A Chbihi

X Sauvage
email: xavier.sauvage@univ-rouen.fr

D Blavette

Atomic scale investigation of Cr precipitation in copper

Keywords: Precipitation, Copper, Chromium, Diffusion, Segregation, Atom Probe Tomography Cu alloy, chromium, precipitation, coarsening, phase transformation

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Atomic scale investigation of Cr precipitation in copper

Abdelahad Chbihi, Xavier Sauvage, Didier

Introduction

Age hardenable Cu-Cr alloys and their minor modifications (including for example Cu-Cr-Zr alloys) are used in numerous applications where an excellent combination of mechanical strength and electrical or thermal conductivity is required. Thus, they are attractive candidates for railway contact wires [1], electrodes for spot welding, heat exchangers and more recently they are being considered for the design of plasma facing components and divertor plates of fusion devices [2][3][4]. The high strength (comparing to pure copper) is achieved thanks to typical precipitations treatment that give rise to the nucleation of a high density of Cr nanoscaled precipitates [1,[3][4][5][6][7]. The high conductivity results from the extremely low solubility of Cr in fcc Cu at temperatures below 500°C [START_REF] Massalski | Binary Alloy Phase diagrams[END_REF].

Dilute Cu-Cr alloys have been investigated since more than a century with the first microscopic study reported by Guillet in 1906 [START_REF] Guillet | [END_REF] and the first data on the binary phase diagram reported by Hindrichs in 1908 [10]. The precipitation kinetic was first investigated using indirect methods like X-ray diffraction or dilatometry [5,11]. These experiments raised some fundamental questions about the shape and the structure of the precipitates. In the 70's and later, nanoscaled Cr precipitates could be observed by Transmission Electron Microscopy (TEM) [5,7,[12][13][14], starting a long controversy on their crystallographic structure and orientation relationships (OR) with the fcc copper matrix. The equilibrium crystallographic structure of Cr is bcc, but in the early stage of precipitation it has been proposed by several authors that they could exhibit a metastable fcc structure [2,7,14]. Part of the enigma was finally solved by Fujii and co-authors 30 years later thanks to High Resolution TEM (HRTEM) observations [15]. They have shown that even if some typical lobe-lobe contrast usually attributed to coherent iso-structural precipitates could be observed, they are all bcc in structure even in the early stage of precipitation. They also demonstrated that two OR may exist (known as the Nishiyama-Wassermann (NW) and the Kurdjumov-Sachs (KS)) but for longer aging time, the particles exhibiting a KS-OR grow at the expense of those having a NW-OR.

In any case, nanoscaled Cr precipitates induce large lattice distortions that make them difficult to observe by TEM. Thus, there are still some ambiguities about their exact shape and size, especially in the early stage of nucleation. Some authors describe them as GP zones (i.e. few atomic planes) [16], others reporting about spherical [3], ellipsoid [15], plate [5,12], rod shaped precipitates [12,13] or even hexagonal [17]. But very recently, Hatakeyama and co-authors have used Atom Probe Tomography (APT) to analyze such precipitates [18,19]. They have shown that this technique can provide some reliable information about the size, the shape, the composition and the density of the Cr precipitates. Unfortunately they mostly focussed their work on large precipitates resulting from coarsening that occurs during reaging treatment at 600°C.

During the past ten years, modified Cu-Cr alloys and especially those from the ternary Cu-Cr-Zr system have been widely investigated because of their improved combination of mechanical and electrical (or thermal) properties [1-4, 6, 18-20]. The addition of Mg or Fe has also been proposed by some authors [17,[21][22]. In such more complex alloys, other phases were observed like Cu 4 Zr or CrCu 2 Zr [3,16,22] and some Zr and Fe segregations along Cu/Cr interfaces were also reported [18]. Moreover, the precipitation kinetic of Cr particles seems modified since Batra and co-authors identified only fcc Cr particles with a cube on cube OR and also a metastable ordered phase with a possible B2 structure [20].

The nucleation of precipitates with a crystallographic structure different from that of the parent matrix is a fundamental problem that has been addressed for a long time by many authors. Geometrical criteria using the invariant-line concept have been successfully adopted to explain ORs between precipitates and matrices [23]. However, one should note that in Cu based systems very different scenario may happen in the early stage of decomposition depending on the solute element. This may indicate that thermodynamics, elastic anisotropy and interfacial energy anisotropy play also a critical role in the precipitation kinetic. For example, in dilute Cu-Be alloy, a complex sequence of metastable phases starting from GP zones has been reported before the formation of the stable B2 phase [24]. In the Cu-Fe system, the coherent metastable fcc -Fe particles that nucleate first can be stable up to 50 nm in size [25,26], while in the symmetric system (dilutes Fe-Cu alloys) copper particles nucleate as bcc but start to progressively transform into the stable fcc structure when they are only few nanometer in size [27]. For this later system, it is also interesting to note that in the early stage, precipitates contain a significant amount of Fe, much higher than the solubility limit given by the phase diagram [28,29]. For the Cu-Cr system, as already discussed, it has been proposed that the phase separation involves the nucleation of the Cr fcc metastable phase, and even a long range ordered structure with an equi-atomic composition has been proposed (namely a B2 phase [20]). Thus, the aim of this work was to get a clear picture of nanoscaled Cr precipitates in the early stages of nucleation using a combination of TEM and APT. This later technique can provide an accurate description of the 3D shape, the volume, the distribution, the density, the volume fraction and the composition of precipitates.

For this study, a Cu-Cr-Zr alloy was chosen since this kind of alloy has attracted a lot of interest during the past decade. Different kinds of precipitate were exhibited and a special emphasis was given on the existing relationships between the size, the morphology and the composition of the precipitates for a better understanding of the precipitation kinetic.

Experimental

The alloy investigated in the present study is a commercial Cu-1Cr-0.1Zr alloy (wt.%) received in the peak-aged hardening state. The material was homogenized at 1323 K for 1 h in Ar atmosphere followed by water quenching and isothermal aging at 713 K. This temperature is slightly lower than conventional treatments (typically in a range of 733 to 753 K [1][2][3][4]) to slow down the precipitation kinetic and to really get the early stage of precipitation. The age hardening behaviour under such condition is reported in Fig. 1, showing that a saturation of the hardness is reached after 10h (155 HV ± 8), in good agreement with published data on similar alloys [5,6]. The Vickers micro hardness of the alloy was measured with a BUEHLER Micromet 2003 machine, using a 300 g load (2.94 N), each plotted value being the average of ten measurements.

As expected, the increase of the hardness is followed by a concomitant increase of the conductivity, up to 76 ± 10 % IACS (see table 1), in good agreement with published data on similar alloys [5,17]. Resistivity measurements were performed by the four-point method using a micro ohmmeter OM21 of AOIP® (Resolution of 0.1 µΩ, precision of 0.03 %).

Provided data are the average of at least 20 measurements.

Precipitates were characterized by conventional TEM using a JEOL 2000FX and by analytical Scanning TEM (STEM) using a JEOL 2100F, both microscopes operating at 200kV. Energy Dispersive X-ray Spectroscopy (EDX) was carried out with a JEOL JED-2300T system.

Electron Energy Loss Spectroscopy (EELS) was performed using a Gatan ENFINA spectrometer. TEM samples were prepared by jet-polishing (TUNEPOL 3 -Struers) at 243K, with a voltage of 15V and using a mixture of 20 % nitric acid and 80 % methanol (in volume) as electrolyte.

APT analyses were performed using a CAMECA Laser Wide Angle Tomography Atom Probe (LAWATAP) [30]. Specimens were prepared using standard electro-polishing methods (10V, electrolyte: H 3 PO 4 + 30% H 2 O,) and field evaporated in ultra-high vacuum conditions at 20 K with femtosecond laser pulses (pulse repetition rate of 100 kHz and wavelength of 515 nm).

In the as quenched state, APT data revealed that during the solution treatment up to 0.78 ± 0.09 at.% Cr atoms are dissolved in the fcc copper matrix (table 1), in good agreement with the equilibrium phase diagram [START_REF] Massalski | Binary Alloy Phase diagrams[END_REF]. Although it was not mentioned in the nominal composition of the commercial alloy, a significant amount of Fe in solid solution was also detected (0.09 ± 0.01 at.%). It is important to note that these two solute elements are homogeneously distributed in the solutionized alloy (APT 3D reconstructed volume not shown here), indicating that any clustering or GP zone formation did not occur during quenching as suggested by some authors [5,21]. The amount of Zr in solid solution after the homogenization treatment was always below the detection limit of the atom probe (i.e. 100 ppm). Large precipitates (i.e. 0.3 to 1 m in diameter) containing a significant amount of Zr do exist in the as received alloy and were observed by TEM and analyzed by EDX (data not shown here). However, such Cu-Zr intermetallic phases are stable up to very high temperatures [START_REF] Massalski | Binary Alloy Phase diagrams[END_REF] and as already reported by Apello and Fenici [START_REF] Apello | [END_REF], they probably do not dissolve during the short homogenization treatment.

It is well known that APT 3D reconstructed volumes can be affected by local magnification effects if different phases with different evaporation fields are evaporated simultaneously [33]. The predicted evaporation fields for chromium (Cr + , 27 V.nm -1 ) and copper (Cu + 30V.nm -1 ) are slightly different [START_REF] Miller | Atom Probe Analysis: principles and applications to materials problems[END_REF]. However, during the present study and under the selected APT analysis conditions, chromium (respectively copper) ions were almost exclusively collected as Cr 2+ (respectively Cu + ). The theoretical evaporation field of Cr 2+ (29 V.nm -1 [START_REF] Miller | Atom Probe Analysis: principles and applications to materials problems[END_REF]) is much closer to that of Cu + , thus non significant local magnification effects are expected. To check this point, Field Ion Microscopy was performed on the alloy in the peakaged hardening state with a high density of Cr precipitates (Fig. 2). Only the typical ring contrast of the Cu fcc crystallographic poles could be observed, even upon the evaporation of a large volume. These data clearly indicate that local magnification effects, if any, would not significantly affect 3D reconstructions and especially precipitate compositions or concentration gradients.

Microscopic features of Cr precipitates nucleated in the fcc Cu matrix

The copper alloy aged during 10h at 713K was first investigated by TEM. Cr precipitates that have nucleated during the aging treatment are very small, typically few nanometers and their orientation relationships with the fcc copper matrix usually induce strong local distortions [23]. This leads to some complex contrasts on bright field TEM images which make the estimation of the precipitate density, morphology, distribution and size complicated.

However, a careful observation of the bright field image obtained near the (002) zone axis (Fig. 3 (c)) revealed three kinds of contrasts associated to these precipitates. First, some typical coffee-bean contrasts (Fig. 3 (d)) with an extinction line along the <001>Cu,fcc direction and corresponding to coherent fcc precipitates [7,14]. Then, some Moiré contrasts (Fig. 3(e)) with fringes aligned along a direction close to <110>Cu,fcc typical of bcc Cr precipitates exhibiting a NW-OR or KS-OR [15]. Last, dark precipitates (Fig. 3 (f)) elongated along a direction close to <110>Cu,fcc which might be attributed to larger bcc Cr precipitates.

The Selected Area Electron Diffraction (SAED) pattern (Fig. 3 (a)) obtained in the (002) zone axis provides further information. For more clarity, spots are indexed in the Fig. 3 (b). First it is interesting to note that all fcc Cu spots are streaked along the <110> direction which is attributed to significant distortions of the crystal lattice. Then, there are some weak spots that can undoubtedly be attributed to the scattering of electrons by (002)Cr,bcc atomic planes. A careful observation reveals that there are three sets of these spots tilted respectively by 0°, 30° and 60° from the <110> direction of the fcc matrix. Such features are consistent with Cr precipitates with a bcc structure having a NW-OR typical of the early stage of precipitation [15]. Last, some additional weak spots also appear close to the transmitted beam. They might be attributed to double diffraction or to (001)Cr,bcc superlattice reflections of an ordered bcc phase as reported by Batra and co-authors [20]. This latter possibility would be discussed in the next section.

From these TEM data collected after 10h of aging at 713K, it seems that three kinds of precipitates co-exist within the microstructure: some of them are most probably fcc and coherent with the copper matrix (coffee-bean contrast, 20% ± 5), others are most probably bcc whether with a NW-OR or a KS-OR (Moiré fringes contrast, 25% ± 5, dark contrast and elongated, 55% ± 5). However, there are obviously some data missing to get a full picture of the precipitation kinetic and among them the size, the composition, the morphology or the density.

Part of this information was collected using analytical TEM. Using EDX mapping in the STEM mode the high density of nanoscaled chromium precipitates was clearly revealed (Fig. 4 (a)). Over the 120 x 120 nm 2 area there are about 40 precipitates, thus assuming that the foil thickness was in a range of 20 to 50 nm, it leads to a precipitate density in a range of 0.5 to 1.5 10 23 m -3 . However, one should note that due to the low spatial resolution and the background noise some of the smallest precipitates are probably not detected leading to an under-estimation of the overall density. High spatial resolution EELS mapping was also performed. Three chromium precipitates, about 5 nm in size, are clearly revealed (Fig. 4 (b,c)). Two of them appear roughly spherical, while the third one exhibits an ellipsoid morphology with a length over width ratio of about two. This later looks similar to the elongated precipitate revealed on the bright field TEM image (Fig. 3 (f)). However, these images provide only a 2D projection of precipitates, thus one can only speculate about the three-dimensional morphology. Moreover, it is impossible to measure accurately their composition because they are fully embedded in the copper matrix. Therefore APT analyses were performed.

The alloy was analyzed after 5 and 10h of aging at 713 K and the nanoscaled Cr rich precipitates are clearly exhibited in the 3D reconstructed volumes (Fig. 5). A quick inspection of these images immediately shows that after 10h of aging the precipitate density is lower while the mean precipitate size is larger. Filtering the data with a Cr concentration threshold of 5 at.% allowed the identification of all precipitates and thus the estimation of the precipitate density (see table 1). The measured density for 10h of aging is twice higher than the estimation performed on the basis of analytical TEM. This might be attributed to the higher capability of APT for the detection of the smallest precipitates. A similar filtering procedure was used to remove atoms linked to precipitates in analyzed volumes and to estimate the amount of Cr in solid solution in the matrix. It is interesting to note that even if the precipitation of chromium is not completed after 5h (0.14at.%Cr left in solid solution versus 0.02at.% after 10h, see table 1), the coarsening regime has already started. As expected, the decrease of the Cr concentration in solid solution in the copper matrix leads to a significant increase of the conductivity (table 1). To the knowledge of the authors, there is no data available in the literature about the Cr solubility limit at a temperature as low as 713 K.

However, according to Nagata and Nishikawa [5] it should be less than 0.05 at.%. Thus, one may reasonably assume that after 10h of aging at 713K the Cr concentration in the matrix has reached the thermodynamical equilibrium.

The 3D reconstructed volumes displayed in the Fig. 5 also clearly show that the precipitate morphology significantly changes between 5 and 10h of aging. Indeed, in the early stage of precipitation (5h), they look more spherical. In fact, three kinds of precipitates with different shapes could be found in these two volumes: spheres (S), ellipsoids (E) and plates (P). The respective abundance and the size distribution of these different precipitates are provided in the Fig. 6 revealing three important features: (i) for all kinds of precipitates, size distributions are large, (ii) the smallest precipitates are spherical while the largest are ellipsoid or plate shaped, (iii) the proportion of spherical precipitates dramatically decreases with the aging time (from 26 to 6 %) while the number of plates strongly increases (from 37 to 64%). One should note that three kinds of precipitates were also observed by TEM, thus these different shapes could be related to different crystallographic structures (fcc or bcc) and / or different OR (KS or NW). No systematic orientation relatively to each others could be found for plate or ellipsoid shaped precipitates in 3D reconstructed volumes. This might be due to the large number of variants offered by both the NW-OR and the KS-OR.

The composition in each precipitate was determined using concentration profiles with small sampling volumes (typical thickness 1 nm). Representative profiles related to spherical, ellipsoid and plate shaped precipitates are displayed in the Fig. 7. The 1nm large Cr gradient appearing at the matrix/precipitate interface can be exclusively related to the thickness of the sampling volume, thus interfaces are chemically sharp. However, all these precipitates contain a significant amount of Cu which is surprisingly much higher than the theoretical prediction of the Cu-Cr binary phase diagram [START_REF] Massalski | Binary Alloy Phase diagrams[END_REF]. Local magnification effects were ruled out in the previous section, thus this feature cannot be attributed to analysis artefact. The chromium concentration is only about 50at.% in spherical precipitates (Fig. 7 (a)), up to about 80at.% in ellipsoids (Fig. 7 (b)) and up to about 90% in plates (Fig. 7 (c)).

Spherical precipitates are the smallest, so it seems that smallest precipitates contain a lowest proportion of Cr. Considering the large size distribution of all kinds of precipitates (Fig. 6), the Cr concentration was plotted as a function of the precipitate size for all precipitates and for 5 and 10h of aging (Fig. 8). The trend is confirmed and appears especially clear for the material aged 5h where the mean precipitate volume is smaller (Fig. 8 (a)): the smaller the precipitates the lower their Cr content.

It is interesting to note that, additionally to Cr, precipitates also contain a significant amount of Fe (Fig. 7). For the smallest precipitates (i.e. spheres), Fe atoms seem homogeneously mixed with Cr (Fig. 7 (a)). But for larger precipitates and especially plate shaped, Fe atoms are segregated along the Cu/Cr interface (Fig. 7 (c)). These observations are in agreement with recently published data from Hatakeyama and co-authors [18]. However, one should note that in the present study, Fe atoms were not found homogeneously distributed on the precipitate surface and there is not a full coverage of the interface.

Discussion about the nucleation and growth of precipitates

Bright field TEM investigations of the early stages of precipitation have revealed the presence of both coherent fcc precipitates as well as bcc Cr-enriched precipitates with two crystal orientation relationships (NW and KS), The proportion of each kind of precipitate is difficult to estimate from this only TEM information. Only HR-TEM of individual precipitates could provide precise information on their OR (NW or KS). The precipitate density was estimated thanks to analytical TEM. APT investigations revealed three morphologies: spherical, ellipsoid and plate shape. Fujii and co-authors have shown that bcc Cr precipitates appear elongated in 2D HR-TEM images [15]. It is also interesting to note that for the NW-OR there is a small misfit (2.3%) only along one direction, namely (111) Cu // (110) Cr , while for the KS-OR there is a second low misfit direction (2.2%), namely [-101] Cu / [-111] Cr . It seems therefore realistic to think that precipitates will favorably grow along these low misfit directions, leading to some ellipsoids for the NW-OR and plate shaped for the KS-OR.

Chromium precipitates with an fcc structure and a cube on cube OR with the fcc matrix should of course grow with a spherical shape resulting from the isotropic distortions.

Moreover, APT data show that spherical precipitates are much smaller than others and they are more numerous upon the early stage of precipitation, but also that the proportion of plate shaped precipitates dramatically increases with the aging time and they are bigger in average than others. Fujii and co-authors have shown that bcc chromium precipitates with a KS-OR grow at the expense of those having a NW-OR [15]. As a consequence, one may conclude that ellipsoid shaped precipitates are bcc with a NW-OR while plate shaped are bcc with a KS-OR. The precipitation sequence in the dilute Cu-Cr system is therefore thought to be as following: (i) nucleation of coherent fcc spherical precipitates, (ii) growth and transformation of fcc spherical precipitates into the bcc structure, (iii) growth of bcc precipitates that have a KS-OR at the expense of those having a NW-OR. APT data suggest that certain precipitates enter into the coarsening stage while nucleation of new embryos go on (i.e. precipitates have not nucleate at the same time).

APT analyses and TEM observations also strongly suggest that precipitation of the Cr-rich equilibrium bcc phase initiates through the nucleation of fcc precipitates that are coherent with the Cu-rich fcc parent phase. Such a precipitation pathway has been observed in numerous systems, in particular in Al base alloys where coherent GP zones nucleate before the stable phase (e.g. GP zones in AlCu). The physical reason behind this process is well i.e. 100% of Cr). Diffusion arguments as well as thermodynamics have both to be considered in order to account for this non-classical nucleation behaviour.

Let us first compute the order of magnitudes of quantities (driving force, elastic energy, interfacial energy) that come into competition in the classical nucleation theory. In the following, we shall compare the classical nucleation barrier (i.e. precipitate composition ~ 100 at.% Cr) to that related to non-classical nucleation (i.e. precipitate composition ~ 50 at.% Cr). The driving force for classical nucleation in dilute alloys reads [36,37]:

∆g n = (RT/V  )X  ln (X 0 /X  ) (4)
with X 0 the atomic fraction of solute (Cr) in Cu, X  the coherent solubility of Cr in the fcc copper phase , X  the Cr concentration in the fcc or bcc chromium phase (close to 100 at.% in theory) and V  the molar volume of the chromium phase. We have assumed this latter to be the same for both bcc and fcc structures of Cr phase.

It must be kept in mind that for a regular solution, X  = 1-X  (G(X) is symmetrical). The coherent solubility limit of Cr in Cu is X  << 1 so that X  ~ 1, leading to g n fcc = 6 10 9 J/m 3 for the nucleation of fcc Cr and to g n bcc = 10 10 J/m 3 for the nucleation of bcc Cr (with X 0 = 0.7%). Let us first compare the "classical" driving force to both the elastic energy resulting from misfits and to the interfacial energy. The classical nucleation barrier will then be computed and compared to the non-classical barrier.

The elastic energy of a coherent fcc precipitate in the fcc copper matrix may be estimated using the classical theory for a coherent inclusion as described by Eshelby [START_REF] Christian | The theory of Transformations in Metals and Alloys[END_REF]: 5)

∆g e = (E/(1-)) (a/a) 2 (X  -X  ) 2 (
Where E and  are respectively the elastic modulus (E = 130 GPa) and the Poisson ratio (=0,34) of copper and a/a the misfit between pure fcc-Cu and fcc-Cr phases (1.9% [START_REF] Pearson | A Handbook of lattice Spacings and structures of Metals and alloys[END_REF]).

This leads to ∆g e = 7.4 10 7 J/m 3 (X  = 100%), two orders of magnitude lower than the driving force ∆g n . The elastic energy is negligible compared to the driving force for pure nuclei. It is important to keep in mind that because of the dependence of both the driving force and elastic energy on X  , this may be not the case for embryos containing less chromium. This will be discussed latter.

Bcc precipitates exhibit a NW or KS orientation relationship that minimizes the elastic energy, thus for the sake of simplicity, this elastic energy for bcc precipiates will be neglected in the following estimations.

In a first approach, the coherent interfacial energy  is due to unlike first nearest pair interactions and may be written as proportional to the concentration gradient squared:

 =  ((X  -X  )/a) 2 (6)
where a is the lattice parameter (abrupt interfaces were here considered) and  the ordering effective energy that can be written as follow:

  

Where N is the Avogadro constant (N=6.02 10 23 mol -1 ) and with Z the coordinance (Z=12 for fcc and 8 for bcc). Again  depends on the composition of the nucleus. For pure nuclei,  = 86 mJ/m 2 (coherent Cr fcc /Cu fcc interface). One should note that it is much smaller than the Cr bcc /Cu fcc interfacial energy reported in the literature for non-coherent Cr-Cu fcc/bcc interfaces (625 mJ/m 2 [START_REF] Aguilar | [END_REF]). The nucleation barrier (∆G*) for fcc precipitates will therefore be lower than that of bcc. This latter writes:

2 el n 3 ) g g ( 3 16 * G       



For an fcc nucleus, this leads to G* fcc  3 10 -22 J, thus two order of magnitude smaller than for a bcc nucleus G* bcc  4 10 -20 J. This clearly confirms that the nucleation of fcc Crenriched nuclei is much easier than that of non-coherent bcc nuclei.

Let us now consider non-classical nucleation [41][42][43][44][45]. As shown in figure 10, lower Cr content in embryos significantly reduces the driving force for the nucleation. However, nuclei containing less Cr will give rise to lower interfacial energy because  is proportional to the square function of the concentration gradient (X) (equation 6). Lower solute (Cr) content in embryos also reduces the elastic energy (equation 5) that also acts as a barrier against nucleation. The expected composition of critical nuclei will be therefore that which minimizes the nucleation barrier (saddle point of the barrier). This situation is schematically represented in Fig. 10. Considering again a regular solid solution, it is easy to show that the non-classical driving force for the nucleation of precipitates that have a solute content X  writes as [44]:

                                          0 β β 0 β β 2 0 β β NC n X 1 X 1 ln ) X 1 ( X X ln X X X η 2 V RT ) g ( (10)
with η = T/T c and T c the critical temperature, T c = Z/2k (T c = 4870K for  = 70 meV and Z=12 (fcc),  = 0.15 for T = 713K).

The driving force yields (g n fcc ) NC = 1.35 10 9 J/m 3 , a value that is 5 times smaller than the classical value (g n fcc = 6 10 9 J/m 3 ). The elastic energy, proportional to (X  -X  ) 2 , is close to one fourth of the previous value for 50% of Cr instead of 100% : ∆g e = 1.85 10 7 J/m 3 . This value is two orders of magnitude smaller than the driving force. Elastic energy due to misfit cannot therefore account for the lower Cr content in nuclei. In the framework of classical nucleation theory, a lower solute content in nuclei is expected to favour a lower misfit, thus reducing the elastic energy and the nucleation barrier. However, if we instead consider nonclassical theory of nucleation, increasing the elastic energy unexpectedly increases the solute concentration in nuclei [45]. In addition, the above estimations show that the elastic contribution can be ignored and anyway this would not lead to an underestimation of Cr content in precipitates.

In contrast, the influence of the interfacial energy is considerable. For X  = 0.5, the interfacial energy is (0.5) 2 smaller ( = 21.5 mJ/m 2 , equation ( 6)) so that this contribution in the barrier expression (equation ( 8)) is considerably smaller (1/2 2 ) 3 = 1/64 times smaller (compared to the decrease of the driving force g n fcc ~ (1/5) 2 . This leads to a nucleation barrier ∆G* NC of only 9 10 -23 J for fcc embryos containing 50at.%Cr. This value of the barrier height is close to ten times smaller than the classical value that we have previously computed for pure Cr embryos (3 10 -22 J). These thermodynamical considerations based on the non-classical theory of nucleation indicate that it is much easier to nucleate precipitates containing 50at% of Cr compared to pure Cr precipitates. The related critical radius can be computed as following: r* = 2 / ∆g n (11) This leads to r* fcc = 2.9 10 -11 m for non-classical fcc nuclei and r* bcc = 1.2 10 -10 m for bcc nuclei. These latter values are unfortunately smaller than the atomic size. This clearly raises the problem of the validity of nucleation theories for such large supersaturations and driving forces.

In summary, the equilibrium structure of Cr precipitate is bcc, and their equilibrium Cr content is close to 100%. However, the nucleation barrier for fcc precipitates is lower than that of bcc precipitates. For low Cr content in nuclei, the nucleation barrier is even lower. For 5h and 10h at 713K, both bcc and fcc precipitates were observed. Because of their much smaller nucleation barrier, dilute fcc precipitates first appear. bcc precipitates then appear at the expense of fcc nuclei. The proportion of each is controlled by the respective values of both the nucleation rate and the nucleation latent times.

The nucleation pathway of incoherent phases is a generic issue that is common for many systems where cascades of metastability are observed (AlCu, AlMgSi, FeCu…). Fcc nuclei being not the stable form of the Cr phase, their "stability" decreases in proportion as they grow so that they finally disappear at the expense of the stable bcc Cr-rich phase. The driving force is of course the misfit elastic energy of coherent fcc precipitates that dramatically increases with size while the interfacial energy becomes less important in the overall energy balance. Then, three mechanisms might be considered:

i/ Heterogeneous precipitation of the bcc phase on fcc nuclei. The driving force for such a heterogeneous nucleation is not the reduction of interfacial energy as interfaces are coherent.

Instead it may originate from the presence of the large amount of Cr present in fcc nuclei.

Nucleation is thus easier and faster. However, APT data never shown coagulated particles, thus this mechanism can be ruled out.

ii/ Dissolution of fcc nuclei, leading to an increase of the supersaturation and the subsequent homogeneous nucleation of the bcc phase that has a larger latent time for its nucleation. The calculation conducted above shows that the driving force remains very large even for 50% of Cr. This mechanism is therefore quite plausible and also consistent with our experiments that clearly show that bcc and fcc precipitates co-exist.

iii/ Transformation of the whole structure of nuclei from fcc to bcc during the course of their growth. This process involves the massive transformation of the precipitate structure as observed in diluted Fe-Cu alloys.

Up to now, the influence of iron was not considered. This element was found to partition preferentially to precipitates (fig. 7 Kinetics effects have also to be considered. The diffusion coefficient of Cu in bcc Cr at 713K (4.1 10 -17 cm 2 s -1 [START_REF] Weast | Hand Book of Chemistry and Physics[END_REF]) is two order of magnitude smaller than that of Cr in fcc Cu (1.5 10 - 15 cm 2 s -1 [START_REF] Kaur | HandBook of Grain and Interphase Boundary Diffusion Data[END_REF]). Thus, compared to the rate at which Cr atoms migrate and agglomerate to form nuclei, the rejection of Cu atoms from bcc Cr-rich precipitates will take more time and precipitates are likely to grow and reach larger sizes (a few nm) before reaching their equilibrium composition (i.e. close to 100% Cr).

The presence of Fe at the interface is also intriguing and deserves to be discussed further. Fe is mainly located in the core of precipitates in the early stages of aging. A rejection of this element to Cu/Cr interfaces is observed during coarsening (Fig. 7). This thin Fe-rich layer formed at the interface leads to the formation of two new interfaces (Cu/Fe and Fe/Cr) and the elimination of Cu/Cr interfaces. This raises the question of whether this transformation is favorable energetically. Cu/Fe interfacial energy γ Cu/Fe is in the range 0.121 to 0.318 J m -2 [START_REF] Kozeschnik | [END_REF] whereas γ Fe/Cr is reported to lie between 0.047 to 0.213 J m -2 [49]. These interfacial energies are much smaller than that of Cu fcc /Cr bcc interfaces (0.625 J/m 2 [START_REF] Aguilar | [END_REF]). Even if the core-shell structure observed for such precipitates (Fe/Cr/Fe) leads to the formation of two interfaces instead of one, segregation of Fe seems quite favorable even when adding interfacial energies.

Conclusions

The precipitation of Cr in fcc Cu has been investigated using TEM and APT to characterize nanoscaled precipitates that nucleate during the early stage of aging. Three kinds of precipitate were observed: spheres (S), ellipsoids (E) and plates (P). Spheres are the smallest in size, they contain between 30 and 60 at.% Cr and are attributed to fcc coherent precipitates.

Ellipsoids are somewhat bigger in volume than spheres, they exhibit a higher Cr content (between 60 and 80 at.%), and are attributed to bcc precipitates with a NW-OR. Plate shaped precipitates are the biggest in size, they exhibit the highest Cr content (in a range of 85 to 100 at.%) but are almost never pure as expected from the phase diagram ; they are attributed to bcc precipitates with a KS-OR.

The observation of fcc Cr rich precipitate was rather surprising because only bcc precipitates were expected from the phase diagram. However, using the classical nucleation theory, it was found that the nucleation barrier for fcc precipitates is much lower (2 order of magnitude) than for bcc precipitates. However, the classical theory could not account for the rather high amount of Cu measured by APT in the fcc precipitates. Thus, the non-classical nucleation theory was applied taking into account both the elastic and the interfacial energies. It was found that a precipitate composition of about 45at.%Cr was minimizing the nucleation barrier.

Then, during the coarsening stage, precipitates transform into bcc and progressively increase their Cr content.

  (a)). Even if this third element is present with a low content, it may have a drastic influence on the low Cr content observed in precipitates. Because of the low miscibility between Fe and Cr (large miscibility gap), the presence of Fe in the core observed in the early stages (fig 7(a)) may reduce significantly the amount of Cr in particles especially in the early stages. This interpretation is reinforced by the fact that the Cu concentration in precipitates decreases (i.e. the Cr content decreases) when iron is rejected at the matrix/precipitate interface during their growth (fig. 7(c)).

FiguresFigure 1 :

 1 Figures

Figure 2 :

 2 Figure 2: Field ion microscopy image of the Cu alloy aged during 10h at 713K. There is no visible contrast between the copper matrix and the chromium precipitates indicating that these two phases have similar evaporation fields.

Figure 3 :

 3 Figure 3: TEM images of the Cu alloy aged 10h at 713K: Selected area electron diffraction pattern collected in the (002) fcc Cu zone axis (a), indexation of the full patternsee text for details (b), bright field image obtained with a specimen orientation slightly tilted from the (002) fcc Cu zone axis (c), zooms on nanoscaled contrasts attributed to three different kind of precipitates (d), (e) and (f)see text for details.

Figure 4 :Figure 5 :

 45 Figure 4: Analytical STEM images of the Cu alloy aged 10h at 713K: EDX Cr map (Cr kalpha signal) showing the high density of nanoscaled Cr rich precipitates (a); EELS Cr (b) and Cu (c) maps showing three Cr rich precipitates. The biggest is a 5nm long ellipsoid

Figure 7 :

 7 Figure 7: 3D reconstruction of typical precipitates (Spheres (a), Ellipsoid (b) and Plate (c)) and corresponding concentration profiles computed across (the arrows indicate the location of the profiles, sampling volume thickness: 1 nm). In the reconstructed volumes only Cr (red) and Fe (green) atoms are displayed.

Figure 8 :

 8 Figure 8: Chromium concentration of the precipitates plotted as a function of their size for the alloy aged 5h (red circles) and 10h (blue squares) at 713K. Obviously the bigger the precipitate the higher is the Cr content.

Figure 9 :Figure 10 :

 910 Figure 9: Formation enthalpy of non equilibrium super saturated solid solution in the Cu-Cr system (from [35])

Acknowledgements

The authors would like to thank Dr. C. Genevois for assistance in TEM. Dr Eiji Okunishi from JEOL Ltd is also gratefully acknowledged by the authors for the EDS and EELS STEM analysis.

known and is associated with the much lower interfacial energy of coherent interfaces that reduces the nucleation barrier compared to incoherent or semi-coherent interfaces.

Experiments show that in the early stage of precipitation most of precipitates contain a significant amount of Cu. Similar features were reported for the precipitation of Cu in bcc-Fe where non equilibrium bcc Cu precipitates nucleate with a significant content of Fe [28,29].

Present researches reveal that in CuCr the transient metastable fcc phase have a Cr content close to 50 at.%. This is rather surprising if we consider the very high enthalpy of mixing of Cu with Cr in the fcc structure for such a composition (∆H m close to 20 kJ/mol, see figure 9) [START_REF] Michaelsen | [END_REF]. Because of this large enthalpy of mixing and the rather low temperature (713K, low configuration entropy), very small mutual solubility is expected (i.e. small Cu amount in fcc Cr-rich precipitates). Let us compute an order of magnitude of the solubility of Cu in Cr precipitates. The parabolic shape of ∆H m (~X (1-X), see figure 9 and equation 2) suggests that the CuCr system may be modelled as a regular solid solution. Ignoring the presence of Fe, the solubility of Cu in the fcc Cr phase can thus be expressed as:

 fcc may be derived from the enthalpy curve (H fcc , figure 9) as following:

Where X is the Cr concentration (see Fig. 9). H Cr fcc -H Cu fcc was found close to 7 kJ/mol (figure 9).  has been derived from the first derivative of H fcc . This latter is null when:

∆H fcc exhibits a derivative equal to zero for X = 0.55 (fig. 9) so that  cfc = 7 10 4 J/mol and X  fcc = 7 10 -6 at 713K. A similar fit of H bcc leads to  cc = 10 5 J/mol and X  bcc = 4 10 -8 at 713K

(H Cu bcc -H Cr bcc = -4 kJ/mol and dH bcc /dX = 0 pour X = 0.48, see Fig. 9)

In contrast to these predictions (almost pure Cr-precipitates), precipitates are observed with 50% of Cr. This is clearly in disagreement with the well known classical nucleation theory that states that nuclei have the "equilibrium" composition (here that of the metastable phase,


Guy Fayolle

Arnaud De

La Fortelle

Jean-Marc Lasgouttes

Laurent Massoulié

James Roberts

Best-effort networks: modeling and performance analysis via large networks asymptotics

Keywords: best-effort service, max-min fairness, min policy, mean field, star-shaped network

In this paper we introduce a class of Markov models, termed best-effort networks, designed to capture performance indices such as mean transfer times in data networks with best-effort service. We introduce the so-called min bandwidth sharing policy as a conservative approximation to the classical max-min policy. We establish necessary and sufficient ergodicity conditions for best-effort networks under the min policy. We then resort to the mean field technique of statistical physics to analyze network performance deriving fixed point equations for the stationary distribution of large symmetrical best-effort networks. A specific instance of such networks is the star-shaped network which constitutes a plausible model of a network with an overprovisioned backbone. Numerical and analytical study of the equations allows us to state a number of qualitative conclusions on the impact of traffic parameters (link loads) and topology parameters (route lengths) on mean document transfer time.

I. INTRODUCTION

Consider a network handling data flows from several users, and assume no quality of service commitments (such as minimum bandwidth allocations) have been made by the network to the users. Such a situation has been prevalent in the Internet until now, and is likely to remain so for another few years.

The preferred service model in that situation, known as best effort service, consists in allocating a fair proportion of bandwidth to contending users; see, e.g., Bertsekas and Gallager [START_REF] Bertsekas | Data Networks[END_REF]. There are actually several possible notions of fairness available for this bandwidth allocation problem (see, e.g., Mo and Walrand [START_REF] Mo | Fair end-to-end window-based congestion control[END_REF] for a parametric family of fairness criteria covering all other notions proposed so far), although the classical notion proposed in [START_REF] Bertsekas | Data Networks[END_REF] is the so-called max-min fairness.

Recent work has led to a relatively good understanding of how bandwidth is shared between network users when a given congestion control algorithm is used; see, e.g., Massoulié and Roberts [START_REF] Massoulié | Bandwidth sharing: objectives and algorithms[END_REF] and references therein. The question of what type of fairness is achieved in the current Internet, where Jacobson's congestion avoidance algorithm-as implemented in TCP-is responsible for congestion control, has been studied in depth by Hurley et al. [START_REF] Hurley | A note on the fairness of additive increase and multiplicative decrease[END_REF]. These studies all assume the number of flows remains fixed.

In comparison, there is little work accounting for the random nature of traffic and its impact on user perceived quality of service. Consider for instance the transfer of digital documents (Web pages, files, emails,. . . ) using a transport protocol like TCP. This constitutes the bulk of Internet traffic today. The performance criterion relevant to such transfers is the overall document transfer time. This time is clearly highly dependent on the number of ongoing transfers on the links shared by the considered connection. This number varies as a random process as new connections are established and existing ones terminate in a way which depends on how bandwidth is allocated, as well as on the underlying traffic parameters.

In the case of a single bottleneck resource shared perfectly fairly, simple traffic assumptions of Poisson arrivals and identically and independently distributed document size lead to a processor sharing queueing model [START_REF] Roberts | Bandwidth sharing and admission control for elastic traffic[END_REF]. This fluid flow model provides useful results on expected response times as a function of the load of an access link or a Web server, for instance. It also shows how a form of congestion collapse can occur when demand (arrival rate × mean document size) exceeds capacity. The processor sharing queue is then no longer ergodic leading to unbounded response times. To understand the impact of multiple bottlenecks and to investigate the effect of different sharing strategies, one would like to dispose of similar analytical results for multiple resource systems.

To the best of our knowledge, the only analytical results available so far are in Massoulié and Roberts [START_REF] Roberts | Bandwidth sharing and admission control for elastic traffic[END_REF], where the so-called linear network topology is investigated. Simulation results for the linear network can be found both in [START_REF] Roberts | Bandwidth sharing and admission control for elastic traffic[END_REF] and in de Veciana et al. [START_REF] Veciana | Stability and performance analysis of networks supporting services with rate control -could the internet be unstable?[END_REF]. The main motivation for the present paper is to study the performance of best-effort networks with alternative topologies, the ultimate objective being the derivation of heuristics enabling the performance evaluation of bandwidth sharing in a general network.

In the present paper we report the results of our preliminary investigations. These include an analysis of the stability conditions under which the expected response time remains finite in a general network. We also apply mean field techniques to evaluate the performance of large symmetrical networks. Numerical results derived from the model illustrate how response times depend on the number of bottleneck links and their utilization. These results are of some practical interest and aide our understanding of the behavior of best effort networks. A further significant contribution is the insight provided into the inherent difficulty of deriving performance estimates when more than one bottleneck limits throughput.

Section II introduces a general class of Markov models for best-effort networks which is intended to capture the impact of network topology, traffic parameters and bandwidth sharing (fairness) criteria on document transfer times. A brief account of the results obtained in [START_REF] Roberts | Bandwidth sharing and admission control for elastic traffic[END_REF] is given, and the so-called "min" bandwidth allocation is introduced as a conservative approximation to max-min fairness. Section III then establishes the necessary and sufficient ergodicity criteria for best-effort networks under the "min" policy. Section IV introduces the so-called "star topology". Its relevance as a model of real networks is discussed and a mean field heuristic is proposed. This heuristic is expected to be accurate in the asymptotic regime where the number of star branches is large. The derived fixed point equations are investigated numerically in Section V. Simulation is used to verify the accuracy of the heuristics. Extensions to the star-shaped network are also considered in Sections IV and V, which notably allow an evaluation of the impact of the number of bottlenecks on the mean transfer time.

II. BEST-EFFORT NETWORKS

Consider the following network model: a set L of links is given, where each link ℓ ∈ L has an associated capacity or bandwidth C ℓ > 0. A set R of routes is given, each route being identified with a subset of links. Fig. 1 illustrates the so-called linear network: it consists of L links with equal capacity, route 0 which crosses each link, and routes r = 1, . . . , L which cross a single link.

To each route r are associated two parameters: λ r is the arrival rate of new transfer requests along route r, and σ r is the average document size. We make the following standard simplifying assumptions: requests for document transfers along route r arrive at the instants of a Poisson process with intensity λ r , while the corresponding document sizes are mutually independent, independent of the arrival times, and drawn from an exponential distribution of mean σ r .

These traffic assumptions make the process specifying the number of transfers in progress on different routes Markovian (see below) and thus greatly simplify analysis. The Poisson arrivals assumption is not unreasonable in a large network. In view of the insensitivity of performance results for an isolated link to the exact document size distribution, we do not expect divergence of the real distribution from the exponential size assumption to invalidate the derived conclusions. However, the main reason for assuming an exponential distribution is clearly one of analytical tractability.

The network state is summarized by the variables X def = {x r , r ∈ R }, where x r denotes the number of transfers in progress along route r. It remains to specify at what speed documents are transmitted in any given state X in order to turn X into a Markov process with well defined dynamics. Indeed, given the rate ζ r (X) at which documents along route r are transferred when the network state is X, X is a Markov process with non-zero transition rates given by

x r → x r + 1 : rate λ r , x r → x r -1 : rate x r ζ r (X)/σ r .
A natural assumption would be to consider that each document receives its fair share of bandwidth. For instance, if as in [START_REF] Bertsekas | Data Networks[END_REF] fairness is understood as max-min fairness, each transfer along route r receives a bandwidth share ζ mm r , where

∑ r∋ℓ x r ζ mm r ≤ C ℓ , ∀ℓ ∈ L, (1)
and for every route r, there is at least one link ℓ ∈ r such that

∑ r ′ ∋ℓ x r ′ ζ mm r ′ = C ℓ , and ζ mm r = max r ′ ∋ℓ ζ mm r ′ . (2)
These two conditions uniquely determine the bandwidth shares ζ mm r . Having specified the Markov process X, one can then attempt to study its steady state properties, identifying the conditions on the load parameters

ρ ℓ def = 1 C ℓ ∑ r∋ℓ λ r σ r
under which it is ergodic and, when it is, determining the stationary distribution. Mean transfer times T r along each route r can then be computed using Little's law: T r = IEx r /λ r . It turns out that explicit formulas for steady state distributions are typically beyond reach. A notable exception is the linear network, with bandwidth shares being allocated to realize proportional rather than max-min fairness; see [START_REF] Roberts | Bandwidth sharing and admission control for elastic traffic[END_REF]. In order to obtain formulas in other cases, one therefore has to resort to asymptotics on various parameters. For instance, for the linear network with max-min fair rate sharing, the regime where the arrival rate λ 0 along route 0 goes to zero (essentially, a form of light traffic analysis) is considered in [START_REF] Roberts | Bandwidth sharing and admission control for elastic traffic[END_REF]; this leads to approximate formulas for T 0 . It can be shown, in particular, that T 0 increases as the logarithm of the number of links L when L increases. This is in contrast to the case of proportionally fair sharing where it increases linearly in L.

The main purpose of this paper is to investigate an alternative asymptotic regime where it is the network topology which evolves. The precise description of this limiting regime will be given in Section IV.

In the following sections, we consider bandwidth allocations according to the following "min" policy: given the network state X, each transfer along route r receives a bandwidth share ζ min r given by

ζ min r def = min ℓ∈r C ℓ X ℓ , (3)
where we have introduced the notation X ℓ = ∑ r ′ ∋ℓ x r ′ to represent the total number of transfers making use of link ℓ.

It is easy to check that this policy satisfies the capacity constraints [START_REF] Bertsekas | Data Networks[END_REF]. Moreover, it is sub-optimal with respect to the maxmin fairness policy, as shown in the next theorem.

Theorem 1: Under the same initial conditions, the vector X mm (t) for the system under the ζ mm allocation policy is stochastically smaller than X min (t), corresponding to the ζ min allocation.

Proof: Assume that, for some t, X mm (t) ≤ X min (t). Then, with the notation of (2),

ζ mm r = max r ′ ∋ℓ ζ mm r ′ ≥ 1 X mm ℓ (t) ∑ r ′ ∋ℓ x mm r ′ (t)ζ mm r ′ = C ℓ X mm ℓ (t) ≥ C ℓ X min ℓ (t) ≥ ζ min r .
Thus, using a coupling argument, one can define the processes X mm and X min (t) in such a way that X mm (t) ≤ X min (t) for all t > 0.

The previous theorem motivates the study of the min policy, as it implies for instance that mean transfer times T r under the min policy provide upper bounds on the corresponding transfer times under the max-min policy.

III. ERGODICITY CONDITIONS

In the following we demonstrate that min and max-min bandwidth sharing policies have a stationary regime under the usual conditions, i.e. when the load on each link ℓ is less than 1:

Theorem 2: Under the allocation policies ζ mm and ζ min , the network is

(i) ergodic if max ℓ∈L ρ ℓ < 1; (ii) transient if max ℓ∈L ρ ℓ > 1.
This result has already been proven for the max-min policy in [START_REF] Veciana | Stability and performance analysis of networks supporting services with rate control -could the internet be unstable?[END_REF]; we note that, by Theorem 1, ergodicity under the min policy implies ergodicity under the max-min policy, thus the treatment of the min policy given below provides an alternative proof to that of [START_REF] Veciana | Stability and performance analysis of networks supporting services with rate control -could the internet be unstable?[END_REF]. However we feel that, since the proof below is simpler and uses only elementary Lyapunov functions results, it should be easier to adapt to a more complicated situation. Transience under condition (ii) is in fact valid for any allocation policy which meets the capacity constraints [START_REF] Bertsekas | Data Networks[END_REF].

Proof: Consider the discrete time chain ( X(n), n ∈ IN) describing the sequence of states visited by the continuous time

jump process X. Transitions from a given state X = xr , r ∈ R satisfy IP ∆ xr (n) = 1 X(n) = X = λ r D , IP ∆ xr (n) = -1 X(n) = X = 1 D xr σ r min ℓ∈r C ℓ Xℓ , where ∆ xr (n) def = xr (n + 1) -xr (n) and D def = ∑ r∈R λ r + xr σ r min ℓ∈r C ℓ Xℓ ≤ |R | • max r∈R λ r + max r∈R 1 σ r • max ℓ∈L C ℓ def = D ′ .
Ergodicity of the continuous time process X will follow from that of X and from the fact that the mean sojourn times in each state X are bounded from above uniformly in X (or equivalently, that the jump rates out of each state X are bounded away from zero), a property which is easily verified.

Sufficient condition. Assume ρ

M def = max ℓ∈L ρ ℓ < 1, and define the Lyapunov function f ( X) def = ∑ r∈R ∑ 1≤k≤ xr γ k r ,
where γ r > 1 will be chosen later. The structure of the function, which may seem unnatural, has been chosen for the sake of computation; it is in fact of the general form ∑ r∈R β r γ xr r + K, for appropriate constants β r and K.

In order to express the transition rates in terms of xr and ρ ℓ , remark that

Xℓ = ∑ r∋ℓ xr = ∑ r∋ℓ λ r σ r xr λ r σ r ≤ ρ ℓ C ℓ max r∋ℓ xr λ r σ r .
Then, using the notation

x * r def = xr λ r σ r , x * M def = max r∈R x * r , we have, ∀r ∈ R , IP ∆ xr (n) = -1 X(n) = X ≥ λ r Dρ M x * r x * M . Thus, IE f ( X (n + 1)) -f ( X(n)) X(n) = X = ∑ r∈R γ xr +1 r IP ∆ xr (n) = 1 X(n) = X -γ xr IP ∆ xr (n) = -1 X(n) = X ≤ ∑ r∈R λ r ρ M D γ xr r ρ M γ r - x * r x * M Let γ r def = γ 1 λr σr
, where γ is such that

ρ M γ r = ρ M γ 1 λr σr < θ < 1, r ∈ R ,
for some real number θ satisfying ρ M < θ < 1. The following inequality sums up what we have so far:

IE f ( X(n + 1)) -f ( X(n)) X(n) = X ≤ ∑ r∈R λ r γ x * r ρ M D θ - x * r x * M .
Let α be a real number such that θ < α < 1. The following quantities will be evaluated separately

           Σ 1 def = ∑ r: x * r >α x * M λ r γ x * r ρ M D θ - x * r x * M , Σ 2 def = ∑ r: x * r ≤α x * M λ r γ x * r ρ M D θ - x * r x * M .
Since Σ 1 is a sum of negative terms, the following bound holds, for any r 0 is such that x *

r 0 = x * M , Σ 1 ≤ λ r 0 γ x * M ρ M D (θ -α) ≤ γ x * M ρ M D (θ -α) min r∈R λ r < 0.
Bounding Σ 2 is straightforward:

Σ 2 ≤ ∑ r: x * r ≤α x * M λ r γ α x * M ρ M D θ ≤ γ x * M ρ M D γ (α-1) x * M |R |θ max r∈R λ r .
(θ -α) min r∈R λ r + γ (α-1)C |R |θ max r∈R λ r ≤ -ε, we have, ∀ X ∈ { x * M > C}, IE f ( X (n + 1)) -f ( X(n)) X(n) = X = Σ 1 + Σ 2 ≤ -ε γ x * M ρ M D ≤ -ε γ C ρ M D ′ < 0. Since { x *
M ≤ C} is a compact set, Foster's theorem applies (see e.g. [START_REF] Fayolle | Topics in the Constructive Theory of Countable Markov Chains[END_REF]) and the Markov chain is ergodic.

Necessary condition. Assume now that there exists ℓ 0 such that

ρ ℓ 0 > 1. Defining g( X) def = ∑ r∋ℓ 0 σ r xr , we immediately have, IE g( X(n + 1)) -g( X(n)) X(n) = X = ∑ r∋ℓ 0 σ r IP ∆ xr (n) = 1 X(n) = X -IP ∆ xr (n) = -1 X(n) = X ≥ 1 D ′ [C ℓ 0 ρ ℓ 0 -C ℓ 0 ] > 0.
Since the jumps are bounded, the chain is transient.

IV. MEAN FIELD ANALYSIS OF LARGE NETWORKS

It does not appear possible to obtain closed form expressions for the stationary distribution of the best-effort network state under the min policy. We therefore turn to the study of these stationary distributions under a limiting regime on network size and topology. A similar approach has previously been successfully applied to loss networks (see [START_REF] Kelly | Loss networks[END_REF], [START_REF] Graham | Chaos hypothesis for a system interacting through shared resources[END_REF], and references therein), and to queueing networks in [START_REF] Vvedenskaya | A queueing system with a choice of the shorter of two queues-an asymptotic approach[END_REF], [START_REF] Delcoigne | Thermodynamical limit and propagation of chaos in polling systems[END_REF]. It is inspired by the so-called mean field models of statistical physics.

Mean field analysis in the present context is best illustrated by the star-shaped network of Fig. 2. This network has N/2 branches, each consisting of one inbound and one outbound link (thus implicitly N is an even number), and all links have unit capacity. Each route r connects the endpoints of two branches via the center node. It has an associated arrival rate 2λ/N and mean message length σ. The factor 2/N is introduced to make the total load on each link, ρ = λσ, independent of the number of links N. As discussed below, when N goes to infinity the number of ongoing transfers on any link becomes independent of the number of ongoing transfers on any finite collection of the other links (this was termed the "chaos propagation" property in [START_REF] Graham | Chaos hypothesis for a system interacting through shared resources[END_REF]). This allows us to derive fixed point equations for the probability distribution of the number of transfers in progress on any link.

A. Symmetrical star-shaped networks

Although amenability to a mean field analysis is a significant motivation for considering the star-shaped topology, it should be noted that it is also relevant to the study of real networks. Any overprovisioned links in a real network are largely transparent to the throughput of elastic best effort flows. Only bottleneck resources, typically located in the access network and within Web servers, need to be included in the network model. The star-shaped network may thus be considered to represent any network with a well provisioned backbone where throughput is limited by bottlenecks at the source and destination edges. For example, inbound links might represent Web server CPU, while outbound links correspond to the last hop of an ISP's interconnection network. This discussion not only motivates the consideration of such a topology, but also suggests that letting N go to infinity might indeed be realistic if N represents the number of Web servers over the Internet. Of course, there would be no reason in practice to assume symmetry. This assumption is introduced solely for reasons of tractability.

Although our focus is on the star-shaped topology, the mean field approach can be applied to other symmetrical topologies. It thus allows one to consider routes with more than two hops. The corresponding extended model is described in detail in Section IV-B below, where the corresponding fixed point equations are derived. Section IV-C then presents analytical results for the star-shaped network. Results of numerical investigation of the fixed point equations are reported in Section V.

B. Fixed point equations for large symmetrical networks

We use the following notation in the sequel.

• N: total number of links;

• L: length of a route through the network;

• R (N) : number of routes going through a given link; • X (N) ℓ : number of active connections on link ℓ ∈ L, in stationary state;

• x (N) r : number of active connections on route r, going through links r(1), . . . , r(L);

• λ: arrival rate on a link; • σ: mean message length;

• ρ def = λσ: load of a link. We have implicitly assumed here that the number of routes going through a link, R (N) , is the same for all links. We shall in fact assume further that the network topology is the same, as seen from any route. We do not attempt to give a formal (0,0,0,1,0) (0,1,0,0,0) (0,0,0,0,0)

(1,0,0,0,0) (0,0,0,0,1) (0,0,1,0,0) Fig. 3. A hypercube-shaped network with dimension 5. definition of this symmetry assumption here. The reader is referred to [START_REF] Graham | Chaos hypothesis for a system interacting through shared resources[END_REF] for a thorough discussion on the minimal symmetry assumptions required. Symmetry implies notably that each route has the same number of hops and the same traffic parameters. The star-shaped network discussed above constitutes an example of such a symmetrical network when L = 2 (with

R (N) = N/2).
It is more difficult to come up with meaningful examples of symmetrical networks supporting routes with L > 2: in particular, the network should not be fully connected, since routes longer than one hop would then be pointless. One reasonable model is a hypercube (Fig. 3) of large dimension, in which each edge contains two one-way links.

The hypercube is a classical structure with many symmetries. It is characterized by its dimension d. Its vertices are represented by d-tuples of 0s and 1s (e.g., (0, 1, 1, 0, 1)) and its edges connect two vertices differing in only one coordinate.

The total number of links in such a network is N = d2 d . The number of routes going through any link is

R (N) = L (d -1)! (d -L)! ,
where the only routes considered are the shortest paths between two vertices which differ in exactly L coordinates. Note that the results below do not depend on the precise topology of the network.

We now derive the fixed point equations. It should be stressed that this derivation is heuristic. We clearly mention which steps need further justification in the course of the derivation. We do believe that the equations are very good approximations, however, especially in view of the numerical and simulation results presented in the following section.

Assume now that ρ < 1 and that the system is in stationary state

X (N) = (x (N) r , r ∈ R ). For any k ≥ 0, the proportion of links in state k is α (N) k def = 1 N ∑ ℓ∈L 1 1 {X (N) ℓ =k}
.

By symmetry, it holds that

IP(X (N) ℓ = k) = IEα (N)
k , ∀ℓ ∈ L.

The chaos propagation assumption1 implies that α

(N)
k obeys a law of large numbers:

α k def = lim N→∞ α (N) k = lim N→∞ IP(X (N) ℓ = k) def = IP(X = k).
It appears that the dynamics of the system are driven by α

(N)
k , traditionally referred to as the mean field. The following notation will also be useful:

ᾱ(N) def = ∑ k>0 kα (N) k , ᾱ def = IEX ℓ = ∑ k>0 kα k .
In order to derive the equation satisfied by the limit stationary distribution α k , we must first describe the possible transitions for α (N) k . The two cases of interest are • arrival on a link with k connections:

α (N) k → α (N) k - 1 N , α (N) k+1 → α (N)
k+1 +

1 N

• departure from a link with k > 0 connections:

α (N) k → α (N) k - 1 N , α (N) k-1 → α (N) k-1 + 1 N
The transition corresponding to a new connection arrival has rate λ. The main problem is to compute the departure rate from a link ℓ, given that it has X (N) ℓ = k ongoing transfers. This can be written as

1 σ ∑ r∋ℓ IE x (N) r min ℓ ′ ∈r 1 X (N) r(ℓ ′ ) X (N) ℓ = k . (4)
Since the total number of routes is much larger than the size N of the network, we assume that the probability of having more than one connection on a route r is negligible 2 , and that the links on route r are independent, conditioned on {x (N) r = 1} 3 . The first property allows to rewrite (4) as

1 σ ∑ r∋ℓ IE[x (N) r | X (N) ℓ = k] IE min ℓ ′ ∈r 1 X (N) r(ℓ ′ ) x (N) r = 1 . (5)
Let j be a given link and let r be one route using link j, i.e., r ∋ j. The distribution of X (N) j , conditioned on there being one connection on r, is then

IP X (N) j = k ′ x (N) r = 1 = IE 1 1 {X (N) j =k ′ } 1 1 {x (N) r =1} IE 1 1 {x (N) r =1}
.

By symmetry, it is possible to sum both sides of the above fraction over all the routes going through link j,

IP X (N) j = k ′ x (N) r = 1 = IE X (N) j 1 1 {X (N) j =k ′ } IEX (N) j = k ′ IEα (N) k ′ IE ᾱ(N) .
Departure rate (5) then becomes, in view of the assumed independence property between the X r(i) given

x r = 1, k σ ∞ ∑ k 2 ,...,k L =1 1 k ∧ 1 k 2 ∧ • • • ∧ 1 k L L ∏ i=2 k i IEα (N) k i IE ᾱ(N) .
Taking the limit N → ∞, the invariant measure equations follow. We have:

-λα 0 + 1 σ α 1 u 1 ᾱL-1 = 0 (6)
and

λ(α k-1 -α k ) + 1 σ α k+1 u k+1 ᾱL-1 -α k u k ᾱL-1 = 0 (7)
for k ≥ 1, where

u k def = k ∞ ∑ k 2 ,...k L =1 1 k ∧ 1 k 2 ∧ • • • ∧ 1 k L L ∏ i=2 k i α k i . (8)
Equations ( 6) and ( 7) can be rewritten in a more concise form as

α k+1 u k+1 = ρ ᾱL-1 α k , ∀k ≥ 0. (9)
The two sets of equations ( 8) and ( 9) together constitute the fixed point equations we require. As noted in the introduction of this section, these equations do not depend on the topology of the network. The expression for u k can be simplified.

Let Y 2 , . . . ,Y L be random variables with distribution

IP(Y i = y) = yα y ᾱ , 2 ≤ i ≤ L,
and let Y def = max(Y 2 , . . . ,Y L ). Then (8) reads

u k = k ᾱL-1 IE 1 k ∧ 1 Y . (10)
Straightforward calculations yield

IE 1 k ∧ 1 Y = ∞ ∑ y=1 1 k ∧ 1 y IP(Y = y) = ∞ ∑ y=k 1 y(y + 1) IP(Y ≤ y).
The simplified form for u k is thus, from the basic properties of the minimum of independent random variables:

u k = k ∞ ∑ y=k 1 y(y + 1) y ∑ m=0 mα m L-1 . (11)
Note that, in the case L = 2 (i.e., for the star-shaped network), the original equation ( 8) is perhaps simpler than the equivalent expression [START_REF] Delcoigne | Thermodynamical limit and propagation of chaos in polling systems[END_REF]. It yields the following form for the fixed point equations:

α k+1 u k+1 = ρ ᾱα k , (12)
u k = ∑ y>0 (k ∧ y)α y . (13)
Inbound links Outbound links Fig. 4. The asymmetrical star-shaped network

Remark 1: When considering the star-shaped network as a model for Web transfers over the Internet, as suggested in Section IV-A, inbound links could be seen as the CPU of Web servers and outbound links as the last hop between the ISP's backbone and the end customers. It thus makes sense to relax the symmetry assumption we had made between inbound and outbound links, as the two types of bottlenecks are of a different nature. We might thus consider a star-shaped network with N in inbound links, N out outbound links, inbound (resp. outbound) links having capacity C in (resp. C out ), see Fig. 4. Assume the mean message length σ is the same for each two-hop route, and the link capacities C out are fixed. The arrival rate on each route has the form λ r def = λ/N in , and the load ρ out def = λσ/C out is less that 1. The capacity C in of a "backbone" link is chosen to ensure ρ in def = λσN out /C in N in < 1 is fixed when the size of the system grows. Then, as N in and N out increase, with N in /N out small, the inbound links have many active connections and a large capacity, while the outbound links remain in "normal" utilization. The same approach as above can then be applied, to yield the set of fixed point equations

α in k+1 u out k+1 = ρ in ᾱout α in k , α out k+1 u in k+1 = ρ in ᾱout α out k ,
where

u in k = ∑ y>0 α in y min k, C out C in y , u out k = ∑ y>0 α out y min C out C in k, y ,
and α in k (resp. α out k ) represents the proportion of inbound (resp. outbound) links with k ongoing transfers.

C. Analytical results for L = 2

While equation ( 9) looks superficially like a "birth and death process" equation, it is in fact non-linear due to the fact that u k and ᾱ both depend on the α y .

From [START_REF] Vvedenskaya | A queueing system with a choice of the shorter of two queues-an asymptotic approach[END_REF], one clearly sees that u k is increasing in k, and tends to ᾱL-1 when k → ∞. Therefore, α k is increasing as long as u k < ρ ᾱL-1 , and decreasing after that. This means that the α k form a modal distribution, which maximal value is attained at

k 0 = max{k > 0, u k < ρ ᾱL-1 }.
We now present analytical results on the solution of the fixed point equations for L = 2. The proof of these results can be found in [START_REF] Fayolle | A nonlinear integral operator encountered in the bandwidth sharing of a star-shaped network[END_REF]. It relies heavily on functional analysis. Equations ( 12)-( 13) have an unique solution for ρ < 1 and, when ρ → 1, the following asymptotic expansions hold:

IEX ≈ 1 (1 -ρ) 2 A , lim k→∞ ρ -k IP(X = k) ≈ (1 -ρ)B exp 1 (1 -ρ)A ,
where A and B are non-negative constants.

Thus, under the min policy, any link in the star-shaped network has a mean queue length which is one order of magnitude larger than for a single server queue with the same load (ρ/(1 -ρ)). Its tail distribution is still geometrical with factor ρ.

It is possible to give an expression for the constant A: if c and v are solutions of the following system of differential equations,

     zc ′ (z) + c(z)v(z) = 0, zv ′′ (z) + v ′ (z) = c(z), v(0) = 0, v ′ (0) = 1, c(0) = 1,
then A can be written as follows:

A = ∞ 0 c(z)dz = lim z→∞ zv ′ (z) ≈ 1.30.
Since this system is numerically highly unstable, it has proven difficult (with the "Livermore stiff ODE" solver from MAPLE) to derive a better estimate for A.

It is interesting to note that the function w(y) def = v(e y ) + 1 satisfies the so-called Blasius [START_REF] Blasius | Grenzschichten in flüssigkeiten mit kleiner reibung[END_REF] equation w ′′′ (y) + w(y)w ′′ (y) = 0, which describes a laminar boundary layer along a flat plate (see, e.g., [START_REF] Schlichting | Boundary layer theory[END_REF]).

V. NUMERICAL ANALYSIS AND SIMULATIONS

While the analytical results of Section IV-C give some good estimates, they are only valid in the heavy traffic regime ρ → 1. In addition, similar results for L > 2 are not available. We thus resort to numerical resolution of the equations to gain a better understanding of the performance of transfers across large symmetrical networks. The very form of the equations suggests the use of a fixed-point method for this numerical resolution: starting from a priori values (α (0) k , k ≥ 0), the algorithm computes the corresponding u (0) k from (8), and then new values α (1) k from [START_REF] Graham | Chaos hypothesis for a system interacting through shared resources[END_REF]. Provided special care is taken to avoid instabilities, the iteration of this process converges rapidly (less than 100 steps). Sample results are shown in Fig. 5, corresponding to a large symmetrical network with routes of length L = 20. As clearly seen in the figure, the distributions are very different from what would be obtained for routes of length L = 1. In this case, the system consists of a collection of independent M/M/1 queues and the associated distribution {α k } is geometric. For L > 1, the {α k } distributions are markedly modal and the positions of the peak values are roughly proportional to (1 -ρ) -2 , a fact has only been proven in Section IV-C for the case L = 2. Moreover, since the shape of the distribution is rather narrow, this position roughly coincides with the mean number of active connection (as can be seen from the raw data).

The impact of route length is illustrated in Fig. 6. It seems that the mean number of active connections (which is again approximately the peak value of the distribution) is roughly proportional to log L. Note that a logarithmic growth rate is very slow suggesting that, beyond 2 or 3, the number of bottlenecks does not have a significant impact on mean transfer times. They depend much more on the load ρ.

The results presented so far only concern the solution of the fixed point equations. As mentioned earlier, there are gaps in the derivation of these equations. To assess their quality and to investigate the accuracy of the asymptotic approximation for finite size networks, we ran a number of simulations of the starshaped network. Fig. 7 displays the corresponding results when and infinite size (fixed point) for a star-shaped network with load ρ = 0.9.

the load on each link is set to ρ = 0.9 for a varying number of links. The agreement between the simulation results and the fixed point equation results is excellent for N = 100 links and improves as N increases.

VI. CONCLUSIONS

In this paper, we have considered a class of Markov processes called best-effort networks which constitutes a natural probabilistic model for evaluating the performance of document transfers over data networks such as the Internet. Unlike almost all previous work, this model accounts for the random nature of traffic: document transfers begin at the epochs of a certain arrival process and the size of each document is drawn from a given probability distribution. In the interests of tractability we assumed Poisson arrivals and exponentially distributed sizes. We introduced the "min" bandwidth sharing policy as a conservative approximation to the more classical max-min policy. Necessary and sufficient ergodicity conditions for best effort networks under the min and max-min policies have been established.

In order to pursue the analysis of the stationary distributions of the number of transfers in progress, we have resorted to large network asymptotics applying the mean field approach of statistical physics. This enabled us to derive fixed point equations for the probability distribution of the number of ongoing transfers on a given network link. The validity of these equations has been established by comparing their solution with the results of simulations.

Analytical and numerical results show how the mean transfer time depends on the number of bottleneck links and their load. The steady state distribution in networks where routes have several bottlenecks (L > 1) has a marked modal behavior. This is significantly different to the geometric distribution which holds when routes have a single bottleneck (L = 1). Performance is also much more sensitive to link load ρ for multiple bottleneck routes: as ρ → 1, mean transfer time increases like 1/(1 -ρ) 2 in the case L = 2, whereas the dependence is in 1/(1 -ρ) when L = 1. Finally, the impact of the number of hops per route L appears small (given that L > 1) compared to that of parameter ρ. This suggests that the star-shaped network is perhaps a sufficiently complex model, and that the study of symmetrical networks with L > 2 is less relevant.

The work presented here can be pursued in several directions. On the theoretical side, the analytical results presented in Section IV-C constitute a first step to understanding the solution of the fixed point equations which could be taken further. Another challenging theoretical question is to improve the fixed point equations in a rigorous way. On a more practical side, the fixed point equations might be simplified so as to find simple approximate formulas for mean transfer times as a function of key parameters (such as ρ in , ρ out in the case of the asymmetrical star-shaped network described in Remark 1). Such approximate formulas could then lead to engineering rules for capacity planning.

We view the present study as a preliminary investigation into the performance of best effort networks with multiple bottleneck links. A significant result of this investigation is the discovery that the extension of the processor sharing model valid for a single bottleneck proves to be very hard. There appears to be no simple parallel to the familiar fixed point techniques used in loss networks. The problem is, however, of considerable practical importance for providers seeking to engineer their network to ensure adequate throughput for document transfers. We hope therefore that this paper will incite further work and the development of alternative heuristic approaches.

Fig. 1 .

 1 Fig. 1. The linear network

Fig. 2 .

 2 Fig. 2. A star-shaped network

Fig. 5 .Fig. 6 .

 56 Fig. 5. Distribution of the mean-field probabilities IP(X = k) for L = 20 and different values of the load ρ.

Fig. 7 .

 7 Fig. 7. Cumulative distribution IP(X ≤ k) for different values of N (simulation)and infinite size (fixed point) for a star-shaped network with load ρ = 0.9.

We have not proven that this assumption holds. It seems, however, that the techniques developed in[START_REF] Graham | Chaos hypothesis for a system interacting through shared resources[END_REF] could be applied to prove that this is the case, provided the parameter R (N) goes to infinity with N.

[START_REF] Mo | Fair end-to-end window-based congestion control[END_REF] This fact is easy to prove in finite time, but requires more work for the stationary

regime.[START_REF] Massoulié | Bandwidth sharing: objectives and algorithms[END_REF] This is the point where the heuristic is not completely exact; it is however likely to be true when ρ tends either to 0 or 1.

This work has been partly supported by a grant from France Télécom R&D.