Ihor Sobchenko

Josef Pesicka

Dietmar Baither

Werner Stracke

Thomas Pretorius

Lifeng Chi

Rudolf Reichelt

Eckhard Nembach
email: nembach@nwz.uni-muenster.de

Josef Pesicka †+

Werner Stracke#

Thomas Pretorius † ¶

Lifeng Chix

Rudolf Reichelt#

Atomic force microscopy (AFM), transmission electron microscopy (TEM), and scanning electron microscopy (SEM) of nano-scale plate-shaped second phase particles

Keywords: Atomic force microscopy (AFM), transmission electron microscopy (TEM), and scanning , transmission electron microscopy atomic force microscopy, nanoparticles, nanoscale precipitates, Nibased superalloys, nickel alloys, phase decomposition, SEM, transmission electron microscopy magnetic force microscopy, superellipsoid, plate-shaped precipitates Atomic force microscopy (AFM), Magnetic force microscopy (MFM), Transmission electron microscopy (TEM), Field emission scanning electron microscopy precipitates, Second phase particles, Nickel based alloy, Nano-structure, Superellipsoid

come

Introduction

Many structural materials derive their high strength from coherent precipitates of secondary phases. Well known examples are Guinier-Preston zones in aluminium based alloys and γ'-precipitates in nickel based superalloys [1][START_REF] Nembach | Particle Strengthening of Metals and Alloys[END_REF][START_REF]Alloying[END_REF]. The strengthening effects of coherent precipitates are governed by their shape, size, and volume fraction and by intrinsic properties of the two phases involved, e.g. by their crystal structure, their lattice mismatch, and their elastic stiffness. The shape of the precipitates in turn depends on their specific interface energy, their lattice mismatch, and the elastic stiffness of both phases. Anisotropies of these parameters are of great importance. The most common shapes are spheres, cubes, and plates.

Plates are favoured by a low specific interface energy, a high lattice mismatch, a low elastic stiffness of the precipitates, and a high precipitate volume [START_REF] Khachaturyan | Theory of Structural Transformations in Solids[END_REF][START_REF] Barnett | [END_REF][6]. In the case of spheres and cubes their size is described by the average radius r and by the average cube length d, respectively. To characterise plate-shaped precipitates at least two parameters are required: some linear measure 2a of their average areal extension and their average thickness 2b. In addition, the distribution functions Φ of the individual particle radius i ρ , the cube length i δ , the areal extension α i 2 , and the thickness β i 3 cost, more recently they too have been used to determine the size and the volume fraction of spherical and cubic particles [7][8][9][10][11][12][13]. Moreover in contrast to the TEM, also compact specimens can be investigated in the AFM, which facilitates the preparation of the specimens, and thanks to the direct three-dimensional digital output automated computer based evaluation procedures can readily be applied.

In two former investigations [11,12], the present groups successfully used AFMs for the accurate determination of r, Φ ( i ρ ), and the volume fraction of two different types of spherical nano-scale second phase particles. Elaborate quantification procedures were developed. Special attention had been paid to effects of the finite size of the AFM tip and to the preferential attack of the matrix by the electrolyte. Presently the feasibility to characterise plate-shaped precipitates by AFM has been explored. Their lower symmetry renders their characterisation more involved because (i) the two lengths a and b instead of just one have to be measured and (ii) the often high aspect ratio a/b complicates the choice of the magnification. Good statistics necessitate the evaluation of a high number of particles. Hence the relatively high value of a calls for a low magnification, whereas the smaller thickness b calls for a high magnification. To verify the AFM results they are compared with results derived from TEM and high resolution field emission scanning electron microscopy (FESEM). Since the investigated material is ferromagnetic, also magnetic force microscopy (MFM) was performed. For this purpose the standard AFM tips were replaced by hard magnetic cobalt alloy coated ones. All experimental and evaluation procedures reported below are also applicable to cube-shaped particles with rounded edges and corners; in this case a equals b.

The experimental alloy Ni 69 Co 9 Al 18 Ti 4 served as a model system. After ageing below about 1380K, this alloy is two-phase: nano-scale coherent precipitates of the disordered or short-range ordered f.c.c. γ-phase are embedded in the L1 2 -long-range ordered γ'-phase [14][15][16]. Evidently this arrangement of phases is inverse to that of nickel based superalloys, in which the γ-phase forms the matrix and particles of the γ'-phase precipitate [1][START_REF] Nembach | Particle Strengthening of Metals and Alloys[END_REF][START_REF]Alloying[END_REF].

Strengthening of Ni 69 Co 9 Al 18 Ti 4 by γ-precipitates and the relevant dislocation processes were the subjects of two former studies [15,16]. In fact, this alloy had been designed as a model system for the investigation of particle strengthening of intermetallic phases. Since the former specimens had been aged for up to 264h at 973K the γ-precipitates in them were very small and of approximately spherical shape: r was below 7nm. The present specimens were aged for 333h at 1173K, which resulted in larger and hence plate-shaped γ-precipitates. TEM, FESEM, AFM, and MFM images are presented in section 4. The present γ'-phase may be considered as a derivative of the well-known intermetallic phase Ni 3 Al [START_REF]Dislocations in Solids[END_REF]. The γ-precipitates are richer in Co and poorer in all other constituents than the γ'-matrix. This will be detailed in section 3.

After the nucleation process of γ-precipitates is accomplished, the growth of their average size is mainly governed (i) by further depletion of the γ'-matrix of Co and (ii) by dissolution of smaller γ-precipitates and incorporation of their material by bigger ones. The latter process is analogous to Ostwald ripening of spherical precipitates [START_REF] Wagner | Materials Science and Technology[END_REF]: the total amount of interface energy is reduced. Pretorius et al. [15] aged Ni 69 Co 9 Al 18 Ti 4 isothermally at 973K

and found that for r<7nm the γ-volume fraction increased with ageing time.

Crystallography and geometry of the γ γ γ γ-precipitates

The crystallography of coherent plate-shaped precipitates determines the optimum imaging geometry. The habit planes of the plate-shaped γ-precipitates in Ni 69 Co 9 Al 18 Ti 4 are of the type {001}. Therefore the Miller indices of the plane of the thin TEM-foils were chosen to be (001) and the incident electron beam in the TEM and FESEM was normal to them. Other imaging geometries would have complicated the evaluations further. The two families of γ-plates with the indices (100) and ( 010) are imaged edge-on, whereas the (001)-family is viewed in-plane. The same foils were studied by the TEM, FESEM, and AFM. The morphology of the γ-precipitates is in accordance with the observation that Young's modulus of γ'-strengthened superalloys is lowest along <001>-directions [START_REF] Fährmann | [END_REF]20].

'[insert figure 1 about here]'

The γ-precipitates are neither flat circular cylinders, nor flat cuboids, nor flat standard ellipsoids of revolution (see section 4), but they can be very well described as superellipsoids [21][22][23][24][25][START_REF] Weisstein | Superellipsoid[END_REF]. The coordinates of points on the γ'-γ-interface of an (001)-in-plane γ-precipitate which is centred at the origin of the coordinate system, are Evidently the mathematical description of a superellipsoid involves four independent parameters. α appears in the denominator of x as well as of y because the extension of in-plane (001)-γ-plates is approximately the same in these two directions (see section 4).

α α β       + + =             1 k n n x y z . ( 1
Deviations from this symmetry will be discussed below and in section 4.1.2. For n=k=2 a standard rotational ellipsoid is obtained and for n → ∞ and k → ∞ a cuboid. The effects of n and k on the shape of the superellipsoid are demonstrated in figure 1 for 5 α β = . The intersections of the superellipsoid with the plane characterised by y ≡ 0, i.e. edge-on view, and with the plane z ≡ 0, i.e. in-plane view, are shown in figures 1(a) and (b), respectively. The quoted intersections are superellipses [25,[START_REF] Weisstein | Superellipsoid[END_REF]. Figure 1(c) gives three-dimensional sketches for four combinations of n and k and thus visualises the effects of these exponents on the shape of the superellipsoid. For the volume V of the superellipsoid holds

V = q V (n,k) 2 α β π . ( 2a
)
The subscript ' V ' of the coefficient q V (n,k) indicates 'volume'. Some values of q V (n,k) are listed in figure 1(a). Equation (2b) is an approximate numerical representation of the function q V (n,k):

q V (n,k) = a 0 + a n n + a nn n 2 + a k k + a kk k 2 + a nk nk (2b)
with a 0 =0.1404, a n = 0.4992, a nn = -0.0383, a k = 0.2369, a kk = -0.0158, a nk = -0.0094. For n and k between 2 and 6, this approximation is accurate within ± 2% [25].

A generalisation of equation ( 1) is possible by allowing (i) for different lengths of the two longer half-axes and / or (ii) for different exponents, i.e. the first two terms in equation

(1) are replaced by ( )

1 1 / n x α and ( ) 2 2 / n y α
, respectively, with α 1 ≠ α 2 and / or n 1 ≠ n 2 [25]. In this case also equation (2a) has to be modified

V = q V (n 1 ,n 2 ,k) 1 2 α α β π . ( 2c
σ α (n) = q C (n,n) α π . ( 3a
)
The subscript ' C ' indicates 'chord'. q C (n,n) equals: 0.5000, 0.5623, 0.5902, 0.6049, 2/π=0.6366 for n = 2, 3, 4, 5, ∞, respectively [25]. n = 2 and ∞ refer to a circle and a square, respectively. Two more analogous average chords are defined:σ β (n,k) , which refers to chords which (i) lie in the (010)-plane (edge-on view) characterised by y ≡ 0 and (ii) are parallel to [001], i.e. to horizontal chords in figure 1(a), and σ α (n,k), which refers to chords which (i) lie in the (010)-plane (edge-on view) characterised by y ≡ 0 and (ii) are parallel to

[100], i.e. to vertical chords in figure 1(a). 3d) is a numerical approximation of the function q C (n,k); for n and k between 2 and 6, it is accurate within ± 1.5%:

σ β (n,k) = q C (n,k) β π (3b) σ α (n,k) = q C (n,k) α π (3c) q C (n,k) is symmetric, i.e. q C (n,k)=q C (k,n). Equation (
q C (n,k) = a 0 + a n (n+k) + a nn (n 2 +k 2 ) + a nk nk (3d)
with a 0 =0.3573, a n =0.0458, a nn = -0.0028, a nk = -0.0027.

With the aid of equations (3b) and (3c) the area ( , , , ) n k α β A of a superellipse which is characterised by the parameters , , , and , n k

α β can be calculated c ( , , , ) 2 ( , ) n k q n k α β α β π = A . ( 3e
)
Applying equations ( 1) -( 3), all relevant geometric parameters of any type of convex voltage: 25V). With the following two exceptions the preparation of the thin TEM-foils was the same as in Refs. [15,16]: (i) the present ageing conditions were 333h at 1173K, which resulted in plate-shaped, instead of spherical (see section 1) γ-precipitates, and (ii) the shape and crystallography of the γ-precipitates required that the indices of the plane of the foils were (001) (see section 2). The present ageing treatment was preceded by a γ-nucleation treatment: 24h at 978K. As described by Baither et al. [16], the aged material contained some widely spaced irregular arrangements of the two phases. This was due to dendritic growth of the single crystals. Since the spacing of these irregularities was around 0.3mm, they could easily be avoided in the microscopes.

Without any additional preparation specific to the applied microscope, the same foils were studied by TEM, FESEM, and AFM. All micrographs were taken at ambient temperature. Two AFMs, AFM1 and AFM2, were used, both with two different types of tips, Tip1 and Tip2. The TEM, the FESEM, and AFM1 were the same ones as in the previous studies of spherical particles [12,16]. AFM1: AP-100 Autoprobe CP (Park Scientific Instruments, Sunnyvale, USA); AFM2: Dimension TM 3000 (Digital Instruments, Santa Barbara, USA); 200kV-TEM: 800NA (Hitachi, Tokyo, Japan); FESEM: S-5000 (Hitachi, Tokyo, Japan). Tip1: SuperSharpSilicon TM (tip radius<5nm, typical ≈ 2nm; half-cone angle of the outermost 200nm of the tip <10°); Tip2: PointProbe® (tip radius<10nm, typical <7nm; half-cone angle ≈ 20°, but <10° at the very end of the tip). Both types of tips are produced by NANOSENSORS TM , Neuchâtel, Switzerland, who also provided the quoted specifications.

The tip radii and the cone angles were checked by FESEM and TEM. TEM yielded the result that for Tip2 the radius is indeed below 10nm and the half-one angle -measured over the outermost 150nm -does not exceed 15°. Since Tip1 is very fragile, its first approach to the specimen required utmost care. Both AFMs are constructed such that the tip of the monolithic were performed along two alternative orthogonal directions: from side to side and from front to back. The quality of the topographs obtained was the same for both directions. The magnifications of the microscopes were calibrated; hence possible remaining errors are estimated to be at most 0.5% for the TEM, 2% for the FESEM, 2% for AFM1, and 5% for AFM2.

FESEM and AFM revealed that the electrolyte attacks the γ-precipitates more strongly than the γ'-matrix (see sections 4.2 and 4.3). Hence they appear as pits in the FESEM and AFM images. In the following the pits left behind by the etched away parts of the γ-precipitates will be referred to as γ-pits. The present etching effect is the same one as that observed for nickel based superalloys, in which the γ-matrix was more strongly attacked than the strengthening γ'-precipitates, which stuck out of the foils [12]. It is emphasised that etching was not intended, but was a mere side effect of the standard electro-polishing procedure of thin TEM-foils. Since the same specimens were to be studied in all three types of microscopes (TEM, FESEM, AFM), the requirements of the TEM had to be fulfilled.

The present material Ni 69 Co 9 Al 18 Ti 4 is ferromagnetic at ambient temperature. Hence also magnetic force microscopy (MFM) could be performed: AFM2 was operated with a tip of the type PointProbe-MFMR, which had a 40nm thick cobalt alloy coating. Its producer NANOSENSORS TM specifies that the typical tip radius is less than 50nm. The preparation of the specimens investigated by MFM was similar to that of those studied by the other microscopes, but they were only mechanically ground and polished, finally with 250nm diamond paste. There was no electro-polish and hence no etching. The root mean square roughness measured by AFM2 equipped with a magnetic tip was less than 2.5nm.

While the TEM produces a parallel projection of the volume of the thin foil into a plane perpendicular to the optical axis, the AFM and the FESEM yield three-dimensional information on the surface profile of the foil and the MFM registers differences in magnetic field above the surface. In the evaluations, allowances are made for the fact that even γ-precipitates whose centres lie outside the thin foil, may yield contrast.

The compositions of both phases of the thin foils were determined by energy dispersive X-ray (EDX) analyses in the TEM. The following atomic percentages were found

F

Data evaluations

Since TEM, SEM, AFM, and MFM images require different evaluation procedures, the respective analyses are presented in separate sections. The same holds for edge-on and in-plane γ-particles. It is stressed that the planes of all micrographs, presented as well as evaluated ones, are parallel to the (001)-plane of the TEM-foil. So far no automated computer based evaluation procedure was implemented, because it would require the exact standardisation of all processes. In routine work, however, such procedures are very helpful.

Transmission electron microscopy

The L1 2 -long-range ordered γ'-precipitates in superalloys are normally dark-field imaged with one of their superlattice reflections. Due to the inverse arrangement of the γand γ'-phase in Ni 69 Co 9 Al 18 Ti 4 (see section 1), presently bright field images were taken. An example is presented in figure 2. The standard magnification was 10,000x and the photographic negatives were scanned with 600dpi. Four to ten scanned neighbouring original micrographs were 'pasted together' in the computer and treated as one micrograph. The evaluation of each pasted together micrograph started with a search for those γ-pits whose entire images were in the micrograph. Subsequently the area A was delimited by straight lines such that the centres of the 'entire' γ-images lay within A. Evidently some margins which contained partially imaged γ-pits, were disregarded. Tracing out the evaluated area A involved some arbitrariness, which is allowed for in the quoted error limits. The percentage of the respective error decreases as A increases. Pasting together helped to use the original micrographs more efficiently because fewer margins containing partially imaged γ-pits had to be disregarded.

Analogous procedures were applied to the FESEM, AFM, and MFM images. The thickness t of the thin TEM-foils was determined by convergent beam electron diffraction. Within the experimental limits of error of 10%, t was constant throughout each pasted together micrograph. Six TEM-foils were studied; they yielded nine pasted together areas taken at different parts of the foils. The same areas were evaluated for edge-on and for in-plane γ-pits. Since the standard direction of the incident electron beam was along the highly symmetric [001]-direction, many beams were excited. In order to study some finer details, a few micrographs were taken under different conditions: (i) higher magnification, (ii) kinematic two beam conditions with [200] or [220] diffraction vectors, (iii) high angles of tilt:

≈ 35°, and (iv) dark-field conditions with γ'-superlattice reflections. Moreover, EDX-line scans across both types of γ-precipitates, edge-on as well as in-plane ones, were performed (see section 3).

Due to a slight dissolution of the γ'-matrix next to the γ-precipitates, the size of the γ-pits exceeds that of the original γ-precipitates to some extent. The γ'-dissolution blunts the edges of the γ-pits because atoms at edges are less strongly bound. Typically the slopes of the blunted edges have the indices {011}. This will be detailed in sections 4.2 and 4.3. Moreover, the strain in the γ'-matrix next to the γ-precipitates may enhance the γ'-dissolution. In the standard (incident electron beam parallel to [001]) micrographs the γ-pits produce mass thickness contrast. The strain in the γ'-matrix next to the not etched away parts of the γ-precipitates may give rise to some strain contrast.

Edge-on γ γ γ γ-precipitates.

Often the strongly bright mass thickness contrast generated by edge-on γ-pits is surrounded by a region of weaker contrast. It is due to the above mentioned dissolution of the γ-matrix, especially at the edges of the γ-pits, and perhaps to some strain.

The area of the original, i.e. not pasted together standard micrograph corresponds to about 60(µm) 2 of the thin foil. In most cases this area is dominated by either the (100)-or the (010)-family of edge-on γ-pits. In figure 2, about 60% of them belong to the (100)-family.

This dominance is probably caused by an elastic interaction between the γ-precipitates [START_REF] Khachaturyan | Theory of Structural Transformations in Solids[END_REF]28],

which gives rise to a growth selection. The interaction in turn is due to their lattice mismatch.

On the basis of their brightness three types of images of edge-on γ-pits are distinguished in figure 2 (i) A: very bright; the edge-on γ-precipitate led to a deep γ-pit in at least one surface of the thin foil. A γ-precipitate which intersects both surfaces gives rise to pits in both of them. In the latter case even holes may appear.

(ii) B: dark; the edge-on γ-precipitate lies entirely within the foil and does not contact either surface; there is only strain contrast due to the elastic distortion of the lattice planes. Since the γand the γ'-phase have nearly the same average atomic number (see section 3) there is hardly any mass thickness contrast.

(iii) C: moderate brightness; the edge-on γ-precipitate led to a shallow γ-pit in one surface of the thin foil. ≈ 60(µm) 2 ). As detailed below, the described procedures do not yield the true half-axes ij α and ij β of the γ-precipitates, but the apparent ones ij α ' and ij β ' .

The γ-precipitates were partially or entirely etched away (see section 3). Not only those of them produced contrast in the TEM whose centres lay within the thin foil (in-foil γ-precipitates), but also those whose centres lay outside the foil (out-of-foil γ-precipitates), however, not more than ( ij α -w) away from a surface of the foil. If and only if the depth of the γ-pit marking the position of the γ-precipitate exceeds the minimum depth w, sufficient contrast is obtained. At F, figure 2 shows the superimposed images of a (100)-and of a (010)-edge-on γ-pit. The contrast of both of them is strong. As communicated above, there is no contact between the two γ-precipitates. At least the depth of one γ-pit cannot exceed half of the thickness t of the foil. In figure 2, t is 352nm. Rather shallow edge-on γ-pits were imaged under standard [001]-conditions and after tilting through about 35°. Comparisons of theses images led to the conclusion that w is around 0.1·t. Hence in all evaluations 0.1·t is inserted for w. The effects of inserting alternative values for w will be discussed towards the end of this section.

In the evaluations, it is assumed (i) that in-foil γ-precipitates are imaged with their true half-axes ij α and ij β and (ii) that the γ-pit images of out-of-foil γ-precipitates yield chords (see equations ( 3)). Hence in the case of w=0, on an average ij α ' and ij β ' of out-of-foil γ-precipitates equal ½σ α (n) and ½σ β (n,k), respectively. With the exception of γ-images of type (ii) (see at B in figure 2) one does not know whether a given γ-image represents an in-foil or an out-of-foil γ-precipitate. Therefore corrections are performed on the basis of equations ( 4), which represent weighted averages over both types of γ-precipitates. The weights are t and [ ij 2( ) w

α -] for in-foil and out-of-foil γ-precipitates, respectively. The factor 2 allows for the two surfaces of the foil.

ij ij ij ij ( ) ( ) 2( ) t w n t w α α σ α α α + - = + - ' (4a) ij ij ij ij ( ) ( , ) 2( )
t w n k t w β α σ β α β + - = + - ' (4b)
If n equals k, the ratios ij

α ' / ij α and ij β ' / ij β are
13 1/ 2 ' ' 2 ' C ij C ij C ij ij C ( ( , ) 2 ) ( ( , ) 2 ) 4 ( , ) ( 2 )

( , )

t q n n w t q n n w q n n t w

q n n α α α α   -- π - + - π - + π -   = π (4c) ij ij ij j C [ 2( )] ( ) ( , ) i t w t w q n k β α β α + - = + - π ' . (4d)
On an average, ij α and ij β are about 7% larger than ij α ' and ij β ' , respectively. σ α (n) and σ β (n,k) represent average chords calculated for w=0. Hence it is not quite correct to insert them into equations (4): analogous average chords calculated under disregard of the outermost parts (of extension w) of the γ-precipitate should be used instead. Equations ( 4) overestimate the corrections slightly.

The depth of a γ-pit left behind by an out-of-foil γ-precipitate exceeds w if and only if the original position of the γ-particle centre was not more than ( ij

α -w), 1<i<Z=9, 1<j<N i ,
away from a γ'-surface. There are Z=9 pasted together micrographs and N i images of edge-on γ-pits in micrograph No. i. Evidently large γ-precipitates are favoured. Therefore the weights

Q ij , which are proportional to {1/[ t i + 2( ij α -w)
]}, are used in the calculations of the average half-axes a i and b i , averaged over micrograph No. i. This weighting procedure is analogous to that in equations ( 4). Inserting the weights Q ij is analogous to the weighting procedures performed for spherical particles [START_REF] Nembach | Particle Strengthening of Metals and Alloys[END_REF]11]. However, presently no corrections are necessary for a deficient resolution of the TEM. Since the overlap of γ-images can easily be discerned no corrections for overlap have to be made either. Equation (2a) yields for the volume Ω ij of the γ-precipitate with the indices ij

i i ij ij 1 N j a Q α = = ∑ , ( 5a
) i i ij ij 1 N j b Q β = = ∑ (5b) with ij i i ij 1 1 2( ) Q S t w α = + - (5c)
and i i m 1 i im 1 2( ) N S t w α = = ∑ + - (5d)
Ω ij = q V (n,k) 2 ij ij α β π . ( 5e
)
The average volume V i of edge-on γ-plates in micrograph No. i is given by

i i ij ij 1 N j V Q Ω = = ∑ . ( 5f
)
Let A i be the evaluated area of micrograph No. i and f i the total γ-volume fraction. 'total' means that the in-plane γ-precipitates are included; this introduces the factor 1.5. Thus one obtains for

f i i i i ij 1 i i i ij 1.5 2( ) N j t f A t t w Ω α = = ∑ + - . (5g)
It is emphasised that the summation is carried out only over edge-on γ-pits. Due to the above mentioned predominance of one family of {001}-γ-precipitates in each original micrograph, the factor 1.5 needs justification. The final result for the γ-volume fraction f is the average over Z=9 pasted together micrographs taken at various areas of six thin foils (see above).

Hence any predominance averages out. To conclude the average aspect ratio R i is defined

i ij i ij 1 ij N j R Q α β = = ∑ . ( 5h
)
The final results for the half-axes a and b, the aspect ratio R, the volume V, and the total γ-volume fraction f are the weighted averages over β is plotted versus ij α , no weighting procedure is performed for this latter diagram. Though the scatter of the data is rather wide, the aspect ratio

a i , b i , R i , V i ,
R ij = ij α / ij β is
β = h 0 + h 1 ij α + h 2 ij α 2 (6)
with h 0 =12.97nm, h 1 = 0.0772, and h 2 = -0.5463•10 -4 (nm) -1 .

'[insert table 1 and figure 3 about here]'

The stability of the final results has been checked by trying alternative input values for n, k, and w. Setting all exponents n and k equal to 3.0 or varying w between 0.0 and 0.2·t (t=thickness of the foil) alters the results for a, b, V, and f only within their standard deviations.

The present number 4889 of evaluated edge-on γ-precipitates is probably higher than necessary. In former TEM-studies of spherical nano-scale particles, about 1000 of them were found to be sufficient [START_REF] Nembach | Particle Strengthening of Metals and Alloys[END_REF]. Presently, however, there must be enough micrographs taken at different parts of the specimen to average out the above mentioned predominance of one family of {001}-γ-precipitates; otherwise the final result for f is inaccurate. (i) at H, (ii) at I, (iii) at J, and (iv) at K.

(i)

H: There is only strain contrast due to the distortion of the lattice planes. This in-plane γ-precipitate lies entirely deep within the foil and has no contact with either surface.

Therefore no etching effects can be discerned. Since the average atomic numbers of the γand the γ'-phase are nearly the same, hardly any mass thickness contrast is generated.

(ii) I: This situation is analogous to the edge-on one at D, but the two γ-precipitates have grown together. As detailed below, both long half-axes ij α are evaluated. This γ-precipitate has no contact with either surface of the thin foil.

(iii) J: This γ-precipitate has been completely dissolved; hence no strain, but only mass thickness contrast is visible. There is overlap of this γ-image with one of type (ii).

Etching of the strain field originally lying between the surface of the foil and an in-plane γ-precipitate may raise the depth of its pit above the actual thickness of the γ-precipitate.

(iv) K: This γ-precipitate has been partially etched away. Two γ-pits appeared; their boundaries are, however, not sharp. The slopes of the upper γ-pit probably have strong {111}-components (see sections 4.2 and 4.3). {111}-planes intersect the surface of the specimen along <011>-directions.There is strain contrast due to the elastic distortion of the lattice planes as well as mass thickness contrast. The slight constriction is reminiscent of the edge-on view at D.

Part of the above information was gained from micrographs taken after tilting the foil through high ( ≈ 35°) angles or under kinematic two beam conditions. These in-plane γ-pit images clearly show that the γ-precipitates are not standard rotational ellipsoids.

The average longer half-axis ij α of in-plane γ-precipitates can be derived from such micrographs. Only type (i) (H in figure 2) and type (ii) ( I in figure 2 3(c)), the disregard of the weights favours small γ-precipitates and thus leads to a slight underestimate of 1-2% of the average half-axis a. Since the growth of each γ-precipitate is influenced by its neighbours, its two half-axes ij α measured along [100] and [010] differ sometimes, e.g. at I in figure 2. Such a case has been mentioned in connection with the introduction of equation (2c). On an average the difference between the two half-axes ij α of in-plane γ-precipitates amounts to ± 10%. Their arithmetic mean is used. This is analogous to the procedures described in section 4.1.1 for edge-on γ-precipitates: for them only one long half-axis ij α was measured and averaged over all evaluated γ-precipitates. Presently the average geometric mean of the two half-axes ij α is only 1.1% smaller than their arithmetic mean. Type (iii) and type (iv) images are not evaluated, because the respective results are very sensitive to the etch and to the choice of the minimum depth w. The final result for the half-axis a in-plane of in-plane γ-precipitates is the weighted average over all evaluated micrographs. The weighting procedure is analogous to that applied in the case of edge-on γ-precipitates (see section 4.1.1).

The result is listed in table 1. Evidently a in-plane agrees within ± 2.6% with its counterpart a edge-on ; this agreement is quite gratifying. The γ-volume fraction cannot be derived from in-plane γ-precipitates alone, because they yield no information about the shorter half-axis b.

The total number of evaluated in-plane γ-precipitates of types (i) and (ii) is 899. This number is compared with an estimate derived from edge-on γ-precipitates:

i i i i i i 1 1 ( 2 ) / 3 Z N f t b A V = = - ∑
in-plane (7) with f i = total γ-volume fraction, V i = average volume of a γ-precipitate, t i = thickness of the foil, b i = average length of the shorter half-axis, and A i = evaluated area, 1 i ≤ ≤ Z =9= total number of pasted together micrographs. The averages are taken over A i . This calculation yields N in-plane =946, which agrees within ± 2.6% with the above quoted experimental datum. The difference is below the error limit 1/ N in-plane , which amounts to 3.3%. This good agreement proves the consistency of all evaluations and that the predominance of one {001}-family of γ-precipitates was effectively averaged out.

'[insert figure 4 about here]'

Scanning electron microscopy

High resolution FESEM images of a thin TEM-foil were recorded with secondary electrons

(SE) at 10kV on photographic film. The tilting angle of the foil was 0° for the standard micrographs, ± 5° for stereo pairs, and ± 30° for some special purposes. Figure 4 shows two micrographs: the edge-on γ-pits are typical, the shape of in-plane ones, however, often strongly differs from that in figure 4 5) are applied with t i =0 and w=0. w is assumed to vanish because even a very shallow γ-pit yields contrast. The factor 2 in equations (5c) and (5d), which stands for the two surfaces of the thin foil, cancels. The total, i.e. edge-on plus in-plane, γ-volume fraction f i is calculated from the fraction of the area A i of the micrograph No. i covered by edge-on γ-pit images. Equation (5g) is replaced by

i i ij ij 1 i 1.5 2 ( , ) N C j f q n k A α β = = π ∑ ' ' . (8)
The area of the evaluated edge-on γ-pits follows from equation (3e). The final averaging procedures over the four micrographs are analogous to those described in section 4.1.1, but f i is weighted by A i instead of by N i . The final results for the averages a, b, R, V, and f are listed in table 1. The quoted statistical error limits represent the standard deviations of the respective averages plus -in the case of a and b -2% for inaccuracies of the magnification of the FESEM (see section 3) and -in the case of f -6% accounting for uncertainties in tracing out A i (see section 4.1). The agreement between the SEM edge-on data and the respective TEM ones is good: a(SEM) and b(SEM) are 8% smaller than a(TEM) and b(TEM), respectively.

Consequently V(SEM) and f(SEM) are 24% and 19%, respectively, below their TEM-counterparts. R(SEM) is very close to R(TEM). The error limit 1/ 63 of f(SEM) amounts to 13%. With the only exception of V(SEM), the edge-on SEM results agree with the respective edge-on TEM ones within the statistical limits of error.

4.2.2

In-plane γ γ γ γ-precipitates. For reasons given above, the shape of the in-plane γ-pits is often rather poorly defined. Moreover, there are only 11 of them. Their evaluation is analogous to that of edge-on ones (see section 4.2.1). β ' , which is unknown. f i is calculated with the aid of equation ( 8), but the factor 1.5 is replaced by 3.0 because -in the case of in-plane γ-precipitates -the summation allows only for one of the three families of {001}-γ-plates. The overall arithmetic mean a of the two long half-axes and the volume fraction f are listed in table 1. The weighting procedures and the derivations of the error limits are the same as in section 4.2.1. The result for a is in excellent agreement with the respective TEM edge-on one, the error limit, however, is high. f(SEM, in-plane) turns out to be much higher than f(TEM, edge-on). The reason is that the number N in-plane =11 of FESEM imaged in-plane γ-pits happens to be exceptionally high. The statistical average number <N in-plane > can be estimated on the basis of N edge-on =63 of imaged edge-on γ-pits. The ratio of the fractions of area covered by edge-on and by in-plane γ-pits is 2.0 because there are two edge-on {001}-families, but only one in-plane family. Allowing for the average aspect ratio R ij =7.8 (see table 1) one expects σ=N edge-on /N in-plane =2.0·7.8=15.6. Evidently the present ratio 63/11=5.7 happens to be far below the expected one and f(SEM, in-plane) is very high. The error limit in-plane 1/ N = 1/11 of f(SEM, in-plane) alone amounts to 30%.

Atomic force microscopy

Magnetic force microscopy (MFM) will be discussed in section 4.4. Both AFMs (AFM1 and AFM2, see section 3) were operated in the tapping mode. They were alternatively equipped with both types of tips (Tip1 and Tip2, see section 3). Figure 7 is a TEM bright field image of a Tip2 before its use. Each standard individual AFM image covers 5µmx5µm of the TEM-thin foil and has 512x512 pixels. The acquisition time for the image was 10-20min. To get more in-plane γ-pits, micrographs were also taken at lower magnification: 15µmx15µm with 512x512 pixels each. The scan direction was chosen to be close to <011>, not <001>, because thus the shorter half-axes ij β ' of the superellipses could be measured more accurately.

In sections 4. Basically the numerical evaluation procedures are analogous to those applied to the FESEM images (see section 4.2). However, the derivation of the apparent half-axes ij α ' and ij β ' from the three-dimensional numerical AFM output data z'(x',y') requires detailed analyses of the three issues (i)-(iii) listed below. Whereas the contrast evaluated for TEM and FESEM images stems mainly from the lower parts of the etched γ-pits, the AFM yields information on their upper parts. This will be discussed below. It depends on the surface profile of the specimen which point of the tip is closest to the specimen during the AFM scan.

This point may be in the outermost sphere-shaped end of the tip or in its pyramidical shaft. At the point of closest approach the slopes '/ ' z x ∂ ∂ of the tip and of the specimen surface are the same [11,[START_REF] Keller | Deleted: 5 Deleted: 6 Deleted: 7 Deleted: 8 Deleted: 29 Page 34 of[END_REF]; here the scan is supposed to be along the x'-axis. The AFM computer registers the nominal coordinates (x', y', z'). If the tip were infinitely slender and not inclined (see section 3), (x', y', z') would represent the profile of the specimen exactly. The function z'(x')

actually registered in a linear scan depends on (i) the original shape of the γ-precipitate, (ii) the amount of γ'-matrix which dissolved next to the γ-precipitate, and (iii) the shape and inclination of the tip. Issues (i) and (ii) determine the shape of the γ-pit.

'[insert figure 10 α . Figure 10 shows the profiles z(x, sim).

Due to the finite opening angle (half-cone angle: ≈ 12°) and the inclination of the tip, the

Determination of ij

α ' : As stated above, ij α ' can easily be measured. ij α ' is visually derived from those points in the micrographs (see figures 8(a) and (b)) where the contrast changes rapidly; this is analogous to the measurement of ij β ' , but there is no need to establish the profiles z(x) and z(y). The level z at which ij α ' is measured, is about one half of 0 ζ used for ij β ' .

Exponents n and k: Since ij α ' and ij β ' are determined at different levels z, n and k cannot be obtained by AFM. Based on the TEM results (see section 4.1.1), throughout the evaluations 3 is inserted for n and k.

Ten pasted together micrographs taken for four thin TEM-foils are evaluated; they contain 1829 edge-on γ-pits. The calculations are the same as described in section 4.2.1 for edge-on γ-pits imaged by FESEM. The only differences concern the calculations of the statistical limits of error: that of the magnification has been given in section 3 and that of the area A i appearing in equation ( 8) is 4.5%.The final results are listed in table 1. The agreement [insert figure 11 about here] About every other pasted together area had to be disregarded because 0 ζ could not be accurately determined. About the same fraction of thin foils proved to be unsatisfactory for TEM work. Moreover, it must be kept in mind that one searches each foil already in the TEM for electron transparent areas and disregards thick ones. In the AFM investigation, however, areas with unsatisfactory γ-pits are only recognised during their evaluation. If the preparation of the present Ni 69 Co 9 Al 18 Ti 4 -specimens had been geared to the AFM instead of to the TEM the fraction of good AFM images would have been higher. Twenty-eight 15µmx15µm AFM-images are evaluated, they are not pasted together.

The micrographs were taken for four different TEM-foils. A total of 402 in-plane γ-pits is evaluated. The half-axes ij α ' are measured as described in section 4.3.1. The further numerical evaluations are the same as in section 4.2.2. The results for the average longer half-axis a(AFM, in-plane) and for the γ-volume fraction f(AFM, in-plane) are listed in table 1. The error limit of the area A i (see equation ( 8)) is 5%. The results agree satisfactorily with the respective edge-on TEM ones: a(AFM, in-plane) is 15% too high and f(AFM, in-plane) is 7% too low.

[insert figure 12 about here]

Magnetic force microscopy

As stated in section 3, AFM2 was equipped with a tip which had been coated with a 40nm thick hard magnetic cobalt alloy layer. As in section 4.3, the coordinate system (x',y',z') is that of the AFM scan and (x,y,z) is determined by the crystallographic axes of the specimen;

z' equals z. Since the specimen had only been polished, but not etched (see section 3), it could not be orientated relative to the scanning direction by using the optical microscope attached to AFM2. Hence there is no general correlation between (x,y) and (x',y'). The tip and the specimen were magnetised with a small permanent magnet such that different poles meet during the measurements. Each line was scanned twice: first the profile z'(x', y'=const.) was registered in the standard tapping mode and subsequently the tip was raised to the level z 0 and scanned at the constant level z 0 , i.e. during the second scan the tip did not contact the specimen. The magnetic interaction of the tip with the specimen alters the resonant frequency of the cantilever and thus leads to a phase shift ϕ between it and the driving frequency.

During the second scan, ϕ (x',y'=const.) was registered. In figure 12(a) a standard tapping mode topography z'(x',y')-image and in figures 12(b), (d), and (f) ϕ (x', y')-images are shown; ϕ varies between +0.1° and -0.15°. Since neither details of the magnetic properties of the γand γ'-phase nor of the tip are known, ϕ could not be calculated as a function of the shape and the size of the γ-precipitates. The same two areas A 1 and A 2 of one specimen were 27 evaluated for edge-on and in-plane γ-pits; A 1 consisted out of one and A 2 out of nine pasted together 5µmx5µm micrographs. The error limit of A i is 4.5%.

Edge-on γ γ γ γ-precipitates.

In total there are 131 of them. The superellipses are drawn such that they include the whole area in which ϕ is strongly negative, i.e. below some limit 0 0 0.02 ω ≈ -(see figure 12(d)). 0 ω varies slightly with z 0 . On the basis of the TEM results (see section 4.1.1) the exponents n and k are set equal to 3 . A superellipse is outlined in figure 12(d). The further evaluations follow those applied to the FESEM images of edge-on γ-pits (see section 4.2.1). The results have been compiled in table 1. That for the longer half-axis a(MFM) is in excellent agreement with that obtained by TEM, but the result for the shorter half-axis b(MFM) exceeds b(TEM) by 48%. Consequently the average aspect ratio R(MFM) is too low and the average volume V(MFM) too high. The volume fraction f (MFM) in turn agrees perfectly with f(TEM). But since b(MFM) is much too high, the latter agreement must be considered as fortuitous. The statistical error limit of f (MFM) amounts to 42%.

In spite of the low number of edge-on γ-precipitates studied by MFM and the uncertainties concerning the interpretation of the ϕ -contrast, the result for a is surprisingly accurate. The strong deviation of b(MFM) from the three other data listed in table 1 for b of edge-on γ-pits, indicates systematic short comings of the present MFM evaluations. Since the main components of the magnetic stray fields produced by edge-on γ-precipitates lie probably in planes normal to them, the evaluation of a is expected to be far less critical than that of b.

Moreover, the smaller size of b makes it more sensitive to all types of errors.

In-plane γ γ γ γ-precipitates.

There are only 20 of them. The superellipses are drawn at the level ϕ ≈ -0.02° (see section 4.4.1). An example is outlined in figure 12(f). The further numerical evaluations are analogous to those performed for FESEM images (see section 4.2.2). The in-plane MFM results for the average longer half-axis a and for the γ-volume fraction f have been compiled in table 1. a and f exceed the respective edge-on TEM results by 11% and by 66%. Evidently the agreement concerning a is good. Hence the large discrepancy concerning f must be due to errors in the number of evaluated in-plane γ-pits.

On the basis of the TEM investigations the ratio σ of the numbers of edge-on to in-plane γ-pits is expected to be 15. A possible reason why there were so many in-plane γ-pits is that even those of them which lay entirely below the surface of the specimen may have produced magnetic stray fields which were registered by the MFM.

Discussion

The microscopic characterisation of second phase particles involves two steps:

(i) generation of some contrast and (ii) relating it to the size and shape of the particles. In most cases step (ii) is quite complex. One of the few exceptions are L1 2 -long-range ordered γ'-precipitates embedded in the disordered γ-matrix of nickel based superalloys (see section 4.1): by TEM these γ'-precipitates are dark-field imaged with one of their superlattice reflections. The difficulties encountered in measuring the γ'-volume fraction f γ' by AFM have been demonstrated by Hazotte et al. [8], who determined f γ' of cuboidal γ'-precipitates in an industrial nickel based superalloy. Their edge-length was about 500nm. The experimental results for f γ' were found to decrease as the depth of etching increased. The correct value for f γ' was obtained by extrapolating this depth to zero. Even in the relatively easy case of dark-field TEM of spherical γ'-precipitates with average radii above 5nm, the error limit of f γ' is at best around 20% [START_REF] Nembach | Particle Strengthening of Metals and Alloys[END_REF].

For the time being magnetic force microscopy is left aside; it will be discussed towards the end of this section. The present plate-shaped γ-precipitates in Ni 69 Co 9 Al 18 Ti 4 have nearly the same chemical composition as the γ'-matrix and both phases have the f.c.c. crystal structure. Hence in order to image the γ-precipitates by SEM and AFM, surface profiles had to be generated. In the TEM these profiles yield mass thickness contrast and the γ-parts remaining embedded in the γ'-matrix give rise to some strain contrast. The present surface profiles were created unintentionally by etching during the standard electro-polish of thin TEM-foils. Since the same specimens were meant to be investigated by TEM, SEM, and AFM, their preparation had to be geared to TEM. During electro-polishing the γ-precipitates dissolved faster than the γ'-matrix; hence γ-pits appeared (see section 3 First edge-on γ-pits are discussed. The present description of the plate-shaped γ-precipitates as superellipsoids turned out to be very convenient. All four relevant parameters, i.e. the two average half-axes a and b and the two average exponents n and k (equation ( 1)), can be derived from micrographs of edge-on γ-pits alone [25]. They also yield the γ-volume fraction. For reasons given in section 4.3.1, n and k were presently not determined by AFM. The contrast observed by TEM and SEM stems mainly from the deeper parts of the γ-pits, whereas the AFM 'sees' only their surface-near parts. Therefore specimens meant to be studied by AFM should be etched very lightly. In this case blunting of the γ-pits is slight; hence the parameter 0 ζ introduced in section 4.3.1 is close to zero and the exponents n and k can be determined by AFM.

The results obtained by TEM for edge-on γ-pits are considered as the most reliable ones. The corresponding SEM and AFM results for the half-axes a and b and the γ-volume fraction f agree well with the respective TEM data (see table 1 and sections 4.2 and 4.3). This proves that AFM is well suited for the characterisation of nano-scale plate-shaped second phase particles. No systematic short comings of this technique became evident. For spherical particles analogous results were reported earlier [11,12].

Since the present preparation schedule had been geared to TEM, the AFM images had to be evaluated as detailed in section 4.3.1. If, however, standard metallographic polishing and etching procedures are carried out, the evaluations of AFM images can be automated. This is a great advantage, especially in routine work. In contrast to TEM and SEM, AFM images can be taken almost automatically. This compensates the advantage of the TEM, which images many γ-pits and γ-precipitates simultaneously. Because of the abrupt changes in the surface profile at the γ-pits, it is unlikely that the scan rates can be raised significantly while a high resolution is kept. All procedures described in section 4 can also be applied to particles which have the shape of cubes with rounded edges and corners. The only difference is that the two half-axes α and β are the same.

In-plane γ-pits yield only ancillary information: they show possible deviations of the plate-shaped γ-precipitates from a four-fold rotational symmetry around the shorter axis (see section 4.1.2). From edge-on γ-pits only the average longer half-axis is obtained. The present average difference between the two longer half-axes is, however, rather small: it amounts to only ± 10% and the linear average over the two longer half-axes exceeds their geometric mean by only 1.1% (see section 4.1.2). Hence no appreciable error is made if the γ-volume fraction is derived from edge-on γ-pits alone.

The great advantage of magnetic force microscopy (MFM) is that no etching is needed. The applicability of MFM to the characterisation of second phase particles is naturally limited to materials for which the magnetic properties of the matrix and of the particles differ sufficiently. In spite of the present lack of information on the magnetic properties of the γ'and the γ-phase and on the interaction of the MFM-tip with the specimen, all results obtained by MFM are acceptable; the result for the average longer half-axis of edge-on γ-pits agrees excellently with the respective TEM datum.

Most (96%) exponents n and k (see equation ( 1)) of the γ-plates in Ni 69 Co 9 Al 18 Ti 4 equal 3. This expresses the observation (see figures 2, 4, 8, and 9) that the shape of the γ-precipitates deviates from that of a standard rotational ellipsoid with n=k=2 in three aspects:

(i) the rim is less sharp, (ii) the large γ-γ'-interfaces which are approximately parallel to {001}, are nearly planar, and (iii) the cross sections of in-plane γ-precipitates deviate from circles towards squares. Their growth and thus their final shape are governed by the parameters listed at the beginning of section 1: specific interface energy, lattice mismatch, elastic stiffnesses, anisotropies, and precipitate volume. Only at the beginning of their growth, the γ-precipitates are of spherical shape. Later their lattice mismatch limits their growth along one <001>-direction and they become plate-shaped: they grow mainly at their rather blunt (see figures 2, 4, and 5) rim. Anisotropies of the stiffnesses and / or of the specific interface energy render the γ-plates nearly square-shaped.

2.

If the magnetic properties of the nano-scale plate-shaped particles differ sufficiently from those of the matrix, they can also be characterised by magnetic force microscopy.

In this case no etching is needed.

3.

Nano-scale particles of any convex shape, e.g. γ'-cubes with rounded edges and corners in nickel based superalloys, can be elegantly described as superellipsoids [25].

A superellipsoid is characterised by four independent parameters, all of which can be measured in just one micrograph. With the aid of equations ( 1 The Miller indices of the plane of the thin foil are (001) and the electron beam is normal to it.

The x-and y-axes are parallel to <001>-directions. Thickness of the foil: 352nm. For A -K see section 4.1. For one edge-on (L) γ-pit and one in-plane (H) γ-precipitate the superellipses are outlined; the magnification of these two small micrographs is 2.1times that of the large one.

5 α

 5 and β are the half-axes with α > β . The exponents n and k are integers exceeding 2.

) 6 For

 6 later reference the average length σ α (n) of chords through the superellipsoid are defined. The average is taken over chords which (i) lie in the (001)-plane (in-plane view) characterised by z ≡ 0 and (ii) are parallel to [010], i.e. σ α (n) equals the average vertical extension of the sketches in figure 1(b):

  : (i) at A, (ii) at B, and (iii) at C. Tilting through high angles ( ≈ 35°) and taking micrographs under two beam conditions yielded the information communicated below.

At A in figure 2 ,

 2 two parallel γ-pits are very close together; at D the two images are nearly in line. The intersections between the edge-on γ-precipitate at E with the two surfaces of the foil have different lengths. The longer one is evaluated. At F, two γ-pit images cross each other. In this case one edge-on γ-precipitate intersects the upper and one the lower surface of the foil. On the basis of detailed analyses of the strain contrast patterns it is concluded that the two γ-precipitates do not contact each other. At G, the images of two parallel edge-on γ-pits superimpose with the image of an in-plane one.The full axes ij 2α ' and ij 2β ' of the individual γ-precipitate images are measured by manually marking the γ-pits. The subscripts i and j refer to the number of the evaluated pasted together micrograph (i<Z=9) and to the number of the individual γ-precipitate (1<j<N i ) in micrograph No. i, respectively. Great care is exerted to only allow for the strong central mass thickness contrast and to disregard the weaker one reaching out further. The integer exponents n and k are chosen individually for each γ-pit such that the superellipse drawn according to equation (1) follows the TEM image well. In most (96%) cases, n=k=3 yield a very good representation. The overall averages over n and k are 2.99 and 3.03, respectively. A total of 4889 edge-on γ-pits is evaluated, on an average about 100 per original micrograph (area:

  the same. The centres of all visible γ-precipitates with ij α <w lie within the thin foil. Hence ij α = ij α ' and ij β = ij β ' hold for them. Inversion of the two above equations yields:

3 .

 3 figure 3(c), ij

  Evidently the aspect ratio Ri = ij α / ij β increases with ij α . Deleted: Since the strain energy of γ-precipitates increases approximately as ij 2 β , ij β of only few of them exceeds 40nm and the distribution function ij . In figure 2 four types of images of them can be distinguished:

8 F 4 . 2 . 1

 8421 photographic processes are performed according to a standardised protocol; hence all recorded negative films and the prints made from these films transfer I SE into the brightness of the prints reproducibly. I SE strongly depends on[START_REF] Reimer | Scanning Electron Microscopy[END_REF]: (i) the angle between the incident electron beam and the normal of the SE emitting area, (ii) the inclination of this area relative to the SE-detector, and (iii) the thickness t of the foil; this, however, only if t is smaller than the range of the incident electrons. Moreover, I SE is strongly enhanced at edges. Three extended regions are distinguished in figures 4(a) and (b):

Figure 9

 9 figure8(a). The difference in contrast between edge-on and in-plane γ-pits is due to the fact that those areas which lie more than 52nm below the γ'-surface are shown in white and only

  figure 8(a).

  figure3), log-normal distribution functions are plotted in figure11with the same parameters as in figure3. Evidently there is very good agreement between the distributions measured by AFM and TEM.

4. 3 . 2

 32 In-plane γ γ γ γ-precipitates. Three types of in-plane γ-pits are distinguished in figure8(a):(i) at B, (ii) at C, and (iii) at D.(i)B: This γ-pit is similar to that marked K in figure2, but that at B has been etched more strongly.(ii) C: This γ-pit is rather shallow and approximately circular. Its ascents and descents have strong {111}-components. {111}-planes intersect the surface of the specimen along <011>-directions. The apparent longer half-axis ij α ' is measured along <001>-directions.(iii) D: Regions (2) (see section 4.2 and figure 6(b)) of this γ-pit are parallel to Deleted: its value for . This γ-pit is very deep. ij α ' is measured along the <001>-diagonals.

  ). The habit planes are of the type {001}. The upper parts of the edge-on γ-pits are blunted: they are bounded by {011}-planes or -in the case of strongly etched specimens -by {111}-planes (see sections 4.2 and 4.3). The contrast produced by the γ-pits was successfully related to the size and γ-precipitates (see section 4). The procedures are different for TEM, SEM, and AFM and for edge-on and in-plane γ-pits.

Fig. 1 :Fig. 2 :

 12 Fig. 1: Superellipsoids, the ratio

Fig. 3 :

 3 Fig. 3: Distribution functions (a)

Fig. 4 :Fig. 5 :

 45 Fig. 4: High resolution FESEM secondary electron images of (a) edge-on and (b) in-plane γ-pits in Ni 69 Co 9 Al 18 Ti 4 . The x-and y-axes are parallel to <001>-directions. (c) The superellipse for γ-pit A is outlined, (d) intensity I(x) of the brightness across A, (e) I(y) across A, (f) superellipse for B, (g) I(x) across B. I is in arbitrary units.

Fig. 6 :

 6 Fig. 6: Schematic sketch of the surface profile at (a) an edge-on and (b) an in-plane γ-pit. The numbers indicate those of the 'Regions' defined in section 4.2.

Fig. 7 :Fig. 8 :

 78 Fig. 7: TEM bright field image of an AFM Tip2 before its use.

Fig. 9 :

 9 Fig. 9: High magnification AFM (AFM2 with Tip1) topography image of an edge-on γ-pit. Lines of equal height are indicated; the difference in height between neighbouring lines is 15nm. Areas which lie deeper than z= -110nm, are shown in black. The γ'-surface is at the level z=0. The arrow in (c) points at a (001)-terrace.

Fig. 10 :

 10 Fig. 10: Computer simulated surface profiles. The actual geometry of Tip2 (see figure 7) with 10° inclination is used. ----Actual surface profile z(x), -----simulated AFM-profile z(x, sim).

Fig. 11 :

 11 Fig. 11: Distribution functions (a)

Fig. 12 :

 12 Fig. 12: (a,b) Images of a just polished Ni 69 Co 9 Al 18 Ti 4 -specimen taken with AFM2 equipped with a magnetic tip: (a) tapping mode, (b) ϕ -shift mode, the x-and y-axes are parallel to <001>-directions. (c) Height-profile z'(x') through (a). (d) Outline of the superellipse of the edge-on γ-pit A in (b), (e) its profile ϕ (x), (f) and (g) as (d) and (e), respectively, for the in-plane γ-pit B in (b). Areas with ϕ < -0.1° are black in (b) and (d).

  /mc.manuscriptcentral.com/pm-pml ilosophical Magazine & Philosophical Magazine Letter

  ) images are evaluated; there are 899 of them. The superellipses are drawn in the middle of the dark contrast (see figure2, right small micrograph). They directly yield ij α , not only the apparent axis ij α in equation (5a) is performed. This is actually not quite correct: since the γ-precipitates which produce these types of images, do not contact either surface, weights approximately proportional to [1/(t i -2 ij β )] would be appropriate. However, ij β of in-plane γ-precipitates is not known. Since ij β increases with ij α (see figure

	Philosophical Magazine & Philosophical Magazine Letters
	weighting procedure as
	F o r
	P
	e
	e r
	R
	e
	v i e
	w
	O n l
	y
	17

' . No Page 17 of 47 http://mc.manuscriptcentral.com/pm-pml

  Some AFM profiles z(x, sim) were computer simulated for edge-on γ-pits and Tip2 shown in figure7. It is inclined by 10° relative to the normal of the agrees with the actual geometry of AFM2. The full width of the γ-pit equals the present average full smaller axis 2b=52.2nm (see table1). In the simulations the scan direction is parallel to the x-(not x'-!)direction. In figure10(a) the γ-pit/γ'-matrix interfaces are exactly normal to the x-direction. The profile in figure 10(b) is similar, but the top edges of the γ-pit are blunted. Their slopes have the indices {011}. Such blunting is brought about by a partial dissolution of the γ'-matrix because atoms at sharp edges are less strongly bound

	Page 23 of 47	Philosophical Magazine & Philosophical Magazine Letters
	specimen; this (see section 4.1). The micrographs presented in figures 2, 4, 5, 8, and 9 clearly show such
	blunting, mainly in the direction of the smaller half-axis ij β . Along the longer half-axis
	ij α blunting is less pronounced, moreover ij α is close to 8 ij β (see table 1). Hence blunting F r o causes hardly any problem in the determination of ij
		P
		e
		e r
		R
		e
	about here]' γ-precipitates. Deleted: 29 4.3.1 Edge-on γ γ γ v i e w O n l y
		23

http://mc.manuscriptcentral.com/pm-pml

  The γ'-precipitates in advanced nickel based superalloys are such rounded cubes[START_REF] Nembach | Particle Strengthening of Metals and Alloys[END_REF][7][8][9].

	Deleted: 4
	F o r
	P
	e
	e r
	R
	e
	v i e
	w
	O n l
	y

Table 1 :

 1 ) -(3) all relevant geometric parameters of such particles can easily be calculated. ACKNOWLEDGEMENTS. The authors thank Mrs. A. Ricker (Institut für Medizinische Physik und Biophysik ) for technical assistance in the AFM work, Mrs. L. Rettich (Institut für Materialphysik), and Mrs. G. Kiefermann (Institut für Medizinische Physik und Biophysik) for photographic work. Financial support by the Deutsche Forschungsgemeinschaft and by the research program MSM 0021620834 of the Ministry of Education of the Czech Republic is gratefully acknowledged. Results for the half-axes a and b, the aspect ratio R, the volume V, and the volume fraction f ( total, i.e. edge-on plus in-plane ) of the γ-precipitates.

	Page 32 of 47
	Deleted: 4

http://mc.manuscriptcentral.com/pm-pml http://mc.manuscriptcentral.com/pm-pml

http://mc.manuscriptcentral.com/pm-pml Philosophical Magazine & Philosophical Magazine Letters


J Gastaldi
email: gastaldi@crmcn.univ-mrs.fr

G Reinhart

H Nguyen-Thi ‡

N Mangelinck-Noel

B Billia

T Schenk ¶

J Härtwig

B Grushko#

H Klein||

A Buffet §

J Baruchel

H Jung

P Pino

B Przepiarzynski# )

In situ study of the quasicrystal growth by synchrotron X-ray imaging

Keywords: quasicrystals, growth, synchrotron radiation user supplied): in situ X-ray imaging

. Noticing that most of the present knowledge on the growth of quasicrystal grains has been collected after their cooling at room temperature, we have carried out the first in situ and real time observation of this peculiar process which has clearly disclosed both, the shape of the growing interface and its defective state. Therefore we have studied the solidification of an AlPdMn alloy giving quasicrystal grains by synchrotron X-ray imaging, combining thereby the radiography and X-ray topography techniques. Radiography allowed us to clearly evidence a facetted growth proceeding by lateral motion of ledges at the solid-melt interface and controlled by the interface kinetics rather than by the local heat flow as widely thought. Thus a realistic estimate of the kinetic coefficient was deduced from the solid-melt interface undercooling which indicates that the quasicrystal growth is more likely comparable to both semiconductor and oxide growths than to pure metal growth. The X-ray topographs recorded simultaneously with radiographs, revealed that a lot of strains and defects are generated in the quasicrystal grains during their growth, which could be related to the growth process itself and very informative on the origin of the stability of the quasicrystal lattice.

How the quasiperiodic order can extend up to the several centimetre size, reached by the single grains of various alloys routinely grown nowadays [1,2,3], remains a mystery. Thus, whether these peculiar solids have to follow local rules during their growth [4], or need longrange atomic correlations to form their stable structures [5] is still undetermined. Because they lack translation symmetry, it seems impossible that quasicrystals can be obtained, like crystals, by an attachment of single atoms, repeating unit cells and generating thereby a network. However, as icosahedral clusters have been identified recently in undercooled intermetallic liquids [6], the old idea of an aggregation of these clusters during the liquid-solid transition [START_REF] Elser | Extended Icosahedral Structures[END_REF], which could be the basis of both the nucleation and the growth mechanisms of the quasicrystalline structure, is reviving. Besides, as quasicrystal grains often display, after growth, facets perpendicular to the symmetry axes (5, 3, 2 for icosahedral quasicrystals) it has also been suggested, from room temperature observations, that quasicrystals could grow by a forward movement of facets swept by terraces and steps [START_REF] Beeli | [END_REF], the latter being predicted as abnormally high by theory [9]. But this is conflicting with Shroers et al. conclusions [10] since they interpreted the results they obtained, from their low resolution observation of the growth of quasicrystalline phases from undercooled melt, within the current theories of dendritic growth. Accordingly, whether the quasicrystal growth has a facetted character or a dendritic one remains controversial. Just as the necessity this growth has to incorporate defects in order to ensure the stability of the quasicrystal lattice, which set supporters of the energy influence [START_REF] Steinhart | The Physics of Quasicrystals[END_REF] against those of the entropy effect [START_REF] Shechtman | [END_REF].

To address this problem, we have performed, by combining synchrotron X-ray radiography and X-ray topography techniques, the first in situ and real time observation of the growth of quasicrystals which has both disclosed clearly the shape of the solid-melt interface and given information on strain and defects generated in the growing grains. The results we acquired confirmed the facetted character of the quasicrystal growth and revealed that this growth is accompanied by a lot of strains and defects which could be related to the growth process.

Experiments:

Synchrotron X-ray radiography and X-ray topography experiments were carried out at the ID 19 beamline of the European Synchrotron Radiation Facility (ESRF) in Grenoble, France. Both techniques were combined using the polychromatic incident beam delivered by a wiggler source. Synchrotron White Beam X-Ray Topographs (SWBXRT) were recorded discontinuously on High Resolution films after the incident beam has crossed the sample solidifying in a graphite crucible. Synchrotron X-Ray Radiographs (SXRR) were obtained simultaneously, in the live mode, using a CCD camera [13] oriented toward the incident beam, which was monochromatized and thereby deviated parallelly after crossing the solidifying sample. This combination based on a polychromatic incident beam had the advantage to allow recording several X-ray topographs of good quality, at once, but offered only poor resolution radiographs because these were acquired from a beam monochromatized after crossing the solidifying samples. In order to get radiographs of better quality, displaying sharper absorption and phase contrasts, we carried out, in addition, in situ synchrotron X-ray radiography experiments only, using an incident beam monochromatized before striking the solidifying samples. The solidification set-up based on the Bridgman technique and allowing to control both the pulling velocity and the temperature gradient (unidirectional solidification) was previously described in [14].

Five Al-Pd-Mn samples (flat polygrained sheets 700 µm thick) prepared at the composition Al 72.4 Pd 20.5 Mn 7.1 known to grow easily icosahedral quasicrystal grain [1] were firstly melted then solidified at various pulling rates (0.4 -3.6 µm/s) under the same temperature gradient of 35 K/cm.

Results:

Facetted grains:

It can be seen in synchrotron X-ray radiographs, recorded in situ during the solidification of the five investigated Al-Pd-Mn samples, that all quasicrystal grains remained distinctly facetted all along their growth. Figure 1 shows three stages of the growth of two grains in one of the five samples. Figure 1a corresponds to the end of the growth sequence with a pulling rate of 0.4 µm/s. The interface between the liquid and the two grains exhibits a cusp at the level of the grain boundary. Two facets are merely visible on grain 1 and three on grain 2. During the growth of grains, some facets disappeared and new ones appeared. In both cases the inclination of moving facets varied continuously. Therefore it was difficult to measure the velocity of each individual facet (along their normal direction), which often forced us to estimate it from the displacement of edges of projections of the facets on radiographs (radiograph images are orthogonal projections of samples features along the incident beam). On average this velocity was close to the applied pulling rate.

As soon as we increased substantially the pulling rate (about three times) the solidification front moved back to the coolest part of the furnace (Figure 1b) while new facetted grains were nucleated in the liquid just above the solid-melt interface (Figure 1b,c) and transverse striations developed on facets. At the same time, the velocity of the former growing grain increased up to the value of the new pulling rate ((Figure 2) unless they meet newly nucleated grains. In this case the growth velocities of all grains (former and new) started decreasing and fell down to zero when impingement occurred (Figure 2). The new facetted grains were identified as dodecahedrons according to both, their images on radiographs and their Laüe diagrams recorded in situ.

Strained growing grains:

Synchrotron X-ray topographs have revealed that quasicrystal grains are strained during their facetted growth. This can be seen in Figure 3 which displays synchrotron X-ray radiographs and synchrotron X-ray topographs recorded simultaneously during their growth. Whatever the growing grains (former or new ones) it is clear that their facets appear not so sharp in topographs (Figures 3b and3d) as in radiographs (Figures 3a and3c) which means that they are distorted and they contain numerous defects. As these defects can not be resolved it can be said that their density is higher than the resolution limit of X-ray topography which is of about 10 4 -10 5 cm/cm 3 .

Discussion:

This in situ observation of the evolution of quasicrystal grains during solidification provides the first unambiguous evidence of their facetted growth and reveals that they are constrained all along the growth process. Actually this observation corroborates both the predictions worked out from the faceting usually displayed by grains after growth [START_REF] Beeli | [END_REF] and the assumption made by Dong et al. of a growth of grains "in the form of regular polyhedrons" [15]. Moreover, in this latter case, it specifies that polyhedrons are dodecahedrons (at least for the icosahedral phase of Al-Pd-Mn) and that the maximums of the plots of volume fraction growth velocities as a function of time, they reported in their figure 7, correspond very probably to the beginning of a slowing down due to the meeting of all facetted growing grains (Figure 1c).

Despite it has been impossible to determine the true velocities of each facet (section 2), the measurements of their average velocities in the growth direction, we performed at each jump in pulling rate, allow us to value the growth kinetic coefficient µ µ µ µ. More precisely, taking into account that at each jump the growth velocities reached progressively the new pulling rate and the solidification front globally receded to a lower temperature (section 2), we had access at two important parameters of the solidification transient which took place in these conditions: the growth velocity jump ∆V and its corresponding undercooling ∆T. In addition, by considering, on the one hand that the transverse striations which develop on facets in the same time, have to be related to ledge growth (lateral motion of macrosteps or bunches of macrosteps) [16] and on the other hand that the ledges are generated at facet edges and vertices where undercooling and solid-liquid interface roughness are locally higher, it was possible to relate ∆V and ∆T linearly: ∆V =µ µ µ µ ∆T [START_REF] Tiller | The Science of Crystallization: Microscopic Interfacial Phenomena[END_REF]. Using the results of our experiments, a value of µ µ µ µ = 0.9 µm.S -1 .K -1 was inferred from this formula [START_REF] Nguyen Thi | [END_REF]. Therefore, the resulting quasicrystal growth kinetic coefficient appears two orders of magnitude larger than the value obtained by Dong et al. [15] who used the Avrami equation in the case of an isothermal growth. This disagreement can be explained by considering that Dong et al. measurements [15], were performed during the latest stage of the growth, when facetted grains were meeting as above-mentioned. Nevertheless, a discrepancy with the kinetic coefficients of pure metals, which are much more higher (µ µ µ µ = 1.25. 10 4 µm .s -1 .K -1 for pure Sn), remains. The value we obtained is more comparable to that of semiconductors or oxides (µ µ µ µ = 0.826 µm.S -1 .K -1 for Bi 3 Ge 3 O 2 [START_REF] Golyshev | [END_REF]) which are known to have a facetted growth. Accordingly, it can be stated that the quasicrystal growth is facetted and not dendritic as it was concluded by Shroers et al. [10]. If some doubts were remaining, the radiographs we recorded at the beginning of the growth, when quasicrystals appeared, could allow to get rid of them. This is clearly illustrated in the Figure 4 which shows that quasicrystals grew, immediately facetted, at the expense of the dendritic grains of the primary phase which has a similar chemical composition. Therefore, from an atomistic point of view, it can be concluded that models based on a thick diffuse interface region [4,5,[START_REF] Elser | Extended Icosahedral Structures[END_REF] are not suited to describe the quasicrystal growth and that the attachment of atomic species is uneasy as indicated both by the slow growth kinetic and the solidification front recoil caused by the sudden increase of the pulling rate.

Is the quasicrystal growth constrained?:

When synchrotron X-ray topographs of facetted growing grains (both the former ones which are columnar and the latter ones which are dodecahedral) are compared to those of the other grains we recorded previously, after cooling at room temperature [20][21], it is surprising to note that they display much more strain and defects. Lattice planes reflecting Xrays (X-ray topography is a diffraction imaging technique) are so strained that the facetted shape of these growing grains can not be distinguished in topographs (Figure 4) whereas, in the ones of as-grown grains, the curved boundaries can be recognized [21]. That is the opposite of what we observed during the solidification of pure metals and alloys for which only very few defects were generated by the growing melt-solid interface while most of them appeared during cooling [22] [23].

Two origins of this strained state of the growing quasicrystal grains can be put forward:

1°) it could be ascribed to the sticking of growing grains on the walls of the graphite crucible and to the related thermal stresses. 2°) it could also result from the growth process itself according to Joseph et al. … who suggested that "tears" (dislocations) could be embedded in the growing interface so as to insured an entropic stabilisation of the quasicrystalline structure [7][24]. We have to check the sticking possibility and eventually to try to diminish it, before invoking a phenomenon intrinsically related to the growth process. In the case of consideration of this latter event, we would have to conciliate the systematic generation of defects with a facetted growth because Joseph et al. … supposed a diffuse interface and a dendritic growth, which completely disagrees with our observations.

Conclusion:

This first in situ and real time observation, by synchrotron X-ray imaging combining radiography and X-ray topography techniques, of the solidification of an AlPdMn alloy giving quasicrystal grains, allowed us to clearly evidence a facetted growth. It has been determined that this growth proceeds by lateral motion of ledges at the solid-melt interface and is controlled by the interface kinetics rather than by the local heat flow as widely thought. Thus a realistic estimate of the kinetic coefficient was deduced from the solid-melt interface undercooling, which indicates that the quasicrystal growth is more likely comparable to both semiconductor and oxide growths than to pure metal growth. The X-ray topographs, we recorded simultaneously with radiographs, revealed that a lot of strains and defects are generated in the quasicrystal grains during their growth, which could be related to the growth process itself and very informative on the origin of the stability of the quasicrystal lattice.

1 .

 1 Facetted growth:

Figure 1 :

 1 Figure 1

Figure 2 :

 2 Figure2: Plot of the time evolution of the growth velocity of a quasicrystalline grain after an increase of the pulling rate from ? to 3,6 µm/s. The decrease to zero of the growth velocity is due to an induced nucleation and growth of new dodecahedral grains ahead of the growth front.

Figure 3 :

 3 Figure3: synchrotron X-ray radiographs and their corresponding synchrotron X-ray topographs, respectively for dodecahedral grains a), b) and for columnar grains c), d). Note that the deformed shapes of the synchrotron X-ray topographs indicate that growing icosahedral grains are distorted.

Figure 4 :

 4 Figure4: synchrotron X-ray radiographs showing icosahedral quasicrystal grains growing at the expense of the dendritic grains of the primary phase which has a similar chemical composition.

http://mc.manuscriptcentral.com/pm-pml Philosophical Magazine & Philosophical Magazine Letters

Acknowledgments:

The authors are very grateful to M. Kleman, V. Dmitrienko, M. de Boissieu for helpful discussions and to both the technical staff of the ESRF and T. Bactivelane for their assistance.


L Pizzagalli
email: laurent.pizzagalli@univ-poitiers.fr

P Beauchamp

H J Ónsson

Calculations of dislocation mobility using nudged elastic band method and first principles DFT calculations

Keywords: ab initio, dislocation structures, semiconductors Nudged Elastic Band

We present a new technique which makes it possible to determine mobility properties of dislocations with first principles accuracy without having to apply corrections for the influence of boundary conditions. The Nudged Elastic Band method is used together with periodic boundary conditions and all dislocations included in the simulated cell are coherently displaced during the calculations. The method is applied to the displacement of a non-dissociated shuffle screw dislocation in silicon along two different directions. Peierls energies as well as dislocation structure as a function of the dislocation position in the lattice have been obtained. We have determined the Peierls stresses for both directions, in excellent agreement with previous determinations. Finally, we discuss the advantages of the technique over other methods.

Introduction

Dislocations modelling has first been developed in the framework of linear elasticity theory, not only because of historical reasons, but also because this approach has been able to capture most of their physical properties [START_REF] Hirth | Theory of dislocations[END_REF]. However, this approach is inadequate to deal with the severe lattice distortion near the core of a dislocation. In consequence, it is not suited for describing important properties governed by the structure of the core, such as dislocation mobility or dislocation interactions at short distances. Major improvements have been obtained thanks to the Peierls-Nabarro (PN) model and its recent developments [START_REF] Lu | I: Methods and Models[END_REF][START_REF] Joós | [END_REF][4], since it combines elasticity theory with the discrete nature of the lattice. This model removes the artificial divergence at the core originating from the continuum description, allowing to determine several key quantities such as the core width, the Peierls energy and the Peierls stress. Nevertheless, while this approach has be proved to be successful in many cases, it is not accurate enough for several classes of materials such as covalent or ionic materials, in which dislocation cores are very narrow or reconstructed. Although the PN model has been recently improved [4], it intrinsically includes some limitations by construction [5].

A full atomistic treatment of the dislocation is expected to yield better results. Hence, accurate values can be determined in case of f.c.c. metals, for which reliable classical interatomic potentials are available. However, for other materials such as semiconductors or ceramics, energetic properties of dislocation cores are poorly described by potentials. This point has been brought to light in the case of silicon for example [6]. Another option is to use first principles density functional theory (DFT) calculations which give an accurate description of most materials even those including unusual and unfavorable bonding configurations. The problem with these methods, however, is that the simulated systems are limited in size, and as a consequence boundary conditions require careful and special treatments [7][8][9], or the use of multiscale modelling methods [10,11]. This is especially true for Peierls stress determination. Recently, Pizzagalli

Position

Peierls energy

E p E f 2 E f 1 E p E f 1 E p

Periodic conditions

Fixed conditions

Figure 1. (color online) Effect of boundary conditions on Peierls energy variation as a function of the dislocation core position. For fixed conditions, the interaction between the single dislocation and the surface cluster ranges from E 1 f to E 2 f (dash-dotted line). By combining periodic conditions and NEB, and by displacing the dislocations coherently, the interaction Ep between dislocations remains constant (dashed line) and does not affect the deduced Peierls stress and activation energy.

and Beauchamp have calculated the Peierls stress of screw dislocation in silicon using reduced systems and both periodic and fixed conditions [12]. This work shows that Peierls stress determination using first principles accuracy is possible, but that corrections are required to compensate for boundary effects.

In this paper, we propose a method allowing an accurate determination of the Peierls stress, without the need for boundary corrections. Also, the calculations provide the full energy curve for the movement of the dislocation through lattice, i.e. the whole Peierls barrier, as well as the associated core structure configurations. The method is based on the combination of the nudged elastic band (NEB) method for finding a minimum energy path (MEP) [START_REF] Jónsson | Classical and Quantum Dynamics in Condensed Phase Simulations[END_REF] of a transition and first principles density functional theory (DFT) calculations of the atomic forces subject to periodic boundary conditions. Previously, the NEB method has been used in a few studies of dislocation mobility but only in combination with empirical potentials and fixed atom boundaries in finite systems [START_REF] Rasmussen | [END_REF][15][16][17]. By using periodic boundary conditions, as we do here, smaller systems can be used in the calculation and it becomes easier to make use of first principles DFT methods. In the following we first describe errors coming from fixed boundaries in small system, and we show how this effect is greatly reduced using NEB and periodic boundaries. Then, the Peierls energy and stress are determined with first principles accuracy in the case of a screw dislocation in silicon, moving along two different directions. Finally we discuss the advantages and the limitations of this method, compared to previous models.

The straightforward approach to computational studies of dislocations is to construct a system that contains one dislocation and fixed atoms at the boundaries of the system. In that case, boundary conditions will introduce errors in the energy variation as a function of dislocation position, especially in the case of small systems. This is schematically depicted in figure 1. The initial and final configuration states include the dislocation-surface interaction energy E 1 f , positive (negative) for a fixed (free) surface. For the transition state, ideally located in the center of the system, the interaction energy E 2 f is different, i.e. E 2 f = E 1 f . As a result, the calculated curve will be different from the real one (Fig. 1). For a screw, the energy change ∆E due to fixed cylindrical boundaries is proportional to the square of the ratio between the dislocation distance from the cell center and the cell size, with a factor depending on the shear modulus and the anisotropy of the material [18]. For silicon and cluster sizes considered in first principles DFT calculations, ∆E is about 50 meV. This effect could be reduced by considering larger cluster systems [17].

Here, we propose to use periodic boundary conditions, which is well suited for standard plane-waves first principles DFT calculations, and as such have been widely used for investigating dislocations [19][20][21][22][23][24]. The initial configuration is a system including an even number of dislocations, with a zero Burgers vectors sum (Figure 1). The final configuration is then obtained by a simple translation equal to the investigated dislocation displacement. Finally, a standard NEB calculation between these two configurations is performed. The important point here is that distance between dislocations remain constant along the whole path. This condition is fullfilled at the beginning, if the initial MEP is built from linear interpolation between the end point configurations. We have found that if the springs linking adjacent replicas in the NEB are chosen to be stiff enough and a direct minimization method is used to relax the system to the MEP closest to the initial guess, then the the distance remains constant during the full relaxation until convergence has been reached. The advantage of the method is illustrated in figure 1. Since distances between all dislocations remain constant throughout the NEB calculation, the energy variation as a function of position is simply shifted by a constant term, provided that the total interaction between dislocations E p is elastic. As a result, the shape of the curve is the same as the real one.

Model and methods

We apply this method to a non-dissociated screw dislocation in silicon, with an initial stable shuffle configuration [6]. This choice has two motivations. First, in a previous study, the Peierls stress has been determined with first principles DFT and corrections due to periodic and fixed boundary conditions [12]. Second, in the same work, it has been shown that interatomic potentials for silicon are not reliable enough for this particular problem. We have then an interesting system, for which first principles accuracy is required, and with available data for comparison. We considered a system similar to that used in [12], including a dipole of dislocations, a cell geometry of 12 × 12 × 1 (144 atoms) corresponding to an infinite quadrupolar arrangement [12]. First principles DFT calculations have been performed with the VASP distribution [25,26], with 2 special k-points along the dislocation line direction, an ultrasoft pseudopotential for Si [27], a plane waves energy cutoff of 140 eV, and the GGA functional PW91 [START_REF] Perdew | Electronic Structure of Solids '91[END_REF]. System relaxation is stopped when all forces are lower than 3×10 -3 eV Å-1 . With these parameters, a lattice constant of 5.475 Å and a bulk modulus of 99.7 GPa have been obtained, in good agreement with experiments [START_REF] Hull | Properties of crystalline silicon[END_REF]. The NEB calculations have been performed with both improved tangent and climbing images algorithms [START_REF] Henkelman | [END_REF]31]. Figure 2 shows the zinc-blende structure, and two dislocation displacements, investigated in this work. D' corresponds to the easiest path for displacing a non-dissociated screw, while D" is the direct path allowing the screw to cross-slip from shuffle to glide {111} planes.

In order to obtain the Peierls stress from the energy variation, an accurate determination of the dislocation position relative to the lattice is required. Distances between images computed in configuration space, as output by NEB, are useless in that case because the dislocation position does not linearly follow the configuration space coordinates. As stated in [32], dislocation centers can not be derived uniequivocally from the atomic positions. Dislocation centers are then determined according to a well-known procedure. Relative displacements of atoms on both sides of the plane perpendicular to { 101} and including the dislocation displacement vector are first calculated. Figure 2 shows the bonds linking these atoms for D' and D". Then, the dislocation centers are determined by fitting relative displacements using the elastic expression b/2π arctan((x -x 0 )/∆), ∆ allowing to define a dislocation core width [START_REF] Hirth | Theory of dislocations[END_REF]. This approach assumes that the whole dislocation can still be described as a Peierls dislocation throughout its displacement in the lattice. As a check, the distance between the two dislocations in the dipole has been determined for all relaxed NEB images. The maximum deviation is 3 × 10 -2 Å, i.e. less than 1% of the total displacement. This shows that the two dislocations have been coherently displaced, as assumed.

Results

The Peierls energy, i.e. the energy variation associated with the displacement of a dislocation in a crystalline lattice, is represented in figure 3 for two paths, D' and D". Note that a multiplying factor of 1/2 has been applied since two dislocations are included in our cell. For both paths, starting from the stable configuration, the energy is smoothly increasing until a halfway maximum, then decreasing to the next stable configuration. In the initial configuration A, the bond linking atoms 1 and 2 is markedly stretched (Fig. 2 and Fig. 4). When the dislocation moves along D', this stretching intensifies until the bond breaks at the saddle configuration (Fig. 4), which is a high symmetry position at the intersection of shuffle and glide planes. A simple force relaxation of this configuration indicates it is metastable. In previous, DFT/LDA calculations, we have found that this configuration was weakly stable [6], with an energy difference of 0.32 eV per Burgers vector, in good agreement with the value of 0.41 eV/b obtained in this work. Finally, atoms 1 and 2 form bonds with neighbor atoms to displace the dislocation core in A'. The D' migration is then done by the breaking and reforming of a stretched bond, explaining why this mechanism is the easiest. For D", we found that the migration mechanism is totally different. Initially, the bond between atoms 2 and 3 is slightly stretched and out of the (101) plane (Fig. 2 and Fig. 4). This bond is progressively twisted when the dislocation moves along D", until it belongs to the (101) plane at the saddle point (Fig. 4). This corresponds to a screw in the glide set, with a sp 2 hybridized core [6]. Actually, it is rather a weakly stable configuration, not seen here due to our limited NEB images number. We have then two totally different mechanisms, in one case, the dislocation is displaced via bond breaking and reforming in one case (D'), and via severe bond twisting in the other (D"). It is also interesting to compare calculated Peierls energies with classical models. In fact, it is common to use a phenomenological expression W sin 2 (x) for approximating the Peierls energy [START_REF] Hirth | Theory of dislocations[END_REF]. Introducing our calculated maximum Peierls energy in this simple model, we obtain a correct estimation of our computed energy curves, in particular for the D' path.

The Peierls stresses can be accurately obtained as the extrema of the derivative of the spline-fitted energy variations (Figure 3), divided by b 2 , b being the Burgers vector. For the D' path, the Peierls stress is 2.47 eV Å-3 , in excellent agreement with the value 2.6 ± 0.2 eV Å-3 , obtained from DFT calculations of sheared systems combined with boundary corrections [12]. For the D" path, the calculated value is 4.05 eV Å-3 . Obviously, comparing energy barriers and Peierls stresses shows that dislocation displacement would be easiest along the D' path, so that the dislocation remains in the shuffle plane. Shuffle-glide transition by dislocation cross-slip along the D" path is then unlikely. This is also in agreement with previous calculations showing that even if a shear stress favoring the D" displacement is applied on the system, the dislocation is always displaced along the D' path [12]. Figure 3 also represents variation of the core width ∆ as a function of the dislocation displacement in the lattice. In both cases D' and D", the core is widened, with a maximum increase of about 20%, corresponding approximately to the maximum Peierls energy. The D" curve shows a local minimum compared to the D' curve. A possible explanation is related to the weak stability of the transition configuration along the D" path, for which the dislocation core is slightly narrowed, whereas for D', the transition configuration is a true saddle point.

Concluding discussion

Our approach combining NEB and first principles DFT calculations allows to determine the energy of a dislocation translated in a crystalline lattice, as well as the associated Peierls stress, without the need for any corrections. In particular, the calculated Peierls stress for a screw dislocation in silicon is in excellent agreement with published results [12]. In addition to the Peierls energy determination, our approach has several other advantages compared to previous models. In particular, we have access to the dislocation structure during its displacement. Another advantage is the possibility of easily investigating several pos- sible paths, such as the D" path corresponding to a shuffle-glide cross-slip. We have shown recently that the latter could not be obtained by simply stressing the system [12]. Also, it has been shown recently that the more stable structure of a non-dissociated screw dislocation in the glide set exhibits a 2 × 1 reconstruction along the dislocation line [33]. Using our approach, it should be possible to investigate a transition between the single-period shuffle to the double-period glide screw dislocation, provided that the initial system dimension along the dislocation line is 2b, and that the reconstruction path is simple enough to be obtained with a small number of images in the NEB band.

In our approach, what is really calculated is the coherent displacement of a periodic arrangement of dislocations in a crystalline lattice. One may wonder whether this situation is equivalent to the displacement of a single dislocation. Obviously, even if the distance between dislocations remains constant, the core structure, and then the dislocation core field, is varying during the displacement. So the interaction between dislocations is not constant during the displacement. However, this effect is expected to be negligible and lower than surface-dislocation interactions present in fixed boundary calculations, even in small cell. This is the same assumption done in standard calculations when comparing the stability of several dislocation core configurations, where a constant elastic interactions is assumed between dislocations, independently of the core. In order to get an estimation of this effect, we have performed dislocation calculations with classical potentials and periodic boundary conditions, comparing energy differences between two core configurations as a function of cell size. In all cases, we have found that for the cell size considered here, i.e. 12 × 12, the core field interaction energy is lower than 1% of core energy differences. The error is then in the range of standard first principles accuracy, and can be safely neglected.

Another question concerns the fact that during the NEB calculations the coherency of the collective displacement is retained. In fact, it is likely that it is energetically more favorable to first displace one dislocation in the dipole system, then the other. This is especially true in hard materials with covalent bonds for which a strong interaction is expected between close dislocations. However, we usually found in our calculations for silicon that the dislocations were coherently displaced. The key explanation lies in the the fact that a relaxation algorithm such as steepest descent or the velocity Verlet algorithm with damped dynamics [START_REF] Jónsson | Classical and Quantum Dynamics in Condensed Phase Simulations[END_REF] brings the NEB to the minimum energy path that is nearest to the initial guess even though this may not be the global minimum energy path. While such convergence to the nearest local minimum rather than the global minimum is often a problem when dealing with systems represented by a complex energy surface, it is an advantage here because one can select out the right minimum energy path by choosing the initial path correctly. In order to keep the coherency, it is important that initial and final configurations in the band are well relaxed, with symmetric dislocation cores. We found that if this condition is not fulfilled, a non-coherent displacement could occur. It is also better to use systems including two dislocations instead of four, since it is easier to keep the coherency in the former case. In NEB calculations, we used standard spring constants (i.e. 5.0 eV Å-2 ). It is possible that using slightly stiffer springs would also help for keeping coherent displacements.

In conclusion, we propose a new method for investigating the mobility of dislocations using periodic boundary conditions and first principles accuracy. With this method, the Peierls energy and stress, and the dislocation structure during the displacement could be investigated, as well as high energy barriers, out of reach of standard calculations. In principle, the approach could be used for all kind of dislocations and for all materials, as long as dislocations in the computational cell are stable towards each other.

3 Figure 2 .

 32 Figure 2. (color online) Ball-and-stick representation of the zinc-blende structure (( 101) plane). The path D' (D") is defined by a dislocation displacement from A to A' (A to A"). Dashed lines show bonds linking atoms used for calculating the disregistry due to the dislocation core. In particular, atoms located in dislocation cores for saddle configurations are marked by numbers (1 and 2 for D', and 2 and 3 for D").

  , P. BEAUCHAMP and H. J ÓNSSON

Figure 3 .

 3 Figure 3. (color online) Energy (in eV/Burgers vector, top graph), stress (derivative of the plotted energy curves, middle graph), and core width variation (bottom graph) as a function of the dislocation core position in the lattice for paths D' (full lines) and D" (dashed lines).

Figure 4 .

 4 Figure 4. (color online) Differential displacement maps corresponding to screw dislocation displacements along D' (left) and D" (right), between A and the saddle configurations. The arrows are proportional to the out-of-plane [ 101] shifts between neighbouring atoms introduced by the dislocation.

  Dislocation mobility using NEB and DFT calculations

http://mc.manuscriptcentral.com/pm-pml Philosophical Magazine & Philosophical Magazine Letters

Acknowledgements

The authors are grateful to Dr. Ladislas Kubin for a critical reading of the manuscript. One of us (L.P.) thanks the Poitou-Charentes region and the University of Iceland Research Fund for financial support during his stay in Reykjavik.