stainless steel at low temperatures by means of gaseous thermochemical treatment or plasma/ion-based techniques (see e.g. refs. [2,3]). This system is unique in the way that colossal amounts of C/N atoms can be dissolved interstitially; for nitrogen stabilised expanded austenite the solubility can be as high as yN = 0.61 [4] and the resulting composition induced (compressive) stresses can reach several GPa’s [5]. The repercussions of such high interstitial contents and concomitant chemical stresses on diffusion are addressed in the last part of the paper. Stress-affected diffusion Generally, in interstitial systems concentration variations of the interstitial component induce a relatively large stress variation as compared to substitutional systems. The assessment of diffusion data from interstitial diffusion experiments under the influence of a (steep) concentration gradient will inevitably be the combined influence of composition and a composition-induced stress on the chemical potential gradient. The formalism applied by Chu and Lee [6] for a system wherein diffusion occurs under the influence of self-induced chemical stress is adopted for the calculations in this work. Hence, the diffusive flux, J, is given by:

xC

CCD

xC

DJ heff ∂

∂

∂∂

+−=∂∂

−=σ

10 (1)

where

∂∂

+=C

CDD heff

σ10 , (2)

Defect and Diffusion Forum Vols. 297-301 1409

( )RTVE

Ch

υσ

−=

∂∂

192 2

. (3)

E is Young’s modulus, υ is Poisson’s ratio, R is the gas constant, hσ is the hydrostatic stress

component, V is the partial molar volume (in 13 −atomm ) and Deff and D0 are the diffusivity of solute in a stressed isotropic solid and in a stress-free isotropic solid, respectively [6]. C is the concentration given in 3−⋅matoms . For the case of austenite (f.c.c.) where the lattice parameter, a, as a function of interstitial content obeys Végard’s law, i.e. XByAa += , with yX the occupancy of the interstitial sublattice and X the interstitial element, the concentration of interstitials (in 3−⋅matoms ) is given by:

3)(4

X

XX ByA

yC

+= . (4)

The partial molar volume (in 13 −atomm ) is given by:

( )243

4 XX

X ByABdydV

V +== (5)

Hence the stress affected diffusion coefficient can be calculated provided that the above mentioned physical parameters are known. Modelling set-up For the simulation of diffusion profiles a numerical finite difference model as developed in ref. [7] was applied. For an exhaustive description of this model the reader is referred to ref. [7], only the key features of the applied model will be mentioned here. The model handles concentration dependent diffusion coefficients and incorporates the equilibrium solubility product, K, which determines the concentration for the on-set of precipitation or trapping of solute atoms. The formalism for stress affected diffusion (cf. above) is readily incorporated into the model. The experiment by Wells et al. for assessment of the concentration dependent diffusion coefficient of carbon in Fe-austenite was carried out at 1127°C.The temperature dependence of the lattice parameter of carbon Fe-austenite was determined by Ridley and Stuart [8] at temperatures up to 1200°C. By interpolation among these data, the relationship between lattice parameter and carbon content in Fe-austenite at 1127°C was calculated as: Cya 6051.06622.3 += . The elastic parameters, E and υ, were assumed to be temperature independent. For the system of nitrogen expanded austenite the lattice parameter as a function of nitrogen content has only been determined at room temperature. However, the temperature regime for low temperature nitriding of stainless steel is approximately 400-450°C, so it is assumed that the room temperature relationship applies at the nitriding temperature. For stainless steel AISI 316 the lattice parameter as a function of nitrogen content in the composition range y� = 0.18 - 0.61 has been determined as [4]: �ya 59724.06396.3 += .This relationship is in the simulations extrapolated to lower nitrogen contents and is assumed to be valid at 445°C.

1410 Diffusion in Solids and Liquids V

Results and discussion Carbon diffusion in Fe-austenite Fig. 2a depicts the original data points for the concentration dependent diffusion coefficient for carbon in austenite obtained by Wells et al. These data were calculated (by Wells et al.) based on the carbon concentration profile shown in Fig 2b by the Boltzmann-Matano analysis. In order to incorporate the concentration dependent diffusion coefficient into the numerical model the data was fitted with a second order polynomial as shown in Fig 2a. Based on the fitted data for the concentration dependent diffusion coefficient the experimental carbon concentration profile in the two specimens was reconstructed. Obviously, this profile follows the discrete experimental data points very closely. As described above the redistribution of carbon atoms gives rise to composition induced compressive stresses in the right part of the welded diffusion couple and tensile stresses in the left. The transition from a tensile to a compressive stress state is located in the vicinity of the weld interface. However, for computational ease it is assumed that the transition from tensile to compressive stress coincides with the weld interface.

(a) (b)

(c) (d)

Figure 2. a) The concentration dependent diffusion coefficient of carbon in Fe-austenite as obtained by Wells et al [1] from evaluation of the data in Fig. 2B. The discrete data points have been fitted with a second order polynomial for use in the simulations. b) Experimental and calculated carbon diffusion profiles in the two specimens. c) Reconstructed concentration dependent diffusion coefficient for carbon in Fe-austenite with and without correction for composition induced stresses. d) Deviation of the concentration dependent diffusion coefficient without correction for composition induced stress from the stress-corrected diffusion coefficient. Correcting the concentration dependent diffusion coefficient for composition induced stress does not dramatically affect the shape of the carbon concentration profile (Fig. 2b). However, when comparing the concentration dependent diffusion coefficient with and without stress correction as shown in Fig. 2c and 2d it is obvious that stresses do have a major effect. The uncorrected diffusion coefficient is overestimated by more than 40% compared to the stress-corrected one (Fig. 2d). It should be noted that both in the zone of tensile and compressive stress diffusion is enhanced and

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that the discontinuity corresponds with the position of the weld, where the sign of the stress changes. The simulations clearly show that the reported diffusion data by Wells et al. should be considered as effective diffusion coefficients and not as pure chemical diffusion coefficients. �itrogen diffusion in expanded austenite

(a) (b)

Figure 3. a) The influence of composition induced stress on the diffusion of nitrogen in expanded austenite expressed as the ratio Deff/D0 as a function of nitrogen content at two different temperatures. b) Calculated nitrogen diffusion profiles based on a numerical finite difference model with and without the influence of composition induced stress. The calculations assume a constant surface concentration, Cs, of 0.44, diffusion time of 22 hours at 445°C, and a value of the equilibrium solubility product of 0.001 (see ref. [7] for further details). The concentration dependent diffusion coefficient was taken from ref. [9] The ratio Deff/D0 as a function of nitrogen content is given Fig. 3a for the temperatures 420 and 445°C. Clearly, the effect of compositional induced stresses is dramatic in this system, for high nitrogen contents the stress-affected concentration dependent diffusion coefficient is more than an order of magnitude higher than the pure chemical diffusion coefficient. The corresponding stresses amount to several GPa’s, and were indeed determined quantitatively [5,9]. Obviously, this effect cannot in any way be neglected. This is exemplified in Fig. 3b where a surface hardening process by gaseous nitriding is mimicked. It is assumed that the surface concentration is identical for the unstressed and stressed cases1. ‘Trapping’ of nitrogen atoms, due to a strong interaction between Cr and N, is incorporated by use of the equilibrium solubility product, K, arbitrarily chosen as 0.001. The concentration dependent diffusion coefficient of nitrogen in expanded austenite at 445°C as reported in ref. [10] was applied. Clearly, the effect of chemical stress has a tremendous influence on the depth of the nitrogen diffusion profile for this case. General discussion Usually, the presence of (composition-induced) stress is not taken into consideration on the determination of the (composition-dependent) diffusivity of interstitials from experiments involving a composition profile. Consequently, a vast number of published diffusion data on interstitial systems are potentially flawed by these composition induced stress effects. It is shown that neglecting composition-induced stress leads to erroneous values for the concentration-dependent diffusion coefficient of carbon in austenite. The effect of chemical stresses

1 Under practical conditions composition-induced stresses will also affect the solubility of interstitials as imposed by para-equilibrium with the gas phase (see refs. [4] and [7] in this respect).

1412 Diffusion in Solids and Liquids V

on diffusion in a ‘normal’ interstitial system such as the C-Fe austenite is clearly present, however the shape of the carbon profile is only slightly affected by the maximal 40% deviation in the concentration dependent diffusion coefficient. Although the validity of the diffusion data is clearly flawed by the stress effect, it should also be emphasised that the data is also likely to be flawed by experimental uncertainty, e.g. determination of carbon concentrations. It is expected that the effect of chemical stresses on diffusion in substitutional systems can be considered less dominant as compared to interstitial systems. Hence, for most systems the effect can probably be neglected and only for special cases the effect has to be considered. For the ‘abnormal’ interstitial system of expanded austenite the effect of chemical stresses cannot in any way be neglected. The effective diffusion coefficient can be more than 10 times higher than the stress-free diffusion coefficient, which has a major impact on the penetration depth. Conclusions The validity of diffusion data for carbon in Fe-austenite has been shown to be flawed by the occurrence of composition induced stresses, this effect may give rise to an overestimation of the concentration dependent diffusion coefficient by more than 40%. The effect of stress on diffusion is obviously something that should be considered when dealing with diffusion in interstitial systems. For the special system of expanded austenite, with very high contents of interstitials, the effect of stress on diffusion is dramatic; the stress-affected diffusion of nitrogen can be more than 10 times faster than stress-free diffusion. Acknowledgement Financial support from the Danish Research Council for Technology and Production Sciences under grant number 274-07-0344 is gratefully acknowledged. References [1] C. Wells, W. Batz and R.F. Mehl: Trans. AIME Vol. 188 (1950), p. 553. [2] Y. Sun, X. Li and T. Bell: Mater. Sci. Techn. Vol. 15 (1999), p. 1171. [3] T. Christiansen and M.A.J. Somers: Surf. Eng. Vol. 21(5-6) (2005), p. 445. [4] T. Christiansen and M.A.J. Somers: Mater. Metall. Trans. A. Vol. 37 (2006), p. 675. [5] T.L. Christiansen, M.A.J. Somers: Metallurgical and Materials Transaction A, in press. [6] J.L. Chu and S. Lee, J. Appl. Phys. Vol. 75(6) (1994), p. 2823. [7] T. Christiansen, K.V. Dahl and M.A.J. Somers, Mater. Sci. Techn. Vol. 24(2) (2008), p. 159. [8] N. Ridley and H. Stuart: Met. Sci. Vol. 4 (1970), p. 219. [9] T. Christiansen and M.A.J. Somers: J. Phase Equil. Diff. Vol. 26 (2005), p. 520. [10] T.L. Christiansen and M.A.J. Somers: Int. J. Mater. Res. (2008), in press.

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