Alexis Front

Bernard Legrand

Guy Tréglia

Christine Mottet
email: mottet@cinam.univ-mrs.fr

Bidimensional phases in Co-Pt surface alloys: A theoretical study of ordering and surface segregation

Keywords:

) one (Co segregation). Among the different surface superstructures, most are simply the surface termination of bulk ordered phases and have been observed experimentally, but we predict also purely bidimensional phases such as the ( × 3

) or the c(2 × 2)(110) ones with no equivalence in the bulk alloy. Such ( × 3

) phase should be stable on nanoalloys where the (111) facets are prevailing. Finally, we show that the c(2 × 2) superstructure of the CoPt(100) surface, which represents the mixed variant of the L1 0 phase, remains stable above the bulk order/disorder critical temperature.

Introduction

Since the compilation of the thermodynamic properties of binary alloys written by [START_REF] Hultgren | Selected Values of the Thermodynamic Properties of Binary Alloys[END_REF] [START_REF] Hultgren | Selected Values of the Thermodynamic Properties of Binary Alloys[END_REF], there have been a lot of studies performed on the Co-Pt bulk phase diagram in the eighties [START_REF] Leroux | [END_REF][3][4] up to the beginning of the 21st century [5][6][7][8]. More recently, a review on Co-Pt nanoalloys [9] shows the huge amount of experimental and theoretical works on Co-Pt-based nanostructures. Such interest is due to the magnetic and catalytic properties of this alloy under the shape of nanoparticles. Indeed, small magnetic nanoparticles consist in a single magnetic domain which can be viewed as a macrospin and measured on individual nanoparticles [10], giving rise to potential applications in ultra-high density data storage, where CoPt nanoparticles as FePt ones are furthermore able to maintain their magnetization thanks to their high magneto-crystalline anisotropy [11][12][13]. This anisotropy competes with superparamagnetism effect leading to the fluctuation of the magnetic moment orientation with the temperature as described by the Néel-Brown model. The other interest comes from the electrocatalytic properties in fuel cells. It has been shown that Co-Pt system presents higher activity in the electrochemical oxygen reduction reaction as compared to pure Pt. Furthermore, the L1 0 ordered phase of the CoPt alloys has been measured to be seven times more active than the A1 disordered alloy [14].

Knowing the strong relationship between structure and properties, it is crucial to know and control the atomic structure and chemical redistribution of the different metal species in the core and at the surface of the nanoparticles. Therefore, there have been a lot of experimental [15][16][17][18][19][20][21][22][23][24][25] and theoretical [26][27][28][29][30][31] investigations to characterize the atomic structure and chemical ordering in Co-Pt nanoalloys. From a theoretical point of view, fully ab initio methods are too heavy to be implemented for large systems and even density functional theory (DFT) calculations, which can treat larger systems because of their approximation concerning the treatment of exchange-correlation features, are limited in the exploration of the configurations space. However they can give reliable results as reference for relatively small systems [26][27][28]. Statistical methods such as Monte Carlo simulations using Tight-Binding Ising model [30] or Cluster Expansion method [31] suffer from a rigid lattice approximation even if the cluster expansions obtained from first principle calculations are initially relaxed. An other approach to couple atomic relaxation and chemical ordering is to build interatomic potentials.

There exists only a few interatomic potentials for Co-Pt systems in the literature [32][33][34] with one developed within the Embedded Atom Method [35] and the other two within the second moment approximation in the Tight Binding framework [36]. We used the last one to study the structure and chemical ordering in CoPt nanoalloys [16,20,29,34]. However, this parameterization, focused on the equiconcentration, took into account neither the asymmetry of the phase diagram nor the experimental reversal in surface segregation as a function of the orientation.

To correct these drawbacks we build a new interatomic potential for Co-Pt systems and test it on the bulk alloy phase diagram and on the segregation profiles for the three low indices surfaces. In particular, for the Co-Pt system, similarly with the well studied Pt-Ni one, a surface segregation reversal is observed experimentally between the more compact surfaces, the (111) and (100) surfaces, where Pt segregates [37][38][39][40][41][42] and the (110) one where Ni [43] or Co [44,45] segregates forming a sandwich structure with oscillating profile. Many questions arise concerning the competition or synergy between the surface segregation of one species and the chemical ordered phases coming from the bulk. These questions are even more complex inside a nanoparticle where different facets with different orientations coexist in the same system [46]. The theoretical description of the structures for semi-infinite systems with (111), (100) and (110) surfaces as a function of temperature and concentration will help to the understanding of the structures of finite nanoalloys. In particular, we characterize a well defined 2D surface alloy with the ( ×

3)R30(111) which has no equivalence in the bulk, and which is mainly observed in intermetallic systems with strong ordering and surface segregation tendencies: Sn/Cu (111) [47], Al/Cu(111) [48,49], Sn/Pt(111) [50,51], Sn/Ni(111) [52] or Te/Cu(111) [53]. Such ( ×

3)R30(111) has also been observed on immiscible systems like Bi/Cu(111) [54,55], Bi/Ag(111) [56], Pb/Ni (111) [57] and Pb/Ag(111) [58][59][60]. In theory, we already obtained this surface chemical ordering on the (111) facets of Pd-Au nanoalloys [61], which was never pointed out before on nanoalloys. We predict the same 2D phases on the (111) facets of Co-Pt nanoalloys, which was not stabilized using on-lattice Ising model [30] or cluster expansion method [31]. We can guess that atomic relaxations and strain effects could have an influence on the stabilization of this 2D surface superstructure.

Our theoretical study is based on a semi-empirical Tight-Binding interatomic potential fitted to ab-initio calculations and experimental data. Numerical simulations are performed using Monte Carlo simulations in canonical and semi-grand canonical ensemble in order to describe systems at a given concentration (canonical) or at a given difference of chemical potential (semi-grand canonical) enabling to vary the concentration and temperature to describe phase diagrams.

The paper is devised in five parts including introduction and conclusion. The second part gives a description of the theoretical energetic model and atomistic simulations. The third and fourth parts describe the results respectively on the bulk and on the surfaces.

Energetic model and Monte Carlo simulations

We use Monte Carlo simulations in canonical and semi-grand canonical ensemble in order to characterize bulk and surface equilibrium phases along the whole range of concentration c in the Co -c

1 Pt c alloys from pure Co (c = 0) to pure Pt (c = 1). The energetic model is a many-body interatomic potential based on a Tight Binding model in the framework of the second moment approximation of the density of states (TB-SMA) for transition metals, fitted to ab-initio calculations in the density functional theory (DFT). The potential is also fitted in order to reproduce experimental results such as the order/disorder critical temperatures for Co-Pt bulk alloys and the surface segregation for low indices surfaces.

Interatomic many-body potential in the TB-SMA approximation

The energetic model is a semi-empirical many-body potential derived from the density of states of the d-band metals (transition metals) approximated to its second moment (width of the d-band) in the tight binding framework [36]. It leads to an attractive term with a square root dependence on the coordination number:

∑ = - < - ⎛ ⎝ ⎜ - ⎞ ⎠ ⎟ E e ξ i band j r r ab q r r , 2 2 1 ij ab c ab ij ab 0 (1)
for the band term at site i. a, b=(A,B) are the nature of the metal atom. r ij is the distance between the atom at site i and one neighbor at site j, r ab c is the cut-off distance and r ab 0 is the first-neighbor distance depending on the nature of the atoms. A repulsive term of Born-Mayer type is added to simulate the core repulsion:

∑ = < - ⎛ ⎝ ⎜ - ⎞ ⎠ ⎟ E A e .
The homoatomic parameters p ab , q ab , A ab , ξ ab with = a b are fitted to the cohesive energies, lattice parameters and elastic constants. The two elements have been stabilized in the fcc structure which is natural for Pt but not for Co which is hcp in bulk. The justification for Co is that when alloyed with Pt, the structure becomes fcc in the major part of the concentration range. The hcp structure with the present TB-SMA potential presents an energy excess of 30 meV/at. for Co with a = 2.54 Åand c = 4.14 Å, and of 10 meV/at. for Pt with a = 2.81 Åand c = 4.60 Å. The vacancy energy calculated within the present potential is equal to 1.5 eV for Co and 1.61 eV for Pt which are in good agreement with experimental results [START_REF] De | Cohesion in Metals: Transition Metal Alloys[END_REF] giving 135 KJ/mole (1.4 eV) for Co and 160 KJ/mole (1.66 eV). In the Co case, we notice that the DFT calculation overestimates the absolute value of the cohesive energy in the face centered cubic (fcc) structure by more than 20% as compared to the experimental value in the hexagonal compact packing (hcp) structure since the hcp calculated energy should be even larger (5.35 eV/at. calculated in fcc structure compared to 4.45 eV/at. in experiment with hcp structure). Better calculations including properly electronic correlations within the GGA+U approach reduce significantly the cohesive energy but too much since they underestimate the experimental value by 17% [START_REF] De La Pe | [END_REF]. In this exceptional case, we preferred to fit the cohesive energy and lattice parameter of Co relatively to the ones of Pt, keeping roughly the same difference as compared to the experimental data. This is a way to respect the fact that for transition metals and noble metals the GGA functional has the tendency to overestimate slightly the lattice parameter (around 2%) and to underestimate the cohesive energy (less than 10%). Concerning the surface energies, the experimental value representing the average (100) surface shows that when expressed in J/ m 2 the Pt has the lower surface energy as compared to Co but it is reversed when expressed in eV/at. In DFT, the Pt surface energies are always lower for any orientation and any units, which is in favor of the Pt surface segregation, as already obtained by Dannenberg et al. [64] and in agreement with the observed Pt surface segregation on (111) and (100) surfaces [38][39][40][41][42]. However, on the (110) orientation, the experiments show Co surface segregation [44,45], which has been a strong constraint to be fulfilled by the present potential. This explains why the TB-SMA surface energies are rather very similar for Pt and Co (at least when expressed in eV/at.) whereas a slight discrepancy arises when expressed in J/m 2 .

The heteroatomic parameters (a ≠ b) are fitted to the mixing energies calculated using the density functional theory (DFT-GGA/PW91) (see Tab. 3 and discussion in the following), and to experimental data such as the order/disorder critical temperatures [START_REF] Hultgren | Selected Values of the Thermodynamic Properties of Binary Alloys[END_REF]4] of the bulk alloys and the surface segregation of alloys surfaces [38][39][40][41][42][43][44][45].

The TB-SMA parameters of the new potential are given on Table 2 together with the ones of the previous potential [33] for comparison. We notice that the parameters of the pure metals are lower in the new parameterization whereas the A and ξ of the mixed interactions are higher than in the old parameterization. This leads to an increase of the mixing enthalpy as compared to the experimental one, to which was fitted the old parameterization (see Fig. 1).

The DFT calculations are performed using the VASP code with the generalized gradient approximation (GGA) in the PW91 version [65] and the projector augmented wave (PAW) [66,67] interaction potential between the ions and the electrons. The choice of the functional is crucial in DFT calculations and the GGA using either PW91 or PBE [68] functionals provide the best results among other functionals including meta-GGA and hybrid functionals as compared to experiments for what concerns structure and energy data [69,70] and also for what concerns their surfaces [71] if we want to get both surface and bulk properties. The s and d valence electrons are considered for each metal with a cutoff energy equal to 600 eV for plane wave basis set. The convergence criterion for electronic and ionic steps are respectively 10 -6 and 10 -5 eV. The Brillouin zone integration is performed with the Monkhorst-Pack scheme with k-point mesh comprising 18 subdivisions along each reciprocal lattice vector for pure bulk calculations when reduced to a mesh of 4 atoms in the slab. This mesh is sufficient to calculate the cohesive energies of pure metals and alloys with simple L1 0 and L1 2 phases. When considering impurities in the bulk or at the surface, the slab is duplicated two times in x and y directions and three times in the z direction so that the k-point mesh is reduced to 9 × 9 × 6 for impurities in bulk (48 atoms in the slab) and 9 × 9 × 1 for impurities at surface (same slab with six "empty" layers in the z direction). The cohesive energy, lattice parameter and surface energy have been calculated for Co and Pt, using spin polarized calculations in the presence of Co (see the results in Table 1). The mixing enthalpies are defined as:

= - - + H E n E m E n m Δ A B mix A B slab coh A coh B n m n m (3)
as a function of the concentration = + c m n m /( ) and are reported in Table 3 and Fig. 1. It is worth noticing that magnetism is the driving force of the chemical ordering in CoPt alloys [72], which justifies that our model implicitely takes into account the magnetism of the system. The dissolution enthalpies correspond to the limit of the dissolution of one impurity whereas mixing enthalpies for Co

= - - H E E Δ A B perm A B slab A slab ( ) n n 1 1 (4)
Then the dissolution enthalpy ΔH dis is equal to:

= + - H H E E Δ Δ ( ) A B dis A B perm coh A coh B ( ) ( ) (5)
The mixing enthalpies presented on Table 3 and Fig. 1 show that the Fig. 1. Mixing enthalpies of the bulk Co-Pt alloys at stoichiometries Co 3 Pt, CoPt and CoPt 3 for the DFT calculations (black circles) at 0 K, the experimental data [START_REF] Hultgren | Selected Values of the Thermodynamic Properties of Binary Alloys[END_REF] (black stars) at 900 K but still in the ordered phase (T c equal 1100 K for CoPt and 1000 K for CoPt 3 and some extrapolated points at c Pt =0.1 and 0.9), the TB-SMA potential (red diamonds) at 0 K and finally TBIM calculations at 0 K (green squares) used in ref. [30]. Full symbols represent the L1 0 and L1 Table 3 Mixing enthalpies for concentred alloys or dissolution enthalpies in the diluted limits Co(Pt) and Pt(Co), and lattice parameters calculated by DFT-GGA/PW91, TB-SMA potential and experiments [START_REF] Hultgren | Selected Values of the Thermodynamic Properties of Binary Alloys[END_REF]. In the diluted limits, the lattice parameters are the ones of the pure metal and Co is considered in the fcc structure. DFT-GGA/PW91 using spin polarized calculations underestimate the mixing enthalpies of the Co-Pt alloys as compared to the experiments at 900K. It would have been even worth if compared to the experimental values at low temperature (0K). On the contrary, the TB-SMA potential overestimates the mixing enthalpies in order to fit the critical temperatures. However, we notice on Table 4 that the overestimation by twice of the mixing enthalpies is not sufficient to get fully accurate critical temperatures. These temperatures are determined by canonical Monte Carlo simulations and are presented in the next section. If there is no analytical relation between the mixing enthalpies and the critical temperatures as in the TBIM model [73], there is a correlation between them. So we can't increase significantly the critical temperature without increasing also the mixing enthalpy. This is why we choose a compromise which is to get an overestimation by two of the mixing enthalpy and an underestimation also by two of the critical temperatures, while respecting the asymmetry between Co 3 Pt and CoPt 3 .

We finally compared the TB-SMA potential to the Tight Binding Ising Model that we have recently developed to study Co-Pt nanoalloys [30]. In that model, the effective pair interactions are evaluated in the diluted limit as the energy difference between two impurities separated by a given distance (first, second or third neighbors) and these two impurities at an infinite distance. We obtained values of V 1 , V 2 or V 3 which are in good agreement with the TBIM values (see Table 5). Notably we find a positive value for V 3 , which is necessary to stabilize the L1 0 phase as compared to the A 2 B 2 , knowing the V 2 are positive according to DFT calculations [30,31]. The mixing enthalpies in each model are plotted in Fig. 1 where we see that the DFT calculations underestimate the experimental values whereas the TB-SMA and the TBIM models overestimate them.

Monte Carlo simulations

We performed Monte Carlo simulations in canonical (N Co , N Pt , P and T constant) and semi-grand canonical ensembles (

= - μ μ μ Δ , Co Pt
N tot =N Co +N Pt , P and T constant) where atomic displacements are proposed and accepted according to a Metropolis sampling [74] insuring to reach a Boltzmann distribution of the chemical configurations at equilibrium. In canonical ensemble, the concentration remains constant and the Monte Carlo trials consist to exchange the positions of two atoms of different nature. In the semi-grand canonical ensemble, the difference of the chemical potential of the two elements remains constant, which determines the average concentration at equilibrium and the Monte Carlo trials consist to permute the chemical nature of one atom. In each case we keep the same number of atoms in the simulation box.

For such simulations, we performed five to ten thousands macrosteps after leaving the system being equilibrated during two to five thousands macrosteps. Each macrostep consists of proposing randomly to any atom of the box, either a displacement or a chemical exchange (exchange or permutation), and a certain number of variations of the size of the box in case of bulk or surface simulations. In the bulk case, where periodic conditions apply on each direction, the lattice parameter of the alloy lattice increases from the one of the pure Co to the one of the pure Pt. In the case of the surfaces, which are simulated by a slab with periodic conditions in two directions parallel to the surface, the core of the layer (60% of its thickness) is equilibrated according to the bulk alloy lattice parameter, fixing the lateral surface lattice parameter whereas the remaining planes at the vicinity of the surfaces can relax perpendicularly to the surface. If M steps is the number of macrosteps, n dep and n ex the numbers of displacements and chemical exchanges per atom, and n box the number of trials for box size variations, the number of microsteps N steps is equal to:

= ⎛ ⎝ + + ⎞ ⎠ N M N n n 2 n steps steps atom dep ex box (6)
We take n dep =10, n ex =1 and n box =100. For each statistical ensemble, we performed a series of simulations either with increasing (or decreasing) slowly the temperature in canonical ensemble or the chemical potential difference in semi-grand canonical ensemble, starting each time from the last configuration.

Co-Pt bulk phase diagram

Order/disorder transition temperatures

We characterize the bulk order/disorder transition temperatures using canonical ensemble. Heating/cooling Monte Carlo simulations start respectively with ordered/disordered phases. The chemical ordered phases are characterized by the occupancy of the four sublattices constituting the face centered cubic lattice mesh. On the Fig. 2, we see that starting with the Co 3 Pt ordered phase, the heating simulation

Table 5

Effective paire interations in both dilute limits calculated by TBIM [30], DFT and TB-SMA. In the TBIM model, the effective paire interactions do not depend on the concentration. In the DFT calculations, only the Pt dilute limit, Co(Pt), is reported since in the other limit, the two Co impurities interact with a strong magnetic coupling which makes confused the significance of the effective paire interaction in that case. curves present one over four sublattice fully filled with Pt and the other three fully filled with Co atoms. This Co 3 Pt L1 2 phase becomes fully disordered at 500 K (following the heating curve on Fig. 2). Looking at the cooling simulations, starting with the disordered phase at 25 % of Pt and high temperature, we get the ordering at = T 400 K with one sub- lattice fully occupied by Pt, the other three with Co. The order/disorder transition temperature of the Co 3 Pt L1 2 phase lies in between, around 450 K, taking the average between heating and cooling transition temperatures. For CoPt as for Co 3 Pt, we notice a slight hysteresis when heating and cooling which is not surprising as the order/disorder transition for infinite system is typically a first order transition. However, there is no hysteresis for the CoPt 3 , with the order/disorder transition being equal to 500 K. Finally in the case of CoPt, the cooling simulation leads to partially occupied sublattices which comes from an anti-phase boundary with two disoriented L1 0 phases, which can happen in Monte Carlo simulations. The critical temperature of the CoPt phase is estimated to be around 540 K. These temperatures are compared to the experimental ones in the Tab. 4. We notice that the anisotropy between the Co 3 Pt and CoPt 3 phases is respected even though all the temperatures are underestimated by 50% as compared to the experimental ones. There is really a weakness of the model to fit in the same time the mixing energies and the critical order/disorder temperatures that we have already noticed for example for the Pd-Au system [61]. The mixing enthalpy has to be overestimated in order to get reasonable critical temperatures as compared to experiment. The right critical temperature is achieved only in the TBIM model (see Fig. 1) for which the mixing enthalpy is the larger.

(in meV) V 1 V 2 V 3 V 4

Phase diagram

The characterization of the Co-Pt bulk phase diagram is performed using semi-grand canonical isotherms simulations which means that we increase (inversely decrease) step by step Δμ starting from pure Co (inversely Pt) system and going to pure Pt (inversely Co) one. A typical example is illustrated on Fig. 3 where the sublattice occupations are plotted as a function of the concentration of Pt in the box. We notice that around the stoichiometric compositions corresponding to Co 3 Pt, CoPt and CoPt 3 , the system is able to stabilize non-stoichiometric phases in a certain range of concentration. There are also concentration ranges notably around 0.20 and 0.8 where no phase is stabilized. This corresponds to phase-coexistence domains or biphasic domains. Indeed, the system does not find any equilibrium concentration corresponding to equilibrated phases in a given range of chemical potentials, except in the case where the simulation box displays itself two phases with an antiphase boundary inside the box, but this would represent an energy excess meaning that the Monte Carlo did not converge towards the minimum energy.

The complete bulk phase diagram is obtained from a series of isotherms at different temperatures, as shown in Fig. 4. The biphasic domains are depicted by hatched domains. We also observe new ordered phases at c=0.375 (Co 5 Pt 3 ), 0.625 (Co 3 Pt 5 ) and 0.875 (CoPt 7 ) which are the same as revealed by the TBIM study [30]. However the present phase diagram obtained within the new TB-SMA potential differs sensibly from the TBIM bulk phase diagram (cf. Fig. 6 of [30]) at least by three important points. The first one is the critical order/disorder transition temperatures which are underestimated by the TB-SMA potential as compared to the TBIM where they have been properly fitted. The second point is the asymmetry of the TB-SMA bulk phase diagram which is more similar with the experimental one. Third, the Co 7 Pt is not stabilized by the TB-SMA potential at very low temperature (less than 300 K) as compared to the TBIM, noting that none of these additional low temperature ordered phases at c=1/8, 3/8, 5/8 and 7/8 have been observed experimentally.

Co-Pt alloys surfaces

One particularity of the Co-Pt system, as the Ni-Pt one [37,43], is to present a surface segregation reversal between the dense (111), (100) alloys surfaces and the more open (110) one. Indeed, it has been observed experimentally by Low Energy Electron Diffraction (LEED) that on (111) Co 3 Pt and CoPt 3 alloys surfaces [38,39], and on (100) Co 3 Pt and CoPt 3 ones [40][41][42], the surface is enriched in Pt whereas on the (110) Co 3 Pt one [44,45] the top surface is almost pure in Co followed by a Pt enrichment in the subsurface layer and again a Co enrichment in the third layer. This structure has been called the sandwich structure with a stacking of almost pure monoatomic layers Co/Pt/Co [43,44]. This face-related segregation reversal when going from the (111) to the (110) face has been reproduced theoretically by different energetic models going from the empirical embedded atom method (EAM) [75], to a density functional theory calculation coupled with the coherent potential approximation (CPA) [76], through the semi-empirical Tight-Binding Ising model (TBIM) [77] in the case of the Pt-Ni system, but essentially in a mean-field approximation. Here we use a TB-SMA semiempirical potential fitted to DFT calculations and Monte Carlo simulations giving both long-range and short-range ordering.

(111), (100) and (110) Co-Pt surface at high temperature

We recall that the parameters of the TB-SMA energetic model have been fitted to reproduce the surface segregation reversal for the Co-Pt system. In Fig. 5, we plot the concentration of the top surface layer (c 0 ) and three layers below (c 1 , c 2 and c 3 ) as a function of the Pt concentration in the slab for the disordered alloy (high temperature phase). The slabs for the different surfaces consist of twenty atomic planes and around forty atoms per plane. In order to get the segregation reversal, we constrain surface energies of the pure metals to be almost identical (at least in eV/at., see Table 1) on each surface orientation. Therefore, the difference in lattice parameter (size effect) and the alloy effect are the main driving forces for surface segregation. This leads to a weak Pt surface segregation on the (111) and (100) alloys surfaces, stronger in the dilute limit because the two driving forces are more important, and a clear Co surface segregation on the (110) surface in the range of composition between 30% and 75% of Pt. The segregation is accompanied by an oscillation profile well established in the (110) surface in the composition range 30-75% of Pt which is in good agreement with the sandwich profile obtained in the experiments.

The volume of the slab is optimized during the Monte Carlo simulation in order to keep the constant pressure equal to zero. As the slab has to reproduce a surface i.e. a semi-infinite crystal we need to optimize its thickness in order to get the same lattice constant in the middle of the slab as in the bulk. This is what is illustrated in the Fig. 6 for the (100) surface of the CoPt 3 alloy at 800 K. Depending on the thickness of the slab (going from 20, 40 to 80 layers), we see that the lattice mesh in the slab converges towards the one of the bulk. We considered that the 20 layers slab (for the (100) slab) is sufficient to give the correct lattice parameter and surface stress at the surface. We get the same results if we fix the lateral dimension of the slab to the one of the bulk alloy with same composition and same temperature. Apart from the core of the slab, the surface undergoes an atomic contraction illustrated at the bottom of Fig. 6 with the relaxation profile of the CoPt 3 alloy (111), (100) and (110) surfaces at 300 K.

(111) Co-Pt surface at 300 K

The (111) surface is simulated by a slab of twenty atomic planes with thirty six atoms (six by six) per atomic layer as illustrated on the Fig. 7, where the snapshots represent the top layer. The slab is divided in three regions in its thickness: the core comprises twelve layers centered in the middle of the slab whereas each surfaces on top and bottom Fig. 5. Segregation isotherms at 650 K for the (111), ( 100) and (110) surfaces. The first diagonal corresponds to no segregation. The four curves with different colors represent the concentration of different atomic layers: the surface layer, c 0 (in black), the subsurface layer, c 1 (in orange), and so on for c 2 (in cyan) and c 3 (in violet). The Monte Carlo simulations for the (110) surface are less regular than for the more compact surfaces so we used dotted line showing the results are more dispersed. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.) Fig. 6. Top: lattice parameter in the bulk and (100) surface slabs of 80, 40 and 20 atomic layers in the x, y, z directions for the CoPt 3 alloy. The z directions for the surface slabs are in the middle of the slab to be compared to the bulk. The x and y are averaged on the whole slab. Bottom: vertical relaxation profile in the CoPt 3 alloy surfaces at 300 K.

comprises four layers each. The central region of the slab has the same properties as the bulk. In particular, the lattice parameter is the one of the bulk of the same composition (obtained by the variation of the size of the box in the Monte Carlo simulations). We checked also that we recover the same ordered phases as in the bulk calculations. In the following and for all the surfaces, the "bulk" Pt concentration c Pt refers to the Pt concentration in the core of the slab (i.e. in the corresponding bulk concentration of a semi-infinite system). In Fig. 7, the top layer concentration c 0 presents different plateaus (a), (b), (c) and (d) which characterize different surface ordered superstructures depending on the bulk Pt concentration. Below 15% of Pt, the bulk alloy is disordered and there is a slight Pt surface segregation illustrated by c 0 slightly higher than the other layer concentrations which follow the first diagonal. From 25% of Pt, the first plateau (a) corresponds to a Pt surface segregation of 33% for the Co 3 Pt alloys with the L1 2 phase. This result is consistent although not fully in agreement with an experimental observation on the Pt 25 Co 75 (111) alloy surface [38] where the Pt enrichment is estimated to be around 50% but on a disordered alloy. The experiments measure also a depletion of Pt in the second layer and a slight enrichment of the third layer, the rest of the layers being homogeneous. This is in qualitative agreement with our model although the oscillations of composition in our model are weaker. However, the experiments performed on the disordered alloy do not observe any long-range ordering at the surface, but a short-range ordering locally arranged with a (1 × 2) unit cell. Here we obtain the

× ( 3
3)R30 bidimensional ordering characterized by nine different surface sublattices plotted in Fig. 7 where three of them are filled with Pt and the other six, with Co. The × ( 33)R30 superstructure is not a pro- longation of the L1 2 bulk ordered phase but it takes place all over the L1 2 bulk phase domain, as can be seen with the correspondence of the surface sublattices and the bulk sublattices (on Fig. 7).

To illustrate the correspondence between the surface superstructures and the bulk phases, we show on a scheme (Fig. 11) the succession of the 2D surface phases in parallel with the 3D bulk phases as a function of the Pt bulk concentration. At the beginning of the L1 0 bulk phase domain (at about 40% of Pt), the p(2 × 1) surface structure takes place, which is characterized by two surface sublattices occupied by Pt and the other two by Co (on Fig. 7), but c 0 still increases up to c 0 =0.5 reaching a small plateau in (b) for the stoechiometric L1 0 phase. At the equiconcentration, the p(2 × 1) superstructure is basically the Fig. 7. Segregation isotherms at 300 K for the (111) surface. The first graph represents the surface layers Pt concentrations with c 0 (in black) for the top surface layer, c 1 (orange) for the subsurface layer, c 2 (cyan) for the third layer and c 3 (violet) for the fourth one. In the middle row are plotted the surface sublattices with 2x2 i.e. 4 sublattices (graph on the left) and 3x3 i.e. 9 surface sublattices (graph on the right). Bottom graph recalls the bulk sublattices as in Fig. 3. All these graphs are reported as a function of c Pt which represents the bulk Pt concentration equivalent to the one in the core of the slab. Finally, some snapshots of the top of the slab illustrate some peculiar surface structures (a), (b), (c) and (d). Pt atoms are in blue and Co ones in red. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.) Fig. 8. Segregation isotherms at 300 K for the (100) surfaces. Same graphs as Fig. 7 except there are only two surface sublattices (c α 0 and c β 0 ) so we plotted also the two equivalent sublattices of the subsurface layer (c γ 1 and c δ 1 ) to recall the ones of the bulk.

cut of the L1 0 bulk phase along a (111) plane. We also notice that the concentration of each layer near the surface (c 0 , c 1 , c 2 and c 3 ) are precisely equal to 0.5. By increasing the Pt concentration, in the over stoechiometric part of the L1 0 phase, c 0 increases up to a large plateau for c 0 =0.66 (c) corresponding to the Pt-rich

× ( 3
3)R30 2D phase. Such superstructure is stabilized allover two bulk phases: the over stoechiometric L1 0 phase and the under stoechiometric L1 2 one. Around 66% of Pt in the bulk, the surface concentration increases up to an other step for c 0 =0.75, the last remarquable plateau (d), where all the surface layers concentration are equal to 0.75 and the surface sublattices are characteristic of the Pt-rich p(2 × 2) (d). This surface ordering is the perfect termination of the L1 2 stoechiometric bulk phase along the (111) orientation.

Finally, just above 75% of Pt, there is a noticeable Pt enrichment corresponding to the over stoechiometric CoPt 3 phase and forming an almost pure Pt surface layer after 80% of Pt in the disordered alloy. This last result is in good agreement with LEED measurements [39] except that once more the oscillating profile measured experimentally is more accentuated than in the model. Indeed the experiment gives a subsurface depletion of 50% of Pt whereas we get only a very slight effect within the model. The surface and the bulk below are chemically disordered according to the solid solution in the bulk phase diagram. Now, if we compare the present model with a more simple on-lattice Ising model [30], we notice two main differences. With the symmetric Ising model we get essentially two types of superstructure: the p (2 × 2) for Co 3 Pt and CoPt 3 alloys and the p(2 × 1) for the CoPt alloy. Within the TB-SMA, the

× ( 3
3)R30 type of superstructure appears on both sides, which could be the result of a coupling between chemical ordering and atomic relaxations. Such structure has already been reported in a theoretical study on the Pd-Au system [61], with an interatomic potential allowing atomic relaxations. The second difference is that the Pt surface segregation was rather amplified with the Ising model. But in both models, the oscillating profiles are less pronounced than what is observed experimentally.

The

× ( 3
3)R30 surface structure is a 2D chemical ordering which has no relation with the underlying bulk alloy and has been seen by atom deposition in ultra-vacuum conditions on pure metal surfaces in systems like Sn on Pt(111) [50,51,[78][79][80], Sb on Cu(111) and Ag (111) [81], Bi on Cu(111) [54,55] or Ag(111) [56], Te on Cu(111) [53], Sn on Ni(111) [52], and Pb on Ni(111) [57] or Ag(111) [58][59][60]. It has also been obtained from surface segregation of solid solutions of Sn or Al in copper-based alloys single crystal surfaces: α-Cu .95 Sn .05 (111) [47] and α-Cu .84 Al .16 (111) [48,49]. The characteristics of the

× ( 3
3)R30 surface ordering is that it has no connection with long-range order in the bulk. The substitutional alloys are stable against the adsorbed systems as soon as the segregated atoms are in minority. These systems are mainly composed of one transition metal and one semiconducting or bad-conducting elements: Sn, Sb, Te, Pb, Bi or Al, forming either Fig. 10. Segregation isotherms at 300 K for the (110) surfaces. Same as Figs. 7 and8. intermetallic compounds with strong ordering tendency leading to well defined ordered compounds as the well known Cu-Al system but not necessarily since there are also immiscible systems mainly with Pb and Bi. To our knowledge, there is no reported experimental observation of the × ( 33)R30 surface ordering on purely metallic alloys. And yet, this structure is theoretically predicted using mean-field Bragg-Williams approximations in systems with an ordering tendency (first neighbors interactions) and surface segregation of one element (the bigger one) [51,80,82].

(100) Co-Pt surface at 300 K

The dimensions of the slab are a three by three lattice mesh which means thirty two atoms per atomic plane and twenty one atomic planes. The surface concentration c 0 , plotted in the first graph of the Fig. 8, presents two plateaus and starts with a Pt surface segregation in the Corich domain. The surface is enriched in Pt as compared to the Pt concentration in the bulk (core of the slab). We notice a slight oscillating profile which concerns essentially the top surface and subsurface layers. The layers below are homogeneous and equal to the bulk sublattices concentrations, up to a global composition of 15% after which the surface displays a miscibility gap as in the bulk between 15% and 25% of Pt.

At the right limit of the miscibility gap, corresponding to 25% of Pt (a), the bulk is ordered according to the L1 2 stoechiometric Co 3 Pt phase, and the surface layers concentrations oscillate between 0.5 and 0 extending the L1 2 phase up to the surface. The surface structure in (a) is the c(2 × 2) which is well characterized by the surface sublattices (c α 0 =1 and c β 0 =0). Increasing c Pt from 0.25 to 0.40, the surface concentration remains constant equal to 0.5 with the c(2 × 2) while c 1 follows the stoichiometric deviation of the L1 2 phase. The comparison with the experimental results performed by quantitative LEED analysis [40] is good for the four surface layers although the experiments are performed on a disordered Co 3 Pt(100) alloy. The surface reconstruction observed by LEED and chemically resolved STM is complex with c(2 × 2) domains limited by shifted rows producing the (1 × 5), (1 × 6) and (1 × 7) periodicity. The model reproduces only the c(2 × 2) superstructure without the right simulation conditions to get longer range periodicities (the slab dimensions should be extended beyond the three by three lattice mesh). A deeper investigation should be performed to look for the shifted-row reconstruction observed experimentally, but this question is beyond the scope of the present study. At 40% of Pt and beyond, the c(2 × 2) superstructure in (b) remains the same but the bulk phase underneath changes to the L1 0 which is naturally stoechiometric at 50% in which case the surface is made of the mixed variant of the L1 0 phase. Around 50% of Pd, in the out of stoechiometric L1 0 stability domain, c 1 , c 2 and c 3 accommodates the stoechiometry variations except c 0 which remains the same with the c(2 × 2) structure at the surface. The same behavior arises with increasing the composition from CoPt to the slightly under stoichiometric CoPt 3 phase: the surface remains the same whereas the subsurface layer follows the sublattice occupation of the Pt-rich L1 2 bulk ordered phase. All over this plateau of the c 0 curve, we can notice that the surface remains perfectly ordered with the c(2 × 2) superstructure whereas the layers below follow the bulk ordered phases with out of stoichiometry phases. The coexistence domains are so narrow in that side of the bulk phase diagram that we can not distinguish it on the alloy surface. The c(2 × 2) superstructure as shown in the Fig. 8 coexists with the three kinds of bulk ordered phases: the L1 2 Co-rich in the over stoichiometric part, the whole L1 0 domain and the L1 2 Pt-rich, in the under stoichiometric part. Such result is illustrated on the Fig. 11.

At the stoichiometric CoPt 3 composition, the surface concentration switches from 0.5 to 1 whereas in the same time the subsurface falls to 0.5, the rest of the slab remaining with the L1 2 ordered phase all over the over stoichiometric part, only the termination of the surface changed from the mixed to the pure layer in the alternation of mixed/ pure planes of the L1 2 Pt-rich phase. This ordered phase is stable up to about 83% of Pt, then all the layers below the surface become homogeneous to form the disordered bulk phase with Pt surface segregation. In that composition range, the layers beyond the surface are homogeneous and their concentration is equal to the bulk one in the disordered phase. The Pt segregation on the first surface layer of an alloy with 80% of Pt has been observed by Low Energy Ion Scattering Spectroscopy (LEISS) and X-ray Photoelectron Spectroscopy (XPS) [41,42]. The experiments show that the subsurface is enriched in Co following an oscillating profile. The comparison is not direct with our simulations because their sample was disordered (annealed at 900 K) whereas in the present case, at 300 K, the bulk is ordered with the L1 2 structure up to the alloy composition of 80% of Pt. However, just above 80%, the bulk becomes disordered and the subsurface composition is depleted in Pt as compared to the bulk composition (60% of Pt instead of 80%) just in agreement with the experimental results.

If we compare these results with the TBIM study on rigid lattice [30], the main difference arises at the equiconcentration alloy where the TBIM model preferred the pure variant (alternation of pure layers along the (100) direction), whereas the TB-SMA interatomic potential displays the mixed variant. The shift from mixed to pure surface happens at 75% of Pt, in coexistence with the L1 2 over stoichiometric CoPt 3 phase up to 83% of Pt and then in coexistence with the solid solution.

Finally, we have simulated the evolution of the different surface ordered phases (a), (b) and (c) of Fig. 8 as a function of the temperature in a canonical ensemble at constant concentration. The results are shown on Fig. 9 where we see the c(2 × 2) superstructure on the L1 2 phases, either Co-rich or Pt-rich, gets disordered at the same temperature as in the bulk alloy (see Fig. 2) i.e. around 500 K. This result is in agreement with other theoretical studies using mean-field Bragg-Williams theory [83,84]. However, we find that the c(2 × 2) on top of the L1 0 phase remains ordered until 800 K which means at a temperature 200 K higher than the bulk critical order/disorder temperature. This is a quite interesting result which should have important implications in the nanoalloys ordering at the equiconcentration, even if it has been shown experimentally that the order/disorder temperature of small nanoalloys of CoPt are well lowered [20] as compared to the bulk one.

(110) Co-Pt surface at 300 K

The (110) surface is simulated by a slab of forty one atomic planes with 36 atoms (6 by 6) per atomic layer as illustrated on the Fig. 10.

The surface segregation reversal between the dense (111) and (100) faces, and the more opened (110) one is clearly evidenced at low temperature up to 50% of Pt. The main experimental results showing such reversal concerns the Co 3 Pt(110) surface alloy [44] or more precisely the Co 65 Pt 35 (110) alloy [45] where the LEED analysis leads to a sandwich-like profile with a Co-pure surface followed by an almost Ptpure subsurface and still after a Co-pure third layer. The composition of 35% corresponds to the maximum of Co segregation in the present model at 650 K i.e. in the disordered state (see Fig. 5). At low temperature (300 K), the model predicts a sandwich profile in the concentration range between 30% to 50% of Pt as can be seen on the Fig. 10, which coexists with both the L1 2 Co-rich phase from 30% to 40% of Pt and with the L1 0 phase from 40% to 50% of Pt. In this last case, the sandwich at the surface is nothing else that the prolongation of the bulk ordered phases up to the surface (Fig. 10-(b)).

Looking at the curves in more details, we observe that between 25% and 30% of Pt in the bulk, the concentration of the subsurface (c 1 ) increases from 0.5 to 1, whereas c 0 and c 2 remain at 0 and c 3 makes a plateau at 0.5 (see Fig. 10 andstructure (a)). This alternation of pure and mixed layer near the surface propagates in the slab, which characterizes well the Co 3 Pt L1 2 phase in the bulk up to 40% of Pt. Between 30% and 40% of Pt, we get the sandwich structure in the first three layers, the next layer with concentration c 3 increasing from 0.5 to 1, together with the appearance of the L1 0 phase in the bulk (c 3 seems to vary according to the off stoichiometry of the L1 0 phase) up to the perfect sandwich structure prolongated in the bulk via the stoechiometric L1 0 phase at 50% (Fig. 10-(b)).

Beyond 50% of Pt, c 0 changes abruptly from 0 to 0.5 reaching an other large plateau consisting of a mixed surface. Looking at the surface sublattices in the middle graphs of the Fig. 10, there are two different surface structures with the same composition. The first one, from 50 to 65% of Pt in bulk, is characterized by four surface sublattices occupied two by two and corresponding to a c(2 × 2) superstructure (Fig. 10-(c)). This superstructure takes place on top of the L1 0 ordered phase, but without stacking relation with the bulk ordered planes. The second one, from 65 to 75% of Pt, is characterized by a p(2 × 1) (Fig. 10-(d)) which is the continuity of the L1 2 phase below and characterized by two surface sublattices: c α 0 =1 and c β 0 =0. The surface structure transition from the c(2 × 2) to the p(2 × 1) is therefore correlated with the bulk phase transition from L1 0 CoPt to L1 2 CoPt 3 .

Finally, at the end of the plateau, there is a continuous transition between the p(2 × 1) surface structure and an almost Pt-pure layer, the rest of the slab remaining with the L1 2 over stoichiometric ordered phase until 80% of Pt. Then, there is a Pt segregation with an oscillating profile, decreasing as a function of the depth, and all the sublattices are equally occupied to form the disordered bulk phase. The (110) surface structure is schematically represented on Fig. 11 together with the bulk phases.

The c(2 × 2) is typically a surface alloy since it does not have an equivalent in bulk layers, except in a DO 22 phase but which is not stable as compared to the L1 2 phase for the Co-Pt system. We also notice an important lattice relaxation perpendicularly to the surface which is correlated the surface segregation reversal in the (110) surface.

Conclusions

We presented a theoretical study of the three low indices surfaces of Co-Pt alloys as an interesting study to foresee the chemical ordering of nanoalloys terminated by ( 111) and (100) facets. The new parameterization of the TB-SMA interatomic potential, which could apply to a more extended categories of alloys such as Pt-Ni, represents a family of systems with a pronounced ordering tendency but a weak surface segregation leading to the segregation reversal with a Pt segregation on the compact (111) and (100) surfaces and a Co or Ni segregation on the (110) one. Using Monte Carlo simulations, we studied specifically each infinite surface in the whole range of concentration of the Co -c

1 Pt c alloys at room temperature. The alloy surfaces can present some new 2D surface alloys, especially the ( ×

3)R30 on the (111) and the c (2x2) on the (110), with no equivalent ordering in bulk, but in general the surface structures are essentially a bulk ordered phase termination, as the p(2 × 1) and p(2 × 2) on the (111) surface or the p(2 × 1) (110).

However, the phase transitions occurring at the surfaces, although correlated to the one in the bulk, are not fully trivial: for example on the (100) surface, the surface composition remains constant with the c (2 × 2) structure on a wide range of bulk concentration whereas the subsurface composition varies as a function of the variation of the off stoichiometry of the ordered phase below. The c(2 × 2) structure remains up to the under stoichiometric CoPt 3 phase. Then, exactly at the CoPt 3 stoichiometric phase, there is an abrupt transition to a Pt-pure surface which coexists with the over stoichiometric CoPt 3 phase which becomes disordered when increasing the Pt concentration.

In the (110) surface at low temperature there is a Co-pure surface in the composition range 25-50% of Pt in the slab, in coexistence with the over stoichiometric Co 3 Pt L1 2 phase and the under stoichiometric L1 0 one, up to the stoichiometric CoPt surface with an alternation of pure layers. Increasing slightly the Pt concentration, Pt atoms occupy half of the sites of the surface forming the original c(2 × 2) 2D surface ordering in coexistence with the over stoichiometric CoPt L1 0 phase, up to the transition with the L1 2 under stoichiometric CoPt 3 phase with the surface termination forming the p(2 × 1). As for the (100) surface, the over stoichiometric L1 2 phase forms a Pt-pure surface layer before the transition towards the solid solution.

Concerning the (111) surface, the situation could have been simple as the four bulk sublattices lie in the surface and we could have observed the succession of the p(2 × 2) Co 3 Pt, the p(2 × 1) CoPt and the p(2 × 2) CoPt 3 as obtained on a rigid-lattice model [30]. However, in the Co-rich part, the p(2 × 2) is not stable, instead we obtain the ( ×

3)R30 in all the composition range of the over stoichiometric L1 2 Co 3 Pt phase. More surprisingly, the ( ×

3)R30 in the Pt-rich domain coexists with both a part of the over stoechiometric L1 0 phase and a part of the under stoichiometric L1 2 phase. Thus, the transitions from the p(2 × 1) to the ( ×

3)R30 and then to the p(2 × 2) do not correspond to chemical ordering transition in the bulk. This is why we guess it could be driven by surface stress.

This study suggests new first principle calculations to investigate the stability of the 2D ( ×

3)R30(111) phase as compared to others [64]. We could also include temperature effects in DFT calculations as performed by Reuter and Scheffler [85] to compare different finite temperature structures obtained by TB-SMA Monte Carlo and by the so-called ab initio thermodynamics. We also expect new experimental studies for concentration ranges which were not investigated before. We rely on the extraordinary progress of the transmission electron microscopy with chemical species sensitivity to be able to characterize the facet composition of nanoalloys [46]: 1/3 or 2/3 could be a first indication of the ( ×

3)R30(111) phase as compared to 1/4-3/4 for the L1 2 termination.

Finally, we show for the first time that the CoPt c(2 × 2)(100) surface remains ordered at higher temperature than the L1 0 bulk phase. This phenomenon could influence the ordering of nanoalloys with (100) facets.

  3 Pt, CoPt and CoPt 3 correspond to the formation energy of the compounds at = c 0.25, = c 0.50 and = c 0.75. The dissolution enthalpy of B in a bulk of A is deduced from the calculation of the permutation enthalpie ( H Δ ) A B perm ( ) i.e. the energy of the large box (48 atoms with 47 of type A and one of type B equivalent to the permutation of one atom of type A with an atom of type B) as compared to the one of the same box with only atoms of type A:

TBIM 69 16 16 -Fig. 2 .

 162 Fig. 2. Sublattice occupations as a function of the temperature showing the order/disorder transitions obtained from canonical Monte Carlo simulations for stoichiometric Co 3 Pt, CoPt and CoPt 3 alloys; heating curves (lines) and cooling curves (dotted lines). The four colors refer to the four sublattices in the fcc structure.

Fig. 3 .

 3 Fig. 3. Isotherm at 300 K of bulk Co -c1 Pt c alloys obtained by semi-grand canonical Monte Carlo simulations. In such simulations, the Pt concentration at equilibrium is obtained as a function of the chemical potential difference= -μ μ μ Δ Pt Co .Here is plotted the chemical potential as a function of Pt concentration to illustrate the correspondence with the sublattice occupations in the graph below.

Fig. 4 .

 4 Fig. 4. Co-Pt bulk phase diagram. Beside the L1 2 , L1 0 and L1 2 phases referred in the graph, there are three other ordered phases at low temperature (below 200 K) which are the Co 5 Pt 3 at c=0.375, the Co 3 Pt 5 at c=0.625 and the CoPt 7 at c=0.875.

Fig. 9 .

 9 Fig. 9. Sublattices of the (100) surface as a function of the temperature in canonical Monte Carlo simulations for the three ordered phases: Co 3 Pt, CoPt and CoPt 3 from top to bottom, full line for heating and dotted line for cooling.

Fig. 11 .

 11 Fig. 11. Schematic representation of the surface structures on the (111), (100) and (110) orientations detailed in the Figs. 7-10.

Table 1

 1 DFT and TB-SMA calculations of the lattice parameters, cohesive energies, elastic constants and surface energies of the pure metals. Experimental values are taken from C. Kittel, Introduction to solid state physics, Wiley, New-York (1996), G. Simmons and H. Wang, Single crystal elastic constants and calculated aggregates properties, MIT, Cambridge (1971) and W.R. Tyson and W.A. Miller, Surf. Sci.62, 267 (1977). The structure is fcc in general except for the experimental Co which is hcp.

		a	E coh	B	c 44	c'	γ 111	γ 100	γ 110
		(Å)	(eV/at.)		(GPa)			(eV/
								at.)/(J/
								m 2 )
	Co
	DFT-GGA/PW91	3.52 -5.35				0.68 /	0.96 /	1.31 /
							2.03	2.47	2.40
	DFT-GGA+U[63] 3.55 -3.79
	TB-SMA	3.60 -4.90	196 88	26 0.54 /	0.67 /	1.00 /
							1.54	1.66	1.75
	Exp. (hcp)	3.54 -4.45	195 82	-	-	1.00 /	-
								2.59
	Pt
	DFT	3.98 -5.53				0.64 /	0.91 /	1.31 /
							1.50	1.85	1.87
	TB-SMA	3.98 -5.53	203 105	37 0.51 /	0.67 /	1.02 /
							1.20	1.36	1.46
	Exp. (fcc)	3.92 -5.86	288 77	52 -	1.19 /
								2.48
	a-b	p ab			q ab		A ab		ξ ab
	Co-Co	8.642		2.300		0.158		1.844
	Co-Co [33]	8.80		2.96		0.189		1.907
	Pt-Pt	10.796		3.198		0.199		2.232
	Pt-Pt [33]	11.14		3.68		0.242		2.506
	Co-Pt	9.719		2.749		0.275		2.488
	Co-Pt [33]	9.97		3.32		0.245		2.386

2 phases whereas empty symbols represent the DO 22 and A 2 B 2 phases. The curve (red line) represents the TB-SMA potential in the disordered state at 800 K. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.) -Table 2

TB-SMA parameters: the parameters of the present study compared with the published ones

[33]

.

Table 4

 4 Order/disorder critical temperatures obtained in our model as compared to the experimental ones[START_REF] Hultgren | Selected Values of the Thermodynamic Properties of Binary Alloys[END_REF][START_REF] Leroux | [END_REF]4,5].

	(in K)	Co 3 Pt	CoPt	CoPt 3
	TB-SMA	450	540	500
	Exp.	800	1100	1000

Acknowledgments

This work was granted access to the HPC resources of IDRIS under the allocation 2017-096829 made by GENCI.


V Chapurlat
email: vincent.chapurlat@ema.fr

B Kamsu-Foguem

F Prunet
email: prunet@lirmm

ENTERPRISE MODEL VERIFICATION AND VALIDATION: AN APPROACH

Keywords: Enterprise Modeling, Verification, Validation, Proof

This article presents a Verification and Validation approach which is used here in order to complete the classical tool box the industrial user may utilize in Enterprise Modeling and Integration domain. This approach, which has been defined independently from any application domain is based on several formal concepts and tools presented in this paper. These concepts are property concepts, property reference matrix, properties graphs, enterprise modeling domain ontology, conceptual graphs and formal reasoning mechanisms.

Introduction

Enterprise Modeling and Integration is now considered as an important research and application domain by industrial users. They dispose of several approaches (AMICE 1993, Loucopoulos 1995[START_REF] Petit | Formal Semantics of Enterprise Models[END_REF], Crestani 1997, Bernus 1998, Sheer 1998, Menzel 1998[START_REF] Geram | GERAM : Generalised Enterprise Reference Architecture and Methodology Version 1.6.1[END_REF], Tissot 1999, Vernadat 1999, Chen 2001, Chorafas 2002, Revelle 2002, Chen 2002) consisting of modeling languages, norms, architecture reference models and derivative tools allowing them to describe their processes, their information systems, their knowledge and knowhow, to better understand and to test the organization behavior by using for example simulation mechanisms, to communicate more efficiently in the enterprise, to exchange information and data without loss of sense with each other partners, to decide several system and organization improvements and so on.

Nevertheless, the user can doubt of the amount of confidence he can have in the different built models such as processes models or human resources ability models for example. Indeed, before considering a model is well suited to be use, this one must be bound by some phases during which it has to be verified (« is the model correctly built ? ») and to be validated (« is the model corresponds accurately to the reality ? is it the good one taking into account the needs and the context ? ») summarized Figure 1. Results of this global step of verification / validation (V&V) depends on two causes. First, the previous modeling step induces several problems on account of loss quality of the model which have to be taken into account during verification and validation:

• The user may not be clear in one's mind and his point of view may evolve during modeling process,

• Modeling hypothesis (temporal, behavioral rules and so on) limit model's expressiveness and accuracy. They are caused by system typology, system complexity level, by modeling concepts or formalisms which may be restricted to particular kinds of systems,

• The verification and validation need to highlight a user's defined analysis perspective (performance, temporal independence, functionalities, reliability and so on). The model may be unsuitable to respond correctly to some of the possible analysis perspectives,

• It stay often difficult to take into account the modeled system environment, its own dynamics, the different interaction between components and possible unforeseen events which may be the origin of some emerging and unexpected phenomena,

• Furthermore, the lack of knowledge and the oversight about particular characteristics, data and information page 2/16

from the system itself or from its environment must limit again the model V&V process.

What comes out of these causes is that a part of the knowledge needed for assuming the quality and the relevance of a model (from a static point of view in a first way then taking into account dynamic evolution rules) remains informal and misunderstands during modeling process.

Model Use

Model must be used for decision process, system improvement, communicating process, …

Figure 1 : A model life cycle vision highlighting verification and validation phases

This lack of knowledge may be justified by taking into account the user's behavior during modeling step. He can objectively rule out some information, data and events because of their nature, the time needed to describe them, their relevance considering his own point of view and so on. He can also forgot them because of its limited vision of the system or the different situations in which this system must evolve. It may be then interesting to develop some mechanisms allowing us to show the user the relevance of some of these information and data because of their possible interaction with the information and data gathered in the model it self in order to complete and/or to improve this model.

Second, each V&V phases shown Figure 1 set to work concepts and tools allowing syntactic checking (not considered here), semantic checking and behavioral analysis which use mainly simulation, emulation, human expertise of model execution results in particular in enterprise modeling and integration domain. The analysis results must be altogether questionable considering several points such as formalization and detail levels of the model for example. The way consisting on developing some other kind of tools allowing us to employ formal properties proof mechanisms seems to be interesting to use in this domain such as proposed in [START_REF] Van Der Aalst | Workflow Verification : Finding Control-Flow Errors Using Petri-Nets-Based Techniques, Business Process Management : Models, techniques and Empirical Studies[END_REF], Lemboley 2001, Covès 2000).

So, the research results presented in this paper intent to cover the V&V step by using a formal verification and validation approach allowing to improve user's knowledge about its model and to manage it in order to establish the relevance and the suitability of a model. The modeling and model/system improvements steps are not considered in this article.

Verification / Validation step approach

The proposed approach is based on several concepts detailed in the following. These concepts have been defined independently from a given application domain and, as we intent to show in the example at the end of this article, they have been specialized and putted into concrete form in Enterprise modeling domain. These elements shown in Figure 2 are:

• The property model is the base of a formal modeling language allowing the user to specify a particular knowledge he wants to prove on the model in order to verify and to validate it.

• A domain ontology (Uschold 1996, Fox 1994) describes a given whole application domain by defining the different concepts and relations between these concepts which are necessary.

• This ontology is then translated by using formal rules into two lattices needed for the V&V chosen tool based on conceptual graphs by splitting up concepts and relations.

• In same time, this ontology allows us to define a set of generic properties associated to the chosen domain and are gathered into a data base named Reference matrix. These properties cannot be proved directly on the model. Their goal is to guide the user, to remember him some crucial information which may be forgotten or misunderstand and to help him to specify completely the relevant and useful properties the model must respect.

• These generic properties will be interpreted, instantiated or translated respecting the model and will be gathered in the Property Graph. This one represents all the properties which have to be proven on the model.

• Some reasoning mechanisms have been developed taking into account Conceptual Graphs theory in order to prove the proposed properties on the model and to make emerging when it is possible some additional knowledge.

Property model

A property may be defined in the literature with several points of view (Feliot 2000, Meinadier, 1998, Thome, 1993, Manna 1982[START_REF] Henzinger | Temporal Proof Methodologies for Timed Transition Systems[END_REF], Manna 1982, Manna 1990). We will consider in the following the definition given by (Lamine 2001): a property translates an expectation, a requirement (behavioral, functional, structural or organic, dependent or not of time) or an objective (performance, safety, reliability and so on) which have to be respected, strictly or with a reliable level being enough by a model.

Informal model property

The proposed model property we will use in the following is based on a causal relation named R between two sets named C (the set of causes) and E (the set of effects). Each of these sets is composed by a collection of objects called facts issued directly from the model. As proposed by (Pearl 2000, Pearl 1999, Sowa 2000), the causal relation R indicates how occurrence of elements of E depends on the occurrence or on the interaction between elements of C. The property may be verified by computing in which cases a condition on the causes and the result on the effects are true by respecting temporal hypothesis and causal relation typology between causes and effects. Before presenting the formal model, it is necessary to define two particular concepts named facts and granularity.

Facts

Facts come from several origins:

• From the modeling language concepts (entities of modeling such as process, activities or resources) and relations between these entities (such as the link between activities and resources for example) as the modeling language evolution laws and rules. They are called handle functions. For example, the stateOf(A,t) handling function allows to define what is the state of the Activity A in time t.

• From the modeled system context and application domain. This needs to define a domain ontology allowing us to define the existing concepts, relations and possible situations in the domain which are not completely or already described into the model. All this added knowledge is then translated into new facts allowing us to merge it with the information coming from the model. They are called modeling variables and modeling parameters. It may represent for example the external temperature of the modeled system which is an input of the model but which has been forgotten during modeling step. It may also represent some external events from the environment or internal events such as data evolution.

• From the model itself. This needs to extract all the information contained in the model as new modeling variables and modeling parameters. It may be for example a modeling parameter containing the maximum level of water in a tank or a modeling variable containing a computerized data such as production rates or the input model vector state at each moment.

• From other existing properties in which the user can trust at a given moment: these properties are then considered by new facts as properties. This may be for example a property specifying what is the speed limitation taking into account some constraints at each moment.

At last, fact may be valued quantitatively or qualitatively (a property is true, a data is set to ' 30' or to 'good'). Most formally, the set of facts F is defined as follows:

F = MV ∪ MP ∪ HF ∪ P
where MV, MP, PR and P gather four kinds of facts :

• MV is the set of facts named modeling variables : each ones evolves within the modeled system:

MV={var / var = < name 1 , type 2 , value, Def 3 >}

• MP is the set of facts named modeling parameters : they described data which have constant values:

MP={par / par = <name, type, value>}

• HF is the set of facts named handle functions. They allow to manipulate all the facts and then to characterize 1 name is a tag defining each fact on an unique manner 2 type is the fact's type: , , Boolean, Character or structured type 3 def is the fact's definition domain (Def ⊂ Type)

the model or the environment dynamic and structure. For example, if the model is a transition model such as a Petri Net, it exists function allowing to describe net structure (before(place), follow(place), weight(arc), tempo(transition) and so on), to describe marking evolution (mark(net,t), fire(transition,t) and so on):

HF={hf / hf = < name, paramaters 4 , type >}

• P is the set of user-defined properties

Granularity

The property concerns a given and unique model. It is an interpretation of some user's requests and must be then proven only on this model However, establishing a complex system model need often to define a hierarchy of description by using decomposition or substitution rules. User may then define system's model(s) then sub(system(s) model(s) and so on in order to manage more efficiently the system complexity. In an other hand, in user's minds there exist some properties which may be assignable to only one detail level of the global system description but the interaction between these properties of a given level could be at the origin of the appearance (that is to say the emergence) of some new properties of a lower level. The granularity G is then defined in order either to allow the user to respect the different detail levels which are necessary to manage the system description, whether to allow this user to choose a particular sets of own detail levels. A granularity G is then the resulting set of chosen detail levels, called degrees.

The example presented Figure 3 shows a granularity allowing us to define four levels of decision from strategic one to execution one. Strategic level is the needed set of activities and processes allowing to define or to better understand enterprise finality and its objectives. Tactic level has to take into account some strategic decision and to define piloting orders and planning allowing the organization to reach these objectives. Operational and execution levels have then to schedule different production processes, to define stocks size, to manage production resources and so on. A temporal dimension may be associated to each level in order to help the user when we will define a property of a given level. At the Execution level, there are lot of exchanges of flows and several humans skills needed during the production process execution based on the resources cooperation. For example, in case of a long enough mechanical failure of a given machine (a given property is then not verified) it will be then possible that operational level intents to re define all scheduling orders which cannot be applied at the execution level.

Formal Property Model

A property Pr is defined by a 5-tuple:

Pr = < name, C p , R p , E p , D p >
Where :

• C p = { cause / cause ∈ F} / card(C p )≥0 (set of causes may be empty)

• E p = { effect / effect ∈ F } / card(E p )>0 (a tangible effect must always exist) and C p ∩ E p = ∅

• D p = < Type , G > is the degree of Pr in a given granularity G page 6/16

• R p is the relation defining the causal link between causes and effects. Type of R p may be of:

-Logical: it describes implication and equivalence (a reciprocity between cause and effect) relations.

-Temporal : it describes temporal links such as antecedence in which the cause must be prior to, or at least simultaneous with the effects.

-Influence : the knowledge about some cause modifies the opinion about the verification of the effect (Pearl 2000). It allows then to describe how causes and effects must be linked respecting some particular events or situation. A sense of variation is associated to each influence relation. It can be interpreted as beneficial or at the opposite harmful influence on the effect.

-Emergence : Each modeled system can be described by some characteristics which are not directly deductible from the own characteristics of its components but which result from relations between these components. The explanation of this kind of property needs then to take into account all the interactions and the feedback which connect the referent with its environment or with its context.

The relation R p is defined by the 4-tuple: R p = < Type , θ c , θ e , d > where:

• The Boolean functions θc describes in which conditions (by interpretation of causes), the property is verified. θc is defined as follows :

θc : C → {true, false} ; If there is an empty cause then θc = true

• The Boolean functions θe describes with which results (by release of effects), the property is verified. θe is defined as follows :

θe : E → {true, false}

Short examples

In a manufacturing process, an activity A i transforms one or several of its inputs in order to furnish one or several outputs taking into account constraints, rules and mechanisms as shown in Figure 4. This activity induces then modifications in the time, in the form or in the space of some of these inputs attributes which corresponds then to one several of the outputs attributes (Mayer 1995, Feliot 1997). This implies some relation between the inputs and the outputs. For example, if the output is of energy type, this induces that one of the input may be of material or of energy type in the same time. This kind of property may be wrote as follows: Last, the following property may allow the user to detect an error in the model. The property is the following: 'a transport link ensures continuity of parts flow in manufacturing system between two sub-systems'. Figure 5 shows the model which may respect this property. It shows a manufacturing process composed of a drilling activity and a polishing activity. If the drill station is not near the polisher station or if the two functions (to polish and to drill) are not assumed by the same machine, the user forgets some transportation activity and support as shown in Figure 6. The modeling language for the property specification has been defined and integrated into the LUSP Language support tool (French acronym of Properties Specification Unified Language) used for specifying, managing and formally proving a property (Chapurlat 2000).

Domain ontology

Before representing knowledge as properties, it is necessary to determine an ontology (Uschold 1997) dedicated to what we need to represent in a given application domain. Thus, the formalism chosen here was adapted to take into account the systemic concepts (Mayer 1995, Le Moigne 1999, CEA 1998, Braesch 1995) and the other concepts dedicated for Enterprise Modeling domain (Schlenoff 1996, Tissot 1999, ISO 2002) by the construction of an object class model by using the Unified Modeling Language (UML). This formalism has been used taking into account its readability, the opportunity to represent concepts and relation and the industrial recognition of this modeling language.

Thus constituted, the obtained set of concepts and relations can represent a common base of a modeling language allowing the user to describe each enterprise organization part. This point allows the authors to link and to justify the presented work concerning V&V approach with the current work in progress on Unified Enterprise Modeling Language (UEML) (Vernadat 2002, Vallespir 2003, see also www.UEML.org) which aims on the definition of a common enterprise modeling language. Figure 7 shows a simplified version of this concept lattice composed by a pre-defined set of concepts. It is divided into four parts: abstract types, behavioral types, modeling types and the entity types.

Properties reference matrix

As shown before, specifying a property needs to extract all the facts from the model and to add some new facts extracted from its environment by using the domain ontology. However, the specification work remains difficult for a non specialist user taking into account formal concepts such as those presented in (Van Lamsweerde 2000) and the model presented before.

First it is necessary to fix some idea about the goal of the V&V step which has to be done. There exist three possible perspectives of analysis (CEA 1998) which can be used independently from the other ones or which can be combined:

• Stability which aims on the ability of the system to maintain its viability all along its life (that is to say to maintain a sufficient relation between its structure and its coherence).

• Reliability which aims on the ability of the system to reach its objective (that is to say to ensure its function with a high level of compliance).

• Integrity which aims on the ability of the system to stay coherent and to be able to ensure its function.

Second, the goal is then to develop a properties data base called Properties reference matrix (Chapurlat 2002). It gathers usual knowledge describe by using property model considering a given application domain, a given set of modeling language and taking into account a systemic vision of the pointed out system and/or model to analyze. These properties cannot be proved on a given model because of these generic nature. They must be instantiated (the user create an occurrence of the generic property by filling in its causes, effects and relation) or to be simply interpreted (the user can be inspired by this generic property).

The property reference matrix uses a property typology which has been inspired from a literature and research works analysis such as (Paynter 1961, Lamport 1980, Manna and al. 1990, Manna and al. 1992, Sahraoui, 1994, Feliot 1997, CEA 1998, Lamboley 2001). This typology discerns three kinds of properties :

• System properties : These properties express the constraints and the functional or not functional requirements in which each system of a given application domain is (or will be) subjected and its assigned objectives. They are properties of functioning (temporal or not), of security, of volume, describing needed performance (productivity, availability, reliability and so on).

• Modeling Language properties : These properties describe the model structure (by defining and representing the modeling language construction rules and the possible existing constructs proposed in the modeling language), the behavioral laws (by defining execution rules effects on the model (Chapurlat 1999)) and the possible properties of liveliness, completeness, coherence, of reinitializing, describing the presence or the absence of parallelism, of synchronization mechanisms, of sequence, of temporary or definitive blocking and so on. For example, considering Petri Nets modeling language, the Model Properties describe structural rules concerning places and transitions placement, marking vector evolution, temporal hypothesis of synchronism or parallelism and so on.

• Axiomatic properties: They permit to describe basic knowledge, that is to say a set of information collectively and unanimously recognized and accepted such as laws of nature, norms, standards and also some existing property which is already verified for the user. Thus, they are indisputable and the user may choose them as facts for describing and proving other properties.

Properties graph

The properties reference matrix help the user to choose and to specify the relevant properties he wants to prove. The results of this specification phase is a graph in which:

• each nodes represents the different sets of facts considered as causes or as effects of a given property,

• each arc represents a typed relation between these causes and these effects .

This graph takes into account simultaneously the knowledge about the model, the modeled system and its environment. All this knowledge is described by using the unique property model (and then the modeling language LUSP). That allows to take into account all this knowledge simultaneously and with the same mechanisms of analysis. In order to structure the user's work a properties graph may be represented as a third axes area in which:

• The first axe called target allows to separate into three level of detail what are the different objects (systems or models of these systems) which represents the target of the V&V step or which interact with this target and can influence then the V&V results. These objects are chosen by the user by respecting the selected analysis perspective (stability, integrity, reliability or a combination of them). The central level called referent level highlight the pointed out system which has been modeled and the resulting model representing the V&V goal.

The upper level called upper referent allows to describe the pointed system's environment regarding the user's point of view. For example if the user has chosen stability perspective the upper referent may represent either all the encountered process in the company whether the process's customer inputs/outputs. For each referent level it may then possible to define several upper referent. The lower level is composed of other objects (sub systems, sub models) which compose an interacting network either corresponding to the decomposition whether having the same global behavior from the object belonging to the referent level. Each of these three levels is connected with the above respecting rules. These ones depends on decomposition rules imposed by a given modeling language if the referent level object is a model or user decomposition point of view if it is the system.

• The second dimension called typology permits to separate system and model properties. For each ones, it is then necessary to clarify the properties which are connected to the structural aspect, to the behavioral aspect or to the functional aspect expected from the object O chosen considering the target axe. Systems properties may be interpreted or instantiated taking into account the properties reference matrix. They represent the current state of the system or the needs at which the system must respond in term or performance and constraints. Model properties are issued from the translation of modeling language properties.

• The third axe is called time. Past, present and future of an object O must be taken into account in order to manage the possible evolution of the properties of O. It allows the user to reuse part of existing properties and to complete them during life cycle evolution of the target.

Figure 8 summarizes these three axes and each case may be considered as a property graph. Figure 9 shows an example of these different object. The referent level is then defined by a given process to improve and the user wants first to verify its model. Doing this needs to define what is the environment of this process and what the possible sub systems. In the same idea it may exists some model describing this environment or each of these sub systems. The information gathered into these new models must be then employed for the V&V phase.

Conceptual graphs

It is necessary to dispose of a set of mechanisms allowing us:

• To manage all the previous highlighted knowledge for V&V steps. As shown before, this knowledge occurs from user's specified properties which are gathered into the properties graph. However additional knowledge depending directly from the system or from its environment but which may be absent and not already defined into the property graph have to be added again.

• To prove formally the proposed properties. This phase may use different kinds of concepts and tools [START_REF] Rushby | Theorem proving and model checking for software, Tutorial of CSR/ENCRESS, Bruges, Belgium L.Lamport[END_REF]. It may be based on some theorem prover such as Zeves (dedicated only for the Z modeling (Barden

page 10/16
1994) language which is more appropriate for proving the structure coherence and adequacy of a given modeling language even a model than properties as proposed below). It may also use model checkers such as PVS (Owre 1999), Stanford Temporal Prover STeP (Bjorner 2001), SMV (see http://www-2.cs.cmu.edu/~modelcheck/smv.html), SPIN (Holzmann 2003) and other. All these tools are formal proof tools based mainly on state diagrams or transition models properties verification and data analysis. Their utilization, very limited in Enterprise Modeling domain, can then make easier proof of behavioral properties if the modeling language uses the same concepts of state, transition and temporal hypothesis. Last, this phase may use simulation or emulation tools allowing the user to execute the model by taking into account some pre-defined scenarios. This kind of solution is now a well known one in the industry but the problem comes first from the scenarios, second from the execution mechanisms used. Scenarios may be subjectively forgotten or wrong defined (for example, some particular and relevant events or system's situations may be ignored). Execution mechanisms i.e. the set of evolution rules are not sufficiently formally defined [START_REF] Petit | Formal Semantics of Enterprise Models[END_REF], Panetto 2001) in an indisputable way. So, the simulation may give some questionable results.

•

To make emerge new knowledge allowing the user to complete again and again its system's representation.

This research work intents to ameliorate and to extent the existing V&V concepts in Enterprise Modeling domain. So, the Conceptual Graphs have been chosen for the followings reasons (Kamsu-Foguem 2003). First, it allows to conceptualize the pre-defined domain ontology as a formal vocabulary. Second, it disposes of a range of formal reasoning mechanisms (rules, projection, first order logic isomorphism and other) permitting to verify facts and to complete (or to make emerging) a new knowledge by combination of existing knowledge.

The goal now consists to define the adapted vocabulary and to translate the model and the properties graph obtained before in order to use the reasoning tools associated to conceptual graphs.

Formalism Presentation

A simple conceptual graph is a finite, connected, directed, and bipartite graph composed of two kinds of nodes called concepts and conceptual relations. A concept is composed of a type and a marker:

[<type>: <marker>] where:

• type represents the concept typology which is necessary to describe a given domain. They are grouped into a hierarchical structure called concept lattice (see a simplified version of it in).

• marker specifies the meaning of a concept by specifying an occurrence of the type of concept. They can be of various natures, in particular individual or generic.

For example [Process: Customer needs definition] means that Customer needs definition is one the numerous processes of the enterprise. Its represents then an occurrence of the Process concept into the modeled domain.

A conceptual relation binds two or many concepts according to the following diagram: The pre-defined ontology is then translated into two needed lattices allowing to organize all the needed concepts and relations by using a hierarchical approach. The translation rules used are summarized in Figure 10 and Figure 11 and Figure 12 show a simplified version of each of these lattices. The Figure 13 illustrates how a given model is then translated by using these two lattices into a conceptual graph. Due to the paper size limitations, the translation rules are not presented here. The same rules are then used to translate each of the graph properties into several little conceptual graphs allowing the user to use now the formal mechanisms associated to the conceptual graphs in order to prove each of these property. (Sowa 1984) has established the existence of an isomorphism between the Conceptual Graphs and first order Logic. For that, an operator Φ has been defined in order to translate conceptual graphs into a set of first-order formulas manipulating predicate and logical relations:

Reasoning concepts and mechanisms : First order logic isomorphism and canonical formation rules

-Each conceptual relation between two concepts is translated into an n-ary predicate -Each concepts is transformed into an unary predicate -Each individual marker becomes a constant -Each generic marker becomes a quantified variable.

Thus, this operator provides a formal semantics to the conceptual graphs. In the following example, the sentence "James, the employee drills a part" which is represented by the conceptual graph G will have, as an equivalent logical formula Φ(G) : In addition, Sowa defined four elementary operations on the conceptual graphs, called canonical formation rules, which allows to handle them easily and to derive canonically from other graphs: copy, restriction, simplification and joint. These basic operations allows to handle and to transform graphs containing one information into other graphs that may contain a new unexpected information or which appeared not very interesting to the modeler: a new property can then emerge from the graph of existing properties and be thus proposed to this modeler.

Reasoning concepts and mechanisms : Projection

The handling operation called projection is the basic operation of a reasoning process in conceptual graphs. Projection corresponds to a graph morphism. The projection search of a graph G in a graph H can be seen as the inclusion search of the information represented by G in H. This leads to calculate a specialization between two graphs.

Reasoning concepts and mechanisms : Graph Rules

The conceptual graph rules (Salvat 1996) permit to represent knowledge in the form of inference rules of kind "If information H is in a graph, information C can be added to this graph". H and C are expressed as conceptual graphs related by co-reference links between some concept nodes. In other words a rule is composed of an hypothesis and a conclusion, and is used in the following classical way: given a simple graph, if the hypothesis of the rule projects to the graph, then the information contained into the conclusion is added to the graph. In this page 13/16 way, we define static rules to express some immutable domain laws and dynamic rules such as shown in Figure 14 to model the world evolutions with the change conditions of states.

Reasoning concepts and mechanisms : Constraints

A constraint defines conditions for simple graph to be valid. It is composed of a condition part and a mandatory part. The condition must be a simple graph. In particular, a condition can be an empty graph. Roughly said, a graph satisfies a constraint if for every projection of its condition part, its mandatory part also projects to the graph. We consider positive and negative constraints. A positive constraint expresses a property such as 'if information A is present, then information B must also be present'. A negative constraint expresses a property such as 'if information A is present, then information B must be absent'.

Hence, constraints are used to check validity of world. For example, let G be a simple graph. The graph G satisfies a positive constraint Pc if every projection of the condition of Pc into G can be extended to a projection of Pc as a whole. The graph G satisfies a negative constraint Nc if no projection of the condition of Nc into G can be extended to a projection of Nc as a whole. It allows to verify the graph G validity that is to say all the properties modeled as constraints are satisfied.

Application example

This way, conceptual graphs provides inference mechanisms for proving properties by using projection, rules and constraints. These demonstrative abilities make validation knowledge possible. For example, we consider an enterprise composed of some departments and of several workshops. The goal is to verify the property P 1 :

P 1 "a department and a workshop have one person in common at the most ".

For doing this, it is possible to use the negative constraint Nc (see) and the rule R 1 presented below: Nc "incompatibility of membership relation with non-membership relation" The proof by reductio ad absurdum of P 1 (by stating a proposition and then showing the resulting contradiction, thus demonstrating the proposition to be false) is as follows : supposing that in the company, there are two different persons x and y which both are the members of a department D and a workshop C, with C is not a subset of D. The previous rule R 1 tells us that all the persons of the workshop C are members of this department D, therefore C is a subset of D. This result belies our starting hypothesis, so it all goes to prove property P 1 . This proof mechanism is formalized in conceptual graphs as follows:

• The starting hypothesis of proof is represented by the conceptual graph Gh in Figure 16.

• The result of the application of rule R 1 to graph Gh is represented by the conceptual graph Gc in Figure 17. page 14/16

• The condition of negative constraint Nc is the empty graph, which can be projected into Gc. And this constraint is violated since there exists a projection of Nc as a whole into Gc. It' s a contradiction in terms, so the proof of property P 1 is established.

Conclusion and Perspectives

The property is an opportunity allowing an user to complete knowledge about a complex system and to formally investigate about it by using some formal mechanisms, concepts, ontology and tools supporting Verification and Validation step.

Evolution perspectives of this work are numerous but we will focus on the two following axes. First the proposed approach stays for the moment a little bit disturbing for an engineer or a consultant in charge of several processes. It is then necessary to develop some supporting and automated tools. Second it is necessary to integrate this approach in a global 'Enterprise Modeling and Analysis' method in which it will be possible to improve the V&V abilities of the main modeling approaches used in industry.

Figure 2 :

 2 Figure 2 : Global approach and components

Figure 3 :

 3 Figure 3 : Granularity and degrees

Figure 4 :

 4 Figure 4 : Activity seen as a transformation scheme

  [∀Activity.OutputInformation / OutputInformation.OperationalDomain = Energy] implication ( ) [ ∃ Activity.InputInformation, (InputInformation.OperationalDomain = Material ∨ InputInformation.OperationalDomain = Energy) ]

Figure 6 :

 6 Figure 5 : A part of a process model to check

Figure 7 :

 7 Figure 7 : Extract from one of the object diagrams showing the ontology

Figure 9 :

 9 Figure 9 : Property graph objects example

[

  C1] → (relation) → [C2] (means "C1 interact with C2 by relation")For example, [Activity: 'To specify needs list'] → (usage) → [Actor: Specialist] means that Specialist who is an Actor uses the activity named 'To specif needs list'. Each relation is characterized by a signature which fixes its arity and gives the types of concepts which are in relation.

Figure 12 :

 12 Figure 12 : Extract from Relations lattice

Figure 13 :

 13 Figure 13 : Use of lattices for model translation into a conceptual graph

G

  : [Employee : James] →(agent) →[Machine : drill] ←(object) ←[Part : *] Φ(G) : ∃x, (Employee(James) ∧ Machine(drill) ∧ Part(x) ∧ Agent (James, drill) ∧ Object(x, drill))

Figure 14 :

 14 Figure 14 : Example of dynamic rule

R 1 "Figure15:

 1 Figure15 :The negative constraint Nc

Figure 17 :

 17 Figure 16 : Graph G h of the starting hypothesis to proof

Concept (Generic)

Instance of concept issued from model

Relation

Concept (Generic)

Instance of concept issued from model Relation