N Lermé

F Malgouyres

Simultaneous Segmentation and Filtering via Reduced Graph Cuts

Keywords: graph cuts, reduction, image segmentation, filtering

Recently, optimization with graph cuts became very attractive but generally remains limited to small-scale problems due to the large memory requirement of graphs, even when restricted to binary variables. Unlike previous heuristics which generally fail to fully capture details, [8] proposes another band-based method for reducing these graphs in image segmentation. This method provides small graphs while preserving thin structures but do not offer low memory usage when the amount of regularization is large. This is typically the case when images are corrupted by an impulsive noise. In this paper, we overcome this situation by embedding a new parameter in this method to both further reducing graphs and filtering the segmentation. This parameter avoids any post-processing steps, appears to be generally less sensitive to noise variations and offers a good robustness against noise. We also provide an empirical way to automatically tune this parameter and illustrate its behavior for segmenting grayscale and color images.

Motivation and scope

Graph cuts have become increasingly popular due to their ability to efficiently compute the Maximum A Posteriori of Markov Random Fields (MRF). This popularity is notably driven by the introduction of a fast maximum-flow (max-flow) algorithm [START_REF] Boykov | An experimental comparison of min-cut/maxflow algorithms for energy minimization in vision[END_REF] making near real-time performance possible for solving numerous labeling problems such as denoising, image segmentation, stereovision, etc.

In parallel, technological advances in image acquisition have both increased the amount and the diversity of data to process. As an illustration, in the satellite SPOT-5 launched by Arianespace in 2002, the embedded high resolution sensors can capture multispectral and panchromatic images of about 1GB.

Processing this type of data amounts to solve large scale optimization problems. In the image segmentation context, almost all graph cuts-based methods are impractical to solve such problems due to the memory requirements for storing the underlying graphs. To overcome this situation, some amount of work has been done in this direction and a number of heuristics [START_REF] Li | Lazy Snapping[END_REF][START_REF] Lombaert | A multilevel banded graph cuts method for fast image segmentation[END_REF][START_REF] Sinop | Accurate banded graph cut segmentation of thin structures using laplacian pyramids[END_REF][START_REF] Kohli | Uncertainty driven multi-scale energy optimization[END_REF] and exact methods [START_REF] Lempitsky | Global optimization for shape fitting[END_REF][START_REF] Delong | A scalable graph-cut algorithm for N-D grids[END_REF][START_REF] Strandmark | Parallel and distributed graph cuts by dual decomposition[END_REF] have been proposed in recent years.

To our best knowledge, this problem seems to be first addressed in [START_REF] Li | Lazy Snapping[END_REF] where the underlying graph is built upon a pre-segmentation. Although this approach greatly reduce the computational burden of graph cuts, the results strongly depend on the algorithm used for computing the pre-segmentation. Also, better results are obtained when over-segmentation occurs, losing in this way the main benefit of such a reduction.

Others have also reported band-based methods [START_REF] Lombaert | A multilevel banded graph cuts method for fast image segmentation[END_REF][START_REF] Sinop | Accurate banded graph cut segmentation of thin structures using laplacian pyramids[END_REF][START_REF] Kohli | Uncertainty driven multi-scale energy optimization[END_REF]. A low-resolution of the image is first segmented and the solution is propagated to the finer level by only building the graph in a narrow band surrounding the interpolated foreground/background interface at that resolution. While such an approach drastically reduce time and memory consumption, it is limited to segment roundish objects. This problem is notably reduced in [START_REF] Sinop | Accurate banded graph cut segmentation of thin structures using laplacian pyramids[END_REF] but still present for low-contrasted details. In [START_REF] Kohli | Uncertainty driven multi-scale energy optimization[END_REF], finer bands are obtained using an uncertainty measure associated to each pixel.

Exact methods have been also investigated [START_REF] Lempitsky | Global optimization for shape fitting[END_REF][START_REF] Delong | A scalable graph-cut algorithm for N-D grids[END_REF][START_REF] Strandmark | Parallel and distributed graph cuts by dual decomposition[END_REF]. In [START_REF] Lempitsky | Global optimization for shape fitting[END_REF], binary energy functions are minimized for the shape fitting problem with graph cuts in a narrow band while ensuring the optimality on the solution. One makes a band evolve around the object to delineate by expanding it when the minimum-cut (min-cut) touches its boundary. This process is iterated until the band no longer evolves. Although the algorithm generally converges in few iterations toward the optimal solution, an initialization is required and no bound on the band size is given.

A parallel max-flow algorithm yielding a near-linear speedup with the number of processors is described in [START_REF] Delong | A scalable graph-cut algorithm for N-D grids[END_REF]. Nevertheless, the algorithm is relatively sensitive to the available amount of physical memory and remains less efficient on small graphs.

The approach used in [START_REF] Strandmark | Parallel and distributed graph cuts by dual decomposition[END_REF] is different: instead of reducing the graphs, the problem is decomposed into optimizable sub-problems, solved independently and updated according to the results of the adjacent problems. This process is iterated until convergence and optimality is guaranteed by Lagrangian decomposition.

Finally, another band-based method called Reduced Graph Cuts (RGC) was proposed for reducing graphs in binary image segmentation [START_REF] Lermé | Reducing graphs in graph cut segmentation[END_REF]. The graph is progressively built by only adding nodes which locally satisfy a condition. In the manner of [START_REF] Sinop | Accurate banded graph cut segmentation of thin structures using laplacian pyramids[END_REF][START_REF] Kohli | Uncertainty driven multi-scale energy optimization[END_REF], the graph nodes are typically located in a narrow band surrounding the object edges to segment. Empirically, the authors show in [START_REF] Lermé | Reducing graphs in graph cut segmentation[END_REF] that the solutions obtained with and without reduction are identical and the time for reducing the graph is even compensated by the time for computing the min-cut in the reduced graph when the regularization is of moderate level.

The rest of this paper is organized as follows. We first briefly review the graph cuts framework in Section 2 as well as the band-based strategy of [START_REF] Lermé | Reducing graphs in graph cut segmentation[END_REF] in Section 3 for reducing graphs. Afterwards, a new parameter is introduced in Section 4 for further reducing the graphs and removing small undesired segments in the segmentation due to noise. The sensitivity of this parameter as well as its robustness against noise are also evaluated through experiments for segmenting grayscale and color images.

Preliminaries

Consider a multi-channels image I : P ⊂ Z d → [0, 1] c (c > 0) as a function, mapping each pixel p ∈ P to a vector I p ∈ [0, 1] c 1 . We define a binary segmentation as an application u affecting to each pixel p ∈ P either 0 (background) or 1 (object) and we write u ∈ {0, 1} P . A popular strategy to segment I is to minimize a MRF of the form [START_REF] Boykov | Interactive graph cuts for optimal boundary and region segmentation of objects in N-D images[END_REF]:

E(u) = β p∈P E p (u p ) + (p,q)∈N E p,q (u p , u q ), (1)
among u ∈ {0, 1} P and for a fixed parameter β ∈ R + . The neighborhood system N ⊂ (P × P) is a subset of all pixel pairs. In the sequel, "connectivity 0" will denote 4 and 6 neighbors in 2D and 3D images while "connectivity 1" will denote 8 and 26 neighbors for the same images. In [START_REF] Bouman | Cluster: An unsupervised algorithm for modeling Gaussian mixtures[END_REF], the data term E p (.) is defined as the negative log-likelihood of a label being assigned to pixel p and is computed from its color and the appearance models of the object and background seeds 2 :

E p (1) = -log P(I p |p ∈ O) E p (0) = -log P(I p |p ∈ B) (2)
For any pixels pair (p, q) ∈ N , the corresponding smoothness term in ( 1) is defined as a contrast-sensitive Ising model:

E p,q (u p , u q ) = 0 if u p = u q , 1 p-q 2 exp - Ip-Iq 2 2 2σ 2 otherwise, (3)
where . 2 is the Euclidean norm (either in R d or R c ) and σ is a free parameter generally related to noise acquisition. As an illustration, when pixels p and q belong to a uniform area, we have I p -I q 2 < σ. This implies a large cost in the exponential and discourages any cut between pixels p and q. Conversely, when these pixels are on both sides of a contour, we have I p -I q 2 > σ. This encourages any cut between pixels p and q due to low value of the exponential.

When the smoothness terms are submodular [START_REF] Kolmogorov | What energy functions can be minimized via graph cuts[END_REF], the minimizer of (1) can be efficiently obtained by computing a min-cut in a weighted digraph G = (V, E) with a set of nodes V = P ∪ {s, t}, a set of edges E ⊂ (V × V) and edge capacities c : (V × V) → R + . The terminal nodes s and t are called the source and the sink, respectively. The set of edges E is split into two disjoint sets E n and E t denoting respectively n-links (edges linking two nodes of P) and t-links (edges linking a node of P to the terminal s or t). Once the min-cut is computed in G, we set u p = 1 if a node p is connected to the source s and u p = 0 if p is connected to the sink t.

1 Usually, P corresponds to a rectangle. 2 In this setting, the distributions are estimated using a Gaussian Mixtures Model.

The number of Gaussians is automatically computed using a statistical criterion [START_REF] Bouman | Cluster: An unsupervised algorithm for modeling Gaussian mixtures[END_REF].

Reduction

As explained before, the memory consumption of graph cuts for segmenting highresolution data can be very large. As an illustration, the max-flow algorithm of [START_REF] Boykov | An experimental comparison of min-cut/maxflow algorithms for energy minimization in vision[END_REF] v3.01 used in the experiments of Section 4, allocates 25 P + 16 E n bytes3 . For a fixed amount of RAM, one clearly see that the maximum image size quickly decreases as the dimensionality d of P increases. As shown in [START_REF] Lermé | Reducing graphs in graph cut segmentation[END_REF], most of the nodes in the graph are however useless during the max-flow computation since they are not traversed by any flow. Ideally, one would like to extract the smallest possible graph G = (V , E ) from G = (V, E) while keeping the max-flow value f * in G identical or very close to the max-flow value f * in G. This corresponds to an ideal optimization problem which we will not try to solve since the method for determining G also needs to be (very) fast.

Let us first introduce some terminology before reviewing the method of [START_REF] Lermé | Reducing graphs in graph cut segmentation[END_REF] for building G . For the sake of clarity, the same notations are used as in [START_REF] Lermé | Reducing graphs in graph cut segmentation[END_REF]. In accordance with the construction given in [START_REF] Kolmogorov | What energy functions can be minimized via graph cuts[END_REF], we consider (without loss of generality) that a node is connected to at most one terminal

(s, p) ∈ E t ⇒ (p, t) ∈ E t , ∀p ∈ P. (4)
We also summarize t-links capacities by

c(p) = c(s, p) -c(p, t), ∀p ∈ P. (5)
For any B ⊂ Z d4 and a node p ∈ P, we denote by B p the set translation of B at p

B p = {q + p | q ∈ (B ∩ P)}. (6)
For Z ⊂ P and B ⊂ Z d , we denote by Z B the dilation of Z by B as

Z B = {p + q | q ∈ (B ∩ P), p ∈ Z} = p∈Z B p . (7)
From here, the idea developed in [START_REF] Lermé | Reducing graphs in graph cut segmentation[END_REF] for building G is to remove from the nodes of G any Z ⊂ P where all nodes are linked to s (resp. to t) and such that all the flow that might get in (resp. out of) the region Z B does so by traversing its boundary and can be absorbed (resp. provided) by the band Z B \ Z. Building such Z is done by testing each individual pixel p ∈ P. In the manner of [START_REF] Sinop | Accurate banded graph cut segmentation of thin structures using laplacian pyramids[END_REF][START_REF] Kohli | Uncertainty driven multi-scale energy optimization[END_REF], the remaining nodes are therefore located in a narrow band surrounding the object edges to segment. In practice, the authors of [START_REF] Lermé | Reducing graphs in graph cut segmentation[END_REF] use a more conservative test by testing each node p ∈ Z in a square window B of size (2r + 1) d centered in p:

   either ∀q ∈ B p , c(q) ≥ +δ r , or ∀q ∈ B p , c(q) ≤ -δ r . (8)
where δ r = P (B) (2r+1) d -1 and P (B) is defined as P (B) = max( {(p, q) : p ∈ B, q ∈ B and (p, q) ∈ N }, {(q, p) : p ∈ B, q ∈ B and (q, p) ∈ N }).

In words, for any node p ∈ Z satisfying the first (resp. second) condition of ( 8), all its neighbors q ∈ B q are only linked to s (resp. t) and the flow that might get in (resp. out) through t-links in B p \ {p} suffices to saturate the n-links going out (resp. in) B p . Thus, p becomes useless and need not to be added to G . An algorithm of complexity O(1) (i.e. independent of r) is also mentioned in [START_REF] Lermé | Reducing graphs in graph cut segmentation[END_REF] for computing [START_REF] Lermé | Reducing graphs in graph cut segmentation[END_REF]. Additionally, the extra memory storage required by this algorithm is a table of dimensionality (d -1) and is therefore negligible over the image and the graph size. A key point of [START_REF] Lermé | Reducing graphs in graph cut segmentation[END_REF] is that the pixel error between segmentations obtained with and without reduction remains extremely low, hence preserving thin structures which are ubiquitous in some applications.

The experiments presented in [START_REF] Lermé | Reducing graphs in graph cut segmentation[END_REF] confirm the intuitive dependence between the size of the reduced graph G and the model parameters. Indeed, when minimizing (1) by graph cuts, the t-links capacities are all multiplied by β. It is therefore straightforward to observe that the test ( 8) is harder to satisfy as β decreases. In such a situation, we need a larger window radius for decreasing δ r in order to reduce the size of the reduced graph G . This results in wider bands around the object contours. Conversely, when β is large, we can afford a large δ r and therefore a small window radius to decrease the size of the reduced graph G . An ideal situation therefore consists of large area of nodes connected to the same terminals separated by rough borders. As opposite, a less favorable situation occurs when these area consist of nodes connected to different terminals. This is typically the case when dealing with noisy images. In the next section, we embed a new parameter in [START_REF] Lermé | Reducing graphs in graph cut segmentation[END_REF] to both further reducing G and filtering the segmentation while keeping β large.

Simultaneous segmentation and filtering

A naive approach to filter the segmentation is to apply morphological operators (e.g. opening or closing). Nonetheless, when the amont of noise is large, such an approach fail during the reduction since a lot of nodes would be added to the reduced graph. Another approach would consist in denoising the image first and then applying [START_REF] Lermé | Reducing graphs in graph cut segmentation[END_REF]. This would lead to unsatisfactory results in the case for instance of echographic images since contours of objects would be over-smoothed.

Another way to filter the segmentation is to relax ( 8) is to allow some nodes in B p to fail complying the test. The proportion of nodes satisfying (8) can be controlled by a parameter η ∈ [0, 1]. As η decreases, the test (8) becomes easier to satisfy since a larger proportion of nodes can be connected to opposite terminals. Embedding η in (8) leads to

   either {q ∈ B p | c(q) ≥ +δ r } ≥ η B p , or {q ∈ B p | c(q) ≤ -δ r } ≥ η B p . (10)

Further reducing graphs

The parameter η can be used for decreasing the memory consumption of graph cuts. The Figure 3 illustrates how far the test (8) can be relaxed for further reducing graphs while getting nearly the same segmentation. In this experiment, the segmentation as well as the reduced graph are shown for segmenting a 2D noisy image. Since the test ( 10) is easier to satisfy as η decreases, the reduced graph G becomes thicker around the object contours.

Automatic tuning of η

Lower bound For a fixed window radius, notice first that the value of η must be large enough to not increase the number of components in the reduced graph G (see Figure 2). Indeed, below some value (denoted by η min ), the reduced graph G is split into multiple pieces in areas with high-curvature and the min-cut is no longer ensured of being fully embedded into G . This implies that some voxels could be wrongly labeled in the segmentation.

The Figure 1 illustrates a situation where η min can be easily computed with an image consisting of two highly-contrasted areas. Using [START_REF] Lombaert | A multilevel banded graph cuts method for fast image segmentation[END_REF] with a square window of radius r and η = 1, the reduced graph G corresponds to a thin band of size 2r. An easy under-estimation of η min is obtained by imposing that η min permits to segment these two contrasted areas. In order to do so, we want the test [START_REF] Lombaert | A multilevel banded graph cuts method for fast image segmentation[END_REF] to be false for any pixel p at the boundary between these areas. For such a pixel, we have (assuming e.g. that c(p) ≥ +δ r )

{q ∈ B p | c(q) ≥ +δ r } = (r + 1)(2r + 1) d-1 . ( 11
)
As a consequence, if

η ≤ (r + 1)(2r + 1) d-1 (2r + 1) d , ( 12
)
the node p does not belong to the reduced graph G . Since we want to avoid the situation, we must therefore have

η > (r+1)(2r+1) d-1 (2r+1) d = 1 -r 2r+1 = η min . (13)
Remark that [START_REF] Strandmark | Parallel and distributed graph cuts by dual decomposition[END_REF] does not depend on the dimensionality d of P. By observing [START_REF] Strandmark | Parallel and distributed graph cuts by dual decomposition[END_REF], it is straightforward to see that, as the window radius r tends to infinity, the proportion of nodes allowed to be connected to opposite terminals tends to 1 2 . In practice, we also observed that ( 13) is less accurate in connectivity 0 than in connectivity 1 (see Figure 2).

Upper bound For a fixed window radius r and a positive amount of noise ξ, one can observe in Figure 3 that there exists a value of the parameter η for which most of the nodes in noisy regions are removed from the graph G, leading to a diminution of the size of the reduced graph G .

The purpose of this paragraph is to identify, from a statistical point of view, a reliable value of the parameter η for which all nodes of P are very likely to be removed from G. For a fixed amount of noise ξ in the image I, we therefore want to find an upper bound on η by finding the maximum value of η in such a way that we control the proportion of nodes corresponding to noisy pixels in homogeneous areas.

Consider a noisy constant image I with a noise generated by a Bernoulli distribution of parameter ξ ∈]0, 1[, corresponding to the amount of noise in I5 . The two cases where ξ = 0 and ξ = 1 are trivial and are not considered in our analysis. Assume now that the graph G is defined as in Section 3 where the nodes corresponding to noise-free pixels are connected to the sink t with a capacity c(q) ≤ -δ r and the nodes corresponding to noisy pixels have a capacity c(q) > -δ r .

First, let X be a discrete random variable counting degraded pixels in a square window B of size n = (2r + 1) d in the image I. Then, the probability that at least k pixels are corrupted in B is given by

P(X > k) = n i=k+1 n i ξ i (1 -ξ) n-i , (14)
where

n i = n! i!(n-i)! .
For a fixed window radius r, it is straightforward to see that ( 14) is decreasing in k and tends to ξ n if we impose that ξ ∈ ]0, 1[. According to the test [START_REF] Lombaert | A multilevel banded graph cuts method for fast image segmentation[END_REF] and the hypothesis on G: a node p ∈ P can be removed from G if and only if

{q ∈ B p | c(q) ≤ -δ r } = {q ∈ B p | q is noise-free} ≥ ηn (15)
Moreover, we assumed

{q ∈ B p | q is noise-free} ∼ (n -X). (16)
Therefore, we have

P(p is not removed) = P((n -X) < ηn) = P(X > (1 -η)n). (17)
Fixing a proportion ε 0 of wrongly constructed nodes, we choose

η + = max {η ∈ [0, 1] | P(X > (1 -η)n) ≥ ε}, (18)
Considering the lower bound η min defined in (13), we set

η max = max {η min , η + }. (19)
Combining the definitions of the lower and upper bounds (see ( 13) and ( 19)), it now becomes easy to get an estimation of the parameter η * for a fixed window radius by setting

η * = η min + η max 2 . ( 20
)
Let us now analyze the joint behavior of the lower and the upper bounds. When the amount of noise ξ is fixed, one can easily observe that the gap

∆η = (η max -η min ), (21)
grows as the window radius r increases. Indeed, we have previously seen that the lower bound η min tends to 1 2 as the window radius r increases (see [START_REF] Strandmark | Parallel and distributed graph cuts by dual decomposition[END_REF]). The previous observation is also due to the fact that the upper bound η max grows as the window radius r increases.

Similarly, when the window radius r is fixed, remark that (21) decreases when the amount of noise ξ increases. This situation is consistent because η min remains the same but η max tends to 1 2 since it is more likely that the number of degraded pixels increase in the window B. Increasing the window radius r can compensate the augmentation of the amount of noise only up to ξ = 0.5. Finally, we empirically found that setting ε = 0.05 gives best estimate of η max in (18).

Filtering

The parameter η can also serves to filter the segmentation. This behavior is illustrated in Figure 4 for segmenting a 3D noisy image acquired from a confocal microscope. White spots correspond to cell nuclei in a mouse cerebellum. Observe how far the filtering acts for small values of η: small regions in the reduced graph G as well as in the segmentation are progressively removed as η decreases.

The robustness (see Figure 6 and 5) and sensitivity to noise of the parameter η are now (see Figure 7) are now analyzed. Let us describe the experimental setup. The experiment consists in segmenting four grayscale and five 2D color images in connectivity 1 with an increasing noise level ranging from 4 to 48%. For each image, we compute a reference segmentation on the noise-free image by placing the seeds by hand. We set β = +∞ and automatically estimate the σ parameter as explained in [START_REF] Rother | GrabCut": Interactive foreground extraction using iterated graph cuts[END_REF]. Then, for each impulsive noise level, we select the segmentation maximizing the Dice Similarity Coefficient (DSC) between the reference image and all segmentations obtained through a fixed range of window radii and η values. We choose window radii r from 1 to 12 and eight linearly spaced values of η from η min (w.r.t. r) to 1. Then, each segmentation is computed using the same seeds as those used for the computing the reference segmentation. Again, the σ parameter is automatically estimated as in [START_REF] Rother | GrabCut": Interactive foreground extraction using iterated graph cuts[END_REF].

As shown in Figure 6, for an impulsive noise level up to 45%, the parameter η appears to be reasonably robust with a DSC always greater than 94% for all images, except for the image "rice". However, such high and stable noise robustness can only be reached by increasing the amount of seeds (see Figure 6). The reason why the algorithm behaves poorly on the "rice" image is the following. As said earlier, r must be large enough when ξ increases for removing a maximum number of segments due to noise. This implies wider bands in G around the object contours. However, the object contours further oscillate as ξ increases. Another reason is due to the proximity of the objects to segment. As an illustration, consider two circles over a uniform background, separated by a distance d 0 > 0. We clearly see that the test [START_REF] Lombaert | A multilevel banded graph cuts method for fast image segmentation[END_REF] becomes more and more difficult to satisfy when the window radius r increases. When (2r + 1) ≥ d 0 , the reduced graphs of both circles fuse into one component. This is typically the case for the image "rice" because it consists of small assembled rice grains near from each other. Finally, the Figure 7 also illustrates that the parameter η is not very sensitive to the variations of r and η. The DSC does not vary much with respect to noise, except for the image "rice". This exception can be explained for the same reasons as before.

Conclusion

In this paper, we have presented a new parameter to embed in [START_REF] Lermé | Reducing graphs in graph cut segmentation[END_REF] for simultaneously reduce graphs and filter segmentations with the same computational complexity. We have also described an original manner to automatically tune it with the window radius parameter. In the proposed experiments, this new parameter generally appears to be less sensitive to noise but only for a limited amount, typically less than 50%. To overcome this situation, we could for instance inspect larger neighborhoords to further discriminate signal from noise. Fig. 7: Sensitivity of η for segmenting the images in Figure 6 with an impulsive noise level of 36%. The seeds and model parameters are the same than those used in Figure 6.

Fig. 1 :Fig. 2 :

 12 Fig. 1: Toy example for computing the lower bound ηmin.

Fig. 3 :Fig. 4 :

 34 Fig. 3: Memory gain when segmenting a 2D synthetic image corrupted by 10% of impulsive noise (left). Top row shows the nodes of the reduced graph in light gray while bottom row shows the corresponding segmentation. In this experiment, we set r = 3 and use connectivity 1.

Fig. 5 :

 5 Fig.5: Qualitative analysis of the robustness to noise for segmenting the images "f117" (left-most column), "pyramid" (left column), "sunflower" (right column) and "flamingo" (right-most column) in with a fixed impulsive noise level of 36%. Seeds and model parameters are the same than those used in Figure6(top row).

Fig. 6 :

 6 Fig. 6: Quantitative analysis of the robustness to noise for segmenting in connectivity 1 four 2D grayscale images (top-most curves in the list) and five 2D color images with an impulsive noise level ranging from 4 to 48%.

The operator ' ' stands for the cardinality of a set.

In practice, B is a square centered at the origin.

Histogram-based techniques can be for instance used to estimate ξ.


Karim Benouareth

Pascal Tristant

Cedric Jaoul

Christophe Le Niniven

Corinne Nouveau

Christelle Dublanche-Tixier

Abderrahman Bouabellou

Study of the interaction between a Zirconium thin film and an EN C100 steel substrate: temperature effect

Keywords: Zirconium Carbide, Sputtering, heat treatment, Diffusion, Hardness

Zirconium thin films were grown on high carbon steel substrates EN C100 (1 %wt. of carbon) by RF magnetron sputtering. In order to study the reactivity of the film/substrate system as a function of the temperature, one hour vacuum annealing was carried out for different temperatures between 600°C and 1100°C. The films were then analyzed by X-ray diffraction, scanning electron microscopy, glow discharge optical emission spectroscopy (GDOES) and nanoindentation.

The obtained results showed a progressive transformation of zirconium film to zirconium carbide.

Carbon atoms diffusion from substrate toward the film induced this transformation. The sample annealed at 900°C exhibited the best mechanicals properties (H =17 GPa and E = 220 GPa).

Samples treated at higher temperature were affected by oxidation and high microporosity. Even if the conversion is uncomplete, annealing significantly promotes adhesion of the film on the substrate.

Introduction

Nowadays, the materials surface properties have more and more important roles in many industrial applications. For example, for metallic parts of tools (fabricated with steel commonly) undergoing mechanical solicitations or those submitted to aggressive environments, it is possible to increase their resistance to corrosion or wear by modifying the material's surface. One of most applicable techniques is to coat the surface with a film. The metallic matrix provides the structural properties whereas the film brings the surface properties. The thin film (thickness less than 10 µm) can be enough to increase the life expectancy of these elements. However, a good adhesion of the coating to the substrate is essential for a successful application [START_REF] Oyama | The Chemistry of Transition Metal Carbides and Nitrides, Blackie A & P[END_REF]. Hard coatings fabricated by habitual physical vapor deposition have a poor film quality due to porous microstructure and weak adhesion [START_REF] Navinsek | [END_REF]. Also, adhesion problems related to the different thermal expansion coefficients of the coating and substrate material were also observed [3]. Many researches were done to make hard coatings with high adhesion by using controlled arc plasma deposition [4], plasma focus device [5] or through deposition by combined magnetron deposition and ion implantation [6] and by thermoreactive diffusion deposition [7].

Relying on the process of the thermally activated atomic interdiffusion between the chemical elements at the coating/substrate interface, we try to manufacture, in the current work, a zirconium carbide hard coating with good adhesion and mechanical properties by conversion from a zirconium film deposited on high carbon steel. This transformation is likely to occur based on the role of carbon diffusing from the substrate into the film through a post-deposition annealing of the coating/substrate system at different temperatures (600°C -1100°C). The zirconium, which is in the IV group of transition elements, is chosen because its alloys have excellent properties such as a low thermal expansion, and they are strongly heat resistant [8]. Zirconium is widely employed in industrial devices being used in aerospace, optoelectronics, and chemistry [9]. Its high melting point and low thermal neutrons capture cross section allows it to be used in nuclear reactor installations [8,10]. Zirconium is applied as an alloy or ceramic in medical implants because it is biocompatible in the human body [11]. The carbide form, ZrC, is widely employed in industry because of its good properties (thermal, chemical, optoelectronical), and excellent corrosion and wear resistances [10,[12][13][14].

Experimental

Films Preparation

Pure Zr thin films of 3.5 µm thickness were deposited by RF magnetron sputtering. The distance between the target of pure Zr and the substrate holder was fixed at 100 mm. The base vacuum of the chamber was kept at 10 -5 Pa. During deposition, the working pressure was fixed at 0.3 Pa with a carrier gas of Argon (99.99 % purity). The deposition duration is 250 minutes, the RF power is 400 W with a target polarization of -900 V and a substrate temperature of 200 °C.

High carbon steel (EN C100 or AISI 1095), 1 %wt. of carbon, in the spheoridized state was chosen as the substrate. The disk shape substrates, of 30 mm in diameter and 4 mm in thickness, are firstly polished using a BUEHLER EcoMet 250 polishing machine. To obtain an average roughness Ra of about 20 nm, a precision polishing was carried by 2400 SiC papers. Substrates were then ultrasonically cleaned for 10 min in ethanol, followed by 10 min in acetone. Prior to deposition, the surface of the substrate was cleaned 10 min by high purity (99.99 %) Argon plasma at a pressure (1 Pa). The ion bombardment induced by the polarization of the samples (-954 V) allowed to remove impurities and native oxides at the surface.

Post-deposition treatment

Zr/C100 specimens were vacuum annealed for one hour at temperatures ranging from 600 to 1100 °C, in a tubular furnace. The pressure of 10 -2 Pa was ensured by a diffusion pump. Heating rate was set at 10 °C/min, while that of cooling was 20°C/min.

Characterization and analysis

The structure and phase composition of the coatings were assessed by X-ray diffraction (XRD) using a conventional Siemens D5000 diffractometer with Cu Kα radiation operating in the Bragg-Brentano configuration. Measurements were performed at room temperature after thermal treatments. Surface and cross section morphology were examined with a Scanning Electron Microscope (SEM). To investigate the phenomena occurring at the film/substrate interface, glow discharge optical emission spectroscopy (GDOES) experiments were performed using a Horiba Jobin Yvon RF GD Profiler equipped with a 4 mm diameter anode and operating at a typical radio frequency discharge pressure of 650 Pa and a power of 20 W.

Hardness and elastic modulus were determined by load-displacement data obtained using a XP nanoindenter model MTS Systems equipped with Berkovich tip, by means of 40 indents per sample for the surface nanohardness. The depth nanohardness measurements were performed on specimen's cross-sections (perpendicularly to the films surface) going from the film surface to the substrate where each indent is spaced by 0.7 μm from the other. The nanohardness value for each depth position is obtained by averaging 4 indents.

Results and discussion

Morphology

The Zr/C100 pristine sample surface shows a reflecting metallic gray appearance while samples treated at high temperatures have a dull blackish brown aspect. The surface of the deposited film has a smooth uniform morphology, Fig. 1-a, with a microstructure characterized by fine grains in nano-domes form, with diameters of a few tens of nanometers, but polishing scratches are still visible. This morphology is characteristic of a growth developed by agglomeration of atoms. The cross section observation of the same sample, Fig. 2-a, shows the existence of a columnar structure which is typical to the zone II according to Thornton's model. This zone is described by a homogeneous structure along the film thickness composed of straight columns, elongated perpendicularly to the substrate surface, from the bottom to the top of the film without intercolumnar voids [15]. In spite of the very low deposition temperature (200 ºC) compared to the melting point for Zr (1855 ºC), the observed morphology is denser than Thornton prediction. This densification is observed for films deposited by PVD under high energy conditions characterized by an intense ion bombardment, limited mobility conditions and high deposition rates [16,17].

The deposited film suffers from a poor adherence due to the low deposition temperature. This is highlighted in Fig. 2-a where the film is detached from the substrate after the cross section preparation.

After annealing at moderate temperatures, 600-800 °C, the film surface appears smoother with denser microstructure (not shown in this paper). The heat treatment leads to the growth and coalescence of grains, and the annihilation or the displacement to the grain boundaries of the structural defects (voids, gaps). The film adheres perfectly to the substrate due to diffusion that increases the film hooking to substrate. All these observations reflect a better crystallinity of the film after annealing. At higher temperatures, Fig. 1-c, the surface morphology becomes rough and porous with nodular bigger grains. The film surface is more homogeneous by the disappearance of polishing scratches observed for lower annealing temperature. Although the grain size increases after annealing, the grains are formed of clusters composed of small crystallites, specifically for annealing at 1100 °C (greatest magnification of the Fig. 1-b and1-c).

At 900 °C, from the SEM cross-sectional images, a different (darker) contrast zone of 580 nm thick appears in the film close to the interface (film/substrate), especially using backscattered electrons mode (Zone 1, Fig. 2-b). While the other part of the film still with the same aspect as those of lesser annealing temperature (Zone 2, Fig. 2-b). Another area with the same (darker) contrast appears near the film surface at 1000 °C and at the same time, the surface roughness increases. The film surface appears very rough and very porous (Zone 3, Fig 2-c). These two zones increase in thickness when the annealing temperature increases (Table 1). Between the tow peripherals dark zones, the film middle zone appears bright with a decreasing width by rising annealing temperature. These observations will be later discussed.

After annealing, the samples do not show any cracks on the surface or in the thickness for the different temperatures studied. These cracks, usually observed after annealing and cooling of the whole coating/substrate, are related to the mismatch of the heat expansion coefficients between the two components. In our case, the creation of an inter-facial connection zone between the coating and the substrate absorbs the mismatch. This is also connected to the film low grains size and the high concentration of grain boundaries having the effect of relaxing the stress, which prevents cracks propagation.

Qualitative analysis by glow discharge optical emission spectroscopy (GDOES)

The GDOES of unannealed Zr/C100 sample analysis, Fig. 3-a, shows that the raw film does not contain carbon. It is clear that signals of both oxygen and carbon are slightly higher at film surface, this comes from the inevitable contamination by impurities before performing analysis.

After annealing at 600 °C, on the carbon signal, a small hump develops at the interface coating/substrate (in the film side) indicating the beginning of its diffusion. By rising the annealing temperature, the film carbon signal gains in intensity and depth in this region (zone 1), Fig. 3-b, but it also increases in the middle region (zone 2), and reaches progressively, at higher temperatures, the surface region (zone 3) , Fig. 3-c , confirming that the diffusion of carbon atoms from substrate to the Zr film is thermally activated. At the same time, the intensity of zirconium signal decrease by increasing annealing temperature, which suggests its reaction with the carbon to form the carbide.

The interface zone is favoured by diffusion because it is close to carbon atoms source (i.e. substrate), for this reason we observe that this region (zone 1) receive always the greatest amount of carbon atoms. Another factor promoting the carbon diffusion is the film microstructure composed of columnar grains of nanometric size with grain boundaries elongated perpendicularly to film surface.

Each time we elevate the annealing temperature, the film uppermost surface is subject of light contamination by oxygen and carbon, which is testified by the increase of their signals in this region, (Zone 3), Fig. 3-b. However, it is clear that the contamination with carbon is more important in the uppermost surface at 1000 °C and especially 1100 °C, while the oxygen contamination progress in the film thickness and reach gradually the middle of the film (Zone 2), Fig 3-c. It seems that it is due to the higher reactivity of oxygen with zirconium than carbon. After annealing, the GDOES elements signals flatten confirming the interdiffusion and the reaction of the film with the substrate, thus logically the adhesion rises.

The superimposition of these results with those of SEM observations, show that the first darker contrast region (zone 1, Fig. 3-b and , Fig. 3-c) is due to the diffusion of carbon, while the second darker contrast region (zone 3, Fig. 3-c) is the imprint of the contamination by oxygen and carbon.

The bright contrast region in the middle (zone 2, Fig. 3-c) contains less interstitial elements (i.e. carbon and oxygen) than the two first regions. The carbon in this later zone may not be due to the carbon diffusion, as the carbon atoms cannot cross the middle zone and diffuses directly to the surface. We will confirm this conclusion in the next section. The source of carbon contamination may be due to the backstreaming of the oil from the diffusion pump that could decompose in the annealing furnace at high temperature.

XRD analysis

X-ray diffraction patterns of Zr/C100 samples as deposited and annealed at temperatures ranging from 600 to 1100 °C are presented in Fig. 4. For the pristine sample, only the hexagonal zirconium alpha phase is detected, as an additional peak attributed to the substrate (alpha iron). The diffraction peaks are wide, which is associated with a small crystallite size (grain fineness), and large microstrain due to the presence of structural defects. An anisotropic shape is observed that can be caused by stress, this also indicates that the crystallites have an anisotropy shape [18].

The zirconium X-rays diffraction pattern becomes narrower and more intense after annealing at 600 °C. This is explained by the increase in the crystallite size and an improvement in the film crystallinity and density: partial removal of structural defects, and a stress relaxation in the film.

From 700 °C, the zirconium peaks intensities begin to decrease and disappear completely at 1100 °C.

At 600 °C, in addition to the peak of zirconium and substrate, a new small peak appears, identified as the plane (111) of the FCC substoichiometric ZrC 0.6 zirconium carbide phase, which confirms the beginning of carbon atoms diffusing from substrate to the film. This diffusion leads to the formation of grains of the carbide phase at the interface of the film. Two phases, zirconium carbide and zirconium coexist between 600 and 1000 °C. Though the zirconium carbide phase is the minor phase at low temperature annealing, by raising temperature, it becomes the predominant phase at 900 °C and remains the only phase at 1100 °C. This reflects a gradual transformation of the zirconium film to zirconium carbide. Every time we raise the temperature, we stimulate a greater amount of carbon atoms to migrate to the film, which leads to the observed increase of the zirconium carbide peaks intensities. This is in good agreement with other works, where coatings of transition metal carbides have been successfully manufactured from transition metals thin films deposited on high carbon steel [19][20][21].

Typically, it is known that transition metals are highly reactive; they are not stable in chemical environments containing light elements. Since the zirconium carbide is an interstitial compound, the inclusion of carbon in zirconium lattice is produced by its incorporation into interstitial sites of the lattice. It has a much smaller radius compared to the zirconium atom and this is manifested by increased stability and moderate chemical reactivity of the system [22].

By rising the annealing temperature, the carbide diffraction peak shift to lower diffraction angles (towards the stoichiometric ZrC theoretical position); see for example (111) peak presented in Fig. 5. This suggests a cell expansion due to the incorporation of more carbon atoms in the zirconium lattice. From 900 °C, the carbide (111) peak position is between the theoretical position of ZrC 0.6 (2θ=33.186°, according to JCPDS card No. 65-9886) and that of the stoichiometric ZrC (2θ=33.040°, according to JCPDS card No. 35-0784). Even if the (111) peak of the two phases are very close, their overlap show that a mixture of ZrC 0.6 and ZrC phases coexist at high annealing temperature. The same conclusion has been reported by E. Grigore et al. [23]. The total volume of substoichiometric carbide grains in the film is more important than that of stoichiometric carbide, since the position of the (111) diffraction peak is always closer to the substoichiometric phase.

From 600 to 1100 °C, Fig. 4, two peaks from the substrate (the alpha phase of iron) are present with a nearly constant intensity, which means that all the film is analysed.

At 1100 °C, low intensity diffraction peaks are detected at low angles and are attributed to the zirconium oxide ZrO 2 monoclinic phase according to JCPDS card No. 04-4339, with the arrival of the main peak at the angle 28.2° as the (-111) plane in agreement with the work of Kato et al. [24].

The monoclinic phase is the stable zirconium oxide phase at room temperature up to 1170°C. Zirconium substoichiometric carbide is certainly subject to oxidation, which can be deduced from the heats of formation of ZrC (-202.0 kJ mol -1 ) and ZrO 2 (-1101.3 kJ mol -1 ). The reacting oxygen probably comes from the inevitable outgassing of the walls of the vacuum system for high annealing temperatures which provides steam oxygen in the vacuum chamber atmosphere. In the literature, the presence of zirconium oxide is still observed in zirconium films deposited by PVD whatever the vacuum used during the operation of the depositing [12,22,[26][27][28], and after vacuum annealing of bulk zirconium [29]. S.-W. Yeh et al. showed that the deposition under a vacuum of 6.7×10 -2 Pa, leads to the formation of zirconium oxide for a 350°C substrate temperature. At 450 °C, the cubic zirconium oxide phase is observed [25]. Even, small amounts of oxides are detected in the zirconium films after the subsequent annealing at 700°C for 45 min under a pressure of 5×10 -2 Pa, with the injection of a hydrogen gas flow [9]. This could be attributed to the higher affinity of zirconium for oxygen than that for carbon.

For the higher annealing temperature, the zirconium film was not completely converted to stoichiometric zirconium carbide, as the carbon signal is not uniform in the film (GDOES results). This can be explained by the deficiency of the diffusing flux of the carbon from the substrate to occupy all the octahedral sites in the zirconium lattice, and by the oxygen contamination at higher temperatures which could hinder the diffusion of carbon. An annealing treatment at 900°C for a longer time would be necessary to completely convert the zirconium film into zirconium carbide and to avoid the oxide phase formation.

It is difficult to detect oxycarbide phase formation by XRD because both ZrCO and ZrC have the same FCC structures with negligible difference in their lattice parameters [30]. In our case, the formation of ZrCO phase is favoured by the vacant octahedrals sites in ZrC 0.6 cell, which accommodate the oxygen atoms. Researchers have noted that oxygen could dissolve in ZrC, leading to oxycarbide formation in which carbon and oxygen incorporate randomly into the interstitial sites in the zirconium lattice [30][31][32][33][34]. A recent study shown that oxygen atoms are incorporated either in the ZrC lattice or in an amorphous phase at the grains boundaries [32].

By superposition of XRD results with those of SEM and GDOES, we can conclude that the different contrast zone in the interface is the imprint of carbon diffusion, since this zone appears for the region where the intensity of GDOES carbon signal becomes high. This is simultaneous to the predominance of the carbide phase in the X-ray diffraction spectra at 900 °C together with the stoichiometric carbide formation. Despite of the carbide formation between 600 and 800 °C, it was noted that there is no observation of different contrast zone, on the contrary to higher annealing temperature. Thus, the stoiciometric carbide, not the substoichiometric one, causes the darker contrast zone in the interface.

The other different contrast region (zone 3) on the surface is caused by oxygen diffusion from the surface. Because the apparition of this zone, from 1000°C, is simultaneous to the apparition of ZrO 2 peaks at high annealing temperatures, Fig. 4, together with the high contamination by oxygen observed by GDOES in the zone 3, Fig. 3-c , where the oxygen signal is more important than the carbon one. Since an area with light elements appears darker than the heavier ones, the middle film (bright zone) contains less light interstitial elements than external zones, in good agreement with GDOES analysis. As unreacted zirconium was not obviously present in the XRD patterns at high temperature annealing, the film middle zone is composed of ZrC 0.6 and ZrC x O y .

Cell parameters

The lattice constants were calculated according to the well-known relations:

1/ d 2 hkl = 4(h 2 +hk+k 2 )/3a 2 +l 2 /c 2 for the zirconium carbide phase (hexagonal structure), and the relation a = d hkl (h 2 +k 2 +l 2 ) ½ for the zirconium phase (cubic structure), where h, k and l are Miller indices.

We have chosen the (100) and (002) peaks for the calculation of the constants, a and c, of the hexagonal phase, respectively. The (111) peak is selected for the calculation of the cell parameter of the cubic phase.

The d hkl interplanar distance is defined by Bragg's law.

Both lattice parameters (a and c) of the zirconium hexagonal phase evaluated for the unannealed sample are higher than the bulk parameters, a = (3.254±0.008)Å, c = (5.176±0.012)Å, this corresponds to a larger volume lattice, this may be related to the incorporation of foreign atoms in the lattice, or to microstructural defects. It is known that the substrate temperature is the major factor that affects the final structure of a PVD coating during deposition operation. If it is low, there is not enough energy for the mobility of zirconium adatoms to rearrange on the surface [16,17].

Nevertheless, the ratio c/a is lower than the one of bulk zirconium. This could be caused by compressive stress in the perpendicular plane to the film surface (plane c) induced by plasma intense bombardment used during deposition.

Researchers have found, by varying substrate temperature from 25 to 600 ºC for zirconium films prepared by pulsed magnetron sputtering, that lattice parameters were always higher than the bulk lattice parameters which is attributed to the presence of compressive stress in the films [35]. The same interpretation has been reported in other works [36,37].

After annealing, the parameters a and c are reduced due to, stress relaxation, defects elimination and atoms best rearrangement, however they remain higher. For example, at 800°C they become: a = (3.261±0.007)Å and c = (5.184±0.011)Å, this result is due to the inclusion of carbon atoms, coming from substrate (and eventually oxygen contamination) in zirconium lattice. This insertion increases gradually as the annealing temperature rises.

When the temperature reaches 863 °C, the structure of zirconium changes from hexagonal Zr-α to to body-centered cubic Zr-β. The latter structure is not suitable to accommodate the important flow of carbon atoms from the substrate in its small octahedral sites [START_REF] Toth | Transition Metal Carbides and Nitrides[END_REF]. Therefore, to form carbide, the metal lattice must change to a compact structure (FCC or HCP), which provides larger octahedral sites for hosting carbon atoms; the compact structure being in turn stabilized by the interstitials atoms [START_REF] Toth | Transition Metal Carbides and Nitrides[END_REF][START_REF] Pierson | Handbook of Refractory Carbides and Nitrides, Properties, Characteristics, Processing and Applications[END_REF].

The calculated lattice parameter for the FCC zirconium carbide phase for the sample annealed at 800°C is 4.666±0.009 Å, which corresponds to the theoretical value of the carbide ZrC 0.6 (JCPDS file No. 65-9886). The formation of the substoichiometric phase means that the diffusing carbon atoms of the substrate do not fill the zirconium cells already occupied by a few atoms of zirconium, but diffuses to other grains and occupy zirconium lattices that have all, or almost, of their octahedral sites empty.

After annealing, the lattice parameter rises continuously with a value of 4.679±0.003 Å at 1100 °C, showing the expansion of zirconium lattice due to the insertion of more carbon atoms. The latter value is lower than the bulk zirconium carbide and higher than that of substoichiometric (ZrC 0.6 ), which indicates the presence of the two phases at higher temperatures, as discussed previously.

A lower lattice parameter was observed for ZrC film deposited by sol gel [31]. The most credible interpretation here is that there are vacant octahedral sites in the zirconium cell. The same interpretation has been reported for ZrN [START_REF] Ashok | [END_REF] and TiN films [28], which both possess the same FCC structure as ZrC. The ZrC lattice parameter depends considerably on composition, it increases with increasing C/Zr ratio and reaches a maximum for C/Zr~0.83, then it decreases [34]. Moreover, from JCPDS files, the stoichiometric zirconium carbide, which corresponds to the filling of all octahedral sites, has a larger lattice parameter than that of the substoichiometric carbide in ZrC 0.6 . At the same time, inserted oxygen is known to reduces the lattice parameter in ZrC by substituting carbon atoms, as the atomic radius of oxygen (r O = 0.66 Å) is inferior to that of carbon (r C = 0.76 Å). In addition, it has been reported that the amount of lattice vacancies increases together with the oxygen content in zirconium oxycarbide [33].

Grains size

The crystallites size is estimated using Scherrer's formula:

   cos 9 . 0  d (1)
λ is the wavelength of the incident beam; β is full width at half maxima (in radian) of the peak and θ is the angle of the peak.

We have used the (002) peak for Zirconium phase and the (111) peak for Carbide phase. The results

show that the grain size of both phases Zr and ZrC is in nanometrical order, Fig. 6. Annealing leads to an increase in grain size for zirconium. The rise of annealing temperature induces greater mobility of grains and grain boundaries, which results in grain coalescence.

For ZrC grains size, which is less than that of zirconium, annealing does not affect the grains size.

After carbon diffusion to the film, ZrC grains nucleate and begin to grow in Zr grains boundaries.

This growth may be hindered by the subsistence of zirconium grains, by the formation of ZrO 2 phase and by the presence of carbon and oxygen atoms in grains boundaries. Recrystallized nuclei may grow up to a certain critical size in preferred orientations with respect to original grains [17]. A similar phenomenon was reported for ZrC thin films deposited by cathode sputtering with the increasing temperature of the substrate [13], which is explained by the fact that, a high temperature diffusion of carbon atoms is faster than zirconium ones, and can easily move through octahedral sites of the zirconium lattice. The zirconium acts as catalyst and carbon covers the surface of growing grains, serving as new nucleation sites. Thus, grain growth is hindered and growth proceeds by continuous renucleation [13,26,31].

Due to the difference in the trend of grain size evolution between Zr and ZrC, the direct transformation of grains from Zr to carbide cannot be considered. However, we think that the transformation is faster through grain boundaries and then the transformation progresses inside grains.

As discussed above, the grains of carbide phase are present in the interface after annealing at intermediates temperatures, 600 to 900 ºC, while the rest of the film is composed of Zirconium grains. Nevertheless, after annealing at higher temperatures, 1000 and 1100 ºC, the growth of carbide grains displace progressively across the film towards the surface, and the carbide grains are formed at the interface and in the middle zone of the film.

Texture

The texture of the samples was quantified from X-ray diffraction spectra using texture coefficient T(hkl), which is given by the ratio:

  n m m hkl I hkl I n hkl I hkl I hkl T 1 0 0 ) ( / ) ( 1 ) ( / ) ( ) ( (2)
Where I m (hkl) is the integrated intensity of the corresponding (hkl) Bragg's peak, I 0 (hkl) is the standard intensity of the peak for random oriented grains given from JCPDS files and n is the total number of reflections measured. The calculation was made for each phase separately (metal and carbide) using the three more intense peaks. A value close to one corresponds to a random orientation, a value higher than one indicates the orientation degree to the (hkl) plane, and conversely a value between zero to one results from a low (hkl) orientation.

The thin films preferential orientation results from the minimization of total energy and is determined by the competition between two thermodynamic parameters, the free surface energy and the strain energy. At a sufficiently low thickness, the layer exhibits an orientation corresponding to that with the lowest surface free energy. With the increase of thickness, the strain energy in the film increases linearly. Thus, at larger film thicknesses, the strain energy will eventually exceed the surface energy. [31,31] The texture coefficients values calculated for the zirconium phase are shown in the Table 2. They hardly change depending on the annealing temperature; T(002) has a value higher than one, while T(101) and T(100) have values close to zero. This shows that the films have a strong orientation in the (002) planes. For Zrα phase, which has a hexagonal structure, the (002) plane is the most compact and has the lower surface energy [18]. This indicates that the zirconium phase texture is always governed by surface energy minimization. The later conclusion confirms that the microstructure of deposited film is of zone II according to the structure zone model [17]. This confirms that high energy flux was impinging the growing film during the sputtering deposition of Zr. This result is coherent with other works where the (002) orientation is also observed for r.f. ion beam sputtering zirconium films deposited in a vacuum of 6.7×10 -2 Pa, on glass substrates (Ts = 200). However, a quasi-amorphous film was obtained when the deposition was performed at room temperature [25]. Recently, the same texture was reported for polycrystalline zirconium films of 1.2-1.5 μm deposited with a pulse power of 100 W, prepared by pulsed magnetron sputtering at room temperature. This orientation decrease slightly when the substrate temperature increases [35].

The texture of very thin polycrystalline zirconium films was studied as a function of thickness. The substrate temperature does not exceed 50 °C and the magnetron power was fixed at 118 W. The texture of the films is still (002), which is the densest plane parallel to the surface, for thicknesses between 12 and 180 nm. For 240 nm, a development of new fiber axis with majority of the grains having {109} planes parallel to the sample surface [18].

Table 3 shows the values of texture coefficients calculated for the zirconium carbide phase. At 900 °C, the calculated values of texture coefficients show an orientation according to the plan (111), but with a low degree compared to higher temperatures. This means that the carbide grains are almost randomly oriented but tend to orientate within the (111) planes. These observations show that the first carbide germs have almost random orientation, which is due to the fact that these germs begin to grow randomly on zirconium grain boundaries.

By raising the annealing temperature the degree of orientation to (111) plane, which is the more compact plane, became stronger to the detriment of the two planes (200) and (220). This texturation is due to carbide phase thickness increasing in the zirconium coating, and to the increase of the thermal stresses level induced by annealing, which implies that the mechanism that imposes the preferred orientation of carbide phase is the strain energy minimization. The same orientation is observed for nanocrystalline ZrC films deposited by sputtering with a DC power of 100 W deposited at 290ºC, which is attributed to more thermodynamic favorability [13]. The texture of ZrC thin films deposited, at temperatures between 700 ºC -850 ºC, by pulsed laser deposition has been studied. They grew along the [001] axis on (001) Si while they grew along the [111] axis on (111) Si and (001) sapphire [42]. In NaCl-type structure, the (200) plane has the lowest surface energy while the (111) plane has the lowest strain energy [43].

Mechanical properties

Surface mechanical properties

Mechanical properties of zirconium thin films are seldomly treated compared to films of zirconium carbides, nitrides, oxides, or even ternary compounds of the three. Carbon, nitrogen, and oxygen elements with zirconium are widely studied because of their excellent mechanical qualities.

The surface hardness of untreated zirconium film is about 7 GPa, Fig. 7. Recently, hardness values between 6.4 and 10.5 GPa have been reported for Zirconium films grown on Si(111) substrates by pulsed magnetron sputtering, at 500 °C and 300 °C, respectively [35]. This is in good agreement with our result. Our value is far above the value found in the literature for bulk zirconium of 0.84 GPa [10,[START_REF] Brandes | Smithells Metals Reference Book, Seventh Edition[END_REF][START_REF] Khan | [END_REF], this difference comes from the difference in microstructure between bulk zirconium characterized by large grains and the film consists of nanometer-sized grains in agreement with Hall-Petch relationship. During the growth, the zirconium film undergoes a particles bombardment, which contributes to the increase of the stress state and subsequently the film hardness.

From 600 to 800 °C, hardness increases with increasing annealing temperature. At 600° C, the XRD results do not indicate a significant formation of carbide in the film. We connect this increase in hardness to the improvement of film state after annealing, due to the organizing of the microstructure and reduction of structural defects, thus increasing the film compactness. However, the hardening observed at 800°C is caused by the formation of ZrC 0.6 grains in the film.

From 900 to 1100 °C, despite the formation of stoichiometric carbide, the hardness drops a little, this can be attributed to the sample's high microporosity (revealed by SEM) and the high surfaces roughness (Ra ~ 80 nm), which appears very steep, thus disturbing the measures and compromising measurement accuracy as the indenter size is small. This also, can be attributed to oxidation. It is known that zirconium oxide hardness is less than that of carbide [START_REF] Pierson | Handbook of Refractory Carbides and Nitrides, Properties, Characteristics, Processing and Applications[END_REF][START_REF] Khan | [END_REF]. Therefore, zirconium oxide formation in zirconium carbide leads to a reduction in the overall hardness [22,32,46]. It has been reported that ZrC film hardness is lower for deposits made under residual vacuum pressure (<2×10 -4 Pa) than for those deposited under CH 4 atmosphere pressure (2×10 -3 -2×10 -2 Pa) [12].

The modulus of elasticity is a measure of material stiffness, it is about 140 GPa for surface of the unannealed sample, and this value is above the Young's modulus value of 98 GPa for bulk zirconium [START_REF] Brandes | Smithells Metals Reference Book, Seventh Edition[END_REF]. The variation of Young's modulus according to the annealing temperature follows exactly the same trend as the hardness, with 230 GPa at 1100 °C. Usually, E increases with the increase in rigidity due to the substitution of Zr-Zr bonds by more rigid Zr-C bonds (and even Zr-O).

Depth mechanical properties

The depth mechanical properties of the sample annealed at 900 °C are shown in the Fig. 8. The hardness values vary between 17 at the film interface and surface, and 13 GPa in the middle. The replacement of Zr-Zr bonds by stiffer Zr-C and Zr-O bonds caused the film hardening, a consequence of the carbide and oxide or even oxycarbide grains formation. This proves that the film middle zone contains less light interstitial elements than peripheral zones. This is in good agreement with our previous results. The formation of a second phase (carbide) adjacent to the parent phase (Zirconium), hinders dislocation motion also leading to the increase in hardness [47].

Furthermore, the maximum measured hardness (17 GPa) is low compared to the values reported in the literature for ZrC films deposited by different PVD methods. The values ranging between 26 and 36 GPa [13,42,48,49]. This can be explained by the fact that the carbide phase that is formed for our films is mainly a substoichiometric phase (ZrC 0.6 ), while the works found in literature deal with the stoichiometric phase. It is known that hardness is a function of carbon to metal ratio and that increases rapidly as it approaches the stoichiometric phase [START_REF] Toth | Transition Metal Carbides and Nitrides[END_REF]. Likewise, D. Ferro et al. [50] have indicated that a decrease of carbon content can lower the hardness of refractory carbides with group IV elements (ZrC). On the other hand, this lower value can also be related to film contamination with oxygen (formation of oxide and oxycarbide phase). Given that the hardness of zirconium oxide is lower than the carbide, and the oxygen incorporation in ZrC crystal lattice would weaken the covalent character of the Zr-C bonding and favour of the more ionic character of the Zr-O bounding resulting in a decrease in the mechanical properties [30,33].

The elasticity modulus values vary between 180 and 230 GPa. These values are smaller than values reported by several authors for zirconium carbide films, which vary between 325 and 450 GPa [13,42,[START_REF] Brandes | Smithells Metals Reference Book, Seventh Edition[END_REF]. This is related to fewer Zr-C bonds in ZrC 0.6 cell compared to ZrC [33].

The mechanical properties of the samples cross-sections are higher than those of films surfaces, which can be attributed to the difference in crystallographic orientation and grains alignment. The mechanical properties are anisotropic in nature and they heavily rely on structural parameters [START_REF] Oyama | The Chemistry of Transition Metal Carbides and Nitrides, Blackie A & P[END_REF][START_REF] Brandes | Smithells Metals Reference Book, Seventh Edition[END_REF].

The hardness of the substrate for the unannealed specimen is between 3 and 4 GPa. It remains in the same range after annealing between 600 and 800°C, in the contrary to the annealing at higher temperatures, 900 to 1100°C, where the hardness rises to values between 5 and 6 GPa, Fig. 8. On the other hand, there is no observed significant decrease in the level of carbon signal at substrate in GDOES analyses after heat treatment in comparison to the pristine substrate, Fig. 3. This indicates that there is no considerable carbon depletion gradient in the uppermost region of the substrates.

These observations prove that the diffusing carbon atoms in the film does not result in the decreases in mechanical properties of the substrate.

The pristine substrate has a microstructure formed of ferrite and globular cementite, which indicates that the steel has undergone a spheroidizing treatment. This treatment is used to improve the ability to cold deformation of steel. It has a purpose to enhance the ability to use steels by making possible the application of high cutting speeds and insuring a very well state of surface.

By annealing an hypereutectoid steel (content of carbon is upper to 0,77 wt.%), when the temperature rises just above the eutectoid (727°C), the ferrite transforms to austenite by the diffusion of carbon from cementite grains. If the temperature continues to rise, the austenite may dissolve more carbon until a temperature slightly higher than 800°C, where all the ferrite and cementite are transformed in austenite. By cooling, the inverse phenomenon happens; the austenite transforms to pearlite and cementite. The pearlite is the heterogeneous mixture (aggregate) of ferrite and cementite. Consequently, the microstructure of steel changes from globular to lamellar cementite, this later microstructure is harder which results in the noticed rise of the substrate hardness values after the thermal cycle.

Conclusion

The evolution of the structural and mechanical properties of zirconium films deposited on C100 steel as a function of annealing temperature under vacuum was investigated. Results indicate a gradual transformation of the metallic film into zirconium carbide. The two phases coexist between 600 and 1000 °C. At 600 °C, zirconium phase is the predominant phase, but by raising the annealing temperature, the carbide phase becomes the major phase at 900 °C and remains the main crystallized phase at 1100 °C. This transformation is induced by the diffusion of carbon atoms from the substrate into the film. The results show also, the formation of ZrO 2 on the surface for the higher temperature and a small amount of oxycarbide phase at intermediate temperatures could be expected. It seems that it is due to the higher reactivity of oxygen with zirconium than carbon.

The measured lattice parameters (a and c) of Zr phase for the untreated sample are higher than the bulk parameters. They decrease after annealing due to stress relaxation, defects elimination and atoms best rearrangement. The calculated lattice parameter for the carbide phase for the sample annealed at 800°C corresponds to the theoretical value of the substoichiometric carbide (ZrC 0.6 ).

After annealing, it rises continuously, showing the expansion of zirconium lattice due to the insertion of more carbon atoms. The lattice parameter calculated at higher temperatures is between the substoichiometric and the bulk parameters, which indicates the presence of the two phases.

The grain size of both phases Zr and ZrC is in nanometrical order. Annealing leads to an increase in grain size for zirconium, they have always a strong orientation according to the plane (002). For ZrC phase, annealing does not affect the grains size, they are orientated within the (111) planes. The grains of carbide phase are present in the interface after annealing at intermediates temperatures, 600 to 900 ºC, while the rest of the film is composed of Zirconium grains. By rising annealing temperature, the growth of carbide grains displace progressively, across the film towards the surface.

The mechanical properties of the film are closely related to internal structure, density, structural defects and amount of phases present in the film: From the unannealed sample to the one annealed at 600 °C, the hardness rises from 7 GPa to 11 GPa. Simultaneously, Young's modulus changes from 140 to 170 GPa. The film consists of a single phase of zirconium and the hardening (despite larger grains) is caused by film densification and removal of defects. At 800 °C, the film is formed from Zr and ZrC 0.6 grains. The hardness climbs to 14 and Young's modulus grows to 220 GPa. The improvement of film mechanical properties is related to the formation of harder and more rigid carbide phase. Diffusing carbons atoms from the substrate does not result in loses in the mechanicals properties of the substrate.

In order to achieve the whole conversion of the metallic film into stochiometric carbide with interesting mechanical properties, a longer annealing time at 900°C would be necessary.

Fig. 1 :Fig. 2 :

 12 Fig. 1: SEM images of Zirconium films surfaces, in SE mode: (a) untreated, (b) annealed at 900°C

Fig. 3 :

 3 Fig. 3: GDOES specters of Zr/C100 samples as a function of annaling temperature: (a) untreated,

Fig. 4 :

 4 Fig. 4: XRD patterns of pristine Zirconium film (a), and vacuum treated at various temperatures, (b to e).

Fig. 5 :Fig. 6 :

 56 Fig. 5: XRD patterns representing the shifts of (111) peak of carbide phase by rising annealing temperature.

Fig. 7 :Fig. 8 :

 78 Fig. 7: Variation of films surfaces mechanical properties according to annealing temperature.

  Figure 1 Click here to download high resolution image

Table 1 :

 1 Thickness variations of different zones of zirconium film as a function of annealing temperature.

	Table 1
	Temperature (°C)	600	900	1000	1100
	Thickness of film (nm)	3071	4140	3370	3480
	Thickness of film inter-facial zone (nm)	_	580	870	1200
	Thickness of film superficial zone (nm)	_	_	740	880
	Thickness of film middle zone (nm)	_	_	1760	1400

Table 2 :

 2 XRD intensities values of zirconium peaks (002), (100) and (101) according to annealing temperature, and texture coefficients values of each peak.

	Annealing	XRD intensity (a.u)	Texture coefficient
	temperature

Table 2

 2 ZrC 0.6(111)

			ZrC 0.6 (200)	ZrC 0.6 (220)	T(111)	T(200)	T(220)
	700 °C	174	-	-
	800 °C	594	-	-
	900 °C	938	329	130	1.82	0.78	0.41
	1000 °C	2096	436	122	2.23	0.56	0.21
	1100 °C	4086	431	121	2.55	0.33	0.12

Table 3 :

 3 XRD intensities values of zirconium carbide peaks (111), (200) and (200) according to annealing temperature, and texture coefficients values of each peak.

	Annealing	XRD Intensity (a.u)	Texture coefficient
	temperature

Table 3

 3