Phase diagrams of alkali halides using two interaction models - FCUL

THE JOURNAL OF CHEMICAL PHYSICS 126, 024503 共2007兲

Phase diagrams of alkali halides using two interaction models: A molecular dynamics and free energy study Pedro C. R. Rodrigues and Fernando M. S. Silva Fernandesa兲 Department of Chemistry and Biochemistry, Faculty of Sciences, University of Lisboa, Campo Grande, Bloco C8, 1749-016 Lisboa, Portugal

共Received 19 October 2006; accepted 22 November 2006; published online 12 January 2007兲 Phase diagrams for potassium and sodium chlorides are determined by molecular dynamics and free energy calculations. Two rigid-ion interaction models, namely, the Born-Mayer-Huggins 共BMH兲 and Michielsen-Woerlee-Graaf 共MWG兲 effective pair potentials, have been used. The critical and triple point properties are discussed and compared with available experimental and simulation data. The MWG model reproduces the experimental liquid-gas equilibria better than the BMH model, being the accordance very good in the lowest temperature region of the coexistent liquids, particularly for NaCl. However, both models underestimate the critical temperatures of KCl and NaCl. Relatively to the solid-gas equilibria, the models do not reproduce well the experimental data. As for the solid-liquid coexistences either the BMH or the MWG models appear unrealistic. © 2007 American Institute of Physics. 关DOI: 10.1063/1.2423030兴 I. INTRODUCTION

A major thrust in direct simulation of phase equilibria has been the introduction of the Gibbs ensemble Monte Carlo method 共GEMC兲 by Panagiotopoulos1 and Panagiotopoulos et al.2 It avoids the explicit consideration of the interface and enables the direct determination of the phase coexistence properties of pure components and mixtures, from a single simulation, without the need of calculating or specifying the chemical potential and pressure. One of the limitations of the original GEMC is the poor convergence of the averages when dealing with polyatomic molecules and dense phases. In these cases successful particle exchange becomes rare and the phases are slow to equate the chemical potentials. There are a number of techniques to remedy those situations as, for example, the use of biased methods.3,4 The other limitation is that it cannot be used to study fluid-solid coexistence. In fact, mass exchange between the fluid and the solid phases requires addition or removal of molecules in an otherwise perfect crystal and this would result in the creation of point defects. However, GEMC has recently been extended by Chen et al.,5 allowing the direct simulation of solid-fluid equilibria. Other alternative, which avoids particle exchange, is the Gibbs-Duhem integration method proposed by Kofke6,7 and Agrawal and Kofke.8,9 It enables the direct simulation of the whole phase diagram of any substance. The conditions of coexistence are calculated according to the Gibbs-Duhem equation and simultaneous N , p , T simulations 共by Monte Carlo or molecular dynamics兲 are performed along the coexistence lines. A major issue of the technique, however, is the definition of starting coexistence states 共to be obtained by other means兲 in order to initiate the integrations. Also, the method requires some sort of coupling between the coexista兲

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ent phases, specially in the approach to a critical point, in order to prevent a catastrophic phase change. As far as ionic systems are concerned the use of the referred to methods is not an easy task namely, due to the required maintenance of electroneutrality in particle exchanges by GEMC 共Refs. 10 and 11兲 or to the determination of starting states and coupling procedures in the GibbsDuhem integration. As an alternative to these methods, Guissani and Guillot12 have combined molecular dynamics simulations with an analytical equation of state to trace the liquidvapor coexistence of NaCl. More recently important progress has been made in the theory and simulation of charged systems. For instance, Camp and Patey13 have studied the coexistence and criticality of fluids with long-range potentials; Anwar et al.14 have reported the melting point of NaCl by molecular dynamics and free energy calculations; Lanning et al.15 have determined the solid-liquid coexistence and the properties of the interface of pure KCl and LiCl by direct simulation of coexisting crystalline and molten regions, including polarization effects on the melting temperature; Bresme et al.16 have looked at the influence of ion size asymmetry on the properties of ionic liquid-vapor interfaces; Valeriani et al.17 have simulated the rate of crystal nucleation of NaCl from its melt at moderate supercooling; and Anwar and Heyes18 have proposed a new approach to eliminate the problems of creation and/or annihilation of atoms in free energy calculations of charged molecular systems. The methods based on the explicit evaluation of free energies are certainly well suited to the determination of phase diagrams, presumably the most robust ones at least for ionic systems. Among them are the thermodynamic integration,4 Widom’s insertion particle scheme,19 Bennett’s overlapping distribution,20 Ferrenberg and Swendsen’s histogram reweighting,21,22 and Torrie and Valleau’s umbrella sampling.23,24

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P. C. R. Rodrigues and M. S. S. Fernandes

Recently, Gelb and Muller25 have also proposed a general method where the efforts to avoid the presence of interfaces are substituted by the usage of large systems in the simulations when the interactions allow an efficient parallelization. It is well known, from experiment, that ionic salts present very low vapor pressures even in the liquid state. The values are so low that, for example, no more than two particle-emission events 共only in exceptional conditions兲 were observed in simulations of solid and liquid ionic clusters at zero external pressure,26 for simulation times of the order of nanoseconds. This fact introduces some further restrictions regarding the choice of the methods to determine ionic salt phase diagrams12 and has lead us to the adoption of the thermodynamic integration method. The main objective of this work is to report the phase diagrams for KCl and NaCl in a wide range of densities and temperatures, based on two interaction models, namely, the Born-Mayer-Huggins 共BMH兲 and the Michielsen-WoerleeGraaf 共MWG兲. The discussion on the strength and limitations of the models is made by comparing our results with available experimental and simulation data. Section II contains the interaction models and the simulation strategies. In Sec. II B an initial diagram sketch of the KCl diagram is presented, which is very useful to guide the simulations and the free energy calculations. Section III gives the free energy results for KCl and Sec. IV reports its phase diagram based on the BMH model. Section V contains the phase diagram obtained from MWG model and a comparative discussion against the BMH model. Sections VI and VII present, respectively, the results for NaCl and the concluding remarks of the work. II. MODELS AND SIMULATION STRATEGIES

Two potential models have been used. The BMH effective pair potential,

␾ij共r兲 =

冋册

␴ij − r z iz j e 2 Cij Dij + cijb exp − 6 − 8, r d r r

共1兲

with the parameters given by Watts and McGee,27 and the MWG effective pair potential,

␾ij共r兲 =

z iz j e 2 b Cij Dij m + n exp关kij共␴m ij − r 兲兴 − 6 − 8 , r r r r

共2兲

with the parameters given by Michielsen et al.28 for n = 4 and m = 1. In order to investigate the coexistence properties of these models, the knowledge of the absolute free energies of both solid and fluid phases is required. As a general strategy the Helmholtz free energy of the system along an isotherm, at temperature T, is evaluated by integrating the pressure, p, as a function of the density according to A共␳ = ␳II,T兲 = A共␳ = ␳I,T兲 + n

冕冉冊 ␳II

␳I

d␳

p共␳兲 , ␳2

共3兲

where n is the number of ions. The free energy along an isochoric, at density ␳, is calculated by

A共T = TII, ␳兲 A共T = TI, ␳兲 = + TII TI

冕

TII

d共1/T兲E共T兲,

共4兲

TI

where 共␳ = ␳I , T兲 and 共T = TI , ␳兲 are the thermodynamic reference states with known free energies 共see below兲, and E is the internal energy of the system. The equilibrium conditions are derived from the equality of the chemical potentials and the pressures of the different phases at a given temperature. The quantities entering Eqs. 共3兲 and 共4兲 have been determined through NVT molecular dynamics simulations using the damped-force and Nosé-Hoover chains methods,4 with samples of 216–1728 ions. As far as the determination of the reference free energies in Eqs. 共3兲 and 共4兲 is concerned, we have used, for the solid phase, the Einstein crystal method.14,29 The interactions in the solid are smoothly transformed, through a coupling parameter ␭, into a corresponding harmonic potential so that the configurational energy is U共rn,␭兲 = UX共rn0兲 + 共1 − ␭兲共UX共rn兲 − UX共rn0兲兲 n

+ ␭ 兺 ␣共ri − r0,i兲2 ,

共5兲

i=1

where r0,i is the lattice position of ion i, UX共rn0兲 is the static contribution to the potential energy 共X stands for KCl or NaCl兲, and ␣ is the spring constant of the Einstein crystal. The free energy of the solid reference state at 共␳ , T兲 is then A共␳,T兲 ⬅ AX = AEinstein +

冕

0

1

d␭

冓冔 ⳵U共␭兲 ⳵␭

, ␭

共6兲

where AEinstein is the free energy of the Einstein crystal 共see Appendix兲. As for the absolute free energies of the fluid phases the use of Widom’s insertion particle schemes,4,18 for example, is inaccurate and imprecise for ionic salts due to the poor statistical convergence. As such, considering the well known peculiarities of the present potential models and their derivatives 共infinitely attractive at short distances and singular at the origin兲 the simplest path we have found, to attain an ideal reference state, involves two steps by means of the coupling parameter technique: • Conversion of the BMH and MWG in half-wing repulsive Gaussian potentials,

␾共r兲 = ␥e−␦r , 2

共7兲

with the repulsion barriers as close as possible to the original ones. The ␥ and ␦ parameters are also chosen in order to avoid boundary condition discontinuities. • Conversion of the Gaussian potential in a null potential in order to reach the ideal gas reference state 共see Appendix兲. To assess the consistency of the results, several values for the absolute free energy, along different isochores and isotherms of the solid and fluid phases, have been calculated by the coupling parameter technique through Eqs. 共6兲 and 共7兲. Then, taking some of those values as reference states,


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Phase diagram of alkali halides

FIG. 2. Vapor pressure of KCl as a function of temperature.

p=

FIG. 1. Differences of pT and pS 共relatively to pT calculated with 1728 ions兲 as a function of density, for different number of KCl ions. 共a兲 T = 1100 K; 共b兲 T = 2900 K.

thermodynamic integrations have been performed by Eqs. 共3兲 and 共4兲, and the two sets of results have been compared. Cubic periodic boundary conditions have been applied. Apart from Ewald’s sum, by the method of Adams and Dubey,30 for the Coulombic energies and forces, no longrange corrections or cutoff, rc, have been used for the potential dispersion terms but the minimum image convention. Indeed, such corrections4 presuppose that the radial distribution function, g共r兲, is ⬃1 for r ⬎ rc which is not the case for highly ordered systems like the present ones but at very low densities or with large simulation boxes. Nonetheless, as the potential is not constant on the surface of a cube around a given particle, we have checked out that no serious instabilities are observed in the dynamics of the systems. Verlet’s leapfrog algorithm3 for the numerical integration of Newton’s equations of motion, with a time step of 5 ⫻ 10−15 s, has been used in all simulations. Equilibration runs with a number of steps between 5 ⫻ 103 and 5 ⫻ 104 have been followed by production runs with 5 ⫻ 104 – 5 ⫻ 105 steps for the thermal properties and with 106 – 107 steps for the calculation of the free energies.

A. Pressure computation

In the calculation of the pressure from the virial theorem31

1 共2n具Ek典 + 具⌿典兲, 3V

共8兲

n−1 n 兺 j⬎1Fij · rij, it is common practice to avoid where ⌿ = 兺i=1 the explicit computation of ⌿. In fact, the Coulombic, dipole-dipole and dipole-quadrupole contributions to the pressure are directly related to the respective potential energy 具⌿dd典 = −6具⌽dd典, 具⌿dq典 contributions: 具⌿e典 = −具⌽e典, dq = −8具⌽ 典. We designate the pressure calculated by this approach as pT. Additionally, we have also determined the pressure by a direct computation of the virial only using the minimum image convention and not correcting the forces with Ewald’s sum. This was made, of course, with no interference in the time evolution of the system for which Ewald’s sum is always applied. We designate the pressure calculated by this approach as pS. Figure 1 shows the deviations of the two approaches as a function of density 共at two different temperatures兲 relatively to pT calculated with 1728 ions. The convergence is remarkable. At first sight, the last approach could seem redundant and laborious. Nonetheless, it may be useful, for example, when stress-tensor components are needed since the use of Ewald’s sum for their calculation it is not a straightforward task.32,33

B. Initial diagram sketch

As we have referred to in the Introduction ionic salts present very low vapor pressures even in the liquid state 共see Fig. 2兲. This fact is enough, by itself, to make a first sketch of the phase diagram10,12 which is very useful to guide the simulations and free energy calculations. To this end, the pressure can be computed on a grid covering the region of interest and fitted to a polynomial. From the obtained profile, at p = 0, a glimpse of the coexistence regions turns out and a first estimate of the critical and triple points is possible. Figure 3 shows the T共p , ␳兲 diagram for p 艌 0. Since the first isotherm where the pressure always remains positive is at 2600 K, this is the first estimate for the critical temperature. The critical density is between 5 and 7 nm−3 in the diagram resolution. As the lower isotherm where liquid is present is at 1000 K and the inflexion in solid density, as a


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TABLE I. Differences between Helmholtz energies of KCl: 共a兲 216 and 512 ions; 共b兲 512 and 1000 ions.

FIG. 3. 共Color兲 First sketch of KCl phase diagram. Temperatures in K, densities in nm−3, pressures in GPa.

function of temperature, is at ⬃1200 K, the triple point temperature is expected to be in this interval. The triple point densities are estimated between 27 nm−3 and 28 nm−3, 22 nm−3 and 24 nm−3, and Ⰶ1 nm−3 for the solid, liquid, and gas, respectively. The sketch profile is in good qualitative agreement with the predicted, by Ashcroft34 and other authors,13,35–37 for the liquid-gas equilibrium when longrange interactions are present.

III. FREE ENERGY COMPUTATIONS A. Solid KCl

Figure 4 presents absolute Helmholtz energies, computed with 216, 512, and 1000 ions, over isotherms at 1100, 1300, 1500, 1700, and 1900 K and for densities between 28 and 39 nm−3. Symbols represent values obtained, by the coupling parameter technique, using Eqs. 共5兲 and 共6兲 with ␣ = 32.043 J m−2. Full lines were calculated through Eq. 共3兲 taking the values for 1000 ions at density of 38 nm−3 as the reference states.

T/K

共a兲

共b兲

1100 1300 1500 1700

1.7 1.7 1.8

0.52 0.39 0.40 0.47

Table I contains the averages of the free energy differences between 216 and 512 ions and between 512 and 1000 ions taken at the different densities over each isotherm. Since these differences are nearly independent of density and temperature there is no need of making individual fittings to the sequences of the free energies obtained with 216, 512, and 1000 ions in order to obtain the corresponding extrapolated bulk values. The averages of the values in columns 共a兲 and 共b兲 are used to form the ordered pairs 共1 / 216, 2.2兲, 共1 / 512, 0.45兲, 共1 / 1000, 0兲 containing the inverse of the number of ions and the average differences to the value obtained with 1000 ions. For example, the extrapolation of the fit to n → ⬁ leads to the correction of −0.38 kJ mol−1 for the values obtained with 1000 ions. Then, the value −795.652 kJ mol−1 共obtained at temperature of 1100 K and after correction becomes density of 38 nm−3兲 −1 −796.032 kJ mol . U共rn0兲 was initially computed with 512 ions in the simulation box, but a more careful computation accounts for the dependence of this value on the number of ions. Table II shows the behavior obtained up to 10 648 particles. The sequence can be used to estimate the value U共rn0兲 = −708.501共2兲 kJ mol−1 in the limit n → ⬁ 共a correction of 0.551共2兲 kJ mol−1 relatively to the initial value computed with 512 ions兲. The corrected free energy of the previous example 共−796.032 kJ mol−1兲 becomes now −795.481 kJ mol−1.

B. Fluid KCl

Figure 5 shows the results of the Helmholtz energy as a function of the density, obtained by the coupling parameter technique, using ␥ = 3.4516⫻ 10−18 J and ␦ = 0.7 Å−2 in the Gaussian potential 共7兲, and including the free energy of the ideal gas reference states 关obtained from Eq. 共A11兲 of the Appendix兴. Additionally, the free energies were also calculated through Eq. 共3兲. Since the transformation begins at liquid phase and ends at an ideal gas phase, careful consistency checks have been TABLE II. Static energy U共rn0兲 / kJ mol−1, for KCl, as a function of the number of particles.

FIG. 4. Helmholtz energy of solid KCl as a function of density for isotherms at 1100, 1300, 1500, 1700, and 1900 K, for 216, 512, and 1000 ions.

n

U共rn0兲

n

U共rn0兲

512 1000 1728 2744

−709.052 −708.118 −708.596 −708.394

4096 5832 8000 10648

−708.526 −708.458 −708.510 −708.480


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TABLE III. Helmholtz energy of KCl as a function of the number of particles, calculated at temperature of 2900 K and density of 20 nm−3.

FIG. 5. Helmholtz energy of fluid KCl as a function of the density 共with 512 ions in the box兲 over isotherms 2600– 2900 K. Full curve was obtained using Eq. 共3兲. At the used scale, values computed using 216 and 1000 ions are indistinguishable.

performed to certify that direct and reverse scaling paths are coincident, and no “phase change” boundary is being crossed. The values obtained by the coupling technique are in an overall good agreement with the ones from the integration of Eq. 共3兲共full line in Fig. 5兲. However, their comparison with the integration of Eq. 共4兲共see Fig. 6兲 reveals a progressive deviation which reaches a few kJ mol−1 共⬃2% of error兲 near to the triple point. The sequence obtained with 216, 512, and 1000 ions in the simulation box 共see Table III with example values for temperature of 2900 K and density of 20 nm−3兲 presents a faster convergence of the coupling technique relatively to one of the solid phase. This behavior remains even for high fluid densities, near the coexistence with the solid. On the whole, the results from the coupling technique and the thermodynamic integration have a remarkable consistency, either for solid or fluid phases. IV. POTASSIUM CHLORIDE PHASE DIAGRAM A. Solid-liquid coexistence

From the Helmholtz energy and pressure values, the chemical potential was computed as a function of density

FIG. 6. Helmholtz energy of KCl 共at density 20 nm−3兲 as a function of temperature. 共full line兲 from Eq. 共4兲; 共circles兲 from coupling technique.

n

A 共kJ mol−1兲

216 512 1000 ⬁

−1205.54 −1204.76 −1204.50 −1204.21

共see Fig. 7兲. The chemical potential and pressure at coexistence are obtained from the intersections of the solid and liquid branches of ␮共p兲 in Fig. 8. The coexistence densities can then be obtained from Fig. 7 and are marked as asterisks in Fig. 9. The most significant error sources are related to the need of extrapolations to obtain p 共and consequently A and pV兲 for the liquid at coexistence densities. This limits the precision of the coexistence densities to a bit less than 0.5% despite, outside this region, A and pV values are computed with less than 0.1% of error 共after corrections referred at Secs. III A and III B兲. 1. Triple point

Polynomials fitted to the solid in coexistence with the gas and to the solid in coexistence with the liquid are used to compute the solid triple point conditions. Similarly, polynomials fitted to the liquid in coexistence with the solid and to the liquid in coexistence with the gas are used to estimate the liquid triple point properties. From this procedure, the estimated temperature at triple point is 1041± 9 K and the estimated pressure, obtained by application of Clapeyron’s equation to the solid-liquid boundary, is 140± 150 bars. Taking into account that from the same equation, there is only a deviation of ⬃0.1 K / bar, the triple point temperature compares well with the experimental melting point of 1044 K. This is in accordance with the value of 1038.7 K estimated by Lanning et al.15 The estimated densities are 28.3共1兲 nm−3 关1.751共6兲 g cm−3兴, for the solid, and 22.3共1兲 nm−3 关1.380共6兲 g cm−3兴, for the liquid, while their experimental values are 1.827共1兲共Ref. 38兲 and 1.494共3兲 g cm−3.39 These

FIG. 7. Chemical potential of KCl in the solid-liquid coexistence region over isotherms at 1100, 1300, 1500, and 1700 K as a function of density.


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TABLE IV. Volume change of KCl on melting. ⌬V / Vs共%兲

Reference This work 共BMH兲 Kirshenbaum et al. 共Ref. 39兲 Bockris et al. 共Ref. 38兲 Landon et al. 共Ref. 59兲 Eucken 共Ref. 60兲

26.91± 0.1 22.27± 0.08 20.20± 0.05 21.0± 0.5 23.0

p = 0兲 in nearly half of the liquid pocket height. The results obtained by this method are marked as ⫻ in Fig. 9 and the ones obtained by equating p, T, and ␮ in the two phases are represented as circles. FIG. 8. KCl chemical potential as a function of pressure along different isotherms in the solid-liquid coexistence region. 共←兲 Crossing points of the liquid and solid branches.

values give a volume change of 26.91% that is in fairly good agreement with the experimental results 共see Table IV兲. Apart from extrapolations using Clausius-Clayperon’s equation, the gas phase density at the triple point cannot, for now, be directly estimated better than ⱗ0.01 nm−3 共 ⱗ0.0006 g cm−3兲 as far as the values below ⬃1800 K 共see next section兲 indicate. B. Liquid-gas coexistence

There are two regions in the gas-liquid coexistence region where gas density evaluation presents special difficulties. • Near critical point due to the typical large fluctuations. • For temperatures under 2000 K, where pressure computation with good accuracy is prevented by the very low gas density values observed in ionic salts. However, the low vapor pressure values, in conjunction with high slopes in the pressure variation with density, are sufficient by themselves to compute three significant digits for the liquid coexistence density 共finding the densities where

1. Critical point

In order to obtain critical values a tentative fitting was made to the order parameter scaling law,40

冉冊

␳l − ␳g T =B 1− 2 TC

␤

共9兲

,

and then, using the resulting Tc, to the rectilinear diameter law,

冉冊

␳l + ␳g T = ␳C + A 1 − , 2 TC

共10兲

␤ was relaxed in order to find its best fitting value. The critical temperature and density, as well as the fit parameters A, B, and the ␹2 statistical test, for two ␤ values are summarized in Table V. The value ␤ = 0.326 that is the best choice for three dimensional spin systems41,42 and Lennard-Jones systems43 differs, significantly, from the best choice in the present case ␤ = 0.488. This result puts potassium chloride in the same group as sodium chloride and ammonia chloride and corresponds to the classical density expansion:12,44 共␳l − ␳g兲/2 = B0␶1/2 + B1␶3/2 + B2␶2 + . . . ,

共11兲

共␳l + ␳g兲/2 = ␳C + D0␶ + D1␶3/2 + D2␶2 + . . . ,

共12兲

where ␶ = 共1 − T / TC兲, up to the terms in B0 and D0 共see Fig. 10兲. As such, we have further carried out the fittings to Eqs. 共11兲 and 共12兲 up to powers of 3 / 2 共see Fig. 10兲. The results for this new fitting are in Table VI. The density values obtained from

␳l = ␳C + B0␶1/2 + D0␶ + B1␶3/2 + D1␶3/2 ,

共13兲

with ␳C, B0, D0, B1, D1, and TC, in Table VI, agree well with the liquid simulation data down to triple point, though that is not so for the gas, below ⬃1800 K, due to the very low densities. TABLE V. Critical values of temperature 共K兲 and density 共g cm−3兲 for two values of ␤.

␤ FIG. 9. KCl phase diagram using BMH model. 共a兲 and 共b兲 are obtained equating p, T, and ␮ in the two phases: 共c兲 and 共d兲 are obtained equating pressure to zero. Dotted lines are a guide for the eye.

0.326 0.488

␹2 2.3 0.085

TC

␳C

A

B

2543 2687

¯ 0.204

11.42 13.81

¯ 18.09


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FIG. 10. Fitting of Eqs. 共9兲 and 共11兲共considering power terms up to 1 / 2 and 3 / 2兲.

The estimated pressure at the critical conditions is 0.0052 GPa 共52 bars兲. The comparison of the simulated critical values with experimental ones39 should be carefully considered due to the inherent model limitations, which does not account, for example, for the existence of neutral atoms.12,45–47 Nevertheless, the present results are in fairly good agreement with the experimental data obtained by Kirshenbaum et al. 共see Table VI兲.

2. Boiling point and enthalpy of vaporization

Since direct computation of values near to the boiling point was not possible, they can be extrapolated by a fitting of Clausius-Clayperon equation: ln共p/pC兲 =

冉冊

FIG. 11. Transform of diagram 9 by a change of variable to ␼ = ␳1/2.

3. Revising TC and ␳C computations

As referred to before, very small gas densities compromise their computation as well as of the pressure and energies with good statistics. However, reasonable values were obtained down to 1800 K where the density value is ⬃0.01 nm−3. This also makes the application of Eqs. 共11兲 and 共12兲 sensitive to liquid density errors. Since most of these problems are due to the diagram axe declination to the left, a variable transform that promotes balance can contribute to better fitting results. Using ␼ = ␳1/2 and similar developments as before 共␼1 − ␼g兲/2 = E0␶1/2 + E1␶3/2 ,

共15兲

共␼1 + ␼g兲/2 = ␼C + F0␶ + F1␶3/2 ,

共16兲

it is possible to obtain new density expansions

⌬vaph TC −1 , RTC T

共14兲

␼2l = 关␼C + E0␶1/2 + F0␶ + 共E1 + F1兲␶3/2兴2 ,

共17兲

as presented in Fig. 2, in comparison with experimental results. Due to poor statistics in their computation, pressures at 1800 and 1900 K were not included in the fitting. The estimated value of 1720 K is in fair agreement with the experimental results, 1678 K from Refs. 48 and 49 and 1680 K from 共Ref. 50兲. Extrapolating in the reverse way, a new critical pressure estimation of 0.0046 GPa 共46 bar兲 is obtained, which is consistent with the one in Table VI. From Eq. 共14兲 the estimated value of the enthalpy of vaporization is ⌬vaph = 153 kJ mol−1, which underestimates the experimental result 216.47± 0.5 kJ mol−1,49 presumably because the global pressure was used instead of partial pressures of KCl monomers and K2Cl2 dimers,51 existing in the gas at this temperature region.

␼2g = 关␼C − E0␶1/2 + F0␶ + 共F1 − E1兲␶3/2兴2 ,

共18兲

from where

␳C = ␼C2 .

共19兲

It must be stressed that the new expansions include terms up to ␶3 while the original ones only go up to ␶3/2, but the number of parameters to fit is exactly the same. Figure 11 indicates a much better fitting with the transformation, in the approach to zero density, than the original fitting 共note the small wing in the pretransformation fitting at gas side兲. The temperature, TC = 2681 K, and density, ␳C = 0.225 g cm−3, obtained show moderate changes relatively to the original fitting. These differences are used to estimate average values and errors, presented in Table VII.

TABLE VI. Critical temperature 共K兲, density 共g cm−3兲, and pressure 共bar兲 values obtained from Eqs. 共11兲 and 共12兲共considering powers up to ␶3/2兲. The third row contains experimental values 共Ref. 39兲. TC

␳C

pC

B0

D0

B1

D1

2713 3200± 200

0.207 0.175± 0.05

52 +100 200共 −70 兲

13.167 ¯

16.65 ¯

1.578 ¯

−5.162 ¯


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TABLE VII. Summary of the BMH model KCl results. Temperature in K, density in g cm−3, and pressure in bars. TC This work BMH 2697共16兲 This work MWG 2860 Expt 3200共200兲

␳C

pC

Ttrip

0.216共9兲 0.208 0.175共50兲

49共3兲

1041共9兲 1096共4兲 1044

+100 兲 220共 −70

共s兲 ␳trip

1.751共6兲 1.749 1.827共1兲

共l兲 ␳trip

1.380共6兲 1.416 1.494共3兲

共g兲 ␳trip

Ⰶ0.0006 Ⰶ0.0006

Tboil 1720 1678

C. Solid-gas coexistence

A. Michielsen-Woerlee-Graaf

At solid-gas coexistence conditions, the difficulties due to the gas phase properties are strongly emphasized. However, solid coexistence density computation is much easier than in other regions of the diagram. The obtained results are represented as diamonds in Fig. 9, together with a plot of equation

Following the same strategies as described before the diagram of the MWG model has also been traced out. The triple point densities 关22.88 nm−3 共1.416 g cm−3兲 for the liquid and 28.26 nm−3 共1.749 g cm−3兲 for the solid兴 are, particularly for the liquid, closer to the experimental results than with the BMH model and the correspondent volume change 共23.5%兲 becomes also better 共see Table IV兲. Yet, the predicted triple point temperature, 1096± 4 K, is overestimated. Liquid densities in equilibrium with gas phase given by this model 共see Fig. 12兲 agree significantly better with experimental data and give a higher critical temperature TC = 2860 K 关at density ␳ = 3.36 nm−3 共0.208 g cm−3兲兴. The predicted solid densities coexisting with the gas, as well as the solid liquid behavior, are, however, practically the same as the ones predicted by the BMH model. Table VII summarizes the critical and triple point properties for the two potential models and their comparison with experiment.

␳s = 1.985 − 5.495 ⫻ 10−5t − 1.836 ⫻ 10−7t2共±0.001兲 g cm−3 ,

共20兲

where t = T − 273.15, obtained by Bockris et al.38 from experimental data. As expected, a coincidence is observed at normal temperature, 298.15 K, at which the potential model parameters were fitted,52 but under- and overestimations take place for higher and lower temperatures, respectively.

V. OTHER POTENTIAL MODELS

The analysis of the results indicates the limitations of the BMH model regarding the prediction of some regions of the KCl phase diagram. Such limitations have been discussed in literature and alternative potential models have been proposed. They are essentially grouped in four types: 共a兲 other rigid-ion models;28 共b兲 nonrigid-ion models 共polarization兲;52 共c兲 ab initio pseudopotentials;53 and 共d兲 three body interaction models54,55 共parametrized using ab initio computations兲. While the first two 关共a兲 and 共b兲兴 are built to refine results obtained at normal conditions, the remaining 关共c兲 and 共d兲兴 were introduced to allow reproduction of high pressure phase transitions experimentally observed.56

FIG. 12. Comparison of the BMH and MWG models for KCl.

VI. SODIUM CHLORIDE PHASE DIAGRAM

As for sodium chloride, part of the phase diagram was obtained using the same strategies as described above. Figure 13 shows the results in comparison with the ones obtained by Guissani and Guillot12 for the liquid-gas coexistence and with experimental data from Kirshenbaum et al.39 and Bockris et al.38 The agreement between the results from the MWG model and the available experimental data for the liquid is excellent. However, the critical temperature predicted using this model 共3114 K兲 is less than the experiment data projections by Kirshenbaum et al. 共3400± 200 K兲共Ref. 39兲 and

FIG. 13. Predicted and experimental values for the NaCl phase diagram.


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Pitzer 共3800± 200 K兲.57 The critical density calculated with the MWG model, 4.44 nm−3 共0.215 g cm−3兲, compares very well with the experimental value 0.22± 0.05 g cm−3. Our results are also in excellent agreement with the simulations of Guissani and Guillot, which were performed by a different method. Finally, despite we have not explicitly estimated the triple point for NaCl, preliminary calculations of some properties at the experimental melting temperature are consistent with results obtained by Anwar et al.14

␾i共ri兲 = ␣共ri − ri,0兲2 ,

where ␣ is the spring constant, r0,i are the equilibrium coordinates in the actual thermodynamic conditions and n

U共rn0兲

ACKNOWLEDGMENTS

One of the authors 共P.C.R.R.兲 gratefully acknowledges the institutional support of the Department of Chemistry and Biochemistry, FCUL, during his Ph.D. work. They also thank Intel Corporation for the free access to their compilers and the GNU and Linux communities for all the invaluable tools they offer. APPENDIX: REFERENCE STATES 1. Einstein solid

For an Einstein solid system reference,58

= 兺 U共r0,i兲

共A2兲

i=1

is the energy of the system studied system at the equilibrium coordinates and temperature zero.4 Excluding, for now, the static energy the Hamiltonian reads

VII. CONCLUDING REMARKS

Phase diagrams for potassium and sodium chlorides have been determined by molecular dynamics and free energy calculations. Two rigid-ion interaction models, namely, the Born-Mayer-Huggins 共BMH兲 and Michielsen-WoerleeGraaf 共MWG兲 effective pair potentials, have been used. The critical and triple point properties have been discussed and compared with available experimental and simulation data. The MWG model reproduces the experimental liquidgas equilibria better that the BMH model, being the accordance very good in the lowest temperature region of the coexistent liquids, particularly for NaCl. However, both models underestimate the critical temperatures of KCl and NaCl. It is noteworthy that the comparison between the liquidgas diagram profiles obtained from the two models for KCl and NaCl seems to indicate that a regular pattern is not followed. This may have implications regarding a corresponding state principle in order to estimate critical values. Further work is in progress to clarify the contributions responsible for the phase equilibria near critical conditions. Relatively to the solid-gas equilibria the models do not reproduce well the experimental solid densities but at normal conditions. However, the estimated values at triple point compare well with experiment. Due to the solid-gas coexistence behavior, the failure of the rigid-ion models in reproducing the high pressure solidsolid transitions, and the predicted solid-liquid coexistence profiles, either the BMH or the MWG models appear unrealistic in this region. The introduction of three body interactions and polarizable ion models may contribute to the improvement of the present results. Work along these lines is in progress.

共A1兲

n

n

p2i + 兺 ␣i共ri − r0,i兲2 , 2mi i=1

HES = 兺 i=1

共A3兲

and the resulting partition function is QES =

1 h3n

冕

+⬁

−⬁

n

e−HES/kBTdpdr = 兿 i=1

冋冉冊册 2mi ␲kBT ␣i h

2 3/2

, 共A4兲

leading to AES = U共rn0兲 − kBT ln共QES兲,

共A5兲

with the static energy included. Considering the present case 共two different ions兲 the Helmholtz free energy per KCl molecule is

兺

AES =

␨=K+,Cl−

再

冋冉冊册冎

3 2m␨ ␲kBT U共r0,␨兲 − kBT ln 2 ␣␨ h

2

, 共A6兲

where ␨ is an index over the species in presence.

2. Ideal gas

For the ideal gas, n

p2i , 2mi

HIG = 兺 i=1

共A7兲

and the canonical partition function is QIG =

1 h 兿n␨! 3n

=兿 ␨

再

冕冕 +⬁

−⬁

n

V

e−HIG/kBTdpdr

冎

1 ␨ V , 兿 n␨! i␨=1 共⌳␨兲3

共A8兲

共A9兲

with ⌳␨ =

h , 共2␲m␨kBT兲1/2

共A10兲

where ␨ is an index over the species in presence. Helmholtz energy, in this case, is


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AIG = kBT

兺

␨=K+,Cl−

冉

ln

e ␳␨共⌳␨兲3

冊

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共A11兲

30
