Nicolas Bourasseau

O Desbiens

M Wilhelmsen

Hammer

V Lachet

B Creton

T De Bruin

E Bourasseau

N Desbiens

Ø Wilhelmsen

M Hammer

Equilibrium and transport properties of CO 2 +N 2 O and CO 2 +NO mixtures. A molecular simulation and equation of state modelling study

Keywords: CO 2 capture and storage, VLE data, Viscosity, Molecular simulation, Equation of state

Equilibrium and transport properties of CO2+N2O and

CO2+NO mixtures : a molecular simulation and equation of state modelling study.

Introduction

In Carbon dioxide Capture and Storage (CCS) operations, the captured CO 2 stream from industrial installations is not a pure CO 2 : it contains some associated compounds, also called contaminants, such as N 2 , O 2 , Ar, SO x , NO x ... [1]. This mixture of gases may have significantly different thermo-physical properties as compared to a pure carbon dioxide. This may have impacts on the different stages of the CCS chain: capture, transportation, compression, injection and storage. To globally account for this impact and for a precise specification of maximal amounts of contaminants that can be tolerated in CO 2 flues, further investigations are strongly desired. Obtaining accurate knowledge of the thermodynamic and transport behaviour of CO 2 +contaminant mixtures is part of the studies that are necessary in order to develop optimized carbon dioxide capture and storage processes. Some of these CO 2 +contaminant mixtures, like CO 2 +O 2 and CO 2 +N 2 for instance, have already been studied over a large range of pressures and temperatures. A comprehensive review of available equilibrium and transport properties of CO 2 +contaminant mixtures is proposed in two recent papers by Li and co-workers [2,3]. In the case of CO 2 +NO x mixtures, data are very scarce or nonexistent. As far as the CO 2 +NO mixture is concerned, to the best of our knowledge, no experimental data are available in the literature. Concerning liquid-vapour equilibrium of the CO 2 +N 2 O mixture, some experimental data are reported in the works of Caubet [4,5], of Cook [6] and of Rowlinson [START_REF] Rowlinson | Proceedings, International Conference Thermodynamics and Transport Properties Fluids[END_REF]. These liquid-vapour equilibrium data cover some temperatures between 277 and 311 K and pressures between 3.5 and 7.5 MPa. Excess properties of CO 2 +N 2 O mixtures are also reported in the literature: excess enthalpies and excess volumes have been measured by Wormald [START_REF] Wormald | [END_REF] and by Cabanas [9] for both liquid and vapour CO 2 +N 2 O mixtures. Concerning CO 2 +N 2 O viscosities, some values at atmospheric pressure are reported by Kestin [10] for temperatures in the range 298 -473 K and for N 2 O contents of 31 and 65 mol%.

The objective of the present work is to compensate the lack of experimental data by generating so-called pseudo-experimental data based on numerical experiments using molecular simulation calculations. Molecular simulation is a widespread technique which consists of performing a detailed simulation of microscopic systems and of calculating appropriate averages in order to derive macroscopic fluid properties. Phase compositions, phase densities and phase viscosities of CO 2 +N 2 O and CO 2 +NO mixtures have been determined using Monte Carlo and/or Molecular Dynamics simulations for temperatures in the range 253 -293 K. These calculations have been performed using force fields based on pure component properties and no adjustment on mixture properties was required. In order to achieve accurate predictions, new optimized force fields were proposed for both nitrous oxide and nitric oxide molecules. Finally, standard cubic equations of states have been optimized in order to reproduce the thermodynamic behaviour of both mixtures. Available experimental data or our new simulated data were taken into account in the parameter regression procedure. This article is organized as follows. In Section 2, the description of the simulation methods and expressions used to compute intermolecular potential energy are given. The new force fields developed to model N 2 O and NO molecules are presented in Section 3.1 where they are tested with respect to their ability to reproduce thermodynamic and transport properties of pure compounds. In Sections 3.2 and 3.3, we present and discuss the simulated results obtained for the two studied binary mixtures. Section 3.4 is devoted to the calibration of standard thermodynamic models. This paper ends with a fourth section which contains our conclusions.

Simulation methods

Monte Carlo method

The liquid-vapour equilibrium data of pure compounds and binary mixtures were obtained with Monte Carlo (MC) simulations using the GIBBS Monte Carlo code [START_REF] Ph | Applications of molecular simulation in the oil and gas industry -Monte Carlo methods[END_REF]. These simulations were performed in the Gibbs NVT ensemble (constant number of particles N, constant total volume V and constant temperature T) or Gibbs NPT ensemble (constant number of particles N, constant pressure P and constant temperature T) [START_REF] Panagiotopoulos | [END_REF]. In such simulations, the two phases are simulated in individual boxes without an explicit interface, and particle transfers between boxes are performed in order to ensure phase equilibrium. In most cases, the following Monte Carlo moves were used in order to sample the configurational space: -internal moves (rigid body translation and rotation of a molecule), -volume changes (opposite volume changes +∆V and -∆V are applied to the two boxes in the case of Gibbs NVT simulations in order to keep the total volume constant), -transfers between phases based on the pre-insertion bias. This pre-insertion bias algorithm, used to improve the efficiency of the sampling [13], involves two steps. The first step consists in the selection of a suitable location for inserting a new molecule by testing several places with a simple Lennard-Jones particle. The second step involves the test of several molecular orientations with the centre of mass at the location selected in the first step. In the case of NO-containing systems, one additional Monte Carlo move, the reaction move, has been used in order to account for the possible dimerization of two NO molecules into N 2 O 2 . These particular simulations are described in section 3.1.2. For all other studied systems, the selected probabilities for the different Monte Carlo moves are 0.3 for translations, 0.3 for rotations, 0.395 for transfers, and 0.005 for volume changes. A total number of 500 to 800 molecules has been considered in our MC simulations, depending on the proximity of the studied temperature to critical temperatures of involved species. A total number of at least 50 million Monte Carlo iterations have been performed for each studied condition.

For the different studied molecules, the molar enthalpy of vaporization was computed at different temperatures as the difference between the average molar enthalpies of the vapour and liquid simulation boxes. The statistical uncertainties of the calculated phase properties were estimated by the block averaging technique [START_REF] Allen | Computer simulation of liquids[END_REF]. The statistical uncertainties on the molar enthalpy of vaporization are typically 2-3%. The saturated vapour pressure was computed using the Virial equation, and the associated statistical uncertainty was less than 5%. The average liquid density was generally determined with a statistical uncertainty of 1-2% for both pure compounds and mixtures, but higher values are found at near-critical temperatures as a result of larger fluctuations.

Molecular Dynamics

Properties such as the density and the dynamic viscosity have been calculated using Molecular Dynamics (MD) simulations with NEWTON, a MD code developed for flexible and rigid molecules [START_REF] Van-Oanh | [END_REF]. In contrast to the MC procedure, MD simulations follow the time evolution of a molecular system by numerically integrating Newton's equations of motion. Simulations in the isobaric-isothermal ensemble (NPT) were performed to compute the density and the viscosity of the considered systems. Up to 1000 molecules were considered in our MD simulations. The velocity Verlet algorithm was used to integrate the equations of motion, with a time step of 1 fs. Practically, MD simulations were split in two parts: (i) one having a 0.2 ns length (2.10 5 steps) intended to the equilibration of the system, (ii) one with a 5 ns length (5.10 6 steps) that is dedicated to the calculation of the desired properties.

As described in previous works [16,17], the viscosity can be calculated either in the canonical ensemble (NVT) or in the isobaric-isothermal ensemble (NPT). The Einstein relation was employed to compute the viscosity η of systems of interest using the following expression

      ∆ + ∆ = ∑ ∑ > ∞ → α β α η 2 αβ 2 T αα t B (t) P 2 (t) P dt d lim T k V 20 1 (1)
where, α and β run over the three cartesian coordinates, k B is the Boltzmann constant, V is the volume, T the temperature and ∆P αβ denotes the displacement of the elements of the pressure tensor P αβ .

Interaction potential and calculation procedure

All the molecules involved in this study have been considered to be rigid, thus no intramolecular energy has been accounted for. Only intermolecular energy has been calculated with the two following contributions: dispersion -repulsion and electrostatic. Dispersion-repulsion energy between two force centres is represented as a function of their separation distance r with a Lennard-Jones (LJ) potential:

              -       = + = 6 12 4 r r U U U disp rep LJ σ σ ε (2)
In the case of binary interactions involving different force centres i and j, two combining rules have been used following requirements of intermolecular potentials. Lorentz-Berthelot combining rules are defined by:

j i ij ε ε ε = (3) 2 j i ij σ σ σ + = (4)
The geometric combining rules are defined by:

j i ij ε ε ε = (5) j i ij σ σ σ = (6)
Unless specified otherwise, Lorentz-Berthelot combining rules have been used to be consistent with previous studies [18][19][20].

Electrostatic energy is computed from the Coulomb law, assuming that the molecules bear electrostatic point charges:

ij j i elec r q q U 0 4πε = (7)
where q i and q j are two charges belonging to different molecules, r ij is the distance separating the charges and ε 0 is the vacuum permittivity.

Numerical values of the involved parameters for all studied molecules, i.e. Lennard-Jones diameters, Lennard-Jones well depths and electrostatic charges, will be discussed and given in the following sections.

Calculation of intermolecular energy is made by applying periodic boundary conditions, following classical procedures of molecular simulations [START_REF] Allen | Computer simulation of liquids[END_REF][START_REF] Frenkel | Understanding Molecular Simulation[END_REF]. Lennard-Jones interactions have been computed by applying a cut-off distance set to half of the box length. A standard long distance correction was used to account for interactions beyond the cut-off distance. The calculation of electrostatic interactions has been done using the Ewald summation method with 7 vectors in each space direction and a Gaussian width set to 2π/L, where L is the size of the simulation box. [23]. The parameters of these two potentials are summarized in Table 1. The accuracy of the models can be evaluated from Fig. 1 to 3 where calculated equilibrium properties are compared to available experimental data. Note that for simulations employing the Costa Gomes force field geometric mixing rules are used for both the size and the energy parameters, whereas classical Lorentz-Berthelot combining rules are used in the Hansen potential. Although both models have been fitted to vapour-liquid equilibrium data, the degree of agreement between experimental and simulated values is not fully satisfying. The first step of the present work was thus the determination of a new interaction potential for nitrous oxide that allows a better quantitative agreement between simulation and experiment over a wide range of temperatures. In the proposed model, the N-N and N-O distances, as well as the N-N-O angle, were fixed to their experimental values [24] (lN 1 -N 2 = 1.1282 Å; lN 2 -O = 1.1842 Å and N 1 -N 2 -O = 180°). Three Lennard-Jones and three point charges have been considered located on the three atoms of the molecule. In order to achieve a good degree of accuracy, the two nitrogen atoms of the molecule have been modelled by two different force centres as previously proposed by Hansen et al. The point charges and the Lennard-Jones parameters of the two nitrogen atoms and of the oxygen atom have been optimized using the following objective function [25]:

Results

3

( )

∑ = - = n i i i sim i S X X F 1 2 exp , (8)
where the sum runs over saturation properties (density, vaporization enthalpy and saturated vapour pressure), exp i X stands for experimental measurements, sim i X denotes the associated computed properties, and S i are the estimated statistical uncertainties on sim i X calculated by the block averaging technique. The minimization of the F function was realized as described in previous works [13,26] using a first-order Taylor expansion of thermodynamic properties versus potential parameters. For this purpose the gradient of F versus the potential parameters was obtained by a fluctuation method [START_REF] Ph | Applications of molecular simulation in the oil and gas industry -Monte Carlo methods[END_REF]27]. It was found that any direct optimization attempt involving more than two parameters failed to converge towards a better optimum, which is a known result from statistical uncertainties in a context where some parameter variations may compensate for each other (this is for instance the case between the Lennard Jones diameters of oxygen and nitrogen atoms). Thus the optimization has been performed by successive steps using 1D-and 2D-subspaces of the full parameter set. Moreover, due to the similar electronic environments of the N 1 and the O atoms in the N 2 O molecule, identical charges on these two atoms have been imposed. The optimization procedure leads to the charges and Lennard-Jones parameters presented in Table 1. The respective average absolute deviations for the densities, the vaporization enthalpies and the saturated vapour pressures are 2.6, 9.5 and 24.7% for Costa Gomes intermolecular potential, 0.9, 7.4 and 6.7% for Hansen intermolecular potential and 1.0, 3.5 and 5.8% for our intermolecular potential. Fig. 4 presents the results obtained for nitrous oxide transport properties using MD simulations. The liquid viscosities are obtained with average absolute deviations from experiments of 35.2%, 22.3%, and 11.8% using the Hansen force field, the Costa Gomes force field, and our new force field, respectively. These comparisons show that our new force field allows a better accuracy also for transport properties compared to existing force fields. Note that densities obtained by MD are consistent with those computed using MC simulations. The accuracy of our force field in the vicinity of the critical point has also been investigated.

Because of the large characteristic size of the density fluctuations, Gibbs Ensemble simulations cannot be performed in the close vicinity of the critical point. Therefore, the critical temperature T c and the critical density ρ c have been obtained by extrapolation from simulation points sufficiently far away from the critical point, assuming the following scaling law:

( ) β γ ρ ρ c L V T T - = - ( 9
)
where β is a characteristic universal exponent [31,[START_REF] Barrat | Basic Concepts for simple and Complex liquids[END_REF]. It is also assumed that the densities of the coexisting liquid and vapour obey the so-called law of rectilinear diameters:

) ( 2 c c L V T T - = - + λ ρ ρ ρ (10)
Practically, T c and ρ c have been regressed by a least square method minimizing the average deviation between the above equations and the simulated coexistence densities. Once these properties were obtained, the critical pressure was then estimated by extrapolating the Clapeyron plot (see Fig. 2) to 1/T c . The resulting critical coordinates, as well as the normal boiling point, are gathered in Table 2 where they are compared to experimental values taken from the DIPPR data base [START_REF] Rowley | Design Institute for Physical Properties[END_REF] and to simulated values obtained using the Costa Gomes and the Hansen models. A very accurate restitution of the critical point is obtained using our new force field with deviations from experiments of 1.6% for the critical temperature, 3.2% for the critical pressure and 0.1% for the critical density. dinitrogen dioxide). The composition of the NO-N 2 O 2 equilibrium mixture depends on pressure and temperature conditions: the dimer is favoured at higher pressures and lower temperatures. The chemical equilibrium between N 2 O 2 and 2 NO has been the subject of some experimental works [START_REF] Smith | [END_REF][34][35][36][37][38] and of some theoretical investigations [34,[39][40][41]. In the liquid state, this reaction has been studied by Smith et al. [START_REF] Smith | [END_REF] and Guedes [34] who reported equilibrium constants from 110 K to 170 K. The mole fraction of dimers in saturated liquid nitric oxide was found to vary from more than 90% at 110 K down to ~40% at 170 K. From these studies, it can be inferred that nitric oxide is strongly associated in the saturated liquid phase at low temperatures and that this degree of association decreases with increasing temperatures. In the gas phase, dimerization can also occur but the fraction of associated molecules (N 2 O 2 ) is very small. Equilibrium constants of the reaction in the gas phase along the saturation curve have been reported by Kohler et al. [42] showing a molar fraction of dimer in the vapour phase around 0.64 to 2.89% for temperatures in the range of 115 -177 K.

In addition to these studies of the N 2 O 2 2NO chemical equilibrium, phase behaviours and phase properties of the NO-N 2 O 2 reacting system have also been investigated in the literature from experimental studies [42][START_REF] Guedes | High Temp[END_REF][START_REF] Johnston | [END_REF][45][46][START_REF] Horvath | Physical Properties of Inorganic Compounds[END_REF][START_REF] Cheesman | [END_REF][START_REF] Stull | The Chemical Thermodynamics of Organic Compounds[END_REF] or from empirical modelling approaches [START_REF] Rowley | Design Institute for Physical Properties[END_REF][START_REF] Rice | [END_REF]. Among others, we can cite Johnston and Giauque [START_REF] Johnston | [END_REF] who have reported heat capacity measurements from 14 K up to the boiling point, as well as vaporization enthalpies and vapour pressures.

In the present work, the vapour-liquid equilibrium of the NO-N 2 O 2 reacting system has been simulated at a molecular level using simultaneously the reaction ensemble and the Gibbs NVT ensemble Monte Carlo methods [19,51]. The reaction ensemble Monte Carlo method (ReMC) [52,53] allows the study of chemically reacting mixtures. No detailed description of this ensemble is proposed here and we invite interested readers to investigate the papers cited above for more details. In comparison with classical Gibbs ensemble simulations, two additional constraints are applied:

• The number of atoms is fixed for each atom type, so the number of different molecules in the system is controlled by the chemical equation which defines the chemical equilibrium:

∑ = = s i i i a 1 0 ν (11)
where ν i is the stoichiometric coefficient of chemical species i.

For example, 2NO N 2 O 2.

• The sum of chemical potentials µ i over the different molecular species implied in the chemical reaction, weighted by stoichiometric coefficients, is equal to zero:

∑ = = s i i i 1 0 µ ν (12) For example, 2 2 2 O N NO µ µ = .
In order to satisfy these constraints during the simulation, an additional Monte Carlo move is used: the reaction move. This move consists in firstly choosing a direction to perform the reaction, second in deleting a set of reactant molecules randomly chosen in the system, and finally inserting product molecules. In our case, the reaction move is 2NO → N 2 O 2 where two randomly chosen NO molecules are deleted and one N 2 O 2 is inserted, or N 2 O 2 → 2NO where one N 2 O 2 molecule randomly selected is deleted and two NO molecules are inserted. The acceptance probability of such Monte Carlo move can be found in reference [19], together with some details on its implementation in the GIBBS Monte Carlo software used in this study. The parameterization procedure we have used is similar to the one previously described for N 2 O force field development. The accuracy of the new proposed force field is compared to those obtained using existing models available in the literature [START_REF] Hirschfelder | Molecular Theory of Gases and Liquids[END_REF][START_REF] Kohler | [END_REF]. Since the dipole moment of NO is rather small, electrostatic forces are neglected by all these models. The parameters of these different intermolecular potentials are summarized in Table 3. In order to test these models, we have performed NVT ReMC GEMC simulations at different temperatures in the 120 -160 K range and computed the phase diagram of the reactive NO-N 2 O 2 system. To model N 2 O 2 molecules, we have used two NO LJ sites separated by 2.237 Å; the individual LJ parameters for each NO site in the dimer being the same as those for the monomer. The standard Gibbs free energies of the reaction 2NO N 2 O 2 needed to perform the reaction moves during these simulations were calculated from the Gibbs free energies of formation of NO and N 2 O 2 taken from the TRC (Thermodynamics Research Center) thermochemical tables [START_REF] Pedley | Thermodynamical Data and Structures of Organic Compounds, TRC Data Series[END_REF]. The numerical values of these Gibbs free energies are gathered in Table 4. Tests of these potentials have been performed by calculating thermodynamic properties of the NO-N 2 O 2 system along the saturation curve. The obtained results (coexistence densities, vaporization enthalpies and vapour pressures) are presented in Fig. 5, 6 and7, where they are compared to experimental or modelling data given in the DIPPR database [START_REF] Rowley | Design Institute for Physical Properties[END_REF]. For vapour pressures, results of Lísal et al. [START_REF] Lísal | [END_REF] using the Kohler force field are also plotted in Fig. 7, showing a very good agreement with our data using the same model. For all studied properties, significant deviations between DIPPR and simulated data are observed when using either potentials of Hirschfelder or the potential of Kohler. The degree of agreement is much more satisfying with our new force field. The temperature dependence of liquid densities and vaporization enthalpies is correctly described by this model, although it slightly overestimates the liquid density and the vaporization enthalpy at low temperature and underestimates them at higher temperatures in the vicinity of the critical point. The respective average absolute deviations for the densities, the vaporization enthalpies and the saturated vapour pressures are 3.3, 8.0 and 27.0%. Such results show that this model allows a reasonable agreement between simulation and experiment over a wide range of temperatures. The NO-N 2 O 2 critical point coordinates (T c , ρ c and x c ) have been approximated by the use of extended scaling laws [START_REF] Ph | Applications of molecular simulation in the oil and gas industry -Monte Carlo methods[END_REF]:

β c c c ) ( 2 γ ε ) λ( ρ ρ T T T T - + - + = ϕ ( 13
)
β c c 2 1 ) ( 2 µ ε ) ( 2 λ ε λ x x T T T T c - - -       - + = ϕ
where ε = 1 for the liquid phase (φ = liq) and ε = -1 for the vapour phase (φ = vap). T c , ρ c , x c (critical coordinates) and λ, λ 1 , λ 2 , γ, µ (adjustable coefficients) are regressed by a least square method from a set of coexistence points (T, x liq , x vap , ρ liq , ρ vap ) below the critical point. The critical pressure was then estimated by extrapolating the Clapeyron plot (see Fig. 7) to 1/T c . The resulting critical coordinates, obtained using the different studied force fields, are gathered in Table 5 where they are compared to experimental values taken from the DIPPR data base [START_REF] Rowley | Design Institute for Physical Properties[END_REF]. We have also compared the fractions of dimers in the saturated liquid phase given by the different studied models. Results are shown in Fig. 8 together with the experimental point of Smith [START_REF] Smith | [END_REF] obtained by measurements of magnetic susceptibility. Fig. 8 shows nearly complete dimerization at low temperatures. Further, the degree of dimerization decreases with increasing temperature and at T > T c , the system becomes monomeric. For the vapour phase at a pressure of 0.1 MPa and a temperature of 121.40 K we obtain, using our new model, a mole percent of monomers of 99.6%, which is in agreement with both Guggenheim's estimate based on analysis of second Virial coefficient data [58] and Turner's estimate based on Monte Carlo simulations [59]. To assess the possible effect of pressure on the degree of dimerization, one simulation of the NO-N 2 O 2 system has been conducted in the monophasic liquid region at 253.15 K and 12.0 MPa, showing a mole fraction of monomers of 99.8%.

This study shows that our simulations are consistent with data available in the literature for both the liquid and the gas phases. The obtained results also show that for temperatures above the critical temperature of the NO-N 2 O 2 system (T c exp = 180.15 K), associations have disappeared and only the monomer form (NO) has to be accounted for. Fig. 9 illustrates the results obtained for the transport properties of the NO-N 2 O 2 system using MD simulations. The degree of dimerization for each studied temperature was taken from MC simulation results presented in Fig. 8, with a total number of 1000 NO particles. The viscosity calculations were thus performed at fixed chemical composition. The liquid viscosities are obtained with average absolute deviations from experiments of 43.7%, 39.2%, and 22.0% using the Kohler force field, our new force field, and the Hirschfelder 2 force field, respectively. Such comparisons show that the Hirschfelder force field allows a better degree of accuracy for the liquid viscosity compared to Kohler and our new force fields, especially at low temperatures. However, considering both transport and thermodynamic properties, our new force field appears as the best compromise.

Study of CO 2 +N 2 O mixtures

Vapour-liquid phase diagrams

To model CO 2 molecules, we have chosen the EPM2 rigid version of the force field proposed by Harris and Yung [60]. This force field, which involves three Lennard-Jones sites and three electrostatic charges, is known to represent the liquid-vapour coexistence data of pure CO 2 as well as phase diagrams of mixtures with other gases fairly well [18][19][20]. The parameters of this potential are given in Table 6. Note that Harris and Yung have developed their model using geometric combining rules for LJ sigma and epsilon. Nevertheless, in this work, Lennard-Jones parameters for the unlike interactions have been obtained using Lorentz-Berthelot rules. The use of an arithmetic combining rule for the calculation of sigma instead of a geometric rule is not expected to have a significant impact on the simulation results since O and C atoms do not exhibit a substantial size difference (σO = 3.033 Å and σC = 2.757 Å). The use of an arithmetic combining rule leads to a cross diameter σCO equal to 2.895 Å, whereas a value of 2.892 Å is obtained with a geometric rule. Moreover, the paper of Nieto-Draghi and co-workers in 2007 shows that the combination of the Harris and Yung potential with the Lorentz-Berthelot rules allows a good restitution of both thermodynamic and transport properties. In order to increase the accuracy of the calculations performed for the CO 2 +N 2 O binary system, we have used the reaction Gibbs Ensemble Monte Carlo method proposed by Lísal et al. [61]. This approach uses a modified GEMC technique that treats the phase equilibrium as a special type of chemical reaction and incorporates knowledge of pure component vapour pressure data into the simulations. The present paragraph provides just a brief summary of the involved algorithm, the reader is referred to the paper of Lísal et al. for more details. From a practical point of view, this method is similar to the Gibbs Ensemble method except that the acceptance probabilities of particle transfers from one box to the other are modified according to the equations:

( ) ( ) ( )         ∆ - + Γ = → → U V N V N
( ) ( ) ( )         ∆ - + Γ = → → U V N V N b a P gas liq liq gas i acc liq gas β exp 1 1 , 1 min (15)
where ( )

T k B 1 =
β and k B is Boltzmann's constant, ∆U is the change in configurational energy, V α is the volume of the simulation box α, and N α is the total number of molecules in box α. In equations ( 14) and ( 15), the term i Γ (i = CO 2 or N 2 O) is given by:

sat GEMC i sat i i P P , exp , = Γ (16)
with sat GEMC i P , being the vapour pressure of component i calculated using standard GEMC and

sat i P exp , the corresponding experimental value. If 1 = Γ i
, equations ( 14) and ( 15) correspond to standard GEMC simulations. In our simulations, we have used a constant i Γ factor equal to 0.95 for the two involved compounds at the different studied temperatures. This choice is justified by the fact that the force fields used for N 2 O and for CO 2 both lead to almost the same overestimate of vapour pressures whatever the temperature. All these simulations have been performed at constant total volume and constant temperature. Note that no experimental information concerning the mixtures is required, making this approach totally predictive for mixture studies.

The phase envelopes of the CO 2 +N 2 O binary mixture have been calculated at three different temperatures: 273 K, 283 K and 293 K. These phase envelopes are presented in Fig. 10 together with experimental data of Cook measured at 293 K, 298 K and 303 K [6] and of Rowlinson measured at 283 K and 293 K [START_REF] Rowlinson | Proceedings, International Conference Thermodynamics and Transport Properties Fluids[END_REF]. All these temperatures are below the critical temperatures of both CO 2 and N 2 O, thus none of the phase diagrams exhibits a critical point. Due to the strong similarities between carbon dioxide and nitrous oxide (T c (CO 2 ) = 304.21 K; T c (N 2 O) = 309.57 K; P c (CO 2 ) = 7.38 MPa; P c (N 2 O) = 7.24 MPa), the two-phase region is very narrow and very flat at all studied temperatures. Differences between the compositions of the liquid and the vapour phases at equilibrium are very small which means that the system may be considered to be azeotropic over the whole range of compositions. A good agreement between experimental and simulated data is obtained at 283 K and 293 K which validates our methodological approach. The numerical values obtained from the Monte Carlo simulations are gathered in Table 7. Fig. 11 presents the pressure-density diagrams obtained from our simulations.

Study of CO 2 +NO mixtures

Vapour-liquid phase diagrams

The phase envelopes of the CO 2 +NO binary mixture have been calculated at three different temperatures: 253 K, 263 K and 273 K, using standard Gibbs NPT simulations. As discussed previously, nitric oxide is fully monomeric in this temperature range. The calculated phase envelopes are presented in Fig. 14 (pressure-composition diagrams) and Fig. 15 (pressuredensity diagrams). The corresponding numerical values are gathered in Table 9. As expected, the extend of the two-phase region decreases as the temperature increases. The three studied temperatures are below the critical temperature of carbon dioxide (T c = 304.21 K) and above the critical temperature of nitric oxide (T c = 180.15 K), thus all phase diagrams exhibit a critical point. The coordinates of these critical points and the near-critical phase behaviour have been determined using extended scaling laws [START_REF] Ph | Applications of molecular simulation in the oil and gas industry -Monte Carlo methods[END_REF].

Densities and viscosities

In this section, we report results obtained using MD simulations for a CO 2 +NO binary mixture containing 10 mol% of NO. The density and the viscosity of this mixture have been calculated at two temperatures: 263 K and 273 K. As previously shown from MC simulation results, the nitric oxide is fully monomeric at these temperatures, and no N 2 O 2 has been considered in the MD simulations. The obtained results are presented in Table 10 and illustrated in Fig. 16 and 17, together with experimental and simulated data for pure CO 2 . Experimental data for pure CO 2 were extracted from the NIST website [30].

Densities and viscosities simulated for pure CO 2 are in excellent agreement with respect to experimental data. When considering a CO 2 +NO binary mixture containing 10 mol% of NO at 263 K and 273 K, Molecular Dynamics simulations lead to liquid density values much lower than corresponding pure CO 2 values (about 9%). Similarly, the addition of 10 mol% of NO to CO 2 at 263 K and 273 K leads to a decrease of liquid viscosities of about 24% with respect to corresponding pure CO 2 viscosities.

, mix i j i j ij mi i x i i ij a a a z z k b b z = - = ∑ ∑ (17)
where the summations are over the mixture components i and where i a , i b , and i z are the pure component attractive parameter, co-volume and molar fraction, respectively. For a given binary mixture and EoS, an unique binary interaction parameter has been regressed. The fitting of this interaction parameter has been performed by minimizing the objective function ) ( ij Y k , which is a sum over deviations for calculated data from experimental or pseudoexperimental data, with respect to ij k :

( ) ( ) ( ) ( ) ( ) [ ] 2 1 , 1 2 , exp, , 2 , exp, ,         - + - = ∑∑ = = j i l N r r l ij r l r l ij r l ij y k y x k x k Y (18)
Here, the experimental data consists of N measurements (CO 2 +N 2 O) or data from molecular simulations (CO 2 +NO) of the liquid compositions r x exp, and gas compositions r y exp, , taken at a given point r in the pressure-temperature space. This is the same objective function as used by Wilhelmsen et al. [62].

) ( ij Y k is preferred compared to objective functions involving relative values, because molar fractions x and y may be small in terms of absolute values. A small absolute deviation may then result in a large relative deviation, and a biased regression. The performance of the EoS is measured by the absolute average deviation (AAD) of the liquid and gas compositions:

( ) ∑∑ = = - = j i l N r r l r l ij r l x x x k x N , 1 , exp, , exp, , 2 1 ADD (19)
and

( ) ∑∑ = = - = j i l N r r l r l ij r l y y y k y N , 1 , exp, , exp, , 2 1 ADD (20)
The optimized parameters obtained for the two binary mixtures and the two studied EoS are gathered in Table 11, together with the corresponding performance. The regression of binary interaction parameters for SRK and PR EoS based on experimental data allows a good description of the pressure-composition phase envelopes of the CO 2 +N 2 O mixture for all studied temperatures. For the two studied EoS, very small values of k ij are obtained, which is reasonable since no significant cross interactions are expected in this system. Concerning coexisting densities, both EoS predict the densities well in the gas phase, but only the PR EoS allows a reasonable restitution of the liquid densities. SRK EoS underpredicts the liquid densities by more than 10%. Such behaviour was pointed out in recent work by Wilhelmsen et al. [62] involving different CO 2 -containing systems. Note that no volume shift has been used to correct the obtained densities.

The phase envelopes of the CO 2 +NO mixture calculated by the two cubic equations of state at 253 K, 263 K and 273 K are presented in Fig. 20 (pressure-composition diagrams) and Fig. 21 (pressure-density diagrams), together with the pseudo-experimental data from Monte Carlo simulations. In addition, the critical points of the CO 2 +NO mixture at the three studied temperatures obtained from cubic equations of state are given in Table 12. Fig. 20 shows that the largest deviations between the pseudo-experimental Monte Carlo data and the cubic equations of state may be found at large pressures in the vicinity of the critical region. This overprediction by standard cubic EoS of dew pressures close to the critical point was also found in a previous work [62] for two other CO 2 +contaminant mixtures (CO 2 +N 2 and CO 2 +O 2 ) in a comparison with experimental data. In the present work with the CO 2 +NO mixture, none of the cubic EoS displays any clear advantage over the other for the restitution of the pressure-composition curves. For density calculations however, Fig. 21 shows that the liquid densities calculated with the PR EoS are in better agreement with the Monte Carlo data compared to the values obtained with the SRK EoS. The same behaviour was already observed in the case of the CO 2 +N 2 O mixture. By comparing the data in Table 12, it is evident that the critical coordinates given by the two cubic equations of state are similar, but very different from the critical coordinates given by Monte Carlo simulations. The deviations are largest for the critical pressures, which are on average 16% larger with the cubic equations of state than with Monte Carlo simulations. The critical composition exhibits only on average 10% deviation, while the predicted critical density from the cubic equations of state deviates 2% from the Monte Carlo simulations.

Conclusion

Original force fields for atomistic simulations have been proposed for N 2 O, NO and N 2 O 2 molecules, based on Lennard-Jones parameters fitted to experimental vapour-liquid equilibrium data. We have shown that thermodynamic and transport properties along the saturation curve of both nitrous oxide and nitric oxide can be well reproduced using these new models in Monte Carlo and Molecular Dynamics simulations. Studies have also been carried out on CO 2 -containing mixtures at different temperatures in the range 253 -293 K. Phase compositions, phase densities and phase viscosities were determined for CO 2 +N 2 O and CO 2 +NO binary mixtures. In the case of CO 2 +N 2 O mixtures, the liquid-vapour phase envelopes calculated at 283 K and 293 K were compared with experimental data available in the literature. Experimental and simulation results appear in good agreement, allowing reliable and accurate predictions at other temperatures. In the case of CO 2 +NO mixtures, the performed calculations are pure predictions since no experimental data are available in the literature for this system. The regression of binary interaction parameters for SRK and PR EoS based on the molecular simulation pseudo-experimental data or true experimental data allows good restitution of vapour-liquid equilibrium data for the two studied mixtures and for the two studied EoS. For the prediction of liquid densities in the two-phase area, PR EoS was much closer to the pseudo-experimental data than SRK EoS.

The present study shows that molecular simulations combined with equation of state modelling can play a significant role in the context of CO 2 storage operations where the degree of purity of the captured CO 2 is a key factor for transportation, injection and sequestration. This work illustrates how molecular simulations may be used as a tool to mitigate the lack of experimental data and improve engineering equations of state to enable process simulations where accurate, robust and computationally fast models are required.

List of symbols

Fig. 1 :Fig. 2 :Fig. 3 :

 123 Fig.1: Liquid-vapour coexisting densities of N 2 O obtained by Monte Carlo simulations using the Hansen force field, the Costa Gomes force field and our new proposed force field. Experimental values shown for comparison are taken from the DIPPR data bank[START_REF] Rowley | Design Institute for Physical Properties[END_REF] for the liquid phase, and from the work of Quinn and Wernimont[START_REF] Quinn | [END_REF] for the vapour phase.

Fig. 4 :

 4 Fig. 4: Comparison of vapour and liquid viscosities of N 2 O obtained by Molecular Dynamics simulations using the Hansen potential, the Costa Gomes potential and our new proposed potential. Experimental values are taken from the saturation properties of N 2 O on the NIST website [30].

Fig. 5 :Fig. 6 :Fig. 7 :

 567 Fig. 5: Temperature dependence of density in the liquid-vapour coexistence region of the NO-N 2 O 2 system. The line represents the recommended values of the DIPPR data bank [28] whereas symbols represent our Monte Carlo simulation results obtained using different interaction potentials to model the NO and N 2 O 2 molecules.

Fig. 10 :Fig. 11 :

 1011 Fig. 10: Pressure-composition diagram of the CO 2 +N 2 O mixture at different temperatures. Experimental data at 283.15 K, 293.15 K, 298.15 K and 303.15 K are taken from Cook[6] and Rowlinson[START_REF] Rowlinson | Proceedings, International Conference Thermodynamics and Transport Properties Fluids[END_REF] whereas pure component vapour pressures at 273.15 K and 283.15 K are taken from the DIPPR data base[START_REF] Rowley | Design Institute for Physical Properties[END_REF].

Fig. 14 :Fig. 15 :

 1415 Fig. 14: Pressure-composition diagram of the CO 2 +NO mixture at 253.15 K (blue), 263.15 K (red) and 273.15 K (black). The symbols represent the simulated results. The lines are guides for the eyes.

Fig. 18 :Fig. 19 :

 1819 Fig. 18: Pressure-composition diagram of the CO 2 +N 2 O mixture at different temperatures; symbols: experimental or simulated results; solid lines: PR EoS results; dashed lines: SRK EoS results.

.1. New interaction potentials for N 2 O and NO 3.1.1. Model for nitrous oxide

	Two intermolecular potentials suitable for liquid-vapour equilibrium studies are available in
	the literature to model nitrous oxide (N 2 O) molecules. Both are based on Lennard-Jones plus
	point charge models. The first one has been proposed by Costa Gomes et al. in 2006 [22] and
	the second one by Hansen et al. one year later

Table 1 :

 1 Intermolecular potential parameters for N 2 O.

	Model	Site	ε/ ε/ ε/ ε/k B (K)	σ σ σ σ (Å)	q (e)	Bond length (Å)
	N 2 O (this work)	N1	78.107	3.116	-0.3400	l N1-N2 = 1.1282
		N2	34.647	2.927	0.6800	l N2-O = 1.1842
		O	65.891	3.044	-0.3400
	N 2 O (Costa Gomes	N1	56.527	3.150	-0.2497	l N1-N2 = 1.1280
	et al. [22])	N2	56.527	3.150	0.5159	l N2-O = 1.1840
		O	78.177	3.050	-0.2662
	N 2 O (Hansen et al.	N1	79.167	3.120	-0.3630	l N1-N2 = 1.1282
	[23])	N2	27.000	2.800	0.7130	l N2-O = 1.1842
		O	79.000	3.050	-0.3500
	Fig. 1 to 3 compare experimental data and results obtained by Monte Carlo simulations in the
	Gibbs NVT ensemble using the models of Costa Gomes, of Hansen and our new force field.

Table 2 :

 2 N 2 O critical coordinates and normal boiling temperature. Systems containing nitric oxide are particularly challenging because this compound may exist as a mixture of monomers (NO: nitrogen monoxide or nitric oxide) and dimers (N 2 O 2 :

	Model	T c (K)	P c (MPa) ρ ρ ρ ρ c (kg/m 3 )	T eb (K)

Table 3 :

 3 Intermolecular potential parameters for NO.

	Model	Site	ε/ ε/ ε/ ε/k B (K)	σ σ σ σ (Å)
	NO -Hirschfelder 1 [54]	NO	91.0	3.60
	NO -Hirschfelder 2 [54]	NO	119.0	3.47
	NO -Kohler [55]	NO	125.0	3.17
	NO -New force field (this work)	NO	130.0	3.40
	To model NO and N 2 O

2 molecules, we have parameterized a force field that treats NO as a single Lennard-Jones sphere and N 2 O 2 as a two-site Lennard-Jones molecule.

Table 4 :

 4 Standard Gibbs free energy changes (∆ r G°) of the reaction 2NO N 2 O 2 at the temperatures studied using reaction ensemble Monte Carlo. These values were evaluated by means of the TRC thermochemical tables[START_REF] Pedley | Thermodynamical Data and Structures of Organic Compounds, TRC Data Series[END_REF].

	T (K)	120.00	130.00	140.00	150.00	160.00
	∆ ∆ ∆ ∆ r G° (kJ mol -1 )	5.344	6.726	8.126	9.530	10.936

Table 5 :

 5 Nitric oxide critical coordinates.

	Model	T c (K)	P c	ρ ρ ρ ρ c (kg/m 3 )	x c
			(MPa)		(N 2 O 2 )
	Simulation (New force field)	174.03	6.37	466.60	0.03
	Simulation (Hirschfelder 2 force field [54])	165.80	5.86	442.88	0.06
	Simulation (Kohler force field [55])	171.19	7.64	540.86	0.04
	DIPPR data	180.15	6.48	517.35	-

  Mole fraction of N 2 O 2 as a function of temperature along the saturation line for the bulk liquid phase of the 2NO N 2 O 2 system from Monte Carlo simulations.

		0.9
	2 mole fraction O 2 N	0.2 0.3 0.4 0.5 0.6 0.7 0.8	New force field Kohler force field Hirschfelder 1 force field Hirschfelder 2 force field
		0.1	Exp. data / Smith, 1952
		110 0	120	130	140	150	160	170
				Temperature (K)
	0.0065 New force field / Vap. 0.007 Kohler / Vap. Hirschfelder 1 / Vap. Hirschfelder 2 / Vap. Fig. 8: 0.006 0 0.1 0.5 0.2 0.3 New force field / Liq. 0.4 Kohler / Liq. Hirschfelder 1 / Liq. Hirschfelder 2 / Liq. η (mPa.s) Experimental data	0.0075	0.008
					1/T (K	-1 )
	Fig. 9: Comparison of vapour and liquid viscosities of the NO-N 2 O 2 system obtained by
	Molecular Dynamics simulations. Experimental values are taken from the saturation
		properties of NO on the NIST website [30].

Table 6 :

 6 Intermolecular potential parameters for CO 2 .

	Model	Site	ε/ ε/ ε/ ε/k B (K) σ σ σ σ (Å)	q (e)	Bond length (Å)
	CO 2 -EPM2 [60]	O	80.507	3.033 -0.3256	l C-O = 1.1490
		C	28.129	2.757	0.6512
		O	80.507	3.033 -0.3256

Table 7 :

 7 Calculated vapour-liquid equilibrium pressures, compositions and densities for the CO 2 +N 2 O mixture at different temperatures. In this section, we report results obtained using MD simulations for a CO 2 +N 2 O binary mixture containing 10 mol% of N 2 O. The density and the viscosity of this mixture have been calculated at two temperatures: 283 K and 293 K. The obtained results are presented in Table8and illustrated in Fig.12and 13, together with experimental and simulated data for pure CO 2 . Experimental data for pure CO 2 were extracted from the NIST website[30].Densities simulated for pure CO 2 are in excellent agreement with respect to experimental data. When considering a CO 2 +N 2 O binary mixture containing 10 mol% of N 2 O at 283 K and 293 K, Molecular Dynamics simulations lead to density values almost equal to the corresponding pure CO 2 values at the same temperatures. As far as the viscosity is concerned, data simulated for pure CO 2 show a reasonable agreement with respect to experimental data. Within the computed statistical uncertainties, the obtained viscosities for the mixture are similar to those calculated for pure CO 2 at the same temperature and pressure conditions. Pressure-density diagram of the CO 2 +N 2 O mixture with 10 mol% of N 2 O, at 283.15 K and 293.15 K. Experimental densities for pure CO 2 have been taken from the NIST website [30].

	T (K)	P (MPa)	x vap (CO 2 )	ρ ρ ρ ρ vap (kg/m 3 )	x liq (CO 2 )	ρ ρ ρ ρ liq (kg/m 3 )
	293.15 K	5.14	0.000	140.73	0.000	741.55
		5.23	0.264	143.90	0.255	727.55
		5.50	0.507	155.52	0.496	760.40
		5.51	0.747	155.96	0.739	736.62
		5.53	0.883	154.75	0.878	736.49
		5.66	1.000	162.48	1.000	752.45
	283.15 K	4.14	0.000	110.19	0.000	831.33
		4.26	0.267	113.94	0.255	835.39
		4.31	0.511	116.15	0.496	837.43
		4.42	0.750	120.34	0.739	839.75
		4.44	0.884	119.24	0.878	835.64
		4.59	1.000	125.41	1.000	849.76
	273.15 K	3.19	0.000	82.29	0.000	888.59
		3.38	0.269	88.35	0.255	900.56
		3.39	0.513	87.86	0.497	900.40
		3.51	0.752	92.48	0.739	912.48
		3.55	0.885	93.12	0.879	910.45
		3.59	1.000	94.55	1.000	914.55
	3.2.2. Densities and viscosities

Table 8 :

 8 Molecular Dynamics simulation results computed for the CO 2 +N 2 O mixture with 10 mol% of N 2 O. Simulated data obtained for the mixture are compared to experimental and simulated data for pure CO 2 .These calculations show that both thermodynamic and transport properties of CO 2 +N 2 O mixtures are similar to the properties of pure CO 2 in the whole range of studied temperatures, pressures and compositions. Such behaviour is reasonable due to the strong similarities between carbon dioxide and nitrous oxide in terms of molecular weights, vapour pressures and critical coordinates.

			ρ	η	ρ	err.	η	err.	ρ	η
	T	P	(kg.m -3 )	(mPa.s)	(kg.m -3 )	(%)	(mPa.s)	(%)	(kg.m -3 )	(mPa.s)
	(K)	(MPa)	Exp. data for pure CO 2		Sim. data for pure CO 2		Sim. data for the CO 2 +N 2 O mixture
	293.15	4.5	116.9	0.016	109.3	-6.5	0.015	-6.7	109.6	0.016
	293.15	5.5	172.9	0.018	151.1	-12.6	0.016	-7.5	152.0	0.017
	293.15	6.0	782.7	0.068	777.0	-0.7	0.066	-2.8	777.8	0.065
	293.15	6.5	796.8	0.070	786.4	-1.3	0.067	-4.0	789.6	0.070
	293.15	7.0	808.6	0.072	797.3	-1.4	0.069	-4.6	799.8	0.075
	293.15	7.5	818.7	0.074	808.9	-1.2	0.069	-6.5	809.5	0.080
	283.15	4.0	108.4	0.015	100.9	-6.9	0.014	-10.3	101.2	0.015
	283.15	5.0	868.6	0.084	860.0	-1.0	0.083	-1.3	859.5	0.080
	283.15	6.0	881.8	0.087	871.7	-1.1	0.086	-1.0	871.5	0.084
	283.15	7.0	893.1	0.090	884.9	-0.9	0.090	0.4	881.1	0.088

Table 9 :

 9 Calculated vapour-liquid equilibrium pressures, compositions and densities for the CO 2 +NO mixture at different temperatures.

	T (K)	P (MPa)	x vap (NO)	ρ ρ ρ ρ vap (kg/m 3 )	x liq (NO)	ρ ρ ρ ρ liq (kg/m 3 )
	253.15 K	2.25	0.000	58.85	0.000	1020.5
		3.00	0.218	73.61	0.025	1007.8
		4.00	0.371	94.71	0.059	985.82
		5.00	0.467	118.09	0.095	961.58
		6.00	0.527	142.58	0.135	934.64
		7.00	0.561	171.90	0.172	909.53
		8.00	0.586	204.36	0.215	877.65

Table 10 :

 10 Molecular Dynamics simulation results computed for the CO 2 +NO mixture with 10 mol% of NO. Simulated data obtained for the mixture are compared to experimental and simulated data for pure CO 2 .

		1000
		800
	)
	-3 ρ (kg.m	400 600				CO 2 -NO / Sim. / 263.15 K CO 2 / Sim. / 263.15 K CO 2 / Experimental data
						CO 2 -NO / Sim. / 273.15 K
		200				CO 2 / Sim. / 273.15 K CO 2 / Experimental data
		0	3	4	5	6	7	8
				P (MPa)
	Fig. 16: Pressure-density diagram of the CO 2 +NO mixture with 10 mol% of NO, at 263.15 K
	and 273.15 K. Experimental densities for pure CO 2 have been taken from the NIST website
				[30].

Optimization of thermodynamic models for CO 2 +N 2 O and CO 2 +NO mixtures

  The last part of the work is devoted to the calibration of standard cubic equations of state (EoS). Because of their simplicity and robustness, cubic EoS are often used to calculate the thermodynamic properties in the oil and gas process industry. Such EoS perform well for vapour-liquid equilibrium calculations and may perform well for densities, at least away from the critical region. Because they have few empirical parameters, they are easily fitted to new experimental data. In the present work, the available liquid-vapour experimental data or the pseudo-experimental data generated by molecular simulation have been used to regress binary interaction parameters for both Soave-Redlich-Kwong (SRK) and Peng-Robinson (PR) EoS for the CO 2 +N 2 O and CO 2 +NO mixtures. Binary interaction parameters ij k are required to calculate the attractive parameter a mix and the co-volume b mix of the mixture from a given mixing rule. In the present work, classical mixing rules have been used:

	1 (	, )

Table 11 :

 11 Optimized binary interaction parameters for the two studied mixtures and the two studied equations of state.

		CO 2 +N 2 O		CO 2 +NO
		SRK	PR	SRK	PR
	k ij	0.004	0.007	-0.119	-0.105
	AAD x	5%	1%	18%	19%
	AAD y	5%	3%	14%	14%
	The phase envelopes of the CO 2 +N 2 O binary mixture as calculated by optimized cubic
	equations of state at 273 K, 283 K, 293 K, 298 K and 303 K are presented in Fig. 18
	(pressure-composition diagrams) and Fig. 19 (pressure-density diagrams), together with the
	experimental data available in the literature and the Monte Carlo pseudo-experimental data
	obtained in the present work.

Table 12 :

 12 Critical point predictions given by SRK EoS, PR EoS and Monte Carlo simulations for the CO 2 +NO mixture at different temperatures. Pressure-density diagram of the CO 2 +NO mixture at 253.15 K (blue), 263.15 K (red) and 273.15 K (black); symbols: simulated results; solid lines: PR EoS results; dashed lines: SRK EoS results.

			SRK EoS			PR EoS			Monte Carlo
	T (K) P c (MPa) x NO,c (-)	ρ c (kg/m 3 )	P c (MPa) x NO,c (-)	ρ c (kg/m 3 )	P c (MPa) x NO,c (-)	ρ c (kg/m 3 )
	253.15	13.7	0.562	510.2	13.6	0.554	559.1	11.48	0.512	526.8
	263.15	13.2	0.483	491.4	13.1	0.476	537.5	11.21	0.427	529.9
	273.15	12.4	0.395	471.1	12.3	0.388	513.2	10.95	0.351	515.2

Acknowledgment

The authors would like to acknowledge the support of the seventh framework program of the European Commission via the collaborative project COCATE "Large-scale CCS Transportation infrastructure in Europe" -GA No. 241381. VL, EB and ND also acknowledge financial help from the French Agence Nationale de la Recherche (ANR), under grant SUSHI (Grant No. ANR-07-BLAN-0268) "SimUlation de Systèmes Hétérogènes et d'Interfaces". BC would like to thank Dr. Bernard Rousseau for the use of the NEWTON MD code.


János Körner
email: korner@di.uniroma1.it

Gábor Simonyi
email: simonyi@renyi.hu

ON TYPES OF GROWTH FOR GRAPH-DIFFERENT PERMUTATIONS

We consider an infinite graph G whose vertex set is the set of natural numbers and adjacency depends solely on the difference between vertices. We study the largest cardinality of a set of permutations of [n] any pair of which differ somewhere in a pair of adjacent vertices of G and determine it completely in an interesting special case. We give estimates for other cases and compare the results in case of complementary graphs. We also explore the close relationship between our problem and the concept of Shannon capacity "within a given type".

Introduction

The topic of our paper has its origins in the following mathematical puzzle of Körner and Malvenuto [START_REF] Körner | Pairwise colliding permutations and the capacity of infinite graphs[END_REF]. Call two permutations of [n] := {1, . . . , n} colliding if, represented by linear orderings of [n], they put two consecutive elements of [n] somewhere in the same position. For the maximum cardinality ρ(n) of a set of pairwise colliding permutations of [n] the following conjecture was formulated. .

This conjecture remains open; for the best bounds the interested reader may consult [START_REF] Körner | Graph-different permutations[END_REF] and [START_REF] Brightwell | Permutation capacity of graphs[END_REF]. In this paper we initiate a systematic study of similar problems for all the graphs on the countable vertex set N. Not only do we believe that these problems are interesting on their own, but beyond this we hope that studying them within a unified framework may shed more light also on the initial problem on colliding permutations.

Let G be an infinite graph whose vertex set is the set N of the natural numbers. We call two permutations of the elements of [n], the first n natural numbers, G-different if they map some i ∈ [n] to adjacent vertices of G. (We will often think and write about permutations of [n] as n-length sequences that contain each element of [n] exactly once. In this language, two permutations are G-different if there exists a position where the corresponding two sequences contain the two endpoints of an edge of G.) We denote by T G (n) the maximum cardinality of a set of pairwise G-different permutations of G. The question about pairwise colliding permutations is that special case of our present problem where the graph is the infinite path L defined by

V (L) = N E(L) = {{i, i + 1} | i ∈ N}.
We will concentrate our attention on the special class of distance graphs. Given an arbitrary (finite or infinite) set D ⊆ N we define the graph G = G(D) by setting

V (G(D)) = N E(G(D)) = {{i, i + d} | i ∈ N, d ∈ D}.
Clearly, L = G({1}). We will write

T (n, D) = T G(D) (n).
In the papers [START_REF] Körner | Pairwise colliding permutations and the capacity of infinite graphs[END_REF] and [START_REF] Körner | Graph-different permutations[END_REF] attention was restricted to those cases where the growth of T (n, D) in n is only exponential. Here we consider various speeds of growth and determine the exact value of T (n, D) for every n in a non-trivial case. We are especially interested in the relationship between the values of T (n, D) and T (n, D), where D = N \ D.

Superexponential growth

The determination of T (n, {1}) leads to a surprisingly simple formula.

Theorem 1 T (n, {1}) = n! 2 ⌊ n 2 ⌋
for every n ∈ N.

Proof. First we prove the upper bound

T (n, {1}) ≤ n! 2 ⌊ n 2 ⌋ .
To this end fix n and define σ i,j to be the permutation that exchanges the entries i ∈ [n] and j ∈ [n], that is, for any permutation π, σ i,j π differs from π only in the places where the entries i and j stand, which are exchanged. For any fixed π consider the set of permutations

C(π) := {σ ε 1,2 1,2 σ ε 3,4 3,4 . . . σ ε k,k+1
k,k+1 π : ∀i ε i,i+1 ∈ {0, 1}}, where k equals 2⌊n/2⌋ -1, σ 0 i,j is meant to be the identity permutation, while σ 1 i,j := σ i,j . Let B be a set of permutations of [n] satisfying our condition that for any pair of them there is an i ∈ [n] they map to numbers at distance at least two and observe that the conditions imply

|C(π) ∩ B| ≤ 1, while C(π) ∩ C(π ′ ) = ∅ if π, π ′ ∈ B, π = π ′ . Since |C(π)| = 2 ⌊n/2⌋ for any π, the foregoing implies |B| ≤ n! 2 ⌊ n 2 ⌋
, which is the claimed upper bound.

In order to prove the inequality in the opposite direction, for every n we shall explicitely construct a set of permutations satisfying the requirement. We start by the odd values of n and build our construction in a recursive manner. It will be important for the recursion that for every odd n the construction be invariant with respect to cyclic shifts. For n = 1 the construction consists of the identical permutation. Suppose next to have constructed

t n-2 := (n -2)! 2 ⌊ (n-2) 2 ⌋
permutations yielding a set B n-2 that satisfies the pairwise relation we need and has the additional property of being closed with respect to cyclic shifts. We will construct a set A n of n-1 2 t n-2 permutations of [n] satisfying the same pairwise condition and define B n to be the set consisting of all the cyclically shifted versions of the elements of A n . For an arbitrary permutation π of [n -2] and 1 < j ≤ n we define the transformations Ψ j in the following manner. The permutation Ψ j π is acting on the set [n],

Ψ j π(1) := n Ψ j π(i) := π(i -1) for every 1 < i < j Ψ j π(j) := n -1 Ψ j π(i) := π(i -2) for every j < i ≤ n.
In other words, the permutation Ψ j π is obtained from π by prefixing n in the position preceding the first number in π and inserting n -1 in the j-th position of the resulting permutation. For a set A of permutations we denote by Ψ j (A) the set of the images by Ψ j of all the permutations of [n -2] belonging to A. As a last element of notation, let us denote by S j the set of those permutations τ of [n] for which τ -1 (n -2) < j. Consider

A j := Ψ j (B n-2 ) ∩ S j

and set

A n := ∪ n j=2 A j . (The attentive reader may note that A 2 = ∅ but we felt it more natural not to exclude this set from the above union.) As every permutation in A n has n at its first position no two of them can be cyclic shifts of each other, whence |B n | = n|A n |. Therefore in order to check that we have constructed the right number of permutations it is sufficient to verify that

|A n | = n -1 2 t n-2 (1)
To this effect, recall that by our hypothesis the set B n-2 is invariant with respect to cyclic shifts. This implies that the number of those of its sequences in which a fixed element, in our case (n -2), is confined to any particular subset of the coordinates is proportionate to the cardinality of the coordinate set in question, and thus

|A j | = j -2 n -2 |B n-2 |,
whence

|A n | = n j=2 |A j | = n j=2 j -2 n -2 |B n-2 | = n -1 2 t n-2 ,
which, substituting the value of t n-2 , yields

|A n | = (n -1)! 2 ⌊ n 2 ⌋ .
This settles our claim (1) and proves that B n has the requested number of permutations.

To conclude the proof it remains to show that every pair of sequences from B n represents a G({1})-different pair of permutations. We will first prove that such is the case if both sequences are from A n . If they belong to the same A j then this is obvious since the two permutations in such a pair must differ somewhere in those coordinates where they feature an element of B n-2 and thus the corresponding elements of B n-2 must be different sequences. This implies, by our hypothesis, that they differ in some coordinate by strictly more than 1. If the two sequences, π and τ, do not belong to the same A j , then we must have, say π ∈ A j and τ ∈ A k with j < k. But then in the k-th position τ (k) = n -1, while by definition, π(k) < n -2, settling this case as well.

If π and τ are two permutations that do not belong to A n but have the value n in the same position, then they are clearly in a similar relation as their respective cyclic shifts in A n , thus the above argument still applies.

Finally, we must prove that any two of our sequences having the symbol n in different positions also represent a G({1})-different pair of permutations. Now, unless the symbol n of both of the two sequences meets the symbol (n -1) of the other one, we are done. Otherwise they have their respective subsequences belonging to B n-2 positioned in the very same coordinates and it suffices to see that these subsequences are different. For this purpose suppose that the two sequences have their symbol (n -1) in the j-th and the k-th position, respectively. But then, supposing j < k we can say that they must have their respective symbols (n -2) in different positions since the one having its (n -1) in the k-th position has its (n -2) in a position belonging to the open interval (j, k) while the other one has it in the complement of the closed interval [j, k] by construction. This proves our theorem for every odd n.

In order to prove our claim also for even values of n, it is enough to consider the set A n+1 (now n + 1 is odd) and delete the first entry, which is (n + 1), from each of the permutations in this set. This way we get the right number of permutations of [n] and their pairwise relations satisfy the requirement by the previous part of the proof.

2

Remark 1 The construction presented in the proof above (in fact, a family of similar constructions that includes it) can also be described in the following short way. Let n be odd and take all those permutations of [n] in which the cyclic order within each of the triples {1, 2, 3}, {3, 4, 5}, . . . , {n -2, n -1, n} is the same as in some given fixed permutation; say, the identity. Verifying the relevant properties of the construction is also simple using this description. The case of even n follows the same way as in the above proof.

Remark 2 Consider the graph whose vertex set is the set of permutations of [n] and such that two permutations form an edge if and only if they satisfy the requirement we dealt with in Theorem 1. Denote this graph by H {1} (n). Observe that its clique number

ω(H {1} (n)) = T (n, {1}
) by definition and notice that by the proof above its chromatic number χ(H {1} (n)) has the same value. (The sets C(π) defined in the proof can serve as color classes of an optimal coloring.) This observation will be used in the proof of the subsequent corollary.

With some additional argument the above theorem gives the exact value of T (n, {q}) also for q = 1. We will need the following well-known lemma, the roots of which go back to Shannon [START_REF] Shannon | The zero-error capacity of a noisy channel[END_REF]. We give a short proof for the sake of completeness.

Lemma 1 Let G 1 , . . . , G k be graphs and let G 1 • . . . • G k denote their co-normal product, i.e., the graph with vertex set V (G 1 )×. . .×V (G k ) in which two vertices x, y are adjacent if there is an i such that the respective i-th entries x i , y i of these sequences satisfy {x i , y

i } ∈ E(G i ). If χ(G i ) = ω(G i ) holds for every i, then ω(G 1 • . . . • G k ) = k i=1 ω(G i ). Proof. It is easy to verify that ω(G 1 • . . . • G k ) ≥ k i=1 ω(G i ) always holds. To prove the reverse inequality observe that χ(G 1 • . . . • G k ) ≤ k i=1 χ(G i ). By ω(G 1 • . . . • G k ) ≤ χ(G 1 • . . . • G k ) the conditions χ(G i ) = ω(G i ) imply the statement. 2
Corollary 1 Let q be an arbitrary fixed natural number and let n have the form aq + m, where m ∈ {0, . . . , q -1}. Then

T (n, {q}) = n! 2 ⌊ a 2 ⌋ q-m 2 ⌊ a+1 2 ⌋ m .
Proof. Let S n be the set of all permutations of [n] represented as sequences and consider a largest possible set B n of sequences from S n which satisfies the requirements for D = {q}. Let h : N → {0, . . . , q -1} be the residue map modulo q, or, in fact, any map for which h(k) = h(ℓ) if and only if q divides |k -ℓ|. For sequences x = x 1 . . . x n extend h as h(x) := h(x 1 ) . . . h(x n ). Partition S n according to the image of h, i.e., put x and y into the same partition class iff h(x) = h(y). The number of partition classes so obtained is

t := n! (a!) q-m ((a + 1)!) m = n a, . . . , a, a + 1, . . . , a + 1
We call the classes W 1 , . . . , W t . If two sequences x, y belong to different W j 's then there must be a position i for which |x iy i | is not divisible by q, in particular, it is not equal to q. Thus T (n, {q}) is just the sum of the maximum possible cardinalities of sets of sequences one can find within each W j such that each pair of these sequences satisfies the condition.

Fix any class W j . For each x ∈ W j and each position i the value h(x i ) is the same by definition. Let h j i denote this common value. For k ∈ {0, . . . , q -1} set E k = {i | h j i = k}. Consider the subsequence of each x ∈ W j given by the entries at the positions belonging to E k . Note that the size of |E k | is either a or a + 1. Let H k be the following graph. Its vertex set consists of

|E k |-length sequences of different numbers from [n]∩{ℓ | h(ℓ) = k}. Two such sequences x and y are adjacent in H k iff at some coordinate i we have |x i -y i | = q. It is straightforward that H k is isomorphic to the graph H {1} (|E k |) defined in Remark 2. Whence its clique number is T (|E k |, {1}
), while, by Remark 2, its chromatic number has this same value. Let Ĥj be the graph with vertex set W j where two vertices are adjacent if they satisfy the requirement that at some position their difference is neither 0 nor q. One easily verifies that Ĥj is isomorphic to the co-normal product (for the definition see Lemma 1) of the graphs H 0 , . . . , H q-1 , which is, by the foregoing, isomorphic to q-1 k=0 H {1} (|E k |). We are interested in the clique number of this graph. By Lemma 1 and Remark 2 this value is equal to q-1 k=0 ω(H {1} (|E k |)). Noticing that qm of the sets E k have size a and m of them have size a + 1, this is further equal to a!/2 ⌊ a 2 ⌋ q-m (a + 1)!/2 ⌊ a+1 2 ⌋ m by Theorem 1. The latter value is the same for all sets W j and the number of these sets is n a,...,a,a+1,...,a+1 (with a and a + 1 appearing qm and m times, respectively). Thus we have obtained (We do not need the sharper form of the upper bound here. For an exponential improvement in the above lower bound the reader is invited to consult Brightwell and Fairthorne [START_REF] Brightwell | Permutation capacity of graphs[END_REF].) Using this in combination with Theorem 1 yields

T (n, {q}) = n a, . . . , a, a + 1, . . . , a + 1 a! 2 ⌊ a 2 ⌋ q-m (a + 1)! 2 ⌊ a+1 2 ⌋ m = n! 2 ⌊ a 2 ⌋ q-m 2 ⌊ a+1 2 ⌋ m . 2 3
Proposition 1 0.33 < φ({1}, {1}) ≤ 1 2
We continue with other examples. Denoting by 2N the set of the even numbers, we would like to determine φ(2N, 2N). To this end, notice first that

T (n, 2N) = n ⌊ n 2 ⌋
.

In fact, this easily follows, as in [START_REF] Körner | Pairwise colliding permutations and the capacity of infinite graphs[END_REF], by observing that two permutations differ in every position by an even number if and only if the even numbers occupy the same set of positions in both. Somewhat surprisingly, T (n, 2N) seems hard to determine and we only have some easy bounds.

Proposition 2 n! n 2 + 1 2 n ⌊ n 2 ⌋ ≤ T (n, 2N) ≤ n! 2 ⌊ n 2 ⌋
Proof. The upper bound is a trivial consequence of (the upper bound part of) Theorem 1. Although the lower bound follows from the lower bound on κ(K n ) in [START_REF] Körner | Graph-different permutations[END_REF], yet for the reader's convenience we give the details without explicit reference to said paper. (Those needing more details may however consult [START_REF] Körner | Graph-different permutations[END_REF]). We consider the set [n] as the disjoint union of its respective subsets of odd and even numbers. Correspondingly, we divide the coordinate set in two (with a little twist). In the first ⌈ n 2 ⌉ + 1 coordinates we write the even permutations of the set A consisting of all the odd numbers from [n] with the addition of the extra symbol ⋆. (More precisely, first we represent these ⌈ n 2 ⌉ + 1 many symbols bijectively by the first natural numbers up to their cardinality, then extend this bijection to the permutations of both sets and consider only those permutations of the elements of A that correspond to the even permutations of the first |A| natural numbers). We represent an arbitrary permutation of A in form of a sequence x and similarly let y be an arbitrary permutation of the set B of the even elements of [n]. We will say that y is hooked up to x if we replace the ⋆ in x by the first coordinate of y and concatenate the rest of y as a suffix to the resulting sequence. Let us denote by x ↼ y the permutation of [n] so obtained. Define A ↼ B to be the set of all these permutations as x and y take all of their possible values. Clearly,

|A ↼ B| = 1 2 n 2 + 1 ! n 2 !
which in turn equals the claimed lower bound in the statement of the Proposition. It is very easy to see on the other hand that all the pairs of permutations from A ↼ B differ by an even number in some coordinate. 2

Corollary 2 0 ≤ φ(2N, 2N) ≤ 1 2 2
Next we quickly review the following immediate consequence of our hitherto results on split strength. Proposition 3 Let q be an arbitrary but fixed natural number. Then φ({q}, {q}) is independent of the actual value of q.

Proof. We prove more, namely that the asymptotics of T (n, {q}) is independent of the value of q we fix and the same is true for T (n, {q}). For the latter it follows immediately from the formula given in Corollary 1. Now we turn to T (n, {q}). Consider the distance graph G({q}) of the set {q} and look at the graph it induces on [n]. Since the latter is isomorphic to a subgraph of P n , the path on n vertices that the analogous distance graph G({1}) induces on the same set, we immediately see that T (n, {q}) ≤ T (n, {1}).

(2)

In the reverse direction, we just have to observe that, for every m ∈ {0, 1, . . . , q -1}, the graph G({q}) induces an infinite path on the residue class qN + m of the numbers congruent to m modulo q. This implies

T (n, {q}) ≥ q-1 m=0 T n -m q , {1} (3)
by concatenating the respective constructions of permutations for each fixed m. Whence it is immediate that T (n, {q}) and T (n, {1}) have the same exponential growth rate. 2

We know very little about split strength and thus there are many questions to ask. Is it always true that φ(D, D) is finite and non-negative as it seems by these examples? In order to see the greater picture, we have to look at different kind of growth rates as well.

Intermediate growth

So far we have only seen growth rates at an exponential factor away from either 1 or n! . We intend to show here, however, that in between growth rates are also possible. In particular, we will see that T (n, D) and T (n, D) can have essentially the same growth rate, while their product is still about n! .

Let ex(n) denote the largest exponent s for which 2 s is a divisor of n. We define

E := {n | n ∈ N, ex(n) ≡ 0(mod 2)}. ( 4
)
Theorem 2 If n is a power of 4, then we have

(a) ( √ n)! √ n ≤ T (n, E) ≤ n! ( √ n)! √ n , (b)
( √ n)! √ n ≤ T (n, E) ≤ n! ( √ n)! √ n .
Proof. We prove the lower bound part of (a) first. It will be convenient to consider the elements of [n] as binary sequences of length t := ⌈log n⌉, with each natural number from [n] represented by its binary expansion. (Integer parts could be deleted by our assumption on n, moreover, we also know that t is an even number.) In fact, instead of permuting the n integers in {1, . . . , n}, now we will permute the n numbers in {0, . . . , n -1}. With a shift by 1, the two are obviously equivalent for our purposes. For simplicity, we will index the coordinates of the binary expansions from right to left. Hence in particular m is odd if in its binary expansion x = x t x t-1 . . . x 1 the rightmost coordinate x 1 is 1 and even else. Let further x odd and x even denote the subsequence of the odd and the even indexed coordinates of x, respectively. Finally, let ν(x) be the smallest (i. e., rightmost) index i for which x i = 1. By a slight abuse of notation we will consider the various subsets of {0, . . . , n -1} as subsets of {0, 1} t . Quite clearly, for every x ∈ {0, 1} t we have

ν(x) = ex(x) + 1
and, in particular, x ∈ E if and only if ν(x) ≡ 1 modulo 2. In order to prove the lower bound, let us consider the partition induced on {0, 1} t (i.e., on {0, . . . , n -1}) by the mapping f : {0, 1} t → {0, 1} t 2 where f (x) := x even for every x ∈ {0, 1} t .

(The classes of the partition are the full inverse images corresponding to the various values of f.) It follows by construction that

f (x) = f (y) implies |x -y| ∈ E (5)
where by the difference of the vectors x and y we mean the difference in ordinary arithmetics of the natural numbers they represent. Indeed, executing the subtraction in the binary number system we are using here one sees that both xy and yx have their rightmost 1 in the position where, scanning the binary expansions of x and y from right to left, we find the first position in which they differ. Now, since x even = y even by assumption, the position in question must have an odd index. In other words, ν(|x -y|) ≡ 1 modulo 2. For every z ∈ {0, 1} t 2 we denote by S(z) the set of all the permutations of the elements of the full inverse image f -1 (z) of z. Thus, by our previous argument, all these permutations are pairwise G(E)-different. Consider the Cartesian product

C := z∈{0,1} t 2 S(z). (6)
Note that the elements of C are permutations of the numbers in {0, . . . , n-1}. The above consideration implies that C is a set of pairwise G(E)-different permutations. Further, observing that for every z ∈ {0, 1}

t 2 |f -1 (z)| = 2 t 2 we have |C| = (2 t 2 )! 2 t 2 (7)
proving the lower bound in (a). (One might get a somewhat larger set by using the hookup operation instead of straightforward direct product but we do not intend to increase the complexity of the presentation for this relatively small gain here.)

Next we prove the upper bound part of (b). Notice that the set C we have constructed above has a stronger property than needed so far. In fact, in every coordinate, the absolute difference of our permutations is either 0 or else it belongs to E. This observation will be the basis for our upper bound.

Consider the auxiliary graph H E the vertices of which are the permutations of [n] and two are adjacent if they satisfy the requirement that at some position they have two numbers such that their difference is in E. Clearly, T (n, E) = ω(H E ), the clique number of this graph by definition. The above observation about C implies that its independence number α(H E ) is at least |C|. Note that H E is vertex transitive (any permuting of the coordinates in the vertices gives an automorphism), thus by a well-known fact (cf., e.g., in [START_REF] Scheinerman | Wiley-Interscience Series in Discrete Mathematics and Optimization[END_REF]) its fractional chromatic number χ f (H E ) is equal to the ratio of the number of vertices and the independence number. Using also that the clique number cannot acceed the fractional chromatic number (cf. [START_REF] Scheinerman | Wiley-Interscience Series in Discrete Mathematics and Optimization[END_REF]) we obtained

T (n, E) = ω(H E ) ≤ χ f (H E ) = |V (H E )| α(H E ) ≤ n! |C| = n! (2 t 2 )! 2 t 2
proving the upper bound in part (b).

Exchanging the role of even and odd above we obtain the lower bound in (b) and the upper bound in (a) in a similar way.

2

Theorem 2 shows that the investigated values of T (n, E) and T (n, E) have the same growth rate at around √ n! . The following statement is a straightforward generalization of the above.

Theorem 3 For every rational number α ∈ (0, 1), there is a set E α ⊆ N such that for infinitely many values of n we have

(n α )! n 1-α ≤ T (n, E α ) ≤ n! (n 1-α )! n α and (n 1-α )! n α ≤ T (n, E α ) ≤ n! (n α )! n 1-α .
Remark 3 Notice that taking logarithm and using the estimate log(k!) ≈ k log k the above inequalities give that log T (n, E α ) is about α log(n!), while log T (n, E α ) is about (1α) log(n!).

Proof. Let

E α := {m | m ∈ N, ex(m) ≡ 0, 1, . . . , p -1(mod q)}.

Set α = p/q and suppose n is a power of 2 q . The reasoning is essentially the same as in Theorem 2, which is the case α = 1/2, q = 2. Instead of [n] we again permute the elements of {0, . . . , n -1} and represent each of these numbers by their binary expansion. We collect into one group those numbers of {0, . . . , n -1} whose binary expansion has the very same subsequence in those positions which are indexed by numbers congruent to p + 1, . . . , q modulo q. There are n q-p q

= n 1-α different such groups each containing n α different numbers. Permuting the numbers within a group we get (n α )! permutations of those numbers and these are bound to differ at some position by the difference of two different numbers in the group. Such a value belongs to E α by construction. Putting all permutations of all our groups together we obtain (n α )! n 1-α different permutations altogether that not only satisfy the requirements given by the set E α but no two of which satisfy the requirements prescribed by E α . This gives the upper bound for T (n, E α ) in a similar way as the upper bound on T (n, E) is proven in Theorem 2. The rest is also similar to what we have seen there. 25 Exponential growth and Shannon capacity

In this section we return to the more familiar territory of distance graphs with finite chromatic number. The relevance of this parameter is shown by the following simple observation.

Proposition 4 Let G be an infinite graph with finite chromatic number χ(G). Then 2 In particular, distance graphs of "rare" sets of distances have finite chromatic number. More precisely, by a result of Ruzsa, Tuza and Voigt [START_REF] Ruzsa | Distance graphs with finite chromatic number[END_REF], if the set D := {d 1 , d 2 , . . . d n , . . .} has the density of a geometric progression in the sense that lim inf n→∞ d n+1 dn > 1, then the distance graph G(D) has finite chromatic number. Clearly, this density condition is sufficient but not necessary for the chromatic number to be finite (cf., e.g., the set of odd numbers as differences that result in a bipartite graph.) However, for some graphs G with finite chromatic number one can give a better upper bound on T G (n). This bound is easily obtained once we realize the tight connection of our present problem with the classical concept of Shannon capacity of a graph [START_REF] Shannon | The zero-error capacity of a noisy channel[END_REF].

T G (n) ≤ (χ(G)) n
Given a sequence x ∈ V n we shall denote by P x the probability distribution on the elements of V defined by

P x (a) = 1 n |{i | x i = a, i = 1, 2, . . . , n}|
for every a ∈ V ; here x = x 1 . . . x n . The probability distribution P x is called the type of

x. Let V n (P, ε) denote the set of those x ∈ V n for which

|P x -P | = max a∈V |P x (a) -P (a)| ≤ ε.
Let G be a finite graph with vertex set V = V (G) and edge set E(G). As always, we will say that the sequences x = x 1 x 2 . . . x n ∈ V n and y = y 1 y 2 . . . y n ∈ V n are G-different if there is at least one index i ∈ [n] with {x i , y i } ∈ E(G). Let ω(G, n) and ω(G, P, ε, n) be the largest cardinality of any set C ⊆ V n and C ′ ⊂ V n (P, ε), respectively, of pairwise G-different sequences. The Shannon capacity of G (or of G in the more usual notational convention, cf. [START_REF] Shannon | The zero-error capacity of a noisy channel[END_REF]) can be defined as

C(G) = lim sup n→∞ 1 n log ω(G, n),
while the capacity C(G, P ) of G within the type P is given (cf. [START_REF] Csiszár | On the capacity of the arbitrarily varying channel for maximum probability of error[END_REF]) as

C(G, P ) := lim ε→0 lim sup n→∞ 1 n log ω(G, P, ε, n).
It is immediate from the definitions that C(G, P ) ≤ C(G) always holds and using the methods of [START_REF] Csiszár | Information theory: Coding theorems for discrete memoryless systems[END_REF] it is easy to prove that in fact C(G) = max P C(G, P ).

In what follows we will restrict attention to graphs we call residue graphs. We say that an infinite graph G with vertex set N is a residue graph if there exists a natural number r and a finite graph M = M(G) with vertex set {0, 1, . . . , r -1} such that {a, b} ∈ E(G) if and only if {(a) mod r , (b) mod r } ∈ E(M)

Let Q be the uniform distribution on {0, 1, . . . , r -1}. We have For the reverse inequality let n be a multiple of r and consider any construction achieving T G (n), i.e., a set of permutations of the elements of [n] that are pairwise Gdifferent, while the cardinality of the set is T G (n). Substitute the occurence of the number i in each of these permutations by the unique j ∈ {0, . . . , r -1} which is congruent to it modulo r. Doing this for all i ∈ [n] we get T G (n) different sequences of vertices of M each having type Q that actually form a clique in M n .

Theorem 4 lim n→∞ 1 n log T G (n) = C(M(G), Q). Proof. To prove lim sup n→∞ 1 n log T G (n) ≥ C(M(G), Q) consider, for every n those sequences x ∈ {0, 1, . . . , r -1} n whose type Q n satisfies Q n (a) = 1 n |{m | m ≤ n, (
2

The following corollary is immediate. As an easy consequence of Lovász' celebrated formula [START_REF] Lovász | On the Shannon capacity of a graph[END_REF] for the Shannon capacity of the pentagon graph we obtain, using the last theorem (and also that the Shannon capacity of C 5 is obtained by sequences the type of which is the uniform distribution), that lim n→∞

1 n log T G (n) = 1 2 log 5.
It is an easy observation that for any graph M and any rational probability distribution P on its vertex set one can construct (by simply substituting each vertex by an independent set of appropriate size) a graph M for which C(M, P ) = C( M , Q), where Q is again the uniform distribution. It is then easy to construct an infinite graph G, which is a residue graph with respect to M and thus the asymptotics of T G (n) is in an analogous relationship with the capacity C(M, P ) as the one expressed in Theorem 4. Taking into account the remark that C(M) can be expressed as the maximum of the values C(M, P ) over P , we can conclude that the class of problems asking for the asymptotics of T G (n) for various infinite graphs G contains the Shannon capacity problem of all such graphs for which Shannon capacity is attained as the capacity within a type for some rational distribution.

Conjecture 1 (

 1 [START_REF] Körner | Pairwise colliding permutations and the capacity of infinite graphs[END_REF]) For every n ∈ Nρ(n) = n n 2

4 ≤

 4 Graph pairsIt seems interesting to study the relationship of the values of T (n, D) for pairs of disjoint sets (graphs) and their union, especially in case of pairs of complementary sets. Let us define φ(D, D) := lim sup n→∞ 1 n log T (n, D)T (n, D) n! and call it the split strength of the partition {D, D} of the natural numbers. (All logarithms in the paper are of base 2.) Consider the case D := {1}. We know from [T (n, {1}) ≤ 2 n .

Proof.

  Let c : V (G) → [χ(G)] be an optimal coloring of the vertices of G and let c n : V (G) n → [χ(G)] n be its usual extension to sequences. Notice that none of the full inverse images c -1 n of the elements of [χ(G)] n can contain two pairwise G-different permutations of [n].

1 n

 1 m) mod r = a}| for every a ∈ {0, 1, . . . , r -1}. Let M n be the graph whose vertices are the n-length sequences of vertices of M and whose vertices are adjacent if the corresponding sequences are M-different. For every n let C n denote a complete subgraph of maximum cardinality M n induces on the set of sequences of typeQ n . Notice that lim n→∞ log |C n | = C(M(G, Q)), because any sequence in [V (M)] n (Q, ε) can be extended to one of type Q n ′ , where n ≤ n ′ ≤ n + 2|V (M)|εn, i.e., by adding at most 2|V (M)|εn new coordinates. This shows |C n ′ | ≥ ω(G, Q n , ε, n) and since |V (M)| is constant and ε can be chosen arbitrarily small, it proves that 1 n log |C n | tends to C(M(G, Q)) as n goes to infinity. Now for any fixed n and to any sequence x ∈ V (C n ) we associate a permutation of [n] by replacing the occurrences of a in the sequence by the different numbers congruent to a modulo r, in a stricly increasing order. The result is a set of permutations which is G-different and has cardinality |C n |. With the observation above this proves the desired inequality.

Corollary 3 lim n→∞ 1 n 2

 12 log T G (n) ≤ C(M(G)).Let us consider the followingExample Let G have vertex set N and set {a, b} ∈ E(G) if |a -b| ≡ 1 or 4 (mod 5).

Acknowledgement. Part of this research was done during a visit of the second author in Rome, where he enjoyed hospitality of Dipartimento di Informatica, Università "La Sapienza".

Research partially supported by the Hungarian Foundation for Scientific Research Grant (OTKA) Nos. T046376, AT048826, NK62321, and EU ToK Project FIST 003006.