Roman Gröger
email: groger@ipm.cz

Karleen J Dudeck

Peter D Nellist

Vaclav Vitek

Peter B Hirsch

J H Cockayne

D J H Cockayne

Effect of Eshelby twist on core structure of screw dislocations in molybdenum: Atomic structure and electron microscope image simulations

Keywords: dislocation structures, free surface, HREM user supplied): screw dislocation, thin foil, Eshelby twist screw dislocation, thin foil, Eshelby twist, free surface, HREM

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Effect of Eshelby twist on core structure of screw dislocations in molybdenum: Atomic structure and electron microscope image simulations

Introduction

It is generally accepted that the plastic deformation of pure BCC metals is controlled by 1/2〈111〉 screw dislocations (for reviews see [1][START_REF] Duesbery | Dislocations in Solids[END_REF][START_REF] Takeuchi | [END_REF][START_REF] Duesbery | Understanding Materials: A Festschrift for Sir Peter Hirsch[END_REF][START_REF] Seeger | [END_REF][START_REF] Vitek | Dislocations in Solids[END_REF]). The reason is, as first suggested by Hirsch [START_REF] Hirsch | Proc. 5th Int. Conf. Crystallography[END_REF], that these dislocations possess non-planar cores extending into several planes of the 〈111〉 zone. This core nonplanarity leads to a very high Peierls stress that is responsible for a rapid increase of the flow stress with decreasing temperature and increasing strain rate. The non-planar core also produces a strong dependence of the flow stress on the orientation of the crystal relative to the loading axes and related break-down of the Schmid law. Such non-planar spreading of the core has, indeed, been found in all atomistic studies performed in the last forty years (for reviews see [START_REF] Duesbery | Dislocations in Solids[END_REF][START_REF] Vitek | Dislocations in Solids[END_REF][START_REF] Vitek | [END_REF][9][10][11]). Transmission electron microscopy (TEM) observations of dislocation substructures after deformation [12][13][14][15] and in-situ studies [16][17][18][19][20][START_REF] Vesely | Understanding Materials: A Festschrift for Sir Peter Hirsch[END_REF] of the glide of dislocations reveal the presence of long screw dislocation segments that indicates their very low mobility relative to the dislocations of other orientations. However, no direct experimental observation confirms the non-planar core structure unambiguously.

In recent years, advances in high resolution electron microscopy (HREM) spawned an interest in utilising this technique to investigate in detail the structure of dislocation cores in thin metallic foils of BCC metals. Two attempts had been made to observe the core structure of the 1/2〈111〉 screw dislocation in molybdenum by HREM [START_REF] Sigle | [END_REF]23]. Apart from severe experimental difficulties, these studies are burdened by two fundamental problems. Firstly, the image principally reflects the displacements at the exit surface, and because of interband scattering the image peaks are displaced relative to the actual positions of atoms at the surface [24]. Secondly, the screw dislocation normal to and terminating at traction-free surfaces generates the so-called Eshelby twist. This deformation field, derived theoretically for an isotropic material by Eshelby and Stroh [25], is characterised by counterrotations of the upper and lower parts of the foil and it is much larger than the core displacements in the bulk. In an attempt to separate the core and Eshelby twist displacements, Mendis et al. [23] revisited the concept of the Nye tensor and showed that it captures only the incompatible part of the deformation field. Their Nye tensor plots show that the displacements due to both the Eshelby twist and the dynamical scattering of the electrons behave as compatible deformations and that, therefore, they do not appear in the components of the Nye tensor.

Since the HREM image reflects the displacements at the exit surface, the fundamental questions are: (i) whether the HREM observations relate to the core displacements in the bulk, and (ii) whether the stresses associated with the Eshelby twist produce such significant changes in the dislocation core structure that the relation to the core structure in the bulk becomes obscured. The objectives of this paper are to elucidate these points by carrying out an atomistic simulation of the 1/2〈111〉 screw dislocation in a thin foil of Mo of thickness typical for HREM observations, and to simulate the HREM images of the calculated structure. The atomic displacements of the relaxed structure automatically contain the surface-induced relaxation displacements, and the effects of the surface relaxation stresses on the core structure. These displacements can then be identified as the difference between the relaxed positions of atoms in the foil and their relaxed positions in a block that is periodic in the direction parallel to the dislocation line, i. e. without free surfaces. Finally, the calculated relaxed atomic positions in the foil can be used as input for the image analysis. This provides the positions of image

Atomistic evaluation of the Eshelby twist and its separation from the core displacements

Computational model

The object of the calculations is to determine the atomic positions around a 1/2[111] screw dislocation that is normal to a thin foil of Mo, and compare them with those in an infinite crystal. The procedure adopted in this paper requires two atomistic simulations in which the energy of a cylindrical block (with its axis of revolution coincident with the screw dislocation) is minimised with respect of the positions of the atoms. To obtain the atomistic positions near the core in an infinite crystal we utilise an atomic block which is effectively of infinite thickness. To ensure that the atoms in the block assume positions that approximate to those in a crystal of infinite radius, we impose the boundary condition that the atoms in an outer rim of the block assume the positions given by the anisotropic linear-elastic strain field of the screw dislocation in an infinite lattice. After inserting the dislocation into the block, the two terminating (111) planes contain ledges that arise necessarily due to the screw character of the dislocation. We do not give the atoms at these surfaces any special treatment and let them relax as it is done with any other atom in the block.

To determine the positions of the atoms in a thin foil we utilise a cylindrical block of the same radius but of finite thickness, terminated by traction-free surfaces normal to its axis of revolution. To ensure the correct boundary conditions in the radial direction appropriate to those expected for a foil of infinite radius we need the linear-elastic solution of the Eshelby twist strain field for an anisotropic lattice. Such a solution is not known. Instead, as an approximation, we impose a boundary condition consisting of the anisotropic linear-elastic strain field for the screw dislocation in an infinite crystal, plus the Eshelby-Stroh solution [25] of the additional tangential displacements due to the Eshelby twist in an isotropic crystal. In that way the constraint of the part of the foil outside the simulated block of finite radius is taken into account in an approximate manner. In the former, the elastic moduli of molybdenum are taken from Ref. [26]: C 11 = 464.7 GPa , C 12 = 161.5 GPa and C 44 = 108.9 GPa . Since the anisotropy ratio 2C 44 / (C 11 -C 12 ) is 0.7, molybdenum is not strongly elastically anisotropic.

Moreover, the Eshelby-Stroh formula for tangential displacements is independent of elastic moduli, which avoids the ambiguity of reducing the set of three anisotropic elastic moduli above to their two isotropic counterparts. Both these observations further justify the use of the isotropic Eshelby-Stroh displacement field far away from the dislocation. The effect of the Eshelby twist is then determined by the differences of the displacements computed in the thin foil and infinite crystal.

The reason for making the blocks cylindrical is to avoid complications associated with possible unequal distances from the dislocation centre to the lateral face of the block that arise for other geometries. This is particularly important in this case because the displacements due to the Eshelby twist are long range. In both simulations, the coordinate system was chosen such that the axis of revolution of the cylinders coincides with the z axis that is parallel to the [111] direction and thus to both the dislocation line and the Burgers vector, the x axis parallel to the [ 12 1] direction and the y axis direction. This choice of axes makes the coordinate system right-handed. In both relaxation calculations below we used the Bond Order Potential (BOP) for Mo [27] that captures the mixed nearly free electron and covalent character of bonding in this metal.

In calculations for the infinite crystal we utilise an atomic block that is periodic along the dislocation line and effectively infinite in the plane perpendicular to the dislocation line. The repeat cell is composed of three (111) planes that represent one 1/2[111] period of the BCC lattice with the outer radius equal to 15a, where a=3.1472 Å is taken as the lattice parameter of Mo. The 1/2[111] screw dislocation was inserted in the middle of the block in the direction parallel to the z axis by displacing all atoms according to the anisotropic linear-elastic strain field of the dislocation in an infinite lattice. The total number of atoms in this block was 1221. During the relaxation, 675 atoms at the distance from the core 10 < r / a < 15, were held fixed, while the positions of 546 atoms at the distances r / a ≤ 10 were adjusted. The former group of atoms thus represents an outer rim of the block in which the positions of atoms are determined by the linear-elastic anisotropic solution for an infinite Volterra dislocation. The positions of the atoms that constitute the inner part of the block are found by the relaxation during which all these atoms are allowed to move in all directions. Close to the centre of the dislocation these atomic positions define the core structure. The relaxation was terminated when the magnitudes of forces acting at all atoms fell below 0.005 eV/Å. The relaxed position of an atom i can then be written as

x ∞ i = X i + u dislo i , (1)
where X i is the initial position of the atom i in the perfect lattice, and u dislo i the displacement of this atom due to the dislocation. The positions x ∞ i are the atomic positions in a block that is infinite in all directions and contains a screw dislocation in the middle.

In the second simulation, for the thin foil with the 1/2[111] screw dislocation perpendicular to the traction-free foil surfaces, the simulation cell is constructed by taking the relaxed periodic block of 1221 atoms from the first simulation and copying it 18 times along the z direction. This cell, which represents the foil, had the following additional properties. The origin of the z axis was chosen such that z = ±h / 2 correspond to the positions of the two free surfaces terminating the block in the z direction. Along the z direction the block was composed of 19 complete periods, i.e. 57 atomic planes of the (111) type with the layer-to-layer distance of a / 2 3 . Hence, the total thickness of the simulated foil was 5.09 nm. The diameter of the block in the (x,y) plane was about 30 lattice periods and the block contained 23199 atoms of which 12825 atoms were located in the outer rim (i. e. at 10 < r / a < 15) and the remaining 10374 atoms with positions r / a ≤ 10 in the inner part of the block. We imposed the traction-free conditions on the upper and lower flat surfaces by displacing all atoms in this block by the isotropic Eshelby-Stroh displacement field [25]

u θ (r, z) = - b 2π (-1) n r w n-+ w n- 2 + r 2 - r w n+ + w n+ 2 + r 2         n= 0 ∞ ∑ . ( 2
)
Here, b is the magnitude of the Burgers vector, r the distance from the dislocation, and w n ± = (2n + 1)h / 2 ± z . Only the terms of the series corresponding to 0,10 n ∈ are considered; all higher order terms not accounted for in the sum have negligible effect on the displacement field.

During the subsequent minimisation of the energy, only the atoms in the inner part of the block move while those in the outer rim are held fixed. Consequently, the displacements of the atoms in the outer rim are given by the superposition of the elastic long-range displacement field of a straight dislocation in an anisotropic crystal and the isotropic Eshelby-Stroh field [START_REF] Duesbery | Dislocations in Solids[END_REF]. This block has many more degrees of freedom than the infinite periodic block and thus the relaxation approached the stable equilibrium much more slowly than in the periodic case. Hence we stopped the relaxation of the atomic positions in this block when the forces on all atoms fell below 0.06 eV/Å. Further relaxation is unlikely to affect the positions of atoms near the centre of the dislocation. The calculated position x f i of each atom i in the inner part of the block is then

x f i = x ∞ i + u twist i = X i + u dislo i + u twist i , (3)
where x ∞ i is the position of the atom obtained from the first simulation, and u twist i the displacement due to the Eshelby twist (including the effects of surface relaxation on the core displacements) owing to the traction-free surfaces on the upper and lower flat surfaces of the cylinder. From symmetry, the Eshelby twist vanishes in the middle plane of the foil (i.e. at z = 0 ) and, therefore, the atoms in this plane are displaced in the same way as in the periodic block. This provides a check on the calculation. The field

{x f i ,i = 1..N} ,
where N is the number of atoms, then represents the positions of atoms in a thin foil that is finite in the z direction but (approximately) infinite in the (x,y) plane. These positions are used in the image simulation in Section 3.

Structure of the dislocation core

In Figs. 1 and2 we show the maps of the differential displacements parallel and perpendicular to the dislocation line (and to the Burgers vector), respectively, for three slices taken from the upper and lower surfaces and from the centre of the relaxed block that represents the foil. The atomic arrangement is shown in the projection perpendicular to the direction of the dislocation line. The circles stand for the atoms in the three consecutive (111) atomic planes (differentiated by shading) and the lengths of the arrows are proportional to the magnitudes of the relative displacements of neighbouring atoms either in the direction parallel (screw displacements; Fig. 1) or perpendicular (further called edge displacements; Fig. 2) to the dislocation line, i.e. corresponding to the relative displacements in the z direction and in the (x,y) plane, respectively. The colours of atoms are chosen such that they get darker with increasing position along the z direction in the ideal lattice. For example, when plotting Figs. 1a and 2a (upper surface), black corresponds to the atoms in the (111) plane that terminates the foil at the upper surface, grey are the atoms in the next (111) plane below, and white correspond to the atoms in the third (111) plane from the upper surface. Figs. 1a and 2a correspond to the slice composed of three (111) planes at the upper surface and Figs. 1c and2c to the slice composed of three (111) planes at the lower surface. Figs. 1b and2b show the central slice. The magnitudes of the edge displacements are much smaller than the magnitudes of the screw displacements. However, the edge displacements near the surfaces are about 5 times larger than those in the centre of the foil and, therefore, the displacements in Fig. 2b are scaled by a factor of 10 while those in Figs. 2a,c only by a factor of 2. The atoms in the centre slice, shown in Figs. 1b and2b, experience very similar displacements as in the case of an infinite straight screw dislocation and thus the displacement maps shown in Figs. 1b and2b are practically the same as those presented in Refs. [27,28]. In the centre of the foil the core possesses very closely both the [111] threefold screw axis and [101] diad symmetry, i. e. it is practically the same as the non-degenerate core found in studies of infinitely long straight screw dislocations [27,28].

However, near the surfaces the [101] diad symmetry is removed. This is similar to the situation encountered in the case of infinite dislocations when an external shear stress in the (101) plane or tension along an axis within the standard triangle is applied [28][29][30]. The core extends into the (101) plane on both surfaces, while the displacements in the (110) and (0 11) planes are suppressed.

Moreover, as mentioned above, the edge displacements in the surface slices are much larger than those in the middle of the foil. This means that there are significant atomic relaxations in the direction perpendicular to the Burgers vector on both the upper and lower surfaces.

Further insight into the effect of the free surface on the structure of the dislocation core can be gained by considering the directions of edge displacements. It was shown in [30] that in the case of infinite straight dislocations these displacements are practically parallel to the lines connecting the neighbouring atoms so that the bond angles remain close to those found in the ideal BCC structure. This means that the edge displacements produce a predominantly local expansion. The same is found in the middle of the foil (Fig. 2b) where the displacements due to the Eshelby twist are negligible. On the other hand, near the surfaces (Figs. 2a,c) the Eshelby twist rotates the atomic bonds towards the directions perpendicular to those in the twist-free region (Fig. 2b). Hence, near the surfaces the system minimises its energy by keeping the local density as unchanged as possible and thus the nearest neighbour separations constant, whilst the bond angles are forced to change.

Determination of the anisotropic Eshelby twist

The anisotropic Eshelby twist can now be calculated by comparing the relaxed positions of atoms in the foil with those in the cell that is periodic in the z direction and effectively infinite in the (x,y) plane, both of which were obtained in Subsection 2.1. The difference

u twist i = x f i -x ∞ i , (4)
where x ∞ i and x f i are obtained from Eqs. Eshelby twist evaluated fully anisotropically with the constraint that these displacements approach the isotropic Eshelby-Stroh solution (2) at r / a = 10 . For an isotropic material it was shown in [25] that the Eshelby twist causes opposite tangential displacements on the two free surfaces and thus necessarily counter-rotations of the upper and lower parts of the block. Similar response is expected in anisotropic crystals. This is demonstrated in Fig. 3 where we show the (x,y) component of the The dots represent positions of atoms in three consecutive (111) planes (not differentiated) in the perfect crystal at the upper surface (Fig. 3a) and at the lower surface (Fig. 3b), respectively; the triangles mark the centre of the dislocation in the infinite periodic block. The differences in the displacements of the atoms nearest to the core in the upper and lower surfaces, respectively, result because the depicted atoms have different distances from the middle plane of the foil. One can clearly see that the displacement field u twist i has a predominantly tangential character and represents rotations in opposite directions in the upper and lower parts of the foil. Since these tangential displacements vary along the z direction and become negligible for z = 0, the induced twist is not uniform throughout the block but varies with both the depth in the foil, i. e. coordinate z, and the distance from the centre of the dislocation. It is apparent from this figure that the main differences between u twist i component in the (x,y) plane and the Eshelby-Stroh linear-elastic displacements (Eq. 2) occur close to the core. This is seen more clearly in Fig. 4 which shows these two displacements for the three (111) planes at the lower surface (Fig. 3b) as a function of r / a ; the main deviations occur for r / a < 3 . At larger values of r / a the varying deviations of the values of u twist i at a given r / a relate to the effect of anisotropy. We note that the displacements calculated by the atomistic simulation approach those of the isotropic Eshelby-Stroh displacements with increasing r / a . To check the sensitivity of the displacements near the core to the boundary conditions at r / a = 10 , a calculation was performed with stress-free boundary conditions at r / a = 10 . The differences were less than 10% at r / a < 3 .

We are now in a position to answer the first question stated in the Introduction, i. e. whether or not the Eshelby twist affects the dislocation core. It will be convenient to utilize the concept of the Nye tensor [31] within which an arbitrary displacement field can be represented by a continuous field of infinitesimal (i. e. virtual) dislocations. The density of the Burgers vectors of these infinitesimal dislocations is then represented by a symmetric tensor α . It satisfies the Bianchi identity ∇ ⋅ = α 0 which constrains the dislocation network such that no dislocation begins or ends inside the body [START_REF] Kröner | Continuum theory of dislocations and self-stresses[END_REF].

The component α ij represents the density of the infinitesimal dislocations whose line directions are parallel to x i and their Burgers vectors are parallel to x j . An important property of the Nye tensor is that it describes only the incompatible part of the elastic-plastic deformation field [23,[START_REF] Kröner | Continuum theory of dislocations and self-stresses[END_REF]. Therefore, any compatible distortion of the lattice for which the displacement field remains single-valued and the strain field integrable does not affect the Nye tensor. Consequently, if the Eshelby twist did not affect the dislocation core, the Nye tensor plots calculated for the two surface slices would have to be identical to that taken from the middle of the foil. In Fig. 5 we show the three relevant components of the Nye tensor for the upper and lower surfaces and the middle of the foil. Since the dislocation line is parallel to the z direction, α 33 corresponds to the screw component of the dislocation density.

Similarly, α 31 and α 32 relate to the two edge components, for the Burgers vectors parallel to x and y, respectively. We represent the dislocation core by a region around the dislocation in which the α , are larger than some threshold. Since the Eshelby twist strongly affects the edge component of this field (as seen in Fig. 5), i.e. the relative displacements of atoms perpendicular to the slip direction, the dislocation core close to the two free surfaces is very different from that in the center of the foil. We thus conclude that the Eshelby twist strongly affects the dislocation core near the foil surfaces but not in the bulk of the foil. The Eshelby twist thus induces both compatible deformation away from the dislocation core and the incompatible deformation in the core region. The latter is not surprising since the surface relaxation strains are very large. For example, using the Eshelby-Stroh formula (Eq. 2) [START_REF] Vesely | Understanding Materials: A Festschrift for Sir Peter Hirsch[END_REF], the shear strains at about 3 Å from the core are about 0.1. The existence of these large surface strains is further corroborated by the edge components of the differential displacement maps shown in Fig. 2, where the surface strains (Figs. 2a,c) are a factor of 5 larger than the strains in the middle of the foil (Fig. 2b).

HREM Image analysis

Simulated HREM images were calculated using the multislice method incorporated into the QSTEM software package [START_REF] Koch | QSTEM[END_REF][START_REF] Koch | Determination of core structure periodicity and point defect density along dislocations[END_REF]. A model consisting of atoms of Mo placed at the set of atomic coordinates given by x f i was used, with the illuminating beam taken to be incident parallel to the [111] direction of the sample (i.e. parallel to the dislocation). Reference images were also calculated for a perfect reference crystal with a set of atomic coordinates given by X i . In both cases the imaging conditions were: an accelerating voltage of 300 kV, an incident beam divergence of 0.2 mrad and a Gaussian distributed defocus spread (arising from chromatic aberration) with a standard deviation of 2.9 nm. No lens aberrations were included in the calculation. Although these zero-aberration conditions do not give weak-phase contrast, the sample here is thick enough to be strongly dynamical leading to significant amplitude changes in the exit wave. Furthermore, the presence of aberrations would lead to increased delocalisation in the image and greater complexity in the contrast from atomic columns. The simulated image of the perfect crystal (Fig. 6a) shows that the zero-aberration conditions assumed here gave simple peaks with a single maximum at the location of each atomic column. Introducing aberrations was found to result in a more complex "donut" contrast for each atomic column. With spherical aberration correctors becoming more commonplace, the zero-aberration condition is now feasible to achieve experimentally. The simulated image of the relaxed model including the Eshelby twist (coordinate set x f i ) under these imaging conditions is shown in Fig 6b.

The effect of the Eshelby twist on the image can be clearly seen where the bright peaks are now displaced and asymmetric bright-dark contrast is seen. Closer to the core, the dark regions are not as apparent.

In order to determine the extent to which the images can be compared with the atomistic model, the intensity peak for each atomic column was located, and the displacement from the perfect crystal position computed. The images were simulated with a pixel size of 0.03 Å, and this represents the precision to which the peaks were located. The precision to which peaks in an HRTEM image can be located is a function of image resolution and signal-to-noise ratio, but the precision used in the image simulations here is similar to the highest precisions that have been observed experimentally [START_REF] Hytch | [END_REF].

Results and implications

This paper raises the question of what information on the core structure of screw dislocations can be obtained from HREM images of such dislocations viewed end-on. There are two issues: (i) the extent to which the images reflect accurately the surface core structure of the dislocation, and (ii) the information that can be gained on the core structure in the bulk.

On the first point, the images reflect the displacements at the surface, but modified by interband scattering [24] and the microscope transfer function. Fig. 7 shows a comparison of the absolute displacement (scaled 5 times) at the exit surface around the screw dislocation obtained from atomistic simulations (black) and from the peak positions in the image analysis (red). Close to the core the directions and magnitudes of the displacements obtained in these two ways differ significantly, while further away the directions are similar. The displacements from the image peak positions are slightly larger than those from the atomistic simulations. The discrepancies between the two plots show the effect of interband scattering and the microscope transfer function on the image peak positions, which are largest close to the dislocation core.

Fig. 8 compares the edge component of the differential displacement maps obtained from atomistic simulations (a) and from image analysis (b). The arrows in both plots have been scaled by a factor of 2. While on the whole the magnitudes of the differential displacements are comparable, there are significant differences in detail, again illustrating the effects of interband scattering and the microscope transfer function.

Mendis et al. [23] simulated images of a singular screw dislocation in an elastically isotropic thin foil including the Eshelby twist using the Eshelby-Stroh formula [24]. They found that the Nye plots were featureless, showing that the effects of interband scattering and the microscope transfer function did not introduce apparent incompatible deformation. Fig. 9 shows a comparison of Nye plots of the edge components α 31 and α 32 obtained from our atomistic calculations (Fig. 9a) and from the positions of image peaks (Fig. 9b). While some features are similar, there are clearly some significant differences. We conclude that in these image simulations both the interband scattering and the microscope transfer function do affect the incompatible as well as the compatible deformations.

However the magnitudes of α ij in Fig. 9b reflect those at the exit surface, which are considerably greater than those expected from the atomistic simulations in the bulk. The assumption made in a previous image simulation of such dislocations by Mendis et al. [23], that the interaction between surface relaxation stresses and the core structure is negligible, appears to be invalid. The possibility of such an interaction was raised in [36], but its magnitude was not known.

It should be emphasised that the interband scattering is sensitive to image conditions; e.g. small changes in thickness or orientation can result in considerable changes in interband scattering. The displacements observed in the experimental images of Sigle [START_REF] Sigle | [END_REF] and Mendis et al [23], which do not conform to those expected from Fig. 5, may well have been affected by small changes of crystal [23] were carried under different imaging conditions from those used in the present paper. Repeating our analysis, but using the imaging conditions of Mendis et al. (data not shown), gives displacements about 3 times smaller than those plotted in Fig. 7, and Nye tensor plots with values two orders of magnitude smaller than those in Fig. 9. Changing the microscope transfer function can clearly have a very significant effect on the measurement of the incompatible deformation using HREM.

In conclusion, the atomistic simulation of the core structure of the screw dislocation normal to the surface of a thin foil of Mo has shown that the surface relaxation displacements are large and that the surface relaxation stresses have a large effect on the core structure at the surface, with serious implications on the interpretation of HREM images of such dislocations viewed end-on. The results of the image simulations in this paper suggest that the displacements, differential displacements and Nye tensor plots obtained from the image peak positions do not reflect accurately the structure at the surface. On the other hand, the apparent displacements derived from the peak positions in the image, which include the effects of interband scattering and the microscope transfer function, are large enough to be measured in an HREM experiment. To that extent, therefore, the much larger displacements at the surface expected from the atomistic simulations are helpful, and suitable for comparison with image simulation from assumed models. But to obtain any meaningful information of the core displacements in the bulk, quantitative comparison between experimental images and those obtained from assumed structures would be essential. Furthermore, the magnitude of edge component of the core displacement in the bulk (~0.01 nm) is smaller than the currently demonstrated precision to which peak positions have been measured in a typical HREM experiment. A similar order of magnitude of these edge displacements is obtained also with other boundary conditions, e.g. when all faces of the cylinder are traction-free. Data that would provide sufficient sensitivity for the core displacements to be determined will require improvements in image signal-to-noise ratio, and therefore microscope and sample stability and cleanliness.

  exit surface, which reflect the Eshelby twist and core displacements, plus the displacements due to the interband scattering and the microscope transfer function.

( 1 )

 1 and (3), respectively, then measures the changes in the atomic positions induced by the traction-free boundaries of the foil . Since the interatomic potential used was fitted to reproduce the anisotropic elastic moduli of Mo, the displacements u twist i represent the

  the anisotropic Eshelby twist, u twist i , determined according to Eq. (4) (black arrows) and those of the isotropic Eshelby-Stroh solution (red arrows). The displacements, i.e. the lengths of arrows in this figure, are scaled by a factor of 5.

  relative to their positions in the ideal lattice, and thus also ij

  thickness. The simulations of Mendis et al.

Figure 1 :

 1 Figure captions

Figure 2 :

 2 Figure 2: Map of edge differential displacements for the 1/2[111] screw dislocation calculated for three slices taken from the block representing the foil at: (a) z = +h / 2 (upper surface), (b) z = 0 (centre of the foil), and (c) z = -h / 2 (lower surface). The displacements in (b) are scaled by a factor of 10 while those in (a) and (c) by a factor of 2.

Figure 3 :

 3 Figure 3: The displacements of the Eshelby twist for (a) the upper and (b) the lower surface of the thin foil. The dots represent the positions of atoms in the three successive (111) planes adjacent to the two free surfaces; triangle shows the position of the dislocation centre. The calculated anisotropic Eshelby twist is shown in black; the isotropic Eshelby-Stroh field is in red. Note the opposite rotations in the two slices. The lengths of the arrows are scaled by a factor of 5.

Figure 4 :

 4 Figure 4: Comparison of the decay of the absolute displacements around the dislocation at the lower surface of the foil, u / a , with the distance from the dislocation, r / a . The purely tangential displacements obtained from the Eshelby-Stroh formula (2) are shown in red. The calculated displacements of atoms in the three (111) planes adjacent to the lower surface of the foil are plotted by black. The black dots above the Eshelby-Stroh displacements come from the atoms in the (111) plane that terminates the foil.

Figure 5 :

 5 Figure 5: Plots of Nye tensor components for three slices taken from the upper ( z = +h / 2 ) and lower surfaces ( z = -h / 2 ) of the foil and from the middle of the foil ( z = 0 ). The colours correspond to the density of the Burgers vectors of continuously distributed infinitesimal dislocations. The component α 33 corresponds to the screw component; α 31 and α 32 to the two edge components parallel to the x and y axes, respectively. The circles represent the relaxed positions of atoms in the three consecutive (111) planes.

Figure 6 : 14 Figure 7 :

 6147 Figure 6: Simulated HREM images of: (a) a perfect crystal containing no dislocation; (b) the relaxed crystal containing the screw dislocation with atom positions given by the set of coordinates x f i .

Figure 8 :

 8 Figure 8: Comparison of the edge component of the differential displacement maps obtained from atomistic simulations (a) and from image analysis (b). The lengths of arrows in both plots are enlarged twice. The intersection of the dislocation line with the plane of the figure is marked by the red triangle.

Figure 9 :

 9 Figure 9: Comparison of the Nye tensor plots α 31 and α 32 obtained from (a) atomistically calculated positions of atoms in the three (111) planes adjacent to the upper surface of the foil, and (b) image simulations at the exit surface of the beam.

  Fig. 4 227x148mm (200 x 200 DPI)

  Fig. 5 1235x1235mm (72 x 72 DPI)

  Fig. 7 152x139mm (200 x 200 DPI)

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Acknowledgments

We thank Yuri Mishin for sharing his code for plotting the Nye tensor maps. RG acknowledges support from the Czech Science Foundation (Grant No. P204/10/0255) and from the Academy of Sciences of the Czech Republic (Research Project No. AV0Z20410507). VV acknowledges support from the US Department of Energy, BES, Grant No. DE-PG02-98ER45702. PDN and PBH gratefully acknowledge the provision of facilities by the Department of Materials, University of Oxford. The access to the METACentrum supercomputing facilities provided under the research intent MSM6383917201 is highly appreciated.


Kyung Sub

Ralf E Schacherl
email: r.schacherl@mf.mpg.de

Ewald ; Bischoff
email: bischoff@mf.mpg.de

Eric Jan Mittemeijer
email: e.j.mittemeijer@mf.mpg.de

Dr Ir E J Mittemeijer Schacherl

Dr Ir E J Mittemeijer Mittemeijer

Eric Jan

K S Jung
email: k.s.jung@mf.mpg.de

-based Fe-Cr-Al alloys; the role of the Cr/Al atomic ratio

Keywords: surface treatments, nitrides, phase transformations user supplied): gas nitriding, absorption isotherms, ternary alloy gas nitriding, absorption isotherms, ternary alloy, excess nitrogen

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Normal and excess nitrogen uptake by iron-based Fe-Cr-Al alloys; the role of the Cr/Al atomic ratio

Introduction

Nitriding is a thermochemical surface-treatment process of ferritic steel components in order to improve properties such as the resistances against fatigue, corrosion and wear [START_REF] Lampman | Introduction to surface hardening of steel ASM Handbook[END_REF][START_REF] Ferkel | [END_REF][3][4][5][6][7]. In order to introduce nitrogen into the component, gaseous nitriding, employing NH 3 -based atmospheres, is widely adopted. The application of a NH 3 /H 2 gas mixture, instead of pure N 2 gas, makes it possible to impose, at 1atm and in a temperature range of 500 -580°C, at the surface of the specimen to be nitrided, a chemical potential of nitrogen equivalent to that of thousands of atmospheres of N 2 gas. To this end the slow thermal dissociation of NH 3 under these conditions is crucial [8,9].

Upon nitriding of iron-based binary alloys containing alloying elements with a strong affinity for nitrogen, such as Cr [10][11][12][START_REF] Jack | Proc. Heat Treatment Conf[END_REF][START_REF] Lightfoot | Proc. Heat Treatment Conf[END_REF][START_REF] Pulkkinen | [END_REF][16][17][18][19][20][21], Al [22][23][24][25][26][27], V [28][29][30][31][32][33] and Ti [34][35][36][37], alloying element nitride precipitates can develop in the diffusion zone.

Chromium and aluminum are often used together as alloying elements (Me) in commercial nitriding steels. Only very recently a few studies were devoted to the nitriding of such ternary iron-based Fe-Me 1 -Me 2 alloys. Whereas, upon nitriding, in binary Fe-Cr alloys rapid precipitation of highly coherent, cubic rock-salt crystal-structure type CrN nitrides takes place [12,[START_REF] Jack | Proc. Heat Treatment Conf[END_REF]16], in binary Fe-Al alloys hexagonal, wurtzite, crystal-structure type AlN nitrides precipitate with difficulty, i.e. only slowly [23][24][25][26]. It was shown that the total amount of nitrogen taken up per atom alloying element Me was rather different for Fe-Cr and Fe-Al Using homogeneously nitrided foils of Fe-Cr-Al alloys with different Cr/Al atomic ratios (0.21 ≤ Cr/Al ≤ 2.00), while keeping the total amount of the alloying element at about 1.50at.%, so-called nitrogen-absorption isotherms (cf. section 3.4) were determined and discussed in terms of various types of absorbed nitrogen. The microstructure of the nitride zone and the morphology and composition of the nitride precipitates were investigated by means of X-ray diffraction (XRD), electron probe microanalysis (EPMA), transmission electron microscopy (TEM) and electron energy loss spectroscopy (EELS).

Experimental

Specimen preparation

Ingots of Fe-Cr-Al alloys, containing about 1.50at.% (Cr + Al), with the Cr/Al atomic ratio varying from 0.21 to 2.00 were prepared from pure Fe (99.98wt.%, Alfa Aesar), pure Cr (99.999wt.%, Alfa Aesar) and pure Al (99.999wt.%, Alfa Aesar) using an inductive furnace. The alloys were cast into rods with a diameter of 10 mm and length of 100 mm. The precise composition of the Fe-Cr-Al alloys was analyzed, applying (i) inductive coupled plasma-optic emission spectroscopy (ICP-OES) to determine the content of the alloying elements Cr and Al, (ii) a combustion method to determine the light elements C and S and (iii) hotextraction to determine the light elements O and N. The thus determined chemical compositions of the alloys are shown in Table 1.

The cast rods were cold-rolled to foils with a thickness about 0.2 mm in several steps and were cut into rectangular specimens (16 × 16 mm 2 ). After being polished, the foils were encapsulated in a quartz tube with an argon atmosphere and annealed at 700°C for 2h followed by air-cooling to obtain a recrystallized grain structure (grain size of about 20 -50 µm). Before nitriding the specimens were ground and polished (last step: 1 µm diamond paste) and then cleaned in ethanol with ultrasonic agitation.

Nitriding; determination of nitrogen-absorption isotherms

The foils were suspended at a quartz fiber and placed in the middle of a vertical quartz tube furnace. The composition of the ammonia/hydrogen gas mixture (purity: NH 3 > 99.998 vol.% and H 2 > 99.999 vol.%) within the furnace was precisely controlled by fixing the ratio of the gas flows of both gases, using mass flow controllers. A linear gas velocity of 13.5 mm/s (at room temperature) was established by keeping the gas flow rate at 500 ml/min in view of the inner diameter of 28 mm of the quartz tube furnace. This linear gas flow is sufficient to avoid any significant (thermal) decomposition of ammonia in the nitriding atmosphere (cf. section 1 and Ref. 9).

In order to maintain a homogeneous precipitation morphology within the entire specimen during the determination of the nitrogen-absorption isotherms, pre-and de-nitriding experiments were carried out prior to the absorptionisotherm experiments. The pre-nitriding was performed at 580°C for 48h with a nitriding potential of r N ( de-nitrided in a H 2 atmosphere at 470°C for 72h. By de-nitriding part of the nitrogen taken up by the specimen was removed (for this, see section 3.4).

Nitrogen-absorption isotherms were then determined at a temperature of 560°C. At this temperature (below the pre-nitriding temperature), the precipitation morphology of the samples does not change. The determination of the absorption isotherms was performed by determining the nitrogen uptake of foils after nitriding at 560°C for 72h at each of the following nitriding potentials: r N = 0.140, 0.117, 0.091 and 0.054 atm -1/2 (under these nitriding conditions no iron nitrides can be formed at the surfaces of the specimen). The nitriding and de-nitriding conditions for the different alloys have been gathered in Table 2.

The amount of nitrogen taken up was determined by weight measurements before and after nitriding using a Mettler microbalance with an accuracy of 0.1 µg. In order to obtain an accurate weight, the average value of ten measurements was taken. The error bars and ranges indicated in Figs. 2b, 5-7 and Tables 34567, respectively, represent the maximal deviation from the average value calculated on the basis of the ten weight measurements.

X-ray diffraction

X-ray diffractograms were recorded from the surface of all specimens before and after nitriding using a PANalytical (formerly Philips) X'Pert Multi-Purpose Diffractometer (MPD) in Bragg-Brentano geometry equipped with a graphite- Electron energy loss spectroscopy (EELS) was performed in a Zeiss 912

Omega TEM operating at 120 kV equipped with an in-column omega-type electron spectrometer. For the elemental mappings the "three-window method" was used: Two pre-edge images are recorded, which enables the background to be fitted according to an inverse power law (I = AE -r , where I is the intensity, E is the at the ionization edge of the element of interest) to get the elemental map of interest which is weak as compared to the background contribution [START_REF] Crozier | [END_REF].

Electron probe microanalysis

The homogeneity of the distribution of the alloying elements Cr and Al and of nitrogen in the specimens was confirmed by electron probe microanalysis (EPMA) using a Cameca SX100 instrument. Pieces of the specimen were cut to prepare cross-sections by subsequently embedding of these pieces with Polyfast (Struers; a conductive bakelite resin with carbon filler embedding material), followed by grinding and polishing (last step: 1 µm diamond paste). A focused electron beam at an accelerating voltage of 15 kV and a current of 100 nA was applied. The concentrations of Fe, Cr, Al and N in the specimen were determined by measuring the intensities of the characteristic Fe-K β , Cr-K α , Al-K α and N-K α X-ray emission peaks at points 2 µm apart along lines traversing the entire specimen cross-sections. The concentration of each element was obtained by applying the Φ(ρz)-correction to the ratio of characteristic X-ray emission peak intensities of the specimen and that of a standard specimen (i.e. pure Fe, pure Cr, pure Al and γ'-Fe 4 N). [41].

Results and evaluation

Pre-nitriding

To maintain a homogeneous precipitation morphology in the diffusion zone of the specimen during the nitrogen-absorption isotherm measurements a pre-nitriding treatment has been applied at an elevated temperature, i.e. a temperature higher [42,43] between the Cr 1-

x Al x N nitride precipitates and the ferrite matrix [38]. The diffraction-line broadening (as represented by the full width at half maximum, FWHM) increases with increasing Cr/Al ratio (Figs. 1b andc). In addition to the diffraction-line broadening, a shift of the intensity maximum of the 200 α-Fe reflection towards lower diffraction angle, 2θ, occurs upon nitriding. This shift of the 200 α-Fe intensity maximum also increases with increasing Cr/Al ratio (Figs. 1b andc).

The EPMA elemental-concentration depth profiles presented in Fig. 2a show that the concentrations of both alloying elements and of nitrogen are constant over the cross section of the specimen; distinct segregation at grain boundaries was not observed. For all (pre-) nitrided alloys/specimens the total Evidently, after the homogeneous nitriding, the nitrogen uptake is larger than the amount of nitrogen required for the precipitation of all alloying element as nitride,

[ ] Me Me N N , plus the amount of nitrogen necessary to establish the equilibrium solubility in an unstrained ferrite matrix, 0 [ ] N α . This so called amount of "normal" nitrogen, Cr/Al ratio (Fig. 2b).

1 2 0 ( , ) [ ] [ ] [ ] nor Me Me N N N N α ≡ +

De-nitriding

Subsequent to the pre-nitriding, and also after the determination of each nitrogenabsorption isotherm, each alloy specimen was de-nitrided at 470°C in pure H 2 (500 ml/min) for 72h (see section 2.2). For each alloy, the remaining amounts of nitrogen after de-nitriding in both cases (after pre-nitriding and after nitrogenabsorption isotherm determination) are the same. This demonstrates that no significant aging effects (i.e. agglomeration and/or coarsening of nitride precipitates) occurred during the nitrogen-absorption isotherm measurements. particular for high Cr/Al atomic ratio; in that case the intensity maxima at the 002 type nitride spot positions on the streaks are less pronounced or even absent (cf.

Morphology and crystallography of nitride precipitates

Fig. 3a). The reverse holds for low Cr/Al atomic ratio: less pronounced steaks and clearer intensity maxima (cf. Fig. 3c). Streaks, through 200 and 020 type ferritematrix diffraction spots, instead of separate nitride diffraction spots, are indicative of a highly coherent nature of the nitride/matrix interface and anisotropic misfitstrain in the ferrite matrix (in particular perpendicular to (001) ferrite-matrix lattice planes) due to the ultra-fine nitride precipitates, which can diffract coherently with the matrix (cf. Fig. 1 and its discussion in section 3.1). The decrease of the extent of streaking with decreasing Cr/Al atomic ratio in the specimen, indicates that the degree of coherency of the nitride precipitates with the ferrite matrix decreases, which development culminates with the emergence of 002 type diffraction spots of nitride precipitates of cubic, rock-salt crystalstructure type (Fig. 3c). Hence, the degree of coherency at the nitrideplatelet/matrix interface increases with increasing Cr/Al atomic ratio.

Nitrogen-absorption isotherms

A nitrogen-absorption isotherm shows the dependence of the amount of nitrogen taken up by a (homogeneously) nitrided specimen as function of the nitriding potential, r N (directly related to the chemical potential of nitrogen absorbed in the ferrite matrix at the ferrite/gas interface as imposed by a given nitriding atmosphere [8]). The analysis of nitrogen-absorption isotherms allows distinction of the various kinds of differently (chemically) bonded nitrogen in the specimen.

The amount of nitrogen dissolved in the ferrite matrix upon nitriding by means of an NH 3 /H 2 gas mixture can be described by the equilibrium:

F o r P e e r R e v i e w O n l y 13 3 2 3 [ ] 2 NH N H α ⇔ + ( 1 )
where [ ] N α denotes the amount of interstitially dissolved nitrogen in the octahedral interstices of the ferrite matrix. If K denotes the equilibrium constant of the above reaction it is immediately obtained for the concentration of dissolved

nitrogen, [ ] N α [ ] N N K r α = ⋅ ( 2 )
In view of the relatively small amount of dissolved nitrogen, Henrian behavior can be assumed and thus the equilibrium constant, K, is supposed to incorporate the constant activity coefficient of dissolved nitrogen and thus is adopted as an effective equilibrium constant.

The nitrogen-absorption isotherms, as recorded after subsequent pre-and de-nitriding treatments (see sections 3.1 and 3.2), are shown for Fe-Cr-Al, with

Cr/Al = 0.21, 0.52, 1.04 and 2.00, in Figs. 5a-d, respectively (see also Table 3).

At constant temperature the amount of interstitially dissolved nitrogen in the ferrite matrix should depend linearly on the nitriding potential, r N (cf. Eq. ( 2)).

Indeed, a straight line can be fitted well (using the least-squares method) to the data points of the total nitrogen content as function of the nitriding potential ), according to the contents of alloying elements in the specimen (cf.

Table 1). Values for the nitrogen levels A, B and C for the specimens of different Cr/Al atomic ratio have been gathered in Table 4.

The difference between the nitrogen levels C and A is interpreted as nitrogen taken up (hypothetically at r N = 0) in excess of the amount of nitrogen necessary to precipitate all Cr and Al as Cr

+ = = ( 3 )
The value of y thus obtained contains indirect information on the average thickness of the precipitate platelet. As shown above (see section 3.3), Cr

+ = ( 4
)
where n is the number of Cr 1-x Al x N monolayers comprising the platelet. Thus the thickness t of a Cr 1-x Al x N platelet follows from

1 1 2 1 x x x x Cr Al N Cr Al N a a t n y - - = ⋅ = - ( 5 )
The lattice parameter of mixed Cr 1-x Al x N nitride can be calculated, for the Cr/Al atomic ratio concerned, by assuming that the lattice constant of mixed Cr 1-

x Al x N nitride complies with Vegard's law and taking the lattice parameters of CrN (rock-salt crystal structure) and AlN (rock-salt crystal structure) as 4.14Å [44] and 3.94Å [45], respectively. Then the nitride-platelet thicknesses can be calculated using Eq. ( 5) from the now known values of y (cf. Eq. ( 3)) and , with respect to that of the reference state (i.e.

0 [ ] N α for unstrained ferrite) can be given by [48]:

1 2 0 ( , ) 0 3 [ ] 4 exp [ ] (1 )
y N Me Me N N V G CY N RT α α α ε ε     =     +     (6)
where

0 [ ] [ ] [ ] strain N N N α α = +
and with misfit parameter:

1 2 1 2 1/ 3 1/ 3 ( , ) ( , ) 1/ 3 [ ( 1)
]

Me Me N Me Me N V y fV V V α α ε + - - = (7)
constant:

1 2 1 2 ( , ) ( , ) 3 (3 4
)

Me Me N Me Me N K C K G α = + (8)
volume fraction of (Me 1 ,Me 2 )N y :

1 2 1 2 1 2 1 2 1 2 ( , ) ( , ) 0 ( , ) ( , ) ( , )
[ ](

)

(1 [ ]) [ ]( ( 1)
where V N is the partial molar volume of nitrogen dissolved in the ferrite matrix, V α and V (Me1,Me2)N are the molar volumes of ferrite and the (Me atomic fraction of alloying elements in the specimen. The parameter f describes the extent to which the full misfit due to building out of the lattice of the (Me 1 ,Me 2 )N precipitates by the adsorbed nitrogen atoms, which act as an entity with the particle, is experienced (0≤ f ≤ 1). The following values have been adopted for some of the parameters mentioned above (see Refs. 33 and 48):

V N = 5.12 cm 3 /mol; V α = 7.092 cm 3 /mol; G α = 81.6 GPa; V (Me1,Me2)N = 9.44, 9.67, 9.90 and 10.12 cm 3 /mol for Cr/Al atomic ratios of 0.21, 0.52, 1.04 and 2.00, respectively (using the procedure described below Eq. ( 5)); K (Me1,Me2)N = 285.98, 300.94, 316.43 and 330.67 GPa for Cr/Al atomic ratios of 0.21, 0.52, 1.04 and 2.00, respectively (derived from the bulk modulus data of K CrN = 361 GPa [49] and K AlN = 270 GPa [45] as linear function of the Cr/Al atomic ratio).

An experimental value for

0 [ ] /[ ] N N α α
, at a given nitriding temperature, follows from the ratio of the slopes of the linear parts of the nitrogen-absorption isotherms recorded for Fe-Cr-Al and pure α-Fe:

0 0 0 [ ] ( [ ] / ) [ ] ( [ ] / ) N Fe Cr Al N N N r S N N r S α α α α α -- ∆ ∆ = = ∆ ∆ (10)
where Fe Cr Al S --and 0 S α denote the slope of the linear part of the absorption isotherms for Fe-Cr-Al alloys and pure α-Fe, respectively (cf. Fig. 5; the absorption-isotherms for pure α-Fe have been drawn as straight lines passing through the points C on the ordinates, using data from Ref. 33).

The values of all constants at the right-hand side of Eq. ( 6), except f (cf.

Eqs. ( 7) and ( 9)), are known and thus by comparing Eqs. ( 6) and (10)

General discussion

The equilibrium crystal structures of the nitride precipitates occurring in (recrystallized and subsequently) nitrided Fe-Cr and Fe-Al alloys are the cubic, rock-salt structure for CrN [12,[START_REF] Jack | Proc. Heat Treatment Conf[END_REF]16] and the hexagonal, wurtzite structure for AlN [23][24][25][26]; a metastable rock-salt structure is possible for AlN [50,51]. The formation of the cubic CrN upon nitriding is relatively fast, whereas the formation of the hexagonal AlN, due to the relatively large volume misfit between this nitride and the ferrite matrix, is relatively very slow [24]. At the nitriding temperature, diffusion of Cr and Al is that slow, as compared to diffusion of N, that the Al atoms are "dragged" into the developing cubic, rock-salt type CrN precipitates: Upon nitriding the system accepts the gain of a smaller than maximal amount of energy, released by nitride precipitation, as an intermediate solution:

metastable, mixed Cr 1-x Al x N precipitates develop (see Fig. 4). This has been shown recently to occur for a Cr/Al atomic ratio of 0.52 in Ref. 38. On the basis of this discussion it is likely that a more copious nucleation, i.e. a higher nucleus density of the cubic, rock-salt type, mixed Cr as observed (Fig. 6).

As follows from the position of level B, relative to the positions of levels C and A (cf. Figs. 5a-d), part (or all) of [ ] interface N , and (for the atomic ratio Cr/Al > 1; cf. Fig. 5) a part of [ ] = Ti [52]. This can be understood as a consequence of the dependence of the nitride/matrix misfit on Cr/Me 2 atomic ratio, as follows.

The parameter ε (cf. Eq. ( 7)) describes the overall misfit between a "free", undeformed "inclusion" and an empty, undeformed "cavity" in a matrix. Upon 21 insertion of the "inclusion" (here: the nitride particle) into the "cavity" in the matrix (here: the ferrite matrix) of finite size a hydrostatic stress is introduced in the matrix (of tensile nature if ε > 0). For the following comparative discussion, the effect of nitrogen adsorbed at the surface of the nitride particle can be ignored and thus ε can be taken as

1 2 1/ 3 1/ 3 ( , ) 1/ 3 Me Me N V V V α α ε   - =       (11)
Using the values of the lattice parameter of each nitride (4.14Å for rock-salt crystalstructure type CrN [44], 3.94Å for rock-salt crystal-structure type AlN [45] and 4.24Å for rock-salt crystal-structure type TiN [54]) and assuming that the lattice parameters of the mixed Cr 1-x Al x N and Cr 1-x Ti x N nitrides comply with Vegard's law, In view of the observed Bain orientation relationship and the occurring {001} α-Fe -type habit plane (cf. section 3.3), the linear misfit along the habit plane (i.e.

ε
parallel to the faces of the nitride platelet), // δ is given by 7) and [ ] interface N decreases (cf. Table 5)).

1 2 ( , ) // 2 100 (%) 2 Me Me N Fe Fe a a a α α δ - -   -   = ×     (12a)
Evidently, f will depend on the degree of coherency at the nitride-platelet/matrix interface and thus the increase of f with increasing Cr/Al atomic ratio can be conceived as a direct consequence of // δ decreasing with increasing Cr/Al atomic ratio (cf. the discussion in the previous paragraph).

Conclusions

Figure 2b

The total amount of nitrogen taken up by the specimen, [ ] total N , which can be compared with the amount of normal nitrogen, [ ] nor N (see also Fig. 2a), as function of the Cr/Al atomic ratio (after pre-nitriding of the Fe-Cr-Al alloys).

Figure 8a

The overall misfit, ε (see Eq. ( 11)), between mixed nitride and ferrite matrix as function of Cr/Me 2 (Me 2 = Al or Ti).

  is of interest to quantitatively investigate the uptake of nitrogen by Fe-Cr-Al alloys, in particular as function of the Cr/Al atomic ratio, which is the purpose of the present project.

  p as partial pressure of component i [8]) = 0.104 atm -1/2 . This treatment ensures through nitriding of the specimen, without iron-nitride formation at the surfaces of the specimen. The specimen was quenched into water at room temperature after the pre-nitriding, and subsequently

7 ICDD

 7 diffracted beam monochromator set to Co-K α (λ=1.7889Å) radiation. The specimens were rotated on a spinner around their vertical axis during each measurement, to improve crystal statistics. The diffraction angle 2θ was scanned over a range from 40° until 120° in steps of 0.05°. Detected phases were identified by the 2θ positions of their diffraction peaks in comparison with data from the data base[START_REF]JCPDS-International Center for Diffraction Data[END_REF]. The diffractograms were evaluated by fitting a Pearson VII profile-shape function, using Profile Fit 1.0c software, for the determination of peak position and full width at half maximum (FWHM) in the diffractograms.2.4 Transmission electron microscopy and electron energy loss spectroscopySpecimens for transmission electron microscopy (TEM) were prepared from the middle of the specimen as follows.Discs (Φ = 3 mm) were punched out from sheets produced by removing material mechanically from both sides (faces) of a nitrided specimen. These discs were thinned by applying jet-electropolishing technique using a Struers Tenupol-3 apparatus (bath composition: 85vol.% acetic acid and 15vol.% perchloric acid, current: 26 mA ≤ I ≤ 41 mA, voltage: 15V ≤ U ≤ 20.5V, temperature: 278K ≤ T ≤ 280K, flow rate setting "20" at a Struers Tenupol-3 apparatus and treatment time 160s ≤ t ≤ 275s). After jet-electropolishing, the specimens were subsequently rinsed in ethanol, acetone and isopropanol. To generate a hole in the middle of the sample, the discs were fixed during the jet-electropolishing treatment between two platinum rings. TEM analysis was performed using a Philips CM 200 transmission electron microscope operating at 200 kV. Bright field (BF) images and selected area diffraction patterns (SADPs) were taken by a CCD camera attached to the TEM apparatus.

  A and r are two fitting parameters) at each pixel in the image. The extrapolated background image is then subtracted from the post-edge image (i.e.

  , [N] tot , taken up after pre-nitriding, as determined by EPMA measurements, matches within experimental accuracy the value as obtained by weight-change measurement.

,

  has been indicated by the horizontal, dashed line in Fig. 2a. The difference between the experimentally obtained total amount of nitrogen, [ ] tot N , and the amount of "normal" nitrogen, [ ] nor N , is defined as excess nitrogen, [ ] ex N . Comparing the amounts of nitrogen taken up by the various alloys, it follows that the value of [ ] tot N , and thus the value of [ ] ex N , after pre-nitriding increases with decreasing

11 TEM

 11 bright field (BF) images obtained from pre-nitrided Fe-Cr-Al (Cr/Al = 2.00, 1.04 and 0.52) alloys showed that the nitrides precipitate as fine platelets surrounded by distinct strain-field contrast caused by the elastic distortion of the matrix around the misfitting precipitates (see Figs.3a-c). From the micrographs it follows that the length of the nitride platelets increases (and the platelet density decreases) with decreasing Cr/Al atomic ratio.Selected area diffraction patterns (SADPs) recorded from all specimensshow diffraction spots at locations corresponding to the ferrite matrix. Further, streaks through the 200 and 020 type diffraction spots of the ferrite matrix can be observed. Intensity maxima on these streaks occur (see upper-right insets of Figs.3a-c), at positions expected for 002 type spots of a cubic, rock-salt crystalstructure type nitride precipitate (see what follows). The SADPs, with their streaks and diffraction spots, are compatible with the presence of nitride precipitates of cubic, rock-salt crystal-structure type in the ferrite matrix (bcc) satisfying a Bain orientation relationship: (001) α-Fe // (001) MeN , [100] α-Fe // [110] MeN : Me = Cr, Al The composition of the nitride platelets was investigated by electron energy loss spectroscopy (EELS). Comparing a TEM BF image comprising a part of the ferrite matrix containing precipitate platelets with correspondingly recorded elemental maps for N, Cr and Al (using N-K, Cr-L 2,3 (+ O-K) and Al-L 2,3ionization edges) shows that N, Cr and Al enrichment occurs at the same locations, there where the precipitate platelets are observed in the TEM BF image (see arrows in Figs. 4a-d). Evidently the nitride precipitates developed as a mixed nitride: Cr 1-x Al x N.

( 14 Figs

 14 Figs.5a-d, using literature data for pure ferrite[33]). The dissolved nitrogen in

  in Table6. It follows that the thickness of the nitride platelets increases with increasing Cr/Al atomic ratio. as function of nitriding potential, r N , in Fig.7. Clearly, at constant r N , [ ] strain N increases with increasing Cr/Al atomic ratio.The presence of misfitting second phase particles in a matrix can lead to elastic distortions of the surrounding matrix. The corresponding stress field (characterized by a tensile hydrostatic component[46,47]) influences the thermodynamics of nitrogen dissolution in the ferrite matrix. The enhancement of the lattice solubility, i.e. [ ] strain N

  can be calculated as function of the Cr/Me 2 (Me 2 = Al or Ti) atomic ratio. The results are shown in Fig. 8a. Evidently, ε increases with increasing atomic ratio Cr/Al and decreases with increasing atomic ratio Cr/Ti. It thus is predicted that [ ] strain N increases with increasing Cr/Al atomic ratio and with decreasing Cr/Ti atomic ratio, as observed (cf. Fig. 7 for Fe-Cr-Al alloys and Ref. 52 for Fe-Cr-Ti alloys).

  misfit perpendicular to this habit plane (i.e. perpendicular to the faces of the nitride platelet) , lattice-parameter data as indicated above, // δ and ⊥ δ can be calculated as function of the atomic ratio Cr/Me 2 . The results are shown in Fig. 8b. For Fe-Cr-Ti alloys both // δ and ⊥ δ are positive and decrease with increasing Cr/Ti atomic ratio. For Fe-Cr-Al alloys an antagonistic behaviour occurs: δ ⊥ increases with increasing Cr/Al atomic ratio, whereas // δ decreases with increasing Cr/Al atomic ratio (for the alloys investigated within the range 0.21-2.00). Although the change of δ ⊥ dominates the change of the overall misfit parameter ε (cf. Eq. (11) and thus is responsible for the increase of [ ] strain N with increasing Cr/Al atomic ratio (see above)), the decrease of // δ with increasing Cr/Al atomic ratio suggests that the nitride/matrix coherency at the faces of the nitride platelets increases with increasing Cr/Al atomic ratio and thus explains the observation of the more pronounced streaking through the 200 and 020 type ferrite-matrix diffraction-spots and the disappearing 200 type nitride diffraction spots with increasing Cr/Al atomic ratio (Fig. 3; section 3.3). The value of f describes the extent to which the volume misfit due to the nitrogen, adsorbed at the interface is experienced (cf. section 3.3). Obviously, given a constant microstructure during the nitrogen absorption-isotherm measurements, the values of f determined in this work as function of the Cr/Al atomic ratio are characteristic for the pre-nitriding temperature (here 580°C). Of course, f will not depend in a direct way on the value of [ ] interface N (note that with increasing Cr/Al f increases (cf. Table

Figure 1c

 1c Figure 1cThe full width at half maximum (FWHM) and the position of the

Figure 2a

 2a Figure 2aConcentration-depth profiles for N, Cr and Al as determined by EPMA after pre-nitriding (48h at 580°C with r N = 0.104 atm -1/2 ) of the Fe-Cr-Al (Cr/Al =

Figure 3 38 (

 338 Figure 3 TEM bright field images (left; electron-beam direction [001] α-Fe )

Figure 4

 4 Figure 4 (a) TEM bright field image (taken near to the hole in the jet-

Figure 5

 5 Figure 5 Nitrogen absorption isotherms for (a) Fe-Cr-Al (Cr/Al = 0.21), (b) Fe-

Figure 6 [

 6 Figure6[N] interface as function of the Cr/Al atomic ratio.

Figure 7 [

 7 Figure 7 [N] strain as function of the Cr/Al atomic ratio. The dashed-lines in the

Figure 8b

 8b Figure 8bThe misfit perpendicular ( ⊥ δ ) and parallel ( // δ ) to the {001} α-Fe habit

  1-x Al x N precipitates develop as platelets of cubic, rock-salt crystal-structure type obeying a Bain orientation relationship with the ferrite matrix. With {001} Cr1-xAlxN as habit plane, the thickness of a monolayer of Cr 1-x Al x N is one half of the lattice parameter of the rock-salt crystal structure (i.e.

				1 Cr Al N x x a
				2
	y	n	n	2

-

). If at every octahedral interstice in the ferrite matrix at the nitride/matrix interface one excess nitrogen atom is trapped (i.e. on top of a Me atom), it follows

  )

	Y	Me Me N	y	= -	Me V Me V α +	Me Me N Me V	Me Me N y + -	Me Me N fV + -fV y	Me Me N

  1-x Al x N nitride, occurs for increasing Cr/Al atomic ratio of the alloy. The results obtained in the present work are in agreement with this prediction (cf. section 3.3). CrN and TiN) have the rock-salt crystal structure, suggesting that the ease of nucleation of mixed (Me 1 ,Me 2 )N nitrides is less dependent on Me 1 /Me 2 atomic ratio for Fe-Cr-Ti alloys than for Fe-Cr-Al alloys.The coherent nature of the interface between nitride platelet and ferrite matrix allows adsorption of nitrogen atoms at the octahedral interstices in the ferrite matrix adjacent to the platelet faces, as in this way bonding to Cr and/or Al in the platelet faces is realized, i.e. [ ] interface N . Now consider: (i) The chemical affinity of Al for N is much larger than that of Cr for N. Further, (ii) the thickness of the mixed Cr 1-x Al x N nitride platelets decreases with increasing relative Al concentration (i.e.

	equilibrium nitrides (
	F o r
	P
	e
	e r
	R
	e
	v i e
	w
	O n l
	y
	The distinct dependence of (Me 1 ,Me 2 )N nucleus density (and nitride-
	platelet length and thickness) on Me 1 /Me 2 atomic ratio, as observed here for

nitrided Fe-Cr-Al alloys, has not been observed for nitrided Fe-Cr-Ti alloys, where mixed Cr 1-x Ti x N nitride platelets of rock-salt crystal structure occur as well for a similar range of the Me 1 /Me 2 atomic ratio

[52]

. Indeed, in the latter case both with decreasing Cr/Al atomic ratio; see Table

6

). Both effects (the chemical one (i) and the geometrical one (ii)) explain, that for a constant total atomic alloying element content (Cr + Al), [ ] interface N increases with decreasing Cr/Al atomic ratio,

  similar effect was not observed upon nitriding Fe-Cr-Ti alloys: Nitriding leads to the development of Cr 1-x Ti x N mixed nitrides of rock-salt crystal structure in a nitrogen saturated ferrite matrix. Upon denitriding all dissolved nitrogen from the ferrite matrix and all nitrogen adsorbed at the nitride-platelet faces is removed. But, in contrast with the Cr 1-x Al x N platelets, no nitrogen is removed from the Cr 1-x Ti x N platelets (for Cr/Ti atomic ratio in the range 0.53 to 2.22)[52]. This may be understood as a consequence of the Cr 1-x Ti x N platelets being relatively more stable (i.e. less metastable) than the Cr 1-x Al

	upon renitriding the nitrogen deficiency of the platelet becomes repaired, as
	observed.
	1-x x Cr Al N These removable amounts of nitrogen increase relatively (i.e. with respect to N , can be removed (reversibly!) upon denitriding. [ ] interface N ) with increasing Cr/Al atomic ratio. This suggests that the larger the relative amount of Al, the smaller the relative amount of such removable nitrogen. Indeed the chemical affinity of Al for N is much larger than that of Cr for N. Thus N adsorbed at the nitride platelet faces can be overall stronger bonded at smaller Cr/Al ratios. A similar discussion may be given for the nitrogen incorporated in a o r P e e r R e v i e w O n l y strain N [48]. Again comparing recent results obtained upon nitriding Fe-Cr-Ti alloys with the present data obtained for Fe-Cr-Al alloys, it strikes that [ ] strain N increases with increasing Cr/Me 2 atomic metastable Cr 1-x Al F ratio if Me 2 = Al (Fig. 7) and decreases with increasing Cr/Me 2 atomic ration if Me 2
	20

x N platelet: at high Cr/Al atomic ratio this nitrogen is less strongly bonded, may be partly removed by denitriding during which Cr and Al atoms in the nitride platelet remain immobile (note the low temperature of denitriding), so that A x N platelets (note that the equilibrium CrN and TiN nitrides both have a rock-salt crystal structure, whereas the equilibrium AlN nitride has a hexagonal, wurtzite crystal structure).

The presence of dissolved excess nitrogen in the ferrite matrix upon nitriding of Fe-Cr-Al alloys is a consequence of elastic accommodation of the misfit between nitride platelet and ferrite matrix: such elastic accommodation of misfit induces a tensile hydrostatic stress component in the ferrite matrix

[53,46,47]

. As a consequence, as compared to the unstrained state, more nitrogen can be dissolved (on octahedral interstices) in the ferrite matrix; i.e. [ ]

Table 1 :

 1 1. Upon nitriding of Fe-Cr-Al alloys, metastable, mixed Cr 1-x Al x N nitrides of cubic, rock-salt crystal-structure type precipitate in the ferrite matrix; the system thus avoids the difficult nucleation of stable AlN (hexagonal, wurtzite structure type) precipitates in the ferrite matrix. The ease of mixed nitride nucleation and thus the nucleation density increases with increasing Cr/Al atomic ratio. Such an effect does not occur for nitrided Fe-Cr-Ti alloys as both equilibrium nitrides, The amounts of Cr and Al and light element impurities in the alloy specimens before nitriding.

	CrN and TiN, have the same (rock-salt type) crystal structure as the corresponding
	metastable mixed Cr 1-x Ti x N precipitate.
	2. The amount of excess nitrogen adsorbed at the faces of the nitride platelets,
	[ ] interface N	, increases with decreasing Cr/Me 2 (Me 2 = Al or Ti) atomic ratio
	because the chemical affinity of Me 2 for N is much larger than that of Cr for N
	and because the thickness of the nitride platelets decreases with decreasing
	Cr/Me 2 atomic ratio.
	3. The amount of excess nitrogen dissolved in the ferrite matrix, [ ] strain N	,
	increases with increasing Cr/Me 2 atomic ratio for Me 2 = Al and decreases with
	increasing Cr/Me 2 atomic ratio for Me 2 = Ti. These antagonistic behaviours can

Table 2 :

 2 Summary of the nitriding parameters for the Fe-Cr-Al alloys.

		temp.	time	NH 3	H 2	r N
		(°C)	(h)	(ml/min)	(ml/min)	(atm -1/2 )
	pre-nitriding	580	48	45	455	0.104
	de-nitriding	470	72	•	500	•
				58	442	0.140
	absorption isotherms	560	72	50 40	450 460	0.117 0.091
				25	475	0.054

Table 3 :

 3 [N] tot for the Fe-Cr-Al alloys (nitrogen-absorption-isotherm

	measurements at 560°C for 72h).
	alloy	r N (atm -1/2 )	0.054	0.091	0.117	0.140
	Fe-Cr-Al (Cr/Al = 0.21): [N] tot (at.%)	2.13 (±0.02)	2.27 (±0.03)	2.34 (±0.02)	2.41 (±0.04)
	Fe-Cr-Al (Cr/Al = 0.52): [N] tot (at.%)	2.12 (±0.01)	2.23 (±0.01)	2.32 (±0.01)	2.38 (±0.03)
	Fe-Cr-Al (Cr/Al = 1.04): [N] tot (at.%)	1.90 (±0.01)	2.03 (±0.03)	2.11 (±0.02)	2.20 (±0.03)
	Fe-Cr-Al (Cr/Al = 2.00): [N] tot (at.%)	1.88 (±0.03)	2.02 (±0.02)	2.10 (±0.01)	2.18 (±0.02)

Table 4 :

 4 Nitrogen levels 'A', 'B' and 'C' (see Fig.5), for the Fe-Cr-Al alloys.

	N content (at.%) alloy	level A	level B	level C
	Fe-Cr-Al (Cr/Al = 0.21)	1.46 (±0.015)	1.78 (±0.007)	1.99
	Fe-Cr-Al (Cr/Al = 0.52)	1.48 (±0.014)	1.72 (±0.005)	1.95
	Fe-Cr-Al (Cr/Al = 1.04)	1.45 (±0.015)	1.40 (±0.006)	1.71
	Fe-Cr-Al (Cr/Al = 2.00)	1.48 (±0.010)	1.37 (±0.005)	1.67

Table 5 :

 5 The values of the [N] interface for the Fe-Cr-Al alloys (after prenitriding at

	580°C).
	alloy	N content (at.%)	[N] interface
	Fe-Cr-Al (Cr/Al = 0.21)	0.53 (±0.015)
	Fe-Cr-Al (Cr/Al = 0.52)	0.47 (±0.014)
	Fe-Cr-Al (Cr/Al = 1.04)	0.26 (±0.015)
	Fe-Cr-Al (Cr/Al = 2.00)	0.06 (±0.010)

Table 6 :

 6 The mixed Cr 1-x Al x N y nitride platelet thickness and corresponding y values (cf. Eq. (3)) for the Fe-Cr-Al alloys.

	alloy	thickness (nm)	y
	Fe-Cr-Al (Cr/Al = 0.21)	1.09 (±0.03)	1.36 (±0.010)
	Fe-Cr-Al (Cr/Al = 0.52)	1.25 (±0.05)	1.32 (±0.012)
	Fe-Cr-Al (Cr/Al = 1.04)	2.24 (±0.14)	1.18 (±0.012)
	Fe-Cr-Al (Cr/Al = 2.00)	2.66 (±0.13)	1.15 (±0.008)

Table 7 :

 7 The values for

	S	0 Fe Cr Al S α --	and f (cf. Eqs. (6-9)) at the nitriding potential,
	r N = 0.140 atm -1/2 for the Fe-Cr-Al alloys.
	alloy		S	0 Fe Cr Al S α --	f
	Fe-Cr-Al (Cr/Al = 0.21)		1.35 (±0.009)	0.10 (±0.04)
	Fe-Cr-Al (Cr/Al = 0.52)		1.39 (±0.012)	0.12 (±0.01)
	Fe-Cr-Al (Cr/Al = 1.04)		1.58 (±0.009)	0.84 (±0.04)
	Fe-Cr-Al (Cr/Al = 2.00)		1.65 (±0.009)	0.96 (±0.04)

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Acknowledgements

The authors thank Mr. S. Meka M. Tech. for discussion, Dipl. Ing. P. Kress and Mr. J. Koehler for assistance with the nitriding experiments, Mrs. S. Haug for assistance with the EPMA experiments, Mr. W. D. Lang for TEM sample preparation and Dr. W. Sigle for assistance during the EELS experiments and for discussion (all with the Max Planck Institute for Metals Research).