Computer simulation of liquid-vapor coexistence of confined quantum fluidsVíctor M. Trejos, Alejandro Gil-Villegas, and Alejandro Martinez Citation: J. Chem. Phys. 139, 184505 (2013); doi: 10.1063/1.4829769 View online: http://dx.doi.org/10.1063/1.4829769 View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v139/i18 Published by the AIP Publishing LLC. Additional information on J. Chem. Phys.Journal Homepage: http://jcp.aip.org/ Journal Information: http://jcp.aip.org/about/about_the_journal Top downloads: http://jcp.aip.org/features/most_downloaded Information for Authors: http://jcp.aip.org/authors

THE JOURNAL OF CHEMICAL PHYSICS 139, 184505 (2013)

Computer simulation of liquid-vapor coexistence of confinedquantum fluids

Víctor M. Trejos, Alejandro Gil-Villegas,a) and Alejandro MartinezDivisión de Ciencias e Ingenierías, Campus León, Universidad de Guanajuato, Loma del Bosque 103,Lomas del Campestre, 37150 León, Guanajuato, Mexico

(Received 26 June 2013; accepted 28 October 2013; published online 13 November 2013)

The liquid-vapor coexistence (LV) of bulk and confined quantum fluids has been studied by MonteCarlo computer simulation for particles interacting via a semiclassical effective pair potentialVeff (r) = VLJ + VQ, where VLJ is the Lennard-Jones 12-6 potential (LJ) and VQ is the first-orderWigner-Kirkwood (WK-1) quantum potential, that depends on β = 1/kT and de Boer’s quantumnessparameter � = h/σ

√mε, where k and h are the Boltzmann’s and Planck’s constants, respectively, m

is the particle’s mass, T is the temperature of the system, and σ and ε are the LJ potential parameters.The non-conformal properties of the system of particles interacting via the effective pair potentialVeff (r) are due to �, since the LV phase diagram is modified by varying �. We found that the WK-1 system gives an accurate description of the LV coexistence for bulk phases of several quantumfluids, obtained by the Gibbs Ensemble Monte Carlo method (GEMC). Confinement effects were in-troduced using the Canonical Ensemble (NVT) to simulate quantum fluids contained within parallelhard walls separated by a distance Lp, within the range 2σ ≤ Lp ≤ 6σ . The critical temperature of thesystem is reduced by decreasing Lp and increasing �, and the liquid-vapor transition is not longerobserved for Lp/σ < 2, in contrast to what has been observed for the classical system. © 2013 AIPPublishing LLC. [http://dx.doi.org/10.1063/1.4829769]

I. INTRODUCTION

Confined systems is a major area of basic and appliedresearch with diverse applications in nanotechnology.1 Prop-erties of fluids are strongly modified when they are confined,since atoms and molecules adsorbed onto a surface display adifferent potential energy between themselves as compared totheir energy when they are in a bulk phase;2 in consequence,the phase diagram of the confined fluid is modified and thecritical temperature is reduced; experimental values of the ra-tio of the critical temperatures (adsorbed fluid/bulk fluid) varyaround 0.4 for a wide range of molecular fluids.3–5 In this way,surfaces work as energy reservoirs that allow different con-ditions for the realization of phase transitions as well as theappearance of novel phases.

Another application of confined fluids are sieving pro-cess, where components of a fluid mixture are separated bythe size and shape of molecules, achieved and engineeredthrough molecular-scale filters and pores.6 In the case ofquantum fluids, such as hydrogen and deuterium, i.e., sub-stances of small molecular weight m whose de Broglie’swavelength (λB = h/

√2πmkT , where T, h, and k are the

temperature, Planck’s, and Boltzmann’s constants, respec-tively) has a magnitude comparable to the mean distancebetween particles. Quantum sieving is a novel separationmethod based on isotopes distinction performed in nanoporesthat have a size comparable to λB.7–13 Quantum sieving is apromising low-cost method for recycling fuel from a nuclearreaction and as cryogenic filters that separate mixture compo-nents via equilibrium condensation at low temperatures.14

a)Electronic mail: gil@fisica.ugto.mx

Prediction of thermodynamic properties of quantum flu-ids is of particular relevance in physics, astronomy, and chem-ical engineering,15–18 and specifically hydrogen has deservedgreat attention due to their technological applications inaerospace, electronics, and petrochemical industry.19–21 Thepossibility of developing an economy based on hydrogen22

that could attend energy and environmental problems is basedon optimal methods for hydrogen storaging in cells, that re-quires a better understanding of the properties of quantumfluids under confinement.23, 24 On the other hand, in the lastdecade several studies have shown the possibility of observ-ing quantum effects with molecules of increasing size, suchas matter wave interferometry of buckyballs25, 26 and organicmolecules with 430 atoms,27 as well as measuring entangle-ment effects in macroscopical systems.28

Over the years, different models and methods have beenproposed to study quantum effects in fluids.29–61 The Wigner-Kirkwood (WK) theory29–32 was the first approach developedfor the prediction of thermodynamic and structural propertiesof fluids at a semiclassical level, where quantum correctionsare given by h2-power terms that are functionals of secondand higher-order derivatives of a continuous pair potentialV (r). This approach has been applied for the Lennard-Jonesmodel33, 34 and also for other molecular systems.35 Close-related semiclassical methods to the WK theory have alsobeen used36, 37 and demonstrated the relevance of quantumcorrections at high pressures.38 However, the WK methoddoes not converge for discontinuous potentials and alterna-tive approaches are required, where the quantum correctionsinclude odd powers of h.39, 40 Quantum hard-spheres (HS) andsquare-well (SW) fluids have been studied in this way, eitherfor the calculation of virial coefficients41, 42 or by perturbation

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184505-2 Trejos, Gil-Villegas, and Martinez J. Chem. Phys. 139, 184505 (2013)

theories.43–47 Based on the seminal work by Feynman on pathintegrals62 and on robust molecular-simulation methods de-veloped for classical fluids,63–67 several authors have studiedquantum fluids with computer simulations,49–61 including thereproduction of a metallic phase for hydrogen61 as predictedby Wigner and Huntington.68

We recently have presented a semiclassical equation ofstate valid for quantum fluids60 based on the Statistical Asso-ciating Fluid Theory for potentials of variable range (SAFT-VR)69 and a perturbation theory for quantum fluids;43–46 thisapproach could give very good predictions of the liquid-vaporphases of hydrogen (H2), deuterium (D2), neon (Ne), andhelium-4 (He). In this article, we review the application ofeffective potentials in computer simulations of quantum flu-ids, and we address the effect of confinement studying thequantum corrections to a Lennard-Jones fluid contained be-tween hard-parallel walls, that has been studied classicallyby Liu et al.70 using histogram-reweighting grand canonicalMonte Carlo simulations. The information obtained is a pre-vious step in the development of a molecular-based equationof state for confined hydrogen, based on the SAFT-VR exten-sions to confined71–73 and quantum fluids.60

II. METHOD

We consider a system of N particles interacting via asemiclassical pair potential Veff that consists in a pair poten-tial V (r) with a quantum correction VQ(r),

Veff (r) = V (r) + VQ(r). (1)

In this work, V (r) is given by the Lennard-Jones potential(LJ),

VLJ (r) = 4ε

[(σ

r

)12−

(σ

r

)6]

, (2)

where r is the relative distance between particles, σ is a char-acteristic length parameter defined by VLJ (σ ) = 0, and ε isthe depth of the attractive well. Although Eq. (1) can be ap-plied to other potential models besides the LJ interaction,there is an extensive list of studies performed with this po-tential that allowed us to compare our results. The standardtheory for VQ is the Wigner-Kirkwood potential,29–32 that atfirst-order in β is given by

VQ(r) = D∇2VLJ (r), (3)

where D = λ2B/24π ; we will refer to Eq. (3) as the WK-1

model. Equation (3) agrees with the first-order expressionsfor VQ from Feynman-Hibbs,62 Bohm’s,60 and Jaen-Khantheories.36 In order to assess the effect of considering a sec-ond order contribution in β in VQ to describe the low temper-ature regime, we studied the WK-2 and Jaen-Khan potentials(JK), finding that WK-2 gives non-physical results and, con-sequently, restricting our study to the the JK model, given by

VQ(r) = D∇2VLJ (r) + 9

10D2∇2∇2VLJ (r). (4)

A. Bulk phases

The Gibbs Ensemble Monte Carlo method (GEMC)66, 67

was used to simulate the vapor-liquid phase equilibrium ofquantum fluids. Results were obtained for N = 1000 parti-cles uniformly distributed in a cubic simulation cell dividedinto two boxes of equal volumes N = N1 + N2 = 1000.A random distribution of particles was set up as initial con-figuration in both boxes, and periodic boundary conditions(PBC) were used during the simulations. Standard GEMCparticles movements were considered: displacements of parti-cles within each box, trial changes of volumes and trial inter-changes of particles between the two boxes. The total numberof particles N and the total volume V = V1 + V2 were fixedconstant whereas the volumes V1,2 and the number of particlesN1, 2 per box were varied during the simulations. The numberof cycles used to reach equilibrium and for averaging was 1-2×107 cycles in each case, where a cycle was defined by 90 at-tempts of displacements of particles, 900 attempts of volumechanges, and 1 attempt of interchange of particles. Maximumdisplacement and maximum volume changes were adjustedto give 30%-40% and approximately 50% of accepted movesfor subsystem, respectively. We used the random walk methoddeveloped by Frenkel and Smit65 to generate a new configura-tion in the volume step. Finally, particles interchange betweenthe boxes were adjusted to 9%-11% of the number of parti-cles interchanged per cycle. The mean numbers of particlesN1 and N2, volumes V1, V2, and energies for the two coex-isting phases were obtained as configurational averages overthe accumulated stage of the simulation. The errors in the av-erage properties of interest were estimated by calculating thestandard deviations.

For the estimation of the critical point, we used a scalinglaw and the rectilinear diameters recipe74 as a first estimatefor the Wegner expansion, i.e., the critical parameters ρc andTc are given by

ρ± = ρc + C2|t | ± 1

2Bo|t |β, (5)

where ρ− and ρ+ denote the vapor and liquid densities, re-spectively, and t = 1 − T/Tc. This expression allowed us to fitthe coexistence data and to obtain estimates for ρc, Tc, β, andthe amplitude terms C2 and Bo. We used the apparent criticalexponent obtained in this way in order to describe the overallshape of the coexistence curves. The critical constants wereestimated by fitting the GEMC data to Eq. (5), using a non-linear squares procedure.75

Simulations were performed with a cut-off radius rc forthe total potential and a tail contribution Utail was used to cor-rect the systematic error induced by the truncation,65

Utail

Nρ= 1

2

∞∫rc

VLJ (r) 4πr2dr + D

2

∞∫rc

∇2VLJ (r) 4πr2dr,

(6)where ρ is the average number density. We have truncated thepair potential at cut-off ratios 2.5σ and 4σ . We found that thesize and cut-off effects are less than 1%.

184505-3 Trejos, Gil-Villegas, and Martinez J. Chem. Phys. 139, 184505 (2013)

FIG. 1. Comparison between pair potentials: Lennard-Jones (LJ), Wigner-Kirkwood (WK-1), and Jaen-Kahn (JK). The quantum systems were obtained forhydrogen (� = 1.7378 (Ref. 33)) and temperatures: (a) T* = 0.10, (b) T* = 0.60, (c) T* = 1.0, and (d) T = 5.0. Each potential is given scaled by the LJ energyparameter, i.e., V ∗

eff = Veff /ε.

B. Confined phases

Liu et al.70 presented the phase behavior of a classical LJfluid confined between parallel hard walls, using histogram-reweighting grand canonical Monte Carlo simulations. In or-der to study the modification that quantum effects introducewith respect to a classical system, we selected the samemodel. The canonical ensemble method76, 77 was applied fora WK-1 fluid with a particle-wall interaction given by

u(r) ={∞ if z < σ/2,

0 if σ/2 ≤ z,(7)

where z is the distance between the fluid particles and the wall,and periodic boundary conditions were applied in the x and y

coordinates. The effective quantum potential was truncated atrc = 2.5σ and no long-tail corrections were considered.

In order to perform computer simulations with thismethod, we followed the same procedures and technical de-tails described in our previous studies with classical discon-tinuous potential systems.78, 79 The simulation box consistedof a rectangular parallelepiped with fixed lengths Ly ≥ 10σ ,Lx = 10Ly and 2σ ≤ Lp ≤ 6σ , where Lp is the wall separa-tion. The density of the system is defined by ρ∗ = Nσ 3/V ,where N is the total number of particles and V is the total vol-ume of the confined system defined by V = LxLyLp. We se-lected Lz = Lp as the distance between the hard walls. A densephase formed by N = 1372 with an initial random distribu-tion was placed at the center of the rectangular parallelepiped,

FIG. 2. (a) WK-1 pair potential, V ∗eff = Veff /ε, for hydrogen, deuterium, neon, and helium-4 at T* = 0.60. The LJ parameters σ and ε were taken from

Ref. 33 and given in Table II. (b) GEMC LV coexistence curves for the previous WK-1 systems (squares), solid curves were obtained using Eq. (5). The criticalpoints obtained by a Wegner expansion,74 as explained in the text, are denoted by stars.

184505-4 Trejos, Gil-Villegas, and Martinez J. Chem. Phys. 139, 184505 (2013)

TABLE I. Critical temperature T ∗c , critical density ρ∗

c , and critical exponent β estimated from GEMC data for the WK-1 potential used in this work. Fittedvalues of Bo and C2 are also given. The errors are estimated from the respective errors in the densities of the coexisting vapor and liquid phases. Results wereobtained with a cut-off radius rc = 4σ .

� ρ∗c T ∗

c β Bo C2

0.0000 0.3134 ± 0.0008 1.3019 ± 0.0002 0.3297 ± 0.0026 1.0856 ± 0.0045 0.2307 ± 0.00360.5991 0.3118 ± 0.0013 1.2478 ± 0.0001 0.3220 ± 0.0040 1.0220 ± 0.0085 0.1741 ± 0.00751.2249 0.2850 ± 0.0008 1.1051 ± 0.0020 0.2963 ± 0.0060 0.8708 ± 0.0055 0.1358 ± 0.00311.7378 0.2645 ± 0.0011 0.9787 ± 0.0026 0.2895 ± 0.0079 0.7757 ± 0.0064 0.0979 ± 0.00382.6783 0.2273 ± 0.0006 0.8289 ± 0.0001 0.3048 ± 0.0024 0.6957 ± 0.0028 0.0929 ± 0.0026

obtaining two vapor phases with a liquid phase in between.The acceptance ratio of particle’s displacement was fixed to40%. The equilibration process consisted of (1–8)×107 MCcycles and (1–8)×107 cycles for accumulated averages. Thedensity profile, ρ*(x), was obtained every two cycles, usingthe data values corresponding to 0 < x < Lx/2, since the pro-file is symmetrical in the x coordinate. The LV coexistencedensities were obtained as a function of the x coordinate of thesimulation box, and parametrized using an hyperbolic tangentfunction,80

ρ∗(x) = 1

2(ρ∗

L + ρ∗V ) − 1

2(ρ∗

L − ρ∗V ) tanh

(x − xo

d

), (8)

where ρ∗L and ρ∗

V are the bulk densities corresponding to theliquid and vapor phase, respectively, d is the interface widthand xo is the position in the transition layer.

TABLE II. Optimized GEMC parameters for hydrogen (H2), deuterium(D2), neon (Ne), and helium-4 (He) fluids obtained by fitting to LV coexis-tence experimental data. The Lennard-Jones parameters are σ and ε, obtainedfrom Ref. 33, and de Boer’s quantumness parameter is � = h/σ

√mε.

LJ parameters (Ref. 33) GEMC parameters

Substance (ε/k)/K σ/Å̇ � (ε/k)/K σ/Å̇ �

Ne 37.10 2.670 0.5991 35.283 2.7661 0.5930D2 37.00 2.928 1.2249 34.118 3.0067 1.2417H2 37.00 2.928 1.7378 33.434 3.0450 1.7579He 10.22 2.556 2.6783 6.3141 2.8927 3.0109

FIG. 3. GEMC liquid-vapor coexistence of quantum fluids (triangles) compared to experimental data (circles),81 using a cut-off radius rc = 4σ . The criticalpoints obtained by a Wegner expansion,74 as explained in the text, are denoted by stars. Results correspond to (a) neon, (b) deuterium, (c) hydrogen, and (d)helium, using the optimized molecular parameters reported in Table II.

184505-5 Trejos, Gil-Villegas, and Martinez J. Chem. Phys. 139, 184505 (2013)

TABLE III. Critical temperature Tc and critical density ρc estimated fromGEMC data for the WK-1 fluid studied in this work. Experimental data (T exp

c

and ρexpc ) were taken from the NIST Chemistry WebBook.81 Results were

obtained with a cut-off radius rc = 4σ .

Substance ρexpc /mol L−1 ρc/mol L−1 T

expc /K Tc/K

Ne 23.882 24.465 ± 0.0318 44.491 44.026 ± 0.0035D2 17.327 17.412 ± 0.0139 38.340 37.704 ± 0.0682H2 15.508 15.557 ± 0.0171 33.145 32.722 ± 0.0869He 17.399 15.594 ± 0.0094 5.195 5.2338 ± 0.0006

III. RESULTS

A. Bulk phases

The semiclassical effective potentials WK-1 and JK forhydrogen (H2) are depicted in Fig. 1, using � = 1.7378 asreported by Kim et al.,33 and for temperatures T* = kT/ε= 0.1, 0.6, 1.0, 5.0. Due to the semiclassical approximationused along this work, no nuclear spin effects are taken intoaccount and there is not a distinction between the hydrogenisomers (orthohydrogen and parahydrogen) and we can as-sume that the description obtained corresponds to a mixtureof these isomers. As expected, the effect of VQ is more impor-tant at low temperatures and the potentials have strong varia-tions between them. These differences reduce as temperatureis increased, although quantum effects are still observed atT* = 5. In all the cases, the net effect of VQ is to in-crease the effective size of the particles with respect to the

FIG. 4. GEMC LV coexistence curves for optimized values of the LJ param-eters σ and ε; circles and stars correspond to � values taken from Ref. 33and for the optimized LJ parameters, respectively.

Lennard-Jones fluid parameter σ . We found that the JK modeloverestimates the effective size of the particles due to thesecond-order correction term.

The quantum parameter � modifies the scaling behaviorof the LJ potential, as observed in Fig. 1, since there is not auniversal potential written as a function of r/σ and u/ε as inthe case of VLJ , i.e., the WK-1 system does not follow a cor-responding states behavior. This non-conformal property hasas major effect that the phase diagram is modified by vary-ing �, in similar way to what happens with the range λ fora SW fluid; in Fig. 2, we present the WK-1 pair potentials at

FIG. 5. (a) Density profiles for a LJ fluid obtained by NVT-MC simulations, for temperature and density T* = 0.70 and ρ* = 0.92, respectively. Two setsof results are shown corresponding to simulations using different number of particles (N = 1372 and N = 2048), and a cut-off radius rc = 2.5σ without tailcorrections. (b) LV coexistence curves obtained for bulk and confined LJ fluids with Lp/σ = 1.5, 1.9, 2, 4, 6 (circles) compared with results reported in Ref. 70(stars). Results were obtained with a cut-off radius rc = 2.5σ , N = 1372 particles and without tail corrections.

184505-6 Trejos, Gil-Villegas, and Martinez J. Chem. Phys. 139, 184505 (2013)

TABLE IV. Liquid-vapor coexistence densities ρ∗L and ρ∗

V at temperaturesT* = 0.70 and T* = 0.92 obtained from computer simulation for the WK-1fluid with a cut-off radius rc = 2.5σ .

T ∗ Method ρ∗L ρ∗

V N Reference

0.70 NVT-MC 0.812288 0.00284807 1372 This work0.70 NVT-MC 0.813583 0.00263298 2048 This work0.70 GEMC 0.815991 0.0039047 1000 This work0.70 NVT-MC 0.812000 0.002900 2048 Ref. 770.92 NVT-MC 0.700541 0.0309412 1372 This work0.92 NVT-MC 0.701654 0.0281589 2048 This work0.92 GEMC 0.703820 0.0293623 1000 This work0.92 NVT-MC 0.70650 0.02940 1000 Ref. 77

T* = 0.60 for � = 0.5991, 1.2249, 1.7378, and 2.6783, thatcorresponds to the cases of neon (Ne), deuterium (D2), hydro-gen (H2), and helium-4 (He), respectively.33 The correspond-ing LV phase diagrams were obtained by the GEMC method;the case of the classical LJ fluid (� = 0.0) is also reported forcomparison. The � dependence of the critical values of tem-perature and density, and the corresponding critical exponent,are given in Table I. By increasing �, the critical temperaturedecreases.

A comparison between the molecular parameters re-ported by Kim et al.,33 and the parameters recalculated fromthe GEMC simulation data are given in Table II. The op-timal parameters were determined by fitting to experimen-tal data of vapor pressure and vapor-liquid density for everysystem, obtained from the NIST Chemistry WebBook.81 A

damped least-squares method75 was used, based on previouswork on the LV coexistence of asymmetric mixtures at highpressures.82–84 The objective function required for this calcu-lation was given by

fob =M∑i=1

(ρ

V,cali − ρ

V,expi

ρV,expi

)2

+M∑i=1

(ρ

L,cali − ρ

L,expi

ρL,expi

)2

,

(9)where M is the number of experimental data points, and ρ isthe density. The superscripts exp, cal , L, and V denote theexperimental and calculated values, and the liquid and vaporphases, respectively. The values of the diameter (σ ) and theenergy parameter (ε/k) were optimized according to Eq. (9).As observed in Table II, the GEMC diameter σ is greater thanKim et al. values,33 whereas the energies ε have lower values.

The reparametrized GEMC LV equilibrium data are com-pared with the corresponding experimental values in Fig. 3.GEMC results were obtained using as a cut-off distancerc = 4σ . For Ne, D2, and H2, we can observe that theGEMC and experimental data are in very good agreement.For helium-4, the simulations results are moderately satisfac-tory, taking into account the range of the experimental data(2.5 K–5.3 K), although we could expect a worse performanceof a semiclassical approach in this case. The correspondingcritical values for densities and temperatures for all the sys-tems are given in Table III. The JK system does not show aliquid-vapor coexistence, that can be explained in terms of theincrease of the repulsive potential as observed in Fig. 1.

FIG. 6. LV coexistence of confined WK-1 fluids obtained with the NVT-MC method, using a cut-off radius rc = 2.5σ , N = 1372 particles and without tailcorrections. Results correspond to (a) neon, (b) deuterium, (c) hydrogen, and (d) helium-4, using the LJ parameters reported in Ref. 33. Three wall separationswere considered, Lp/σ = 2, 4, 6, and GEMC results for the bulk system are also included. Solid curves were obtained using Eq. (5), with an estimation of thecritical points denoted by stars, using the Wegner expansion.74

184505-7 Trejos, Gil-Villegas, and Martinez J. Chem. Phys. 139, 184505 (2013)

FIG. 7. (a) Critical temperature T ∗c as a function of the wall separation Lp for confined WK-1 fluids. Results correspond to neon, deuterium, hydrogen, and

helium-4, using the LJ parameters reported in Ref. 33. Three wall separations were considered, Lp/σ = 2, 4, 6. Results for the classical LJ system are included,obtained in this work and from Ref. 70. (b) Thermal de Broglie wavelength as a function of temperature for the systems considered in (a).

Notice that the GEMC fitted parameters values (σ ,ε)modifies the parameter � originally calculated with thevalues reported in Ref. 33. In order to assure a consistent set ofmolecular parameters (σ , ε, �), an iterative procedure can beused to improve the representation of the experimental data,that consists in repeating the GEMC simulations using the re-calculated � parameter, and then to obtain σ and ε by thesame fitting procedure used before. The procedure is repeateduntil a convergence criteria for the determination of � is sat-isfied. We found that this method converges very fast aftertwo iterations; the results for the convergence are presented inFig. 4.

A robust version of SAFT-VR for Mie potentials has re-cently been developed,85 being the LJ interaction a particularcase. This new approach, which has given very accurate pre-dictions for a wide range of substances, can be extended toquantum fluids using the method presented in Ref. 60, andspecific equations of state could be obtained for the WK-1potentials studied here.

B. Confined phases

A first step in the computer simulation of confined quan-tum fluids consisted in the study of the classical fluid with �

= 0. The bulk phase was obtained using the NVT ensemblefor 1372 and 2048 particles with a cut-off radius rc = 2.5σ ,contained into a cell box with dimensions Ly = Lz = 13.41σ

and Lx = 39.81σ . Density profiles for the LV coexistencefor temperatures T* = 0.70 and T* = 0.92 are presented inFig. 5(a). A clear formation of two phases can be observed.To confirm these predictions, results were also obtained us-ing the GEMC method with N = 1000 particles and a cut-offradius rc = 2.5σ , without long tail-corrections. Results ob-tained from both methods are summarized in Table IV, wherewe observe that are in very good agreement. The LV phasediagram is presented in Fig. 5(b) where the case correspond-ing to the bulk system was simulated with GEMC and theconfined diagrams at different values of wall separations Lp

were calculated with NVT-MC. The confined systems wereobtained for five different wall separations, Lp/σ = 1.5, 1.9, 2,

4, 6. Very good agreement was found between the results ob-tained here with the data reported by Liu et al.,70 even for Lp

< 2. However, it is important to bear in mind that the methodfollowed by these authors consisted in histogram-reweightinggrand canonical MC simulations, which is more appropriatefor the detailed study of the critical region.

The LV coexistence curves for WK-1 fluids under con-finement were obtained for three wall separations Lp/σ = 2,4, 6, see Fig. 6. Results correspond to (a) neon, (b) deuterium,(c) hydrogen, and (d) helium-4. The phases diagrams aremodified in similar way by reducing Lp or increasing �, i.e.,confinement and quantumness tend to reduce the LV phaseregion.

The trend of the critical temperature Tc versus the wallseparation Lp is shown in Fig. 7(a), and compared with theclassical behavior obtained by Liu et al.70 Since the LV coex-istence could be preempted by a solid phase, it is clear that thedetermination of the triple point is necessary in order to havea proper determination of the phase diagram; however, wecan conclude that the introduction of quantum effects substan-tially modifies the phase diagram of a confined fluid with a �

value of a typical substance like H2. On the other hand, the deBroglie wavelengths for all the substances reported here areof the same order of magnitude than the confinement distanceLp, as observed in Fig. 7(b). This is a clear indication that thepresence of a liquid phase for a confined fluid requires theintroduction of the quantum potential in order to describe itsproperties when Lp < 6σ , since quantum diffraction becomesvery relevant.

For Lp/σ < 2, we found that the density profiles havestrong variations not observed in the classical system. InFig. 8(a), we present a comparison of the density profiles ob-served for the LJ and WK-1 systems, for � = 0.5991, T*= 0.54, and Lp/σ = 1.9, as well as the corresponding snap-shots of configurations in Figures 8(b) and 8(a) for the LJand WK-1 systems, respectively. Whereas the classical sys-tem clearly shows a liquid-vapor transition, the quantum sys-tem presents a modulated phase with the presence of voids,similar to the phases observed for 2D disks systems.86, 87

The trend observed for the classical LJ fluid is that just forLp = 2σ there is evidence of a 3D-2D crossover in the

184505-8 Trejos, Gil-Villegas, and Martinez J. Chem. Phys. 139, 184505 (2013)

FIG. 8. (a) Comparison between density profiles of confined LJ and WK-1 systems (black and blue lines, respectively), obtained by NVT-MC simulations with1000 particles for temperature T* = 0.54 and Lp/σ = 1.9. (b) and (c) are the corresponding snapshots of configurations for both systems.

behavior of the critical temperature T ∗c .70 Although in the

quantum case was not possible to determine T ∗c when

Lp/σ < 2, the similarity between phases with 2D-systems sug-gest a dimensional crossover although the liquid-vapor transi-tion could be already lost.

IV. CONCLUSIONS

We have presented semiclassical simulations for thermo-dynamic properties of quantum fluids using GEMC and NVT-MC methods, either for bulk or for a specific confinementgeometry — parallel hard walls. Several effective potentialswere considered, based on semiclassical approaches.29, 30, 36

As model system, we considered molecules interactingvia a Lennard-Jones pair potential complemented by atemperature-dependent quantum potential. This effective po-tential determines non-conformal properties to the fluid sys-tem. Very good agreement has been obtained for the predic-tion of bulk phases where experimental data are available, anda simple approach to obtain optimized parameters of the LJpotential from GEMC data has been provided. For the caseof confined fluids, we have compared their LV phase diagramwith respect to the classical model studied in Ref. 70, and theeffect of the reduction of the critical temperature by reduc-ing Lp is now enhanced by increasing the quantumness of thesystem, measured by �. For Lp/σ < 2, the liquid-vapor tran-sition seems to be lost, and this could be as a consequenceof the preempting of a solid phase; however, a dimensionalitycrossover could be present as in the case of the classical LJfluid, since the configurations observed have strong similari-ties with classical 2D-systems at low temperatures.

Although robust methods based on the path-integral for-malism are well known, we have shown that an accurate de-scription of the liquid and vapor phases can be obtained usingsemiclassical potentials. The information presented here is a

valuable tool to be used in combination with thermodynamictheoretical approaches developed previously for classical con-fined fluids,71–73 that require equations of state for three- andtwo-dimensional fluids, as well as a semiclassical version ofthe SAFT-VR approach.60

ACKNOWLEDGMENTS

The authors acknowledge financial support from CONA-CYT through Grant No. 61418 and a Ph.D. scholarship(VMT), as well as computer facilities and useful commentsprovided by Susana Figueroa Gerstenmaier and José TorresArenas (Universidad de Guanajuato, Mexico).

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